**Localized Surface Plasmon Resonances: Noble Metal Nanoparticle Interaction with Rare-Earth Ions**

V.A.G. Rivera, F.A. Ferri and E. Marega Jr.

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/50753

### **1. Introduction**

282 Plasmonics – Principles and Applications

2009; 135 492–498.

Actuators B 2009 138 625–630.

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Particles of sizes between 1 and 100 nm show fascinating properties with unusual characteristics that lead to the formation of unique properties in nanosystems, which are not observed in ordinary materials. These are considered hereby as nanoparticles (NPs). Additionally, metallic NPs with sizes smaller than the wavelength of light show strong dipolar excitations in the form of localized surface plasmon resonances (LSPR). LSPRs are nonpropagating excitations of the conduction electrons of metallic NPs coupled to the electromagnetic field [1]. This effect has been the subject of extensive research, both fundamental and with a view to applications [2,3]. The resonance frequency of the oscillation, i.e., the surface plasmon (SP) energy, it is essentially determined by the dielectric properties of the metal and the surrounding medium, and by the particle size and shape. The collective charge oscillation causes a large resonant enhancement of the local field inside aand near the NP. This field enhancement is used in surface-enhanced Raman scattering (SERS) [4] and is currently discussed for potential applications in nonlinear optical devices [5], in optical tweezers [6], and generally for the manipulation of the local photonic density of states.

Rare-earth (RE) elements are a group of chemical elements known as Lanthanides that occur together in the periodic table. These elements are used in common consumer goods such as: computer memory, DVD's, rechargeable batteries, cell phones, car catalytic converters, magnets, fluorescent lighting and much more. Furthermore, play an essential role in modern national defense, e.g.: lanthanum in night-vision goggles, neodymium in laser range-finders, guidance systems, communications; europium in fluorescents and phosphors in lamps and monitors; erbium amplifiers in fiber-optics data transmission; samarium in permanent magnets stable at high temperatures and, others technological applications. We center our

© 2012 Rivera et al., licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2012 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

attention in the RE ions and their optical properties that are widely employed in photonic. Devices of general interest span RE ions concentrations of tens to several thousand parts per million (ppm), resulting in devices of one to tens of meters long such as channel waveguides and optical fiber, respectively. In optical devices, the RE should ideally be confined as a delta function in the center of the core for maximum gain per unit pump power. Thereby, there is a necessary tradeoff between the confinement and the RE ions concentration, such that the more confined structures require a higher rare-earth ions concentration. An important feature that distinguishes them from other optically active ions: they emit and absorb over narrow wavelength ranges, the wavelengths of the emission and absorption transitions depend the host material, the intensities of these transitions are weak, the lifetimes of metastable states are long, and the quantum efficiencies tend to be high, except in aqueous solutions. These properties lead to excellent performance of RE ions in many optical applications. Devices that provide gain (e.g. lasers and amplifiers), must have low scattering losses, and one is restricted to using single-crystal or glass hosts. Whereas in many applications crystalline materials are preferred for reasons that include higher peak cross sections or better thermal conductivities, the versatility of glasses and the broader emission and absorption spectra they provide have led to the use of RE doped glasses in many applications, eventually running into the clustering limit for the particular host glass composition [7].

Localized Surface Plasmon Resonances: Noble Metal Nanoparticle Interaction with Rare-Earth Ions 285

intensity emission of the RE ions due to long-range electromagnetic interaction between

In this context, it is worth mentioning that the main purpose of this chapter is to show some of concrete concept of *noble metallic nanoparticle interaction with rare-earth ions* exhibiting a field enhancement in transparent medium with noble metal NPs embedded via a *localized* 

First, we know that a surface plasmon resonance (SPR) can be described as the resonance collective oscillation of valence electrons in a solid stimulated by beam of light incident (electromagnetic field – see Figure 1). The resonance condition is established when the frequency of light matches the natural frequency of valence electrons oscillating against from this restoring force. SPR in nanometer-sized structures is called LSPR. For gold and silver NPs, the resonance falls into the visible region of the electromagnetic spectrum. A striking consequence of this is the bright colors exhibited by particles both in transmitted and reflected light, due to resonantly enhanced absorption and scattering. This effect has found applications for many hundreds of years, for example in the staining of glass for

**2. Localized surface plasmon resonance and metallic nanoparticles** 

**Figure 1.** Left: Schematic of the electron charge displacement (valence electrons) in a metallic NP (LSPR) interacting with an incident plane wave, with electric field polarized *E* into a host matrix. Right: The corresponding electric field strength pattern for a 100 nm silver sphere, irradiated at a wavelength

For spherical NPs smaller than compared to local variations of the involved electromagnetic fields (*d*<<λ) in the surrounding medium, the quasi-static approximation [16], we lead to a

2 Localized surface plasmons have been observed since the Romans who used gold and silver nanoparticles to create colored glass objects such as the Lycurgus Cup (4th Century A.D.). A gold sol in the British museum, created by Michael Faraday in 1857, is still exhibiting its red color due to the plasmon resonance at ,530 nm [L. M. Liz-Marzan,

LSPR and the RE ions.

*surface plasmon resonance*.

windows or ornamental cups2.

of 514 nm, this is reproduced from X. Lu et al. [15].

Mater. Today **7**, 26 (2004).].

**2.1. Noble metal nanoparticles in an electric field** 

Most of existing and potential future applications of NPs suffer from damping caused by metal absorption. Sudarkin and Demkovich [8] suggested increasing the propagation length of the surface plasmon polariton1 (SPP) by creating the population inversion in the dielectric medium adjacent to the metallic surface (film). Recently, gain-assisted propagation of the SPPs at the interface between a metal and a dielectric with optical gain has been analyzed theoretically [9,10]. Thus, NPs embedded in a gain media represent a field enhancement sustained that resonant excitation can lead to a reduction in the threshold for achieving inversion in the optically active surrounding medium, and the presence of gain can counteract the inherent absorption losses in the NP [1]. While this enhancement of plasmon resonances in gain medium not is experimentally confirmed, amplification of fluorescence due to field enhancement in gain medium with metal NPs has been observed [11,12,13,14].

In the present chapter, we explore the physics of localized surface plasmons by first considering the interaction of metal NPs with an electromagnetic wave in order to arrive at the resonance condition (LSPR). After, we show studies of plasmon resonances in particles with different shapes and sizes, and the effects of interactions between particles in the ensemble. Then we will focus on the RE ions, discussing the optical properties of the trivalent ions through the principles of quantum mechanics (in terms of oscillator strengths) with special interest in the interactions between the 4f electrons themselves. Since all the other electronic shells are spherically symmetric. Finally we present diverse experimental results of the interaction of RE ions interaction with NPs, resulting in an enhancement of the

<sup>1</sup> A SPP is an electromagnetic wave propagating along the interface between two media possessing permittivities with opposite signs, such as the metal–dielectric interface.

intensity emission of the RE ions due to long-range electromagnetic interaction between LSPR and the RE ions.

In this context, it is worth mentioning that the main purpose of this chapter is to show some of concrete concept of *noble metallic nanoparticle interaction with rare-earth ions* exhibiting a field enhancement in transparent medium with noble metal NPs embedded via a *localized surface plasmon resonance*.

### **2. Localized surface plasmon resonance and metallic nanoparticles**

284 Plasmonics – Principles and Applications

composition [7].

opposite signs, such as the metal–dielectric interface.

attention in the RE ions and their optical properties that are widely employed in photonic. Devices of general interest span RE ions concentrations of tens to several thousand parts per million (ppm), resulting in devices of one to tens of meters long such as channel waveguides and optical fiber, respectively. In optical devices, the RE should ideally be confined as a delta function in the center of the core for maximum gain per unit pump power. Thereby, there is a necessary tradeoff between the confinement and the RE ions concentration, such that the more confined structures require a higher rare-earth ions concentration. An important feature that distinguishes them from other optically active ions: they emit and absorb over narrow wavelength ranges, the wavelengths of the emission and absorption transitions depend the host material, the intensities of these transitions are weak, the lifetimes of metastable states are long, and the quantum efficiencies tend to be high, except in aqueous solutions. These properties lead to excellent performance of RE ions in many optical applications. Devices that provide gain (e.g. lasers and amplifiers), must have low scattering losses, and one is restricted to using single-crystal or glass hosts. Whereas in many applications crystalline materials are preferred for reasons that include higher peak cross sections or better thermal conductivities, the versatility of glasses and the broader emission and absorption spectra they provide have led to the use of RE doped glasses in many applications, eventually running into the clustering limit for the particular host glass

Most of existing and potential future applications of NPs suffer from damping caused by metal absorption. Sudarkin and Demkovich [8] suggested increasing the propagation length of the surface plasmon polariton1 (SPP) by creating the population inversion in the dielectric medium adjacent to the metallic surface (film). Recently, gain-assisted propagation of the SPPs at the interface between a metal and a dielectric with optical gain has been analyzed theoretically [9,10]. Thus, NPs embedded in a gain media represent a field enhancement sustained that resonant excitation can lead to a reduction in the threshold for achieving inversion in the optically active surrounding medium, and the presence of gain can counteract the inherent absorption losses in the NP [1]. While this enhancement of plasmon resonances in gain medium not is experimentally confirmed, amplification of fluorescence due to field enhancement in gain medium with metal NPs has been observed [11,12,13,14].

In the present chapter, we explore the physics of localized surface plasmons by first considering the interaction of metal NPs with an electromagnetic wave in order to arrive at the resonance condition (LSPR). After, we show studies of plasmon resonances in particles with different shapes and sizes, and the effects of interactions between particles in the ensemble. Then we will focus on the RE ions, discussing the optical properties of the trivalent ions through the principles of quantum mechanics (in terms of oscillator strengths) with special interest in the interactions between the 4f electrons themselves. Since all the other electronic shells are spherically symmetric. Finally we present diverse experimental results of the interaction of RE ions interaction with NPs, resulting in an enhancement of the

1 A SPP is an electromagnetic wave propagating along the interface between two media possessing permittivities with

First, we know that a surface plasmon resonance (SPR) can be described as the resonance collective oscillation of valence electrons in a solid stimulated by beam of light incident (electromagnetic field – see Figure 1). The resonance condition is established when the frequency of light matches the natural frequency of valence electrons oscillating against from this restoring force. SPR in nanometer-sized structures is called LSPR. For gold and silver NPs, the resonance falls into the visible region of the electromagnetic spectrum. A striking consequence of this is the bright colors exhibited by particles both in transmitted and reflected light, due to resonantly enhanced absorption and scattering. This effect has found applications for many hundreds of years, for example in the staining of glass for windows or ornamental cups2.

**Figure 1.** Left: Schematic of the electron charge displacement (valence electrons) in a metallic NP (LSPR) interacting with an incident plane wave, with electric field polarized *E* into a host matrix. Right: The corresponding electric field strength pattern for a 100 nm silver sphere, irradiated at a wavelength of 514 nm, this is reproduced from X. Lu et al. [15].

### **2.1. Noble metal nanoparticles in an electric field**

For spherical NPs smaller than compared to local variations of the involved electromagnetic fields (*d*<<λ) in the surrounding medium, the quasi-static approximation [16], we lead to a

<sup>2</sup> Localized surface plasmons have been observed since the Romans who used gold and silver nanoparticles to create colored glass objects such as the Lycurgus Cup (4th Century A.D.). A gold sol in the British museum, created by Michael Faraday in 1857, is still exhibiting its red color due to the plasmon resonance at ,530 nm [L. M. Liz-Marzan, Mater. Today **7**, 26 (2004).].

#### 286 Plasmonics – Principles and Applications

good agreement between theory and experiment. So, one can calculate the spatial field distribution (assuming as a problem of a particle in an electrostatic field), from the Laplace equation for the potential <sup>2</sup> ∇ Φ= 0 , therefore the electric field *E* = −∇Φ . Besides, the harmonic time dependence can then be added to the solution once the field distributions are well known. The solution this problem was obtained by Jackson [17]:

$$\Phi\_{in} = -\frac{3\varepsilon\_{NP}}{\varepsilon\_{NP} + 2\varepsilon\_{m}} E\_0 r \cos \theta \tag{1}$$

Localized Surface Plasmon Resonances: Noble Metal Nanoparticle Interaction with Rare-Earth Ions 287

and n is the unit vector in the direction of the point P of interest. For *kr*<<1

3 2

= + *i* in the quasi-static limit, the extinction cross

(8)

*ikr*

(7)

( ) ( ) ( ) <sup>2</sup>

(near zone) we have the electrostatic result of (5). The magnetic field present has the form

field ( ) *kr* → 0 , the magnetic field vanishes. For *kr>>*1, the dipole fields have spherical-wave

From the viewpoint of optics, it is much interesting to note that another consequence of the enhanced polarization α in which a NP scatters and absorbs light [18]. For a sphere of

> ( ) 3/2 2

εε

For particles with larger dimensions, where the quasi-static approximation is not justified due to significant phase-changes of the driving field over the particle volume, a rigorous electrodynamics approach is required. This way, Gustav Mie solved Maxwell's equations for the case of an incoming plane interacting with a spherical particle [19]. In essence, the electromagnetic fields are expanded in multipole contributions and the expansion coefficients are found by applying the correct boundary conditions for electromagnetic

2

*m*

( ){ }

*n ab*

= ++ (9)

 β

 β

Ψ Ψ −Ψ Ψ <sup>=</sup> Ψ −Ψ (10)

Ψ Ψ −Ψ Ψ <sup>=</sup> Ψ −Ψ (11)

2 1 Re 2 *ext n n*

() ( ) ( ) () () ( ) ( ) () ' ' ' ' *nn n n*

() ( ) ( ) () () ( ) ( ) () ' ' ' ' *nn n n*

 ββ

 βξ

 ββ

 βξ

*m mm*

*m mm*

*nn n n*

 β

 β

*nn n n*

 β

 β

*m mm*

*m mm*

+ +

ε

2 2 1 2

 ε

n

0

ε ε

ε

*C*

fields at the interface between the metallic NP and its surrounding.

σ

ξ β

> ξ β

Here the parameters *an* and *bn* are defined as:

*n*

*n*

*b*

*a*

9

ω

ε

The extinction cross section of a spherical NP is given by the following expression:

∞

0

=

*n*

2

π

λ

β

β

*ext m*

*<sup>c</sup>* <sup>=</sup>

*m E H* μ

> εε

= × .

<sup>×</sup> <sup>=</sup> . Hence, in the near field the fields are predominantly electric and for static

 *r r r* <sup>=</sup> ×× + ⋅− −

*<sup>e</sup> ik E k <sup>e</sup>*

1 1 n p n 3n n p p <sup>4</sup> *ikr*

0

πε ε

<sup>×</sup> <sup>=</sup> and <sup>0</sup>

volume *V* and dielectric function *NP* 1 2

With *k* = 2 / π λ

ω

π

*H*

4 *<sup>i</sup> <sup>H</sup>*

( ) 2 n p

*r*

form: ( ) <sup>2</sup> n p 4

π*r*

section *CCC ext abs sca* = + is:

**2.2. Mie theory** 

*ikr ck e*

*m*

$$\Phi\_{\rm out} = -E\_0 r \cos \theta + \frac{\varepsilon\_{\rm NP} - \varepsilon\_m}{\varepsilon\_{\rm NP} + 2\varepsilon\_m} E\_0 \left(\frac{d}{2}\right)^3 \frac{\cos \theta}{r^2} \tag{2}$$

*E0* is the amplitude of the electric field, *NP* ε and *m*ε are the dielectric permittivity of the NP and of the surrounding medium respectively, both are function the excitation frequencyω , *r* is the module of the position vector. Here Φ*out* describes the superposition of the applied field and that of a dipole located at the NP center, i.e., an applied field induces dipole moment inside the sphere proportional to <sup>0</sup> *E* . Additionally, we can arrive at

$$\alpha = 4\pi \left(\frac{d}{2}\right)^3 \frac{\varepsilon\_{\rm NP} - \varepsilon\_m}{\varepsilon\_{\rm NP} + 2\varepsilon\_m} \tag{3}$$

Here α is the (complex) polarizability of the NP in the electrostatic approximation. Under the condition that 2 *NP m* ε ε + is a minimum, the polarizability shows a resonant enhancement. Thus, we can also write the Frohlich condition Re ( ) 2 *NP <sup>m</sup>* = − εω ε that is the associated mode the dipole surface plasmon of the NP (in an oscillating field). Nevertheless, the distribution of the electric field *in E* inside and *out E* outside the sphere can be written as:

$$E\_{in} = \frac{3\mathcal{E}\_m}{\mathcal{E}\_{NP} + 2\mathcal{E}\_m} E\_0 \tag{4}$$

$$E\_{out} = E\_0 + \frac{3\mathbf{n}\left(\mathbf{n} \cdot \mathbf{p}\right) - \mathbf{p}\left(1\right)^3}{4\pi\varepsilon\_0\varepsilon\_m} \left(\frac{1}{r}\right)^3 \tag{5}$$

Here, 0m 0 p = ε ε *αE* is the dipole moment. Therefore, a resonance in α implies a resonant improvement of both the internal and dipolar fields, resulting in prominent applications of NPs in optical devices. Now consider a plane wave incident with ( ) <sup>0</sup> , *i t Ert Ee*<sup>−</sup> ω = , this induce on NP an oscillating dipole moment ( ) 0m 0 <sup>p</sup> *i t <sup>t</sup> <sup>α</sup>E e* ω ε ε <sup>−</sup> = , i.e., we have a scattering of the plane wave by the NP. In this sense, the electromagnetic fields associated with an electric dipole in the near, intermediate and radiation zones are [17]:

$$H = \frac{ck^2}{4\pi} (\mathbf{n} \times \mathbf{p}) \frac{e^{ikr}}{r} \left(1 - \frac{1}{ikr}\right) \tag{6}$$

Localized Surface Plasmon Resonances: Noble Metal Nanoparticle Interaction with Rare-Earth Ions 287

$$E = \frac{1}{4\pi\varepsilon\_0\varepsilon\_m} \left\{ k^2 \left( \mathbf{n} \times \mathbf{p} \right) \times \mathbf{n} \frac{e^{ikr}}{r} + \left( 3\mathbf{n} \left( \mathbf{n} \cdot \mathbf{p} \right) - \mathbf{p} \right) \left( \frac{1}{r^3} - \frac{ik}{r^2} \right) e^{ikr} \right\} \tag{7}$$

With *k* = 2 / π λ and n is the unit vector in the direction of the point P of interest. For *kr*<<1 (near zone) we have the electrostatic result of (5). The magnetic field present has the form ( ) 2 n p 4 *<sup>i</sup> <sup>H</sup> r* ω π <sup>×</sup> <sup>=</sup> . Hence, in the near field the fields are predominantly electric and for static field ( ) *kr* → 0 , the magnetic field vanishes. For *kr>>*1, the dipole fields have spherical-wave form: ( ) <sup>2</sup> n p 4 *ikr ck e H r* <sup>×</sup> <sup>=</sup> and <sup>0</sup> n *E H* μ = × .

From the viewpoint of optics, it is much interesting to note that another consequence of the enhanced polarization α in which a NP scatters and absorbs light [18]. For a sphere of volume *V* and dielectric function *NP* 1 2 ε εε = + *i* in the quasi-static limit, the extinction cross section *CCC ext abs sca* = + is:

0

ε ε

*m*

$$\mathcal{C}\_{ext} = 9 \frac{a\nu}{c} \mathcal{E}\_m \frac{\varepsilon\_2}{\left(\varepsilon\_1 + 2\varepsilon\_m\right)^2 + \varepsilon\_2} \tag{8}$$

#### **2.2. Mie theory**

π

286 Plasmonics – Principles and Applications

good agreement between theory and experiment. So, one can calculate the spatial field distribution (assuming as a problem of a particle in an electrostatic field), from the Laplace equation for the potential <sup>2</sup> ∇ Φ= 0 , therefore the electric field *E* = −∇Φ . Besides, the harmonic time dependence can then be added to the solution once the field distributions are

3

ε

θ

ε

ε

2 *NP*

 ε

0 0 2

 ε

> ε

*NP m*

*NP m <sup>d</sup> E r <sup>E</sup>*

> and *m*ε

is the module of the position vector. Here Φ*out* describes the superposition of the applied field and that of a dipole located at the NP center, i.e., an applied field induces dipole

3

Here α is the (complex) polarizability of the NP in the electrostatic approximation. Under

associated mode the dipole surface plasmon of the NP (in an oscillating field). Nevertheless, the distribution of the electric field *in E* inside and *out E* outside the sphere can be written as:

> 3 2 *m*

ε

*NP m E E* <sup>=</sup> <sup>+</sup> ε

 ε

0 3n n p p 1

πε ε *r* ⋅ −

improvement of both the internal and dipolar fields, resulting in prominent applications of NPs in optical devices. Now consider a plane wave incident with ( ) <sup>0</sup> , *i t Ert Ee*<sup>−</sup>

the plane wave by the NP. In this sense, the electromagnetic fields associated with an

*r ikr*

=× −

( ) <sup>2</sup> <sup>1</sup> np 1 <sup>4</sup>

*ikr ck e <sup>H</sup>* π

= +

 <sup>−</sup> <sup>=</sup> +

*d* ε

2 2 *NP m NP m*

ε

 ε

> ε

0

ω

( ) <sup>3</sup>

*m*

*αE* is the dipole moment. Therefore, a resonance in α implies a resonant

ε ε

+ is a minimum, the polarizability shows a resonant

*NP m*

+

ε

ε

and of the surrounding medium respectively, both are function the excitation frequency

<sup>−</sup> Φ =− + <sup>+</sup>

0

*E r*

cos

2 2

θ

3

cos

θ

*r*

(1)

are the dielectric permittivity of the NP

(3)

*NP <sup>m</sup>* = −

(4)

<sup>−</sup> = , i.e., we have a scattering of

(6)

(5)

 ε

that is the

ω= , this

εω

(2)

ω, *r*

well known. The solution this problem was obtained by Jackson [17]:

*in*

*out*

*E0* is the amplitude of the electric field, *NP*

the condition that 2 *NP m*

Here, 0m 0 p = ε ε

ε

 ε Φ =−

cos

moment inside the sphere proportional to <sup>0</sup> *E* . Additionally, we can arrive at

4

enhancement. Thus, we can also write the Frohlich condition Re ( ) 2

*in*

0

*E E*

induce on NP an oscillating dipole moment ( ) 0m 0 <sup>p</sup> *i t <sup>t</sup> <sup>α</sup>E e*

electric dipole in the near, intermediate and radiation zones are [17]:

4 *out*

α π

> For particles with larger dimensions, where the quasi-static approximation is not justified due to significant phase-changes of the driving field over the particle volume, a rigorous electrodynamics approach is required. This way, Gustav Mie solved Maxwell's equations for the case of an incoming plane interacting with a spherical particle [19]. In essence, the electromagnetic fields are expanded in multipole contributions and the expansion coefficients are found by applying the correct boundary conditions for electromagnetic fields at the interface between the metallic NP and its surrounding.

The extinction cross section of a spherical NP is given by the following expression:

$$
\sigma\_{ext} = \frac{\lambda^2}{2\pi} \sum\_{n=0}^{\infty} (2n+1) \text{Re}\{a\_n + b\_n\} \tag{9}
$$

Here the parameters *an* and *bn* are defined as:

$$a\_n = \frac{\Psi\_n(\boldsymbol{\beta})\Psi\_n^\prime(m\boldsymbol{\beta}) - m\Psi\_n(m\boldsymbol{\beta})\Psi\_n^\prime(\boldsymbol{\beta})}{\xi\_n(\boldsymbol{\beta})\Psi\_n^\prime(m\boldsymbol{\beta}) - m\Psi\_n^\prime(m\boldsymbol{\beta})\tilde{\xi}\_n^\prime(\boldsymbol{\beta})}\tag{10}$$

$$b\_n = \frac{m\Psi\_n(\beta)\Psi^\*\_{\
u}(m\beta) - \Psi\_n(m\beta)\Psi^\*\_{\
u}(\beta)}{m\tilde{\xi}\_n(\beta)\Psi^\*\_{\
u}(m\beta) - \Psi\_n(m\beta)\tilde{\xi}^\*\_{\
u}(\beta)}\tag{11}$$

The size parameter β is defined as <sup>0</sup> 0 π*dm* β λ <sup>=</sup> , where λ*<sup>0</sup>* is the incident wavelength with Localized Surface Plasmon Resonances: Noble Metal Nanoparticle Interaction with Rare-Earth Ions 289

distribution of 20 % STD, this simulation was extracted from program MiePlot v4.2.11 of Philip Laven. Also, it is possible to observe an increment of bandwidth and the intensity of peak for NPs embedded in the glass when compared with the material bulk. Therefore, LSPR results in enhanced local electromagnetic fields near the surface of the NP (Novotny &

**Figure 3.** Extinction cross section from Mie theory via the program MiePlot v4.2.11 of Philip Laven, for silver and gold NPs with a size distribution of 20 % STD of 20 nm size embedded in a glass with refraction index in function of the wavelength and a bulk with refraction index 2 constant. In both cases, we observed a red-shift (see arrow). Inset figure shows the size distribution of NPs for 50 NPs in both

Hideki Nabika and Shigehito Deki [23], show an experimental evidence of this dependence above mentioned for silver NPs of different sizes were synthesized by reducing AgNO3 with *N*,*N*-dimethylformamide (DMF), Poly(vinylpyrrolidone) (PVP) aqueous solution (10 mL, 10 wt %) was added to DMF (80 mL), followed by an addition of AgNO3 aqueous solution (10 mL). They obtained three sets of spherical silver NPs with a particle diameter varying from 9.7 to 27.1 nm and rod-shaped silver NPs with an aspect ratio of 1.79, its results are show

Thereby, the LSPR properties are sensitive to its environment in the order of its shape and size [24], the dielectric function of the glass host containing the NPs, changes the LSPR

We can be written as the sum of the bound and quasi-free (conduction) electron

 γ

unchanged [26]. A similar expression can be used for the contribution of conduction

= =+ *g V d* . Where the first term, 0 1 /

bulklike electron scattering process in the particle and the second term is a consequence of quasi-electron-free interaction with the surface and, for a sphere, *VF* is the Fermi velocity, and *gs* is the surface factor [25]. Nevertheless, the LSPR dependence on the matrix refractive

=− + *i* , γ is the damping the resonance, *ω* the

ε

is associated to

τ

remains

*<sup>p</sup>* the plasma frequency. The bound electron contribution *ib*

*p*

*s F*

bandwidth and resonance frequency, see Figure 2-4.

