Nanowires Integrated to Optical Waveguides

*Ricardo Téllez-Limón and Rafael Salas-Montiel*

### **Abstract**

Chip-scale integrated optical devices are one of the most developed research subjects in last years. These devices serve as a bridge to overcome size mismatch between diffraction-limited bulk optics and nanoscale photonic devices. They have been employed to develop many on-chip applications, such as integrated light sources, polarizers, optical filters, and even biosensing devices. Among these integrated systems can be found the so-called hybrid photonic-plasmonic devices, structures that integrate plasmonic metamaterials on top of optical waveguides, leading to outstanding physical phenomena. In this contribution, we present a comprehensive study of the design of hybrid photonic-plasmonic systems consisting of periodic arrays of metallic nanowires integrated on top of dielectric waveguides. Based on numerical simulations, we explain the physics of these structures and analyze light coupling between plasmonic resonances in the nanowires and the photonic modes of the waveguides below them. With this chapter we pretend to attract the interest of research community in the development of integrated hybrid photonic-plasmonic devices, especially light interaction between guided photonic modes and plasmonic resonances in metallic nanowires.

**Keywords:** plasmonics, integrated optics, nanowires, optical waveguides, hybrid modes

#### **1. Introduction**

Plasmonics, the science of plasmons, is a research field that has been extensively studied in recent years due to its multiple applications like biosensing, optical communications, or quantum computing, to mention but a few.

Generally, the field of plasmonics is associated with two types of collective oscillations of conductive electrons at the boundaries of metallic nanostructures, known as surface plasmon polaritons (SPP) and localized surface plasmons (LSP). While SPP are referred as surface waves propagating at a dielectric-metal interface, LSP can be regarded as standing surface waves confined in metallic nanoparticles embedded in a dielectric environment [1].

As it is well known, SPP modes can only be excited when appropriate phase match conditions are fulfilled. An option to achieve this condition, is by making use of the electromagnetic near field scattered by a local defect or emitter. To this purpose, the LSP mode of a metallic nanoparticle can be excited and coupled to the SPP of a metallic substrate, giving rise to hybrid plasmon polaritons [2, 3].

In addition to these types of plasmonic oscillations, there are other resonances named plasmonic chain modes. These modes can be generated in linear arrays of

closely spaced metallic nanoparticles, including nanowires, and they result from the near field coupling between adjacent nanoparticles excited at their plasmonic resonances. Due to this coupling effect, light can propagate through the periodic arrays. Thus, these periodic structures can be regarded as discrete plasmonic waveguides [4–6]. When placing a periodic array of metallic nanoparticles in a layered media, under proper excitation conditions, the plasmonic chain modes can also couple to the SPP of a metallic substrate, forming hybrid SPP-chain modes [7].

after a round-trip the wave reproduces itself, preserving the same spatial distribution and polarization along the waveguide. Fields satisfying this self-consistency condition are known as eigenmodes or modes of the waveguide [10]. The schematic in **Figure 1b** shows the profile of the field distribution for the first two modes of the

To determine the propagation constant of the modes supported by the waveguide, let us consider a waveguide with a core of refractive index *n*<sup>2</sup> and thickness *d*, surrounded by two semi-infinite dielectric media of refractive index *n*<sup>1</sup> and *n*3, as

For each medium, the field can be represented as a sum of propagative and counter-propagative waves along the *z* axis, and propagative in the *x* direction that

where *m* ¼ *I*,*II*,*III*, *Am* and *Bm* are the amplitudes of the propagative and counter-propagative waves, respectively, and the propagation constants *α<sup>m</sup>* and *β<sup>m</sup>*

> *ω c* � �<sup>2</sup>

*μ ω*ð Þ can be considered as unit. Req. (2) is obtained from the Helmholtz and

r

�*iω<sup>t</sup>* <sup>þ</sup> *Bme*

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

where *εm*ð Þ *ω* is the dielectric constant of the m-th medium related to the refrac-

At the interfaces *z* ¼ 0 and *z* ¼ *d*, the electromagnetic field should be contin-

*εm*ð Þ� *ω α*<sup>2</sup>

*m*

*<sup>ψ</sup>I*ð Þj *<sup>x</sup>*, *<sup>z</sup>*,*<sup>ω</sup> <sup>z</sup>*¼<sup>0</sup> <sup>¼</sup> *<sup>ψ</sup>II*ð Þj *<sup>x</sup>*, *<sup>z</sup>*, *<sup>ω</sup> <sup>z</sup>*¼<sup>0</sup>, (3)

*<sup>ψ</sup>II*ð Þj *<sup>x</sup>*, *<sup>z</sup>*,*<sup>ω</sup> <sup>z</sup>*¼*<sup>d</sup>* <sup>¼</sup> *<sup>ψ</sup>III*ð Þj *<sup>x</sup>*, *<sup>z</sup>*, *<sup>ω</sup> <sup>z</sup>*¼*<sup>d</sup>:* (4)

*ε ω*ð Þ*μ ω*ð Þ <sup>p</sup> . At optical wavelengths, he magnetic permeability

*iαmz e* �*i<sup>β</sup>mxe* �*iωt*

, (1)

, (2)

�*iαmz e* �*i<sup>β</sup>mxe*

*β<sup>m</sup>* ¼

*Schematic representation of the field components in an asymmetric planar waveguide.*

proposed waveguide.