 ε ω ω ωω

> τ

contributions [25]: () () <sup>2</sup> / *ib*

excitation frequency,

εω

electrons [27]: 0 1/ 1/ 2 / *NP*

γτ

ω

Hecht, 2006) [22], see Figure 1.

cases.

Figure 4.

respect to vacuum, and *m0* represents the refractive index of the surrounding medium. The Ricatti-Bessel functions Ψ andξare defined in terms of the half-integer-order Bessel

function of the first kind (*J*n+1/2(*z*)), ( ) 0.5 1/2( ) <sup>2</sup> *n n rx x Jx* <sup>+</sup> Ψ = and ( ) 0.5 1/2( ) <sup>2</sup> *n n rx* ξ *x Hx* <sup>+</sup> <sup>=</sup> ,

1/2( ) *H x <sup>n</sup>*<sup>+</sup> is the half-integer-order Hankel function of the second kind.

We will focus our attention in silver and gold, since the localized plasmon resonance condition mentioned above is satisfied at visible light frequencies. Additional advantages of these metal NPs include simple preparation methods for a wide range of sizes and shapes and easy surface conjugation to a variety of ligands.

We now evaluate the extinction cross section using the Mie theory from equation (9), Figure 2, we take dielectric constants for silver and gold from Palik [20], and the medium dielectric constant is assumed to be 1.0 (i.e., a particle in a vacuum) and 2.0 for NPs with sizes different.

We can see from Figure 2 the dependence of resonance frequency with size of NPs (silver and gold) and the refraction index (surrounding medium). K. Lance Kelly et al. (2003), show that for spherical NPs clear differences between the quasistatic and the Mie theory results. However, the important features are retained, e.g. the frequency resonance. Although Mie theory is not a very expensive calculation, the quasistatic expressions are convenient to use when only qualitative information is needed [21].

**Figure 2.** Extinction cross section from Mie theory for (a) silver and (b) gold NPs. Both in function of size with refraction index 1 (line curves) and 2 (dot lines). Inset figure shows a red-shift with the increment of size NPs and refraction index.

Now, Figure 3 shows clearly the dependence of resonance frequency embedded in a glass (with refraction index of: 1.99 for 400 nm, 1.88 for 630 nm, 1.75 for 900 nm and 1.70 for 1000 nm) and in a bulk with refraction index 2, in both cases with 20 nm size and a size distribution of 20 % STD, this simulation was extracted from program MiePlot v4.2.11 of Philip Laven. Also, it is possible to observe an increment of bandwidth and the intensity of peak for NPs embedded in the glass when compared with the material bulk. Therefore, LSPR results in enhanced local electromagnetic fields near the surface of the NP (Novotny & Hecht, 2006) [22], see Figure 1.

288 Plasmonics – Principles and Applications

Ricatti-Bessel functions Ψ and

β

function of the first kind (*J*n+1/2(*z*)), ( )

and easy surface conjugation to a variety of ligands.

when only qualitative information is needed [21].

**Ag-NPs Index refraction 1 2**

increment of size NPs and refraction index.

**300 400 500 600 700 800 900 1000**

**Wavelength (nm)**

**Wavelength (nm)**

**Ag-NPs size (nm) index refraction 1 and 2 5 5 10 10 20 20 30 30**

**5 10 15 20 25 30**

**Diamter (nm)**

is defined as <sup>0</sup>

ξ

1/2( ) *H x <sup>n</sup>*<sup>+</sup> is the half-integer-order Hankel function of the second kind.

0

respect to vacuum, and *m0* represents the refractive index of the surrounding medium. The

Ψ =

We will focus our attention in silver and gold, since the localized plasmon resonance condition mentioned above is satisfied at visible light frequencies. Additional advantages of these metal NPs include simple preparation methods for a wide range of sizes and shapes

We now evaluate the extinction cross section using the Mie theory from equation (9), Figure 2, we take dielectric constants for silver and gold from Palik [20], and the medium dielectric constant is assumed to be 1.0 (i.e., a particle in a vacuum) and 2.0 for NPs with sizes

We can see from Figure 2 the dependence of resonance frequency with size of NPs (silver and gold) and the refraction index (surrounding medium). K. Lance Kelly et al. (2003), show that for spherical NPs clear differences between the quasistatic and the Mie theory results. However, the important features are retained, e.g. the frequency resonance. Although Mie theory is not a very expensive calculation, the quasistatic expressions are convenient to use

**Figure 2.** Extinction cross section from Mie theory for (a) silver and (b) gold NPs. Both in function of size with refraction index 1 (line curves) and 2 (dot lines). Inset figure shows a red-shift with the

(a) (b)

**QExt (a.u.)**

Now, Figure 3 shows clearly the dependence of resonance frequency embedded in a glass (with refraction index of: 1.99 for 400 nm, 1.88 for 630 nm, 1.75 for 900 nm and 1.70 for 1000 nm) and in a bulk with refraction index 2, in both cases with 20 nm size and a size

λ

0.5 1/2( ) <sup>2</sup> *n n rx x Jx* <sup>+</sup>

are defined in terms of the half-integer-order Bessel

and ( )

ξ

*<sup>0</sup>* is the incident wavelength with

**300 400 500 600 700 800 900 1000**

**Wavelength (nm)**

**Au-NPs Index refraction 1 2**

**Au-NPs size (nm) index refraction 1 and 2 5 5 10 10 20 20 30 30**

**Wavelength (nm)**

**5 10 15 20 25 30**

**Diamter (nm)**

0.5 1/2( ) <sup>2</sup> *n n rx*

,

 *x Hx* <sup>+</sup> <sup>=</sup>

λ<sup>=</sup> , where

π*dm*

β

The size parameter

different.

**0**

**5**

**10**

σ**ext (a.u.)**

**15**

**20**

**25**

**Figure 3.** Extinction cross section from Mie theory via the program MiePlot v4.2.11 of Philip Laven, for silver and gold NPs with a size distribution of 20 % STD of 20 nm size embedded in a glass with refraction index in function of the wavelength and a bulk with refraction index 2 constant. In both cases, we observed a red-shift (see arrow). Inset figure shows the size distribution of NPs for 50 NPs in both cases.

Hideki Nabika and Shigehito Deki [23], show an experimental evidence of this dependence above mentioned for silver NPs of different sizes were synthesized by reducing AgNO3 with *N*,*N*-dimethylformamide (DMF), Poly(vinylpyrrolidone) (PVP) aqueous solution (10 mL, 10 wt %) was added to DMF (80 mL), followed by an addition of AgNO3 aqueous solution (10 mL). They obtained three sets of spherical silver NPs with a particle diameter varying from 9.7 to 27.1 nm and rod-shaped silver NPs with an aspect ratio of 1.79, its results are show Figure 4.

Thereby, the LSPR properties are sensitive to its environment in the order of its shape and size [24], the dielectric function of the glass host containing the NPs, changes the LSPR bandwidth and resonance frequency, see Figure 2-4.

We can be written as the sum of the bound and quasi-free (conduction) electron contributions [25]: () () <sup>2</sup> / *ib p* εω ε ω ω ωω γ =− + *i* , γ is the damping the resonance, *ω* the excitation frequency, ω*<sup>p</sup>* the plasma frequency. The bound electron contribution *ib* ε remains unchanged [26]. A similar expression can be used for the contribution of conduction electrons [27]: 0 1/ 1/ 2 / *NP s F* γτ τ = =+ *g V d* . Where the first term, 0 1 / τ is associated to bulklike electron scattering process in the particle and the second term is a consequence of quasi-electron-free interaction with the surface and, for a sphere, *VF* is the Fermi velocity, and *gs* is the surface factor [25]. Nevertheless, the LSPR dependence on the matrix refractive index (*n*λ) also can be calculated by [12]: ( ) 2 \* <sup>0</sup> 4 *p d* ω π εε ω = *ne m* ′ , where *n*′ is the electrons density, *e* the electron charge, ( ) ( ) <sup>2</sup> *<sup>d</sup> n*λ ε ω = the dielectric permittivity, *and m\** the electron mass.

Localized Surface Plasmon Resonances: Noble Metal Nanoparticle Interaction with Rare-Earth Ions 291

**n=1.0 / r 1.0 5.0 10.0 20.0 n=1.5 / r 1.0 5.0 1.00 20.0 n=2.0 /** r **1.0 5.0 15.0 20.0**

**12345**

**Aspect ratio** 

**Figure 5.** Resonance wavelength with aspect ratio from Cheng - ping Huang prediction for radius

Figure 6 illustrates this further, via experiments of X. Lu et al. [15] and Mock et al. [32]. Panel (a) the nanobar Ag NPs and the corresponding dark-field light scattering spectra, in broad agreement with the simulated results shown in [15]. Panel (b) shows the dipolar

We can say that the localized plasmon resonance frequency of a single metallic NP can be shifted through of alterations in shape, size and surrounding medium from the Frohlich condition, section 2.1. Nevertheless, in a NP ensembles we can obtain additional shifts due to electromagnetic interactions between the localized modes, see Figure 3 and 4 (c). These interactions are basically of a dipolar nature (when *d*<<λ). So, the NP ensemble can be treated as an ensemble of interacting dipoles (in a first approximation). Those NPs can be embedded into a host matrix ordered or random, in one-, two- or three-dimensional arrays with interparticle spacing *D*. Electromagnetic coupling of those arrays shows interesting localization effects for closely spaced particles such as, enhancement process due to field localization in NP junctions. Assuming a dipolar approximation the NPs can be treated as

interactions with a distance dependence of *D-3* dominate, equation (5), and the NP array is described as an array of point dipoles coupling via their near-field, see section 2.1 and Figure 7. These arrays can serve as hot-spots for field enhancement, e.g. in a context of surface-enhanced Raman scattering (SERS). (ii) For larger NP separations, far-field dipolar

Therefore, the maximum field enhancement is determined by the shortest distance between

λ

, near-field

point dipoles. In this sense, two regimes have to be distinguished (i) For *D<<*

coupling with a distance dependence of *D-1* dominates, see section 2.1.

two equipotential particles.

different and embedded in a bulk with refraction index 1.0 , 1.5 and 2.0.

plasmon lineshapes of colloidal silver NPs of different shapes [32].

**2.3. Coupling between metallic nanoparticles** 

**400**

**600**

**800**

**1000**

**Resonance wavelength (nm)**

**1200**

**1400**

**1600**

**1800**

**Figure 4.** TEM micrographs, size distributions and optical absorption spectra of spherical silver NPs. Hideki Nabika and Shigehito Deki [23]. (a) 90 °C, AgNO3 = 0.125 M, (b) 90 °C, AgNO3 = 0.250 M, and (c) 110 °C, AgNO3 = 0.250 M and (d) of the rod-shaped silver NPs.

On the other hand, in the Gans theory, the LSPR is only a function of the aspect ratio and refractive index. Thus, in certain conditions, a linear relationship between them can be resulted [28]. Nevertheless, numerical results suggest that, even when the aspect ratio is fixed and the retardation effect is weak, the position of longitudinal resonance can still depend strongly with the aspect ratio [29, 30] . Using the model of Cheng - ping Huang et al [31], we can write:

$$
\lambda\_p = \pi n \sqrt{10\kappa \left(2\,\delta^2 + r^2 \ln\left[\,\kappa^\circ\right]\right)}\tag{12}
$$

Where κ *= l/2r* is the aspect ratio of the NPs (with the inner radius *r* and in a cylindrical region with the length *l),* and δ is the skin depth. Cheng - ping Huang considered the problem as using an LC circuit model without solving the Laplace or Maxwell equations, and show that its results overcomes the deficiency of Gans theory and provides a new insight into the phenomenon. This way, we can observe resonance wavelength change with the aspect ratio of NP, Figure 5. This means a breakdown of the linear behavior presenting oscillations electrons originating from the amorphous geometry of the NPs associated with the inertia of electrons, see inset Figure 2 as well.

**Figure 5.** Resonance wavelength with aspect ratio from Cheng - ping Huang prediction for radius different and embedded in a bulk with refraction index 1.0 , 1.5 and 2.0.

Figure 6 illustrates this further, via experiments of X. Lu et al. [15] and Mock et al. [32]. Panel (a) the nanobar Ag NPs and the corresponding dark-field light scattering spectra, in broad agreement with the simulated results shown in [15]. Panel (b) shows the dipolar plasmon lineshapes of colloidal silver NPs of different shapes [32].

#### **2.3. Coupling between metallic nanoparticles**

290 Plasmonics – Principles and Applications

density, *e* the electron charge, ( ) ( ) <sup>2</sup>

) also can be calculated by [12]: ( ) 2 \*

*<sup>d</sup> n*λ

ε ω ω

**Figure 4.** TEM micrographs, size distributions and optical absorption spectra of spherical silver NPs. Hideki Nabika and Shigehito Deki [23]. (a) 90 °C, AgNO3 = 0.125 M, (b) 90 °C, AgNO3 = 0.250 M, and (c)

On the other hand, in the Gans theory, the LSPR is only a function of the aspect ratio and refractive index. Thus, in certain conditions, a linear relationship between them can be resulted [28]. Nevertheless, numerical results suggest that, even when the aspect ratio is fixed and the retardation effect is weak, the position of longitudinal resonance can still depend strongly with the aspect ratio [29, 30] . Using the model of Cheng - ping Huang et al

( ) 2 2 10 2 ln *<sup>p</sup>*

problem as using an LC circuit model without solving the Laplace or Maxwell equations, and show that its results overcomes the deficiency of Gans theory and provides a new insight into the phenomenon. This way, we can observe resonance wavelength change with the aspect ratio of NP, Figure 5. This means a breakdown of the linear behavior presenting oscillations electrons originating from the amorphous geometry of the NPs associated with

 *= l/2r* is the aspect ratio of the NPs (with the inner radius *r* and in a cylindrical

 κ

= + *n r* (12)

is the skin depth. Cheng - ping Huang considered the

 κδ

110 °C, AgNO3 = 0.250 M and (d) of the rod-shaped silver NPs.

λ π

δ

<sup>0</sup> 4 *p d*

 εε ω

= the dielectric permittivity, *and m\**

= *ne m* ′ , where *n*′ is the electrons

the electron

 π

index (*n*

mass.

λ

[31], we can write:

κ

region with the length *l),* and

the inertia of electrons, see inset Figure 2 as well.

Where

We can say that the localized plasmon resonance frequency of a single metallic NP can be shifted through of alterations in shape, size and surrounding medium from the Frohlich condition, section 2.1. Nevertheless, in a NP ensembles we can obtain additional shifts due to electromagnetic interactions between the localized modes, see Figure 3 and 4 (c). These interactions are basically of a dipolar nature (when *d*<<λ). So, the NP ensemble can be treated as an ensemble of interacting dipoles (in a first approximation). Those NPs can be embedded into a host matrix ordered or random, in one-, two- or three-dimensional arrays with interparticle spacing *D*. Electromagnetic coupling of those arrays shows interesting localization effects for closely spaced particles such as, enhancement process due to field localization in NP junctions. Assuming a dipolar approximation the NPs can be treated as point dipoles. In this sense, two regimes have to be distinguished (i) For *D<<*λ, near-field interactions with a distance dependence of *D-3* dominate, equation (5), and the NP array is described as an array of point dipoles coupling via their near-field, see section 2.1 and Figure 7. These arrays can serve as hot-spots for field enhancement, e.g. in a context of surface-enhanced Raman scattering (SERS). (ii) For larger NP separations, far-field dipolar coupling with a distance dependence of *D-1* dominates, see section 2.1.

Therefore, the maximum field enhancement is determined by the shortest distance between two equipotential particles.

Localized Surface Plasmon Resonances: Noble Metal Nanoparticle Interaction with Rare-Earth Ions 293

glasses 1.5–1.6, 2.7, 0.55

fluorides 0.28–0.33

**wavelengths (µm)**

active RE ions and host media together with typical emission wavelength ranges are shown

**Ion Common host media Important emission** 

Neodymium (Nd3+) YAG, YVO4, YLF, silica 1.03–1.1, 0.9–0.95, 1.32–1.35

Thulium (Tm3+) YAG, silica, fluoride glasses 1.7–2.1, 1.45–1.53, 0.48, 0.8

(Pr3+) silica, fluoride glasses 1.3, 0.635, 0.6, 0.52, 0.49

Ytterbium (Yb3+) YAG, tungstates, silica 1.0–1.1

Holmium (Ho3+) YAG, YLF, silica 2.1, 2.8–2.9

**Table 1.** Common laser-active rare earth ions and host media and important emission wavelengths.

The 4f electrons are not the outermost ones. They are shielded from external fields by two electronic shells with larger radial extension (5s25p6), which explains the atomic nature of their spectra absorption/emission, Figure 8. Thus the 4f electrons are only weakly perturbed by the charges of the surrounding atoms. Important characteristics that distinguish them from other optically active ions are: (i) the wavelengths of the emission and absorption transitions are relatively insensitive to host material, (ii) the intensities of these transitions are weak, (iii) the lifetimes of metastable states are long and, (iv) the quantum efficiencies tend to be high. This is why the RE ions are such a useful probe in a solid; the crystal environment constitutes only a small perturbation on the atomic energy levels, and many of the solid state, and hence spectroscopic, properties can be understood from a consideration of the free ions. In turn, the wavefunctions of the free ions constitute a good zero order

The solutions to this problem can then befactored into a product of a radial and angular function. Whereas the radial function depends on the details of the potential the spherical symmetry ensures that the angular component is identicalwith that of a hydrogen atom and can be expressed as spherical harmonics. Except for Ce3+ and Yb3+, which have only one electron, the solutions of the central-field problem are products of one-electron states that are antisymmetric under the interchange of a pair of electrons, as required by the Pauli

In describing the state of a multielectron atom, the orbital angular momenta and the spin angular momenta are added separately. The sum of the orbital angular momenta is designated by the letter *L,* and the total spin is characterized by *S.* The total angular momentum *J* of the atom may then be obtained by vector addition of *L* and *S.* The collection of energy states with common values of *J, L,* and *S* is called a term. Here, the Russell-

Erbium (Er3+) YAG, silica, tellurite, chacogenetos

Cerium (Ce3+) YLF, LiCAF, LiLuF, LiSAF, and similar

approximation for a description of solid-state properties, Figure 8.

in the table 1:

Praseodymium

exclusion principle.

**Figure 6.** Panel (a) scattering spectra. In the insets are SEM images of the individual Ag nanobars. X. Lu et al. [15]. For simple, highly symmetric shapes (sphere, cube and triangular plate), the spectra are dominated by a single peak, but with the peak position sensitive to the shape, and ranging from 400 to up 800 nm. Panel (b) Scattering spectra of single silver nanoparticles of different shapes obtained in dark-field configuration. Mock et al. [32].

Figure 7 shows the near field coupling between NPs. Here, the restoring force acting on the oscillating electrons of each NP in the chain is either increased or decreased by the charge distribution of neighboring particles. Depending on the polarization direction of the exciting light, this leads to a blue-shift of the plasmon resonance for the excitation of transverse modes, Figure 7 (a), and a red-shift for longitudinal modes, Figure 7 (b).

**Figure 7.** Schematic of near-field coupling between metallic NPs for two different polarizations.

Today, there are several theoretical methodologies available to describe the interaction among the metallic NPs with electromagnetic radiation [33,34,35,36,37,38,39,40].

### **3. Rare-earths**

Rare earths ions in solids are either divalent or trivalent. Their electronic configuration is 4fN5s25p6 or 4fN-15s25p6, respectively. By far the most common valence state of the RE ions in solids is the trivalent one. Those ions have a long history in optical and magnetic applications. We have special interest in the devices luminescent using crystal, powders, and glasses. On the other hand, divalent RE ions have also been used in laser devices, but only in relatively exotic ones for cryogenic operation [41,42]. The most frequently used laser-


active RE ions and host media together with typical emission wavelength ranges are shown in the table 1:

292 Plasmonics – Principles and Applications

dark-field configuration. Mock et al. [32].

**3. Rare-earths** 

**Figure 6.** Panel (a) scattering spectra. In the insets are SEM images of the individual Ag nanobars. X. Lu et al. [15]. For simple, highly symmetric shapes (sphere, cube and triangular plate), the spectra are dominated by a single peak, but with the peak position sensitive to the shape, and ranging from 400 to up 800 nm. Panel (b) Scattering spectra of single silver nanoparticles of different shapes obtained in

Figure 7 shows the near field coupling between NPs. Here, the restoring force acting on the oscillating electrons of each NP in the chain is either increased or decreased by the charge distribution of neighboring particles. Depending on the polarization direction of the exciting light, this leads to a blue-shift of the plasmon resonance for the excitation of transverse

**Figure 7.** Schematic of near-field coupling between metallic NPs for two different polarizations.

among the metallic NPs with electromagnetic radiation [33,34,35,36,37,38,39,40].

Today, there are several theoretical methodologies available to describe the interaction

Rare earths ions in solids are either divalent or trivalent. Their electronic configuration is 4fN5s25p6 or 4fN-15s25p6, respectively. By far the most common valence state of the RE ions in solids is the trivalent one. Those ions have a long history in optical and magnetic applications. We have special interest in the devices luminescent using crystal, powders, and glasses. On the other hand, divalent RE ions have also been used in laser devices, but only in relatively exotic ones for cryogenic operation [41,42]. The most frequently used laser-

modes, Figure 7 (a), and a red-shift for longitudinal modes, Figure 7 (b).


The 4f electrons are not the outermost ones. They are shielded from external fields by two electronic shells with larger radial extension (5s25p6), which explains the atomic nature of their spectra absorption/emission, Figure 8. Thus the 4f electrons are only weakly perturbed by the charges of the surrounding atoms. Important characteristics that distinguish them from other optically active ions are: (i) the wavelengths of the emission and absorption transitions are relatively insensitive to host material, (ii) the intensities of these transitions are weak, (iii) the lifetimes of metastable states are long and, (iv) the quantum efficiencies tend to be high. This is why the RE ions are such a useful probe in a solid; the crystal environment constitutes only a small perturbation on the atomic energy levels, and many of the solid state, and hence spectroscopic, properties can be understood from a consideration of the free ions. In turn, the wavefunctions of the free ions constitute a good zero order approximation for a description of solid-state properties, Figure 8.

The solutions to this problem can then befactored into a product of a radial and angular function. Whereas the radial function depends on the details of the potential the spherical symmetry ensures that the angular component is identicalwith that of a hydrogen atom and can be expressed as spherical harmonics. Except for Ce3+ and Yb3+, which have only one electron, the solutions of the central-field problem are products of one-electron states that are antisymmetric under the interchange of a pair of electrons, as required by the Pauli exclusion principle.

In describing the state of a multielectron atom, the orbital angular momenta and the spin angular momenta are added separately. The sum of the orbital angular momenta is designated by the letter *L,* and the total spin is characterized by *S.* The total angular momentum *J* of the atom may then be obtained by vector addition of *L* and *S.* The collection of energy states with common values of *J, L,* and *S* is called a term. Here, the RussellSaunders notation for the energy levels (terms) is used, *2S+1LJ*, this takes into account the spin-spin coupling, orbit-orbit coupling and spin-orbit coupling. The energy levels of a free RE ion are usually interpreted by considering only interactions between the 4f electrons themselves. Since all the other electronic shells are spherically symmetric, Figure 8, their effect on all the terms of a configuration is the same in first order, and therefore do not contribute significantly to the relative positions of the 4f energy levels, we can write:

Localized Surface Plasmon Resonances: Noble Metal Nanoparticle Interaction with Rare-Earth Ions 295

electrons. Details of the matrix element calculations of *HC* and *HSO* can be found in ref.

Figure 8 shows the radial distribution functions 4f, 5s, 5p, and 6s electrons for Gd+ as obtained from Harteww-Fock calculation by Freeman and Watson [46]. We can see that the 4f electrons are inner electrons with relatively small <*rn*> values. As we shall see, the crystal field interaction will be small in REs. It is also evident that the 4f wavefunctions do not extend very far beyond the 5s and 5p shells. Thus, all quantities of a solid depend on an overlap of the 4f

The 4f shell of the RE ions are an unfilled shell and therefore have a spherical charge distribution. If the ion is introduced into a crystal, the ion experiences an inhomogeneous electrostatic field, the so-called crystal field, which is produced by the charge distribution into the crystal. This crystal field distorts the closed shells of the RE ion. Producing an effect on the energy level 4f, i.e., removes to a certain degree the degeneracy of the free ion 4f levels, thus producing a major modification of the energy levels (but this depends on the crystal symmetry). In the luminescence spectrum are observed additional transitions, which originate from the excited crystal field levels of the ground term is evident, and the spectra look more complicated. On the other hand, these spectrums can be used to determine the

We shall now try to understand the behavior of these additional transitions. To do so we draw on our treatment of the Coulomb interaction of the 4f electrons. The crystal field interaction comes by the interaction of the 4f electrons with all the charges of the crystal with all charges of the crystal, except for trivial factors, it is therefore given by one over the radius vector between the 4f electrons and the crystal charges. This function is given, apart from radial factors, of two spherical harmonics *Ykq* , one containing the coordinates of the 4f electrons, the other containing coordinates of the crystal charge. The latter has to be integrated over the whole crystal, whereupon it gives the strength of the crystal potential at the site of the RE ion. Finally, this shows the existence of the splitting telling us that the development of the crystal potential into spherical harmonics contains a second order term.

Crystal field splitting has two aspects: (i) symmetry, namely, the number of the levels into which an ion is free the *J* term are split in a crystal field of a given symmetry; (ii) the actual

This way, the point charge model can be used for illustrative purposes but not for a quantitative description of the crystal field interaction. A realistic description of this interaction must take into account that the crystal field is built up of the spatially extended charge clouds of the individual ions. Thus, these charge clouds can penetrate each other and

thereby interact. Hence, we can considered the following elements:

wavefunctions with those of a neighboring ion have to be small in RE compounds.

**3.1. Trivalent ions in a static crystal field** 

crystal field energies of the ground term.

*3.1.1. Crystal field splitting* 

size of the crystal field splitting.

[44,45].