*Nanowires Integrated to Optical Waveguides DOI: http://dx.doi.org/10.5772/intechopen.95689*

**2.2 Dispersion relation**

depicted in **Figure 2**.

can be represented as

*ψ <sup>m</sup>*ð Þ¼ *x*, *z*, *ω Ame*

along the *z* and *x* axis are related through

tive index by *nm* <sup>¼</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

Maxwell equations [11].

uum, that is to say:

and

**Figure 2.**

**137**

In this same sense, when placing periodic arrays of metallic nanoparticles on top of dielectric waveguides, the plasmonic chain modes can couple to the photonic modes of the waveguide [8]. These integrated structures give rise to the so-called hybrid photonic-plasmonic waveguide modes [9], and they are the main subject of interest in this chapter. We will focus our attention to integrated structures consisting of periodic arrays of metallic nanowires integrated on top of twodimensional dielectric photonic waveguides.

To this purpose, we will bring a comprehensive explanation about the physics behind the dispersion curves of integrated hybrid photonic-plasmonic waveguiding structures. Then will be studied the propagation of electromagnetic fields through the integrated systems varying the geometric cross-section of the metallic nanowires. For a better understanding, this comprehensive study will be accompanied by numerical simulations, making easier to elucidate the potential applications of these outstanding structures.

### **2. Hybrid photonic-plasmonic waveguides**

#### **2.1 Optical waveguides**

From the analysis of the chemical composition of farer stars to imaging of microscopic living cells, information transport through light is one of the main subjects of interest in optical sciences. Among the different ways to transport light can be found optical waveguides, whose principle of operation is based on the total internal reflection effect. This phenomenon consists of the complete reflection of light within a medium surrounded by media with smaller refractive index, as depicted in **Figure 1**.

The schematic in **Figure 1a** represents an asymmetric planar waveguide invariant along the out-of-plane direction, consisting of a dielectric medium of refractive index *n*<sup>2</sup> between two media of refractive index *n*<sup>1</sup> and *n*3, where *n*<sup>2</sup> >*n*<sup>1</sup> >*n*3. As light propagates in the inner medium *n*2, certain rays will present a phase difference of zero or a multiple of 2*π*, when they are twice reflected. This situation means that

#### **Figure 1.**

*Schematic representation of a planar asymmetric waveguide consisting of three dielectric media of refractive index n*1*, n*<sup>2</sup> *and n*3*. (a) Self-consistency condition defining the modes of the waveguide. (b) Profile of the field distributions of the first two guided modes of the waveguide.*

after a round-trip the wave reproduces itself, preserving the same spatial distribution and polarization along the waveguide. Fields satisfying this self-consistency condition are known as eigenmodes or modes of the waveguide [10]. The schematic in **Figure 1b** shows the profile of the field distribution for the first two modes of the proposed waveguide.

#### **2.2 Dispersion relation**

closely spaced metallic nanoparticles, including nanowires, and they result from the near field coupling between adjacent nanoparticles excited at their plasmonic resonances. Due to this coupling effect, light can propagate through the periodic arrays. Thus, these periodic structures can be regarded as discrete plasmonic waveguides [4–6]. When placing a periodic array of metallic nanoparticles in a layered media, under proper excitation conditions, the plasmonic chain modes can also couple to

In this same sense, when placing periodic arrays of metallic nanoparticles on top of dielectric waveguides, the plasmonic chain modes can couple to the photonic modes of the waveguide [8]. These integrated structures give rise to the so-called hybrid photonic-plasmonic waveguide modes [9], and they are the main subject of

To this purpose, we will bring a comprehensive explanation about the physics behind the dispersion curves of integrated hybrid photonic-plasmonic waveguiding structures. Then will be studied the propagation of electromagnetic fields through

nanowires. For a better understanding, this comprehensive study will be accompanied by numerical simulations, making easier to elucidate the potential applications

From the analysis of the chemical composition of farer stars to imaging of microscopic living cells, information transport through light is one of the main subjects of interest in optical sciences. Among the different ways to transport light can be found optical waveguides, whose principle of operation is based on the total internal reflection effect. This phenomenon consists of the complete reflection of light within a medium surrounded by media with smaller refractive index, as

The schematic in **Figure 1a** represents an asymmetric planar waveguide invariant along the out-of-plane direction, consisting of a dielectric medium of refractive index *n*<sup>2</sup> between two media of refractive index *n*<sup>1</sup> and *n*3, where *n*<sup>2</sup> >*n*<sup>1</sup> >*n*3. As light propagates in the inner medium *n*2, certain rays will present a phase difference of zero or a multiple of 2*π*, when they are twice reflected. This situation means that

*Schematic representation of a planar asymmetric waveguide consisting of three dielectric media of refractive index n*1*, n*<sup>2</sup> *and n*3*. (a) Self-consistency condition defining the modes of the waveguide. (b) Profile of the field*

the SPP of a metallic substrate, forming hybrid SPP-chain modes [7].

dimensional dielectric photonic waveguides.