**Figure 8.** Square of the radial wavefunctions for the 4f, 5s, 5p and 6s energy levels from Hartree-Fock. It was a calculation for Gd+ by Freeman e Watson (1962) [46]. Picture inset schematically shows that the 4f orbital is within the 6s, 5p and 5s levels.

$$H = -\frac{\hbar^2}{2m} \sum\_{i=1}^{N} \Delta\_i - \sum\_{i=1}^{N} \frac{Z^\* e^2}{r\_i} + \underbrace{\sum\_{i$$

Where *N*=1…14 is the number of the 4f electrons, *Z\* e* the screened charge of the nucleus because we have neglected the closed electronic shell, *VEF* treats the interaction of the ion with the electromagnetic field, and ( )*<sup>i</sup>* ζ*r* the spin-orbit coupling function [43],

$$\mathcal{L}\left(r\_i\right) = \frac{\hbar^2}{2m^2c^2r\_i}\frac{d\mathcal{U}\left(r\_i\right)}{dr\_i} \tag{14}$$

Where ( ) *U ri* is the potential in which the electron *i* is moving. The first two terms of the Hamiltonian (equation (13)) are spherically symmetric and therefore do not remove any of the degeneracies within the configuration of the 4f electrons, therefore we can neglect. The next two terms, which represent the mutual Coulomb interaction of the 4f electrons (*HC*) and their spin-orbit interaction (*HSO*) are responsible for the energy level structure of the 4f electrons. Details of the matrix element calculations of *HC* and *HSO* can be found in ref. [44,45].

Figure 8 shows the radial distribution functions 4f, 5s, 5p, and 6s electrons for Gd+ as obtained from Harteww-Fock calculation by Freeman and Watson [46]. We can see that the 4f electrons are inner electrons with relatively small <*rn*> values. As we shall see, the crystal field interaction will be small in REs. It is also evident that the 4f wavefunctions do not extend very far beyond the 5s and 5p shells. Thus, all quantities of a solid depend on an overlap of the 4f wavefunctions with those of a neighboring ion have to be small in RE compounds.

### **3.1. Trivalent ions in a static crystal field**

294 Plasmonics – Principles and Applications

orbital is within the 6s, 5p and 5s levels.

with the electromagnetic field, and ( )*<sup>i</sup>*

Where *N*=1…14 is the number of the 4f electrons, *Z\**

Saunders notation for the energy levels (terms) is used, *2S+1LJ*, this takes into account the spin-spin coupling, orbit-orbit coupling and spin-orbit coupling. The energy levels of a free RE ion are usually interpreted by considering only interactions between the 4f electrons themselves. Since all the other electronic shells are spherically symmetric, Figure 8, their effect on all the terms of a configuration is the same in first order, and therefore do not

**Figure 8.** Square of the radial wavefunctions for the 4f, 5s, 5p and 6s energy levels from Hartree-Fock. It was a calculation for Gd+ by Freeman e Watson (1962) [46]. Picture inset schematically shows that the 4f

> 2 \*2 2 1 1 <sup>1</sup> 2

*m rr*

ζ

ζ

*i*

*r*

*NN N N*

*Ze e <sup>H</sup> rs l V*

because we have neglected the closed electronic shell, *VEF* treats the interaction of the ion

( ) ( ) <sup>2</sup> 2 2 2

Where ( ) *U ri* is the potential in which the electron *i* is moving. The first two terms of the Hamiltonian (equation (13)) are spherically symmetric and therefore do not remove any of the degeneracies within the configuration of the 4f electrons, therefore we can neglect. The next two terms, which represent the mutual Coulomb interaction of the 4f electrons (*HC*) and their spin-orbit interaction (*HSO*) are responsible for the energy level structure of the 4f

*mcr dr*

*i i i ij ij i*

== < = =− Δ − + + ⋅ +

( )

(13)

*e* the screened charge of the nucleus

ζ

*r* the spin-orbit coupling function [43],

<sup>=</sup> (14)

*i*

*i i*

*dU r*

*C SO*

*H H*

*i i i i EF*

contribute significantly to the relative positions of the 4f energy levels, we can write:

The 4f shell of the RE ions are an unfilled shell and therefore have a spherical charge distribution. If the ion is introduced into a crystal, the ion experiences an inhomogeneous electrostatic field, the so-called crystal field, which is produced by the charge distribution into the crystal. This crystal field distorts the closed shells of the RE ion. Producing an effect on the energy level 4f, i.e., removes to a certain degree the degeneracy of the free ion 4f levels, thus producing a major modification of the energy levels (but this depends on the crystal symmetry). In the luminescence spectrum are observed additional transitions, which originate from the excited crystal field levels of the ground term is evident, and the spectra look more complicated. On the other hand, these spectrums can be used to determine the crystal field energies of the ground term.

We shall now try to understand the behavior of these additional transitions. To do so we draw on our treatment of the Coulomb interaction of the 4f electrons. The crystal field interaction comes by the interaction of the 4f electrons with all the charges of the crystal with all charges of the crystal, except for trivial factors, it is therefore given by one over the radius vector between the 4f electrons and the crystal charges. This function is given, apart from radial factors, of two spherical harmonics *Ykq* , one containing the coordinates of the 4f electrons, the other containing coordinates of the crystal charge. The latter has to be integrated over the whole crystal, whereupon it gives the strength of the crystal potential at the site of the RE ion. Finally, this shows the existence of the splitting telling us that the development of the crystal potential into spherical harmonics contains a second order term.

### *3.1.1. Crystal field splitting*

Crystal field splitting has two aspects: (i) symmetry, namely, the number of the levels into which an ion is free the *J* term are split in a crystal field of a given symmetry; (ii) the actual size of the crystal field splitting.

This way, the point charge model can be used for illustrative purposes but not for a quantitative description of the crystal field interaction. A realistic description of this interaction must take into account that the crystal field is built up of the spatially extended charge clouds of the individual ions. Thus, these charge clouds can penetrate each other and thereby interact. Hence, we can considered the following elements:


Therewith, we can calculate the crystal potential ( ) , , *iii* Φ *r* ϕ θ at the site of the 4f electrons and the potential energy of the 4f electrons in this potential. If the crystal has charge density ρ( ) *R* and the 4f electrons have radius *ri*, we have:

$$V = -\sum\_{i} \int \frac{e\_i \rho\left(R\right)}{\left|R - r\_i\right|} d\tau = -\sum\_{k,i} e\_i \int \rho\left(R\right) P\_k\left(\cos\left[R\_r r\_i\right]\right) \frac{r\_<^k}{r\_>^{k+1}} d\tau\tag{15}$$

Where *r<* and *r>* are, respectively, the smaller and larger value of *r* and *ri*. Here *Pk*(cos[*x*]) are the Legendre polynomials. Also, the equation (15) can be re-write as:

$$V = -\sum\_{k,q,i} B\_{k,q} \mathbb{C}\_{k,q} \left(\boldsymbol{\theta}\_i, \boldsymbol{\varphi}\_i\right) \tag{16}$$

Localized Surface Plasmon Resonances: Noble Metal Nanoparticle Interaction with Rare-Earth Ions 297

one) define the terms of the configuration, all of which are degenerate in the central-field approximation, as illustrated in Figure 9. Next in the hierarchy is spin–orbit, the strongest of the magnetic interactions. Spin–orbit lifts the degeneracy in total angular momentum and

We can see that the environment provided by silica hosts destroys the spherically symmetric environment that Er3+ ions enjoy in the vapor phase. Thus the degeneracy of the 4f atomic states will be lifted to some degree. This splitting is also referred to as stark splitting, and the resulting states are called Stark components (of the parent manifold). The even-*k* terms in the expansion split the free ion *J* multiplets into Stark components generally separated by 10–100 cm-1 (see Figure 9). The ion–lattice interaction can mix multiplets with different *J*  values (*J* mixing), although it usually remains a good quantum number. The odd-*k* terms admix higher lying states of opposite parity [e.g., 4fN15d1] into the 4fN configuration. This admixture does not affect the positions of the energy levels, but it has a very important

In recent years most of the interest in luminescent RE ions has concentrated on the trivalent erbium (Er3+), and in particular its emission band around 1550 nm. This emission is within the telecommunication windows, and fortuitously coincides with the 1550 nm intra-4f

For this reason exists today a great interest in utilizing erbium-doped materials for gain elements and sources in telecommunications systems [48,49,50]. The development of the erbium-doped fiber amplifier (EDFA) in the late 1980s [51,52] exploited the 4I13/2→4I15/2 transition and allowed the transmission and amplification of signals in the 1530–1560 nm region without the necessity for expensive optical to electrical conversion [53]. Hence, it is originated a driving force for research in RE doped fibers and integrated optics waveguides has been their use for amplifying weak signals in optical communications systems at 1300 and 1600 nm. This may be achieved by simply splicing a section of RE-doped fiber into the transmission one and injecting pump light through a fiber coupler. The signal generated within the RE emission band stimulates emission of radiation at the same frequency, amplifying the optical communication signal with high gain, high efficiency and low noise, which is highly advantageous for optical communications [54]. There are five main RE candidates for use as dopants in fiber or waveguide amplifiers for optical communications systems: Er3+, Tm3+, Nd3+, Pr3+ and Dy3+. The Er3+ and Tm3+ ions are the choice for the 1400– 1600 nm window centered at 1550 nm, based on the 4I13/2→4I15/2 transition of Er3+ ion and the 3H4→3F4 transition of Tm3+ ion. The 4F3/2→4I13/2 emission of Nd3+ ion, the 1G4→3H5 transition of Pr3+ ion and the 6F11/2(6H9/2)→6H15/2 transition of Dy3+ ion are all potentially useful for the

Further, the 3+ ions all exhibit intense narrow-band intra-4f luminescence in a wide variety of hosts, and the shielding provided by the 5s2 and 5p6 electrons (see Figure 8), which means that rare-earth radiative transitions in solid hosts resemble those of the free ions and

splits the *LS* terms into *J* levels, Figure 9.

**3.2. Optical properties** 

4I13/2→4I15/2 transition of the Er3+ ion.

1300 nm telecommunication window.

effect on the strengths of the optical transitions between levels.

Where the crystal field parameter *Bk,q* (*k* ≤ 6 for *f* electrons) and the summation *i* is carried out over all the 4*f* electrons of the ions. *Bk,q* have the form:

$$B\_{k,q} = -e \int (-1)^q \, \rho \left( R \right) \mathbb{C}\_{k-q} \left( \theta, \rho \right) \frac{r\_<^k}{r\_>^{k+1}} d\tau \tag{17}$$

Here *Ck,q* is called a tensor operator and is defined as: 0.5 , , 4 2 1 *C Y k q k q <sup>k</sup>* π<sup>=</sup> <sup>+</sup>

In early treatments of the crystal fiel interaction it was assumed that in RE compounds the point charge contribution would be the dominant part of the crystal field interaction. Then, in equation (16) the integral over the lattice can be replaced by a sum over all lattice points and *<sup>k</sup> r* <sup>&</sup>lt; can be replaced by *<sup>k</sup> i r* . The latter replacement can be performed as long as the charge distribution of the crystal does not enter that of the 4f electrons (as long as *ri* < *R*), which implies that the potential acting on the 4f electrons obeys the Laplace equation ( ) ΔΦ = ( ) *r*,ϕ θ, 0 at the position of the 4f electrons.

Figure 9 further illustrates the effect of spin–orbit and crystal field interactions on the energy levels of the Er3+ ion. This figure were extracted from A.J. Kenyon [47]. A further splitting of the energy levels comes about when the ion is placed in a silica host. The 4fN configuration is composed for a number of states where the quantum numbers (*L*, *S*, *J*, and another arbitrary one) define the terms of the configuration, all of which are degenerate in the central-field approximation, as illustrated in Figure 9. Next in the hierarchy is spin–orbit, the strongest of the magnetic interactions. Spin–orbit lifts the degeneracy in total angular momentum and splits the *LS* terms into *J* levels, Figure 9.

We can see that the environment provided by silica hosts destroys the spherically symmetric environment that Er3+ ions enjoy in the vapor phase. Thus the degeneracy of the 4f atomic states will be lifted to some degree. This splitting is also referred to as stark splitting, and the resulting states are called Stark components (of the parent manifold). The even-*k* terms in the expansion split the free ion *J* multiplets into Stark components generally separated by 10–100 cm-1 (see Figure 9). The ion–lattice interaction can mix multiplets with different *J*  values (*J* mixing), although it usually remains a good quantum number. The odd-*k* terms admix higher lying states of opposite parity [e.g., 4fN15d1] into the 4fN configuration. This admixture does not affect the positions of the energy levels, but it has a very important effect on the strengths of the optical transitions between levels.

### **3.2. Optical properties**

296 Plasmonics – Principles and Applications

ρ

and *<sup>k</sup> r*

( ) ΔΦ = ( ) *r*,ϕ θ

and their effect on the energy levels.

Both contribute to the crystal field interaction.

Therewith, we can calculate the crystal potential ( ) , , *iii* Φ *r*

*i*

out over all the 4*f* electrons of the ions. *Bk,q* have the form:

Here *Ck,q* is called a tensor operator and is defined as:

*i*

, 0 at the position of the 4f electrons.

<sup>&</sup>lt; can be replaced by *<sup>k</sup>*

ρ

*i i k i*

the Legendre polynomials. Also, the equation (15) can be re-write as:

τ

correlation effects play no significant role.

( ) *R* and the 4f electrons have radius *ri*, we have:

1. The ions are considered static in the crystal, i.e., we can neglected the lattice vibrations

2. We regard the 4f electrons of one RE ion, as representative of those of all the RE ions in the crystal and, thus the interaction of 4f electrons of adjacent ions is neglected. 3. The crystal consists of the extended charge distributions. This produces an overlap of the charge distributions of the neighboring ions and the 4f electrons. Aditionally, a charge transfer between 4f electrons and the electrons of the ligands can take place.

4. The 4f electrons of one ion are considered to be independent of each other, that is

and the potential energy of the 4f electrons in this potential. If the crystal has charge density

*e R <sup>r</sup> <sup>V</sup> d e R P Rr d R r r*

Where *r<* and *r>* are, respectively, the smaller and larger value of *r* and *ri*. Here *Pk*(cos[*x*]) are

Where the crystal field parameter *Bk,q* (*k* ≤ 6 for *f* electrons) and the summation *i* is carried

( ) () ( ) , <sup>1</sup> 1 ,

In early treatments of the crystal fiel interaction it was assumed that in RE compounds the point charge contribution would be the dominant part of the crystal field interaction. Then, in equation (16) the integral over the lattice can be replaced by a sum over all lattice points

distribution of the crystal does not enter that of the 4f electrons (as long as *ri* < *R*), which implies that the potential acting on the 4f electrons obeys the Laplace equation

Figure 9 further illustrates the effect of spin–orbit and crystal field interactions on the energy levels of the Er3+ ion. This figure were extracted from A.J. Kenyon [47]. A further splitting of the energy levels comes about when the ion is placed in a silica host. The 4fN configuration is composed for a number of states where the quantum numbers (*L*, *S*, *J*, and another arbitrary

= − = −

, ,

*kqi V BC* = −

*q k q k q k <sup>r</sup> B e RC <sup>d</sup>*

ρ ρ

( ) ( ) ( ) <sup>1</sup> ,

( ) , ,

θ ϕ

, *kq kq i i*

ϕ θ

cos ,

<sup>−</sup> (15)

*k*

*r* . The latter replacement can be performed as long as the charge

τ

, , 4 2 1 *C Y k q k q <sup>k</sup>* π = +

0.5

*r*

 <sup>&</sup>lt; <sup>−</sup> <sup>+</sup> > =− − (17)

θ ϕ

*ik i k*

*k*

τ < + >

(16)

at the site of the 4f electrons

In recent years most of the interest in luminescent RE ions has concentrated on the trivalent erbium (Er3+), and in particular its emission band around 1550 nm. This emission is within the telecommunication windows, and fortuitously coincides with the 1550 nm intra-4f 4I13/2→4I15/2 transition of the Er3+ ion.

For this reason exists today a great interest in utilizing erbium-doped materials for gain elements and sources in telecommunications systems [48,49,50]. The development of the erbium-doped fiber amplifier (EDFA) in the late 1980s [51,52] exploited the 4I13/2→4I15/2 transition and allowed the transmission and amplification of signals in the 1530–1560 nm region without the necessity for expensive optical to electrical conversion [53]. Hence, it is originated a driving force for research in RE doped fibers and integrated optics waveguides has been their use for amplifying weak signals in optical communications systems at 1300 and 1600 nm. This may be achieved by simply splicing a section of RE-doped fiber into the transmission one and injecting pump light through a fiber coupler. The signal generated within the RE emission band stimulates emission of radiation at the same frequency, amplifying the optical communication signal with high gain, high efficiency and low noise, which is highly advantageous for optical communications [54]. There are five main RE candidates for use as dopants in fiber or waveguide amplifiers for optical communications systems: Er3+, Tm3+, Nd3+, Pr3+ and Dy3+. The Er3+ and Tm3+ ions are the choice for the 1400– 1600 nm window centered at 1550 nm, based on the 4I13/2→4I15/2 transition of Er3+ ion and the 3H4→3F4 transition of Tm3+ ion. The 4F3/2→4I13/2 emission of Nd3+ ion, the 1G4→3H5 transition of Pr3+ ion and the 6F11/2(6H9/2)→6H15/2 transition of Dy3+ ion are all potentially useful for the 1300 nm telecommunication window.

Further, the 3+ ions all exhibit intense narrow-band intra-4f luminescence in a wide variety of hosts, and the shielding provided by the 5s2 and 5p6 electrons (see Figure 8), which means that rare-earth radiative transitions in solid hosts resemble those of the free ions and electron–phonon coupling is weak. Although some of the divalent species also exhibit luminescence (e.g., samarium and europium), it is the trivalent ions that are of most interest.

Localized Surface Plasmon Resonances: Noble Metal Nanoparticle Interaction with Rare-Earth Ions 299

Some line emission is not a specific property of RE ions and in addition to that, in the case of RE ions, broad emission spectra can be obtained as well, depending on the optical transitions involved. In this respect, RE ions are not unique. The unique properties of the RE ions originate from the fact that the spectral position of the emission lines is almost independent of the host lattice, in contrast to line emission generate by the emission of metal

The terms shown in equation (13) are time-dependent; thereby, they do not lead to stationary states of the system. This way, they are treated using time-dependent perturbation theory resulting in transitions between the states established by the static interactions. In luminescent devices the most important term is *VEF* which gives rise to the emission and absorption of photons from decay radiative of the RE ions. This involves the interaction between the electron charge and the electric field and, the interaction between

The experimental data on spectra of RE ions show that the radiation is mostly electric dipole (ED) nature, though some cases of the magnetic dipole radiation are also observed. Since the optical transitions take place between levels of a particular 4fN configuration, the electric dipole radiation is forbidden in first order, because the electric dipole operator has uneven parity and the transition matrix element must have even parity (Laporte selection rules). Van Vleck [55] pointed out the dipole electric radiation can only occur because the 4fN states have admixtures of 4fN-1nl configuration (nl will be mostly 5d), thus the 4fN-1nl has to be chosen such that it has opposite parity from 4fN. The admixture is produced via interactios that have odd parity and this depend of the host matrix of the RE ions. However, we can

In free atoms, magnetic dipole is about six orders of magnitude weaker than the electric dipole radiation. This latter occurs only as a consequence of a perturbation, both kinds of radiation shows up in the RE spectra with about intensity. Then, quadrupole radiation are

There are different types of transitions between levels (called electric dipole, electric quadrupole, and magnetic dipole) and therefore the transitions are divided into allowed ones (with a high probability) and so-called forbidden transitions (for transitions where the

For the first order allowed magnetic dipole radiation the free atom selection rules are still quite valid: Δ =Δ = Δ = ± ≠ *SL J* 0 and 0, 1 0 0 ( ) . In the electric dipole case we have: Δ =± *l* 1 , Δ = *S* 0 , Δ*L* , Δ ≤ *J* 2*l* . However, since ED transitions is induced by the crystal field, the free

*3.2.1. Radiative transitions: Intensities of optical transitions* 

have four dominant sources of optical radiation in RE ions spectra: i. Forced electric dipole induced by odd terms of the crystal field. ii. Forced electric dipole radiation induiced by lattices radiation.

less probable still in comparison with the magnetic dipole.

electric dipole transition is quantum mechanically forbidden).

the electron spin and the magnetic field.

iii. Allowed magnetic dipole transition. iv. Allowed electric quadrupole radiation.

ions.

**Figure 9.** The effect of spin-orbit and crystal field splitting on the energy levels of the Er3+ ion in silica matrix. The energy diagram shows the hierarchy of splitting resulting from electron-electron and electron-host interactions. Figure extracted from A.J. Kenyon [47].

As mentioned previously, the intra-4f transitions are parity forbidden and are made partially allowed by crystal field interactions mixing opposite parity wavefunctions, resulting in luminescence lifetimes long (often in the millisecond range), and linewidths narrow. The selection of an appropriate ion with intense and narrow-band emission can be obtained across much of the visible region and into the near-infrared. Figure 10 shows energy level diagrams for the isolated 3+ ions of each of the 13 lanthanides with partially filled 4f orbitals.

Almost all the RE ions their emission is due to optical transitions within the f-manifold (e.g. Tb3+ (4f8), Gd3+ (4f7) and Eu3+ (4f7). The 4f-electrons are well shielded from the chemical environment and therefore have almost retained their atomic character. Nevertheless, for a number of RE ions, also broad emission bands are known. Prominent examples are Eu2+ and Ce3+. Here, the emission is from to 5d-4f optical transitions. As electrons participate in the chemical bonding, the d-f emission spectra consist of broad bands. These transition types are allowed and are consequently very fast (a few µs or less).

Some line emission is not a specific property of RE ions and in addition to that, in the case of RE ions, broad emission spectra can be obtained as well, depending on the optical transitions involved. In this respect, RE ions are not unique. The unique properties of the RE ions originate from the fact that the spectral position of the emission lines is almost independent of the host lattice, in contrast to line emission generate by the emission of metal ions.

### *3.2.1. Radiative transitions: Intensities of optical transitions*

298 Plasmonics – Principles and Applications

electron–phonon coupling is weak. Although some of the divalent species also exhibit luminescence (e.g., samarium and europium), it is the trivalent ions that are of most interest.

**Figure 9.** The effect of spin-orbit and crystal field splitting on the energy levels of the Er3+ ion in silica matrix. The energy diagram shows the hierarchy of splitting resulting from electron-electron and

As mentioned previously, the intra-4f transitions are parity forbidden and are made partially allowed by crystal field interactions mixing opposite parity wavefunctions, resulting in luminescence lifetimes long (often in the millisecond range), and linewidths narrow. The selection of an appropriate ion with intense and narrow-band emission can be obtained across much of the visible region and into the near-infrared. Figure 10 shows energy level diagrams for the isolated 3+ ions of each of the 13 lanthanides with partially

Almost all the RE ions their emission is due to optical transitions within the f-manifold (e.g. Tb3+ (4f8), Gd3+ (4f7) and Eu3+ (4f7). The 4f-electrons are well shielded from the chemical environment and therefore have almost retained their atomic character. Nevertheless, for a number of RE ions, also broad emission bands are known. Prominent examples are Eu2+ and Ce3+. Here, the emission is from to 5d-4f optical transitions. As electrons participate in the chemical bonding, the d-f emission spectra consist of broad bands. These transition types are

electron-host interactions. Figure extracted from A.J. Kenyon [47].

allowed and are consequently very fast (a few µs or less).

filled 4f orbitals.

The terms shown in equation (13) are time-dependent; thereby, they do not lead to stationary states of the system. This way, they are treated using time-dependent perturbation theory resulting in transitions between the states established by the static interactions. In luminescent devices the most important term is *VEF* which gives rise to the emission and absorption of photons from decay radiative of the RE ions. This involves the interaction between the electron charge and the electric field and, the interaction between the electron spin and the magnetic field.

The experimental data on spectra of RE ions show that the radiation is mostly electric dipole (ED) nature, though some cases of the magnetic dipole radiation are also observed. Since the optical transitions take place between levels of a particular 4fN configuration, the electric dipole radiation is forbidden in first order, because the electric dipole operator has uneven parity and the transition matrix element must have even parity (Laporte selection rules). Van Vleck [55] pointed out the dipole electric radiation can only occur because the 4fN states have admixtures of 4fN-1nl configuration (nl will be mostly 5d), thus the 4fN-1nl has to be chosen such that it has opposite parity from 4fN. The admixture is produced via interactios that have odd parity and this depend of the host matrix of the RE ions. However, we can have four dominant sources of optical radiation in RE ions spectra:


In free atoms, magnetic dipole is about six orders of magnitude weaker than the electric dipole radiation. This latter occurs only as a consequence of a perturbation, both kinds of radiation shows up in the RE spectra with about intensity. Then, quadrupole radiation are less probable still in comparison with the magnetic dipole.

There are different types of transitions between levels (called electric dipole, electric quadrupole, and magnetic dipole) and therefore the transitions are divided into allowed ones (with a high probability) and so-called forbidden transitions (for transitions where the electric dipole transition is quantum mechanically forbidden).

For the first order allowed magnetic dipole radiation the free atom selection rules are still quite valid: Δ =Δ = Δ = ± ≠ *SL J* 0 and 0, 1 0 0 ( ) . In the electric dipole case we have: Δ =± *l* 1 , Δ = *S* 0 , Δ*L* , Δ ≤ *J* 2*l* . However, since ED transitions is induced by the crystal field, the free

Localized Surface Plasmon Resonances: Noble Metal Nanoparticle Interaction with Rare-Earth Ions 301

atom selection rules break down almost completely, i.e., the selection rules with regard to the *L, S* and *J* quantum numbers are now essential governed by the crystal field interaction yielding Δ ≤ *J* 6 (approximately). For the electric quadrupole case the rules selection is:

We now outline the Judd-Ofelt theory for the determination of the intensities in RE crystal spectra [56,57]. It is in essence a quantification of the ideas formulated by van Vleck [55] under optical radiation sources. Intensities are often expressed in terms of oscillator strengths, where the total oscillator strength for a transition from an energy level *a* to an

> ( )<sup>2</sup> <sup>2</sup> <sup>2</sup> <sup>2</sup> <sup>8</sup> , 32 1 <sup>9</sup> *ed md m n f ab S nS h J n*

the frequency of the transition *a*

host. *Sed* and *Smd* are the electric dipole and magnetic dipole intensities respectively and are

= + <sup>+</sup>

<sup>+</sup>

<sup>2</sup> <sup>2</sup>

*N N*

<sup>2</sup> <sup>2</sup> \* , ,

( )<sup>2</sup> <sup>2</sup> 4 3

*n n*

λ

2 1 *kq k*

*B Y k*

λ

*N N*

′ ′ =Ω + ′ ′′ ′

→

α

<sup>+</sup> (20)

3

( ) <sup>2</sup>

α

(18)

*b*, *n* the refraction index of

(19)

α*J* to

(21)

( ) ( )

υ

(, )

αα

′ ′*J* . Those are defined as:

α α

α α

*ed*

*md*

2,4,6

λ

=

λ Ω= + λ

( ) ( )

λ

=

 β

2,4,6

(, ) 2

λ

( )

2 1

λ

*S JJ f SL J U f S L J*

α

*S JJ f SL J L S f S L J*

α

) is a tensor operator of rank λ, and the sum runs over the three terms values 2, 4 and 6 of

,

 parameters have so far been assumed to arise solely from crystal field; however, they also contain contributions from admixtures by the lattice vibrations. The asterisk means that

<sup>+</sup> ′ ′ =+ = <sup>+</sup> <sup>+</sup>

*k q*

*k q*, *B* the radial integral is to be taken between states 4f and nl instead of 4f and 4f. Also, it is interesting to determine the coefficient for spontaneous light from state *a* ( )

> 2 <sup>2</sup> <sup>64</sup> , 3 21 <sup>9</sup> *ed md ed md*

Here *c* velocity of light in vacuum. The Judd-Ofelt formalism has been applied to the analysis of a number of systems. In most of these analyses the crystal field splitting of the

*AJ J A A S nS hc J* π ν

′ ′ = Ω ′ ′′ ′

π υ

Δ= Δ Δ≤ *S LJ* 0, , 2 .

energy level *b* is given:

Here *h* is the Planck constant,

defined as:

*U*(λ

λ. With:

The Ωλ

for \*

state *b* ( ) α

**Figure 10.** Energy levels of the triply charged lanthanide ions. Besides, the most technologically important radiative transitions are labeled. Figure adapted from A.J. Kenyon [47].

atom selection rules break down almost completely, i.e., the selection rules with regard to the *L, S* and *J* quantum numbers are now essential governed by the crystal field interaction yielding Δ ≤ *J* 6 (approximately). For the electric quadrupole case the rules selection is: Δ= Δ Δ≤ *S LJ* 0, , 2 .

300 Plasmonics – Principles and Applications

**Figure 10.** Energy levels of the triply charged lanthanide ions. Besides, the most technologically

important radiative transitions are labeled. Figure adapted from A.J. Kenyon [47].

We now outline the Judd-Ofelt theory for the determination of the intensities in RE crystal spectra [56,57]. It is in essence a quantification of the ideas formulated by van Vleck [55] under optical radiation sources. Intensities are often expressed in terms of oscillator strengths, where the total oscillator strength for a transition from an energy level *a* to an energy level *b* is given:

$$f\left(a,b\right) = \frac{8\pi^2 m\nu}{3h\left(2J+1\right)} \left[\frac{\left(n^2+2\right)^2}{9n}S\_{el} + nS\_{md}\right] \tag{18}$$

Here *h* is the Planck constant, υ the frequency of the transition *a*→*b*, *n* the refraction index of host. *Sed* and *Smd* are the electric dipole and magnetic dipole intensities respectively and are defined as:

$$\begin{aligned} \left| S\_{cl}(\alpha \mathbf{I}, \alpha' \mathbf{I}') = \sum\_{\lambda = 2, 4, 6} \mathfrak{Q}\_{\lambda} \left\langle f^N \alpha \left[ S \mathbf{L} \right] f \right| \left\| \mathbf{I}^{(\lambda)} \right\| f^N \alpha' \left[ S' \mathbf{L}' \right] f \right\rangle^2 \\ \left| S\_{md}(\alpha \mathbf{I}, \alpha' \mathbf{I}') = \mathfrak{P}^2 \sum\_{\lambda = 2, 4, 6} \mathfrak{Q}\_{\lambda} \left\langle f^N \alpha \left[ S \mathbf{L} \right] f \right| \left\| \mathbf{L} + 2 \mathbf{S} \right\| f^N \alpha' \left[ S' \mathbf{L}' \right] f \right\rangle^2 \end{aligned} \tag{19}$$

*U*(λ) is a tensor operator of rank λ, and the sum runs over the three terms values 2, 4 and 6 of λ. With:

$$\mathfrak{U}\mathfrak{Q}\_{\lambda} = \left(2\mathcal{X} + 1\right) \sum\_{k,q} \frac{\left|B\_{k,q}\right|^2 \left|Y\_{k,\lambda}\right|^2}{2k+1} \tag{20}$$

The Ωλ parameters have so far been assumed to arise solely from crystal field; however, they also contain contributions from admixtures by the lattice vibrations. The asterisk means that for \* *k q*, *B* the radial integral is to be taken between states 4f and nl instead of 4f and 4f.

Also, it is interesting to determine the coefficient for spontaneous light from state *a* ( ) α*J* to state *b* ( ) α′ ′*J* . Those are defined as:

$$A\left(\alpha\mathfrak{I}, \alpha'\mathfrak{I}'\right) = A\_{ed} + A\_{md} = \frac{64\pi^4\nu^3}{3\hbar c^2 \left(2J + 1\right)} \left[\frac{n\left(n^2 + 2\right)^2}{9}\mathcal{S}\_{ed} + n^3\mathcal{S}\_{md}\right] \tag{21}$$

Here *c* velocity of light in vacuum. The Judd-Ofelt formalism has been applied to the analysis of a number of systems. In most of these analyses the crystal field splitting of the terms is neglected; therefore, the total absorption intensities between the ground term and the excited terms are analyzed with only three empirical parameters ( ) 2,4 and 6 λ Ω = λ.

If *b* is an excited state that decays only by the emission of photons, its observed that relaxation rate is the sum of the probabilities for transitions to all possible final states. The total rate is the reciprocal of the excited-state lifetime τ*<sup>a</sup>*

$$\pi\_a = \frac{1}{A\left(\alpha \mathfrak{J}, \alpha' \mathfrak{J}'\right)}\tag{22}$$

Localized Surface Plasmon Resonances: Noble Metal Nanoparticle Interaction with Rare-Earth Ions 303

decrease the efficiency of the optical device. For example, depending on the phonon energies of the host matrix, some of the level lifetimes can be strongly quenched by multi-phonon transitions. Such effects are minimized in low-phonon-energy host media such as fluoride fibers. The effect of internal loss is most dramatic in distributed amplifiers, where pump light

Additionally, into the glass matrix can happen diverse kinds of interactions, in particular dipole–dipole interactions between the RE ions similar or of different species, allow energy transfer between those. This is exploited e.g. in Er3+–Yb3+-codoped fibers, where the pump radiation is dominantly absorbed by Yb3+ ions and mostly transferred to Er3+ ions [48].

In this way, RE-doped glasses shown are excellent materials that provide gain media. This makes them ideal candidates for embedding metallic NPs, resulting in new materials for the fabrication of devices optical with large enhancements even when the gain is saturated due light localization effects. It has been shown theoretically and experimentally that homogeneous aggregates of structures supporting LSPR can lead to extremely large enhancement of local field amplitudes exceeding those of single structures [60,61]. Besides, inorganic glasses are the host matrix for metallic NPs formation. The wide temperature range of glass viscosity growth provides the possibility to control over the NPs size within the wide range by means of modifying the temperature and duration of thermal processing. In fact, only such kind of matrix makes it possible to control and investigate all the stages of

The LSPR in metallic NPs is predicted to exhibit a singularity when the surrounding dielectric medium has a critical value of optical gain [63]. This singularity is obtained from

Therefore, the strong local electric field induced by the NP (LSPR) can enhance the total electric field, and can also improve the quantum yield of the luminenescence of the RE ions from the NP (this strong enhancement of electromagnetic fields is essential in nonlinear optics effects) [64,65]. Thus, the enhancement obtained with these NPs is due to the

electrons density, and *e* is the electronic charge [12]. Therefore, we obtained a modification in the local electric field by these dipoles (local field correction). Using the results obtained

> ( ) () { ( ) } 2 22 <sup>0</sup> 0 00 *eff* 2 1 / 3 1 /3 / 3 *P P E qq*

*i E* = + + − −+

amplitude *E0*. In the presence of an electromagnetic wave we have LSPR which forms EDs

 ω

is the dielectric constant in the presence of an external electromagnetic field of

 ε ω γω

(24)

 ε

formation of EDs which generates a polarization given by *P qeN x NP* <sup>=</sup> ' , where *<sup>x</sup>*

≈ − , the NP absorption presents a maximum. This is so-

6 is the NP specific volume, *N*' is the conduction

, some of which will contribute to the luminescence

is the

must travel long distances in the process of distributing gain.

NPs formation, including the starting stage [14,62].

εω

 ε

called the Frohlich condition is associated with dipole mode [1].

π

by O.L. Malta et al [66], the effective electric field can be written as:

 ω

the equation (4), for Re ( ) 2 *NP <sup>m</sup>*

distance between each NP , <sup>3</sup> *q d* <sup>=</sup>

separated by different distance *r*

Where 0 ε

enhancement.

ε

The branching ratio *βa,b*, for the transition *b* → *a* is the fraction of all spontaneous decay processes that occur through that channel and is defined as follows:

$$\beta\_{a,b} = \frac{A\left(a,b\right)}{\sum\_{c} A\left(a,c\right)} = A\left(a,b\right)\pi\_a \tag{23}$$

The branching ratio, which has an important influence on the performance of a device based on a particular transition, appears often in the discussion of specific ions. It has a significant effect on the threshold of a laser and the efficiency of an amplifier.

### **4. Metallic nanoparticle embedded in a gain media: LSPR interaction radiative transitions**

The RE-doped laser crystals and glasses are among the most popular solid-state gain media. In order of maintaining the efficiency of these materials in which the RE dopant is uniformily dispersed, as in multi-component glasses. In this sense, glasses have good optical, mechanical, and thermal properties to withstand the severe operating conditions of optical amplifier. Desirable properties include hardness, chemical inertness, absence of internal strain and refractive index variations, resistance to radiation-induced color centers, and ease of fabrication. Furthermore, the matrix host composition affects the solubility of the RE dopant, this affect the lifetime, absorption, emission, and excited state absorption cross sections of the dopant transitions [47]. This competing absorption phenomenon can seriously diminish the efficiency of an optical device. For example, a decrease in the excited-state absorption for Er3+ doped fibers going from a germane-silicate host to an alumina-silicate host has been verified [58], demonstrating the importance of host selection for a given RE ion or laser transition. Host glasses compatible with this relatively high concentration of RE without clustering require the open, chain-like structure of phosphate glasses or the addition of modifier ions (Ca, Na, K, Li, or other) to open the silicate structure and increase solubility [48]. The limitation owing to clustering in a predominantly silica host without modifier ions has been well documented [19]. The maximum erbium concentration in silica for optimum amplifier performance has been suggested to be lesser to 100 ppm [59]. However a 14.4-dB gain, 900-ppm erbium-doped silica fiber amplifier has been reported, indicating that higher concentrations can produce useful devices [48]. Nevertheless, the interactions between host and RE ions, it is necessary to consider background losses from impurity absorption and scattering mechanisms that decrease the efficiency of the optical device. For example, depending on the phonon energies of the host matrix, some of the level lifetimes can be strongly quenched by multi-phonon transitions. Such effects are minimized in low-phonon-energy host media such as fluoride fibers. The effect of internal loss is most dramatic in distributed amplifiers, where pump light must travel long distances in the process of distributing gain.

302 Plasmonics – Principles and Applications

**radiative transitions** 

total rate is the reciprocal of the excited-state lifetime τ*<sup>a</sup>*

The branching ratio *βa,b*, for the transition *b* 

terms is neglected; therefore, the total absorption intensities between the ground term and the excited terms are analyzed with only three empirical parameters ( ) 2,4 and 6

If *b* is an excited state that decays only by the emission of photons, its observed that relaxation rate is the sum of the probabilities for transitions to all possible final states. The

τ

processes that occur through that channel and is defined as follows:

β

effect on the threshold of a laser and the efficiency of an amplifier.

( ) 1 , *<sup>a</sup> A J J*

*A a b*

τ

α α

→

( ) ( ) ( ) , , , , *a b <sup>a</sup>*

*Aac*

The branching ratio, which has an important influence on the performance of a device based on a particular transition, appears often in the discussion of specific ions. It has a significant

The RE-doped laser crystals and glasses are among the most popular solid-state gain media. In order of maintaining the efficiency of these materials in which the RE dopant is uniformily dispersed, as in multi-component glasses. In this sense, glasses have good optical, mechanical, and thermal properties to withstand the severe operating conditions of optical amplifier. Desirable properties include hardness, chemical inertness, absence of internal strain and refractive index variations, resistance to radiation-induced color centers, and ease of fabrication. Furthermore, the matrix host composition affects the solubility of the RE dopant, this affect the lifetime, absorption, emission, and excited state absorption cross sections of the dopant transitions [47]. This competing absorption phenomenon can seriously diminish the efficiency of an optical device. For example, a decrease in the excited-state absorption for Er3+ doped fibers going from a germane-silicate host to an alumina-silicate host has been verified [58], demonstrating the importance of host selection for a given RE ion or laser transition. Host glasses compatible with this relatively high concentration of RE without clustering require the open, chain-like structure of phosphate glasses or the addition of modifier ions (Ca, Na, K, Li, or other) to open the silicate structure and increase solubility [48]. The limitation owing to clustering in a predominantly silica host without modifier ions has been well documented [19]. The maximum erbium concentration in silica for optimum amplifier performance has been suggested to be lesser to 100 ppm [59]. However a 14.4-dB gain, 900-ppm erbium-doped silica fiber amplifier has been reported, indicating that higher concentrations can produce useful devices [48]. Nevertheless, the interactions between host and RE ions, it is necessary to consider background losses from impurity absorption and scattering mechanisms that

*A ab*

**4. Metallic nanoparticle embedded in a gain media: LSPR interaction** 

= =

*c*

λ Ω = λ

<sup>=</sup> ′ ′ (22)

(23)

 *a* is the fraction of all spontaneous decay

.

Additionally, into the glass matrix can happen diverse kinds of interactions, in particular dipole–dipole interactions between the RE ions similar or of different species, allow energy transfer between those. This is exploited e.g. in Er3+–Yb3+-codoped fibers, where the pump radiation is dominantly absorbed by Yb3+ ions and mostly transferred to Er3+ ions [48].

In this way, RE-doped glasses shown are excellent materials that provide gain media. This makes them ideal candidates for embedding metallic NPs, resulting in new materials for the fabrication of devices optical with large enhancements even when the gain is saturated due light localization effects. It has been shown theoretically and experimentally that homogeneous aggregates of structures supporting LSPR can lead to extremely large enhancement of local field amplitudes exceeding those of single structures [60,61]. Besides, inorganic glasses are the host matrix for metallic NPs formation. The wide temperature range of glass viscosity growth provides the possibility to control over the NPs size within the wide range by means of modifying the temperature and duration of thermal processing. In fact, only such kind of matrix makes it possible to control and investigate all the stages of NPs formation, including the starting stage [14,62].

The LSPR in metallic NPs is predicted to exhibit a singularity when the surrounding dielectric medium has a critical value of optical gain [63]. This singularity is obtained from the equation (4), for Re ( ) 2 *NP <sup>m</sup>* εω ε ≈ − , the NP absorption presents a maximum. This is socalled the Frohlich condition is associated with dipole mode [1].

Therefore, the strong local electric field induced by the NP (LSPR) can enhance the total electric field, and can also improve the quantum yield of the luminenescence of the RE ions from the NP (this strong enhancement of electromagnetic fields is essential in nonlinear optics effects) [64,65]. Thus, the enhancement obtained with these NPs is due to the formation of EDs which generates a polarization given by *P qeN x NP* <sup>=</sup> ' , where *<sup>x</sup>* is the distance between each NP , <sup>3</sup> *q d* <sup>=</sup> π 6 is the NP specific volume, *N*' is the conduction electrons density, and *e* is the electronic charge [12]. Therefore, we obtained a modification in the local electric field by these dipoles (local field correction). Using the results obtained by O.L. Malta et al [66], the effective electric field can be written as:

$$\overrightarrow{E}\_{\text{eff}} = \left(\varepsilon\_0 + 2\right) \left[1 + q\alpha\_p^2 \left/\left\{3\varepsilon\_0 \left[\left(1 - q\right)\left(\alpha\_p^2 \mid \, 3\varepsilon\_0\right) - \alpha^2 + i\gamma\alpha\right]\right\}\right] \overrightarrow{E}\_0 \,\, / \,\tag{24}$$

Where 0 ε is the dielectric constant in the presence of an external electromagnetic field of amplitude *E0*. In the presence of an electromagnetic wave we have LSPR which forms EDs separated by different distance *r* , some of which will contribute to the luminescence enhancement.

Assuming that the RE ions may occupy different sites in the host, a direct coupling between the excited states of the RE ions and the NPs modifies the Stark levels energies [12,13,67]. As the NPs just give a contribution to the local field when the light is present, the oscillator strength of a spectral line, corresponding to transition from the ion ground level *i* to the component *f* of the excited level can be re-write as:

$$f\left(a,b\right) = \frac{8\pi^2 m\nu}{3h\left(2f+1\right)} \left[\frac{\left(n^2+2\right)^2}{9n} S\_{cd} + \sum\_{\lambda} \left| \left< i \right| D\_{\lambda}^{(1)} \left| f \right> \right|^2 + nS\_{md} \right] \tag{25}$$

Localized Surface Plasmon Resonances: Noble Metal Nanoparticle Interaction with Rare-Earth Ions 305

(a) (b)

(c) (d)

(e) (f)

(g)

The second term into parenthesis is added to the equation from the Judd-Ofelt theory [56,57], and represents the ED transition due to LSPR of NP. To obtain non vanishing matrix elements of the components (1) *D*λ *NP* it is necessary to admix into *i* and *f* other states of opposite parity.

$$\begin{aligned} \text{Thus, the initial and final states are:}\\ \left\langle \left| f \right> = \left| \phi\_f \right> + \sum\_{\beta} \left\langle \phi\_{\beta} \right| V \left| \phi\_{\beta} \right> \left( E\_f - E\_{\beta} \right) \left| \phi\_{\beta} \right> , \text{ then:}\\ \left\langle i \left| \overline{P} + \overline{P}\_{NP} \right| f \right> = \sum\_{\beta} \left\langle \frac{\left\langle \phi\_i \middle| V \left| \phi\_{\beta} \right> \left\langle \phi\_{\beta} \middle| \overline{P} \right> \left| \phi\_f \right>}{E\_i - E\_{\beta}} + \frac{\left\langle \phi\_i \middle| \overline{P} \middle| \phi\_{\beta} \right> \left\langle \phi\_{\beta} \middle| V \left| \phi\_f \right>}{E\_f - E\_{\beta}} \right> + \frac{\left\langle \phi\_i \middle| V \middle| \phi\_{\beta} \right> \left\langle \phi\_{\beta} \middle| \overline{P} \middle| \phi\_f \right>}{E\_i - E\_{\beta}} \dots \end{aligned} \tag{26}$$

$$\ldots + \frac{\left\langle \phi\_i \middle| \overline{P}\_{NP} \middle| \phi\_{\beta} \middle> \left\langle \phi\_f \middle| V \middle| \phi\_{\beta} \right> \right\rangle}{E\_f - E\_{\beta}}$$

Where *V* is the crystalline field, equation (16), responsible for the Stark levels and treated as a perturbation, φ*<sup>i</sup>* and φ*f* have the same parity, φβ has opposite parity in relation of φ*i* and φ*f*, *P* is the ED from the light interaction. For example, Er3+ ions presented an electronic configuration [Xe]4f11, with *J =*15/2 been the spin-orbit ground state because the f shell is more than half-filled. The order of these Stark levels depends on the strength of the crystal field and the influence of *PNP* .

This can be observed experimentally in the luminescence spectra of different RE ions shown in Figure 11 interaction with silver or gold NPs in different host matrix, as a blue- or redshift (see equation 26) and a pronounced increase in luminescence (see equation 25).

In this scenario, the influence of *PNP* appeared in the luminescence spectra as a blue or a red -shift of the peaks, e.g. a transfer energy from the NPs dipoles to RE ions, thus modifying the Stark energy level's bandwidth, see Figure 11. This energy transfer is obtained from the second term added in equation (25) and can be understood through the following definition: ( ) (1) (1) ' , *q NP m NP m m m D rC*λ <sup>=</sup> θ φ , where ( ) (1) , *C*λ *NP m m* θ φis the Racah tensor

and where *rm* is the distance between the NPs and the REions ( *<sup>m</sup> r x* <sup>=</sup> ). 304 Plasmonics – Principles and Applications

elements of the components (1) *D*

β

β

φ*<sup>i</sup>* and φ

field and the influence of *PNP*

In this scenario, the influence of *PNP*

following definition: ( ) (1) (1)

 φφ

β

*NP*

...

opposite parity.

*f <sup>f</sup>*

= + φ

a perturbation,

*P* 

component *f* of the excited level can be re-write as:

( ) ( )

*V EE f f* /( )

φ φφ

<sup>−</sup> , then:

β β

φ

.

π υ

λ

β β

 φ

*NP i f f*

<sup>+</sup> <sup>−</sup>

 φ

*P V E E*

β β

*f* have the same parity,

' , *q NP m NP m m m D rC*

and where *rm* is the distance between the NPs and the REions ( *<sup>m</sup>*

λ<sup>=</sup>

 φφ

Thus, the initial and final states are: *i ii i V EE*

*iP P f E E E E E E*

+ = <sup>+</sup> <sup>+</sup> −− −

β

ββ

 φ

 φ φφ

*m n*

Assuming that the RE ions may occupy different sites in the host, a direct coupling between the excited states of the RE ions and the NPs modifies the Stark levels energies [12,13,67]. As the NPs just give a contribution to the local field when the light is present, the oscillator strength of a spectral line, corresponding to transition from the ion ground level *i* to the

> ( )<sup>2</sup> <sup>2</sup> <sup>2</sup> <sup>2</sup> (1) <sup>2</sup> <sup>8</sup> , 32 1 <sup>9</sup> *ed NP md*

= ++ <sup>+</sup>

λ

λ

*NP* it is necessary to admix into *i* and *f* other states of

 φφ

> φ φφ

has opposite parity in relation of

appeared in the luminescence spectra as a blue or a

 *NP m m* θ φ

*r x* <sup>=</sup>

).

 , where ( ) (1) , *C*λ

(26)

β

= + φ

> φ

*NP i f i f i f*

*if i*

β β

Where *V* is the crystalline field, equation (16), responsible for the Stark levels and treated as

φβ

 is the ED from the light interaction. For example, Er3+ ions presented an electronic configuration [Xe]4f11, with *J =*15/2 been the spin-orbit ground state because the f shell is more than half-filled. The order of these Stark levels depends on the strength of the crystal

This can be observed experimentally in the luminescence spectra of different RE ions shown in Figure 11 interaction with silver or gold NPs in different host matrix, as a blue- or red-

red -shift of the peaks, e.g. a transfer energy from the NPs dipoles to RE ions, thus modifying the Stark energy level's bandwidth, see Figure 11. This energy transfer is obtained from the second term added in equation (25) and can be understood through the

shift (see equation 26) and a pronounced increase in luminescence (see equation 25).

θ φ

*V P PV V P*

(25)

β

β β

 ββ/( )

> φ

<sup>−</sup> , and

 β  φ

...

φ*i* and φ*f*,

is the Racah tensor

<sup>+</sup>

*f ab S i D f nS h J <sup>n</sup>*

The second term into parenthesis is added to the equation from the Judd-Ofelt theory [56,57], and represents the ED transition due to LSPR of NP. To obtain non vanishing matrix

Localized Surface Plasmon Resonances: Noble Metal Nanoparticle Interaction with Rare-Earth Ions 307

(triangle). Inset Er decay of unpatterned (star) and Fibonacci arrays with different Δmin as specified in

(h). Picture on the left; PL of the TE025-Y samples pumped with diode laser at 980 nm, showing PL enhancement. The inserted figure shows a zoom of the peaks. The vertical dashed line is a reference for showing the blue-shift of the peaks, and the arrow indicates the enhancement due to transfer energy from EDs to Er3+. Picture on the right; same for the samples TE050-Y. The enhancement of luminescence was found to be reproducible for all ours samples. These pictures was extracted and modified from Ref.

(i). Luminescence spectra of Er3+:Au-doped tellurite glass for several annealing times, pumped at 980 nm. Picture on the left; Upconversion, for the range 400<λ< 900 nm. Picture on the right; Energy-levels diagram of Er3+ for the luminescence spectra. ET stands for energy transfer, CET for cooperative energy transfer, Q for quenching, GSA for ground state absorption, NR for non-radiative decay, and ESA for

As explained above, those NPs can be excited by a predefined incident radiation through a direct coupling between the excited states of the both RE ions and NPs, resulting in: (i) a

Joule effect) or (iii) a radiative energy release which depends on the albedo of the NPs. Thereby, the exact response of LSPR will depend on the details of the physical system (e.g. arrangement, shape, host matrix), and usually not strictly symmetric about the resonant frequency. A schematic representation of the interaction process within the RE:NP system is

Also we can elucidate the enhancement (quenching) from the process of energy transfer as following: Er3+ emission promotes energy transfers into a plasmonic mode, which can decay nonradiatively by heat generation (Joule effect) or radiatively by releasing energy that depends on the albedo of the NPs. We consider two types of emission from a system of

> 1 0 *p p I I* ∝ η η

2 10 *LSPR scatt II Q* ∝ η η

Here, *I1* is the intensity emission for a single emitter, *I2* is the intensity emission of the NP, *Ip*

η

( )

ωω

3 4

ω

− +

is the efficiency of the energy transfer (nonradiative) to LSPR modes

4 4

<sup>2</sup> 2 2 64

9

*y*

ω

 ω

*p p*

is the pump efficiency, 0

( )

= =

*scatt scatt*

Where ( ) *Ray Qscatt* is the Rayleigh scattering [67,68] and, *y =* 

Therefore the total intensity *IT*, can be written as [62]:

and the scattering efficiency of the plasmon at the emission wavelength *Qscatt* is:

8

*<sup>y</sup> Q Q*

*Ray p*

*<sup>p</sup>*, (ii) a nonradiatively decay (heat generation by

(27)

is the internal quantum efficiency of the

(28)

(29)

*D/c*, with *c* the speed of light.

ω

[12]. These spectrums are of Er3+-doped tellurite glass, more details see [12].

Nevertheless, an interesting question arises: how these NPs are excited?

the legend. For more details see Ref. [73].

excited state absorption. For more details see Ref. [13].

local field increase (Frohlich condition), at

depicted in Figure 12 (a).

identical dipoles [74]:

the pump intensity, *<sup>p</sup>*

energy transfer, *LSPR*

η

η

**Figure 11.** (a). Fluorescence spectra of the Eu3+ ions in the presence (a) and in the absence (b) of Ag NPs.The silver concentration is 7.5 in weight percent. The matrix glass composition can be found in [66]. (b). Frequency UC spectra for excitation at 980 nm, (for composition see Ref. [68]. After the cooling the samples were annealed for different durations (24, 48, and 72 h) in order to nucleate silver NPs [68]. (c). Upconversion spectra extracted from [69] under excitation wavelength at λex=798 nm (for composition and for amplification ratio see Ref. [69]. The bases of the emission curves (c), (d), and (e) have been uplifted for better visibility.

(d). PL spectra extracted from [70] (i) Eu(III)EDTA, 3H2O complex, (ii) Eu complex with Au nanoparticles and (iii) Eu complex with Au–ZnO nanoparticles.

(e). Upconversion spectra of Tm3+/Yb3+ codoped PbO–GeO2 samples containing silver NPs obtained by pumping the samples with a diode laser operating at 980 nm. For more details see [71].

(f). PL spectra (λexc≈337 nm) of sy-activated with 0.037 mol% Ag containing glasses co-doped with 0.19 mol% Sm (black curve), Dy (red), and Tb (blue), respectively, and of non-activated and non-annealed (n.a.) 0.037 mol% Ag containing samples co-doped with 0.19 mol% Sm (grey), Dy (magenta), and Tb (light blue) [72].

(g). Picture on the left, Integrated PL enhancement in periodic (circle) and Fibonacci (square) nanoparticle arrays of various interparticle separations. The top inset shows the representative PL spectrum of periodic (bottom), Fibonacci (top) nanoparticle arrays with Δmin= 50 nm and unpatterned area (middle),and the inset bottom shown the SEM micrograph of (left) periodic, (right) Fibonacci array Au nanocylinders. Picture on the right, PL lifetime of periodic (circle), Fibonacci (square), unpatterned

(triangle). Inset Er decay of unpatterned (star) and Fibonacci arrays with different Δmin as specified in the legend. For more details see Ref. [73].

(h). Picture on the left; PL of the TE025-Y samples pumped with diode laser at 980 nm, showing PL enhancement. The inserted figure shows a zoom of the peaks. The vertical dashed line is a reference for showing the blue-shift of the peaks, and the arrow indicates the enhancement due to transfer energy from EDs to Er3+. Picture on the right; same for the samples TE050-Y. The enhancement of luminescence was found to be reproducible for all ours samples. These pictures was extracted and modified from Ref. [12]. These spectrums are of Er3+-doped tellurite glass, more details see [12].

(i). Luminescence spectra of Er3+:Au-doped tellurite glass for several annealing times, pumped at 980 nm. Picture on the left; Upconversion, for the range 400<λ< 900 nm. Picture on the right; Energy-levels diagram of Er3+ for the luminescence spectra. ET stands for energy transfer, CET for cooperative energy transfer, Q for quenching, GSA for ground state absorption, NR for non-radiative decay, and ESA for excited state absorption. For more details see Ref. [13].

Nevertheless, an interesting question arises: how these NPs are excited?

306 Plasmonics – Principles and Applications

have been uplifted for better visibility.

(light blue) [72].

nanoparticles and (iii) Eu complex with Au–ZnO nanoparticles.

(h)

(i) **Figure 11.** (a). Fluorescence spectra of the Eu3+ ions in the presence (a) and in the absence (b) of Ag NPs.The silver concentration is 7.5 in weight percent. The matrix glass composition can be found in [66]. (b). Frequency UC spectra for excitation at 980 nm, (for composition see Ref. [68]. After the cooling the samples were annealed for different durations (24, 48, and 72 h) in order to nucleate silver NPs [68]. (c). Upconversion spectra extracted from [69] under excitation wavelength at λex=798 nm (for composition and for amplification ratio see Ref. [69]. The bases of the emission curves (c), (d), and (e)

(d). PL spectra extracted from [70] (i) Eu(III)EDTA, 3H2O complex, (ii) Eu complex with Au

(g). Picture on the left, Integrated PL enhancement in periodic (circle) and Fibonacci (square) nanoparticle arrays of various interparticle separations. The top inset shows the representative PL spectrum of periodic (bottom), Fibonacci (top) nanoparticle arrays with Δmin= 50 nm and unpatterned area (middle),and the inset bottom shown the SEM micrograph of (left) periodic, (right) Fibonacci array Au nanocylinders. Picture on the right, PL lifetime of periodic (circle), Fibonacci (square), unpatterned

pumping the samples with a diode laser operating at 980 nm. For more details see [71].

(e). Upconversion spectra of Tm3+/Yb3+ codoped PbO–GeO2 samples containing silver NPs obtained by

(f). PL spectra (λexc≈337 nm) of sy-activated with 0.037 mol% Ag containing glasses co-doped with 0.19 mol% Sm (black curve), Dy (red), and Tb (blue), respectively, and of non-activated and non-annealed (n.a.) 0.037 mol% Ag containing samples co-doped with 0.19 mol% Sm (grey), Dy (magenta), and Tb

As explained above, those NPs can be excited by a predefined incident radiation through a direct coupling between the excited states of the both RE ions and NPs, resulting in: (i) a local field increase (Frohlich condition), at ω*<sup>p</sup>*, (ii) a nonradiatively decay (heat generation by Joule effect) or (iii) a radiative energy release which depends on the albedo of the NPs. Thereby, the exact response of LSPR will depend on the details of the physical system (e.g. arrangement, shape, host matrix), and usually not strictly symmetric about the resonant frequency. A schematic representation of the interaction process within the RE:NP system is depicted in Figure 12 (a).

Also we can elucidate the enhancement (quenching) from the process of energy transfer as following: Er3+ emission promotes energy transfers into a plasmonic mode, which can decay nonradiatively by heat generation (Joule effect) or radiatively by releasing energy that depends on the albedo of the NPs. We consider two types of emission from a system of identical dipoles [74]:

$$I\_1 \approx I\_p \eta\_p \eta\_0 \tag{27}$$

$$I\_2 \approx I\_1 \eta\_0 \eta\_{LSPR} Q\_{scatt} \tag{28}$$

Here, *I1* is the intensity emission for a single emitter, *I2* is the intensity emission of the NP, *Ip* the pump intensity, *<sup>p</sup>* η is the pump efficiency, 0 η is the internal quantum efficiency of the energy transfer, *LSPR* η is the efficiency of the energy transfer (nonradiative) to LSPR modes and the scattering efficiency of the plasmon at the emission wavelength *Qscatt* is:

$$\mathcal{Q}\_{scatt} = \mathcal{Q}\_{scatt}^{(Rag)} = \frac{8}{3} \frac{\alpha\_p^4 y^4}{\left(\rho^2 - \alpha\_p^2\right)^2 + \frac{4}{9} y^6 \phi\_p^{\*4}} \tag{29}$$

Where ( ) *Ray Qscatt* is the Rayleigh scattering [67,68] and, *y =* ω*D/c*, with *c* the speed of light. Therefore the total intensity *IT*, can be written as [62]:

$$I\_T = I\_1 \left(1 - \eta\_{LSPR}\right) + \eta\_0 I\_2 \tag{30}$$

Localized Surface Plasmon Resonances: Noble Metal Nanoparticle Interaction with Rare-Earth Ions 309

or neutral dimmers and multimers. Consequently, the contribution of the latter species to the RE luminescence enhancement and band shape features cannot be excluded [14], Figure 11. Moreover, the insertion of silver or gold in the samples leads to strong modification of the glassy network and consequently on the RE local environment, this is verified in

This chapter present results where it was demonstrated the simultaneous exploitation of the enhanced local field due to NPs and energy transfer processes in order to enhance the luminescence spectra of a glassy composite material, and others. Besides, from these nanoparticles it is possible to modulate the down/up-conversion emission of the REs with applicability in areas such as optical telecommunication, including biomedical imaging and energy conversion. On the other hand, this significant enhanced fluorescence has high potential for application in photonics, optical displays, lasers and optical memory devices, amongst others. However, the success of new applications of nanoparticles depends on improvement in the understanding of the properties of LSPR and the environment around them. Further efforts and systematic studies must be realized in order to offer new developments to extend the analytical applications field of metallic nanoparticle interaction

We hope that this chapter stimulated our readers for the development of theoretical and

This work was financially supported by the Brazilian agencies FAPESP, CNPq and CEPOF/INOF. V.A.G. Rivera thanks to FAPESP for financial support (project 2009/08978-4 and 2011/21293-0) that allowed my pos-doctoral and my gratefully to Dr. Luiz Antonio Nunes of the Instituto de Fisica de São Carlos – University São Paulo - Brazil and the Dr. Yannick Ledemi and the Dr. Younnes Messaddeq of the Centre d'Optique, Photonique et

[1] Stefan Alexander Maier. Plasmonics Funadamentals and Applications. Springer

[3] Y. Wang, E. W. Plummer and K. Kempa. Advances in Physics 2011;60(5): 799-898.

*Instituto de Física de São Carlos, INOF/CEPOF, USP, São Carlos – São Paulo, Brazil* 

laser – University Laval- Canada for the discussions on this issue.

[2] Mark I. Stockman. Opt. Exp. 2011;19(22): 22029-22106.

reference [62].

**5. Conclusion** 

with rare-earth ions.

**Author details** 

**Acknowledgement** 

**6. References** 

experimental work on plasmonics and rare-earths.

V.A.G. Rivera, F.A. Ferri and E. Marega Jr.

Science+Business Media LLC; 2007.

We consider *I1* as the contribution from the emitter dipole (RE ions) uncoupled to the metal and *I2* the plasmon enhanced emission from coupled dipoles, see Figure 12 (b).

Finally, we can obtain the following conditions: <sup>2</sup> ( ) <sup>0</sup> <sup>1</sup> *Ray Qscatt* η <sup>≥</sup> , where ( ) ( ) , *Ray Q D scatt* ω , i.e. the luminescence enhancement depends on the incident radiation frequency and the NP size. But, as were mentioned, the incident radiation for the NP activation comes from the emitter dipole ( ) <sup>1</sup> *LSPR I* η and not from the pump radiation (*Ip*), i.e., from transition radiative from RE. The coupling efficiency *LSPR* η is defined by: /( ) *LSPR LSPR rad nonrad LSPR* η =Γ Γ +Γ +Γ , where: Γ*LSPR* is the energy transfer rate to the LSPR mode [73,74].

**Figure 12.** (a) Energy level scheme for a resonant and RE ion absorption. The dot line indicates the energy transfer between RE→NP or NP→ RE, and the vertical dot line shows the transition radiative under consideration. The curved arrows indicate non-radiative transitions. (b) Schematic representation of the system RE:NP. A monochromatic plane wave with pump intensity (*Ip*) which is proportional to the pump wavelength λpump, induces the following processes: (i) absorption of RE ion, *Ip*, (ii) activation of NPs, due the coupling RE:NP (transitions levels), 1 *LSPR I* η (iii) NP transmitter, via electric dipole. Such coupling depends on the coupling efficiency *LSPR* η . Where *I1* the intensity emission of emitter, *I2* the intensity emission of the NP, η*LSPR* the efficiency of the energy transfer (nonradiative) to LSPR modes. Equipotential surfaces (electric dipole coupling) with electric potentials -V/2 and V/2 for the NPs and the Er3+ ions respectively.

The strong local electric field induced by NPs (ED), increases the quantum yield η of the RE luminescence, defined by the ratio of the local field *Eloc* and the incident field *Ei* , η =+ = ( ) *r ddE E <sup>m</sup>* / / *loc i* [14], here *EEE eff loc i* = + . Thus, the maximum field enhancement is determined by the shortest distance between two equipotential particles. It corroborates well with the demonstration in the references [12,13,67,75].

Another possible mechanism for the energy transfer is due to that part of silver or gold (small silver or gold aggregates) probably remained under the form of ions, atoms, charged or neutral dimmers and multimers. Consequently, the contribution of the latter species to the RE luminescence enhancement and band shape features cannot be excluded [14], Figure 11. Moreover, the insertion of silver or gold in the samples leads to strong modification of the glassy network and consequently on the RE local environment, this is verified in reference [62].

### **5. Conclusion**

308 Plasmonics – Principles and Applications

RE. The coupling efficiency *LSPR*

dipole ( ) <sup>1</sup> *LSPR I* η

( ) <sup>1</sup> 0 2 1 *<sup>T</sup> LSPR I I* =− + η

We consider *I1* as the contribution from the emitter dipole (RE ions) uncoupled to the metal

luminescence enhancement depends on the incident radiation frequency and the NP size. But, as were mentioned, the incident radiation for the NP activation comes from the emitter

**Figure 12.** (a) Energy level scheme for a resonant and RE ion absorption. The dot line indicates the energy transfer between RE→NP or NP→ RE, and the vertical dot line shows the transition radiative under consideration. The curved arrows indicate non-radiative transitions. (b) Schematic representation of the system RE:NP. A monochromatic plane wave with pump intensity (*Ip*) which is proportional to the pump wavelength λpump, induces the following processes: (i) absorption of RE ion, *Ip*, (ii) activation

η

η

The strong local electric field induced by NPs (ED), increases the quantum yield

modes. Equipotential surfaces (electric dipole coupling) with electric potentials -V/2 and V/2 for the NPs

determined by the shortest distance between two equipotential particles. It corroborates

Another possible mechanism for the energy transfer is due to that part of silver or gold (small silver or gold aggregates) probably remained under the form of ions, atoms, charged

and *I2* the plasmon enhanced emission from coupled dipoles, see Figure 12 (b).

Finally, we can obtain the following conditions: <sup>2</sup> ( )

η

where: Γ*LSPR* is the energy transfer rate to the LSPR mode [73,74].

of NPs, due the coupling RE:NP (transitions levels), 1 *LSPR I*

η

luminescence, defined by the ratio of the local field *Eloc*

well with the demonstration in the references [12,13,67,75].

[14], here *EEE eff loc i* = +

Such coupling depends on the coupling efficiency *LSPR*

the intensity emission of the NP,

and the Er3+ ions respectively.

 =+ = ( ) *r ddE E <sup>m</sup>* / / *loc i*

η

 η

η

<sup>0</sup> <sup>1</sup> *Ray Qscatt*

and not from the pump radiation (*Ip*), i.e., from transition radiative from

η

*I* (30)

=Γ Γ +Γ +Γ ,

(iii) NP transmitter, via electric dipole.

η

and the incident field *Ei*

. Thus, the maximum field enhancement is

of the RE

,

. Where *I1* the intensity emission of emitter, *I2*

*LSPR* the efficiency of the energy transfer (nonradiative) to LSPR

ω

, i.e. the

<sup>≥</sup> , where ( ) ( ) , *Ray Q D scatt*

is defined by: /( ) *LSPR LSPR rad nonrad LSPR*

This chapter present results where it was demonstrated the simultaneous exploitation of the enhanced local field due to NPs and energy transfer processes in order to enhance the luminescence spectra of a glassy composite material, and others. Besides, from these nanoparticles it is possible to modulate the down/up-conversion emission of the REs with applicability in areas such as optical telecommunication, including biomedical imaging and energy conversion. On the other hand, this significant enhanced fluorescence has high potential for application in photonics, optical displays, lasers and optical memory devices, amongst others. However, the success of new applications of nanoparticles depends on improvement in the understanding of the properties of LSPR and the environment around them. Further efforts and systematic studies must be realized in order to offer new developments to extend the analytical applications field of metallic nanoparticle interaction with rare-earth ions.

We hope that this chapter stimulated our readers for the development of theoretical and experimental work on plasmonics and rare-earths.

### **Author details**

V.A.G. Rivera, F.A. Ferri and E. Marega Jr. *Instituto de Física de São Carlos, INOF/CEPOF, USP, São Carlos – São Paulo, Brazil* 

### **Acknowledgement**

This work was financially supported by the Brazilian agencies FAPESP, CNPq and CEPOF/INOF. V.A.G. Rivera thanks to FAPESP for financial support (project 2009/08978-4 and 2011/21293-0) that allowed my pos-doctoral and my gratefully to Dr. Luiz Antonio Nunes of the Instituto de Fisica de São Carlos – University São Paulo - Brazil and the Dr. Yannick Ledemi and the Dr. Younnes Messaddeq of the Centre d'Optique, Photonique et laser – University Laval- Canada for the discussions on this issue.

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**Chapter 12** 

© 2012Suyama et al., licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

© 2012 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution,

**Resonant Excitation of Plasmons in Bi-Gratings** 

Taikei Suyama, Akira Matsushima, Yoichi Okuno and Toyonori Matsuda

Metal gratings have an interesting property known as resonance absorption in the optics region [Raeter 1982], which causes partial or total absorption of incident light energy. This absorption is associated with the resonant excitation of plasmons on a grating surface; incident light couples with surface plasmons via an evanescent spectral order generated by the grating [Nevièr 1982]. Resonance absorption in metal film gratings has been the subject of many theoretical [Nevièr 1982] and experimental investigations focused on various applications including chemical sensing [DeGrandpre 1990, Zoran 2009], surface enhanced phenomena such as Raman scattering [Nemetz 1994], and photonic bandgaps [Barnes 1995,

A thin-film metal grating, which is a corrugated thin metal film, also results in absorption similar to that observed for thick gratings [Inagaki, Motosuga 1985, Chen 2008, Bryan-Brown 1991, Davis 2009]. Absorption in thin-film metal gratings, however, is much more complicated than in thick gratings because of the existence of coupled plasmon modes in addition to those observed in thick gratings. If the metal film is sufficiently thick, singleinterface surface plasmons (SISPs) alone are excited [Raeter 1977, Okuno 2006, Suyama 2009]. However, if the film is sufficiently thin, simultaneous excitation of surface plasmons occurs on both surfaces; these plasmons interfere with each other and produce two coupled plasmon modes, short-range surface plasmons (SRSPs) and long-range surface plasmons

Most previous studies on resonance absorption have mainly dealt with metal gratings whose surfaces are periodic in one direction. Metal bi-gratings, which are periodic in two directions, also yield plasmon resonance absorption, similar to singly periodic gratings [Glass 1982, Glass 1983, Inagaki, Goudonnet 1985, Harris 1996]. In this work, we therefore investigated coupled plasmon modes excited in multilayered bi-gratings [Matsuda 1993, Matsuda 1996, Suyama 2010]. We anticipated interesting behavior in the resonance

and reproduction in any medium, provided the original work is properly cited.

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/50754

(LRSPs) [Chen 1988, Hibbins 2006].

**1. Introduction** 

Tan 1998].


## **Resonant Excitation of Plasmons in Bi-Gratings**

Taikei Suyama, Akira Matsushima, Yoichi Okuno and Toyonori Matsuda

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/50754

### **1. Introduction**

312 Plasmonics – Principles and Applications

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Metal gratings have an interesting property known as resonance absorption in the optics region [Raeter 1982], which causes partial or total absorption of incident light energy. This absorption is associated with the resonant excitation of plasmons on a grating surface; incident light couples with surface plasmons via an evanescent spectral order generated by the grating [Nevièr 1982]. Resonance absorption in metal film gratings has been the subject of many theoretical [Nevièr 1982] and experimental investigations focused on various applications including chemical sensing [DeGrandpre 1990, Zoran 2009], surface enhanced phenomena such as Raman scattering [Nemetz 1994], and photonic bandgaps [Barnes 1995, Tan 1998].

A thin-film metal grating, which is a corrugated thin metal film, also results in absorption similar to that observed for thick gratings [Inagaki, Motosuga 1985, Chen 2008, Bryan-Brown 1991, Davis 2009]. Absorption in thin-film metal gratings, however, is much more complicated than in thick gratings because of the existence of coupled plasmon modes in addition to those observed in thick gratings. If the metal film is sufficiently thick, singleinterface surface plasmons (SISPs) alone are excited [Raeter 1977, Okuno 2006, Suyama 2009]. However, if the film is sufficiently thin, simultaneous excitation of surface plasmons occurs on both surfaces; these plasmons interfere with each other and produce two coupled plasmon modes, short-range surface plasmons (SRSPs) and long-range surface plasmons (LRSPs) [Chen 1988, Hibbins 2006].

Most previous studies on resonance absorption have mainly dealt with metal gratings whose surfaces are periodic in one direction. Metal bi-gratings, which are periodic in two directions, also yield plasmon resonance absorption, similar to singly periodic gratings [Glass 1982, Glass 1983, Inagaki, Goudonnet 1985, Harris 1996]. In this work, we therefore investigated coupled plasmon modes excited in multilayered bi-gratings [Matsuda 1993, Matsuda 1996, Suyama 2010]. We anticipated interesting behavior in the resonance

© 2012Suyama et al., licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2012 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

phenomenon due to the presence of the double periodicity. Further, in view of the fact that layered gratings are interesting structures for optical device applications, we investigated a multilayered bi-gratings, which is a stack of thin-film bi-gratings made of a dielectric or metal. This paper is structured as follows. After formulating the problem in Section **2**, we briefly describe a method for obtaining a solution in Section **3**. Focusing our attention on the resonant excitation of plasmon modes, we then show the computational results in Section **4**, before presenting the conclusions of this study.

Resonant Excitation of Plasmons in Bi-Gratings 315

*i i*

*i* (2)

*μ* (3)

. The symbol *k* (= 2π /

(5)

λ) is the

> θand

*k* (4)

, , *e* (6)

θ

is the wavelength of the incident wave. We define

−

=

. The interface between V and V+<sup>1</sup> is denoted by S ( = 1,2, ,*L* ).

(1)

<sup>1</sup> <sup>2</sup><sup>π</sup> <sup>2</sup><sup>π</sup> sin sin 4 <sup>1</sup>

*<sup>h</sup> <sup>x</sup> <sup>y</sup> <sup>z</sup> <sup>η</sup> x,y <sup>p</sup> p e d d*

= = ++ + −

Here, *p* represents the phase of *S* in each direction, and *e* denotes the average distance between *S* and *<sup>l</sup>* <sup>1</sup> *S* <sup>+</sup> . The value *h* is regarded as the groove depth of the boundary *S* .

It should be noted that we can easily generalize the shape of *S* by distinguishing between *d*  and *p* in the *X*-and *Y*-directions, by writing them as *<sup>x</sup> d* , *<sup>y</sup> d* , *<sup>x</sup> p* and *<sup>y</sup> p* , respectively. A singly periodic grating is a special case where *<sup>y</sup> d* → ∞ in the doubly periodic case. In the present paper, we concentrate our attention on describing the analysis only for the doubly

( ) ( )

i

periodic case, since it reduces to the singly periodic case through a simple procedure.

i i

*H h*

i i exp *E e <sup>P</sup> k P*

= ⋅

( ) <sup>i</sup> i i <sup>1</sup> <sup>0</sup> *h ke* = × ω

Here, *P* is the position vector for an observation point P(*X, Y, Z*), and <sup>i</sup> *k* is the wave vector

<sup>i</sup> <sup>T</sup> = − α , , β γ

as polar and azimuth angles, respectively, as shown in Fig. 1(a), and the superscript "T"

The amplitude of the incident electric field can be decomposed into TE and TM modes

<sup>i</sup> TE TM *ee e* = + cosδ sin

TE <sup>T</sup> = − sin cos 0 ϕ

TM <sup>T</sup> <sup>=</sup> cos cos cos sin sin

ϕ

θ δ

 ϕ

, , θ

ϕ

θ*e* (7)

 , and 0 γ= *n k* cos

ϕ

to that of vacuum, 0

**2.2. Incident wave** 

of the incident wave defined by

 θ= *n k* sin cos

wave number in vacuum and

denotes a transposition.

ϕ , 0 β

with

where 0 α

ϕ

μ

The profile of S is sinusoidal and is given by

( )

*l*

The electric and magnetic fields of an incident wave are given by

 = *n k* sin sin θ

λ

[Chen 1973] (with respect to the *Z*-axis) and is written as

### **2. Formulation of the problem**

Here, we formulate the problem of diffraction from multilayered bi-gratings when an electromagnetic plane wave is incident on it. The time-dependent factor, exp(*−iωt*), is suppressed throughout this paper as customary.

### **2.1. Geometry of the gratings**

Figure 1 shows the schematic representation of multilayered sinusoidal gratings with double periodicity. The grating, with *L*-1 laminated grating layers, has a period *d* in both the *X*- and *Y*-directions. The semi-infinite regions corresponding to the medium above the

**Figure 1.** Schematic representation of bi-gratings. (a) Multilayered bi-gratings; (b) The *l* th thin-film bigrating; (c) A bi-grating

grating and the substrate are denoted by V1 and VL+1, respectively. The individual layers in the grating, beginning from the upper layer (light-incidence side), are denoted by V ,,. ( ) = 2,3 *L* All of the regions V 1,2, , 1 ( ) = + *L* are filled with isotropic and homogeneous media with refractive indices *n* , and the permeability of each region is equal to that of vacuum, 0 μ . The interface between V and V+<sup>1</sup> is denoted by S ( = 1,2, ,*L* ). The profile of S is sinusoidal and is given by

$$z = \eta\_{\ell} \left( x, y \right) = \frac{h\_l}{4} \left\{ \sin \left( \frac{2\pi x}{d} + p\_{\ell} \right) + \sin \left( \frac{2\pi y}{d} + p\_{\ell} \right) \right\} - \sum\_{i=1}^{\ell - 1} e\_i \tag{1}$$

Here, *p* represents the phase of *S* in each direction, and *e* denotes the average distance between *S* and *<sup>l</sup>* <sup>1</sup> *S* <sup>+</sup> . The value *h* is regarded as the groove depth of the boundary *S* .

It should be noted that we can easily generalize the shape of *S* by distinguishing between *d*  and *p* in the *X*-and *Y*-directions, by writing them as *<sup>x</sup> d* , *<sup>y</sup> d* , *<sup>x</sup> p* and *<sup>y</sup> p* , respectively. A singly periodic grating is a special case where *<sup>y</sup> d* → ∞ in the doubly periodic case. In the present paper, we concentrate our attention on describing the analysis only for the doubly periodic case, since it reduces to the singly periodic case through a simple procedure.

#### **2.2. Incident wave**

The electric and magnetic fields of an incident wave are given by

$$
\begin{pmatrix} E^i \\ H^i \end{pmatrix} \begin{pmatrix} P \end{pmatrix} = \begin{pmatrix} e^i \\ h^i \end{pmatrix} \exp\left( \begin{array}{c} ik^i \cdot P \end{array} \right) \tag{2}
$$

with

314 Plasmonics – Principles and Applications

before presenting the conclusions of this study.

suppressed throughout this paper as customary.

**2. Formulation of the problem** 

**2.1. Geometry of the gratings** 

grating; (c) A bi-grating

phenomenon due to the presence of the double periodicity. Further, in view of the fact that layered gratings are interesting structures for optical device applications, we investigated a multilayered bi-gratings, which is a stack of thin-film bi-gratings made of a dielectric or metal. This paper is structured as follows. After formulating the problem in Section **2**, we briefly describe a method for obtaining a solution in Section **3**. Focusing our attention on the resonant excitation of plasmon modes, we then show the computational results in Section **4**,

Here, we formulate the problem of diffraction from multilayered bi-gratings when an electromagnetic plane wave is incident on it. The time-dependent factor, exp(*−iωt*), is

Figure 1 shows the schematic representation of multilayered sinusoidal gratings with double periodicity. The grating, with *L*-1 laminated grating layers, has a period *d* in both the *X*- and *Y*-directions. The semi-infinite regions corresponding to the medium above the

**Figure 1.** Schematic representation of bi-gratings. (a) Multilayered bi-gratings; (b) The *l* th thin-film bi-

grating and the substrate are denoted by V1 and VL+1, respectively. The individual layers in the grating, beginning from the upper layer (light-incidence side), are denoted by V ,,. ( ) = 2,3 *L* All of the regions V 1,2, , 1 ( ) = + *L* are filled with isotropic and homogeneous media with refractive indices *n* , and the permeability of each region is equal

$$\mathbf{h}^{\dot{\mathbf{i}}} = \left(1 \,/ \partial \boldsymbol{\mu}\_0\right) \,\mathbf{k}^{\dot{\mathbf{i}}} \times \mathbf{e}^{\dot{\mathbf{i}}} \tag{3}$$

Here, *P* is the position vector for an observation point P(*X, Y, Z*), and <sup>i</sup> *k* is the wave vector of the incident wave defined by

$$\mathbf{k}^{\mathrm{i}} = \left[ \alpha , \beta , -\boldsymbol{\chi} \right]^{\mathrm{T}} \tag{4}$$

where 0 α θ = *n k* sin cosϕ , 0 β = *n k* sin sin θ ϕ , and 0 γ = *n k* cosθ . The symbol *k* (= 2π / λ ) is the wave number in vacuum and λ is the wavelength of the incident wave. We define θ and ϕ as polar and azimuth angles, respectively, as shown in Fig. 1(a), and the superscript "T" denotes a transposition.

The amplitude of the incident electric field can be decomposed into TE and TM modes [Chen 1973] (with respect to the *Z*-axis) and is written as

$$\mathbf{e}^{\dagger} = \cos \delta \mathbf{e}^{\mathrm{TE}} + \sin \delta \mathbf{e}^{\mathrm{TM}} \tag{5}$$

$$\mathbf{e}^{\text{TE}} = \begin{bmatrix} \sin \varphi\_{\prime} - \cos \varphi\_{\prime} \mathbf{0} \end{bmatrix}^{\text{T}} \tag{6}$$

$$\mathbf{e}^{\text{TM}} = \left[ \cos \theta \cos \varphi , \cos \theta \sin \varphi , \sin \theta \right]^{\text{T}} \tag{7}$$

Here, the superscript TE (TM) indicates the absence of a *Z* component of the electric (magnetic) field.

Resonant Excitation of Plasmons in Bi-Gratings 317

(11)

(12)

*<sup>N</sup> <sup>E</sup> <sup>P</sup>* and ( ) <sup>d</sup> *<sup>H</sup><sup>N</sup> <sup>P</sup>* , we define the

± *P* as

(14)

is the

ψ*mn*

*<sup>e</sup>* <sup>±</sup> ± ± <sup>=</sup> *<sup>P</sup> <sup>P</sup>* (13)

± ±± = × *<sup>P</sup> <sup>P</sup>* (15)

*ik m n* ± ± = ⋅ = ±± *P P* (16)

<sup>±</sup> = ± (17)

=+ =+ (18)

( )

denotes the unit vector normal to the surface *S* ,

i

0 on <sup>−</sup>

2 2

*<sup>P</sup>* and the magnetic modal function ( ) TE,TM

 φ

 ϕ*k*

*P* is the solution of the Helmholtz equation satisfying the periodic condition in

TE , *mn Z mn mn*

*mn Z mn mn*

TE

(,) (,) , ,1

 η

*XY XY X Y*

*XY XY X Y*

 ∂ ∂ + + ∂ ∂

We next explain the mode-matching method [Yasuura 1965, Yasuura 1971, Okuno 1990] for determining the diffracted field produced by the multilayered bi-grating. We introduce vector modal functions in the region ( 1,...,L 1) *V* = + to express the diffracted field in each

> ( ) ( ) TE,TM TE,TM *mn mn mn*

*ki e k*

± ± ±

± ± ± × × = = × × 

*ki e k*

( ) ( ) TE,TM TE,TM 0 1 *mn mn mn*

( ) exp( ) ( ) , 0, 1, 2,... *mn mn*

<sup>T</sup> , , *mn m n mn <sup>k</sup>* αβ

<sup>2</sup> <sup>π</sup> <sup>2</sup> <sup>π</sup> , *m n m n d d*

 ββ

 γ

where the positive and negative signs match on either side of the equation, and *mn <sup>k</sup>*<sup>±</sup>

ωμ

∂ ∂ − − ∂ ∂ <sup>=</sup>

(,) (,) <sup>1</sup>

 η

1 i

 *E E E ν S H H H*

 × + −= 

δ

ν

η

η

1

1

−

is the Kronecker delta, and

**3. Mode-matching method** 

electric modal function ( ) TE,TM

ν

individual region. To construct the wave functions ( ) <sup>d</sup>

ψ

the region ( 1,.2,.., 1) *V L* = + . It is written as

ϕ

wave vector of the (*m,n*)th order diffracted wave given by

αα

ϕ

TE TM

*e e*

*mn mn*

± ±

*mn* φ±

where 1 δ

where φ*mn* ( ) <sup>±</sup>

which is given by

**Figure 2.** Definition of a polarization angle.

The symbol δ is the polarization angle between TE *e* and <sup>i</sup> *e* as shown in Fig. 2; for 0 δ =  ( 90 δ= ), this represents TE-mode (TM-mode) incidence.

#### **2.3. Diffracted wave**

We denote the diffracted fields as *E P*( ) and *H P*( ) in the region *V LL* ( ) = + 1,2, , 1 . These satisfy the following conditions:

(C1) Helmholtz equation:

$$\left(\nabla^2 + n\_\ell^2 k\_\ell^2\right) \begin{pmatrix} \mathbf{E}\_\ell \\ \mathbf{H}\_\ell \end{pmatrix} \begin{pmatrix} \mathbf{P} \\ \end{pmatrix} = \mathbf{0} \quad \left(\ell = 1, 2, \dots, L, L+1\right) \tag{8}$$

(C2) Radiation conditions:

<sup>1</sup> *E* and *H*1 propagate or attenuate in the positive *Z*-direction.

*<sup>L</sup>* <sup>1</sup> *E* <sup>+</sup> and *<sup>L</sup>* <sup>1</sup> *E* <sup>+</sup> propagate or attenuate in the negative *Z*-direction.

(C3) Periodicity conditions:

$$f\left(X+d,Y,Z\right) = \exp\left(iad\right)f\left(X,Y,Z\right)\tag{9}$$

$$f\left(\mathbf{X}, \mathbf{Y} + d, \mathbf{Z}\right) = \exp\left(i\beta d\right) f\left(\mathbf{X}, \mathbf{Y}, \mathbf{Z}\right) \tag{10}$$

Here, *f* denotes any component of *E P*( ) or *H P*( ) .

(C4) Boundary condition ( 0 < < *X d* ; 0 < < *Y d* ; = 1,2,...,*L* ):

#### Resonant Excitation of Plasmons in Bi-Gratings 317

$$\mathbf{w} \times \begin{bmatrix} \left(\mathbf{E}\_{\ell-1} \\ \mathbf{H}\_{\ell-1} \end{bmatrix} + \delta\_{\ell 1} \begin{pmatrix} \mathbf{E}^{\mathrm{i}} \\ \mathbf{H}^{\mathrm{i}} \end{pmatrix} - \begin{pmatrix} \mathbf{E}\_{\ell} \\ \mathbf{H}\_{\ell} \end{pmatrix} \end{bmatrix} = \mathbf{0} \quad \text{(on } \mathbf{S}\_{\ell}\text{)}\tag{11}$$

where 1 δ is the Kronecker delta, and ν denotes the unit vector normal to the surface *S* , which is given by

$$\nu\_{\ell} = \frac{\left[ -\frac{\partial \eta\_{\ell}(X, Y)}{\partial X}, -\frac{\partial \eta\_{\ell}(X, Y)}{\partial Y}, 1 \right]}{\sqrt{\left( \frac{\partial \eta\_{\ell}(X, Y)}{\partial X} \right)^{2} + \left( \frac{\partial \eta\_{\ell}(X, Y)}{\partial Y} \right)^{2} + 1}} \tag{12}$$

### **3. Mode-matching method**

316 Plasmonics – Principles and Applications

**Figure 2.** Definition of a polarization angle.

These satisfy the following conditions:

= ), this represents TE-mode (TM-mode) incidence.

δ

**2.3. Diffracted wave** 

(C1) Helmholtz equation:

(C2) Radiation conditions:

(C3) Periodicity conditions:

(magnetic) field.

The symbol

( 90 δ

Here, the superscript TE (TM) indicates the absence of a *Z* component of the electric

is the polarization angle between TE *e* and <sup>i</sup> *e* as shown in Fig. 2; for 0

We denote the diffracted fields as *E P*( ) and *H P*( ) in the region *V LL* ( ) = + 1,2, , 1 .

( ) () ( ) 2 22 *n k* 0 , 1,2, 1 *L L* ∇ + == +

*f X dYZ i d f XYZ* ( ) ( )( ) + = , , exp , , α

*f XY dZ i d f XYZ* ( ) ( )( ) , + = , exp , , β

*P*

*H* (8)

(9)

(10)

 

*E*

<sup>1</sup> *E* and *H*1 propagate or attenuate in the positive *Z*-direction.

Here, *f* denotes any component of *E P*( ) or *H P*( ) .

(C4) Boundary condition ( 0 < < *X d* ; 0 < < *Y d* ; = 1,2,...,*L* ):

*<sup>L</sup>* <sup>1</sup> *E* <sup>+</sup> and *<sup>L</sup>* <sup>1</sup> *E* <sup>+</sup> propagate or attenuate in the negative *Z*-direction.

δ=  We next explain the mode-matching method [Yasuura 1965, Yasuura 1971, Okuno 1990] for determining the diffracted field produced by the multilayered bi-grating. We introduce vector modal functions in the region ( 1,...,L 1) *V* = + to express the diffracted field in each individual region. To construct the wave functions ( ) <sup>d</sup> *<sup>N</sup> <sup>E</sup> <sup>P</sup>* and ( ) <sup>d</sup> *<sup>H</sup><sup>N</sup> <sup>P</sup>* , we define the electric modal function ( ) TE,TM *mn* φ ± *<sup>P</sup>* and the magnetic modal function ( ) TE,TM ψ *mn* ± *P* as

$$\boldsymbol{\phi}\_{\ell mn}^{\text{TE,TM}\pm} \left( \boldsymbol{\mathcal{P}} \right) = \boldsymbol{e}\_{\ell mn}^{\text{TE,TM}\pm} \boldsymbol{\phi}\_{\ell mn}^{\pm} \left( \boldsymbol{\mathcal{P}} \right) \tag{13}$$

$$\mathbf{e}\_{\ell mn}^{\mathrm{TE\pm}\pm} = \frac{k\_{\ell mn}^{\pm} \times \mathbf{i}\_{Z}}{\left| k\_{\ell mn}^{\pm} \times \mathbf{i}\_{Z} \right|}, \; e\_{\ell mn}^{\mathrm{TM\pm}\pm} = \frac{e\_{\ell mn}^{\mathrm{TE\pm}\pm} \times k\_{\ell mn}^{\pm}}{\left| e\_{\ell mn}^{\mathrm{TE\pm}\pm} \times k\_{\ell mn}^{\pm} \right|} \tag{14}$$

$$\boldsymbol{\Psi}\_{\ell mn}^{\text{TE,TM\pm}}\left(\boldsymbol{\mathcal{P}}\right) = \frac{1}{\alpha \mu\_0} k\_{\ell mn}^{\pm} \times \boldsymbol{\mathcal{o}}\_{\ell mn}^{\text{TE,TM\pm}}\left(\boldsymbol{\mathcal{P}}\right) \tag{15}$$

where φ *mn* ( ) <sup>±</sup> *P* is the solution of the Helmholtz equation satisfying the periodic condition in the region ( 1,.2,.., 1) *V L* = + . It is written as

$$\left(\boldsymbol{\varphi}\_{\ell mn}^{\pm}\left(\boldsymbol{\mathsf{P}}\right)=\exp\left(ik\_{\ell mn}^{\pm}\cdot\boldsymbol{\mathsf{P}}\right)\quad\left(m,n=0,\pm1,\pm2,\ldots\right)\tag{16}$$

where the positive and negative signs match on either side of the equation, and *mn <sup>k</sup>*<sup>±</sup> is the wave vector of the (*m,n*)th order diffracted wave given by

$$k\_{\ell mn}^{\pm} = \left[ \alpha\_{m'} \beta\_{n'} \pm \gamma\_{\ell mn} \right]^{\mathrm{T}} \tag{17}$$

$$
\alpha\_m = \alpha + \frac{2m\pi}{d}, \ \beta\_n = \beta + \frac{2m\pi}{d} \tag{18}
$$

#### 318 Plasmonics – Principles and Applications

$$\gamma\_{\ell mn} = \left( n\_{\ell}^2 k^2 - \alpha\_m^2 - \beta\_n^2 \right)^{\frac{1}{2}}, \text{Re}\left( \gamma\_{\ell mn} \right) \ge 0 \text{ and } \text{Im}\left( \gamma\_{\ell mn} \right) \ge 0 \tag{19}$$

Resonant Excitation of Plasmons in Bi-Gratings 319

Here, ' *<sup>S</sup>* is a one-period cell of the interface *S* , and Γ is the intrinsic impedance of the

To solve the least-squares problem on a computer, we first discretize the weighted meansquare error *NI* by applying a two-dimensional trapezoidal rule where the number of divisions in the *x*-and *y*-direction is chosen as 2(2*N* + 1) [Yasuura 1965, Yasuura 1971, Okuno 1990]. We then solve the discretized least-squares problem by the QR decomposition method. Computational implementation of the least-squares problem is detailed in the

Here we show some numerical results obtained by the method described in the preceding section. After making necessary preparation, we show the results for three the bi-grating,

It is known that the solutions obtained by mode-matching method [Yasuura 1965, Yasuura 1971, Okuno 1990] have proof of convergence. We, therefore, can employ the coefficient

with sufficiently large *N* for which the coefficients are stable in evaluating

The power reflection and transmission coefficient of the (*m*, *n*) order propagating mode in

mn <sup>1</sup> <sup>1</sup> <sup>1</sup> [Re 0], , *mn mn mn mn mn mn A A mn mn mn* γ

mn <sup>1</sup> <sup>1</sup> <sup>1</sup> [Re 0], , *L mn L mn mn mn L mn mn A A L mn mn L mn* γ

The coefficient defined above is the power carried away by propagating diffraction orders normalized by the incident power. We calculate the total diffraction efficiency

Although it is known that the solutions obtained by mode-matching modal expansion method [Yasuura 1965, Yasuura 1971, Okuno 1990] have proof of convergence for problems of diffraction by gratings, we compare our results with other existing theoretical [Glass 1983] and experimental results [Inagaki, Goudonnet 1985] on plasmon resonance absorption in bi-gratings to show the validity of the present method. Figure 3 shows the reflectivity curves calculated by the present method and those from Rayleigh's method [Glass 1983] for three sinusoidal silver bi-gratings with different corrugation amplitudes. As confirmed in Fig. 3, the reflectivity curves from the present method are coincident with those of the

 τ

<sup>=</sup> where ' denotes a summation over the propagating orders.

 ρ

( ) 2 2 TE TM TE 1 1 TE TM TM

( ) 2 2 TE TM TE 1 1 TE TM TM

γ

γ

+ ++ = + ≥ = = (24)

+ + =+ ≥ = = (23)

 γ

> γ

 γ

> γ

ρ

τ

+ + − −

literature [Lawson 1974, Matsuda 1966, Suyama 2008].

thin-film bi-grating and multilayered thin-film bi-gratings cases.

 γ

 γ

medium in *V* .

**4. Numerical results** 

**4.1. Preparation** 

( ) TM,TE *A N mn* ±

diffracted fields.

τ

'total

 ρ

ρ

V and 1 VL 1+ are given by

ρ

 ττ

*mn*

 ρρ

Note that the superscripts + and - represent upwardly and downwardly propagating waves in the positive and negative *Z*-direction, respectively.

In terms of the linear combinations of the vector modal functions, we form approximate solutions for the diffracted electric and magnetic fields in *V* :

$$\begin{split} \left(\boldsymbol{\mathcal{E}}\_{\ell\text{M}}^{\text{d}}\right)(\mathbf{P}) &= \sum\_{m,m=-N}^{N} \boldsymbol{A}\_{\ell\text{nm}}^{\text{TE+}}\left(\boldsymbol{N}\right) \begin{pmatrix} \boldsymbol{\sigma}\_{\ell\text{nm}}^{\text{TE+}}\\ \boldsymbol{\sigma}\_{\ell\text{nm}}^{\text{TE+}} \end{pmatrix} \left(\boldsymbol{\mathcal{P}}\right) \\ &+ \sum\_{m,m=-N}^{N} \boldsymbol{A}\_{\ell\text{nm}}^{\text{TM+}}\left(\boldsymbol{N}\right) \begin{pmatrix} \boldsymbol{\sigma}\_{\ell\text{nm}}^{\text{TM+}}\\ \boldsymbol{\sigma}\_{\ell\text{nm}}^{\text{TM+}} \end{pmatrix} \left(\boldsymbol{\mathcal{P}}\right) \\ &+ \sum\_{m,m=-N}^{N} \boldsymbol{A}\_{\ell\text{nm}}^{\text{TE-}}\left(\boldsymbol{N}\right) \begin{pmatrix} \boldsymbol{\sigma}\_{\ell\text{nm}}^{\text{TE-}}\\ \boldsymbol{\sigma}\_{\ell\text{nm}}^{\text{TE-}} \end{pmatrix} \left(\boldsymbol{\mathcal{P}}\right) \\ &+ \sum\_{m,m=-N}^{N} \boldsymbol{A}\_{\ell\text{nm}}^{\text{TM-}}\left(\boldsymbol{N}\right) \begin{pmatrix} \boldsymbol{\sigma}\_{\ell\text{nm}}^{\text{TM-}}\\ \boldsymbol{\sigma}\_{\ell\text{nm}}^{\text{TM-}} \end{pmatrix} \left(\boldsymbol{\mathcal{P}}\right) \\ &\left(\ell=1,2,\dots,L+1\right) \end{split} \tag{20}$$

with

$$\boldsymbol{\Psi}\_{\ell mn}^{\rm TE, TM \pm} \left( \boldsymbol{\mathcal{P}} \right) = \frac{1}{\alpha \mu\_0} k\_{\ell mn}^{\pm} \times \boldsymbol{\phi}\_{\ell mn}^{\rm TE, TM \pm} \,. \tag{21}$$

Here, () () TE TM 1 1 *A NA N* <sup>0</sup> − − = = *mn mn* and () () TE TM 1 1 0 *A NA N* <sup>n</sup> + + + + = = *L mn L m* because of the radiation conditions (C3) stated in **Section 2.3**.

The approximate solutions ( ) <sup>d</sup> *<sup>E</sup><sup>N</sup> <sup>P</sup>* and ( ) <sup>d</sup> *H P N* already satisfy the Helmholtz equation (C1), the periodicity conditions (C2), and the radiation conditions (C3). The unknown coefficients ( ) TE *A N mn* ± and ( ) TM *A N mn* ± are therefore determined such that the solutions approximately satisfy the boundary conditions (C4). In the mode-matching method [Yasuura 1996, Yasuura 1971, Okuno 1990], the least-squares method is employed to fit the solution to the boundary conditions [Hugonin 1981]. That is, we find coefficients that minimize the weighted mean-square error by

$$\begin{split} I\_{N} &= \int\_{S\_{i}} \left| \mathbf{v} \times \left[ \mathbf{E}\_{1N}^{\mathrm{d}} + \mathbf{E}^{\mathrm{i}} - \mathbf{E}\_{2N}^{\mathrm{d}} \right] \left( s\_{1} \right) \right|^{2} ds \\ &+ \left| \Gamma\_{1} \right|^{2} \int\_{S\_{i}} \left| \mathbf{v} \times \left[ \mathbf{H}\_{1N}^{\mathrm{d}} + \mathbf{H}^{\mathrm{i}} - \mathbf{H}\_{2N}^{\mathrm{d}} \right] \left( s\_{1} \right) \right|^{2} ds \\ &+ \sum\_{\ell=2}^{L} \left\{ \int\_{S\_{\ell}} \right| \mathbf{v} \times \left[ \mathbf{E}\_{\ell N}^{\mathrm{d}} - \mathbf{E}\_{\ell+1N}^{\mathrm{d}} \right] \left( s\_{\ell} \right) \right|^{2} ds + \\ & \left| \Gamma\_{\ell} \right|^{2} \int\_{S\_{\ell}} \left| \mathbf{v} \times \left[ \mathbf{H}\_{\ell N}^{\mathrm{d}} - \mathbf{H}\_{\ell+1N}^{\mathrm{d}} \right] \left( s\_{\ell} \right)^{2} \right| ds \right\} \end{split} \tag{22}$$

Here, ' *<sup>S</sup>* is a one-period cell of the interface *S* , and Γ is the intrinsic impedance of the medium in *V* .

To solve the least-squares problem on a computer, we first discretize the weighted meansquare error *NI* by applying a two-dimensional trapezoidal rule where the number of divisions in the *x*-and *y*-direction is chosen as 2(2*N* + 1) [Yasuura 1965, Yasuura 1971, Okuno 1990]. We then solve the discretized least-squares problem by the QR decomposition method. Computational implementation of the least-squares problem is detailed in the literature [Lawson 1974, Matsuda 1966, Suyama 2008].

### **4. Numerical results**

318 Plasmonics – Principles and Applications

with

Here, () () TE TM

coefficients ( ) TE *A N mn*

conditions (C3) stated in **Section 2.3**.

±

γ

( ) () ()

 αβ

solutions for the diffracted electric and magnetic fields in *V* :

ψ

and ( ) TM *A N mn*

minimize the weighted mean-square error by

1 1 *A NA N* <sup>0</sup> − − = = *mn mn* and () () TE TM

±

1

2

=

'

Γ

*L*

1

'

'

ν E EE

= × +− 

*I s d*

+ × +−

*N NN S*

'

Γ

in the positive and negative *Z*-direction, respectively.

*E*

*H*

1

= −− *n k* ≥

22 2 2 <sup>2</sup> , Re 0 andIm 0 *mn m n mn mn*

Note that the superscripts + and - represent upwardly and downwardly propagating waves

In terms of the linear combinations of the vector modal functions, we form approximate

d TE TE d TE

<sup>=</sup>

 

,

*N N mn mn N mn N mn*

,

,

,

( ) ( ) ()

*P P*

+ <sup>+</sup> =−

*A N*

TM

*A N*

+ <sup>+</sup> =−

+

*<sup>N</sup> mn mn mn N mn <sup>N</sup> mn mn mn N mn <sup>N</sup> mn mn mn N mn*

TE

*A N*

− <sup>−</sup> =−

+

TM

( ) TE,TM <sup>1</sup> TE,TM 0 . *mn mn mn*

ωμ

The approximate solutions ( ) <sup>d</sup> *<sup>E</sup><sup>N</sup> <sup>P</sup>* and ( ) <sup>d</sup> *H P N* already satisfy the Helmholtz equation (C1), the periodicity conditions (C2), and the radiation conditions (C3). The unknown

approximately satisfy the boundary conditions (C4). In the mode-matching method [Yasuura 1996, Yasuura 1971, Okuno 1990], the least-squares method is employed to fit the solution to the boundary conditions [Hugonin 1981]. That is, we find coefficients that

> <sup>2</sup> d id 1 21 <sup>2</sup> <sup>2</sup> d id 1 1 21

ν H HH

*S N N*

ν E E

*S N N*

+ ×− +

*A N*

− <sup>−</sup> =−

+

 γ

<sup>≥</sup> (19)

( ) ()

 

ϕ

ψ

ϕ

ψ

ϕ

ψ

TM TE

 

−

TM

+

*P*

*P*

(20)

*P*

+

ϕ

ψ

( ) ()

 

TE TM

 

( ) ()

TM

 = +

−

( )

 φ*k*

( )

<sup>2</sup> d d 1

+

2 2 d d

× − 

ν H H

*S N N*

( )

*s*

*s d*

*s d*

*s*

( ) }

*s*

(22)

*s d*

*s*

( )

1

+

1 1 0 *A NA N* <sup>n</sup> + +

1,2, 1

,

*L*

± ± <sup>±</sup> = × *<sup>P</sup>* (21)

are therefore determined such that the solutions

+ + = = *L mn L m* because of the radiation

γ

Here we show some numerical results obtained by the method described in the preceding section. After making necessary preparation, we show the results for three the bi-grating, thin-film bi-grating and multilayered thin-film bi-gratings cases.

#### **4.1. Preparation**

It is known that the solutions obtained by mode-matching method [Yasuura 1965, Yasuura 1971, Okuno 1990] have proof of convergence. We, therefore, can employ the coefficient ( ) TM,TE *A N mn* ± with sufficiently large *N* for which the coefficients are stable in evaluating diffracted fields.

The power reflection and transmission coefficient of the (*m*, *n*) order propagating mode in V and 1 VL 1+ are given by

$$\boldsymbol{\rho}\_{1\text{min}} = \boldsymbol{\rho}\_{\text{min}}^{\text{TE}} + \boldsymbol{\rho}\_{\text{min}}^{\text{TM}} \left[ \text{Re} \left( \boldsymbol{\gamma}\_{1\text{min}} \right) \geq 0 \right], \\ \boldsymbol{\rho}\_{\text{min}}^{\text{TE}} = \frac{\boldsymbol{\mathcal{Y}}\_{1\text{min}}}{\boldsymbol{\mathcal{Y}}} \left| \boldsymbol{A}\_{1\text{min}}^{\text{TE}+} \right|^2, \\ \boldsymbol{\rho}\_{\text{min}}^{\text{TM}} = \frac{\boldsymbol{\mathcal{Y}}\_{1\text{min}}}{\boldsymbol{\mathcal{Y}}} \left| \boldsymbol{A}\_{1\text{min}}^{\text{TM}+} \right|^2 \tag{23}$$

$$\tau\_{\rm min} = \tau\_{\rm min}^{\rm TE} + \tau\_{\rm min}^{\rm TM} \left[ \text{Re} \left( \boldsymbol{\gamma}\_{L+1\,\rm min} \right) \geq 0 \right], \\ \tau\_{\rm min}^{\rm TE} = \frac{\mathcal{Y}\_{L+1\,\rm min}}{\mathcal{Y}} \left| A\_{L+1\,\rm min}^{\rm TE-} \right|^2, \\ \tau\_{\rm min}^{\rm TM} = \frac{\mathcal{Y}\_{L+1\,\rm min}}{\mathcal{Y}} \left| A\_{L+1\,\rm min}^{\rm TM-} \right|^2 \tag{24}$$

The coefficient defined above is the power carried away by propagating diffraction orders normalized by the incident power. We calculate the total diffraction efficiency ' total *mn* ρ ρ<sup>=</sup> where ' denotes a summation over the propagating orders.

Although it is known that the solutions obtained by mode-matching modal expansion method [Yasuura 1965, Yasuura 1971, Okuno 1990] have proof of convergence for problems of diffraction by gratings, we compare our results with other existing theoretical [Glass 1983] and experimental results [Inagaki, Goudonnet 1985] on plasmon resonance absorption in bi-gratings to show the validity of the present method. Figure 3 shows the reflectivity curves calculated by the present method and those from Rayleigh's method [Glass 1983] for three sinusoidal silver bi-gratings with different corrugation amplitudes. As confirmed in Fig. 3, the reflectivity curves from the present method are coincident with those of the Rayleigh's method [Glass 1983]. Next, we make comparison with the experimental results [Inagaki, Goudonnet 1985] in which the resonance angle *θ*d, i.e., the polar angle at which the dip of reflectivity occurs, is observed near 55° for a sinusoidal silver bi-grating with *h* = 0.048 µm, *d* = 2.186 µm, *λ* = 0.633 µm, and φ = 45° . The resonance angle calculated from the present metod for these parameters is *θ*d = 54.1° , which is close to the experimental data. These examples show that the present method gives reliable results for the analysis of plasmon-resonance absorption in metal bi-gratings.

Resonant Excitation of Plasmons in Bi-Gratings 321

is fixed. In the efficiency curves we observe

corresponding to the medium above the grating and the substrate are denoted by V1 and V2, respectively. V1 is vacuum (V) with a relative refractive index *n*1=1 and V2 consists of a lossy

Figure 4 shows the total diffraction efficiency of a sinusoidal silver bi-grating as functions of

four dips which occur at the same angles of incidence for both TE and TM polarized incident light. In this subsection, we demonstrate that the dips are associated with absorption that is caused by the coupling of surface plasmons with an evanescent mode diffracted by a sinusoidal bi-grating. For convenience, the four dips in Fig. 4 are labeled as A, B, C, and D.

> θ(*L*=1)*.*

1( 1,0) at the angles of incidence *θ* = 9.5 º, i.e., dip A, and *A* <sup>−</sup>

TM 1(0, 1) at

In Fig. 5 we plot the expansion coefficients of the (0, -1)st-order and (-1, 0)th-order TM vector modal function, which are two evanescent modes, as a functions of *θ* under the same parameters as in Fig. 4. Solid curves in Fig. 5 represent the real part of the expansion coefficient, and dashed curves for the imaginary part. We observe a resonance curve of the

the angles of incidence *θ* = 23.3 º, 41.5 º and 49.5 º, i.e., dips B, C, and D in Fig. 4 for both TE

φ= 30º

metal characterized by a complex refractive index *n*2.

**Figure 4.** Total diffraction efficiencies *ρ*total as functions of

TM

*4.2.2. Expansion coefficients* 

expansion coefficient *A* <sup>−</sup>

and TM incidence.

*4.2.1. Diffraction efficiency* 

a polar angle *θ* when the azimuthal angle

**Figure 3.** Comparison of resonance absorption curves calculated by the present method with other existing theoretical results. Solid curves show our results, and dotted curves are taken from Figure 2. of Ref. [Glass 1983].

In the numerical examples presented here, we deal with a shallow sinusoidal silver bigrating with height *h* = 0.030 µm and period *d* = 0.556 µm. The wavelength of an incident light is chosen as *λ* = 0.650 µm. We take *n*2 = 0.07+*i*4.20 as the refractive index of silver at this wavelength [Hass 1963].

It should be noted, however, that the index of a metal film depends not only on the wavelength but also on the thickness of the film, in particular when the film is extremely thin it may take unusual values if circumstances require. When dealing with a thin metal structure, hence, we should be careful in using the index value given in the literature. As for the value taken in our computation, we assume that *n*2 = 0.07+*i*4.20 is available even for the case of *e*/*d* = 0.02. This is because a similar assumption was supported by experimental data in a problem of diffraction by an aluminum grating with a thin gold over-coating.

#### **4.2. A bi-grating case**

Using the numerical algorithm stated in the previous section, we first investigate the absorption in a metal bi-gratings by *L* = 1 as shown in Fig. 1(c). The semi-infinite regions corresponding to the medium above the grating and the substrate are denoted by V1 and V2, respectively. V1 is vacuum (V) with a relative refractive index *n*1=1 and V2 consists of a lossy metal characterized by a complex refractive index *n*2.

#### *4.2.1. Diffraction efficiency*

320 Plasmonics – Principles and Applications

Ref. [Glass 1983].

wavelength [Hass 1963].

**4.2. A bi-grating case** 

µm, *d* = 2.186 µm, *λ* = 0.633 µm, and

present metod for these parameters is *θ*d = 54.1°

plasmon-resonance absorption in metal bi-gratings.

Rayleigh's method [Glass 1983]. Next, we make comparison with the experimental results [Inagaki, Goudonnet 1985] in which the resonance angle *θ*d, i.e., the polar angle at which the dip of reflectivity occurs, is observed near 55° for a sinusoidal silver bi-grating with *h* = 0.048

These examples show that the present method gives reliable results for the analysis of

**Figure 3.** Comparison of resonance absorption curves calculated by the present method with other existing theoretical results. Solid curves show our results, and dotted curves are taken from Figure 2. of

In the numerical examples presented here, we deal with a shallow sinusoidal silver bigrating with height *h* = 0.030 µm and period *d* = 0.556 µm. The wavelength of an incident light is chosen as *λ* = 0.650 µm. We take *n*2 = 0.07+*i*4.20 as the refractive index of silver at this

It should be noted, however, that the index of a metal film depends not only on the wavelength but also on the thickness of the film, in particular when the film is extremely thin it may take unusual values if circumstances require. When dealing with a thin metal structure, hence, we should be careful in using the index value given in the literature. As for the value taken in our computation, we assume that *n*2 = 0.07+*i*4.20 is available even for the case of *e*/*d* = 0.02. This is because a similar assumption was supported by experimental data

Using the numerical algorithm stated in the previous section, we first investigate the absorption in a metal bi-gratings by *L* = 1 as shown in Fig. 1(c). The semi-infinite regions

in a problem of diffraction by an aluminum grating with a thin gold over-coating.

. The resonance angle calculated from the

, which is close to the experimental data.

φ= 45°

> Figure 4 shows the total diffraction efficiency of a sinusoidal silver bi-grating as functions of a polar angle *θ* when the azimuthal angle φ = 30º is fixed. In the efficiency curves we observe four dips which occur at the same angles of incidence for both TE and TM polarized incident light. In this subsection, we demonstrate that the dips are associated with absorption that is caused by the coupling of surface plasmons with an evanescent mode diffracted by a sinusoidal bi-grating. For convenience, the four dips in Fig. 4 are labeled as A, B, C, and D.

**Figure 4.** Total diffraction efficiencies *ρ*total as functions of θ(*L*=1)*.*

#### *4.2.2. Expansion coefficients*

In Fig. 5 we plot the expansion coefficients of the (0, -1)st-order and (-1, 0)th-order TM vector modal function, which are two evanescent modes, as a functions of *θ* under the same parameters as in Fig. 4. Solid curves in Fig. 5 represent the real part of the expansion coefficient, and dashed curves for the imaginary part. We observe a resonance curve of the expansion coefficient *A* <sup>−</sup> TM 1( 1,0) at the angles of incidence *θ* = 9.5 º, i.e., dip A, and *A* <sup>−</sup> TM 1(0, 1) at the angles of incidence *θ* = 23.3 º, 41.5 º and 49.5 º, i.e., dips B, C, and D in Fig. 4 for both TE and TM incidence.

This implies that the TM component of the (0, -1)st-order and (-1, 0)th-order evanescent mode couples with surface plasmons at dips B and D. We can similarly confirm that dips A and C are associated with the coupling of the TM component of the (-1, 0)th- and (-1, -1)storder evanescent mode, respectively, with surface plasmons, although we do not include any numerical example here.

Resonant Excitation of Plasmons in Bi-Gratings 323

ρ

 ( <sup>00</sup> ρ

 = TE 00 ρ

> ρTE

is

00 ρ

We calculate the energy flow at each point located densely near the grating surface and

Figure 6(b) shows that the energy of electromagnetic fields in the vicinity of the grating surface flows uniformly in the direction that the (0, -1)st-order evanescent mode travels in the *XY* plane. We thus confirm that surface plasmons are excited on the grating surface

**Figure 6.** Field distribution (a) and Energy flow (b) for the total field when plasmon resonance

absorption. In Fig. 7, the TE and TM component of the diffraction efficiency 00

Diffracted fields from a sinusoidal metal bi-grating have both TE and TM component for an arbitrary polarized incident light. We therefore observe polarization conversion that a TM (or TE) component of the incident light is converted into a TE (or TM) component of the reflected light. It has been pointed out [Chen 1973, Inagaki, Goudonnet 1985] that the polarization conversion [Elston 1991, Matsuda 1999, Suyama 2007] is strongly enhanced when the plasmon-resonance absorption occurs in a sinusoidal metal bi-grating. Our study confirms the enhancement of polarization conversion through plasmon-resonance

) of Fig. 4 are shown for the case of the TM incidence. The TM component TM

decreased at the position of the plasmon-resonance absorption, but the TE component <sup>00</sup>

is contrary increased there. That is, the resonant excitation of surface plasmons causes the enhancement of polarization conversion. On the other hand, in the case of the TE incidence the conversion from a TE to a TM component occurs through the plasmon-resonance absorption. It should be noted that the conversion efficiency depends on the azimuthal

*4.2.4. Polarization conversion through plasmon resonance absorption* 

through the coupling of the TM component of an evanescent mode.

show the results in Fig. 6(b).

absorption occurs at

+ TM 00 ρ

angle ϕ φ= 30 º , θ= 23.3 º.

and the depth of the grating surface *h*.

**Figure 5.** TM *<sup>A</sup>*1(0, 1) <sup>−</sup> and TM *<sup>A</sup>*1( 1,0) <sup>−</sup> as functions of θfor both TM- (a) and TE- (b) polarized incident light*.* 

#### *4.2.3. Field distributions and energy flows*

In order to investigate the resonant excitation of surface plasmons, we study field distributions and energy flows in the vicinity of the grating surface when the absorption occurs. Here we consider the case of the dip B at which the TM component of the (0, -1)storder evanescent mode couples with surface plasmons. We calculate the electric field of the TM component of the (0, -1)st-order evanescent mode TM 0 1 *<sup>E</sup>* <sup>−</sup> = ( ) TM *A N* 0 1<sup>−</sup> ( ) TM 0 1 ϕ 1,2 <sup>−</sup> = and the total electric field *E* <sup>t</sup> . The magnitude of these fields along the Z-axis where ( ) Y X 1 0 β α <sup>−</sup> = ∗ is plotted in Fig. 6(a). We observe in this figure that very strong electric fields are induced at the grating metal surface and the fields exponentially decay away from the surface.

Next we show the energy flows **S** that are the real part of Poynting's vectors for the total field. The X and Y components of the energy flows **S** are plotted as the vector (S*X,* S*Y)* in Fig. 6(b). The energy flows are calculated over the region close to the grating surface:

$$\begin{aligned} \{ \mathbf{P}(\mathbf{x}, y, z) \colon 0 \le \mathbf{x} \le d, 0 \le y \le d \} \\ \mathbf{z} = (h \land 4) [\sin(2\pi \mathbf{x} \mid d) + \sin(2\pi y \nmid d)] + 0.01d \}. \end{aligned} \tag{25}$$

The energy flow at a point *P* is given by Re[*S*(*P*)], where *S*(*P*)=(1/ 2) *E*<sup>t</sup> (*P*)× *H*<sup>t</sup> (*P*) stands for Poynting's vector, *E*<sup>t</sup> and *H* t denote total fields, and the over-bar means complex conjugate. We calculate the energy flow at each point located densely near the grating surface and show the results in Fig. 6(b).

322 Plasmonics – Principles and Applications

any numerical example here.

**Figure 5.** TM *<sup>A</sup>*1(0, 1) <sup>−</sup> and TM *<sup>A</sup>*1( 1,0) <sup>−</sup> as functions of

*4.2.3. Field distributions and energy flows* 

the total electric field *E* <sup>t</sup>

( ) Y X 1 0 β α

Poynting's vector, *E*<sup>t</sup>

the surface.

TM component of the (0, -1)st-order evanescent mode TM

{( )0 0

and *H*

P ,, : , ,

The energy flow at a point *P* is given by Re[*S*(*P*)], where *S*(*P*)=(1/ 2) *E*<sup>t</sup>

*xyz x y*

This implies that the TM component of the (0, -1)st-order and (-1, 0)th-order evanescent mode couples with surface plasmons at dips B and D. We can similarly confirm that dips A and C are associated with the coupling of the TM component of the (-1, 0)th- and (-1, -1)storder evanescent mode, respectively, with surface plasmons, although we do not include

θ

In order to investigate the resonant excitation of surface plasmons, we study field distributions and energy flows in the vicinity of the grating surface when the absorption occurs. Here we consider the case of the dip B at which the TM component of the (0, -1)storder evanescent mode couples with surface plasmons. We calculate the electric field of the

 <sup>−</sup> = ∗ is plotted in Fig. 6(a). We observe in this figure that very strong electric fields are induced at the grating metal surface and the fields exponentially decay away from

Next we show the energy flows **S** that are the real part of Poynting's vectors for the total field. The X and Y components of the energy flows **S** are plotted as the vector (S*X,* S*Y)* in Fig.

*z h y*

*d d*

= ++

( / 4)[sin(2πx ) sin(2π )] 0.01 }

/ .

t denote total fields, and the over-bar means complex conjugate.

6(b). The energy flows are calculated over the region close to the grating surface:

≤≤ ≤≤

for both TM- (a) and TE- (b) polarized incident light*.* 

0 1 *<sup>E</sup>* <sup>−</sup> = ( ) TM *A N* 0 1<sup>−</sup> ( ) TM

*d dd* / (25)

(*P*)× *H*<sup>t</sup>

. The magnitude of these fields along the Z-axis where

0 1 ϕ

1,2 <sup>−</sup> = and

(*P*) stands for

Figure 6(b) shows that the energy of electromagnetic fields in the vicinity of the grating surface flows uniformly in the direction that the (0, -1)st-order evanescent mode travels in the *XY* plane. We thus confirm that surface plasmons are excited on the grating surface through the coupling of the TM component of an evanescent mode.

**Figure 6.** Field distribution (a) and Energy flow (b) for the total field when plasmon resonance absorption occurs at φ = 30 º , θ= 23.3 º.

#### *4.2.4. Polarization conversion through plasmon resonance absorption*

Diffracted fields from a sinusoidal metal bi-grating have both TE and TM component for an arbitrary polarized incident light. We therefore observe polarization conversion that a TM (or TE) component of the incident light is converted into a TE (or TM) component of the reflected light. It has been pointed out [Chen 1973, Inagaki, Goudonnet 1985] that the polarization conversion [Elston 1991, Matsuda 1999, Suyama 2007] is strongly enhanced when the plasmon-resonance absorption occurs in a sinusoidal metal bi-grating. Our study confirms the enhancement of polarization conversion through plasmon-resonance absorption. In Fig. 7, the TE and TM component of the diffraction efficiency 00 ρ ( <sup>00</sup> ρ = TE 00 ρ + TM 00 ρ ) of Fig. 4 are shown for the case of the TM incidence. The TM component TM 00 ρ is decreased at the position of the plasmon-resonance absorption, but the TE component <sup>00</sup> ρTE is contrary increased there. That is, the resonant excitation of surface plasmons causes the enhancement of polarization conversion. On the other hand, in the case of the TE incidence the conversion from a TE to a TM component occurs through the plasmon-resonance absorption. It should be noted that the conversion efficiency depends on the azimuthal angle ϕand the depth of the grating surface *h*.

**Figure 7.** Diffraction efficiencies of 00 ρ (a) TE 00 ρ and TM 00 ρ (b) as functions of θ ; parameters are the same as in Figure 4.

#### *4.2.5. Prediction of resonance angles*

We seek to determine a complex incidence angle *θc* for which total or partial absorption occurs, i.e., ( ) total 00 ρ ρ = takes a minimum. This angle relates to the propagation constant of the surface plasmon on the corrugated surface. Here, we denote by SP αˆ and SP ˆ β the *X* and *Y* components of the surface plasmon wave vector normalized by the wave number *k*1:

$$\begin{aligned} \hat{a}\_{S^p} &= \sin \theta\_{\text{c}} \cos \phi + m\lambda \, / \, d, \\ \hat{\beta}\_{S^p} &= \sin \theta\_{\text{c}} \sin \phi + n\lambda \, / \, d. \end{aligned} \tag{26}$$

Resonant Excitation of Plasmons in Bi-Gratings 325

 ) SP θ

d θ

Dip Mode Re( SP

0.03µm, 0.556µm, 0.650µm *<sup>a</sup> h d* == =

**Figure 8.** Resonance angles SP

θ

*4.2.6. Simultaneous resonance absorption* 

as functions of azimuthal angle

From Fig. 8, it is predicted the angle of incidence at which the simultaneous resonance absorption occurs from the position of the intersection of the (-1,0) and (0,-1) curve. At the intersection **E**, the (0,-1)st- and (-1,0)th-order evanescent modes couple simultaneously with two surface-plasmon waves at the same angle of incidence. Thus, two surface-plasmon waves are excited simultaneously in directions symmetric with respect to the plane of incidence and interact with each other. The interference of the surface-plasmon waves causes the standing wave in the vicinity of the grating surface. This is confirmed from Figs. 9, the strong fields along the Z-axis where Y=X, and where the *X* and *Y* components of Poynting's vectors **S** on a surface 0.01*d* above the one-unit cell of the grating surface are

φ.

α

λ

**Table 1.** Propagation Constants and Estimated Resonance Angles*<sup>a</sup>*

ˆ ) Im( SP

α

ˆ ) Re( SP

A (-1, 0) -1.026277 -0.000804 0.087099 -0.001279 9.49 9.5 B (0, -1) 0.343795 -0.004910 -0.971021 -0.002498 23.39 23.3 C (0, -1) -0.595897 -0.001282 -0.837168 -0.000549 41.44 41.5 D (0, -1) 0.668265 -0.004365 -0.778336 -0.005224 50.50 50.5

<sup>E</sup>(0, -1) 0.149639 -0.000679 -1.019425 -0.000679 12.22 12.2 (-1, 0) -1.019426 -0.000679 0.149639 -0.000679 12.22

ˆ β

) Im( SP

ˆ β

In reality, we cannot realize a complex incidence angle <sup>c</sup> θ . We can, however, estimate the real angle of incidence at which the absorption occurs by taking the real part of Eq. (26).

If the wavevector of the surface plasmon ( SP αˆ , SP ˆ β ) is obtained, we can estimate the resonance angle SP θ for each azimuthal angle ϕ from the phase-matching condition for coupling of a surface plasmon wave with the (*m*, *n*)th-order evanescent mode:

$$\begin{aligned} \operatorname{Re}\{\hat{\alpha}\_{\rm SP}\} &= \sin\theta\_{\rm SP}\cos\varphi + m\hat{\lambda} \,/\, d, \\ \operatorname{Re}\{\hat{\beta}\_{\rm SP}\} &= \sin\theta\_{\rm SP}\sin\varphi + n\hat{\lambda} \,/\, d. \end{aligned} \tag{27}$$

We solve the homogenous problem [Nevièr 1982] for a sinusoidal metal bi-grating by present method and then obtain the surface-plasmon wave vector. Table 1 shows the propagation constants of the surface plasmon and the resonance angles SP θ . The data demonstrate that the estimated resonance angle SP θ agrees with d θ which is the absorption peak in Fig. 4. Figure 8 shows the estimated resonance angle SP θ as a function of the azimuthal angle ϕ for the sinusoidal bi-grating considered in Fig. 4. Note that points A, B, C, and D in Fig. 8 are results obtained from the absorption peak of total-efficiency curves in Fig. 4. From this figure, we can find the resonance angle for each azimuthal angle.


0.03µm, 0.556µm, 0.650µm *<sup>a</sup> h d* == = λ

324 Plasmonics – Principles and Applications

**Figure 7.** Diffraction efficiencies of 00

*4.2.5. Prediction of resonance angles* 

 ρ

<sup>c</sup>

If the wavevector of the surface plasmon ( SP

demonstrate that the estimated resonance angle SP

θ

ϕ

same as in Figure 4.

occurs, i.e., ( ) total 00 ρ

resonance angle SP

azimuthal angle

ρ

the surface plasmon on the corrugated surface. Here, we denote by SP

*SP SP*

*α*

for each azimuthal angle

α

β

peak in Fig. 4. Figure 8 shows the estimated resonance angle SP

coupling of a surface plasmon wave with the (*m*, *n*)th-order evanescent mode:

SP SP SP SP

propagation constants of the surface plasmon and the resonance angles SP

Fig. 4. From this figure, we can find the resonance angle for each azimuthal angle.

In reality, we cannot realize a complex incidence angle <sup>c</sup>

 (a) TE 00 ρ

 and TM 00 ρ

We seek to determine a complex incidence angle *θc* for which total or partial absorption

*Y* components of the surface plasmon wave vector normalized by the wave number *k*1:

c

real angle of incidence at which the absorption occurs by taking the real part of Eq. (26).

sin cos sin sin

= + = +

ˆ / , ˆ / .

θφλ

θφλ

αˆ , SP ˆ β

Re{ } sin cos / , ˆ Re{ } sin sin / . ˆ

= + = +

We solve the homogenous problem [Nevièr 1982] for a sinusoidal metal bi-grating by present method and then obtain the surface-plasmon wave vector. Table 1 shows the

C, and D in Fig. 8 are results obtained from the absorption peak of total-efficiency curves in

 θ

 θ ϕ

 ϕλ

 ϕλ

θ

(b) as functions of

= takes a minimum. This angle relates to the propagation constant of

θ

*m d n d*

agrees with d

for the sinusoidal bi-grating considered in Fig. 4. Note that points A, B,

θ

θ

*m d n d*

θ

α

*β* (26)

ˆ and SP ˆ β

. We can, however, estimate the

θ

which is the absorption

as a function of the

. The data

) is obtained, we can estimate the

from the phase-matching condition for

; parameters are the

the *X* and

(27)

**Table 1.** Propagation Constants and Estimated Resonance Angles*<sup>a</sup>*

**Figure 8.** Resonance angles SP θ as functions of azimuthal angle φ.

#### *4.2.6. Simultaneous resonance absorption*

From Fig. 8, it is predicted the angle of incidence at which the simultaneous resonance absorption occurs from the position of the intersection of the (-1,0) and (0,-1) curve. At the intersection **E**, the (0,-1)st- and (-1,0)th-order evanescent modes couple simultaneously with two surface-plasmon waves at the same angle of incidence. Thus, two surface-plasmon waves are excited simultaneously in directions symmetric with respect to the plane of incidence and interact with each other. The interference of the surface-plasmon waves causes the standing wave in the vicinity of the grating surface. This is confirmed from Figs. 9, the strong fields along the Z-axis where Y=X, and where the *X* and *Y* components of Poynting's vectors **S** on a surface 0.01*d* above the one-unit cell of the grating surface are

#### 326 Plasmonics – Principles and Applications

plotted as the vector ( ) S ,S *X Y* . We further observe in Fig. 10 that the simultaneous excitation of the surface plasmons waves causes the strong absorption for both TE- and TM-polarized incident light.

Resonant Excitation of Plasmons in Bi-Gratings 327

in V (Vacuum) (a) and 1

= 0 is fixed; *e* is the thickness of the

= on the *e d*/ = 0.4 curve,

θ

θ

for two

θ=

= and

θ

**4.3. A Thin-film Bi-grating case** 

surface plasmon modes.

*4.3.1. Diffraction efficiency* 

the transmission coefficient 00

8.8 , respectively.

values of *e /* d (*L*=2).

As a numerical example, we consider a sinusoidal silver (Ag) film bi-grating having a common period as shown in Fig. 1(b). The values of the parameters are the same as those in Fig. 4 except for the thickness of the silver film. Using the present algorithm, we calculated the diffraction efficiencies and field distributions to clarify the properties of the coupled

First, we consider a sinusoidal silver-film bi-grating. The bi-grating is denoted by *L* = 2

silver film. We observe partial absorption of the incident light as dips in the efficiency curves in Fig. 11(a), in addition to the constant absorption corresponding to the reflectivity of silver. We assume that the dips are caused by resonant excitation of surface plasmons. If this is the case, each of the dips can be related to one of the three types of

ρ

When the grating is thick ( *e d*/ = 0.4 ), the power can be seen in V alone and no transmitted 1

corresponding to absorption in Fig. 11(a). If the grating is relatively thin (*e d*/ = 0.08 ), the power exists in both V and 1 V . Although the power in 3 V is generally small, it becomes 3 large at the incidence angles for which absorption was observed in Fig. 11. This suggests

θ

that coupled oscillations occur on the upper and lower surfaces of the grating.

 (a) and 00 τ

(b) as functions of

from 0 , we first observe the dip at 8.0

and a SRSP and LRSP corresponding to the dips in the *e d*/ = 0.08 curve at 6.54

φ

ρ

in V (Vacuum) (b) as function <sup>3</sup> s of the incident angle

θ

(V/Ag/V). Figure 11 shows the (0,0)th order power reflection 00

plasmon modes: a SISP that is observed as a single dip at 8.0

τ

for two values of *e /* d when the azimuth angle

**Figure 11.** The (0, 0)-th order diffraction efficiencies 00

power exists in V in Fig. 11. Increasing 3

It has been reported [Barnes 1995, Ritchie 1968] that surface-plasmon band gaps exist at the angles of incidence at which simultaneous excitation of plasmon waves occurs, and that the appearance of the band gaps depends strongly on the surface profile. Hence, there is a possibility that a band gap will be observed at the point *E* in Figs. 9 and 10 provided that the grating profile is appropriately chosen, because two plasmon modes are excited at that point.

**Figure 9.** Field distributions and Energy flows for the total field when plasmon resonance absorption occurs.

**Figure 10.** Total diffraction efficiencies ρtotal as functions of *θ*.

#### **4.3. A Thin-film Bi-grating case**

326 Plasmonics – Principles and Applications

incident light.

occurs.

**Figure 10.** Total diffraction efficiencies

ρ

total as functions of *θ*.

plotted as the vector ( ) S ,S *X Y* . We further observe in Fig. 10 that the simultaneous excitation of the surface plasmons waves causes the strong absorption for both TE- and TM-polarized

It has been reported [Barnes 1995, Ritchie 1968] that surface-plasmon band gaps exist at the angles of incidence at which simultaneous excitation of plasmon waves occurs, and that the appearance of the band gaps depends strongly on the surface profile. Hence, there is a possibility that a band gap will be observed at the point *E* in Figs. 9 and 10 provided that the grating profile is appropriately chosen, because two plasmon modes are excited at that point.

**Figure 9.** Field distributions and Energy flows for the total field when plasmon resonance absorption

As a numerical example, we consider a sinusoidal silver (Ag) film bi-grating having a common period as shown in Fig. 1(b). The values of the parameters are the same as those in Fig. 4 except for the thickness of the silver film. Using the present algorithm, we calculated the diffraction efficiencies and field distributions to clarify the properties of the coupled surface plasmon modes.

#### *4.3.1. Diffraction efficiency*

First, we consider a sinusoidal silver-film bi-grating. The bi-grating is denoted by *L* = 2 (V/Ag/V). Figure 11 shows the (0,0)th order power reflection 00 ρ in V (Vacuum) (a) and 1 the transmission coefficient 00 τ in V (Vacuum) (b) as function <sup>3</sup> s of the incident angle θ for two values of *e /* d when the azimuth angle φ = 0 is fixed; *e* is the thickness of the silver film. We observe partial absorption of the incident light as dips in the efficiency curves in Fig. 11(a), in addition to the constant absorption corresponding to the reflectivity of silver. We assume that the dips are caused by resonant excitation of surface plasmons. If this is the case, each of the dips can be related to one of the three types of plasmon modes: a SISP that is observed as a single dip at 8.0 θ = on the *e d*/ = 0.4 curve, and a SRSP and LRSP corresponding to the dips in the *e d*/ = 0.08 curve at 6.54 θ = and 8.8 , respectively.

**Figure 11.** The (0, 0)-th order diffraction efficiencies 00 ρ (a) and 00 τ (b) as functions of θ for two values of *e /* d (*L*=2).

When the grating is thick ( *e d*/ = 0.4 ), the power can be seen in V alone and no transmitted 1 power exists in V in Fig. 11. Increasing 3 θ from 0 , we first observe the dip at 8.0 θ = corresponding to absorption in Fig. 11(a). If the grating is relatively thin (*e d*/ = 0.08 ), the power exists in both V and 1 V . Although the power in 3 V is generally small, it becomes 3 large at the incidence angles for which absorption was observed in Fig. 11. This suggests that coupled oscillations occur on the upper and lower surfaces of the grating.

#### *4.3.2. Expansion coefficients*

We examined the same phenomena observing the modal expansion coefficients in V ( 1,3) *<sup>l</sup> l* = . Figures 12 and 13 illustrate the (-1,0)th-order coefficients *A*TM<sup>+</sup> 1(-1,0) for *e d*/ = 0.4 and 0.08, respectively. We observe, in Fig. 12(a) ( *e d*/ = 0.4 ), the resonance characteristics (enhancement and rapid change in phase) of the coefficient TM *<sup>A</sup>*1(-1,0) <sup>+</sup> near 8.0 θ <sup>=</sup> . The coefficient <sup>0</sup> TM *<sup>A</sup>*3(-1,0) <sup>−</sup> = remains unchanged, as seen in Fig. 12(b). This means that the incident wave illuminating the grating at this angle causes coupling between the (-1,0)th-order evanescent mode and some oscillation excited on the upper surface of the grating. The oscillation exists locally in the vicinity of the illuminated surface and hence does not have any influence on the field in V3 .

Resonant Excitation of Plasmons in Bi-Gratings 329

the oscillation in the vicinity of the upper surface causes another oscillation on the lower surface at this thickness. The oscillations interfere with each other and result in two coupled oscillating modes: the SRSP and LRSP. This means that the TM component of the (-1,0)th order evanescent mode couples with the surface plasmons simultaneously excited on the upper and lower surface of the film grating. The two surface plasmons interfere with each other and result in symmetric and antisymmetric coupled modes, SRSP and LRSP, as we will see next.

We consider the same phenomena observing the field distributions and energy flows near the grating surfaces. In the former we find that the total field is enhanced. In the latter we observe the symmetric (even) and anti-symmetric (odd) nature of the oscillations, which

θ

θ

= 8.0 , which corresponds to the single dip

= 6.54 , which corresponds to the left dip

*4.3.3. Field distributions and energy flows* 

correspond to the LRSP and SRSP [Raeter 1977].

**Figure 14.** Field distributions (a) and energy flows (b) at

**Figure 15.** Field distributions (a) and energy flows (b) at

on the *e d*/ 0.4 = curve in Fig. 11.

on the *e d*/ 0.08 = curve in Fig. 11.

**Figure 12.** The (-1, 0)th order modal coefficients TM *<sup>A</sup>*1( 1,0) *-* <sup>+</sup> (a) and TM *<sup>A</sup>*3(-1,0) <sup>−</sup> (b) as functions of θ at *e d*/ 0.4 = .

**Figure 13.** The (-1, 0)th order modal coefficients TM *<sup>A</sup>*1( 1,0) *-* <sup>+</sup> (a) and TM *<sup>A</sup>*3(-1,0) <sup>−</sup> (b) as functions of θ at *e d*/ 0.08 = .

In Fig. 13 ( *e d*/ 0.08 = ), we find the resonance characteristics in both *A*TM<sup>+</sup> 1(-1,0) and TM *<sup>A</sup>*3(-1,0) <sup>−</sup> . In addition, they appear around two incidence angles: 6.54 θ = and 8.8 θ= . This means that the oscillation in the vicinity of the upper surface causes another oscillation on the lower surface at this thickness. The oscillations interfere with each other and result in two coupled oscillating modes: the SRSP and LRSP. This means that the TM component of the (-1,0)th order evanescent mode couples with the surface plasmons simultaneously excited on the upper and lower surface of the film grating. The two surface plasmons interfere with each other and result in symmetric and antisymmetric coupled modes, SRSP and LRSP, as we will see next.

#### *4.3.3. Field distributions and energy flows*

328 Plasmonics – Principles and Applications

*4.3.2. Expansion coefficients* 

We examined the same phenomena observing the modal expansion coefficients in V ( 1,3) *<sup>l</sup> l* = .

respectively. We observe, in Fig. 12(a) ( *e d*/ = 0.4 ), the resonance characteristics (enhancement

remains unchanged, as seen in Fig. 12(b). This means that the incident wave illuminating the grating at this angle causes coupling between the (-1,0)th-order evanescent mode and some oscillation excited on the upper surface of the grating. The oscillation exists locally in the vicinity of the illuminated surface and hence does not have any influence on the field in V3 .

<sup>+</sup> near 8.0 θ

<sup>+</sup> (a) and TM *<sup>A</sup>*3(-1,0)

<sup>+</sup> (a) and TM *<sup>A</sup>*3(-1,0)

θ

= and 8.8

θ

1(-1,0) for *e d*/ = 0.4 and 0.08,

<sup>−</sup> =

<sup>=</sup> . The coefficient <sup>0</sup> TM *<sup>A</sup>*3(-1,0)

<sup>−</sup> (b) as functions of

<sup>−</sup> (b) as functions of

θat

> θat

> > <sup>−</sup> . In

1(-1,0) and TM *<sup>A</sup>*3(-1,0)

= . This means that

Figures 12 and 13 illustrate the (-1,0)th-order coefficients *A*TM<sup>+</sup>

and rapid change in phase) of the coefficient TM *<sup>A</sup>*1(-1,0)

**Figure 12.** The (-1, 0)th order modal coefficients TM *<sup>A</sup>*1( 1,0) *-*

**Figure 13.** The (-1, 0)th order modal coefficients TM *<sup>A</sup>*1( 1,0) *-*

In Fig. 13 ( *e d*/ 0.08 = ), we find the resonance characteristics in both *A*TM<sup>+</sup>

addition, they appear around two incidence angles: 6.54

*e d*/ 0.4 = .

*e d*/ 0.08 = .

We consider the same phenomena observing the field distributions and energy flows near the grating surfaces. In the former we find that the total field is enhanced. In the latter we observe the symmetric (even) and anti-symmetric (odd) nature of the oscillations, which correspond to the LRSP and SRSP [Raeter 1977].

**Figure 14.** Field distributions (a) and energy flows (b) at θ = 8.0 , which corresponds to the single dip on the *e d*/ 0.4 = curve in Fig. 11.

**Figure 15.** Field distributions (a) and energy flows (b) at θ = 6.54 , which corresponds to the left dip on the *e d*/ 0.08 = curve in Fig. 11.

Resonant Excitation of Plasmons in Bi-Gratings 331

in V (Vacuum) (a) and the 1

θ

(b) as functions of

θ

<sup>=</sup> and 2 SiO *e d*/ 0.3 <sup>=</sup> .

for two

θ

. For example, in the

for two

θ

**4.4. Multilayered thin-film bi-gratings case** 

Figure 17 shows the (0,0)th order power reflection 00

absorption is characterized by its occurrence over a wider range of

τ

**Figure 17.** The (0, 0)-th order diffraction efficiencies 00

(0,0) TM *l*

(*l* =1,2,...5) along the *Z*-axis are plotted in Fig. 18 where 0

<sup>2</sup> SiO *<sup>n</sup>* <sup>=</sup>1.5, <sup>2</sup> SiO *<sup>e</sup>* / d 0.3 or 0.08 <sup>=</sup> .

*4.4.1. Diffraction efficiency* 

transmission coefficient 00

ranging from 0 to 12 .

values of *e*SiO2/*d* (*L*=4).

and TM *<sup>E</sup>l*(0,0)

*4.4.2. Field distributions* 

order diffracted electric field

Next, we consider multilayered thin-film bi-gratings as shown in Fig. 1(a) indicated by *L* = 4 ( V/Ag/SiO /Ag/V ) that consists of a stack of silver and SiO <sup>2</sup> 2 films pairs. As listed in the figure, the values of the parameters are the same as those in Fig. 11 except for *L* = 4,

different values of 2 SiO *e d*/ . The curve for <sup>2</sup> SiO *e d*/ <sup>=</sup> 0.08 is almost the same as for a sinusoidal silver film bi-grating (Fig. 11): the coupled surface plasmon modes, SRSP and LRSP, are excited. On the other hand, in the curve for 2 SiO *e d*/ <sup>=</sup> 0.3 , we find a new type of absorption of incident light besides the plasmon resonance absorption associated with SRSP or LRSP.

This is related to the resonant excitation of a guided wave supported by the SiO2 film. This

ρ

In order to examine the properties of the wide absorption found in Fig. 17, we investigated the field distributions of the total electric field *E*total and the TM component of the (0, 0)th-

 (a) and 00 τ

*E* in the vicinity of the SiO2 film. The magnitude of *E*total

θ

case of 2 SiO *e d*/ 0.3 <sup>=</sup> , the extinction power is more than 50% for all angles of incidence

ρ

in V5 (Vacuum) (b) as functions of the incident angle

**Figure 16.** Field distributions (a) and energy flows (b) at θ = 8.8 , which corresponds to the right dip on the *e d*/ 0.08 = curve in Fig. 11.

Figures 14, 15, and 16 show the field distributions of the *X*- and *Z*-components of the total electric fields (a) and energy flows (b) in the vicinity of the silver-film grating at the incidence angles at which absorption was observed in Fig. 11. The abscissa and ordinate show the magnitude and distance in the *Z* direction normalized by the wavelength λ . The parallel broken lines represent the grating surfaces.

Figure 14 shows *<sup>X</sup> E* and *EZ* (a), *SX* and *SZ* (b) for the case of *e d*/ = 0.4 at 8.0 θ = , which corresponds to the single dip in Fig. 11(a). Figures 15 and 16 show the same thing for the *e d*/ 0.08 = case. Figure 15 illustrates the results at 6.54 θ = , where the left dip is observed in Fig. 11(a). On the other hand, Fig. 16 depicts the results at 8.8 θ = , corresponding to the right dip in Fig. 11(a). In Figs. 14 to 16, we observe strong enhancement of *<sup>X</sup> E* and *EZ* (note that the magnitude of the incident radiation is 1), which is observed at the incidence angles where absorption occurs.

We observed that the total field above the grating surface decays exponentially in the *Z* direction and the magnitude of the total field is almost *ZE* in Fig. 14(a). The state of affairs is nearly the same in the metal region except for the rapid decay. Because the grating is thick, the oscillation near the upper surface does not reach the lower surface and, hence, the field below the grating is zero. Figure 14(b) illustrates the energy flow **S**, which is magnified by 15 in the metal region. We see that the energy flow is almost in the *X* direction and that it goes in opposite directions in vacuum and in metal. This is commonly observed when a SISP is excited.

In Figs. 15(a) and 16(a), we again see the enhancement of *XE* and *ZE* on the upper and lower surface of the silver-film grating, respectively. The rate of enhancement in Fig. 15(a) is not as large as that in Fig. 16(a). *ZE* is strongly enhanced at both the upper and lower surfaces of the grating and exponentially decays away from each surface. We thus observe the simultaneous excitation of surface plasmons at the surfaces. We can understand the difference of the field distributions assuming that the former and the latter refer to Figs. 15 and 16 are the results of the SRSP and the LRSP mode excitation. Figures 15(b) and 16(b) complement the understanding showing the even and odd nature of relevant oscillations.

#### **4.4. Multilayered thin-film bi-gratings case**

Next, we consider multilayered thin-film bi-gratings as shown in Fig. 1(a) indicated by *L* = 4 ( V/Ag/SiO /Ag/V ) that consists of a stack of silver and SiO <sup>2</sup> 2 films pairs. As listed in the figure, the values of the parameters are the same as those in Fig. 11 except for *L* = 4, <sup>2</sup> SiO *<sup>n</sup>* <sup>=</sup>1.5, <sup>2</sup> SiO *<sup>e</sup>* / d 0.3 or 0.08 <sup>=</sup> .

#### *4.4.1. Diffraction efficiency*

330 Plasmonics – Principles and Applications

**Figure 16.** Field distributions (a) and energy flows (b) at

parallel broken lines represent the grating surfaces.

*e d*/ 0.08 = case. Figure 15 illustrates the results at 6.54

in Fig. 11(a). On the other hand, Fig. 16 depicts the results at 8.8

on the *e d*/ 0.08 = curve in Fig. 11.

angles where absorption occurs.

SISP is excited.

θ

Figures 14, 15, and 16 show the field distributions of the *X*- and *Z*-components of the total electric fields (a) and energy flows (b) in the vicinity of the silver-film grating at the incidence angles at which absorption was observed in Fig. 11. The abscissa and ordinate

corresponds to the single dip in Fig. 11(a). Figures 15 and 16 show the same thing for the

right dip in Fig. 11(a). In Figs. 14 to 16, we observe strong enhancement of *<sup>X</sup> E* and *EZ* (note that the magnitude of the incident radiation is 1), which is observed at the incidence

We observed that the total field above the grating surface decays exponentially in the *Z* direction and the magnitude of the total field is almost *ZE* in Fig. 14(a). The state of affairs is nearly the same in the metal region except for the rapid decay. Because the grating is thick, the oscillation near the upper surface does not reach the lower surface and, hence, the field below the grating is zero. Figure 14(b) illustrates the energy flow **S**, which is magnified by 15 in the metal region. We see that the energy flow is almost in the *X* direction and that it goes in opposite directions in vacuum and in metal. This is commonly observed when a

In Figs. 15(a) and 16(a), we again see the enhancement of *XE* and *ZE* on the upper and lower surface of the silver-film grating, respectively. The rate of enhancement in Fig. 15(a) is not as large as that in Fig. 16(a). *ZE* is strongly enhanced at both the upper and lower surfaces of the grating and exponentially decays away from each surface. We thus observe the simultaneous excitation of surface plasmons at the surfaces. We can understand the difference of the field distributions assuming that the former and the latter refer to Figs. 15 and 16 are the results of the SRSP and the LRSP mode excitation. Figures 15(b) and 16(b) complement the understanding showing the even and odd nature of relevant oscillations.

θ

show the magnitude and distance in the *Z* direction normalized by the wavelength

Figure 14 shows *<sup>X</sup> E* and *EZ* (a), *SX* and *SZ* (b) for the case of *e d*/ = 0.4 at 8.0

= 8.8 , which corresponds to the right dip

λ. The

= , which

θ

= , corresponding to the

= , where the left dip is observed

θ

Figure 17 shows the (0,0)th order power reflection 00 ρ in V (Vacuum) (a) and the 1 transmission coefficient 00 τ in V5 (Vacuum) (b) as functions of the incident angle θ for two different values of 2 SiO *e d*/ . The curve for <sup>2</sup> SiO *e d*/ <sup>=</sup> 0.08 is almost the same as for a sinusoidal silver film bi-grating (Fig. 11): the coupled surface plasmon modes, SRSP and LRSP, are excited. On the other hand, in the curve for 2 SiO *e d*/ <sup>=</sup> 0.3 , we find a new type of absorption of incident light besides the plasmon resonance absorption associated with SRSP or LRSP.

This is related to the resonant excitation of a guided wave supported by the SiO2 film. This absorption is characterized by its occurrence over a wider range of θ . For example, in the case of 2 SiO *e d*/ 0.3 <sup>=</sup> , the extinction power is more than 50% for all angles of incidence θ ranging from 0 to 12 .

**Figure 17.** The (0, 0)-th order diffraction efficiencies 00 ρ (a) and 00 τ (b) as functions of θ for two values of *e*SiO2/*d* (*L*=4).

#### *4.4.2. Field distributions*

In order to examine the properties of the wide absorption found in Fig. 17, we investigated the field distributions of the total electric field *E*total and the TM component of the (0, 0)thorder diffracted electric field (0,0) TM *l E* in the vicinity of the SiO2 film. The magnitude of *E*total and TM *<sup>E</sup>l*(0,0) (*l* =1,2,...5) along the *Z*-axis are plotted in Fig. 18 where 0 θ<sup>=</sup> and 2 SiO *e d*/ 0.3 <sup>=</sup> . We observe in the figure that the field distributions of *E*total inside the SiO2 film indicates a standing wave pattern corresponding to the normal mode of a one dimensional cavity resonator, and that the distribution is almost close to that of 3(0 ,0) TM *E* . Hence, we conclude that the wide absorption observed in the multilayered grating V/Ag/SiO2/Ag/V is associated with resonance of the (0, 0)th-order diffracted wave 3(0,0) TM *E* in the SiO2 film sandwiched by a sinusoidal silver film grating.

Resonant Excitation of Plasmons in Bi-Gratings 333

This work was supported in part by Grant-in-Aid for Scientific Research from Japan Society for the Promotion of Science (Grant number 23560404). The authors thank Shi Bai and Qi

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**Acknowledgement** 

**6. References** 


**Figure 18.** Standing wave pattern of the electric filed in the SiO2 film.

### **5. Conclusions**

We have investigated the resonance absorption associated with the resonant excitation of surface plasmons in bi-gratings. Calculating diffraction efficiency, expansion coefficients, field profiles, and energy flows, we examined the characteristics of the resonant excitation of surface plasmons in detail. Interesting phenomena were revealed, including the conversion of a TM (or TE) component of the incident light into a TE (or TM) component at several different incidence angles, strong field enhancement on the grating surface where surface plasmons are excited, and simultaneous resonance absorption that does not occur in the case of a singly periodic grating in general. The results presented here facilitate a clear understanding of the coupled plasmon modes, SISP, SRSP and LRSP, excited in a thin film doubly periodic metal grating.

### **Author details**

Taikei Suyama, Akira Matsushima and Yoichi Okuno *Graduate School of Science and Technology, Kumamoto University, Kurokami, Kumamoto, Japan* 

Toyonori Matsuda *Kumamoto National College of Technology, Suya, Nishigoshi, Japan* 

### **Acknowledgement**

This work was supported in part by Grant-in-Aid for Scientific Research from Japan Society for the Promotion of Science (Grant number 23560404). The authors thank Shi Bai and Qi Zhao for help in numerical computation and in preparation of the manuscript.

#### **6. References**

332 Plasmonics – Principles and Applications

sinusoidal silver film grating.

**5. Conclusions** 

doubly periodic metal grating.

Taikei Suyama, Akira Matsushima and Yoichi Okuno

*Kumamoto National College of Technology, Suya, Nishigoshi, Japan* 

**Author details** 

Toyonori Matsuda

We observe in the figure that the field distributions of *E*total inside the SiO2 film indicates a standing wave pattern corresponding to the normal mode of a one dimensional cavity

the wide absorption observed in the multilayered grating V/Ag/SiO2/Ag/V is associated with

We have investigated the resonance absorption associated with the resonant excitation of surface plasmons in bi-gratings. Calculating diffraction efficiency, expansion coefficients, field profiles, and energy flows, we examined the characteristics of the resonant excitation of surface plasmons in detail. Interesting phenomena were revealed, including the conversion of a TM (or TE) component of the incident light into a TE (or TM) component at several different incidence angles, strong field enhancement on the grating surface where surface plasmons are excited, and simultaneous resonance absorption that does not occur in the case of a singly periodic grating in general. The results presented here facilitate a clear understanding of the coupled plasmon modes, SISP, SRSP and LRSP, excited in a thin film

*Graduate School of Science and Technology, Kumamoto University, Kurokami, Kumamoto, Japan* 

3(0 ,0)

TM *E* . Hence, we conclude that

TM *E* in the SiO2 film sandwiched by a

resonator, and that the distribution is almost close to that of

**Figure 18.** Standing wave pattern of the electric filed in the SiO2 film.

resonance of the (0, 0)th-order diffracted wave 3(0,0)


**1. Introduction**

http://dx.doi.org/50752

**2. Fundamental of polaritons**

cited.

The polariton is a kind of coupling between electromagnetic waves (photons) and elementary excitations such as phonons, plasmons and magnons, It includes two modes of surface and bulk polariton that can be excited by means of semiconductors, metals, ferromagnets, antiferromagnets, and so on. The coupling causes an intensity enhancement of the electromagnetic field, which is very useful for nano-technology such as biosensing, waveguide applications, nano-antenna device. The majority of the literature is concentrating on the magnetic surface polariton supported by magnetic materials. The outline of paper is as follows. In sections 2 we introduce the fundamental of the polaritons first, and then the general dispersion equation of the magnetic surface polaritons (MSPs) mode by considering the full form of Maxwell's equations in section 3. Next, in sections 4 and 5, we present numerical and experimental results of realizing the MSPs, by naturally existing materials − in section 4, by ferromagnetic and antiferromagnetic materials and in section 5, by the effective media of ferromagnetic and anti-ferromagnetic superlattices, respectively. By employing metamaterials, artificially constructed materials whose properties mainly stem from structures rather than their constitutive elements, one can also achieve MSPs mode with a greater engineering freedom; thus, in section 6, we start from interpreting what metamaterials and their operation rationales are, and then how metamaterials support the MSPs mode. Finally, in section 7, there comes a conclusion. Note that in this study we moderately modify the quoted definitions in the diagrams to correspond to our definitions.

**A Treatise on Magnetic Surface Polaritons:** 

**and Experimental Realization** 

Additional information is available at the end of the chapter

Yu-Hang Yang and Ta-Jen Yen

**Theoretical Background, Numerical Verification** 

**Chapter 13**

The well-known surface plasmon polaritons (SPPs) denote that a collective oscillation of electrons couples with the transverse magnetic wave, and they propagate along the interface between two media (ex: a metal and a dielectric) with an exponential decay into two media. Due to the polarization dependence of SPPs, it is a reasonable perception to realize a magnetic

and reproduction in any medium, provided the original work is properly cited.

©2012 Yang and Yen, licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0),which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly

© 2012 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution,