**2. Hybrid photonic-plasmonic waveguides**

*distributions of the first two guided modes of the waveguide.*

of these outstanding structures.

**2.1 Optical waveguides**

*Nanowires - Recent Progress*

depicted in **Figure 1**.

**Figure 1.**

**136**

interest in this chapter. We will focus our attention to integrated structures consisting of periodic arrays of metallic nanowires integrated on top of two-

the integrated systems varying the geometric cross-section of the metallic

To determine the propagation constant of the modes supported by the waveguide, let us consider a waveguide with a core of refractive index *n*<sup>2</sup> and thickness *d*, surrounded by two semi-infinite dielectric media of refractive index *n*<sup>1</sup> and *n*3, as depicted in **Figure 2**.

For each medium, the field can be represented as a sum of propagative and counter-propagative waves along the *z* axis, and propagative in the *x* direction that can be represented as

$$
\Psi\_m(\mathbf{x}, z, \alpha) = A\_m e^{-ia\_m x} e^{-i\beta\_m x} e^{-ia\alpha t} + B\_m e^{ia\_m x} e^{-i\beta\_m x} e^{-i\alpha t},\tag{1}
$$

where *m* ¼ *I*,*II*,*III*, *Am* and *Bm* are the amplitudes of the propagative and counter-propagative waves, respectively, and the propagation constants *α<sup>m</sup>* and *β<sup>m</sup>* along the *z* and *x* axis are related through

$$
\beta\_m = \sqrt{\left(\frac{\alpha}{c}\right)^2 \varepsilon\_m(\alpha) - a\_m^2},
\tag{2}
$$

where *εm*ð Þ *ω* is the dielectric constant of the m-th medium related to the refractive index by *nm* <sup>¼</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *ε ω*ð Þ*μ ω*ð Þ <sup>p</sup> . At optical wavelengths, he magnetic permeability *μ ω*ð Þ can be considered as unit. Req. (2) is obtained from the Helmholtz and Maxwell equations [11].

At the interfaces *z* ¼ 0 and *z* ¼ *d*, the electromagnetic field should be continuum, that is to say:

$$\left.\psi\_I(\mathbf{x}, z, \boldsymbol{\alpha})\right|\_{\boldsymbol{x}=\mathbf{0}} = \left.\psi\_{II}(\mathbf{x}, z, \boldsymbol{\alpha})\right|\_{\boldsymbol{x}=\mathbf{0}}.\tag{3}$$

and

$$\left.\psi\_{II}(\mathbf{x},\mathbf{z},a)\right|\_{\mathbf{z}=d} = \left.\psi\_{III}(\mathbf{x},\mathbf{z},a)\right|\_{\mathbf{z}=d}.\tag{4}$$

**Figure 2.** *Schematic representation of the field components in an asymmetric planar waveguide.*

From the conservation of the tangential components of the electromagnetic field at the boundaries between two media [12] are obtained the relationships

$$\left. \frac{1}{\nu\_I} \frac{\partial \nu\_I(\mathbf{x}, z, \alpha)}{\partial \mathbf{z}} \right|\_{x=0} = \frac{1}{\nu\_{II}} \frac{\partial \nu\_{II}(\mathbf{x}, z, \alpha)}{\partial \mathbf{z}} \Big|\_{x=0},\tag{5}$$

**2.3 Plasmonic waveguides**

*Nanowires Integrated to Optical Waveguides DOI: http://dx.doi.org/10.5772/intechopen.95689*

propagate through kilometers.

antisymmetric modes, respectively.

even imaging applications [13–15].

simply hybrid plasmonic waveguide.

**Figure 3.**

**139**

**2.4 Hybrid photonic-plasmonic waveguides**

As previously explained, dielectric waveguides guide light modes by using the total internal reflection principle and self-consistency condition. These waveguides are diffraction limited due to the dielectric constant values. However, if the dielectric constant is a complex number, it would be possible to obtain solutions to the dispersion relation (Eq. 8) below the diffraction limit. This is the case of metallic materials. Hence, if at least one of the three media in the waveguide structure is metallic, it is always possible to obtain a propagative mode in the structure. The price to pay for this solution is that due to ohmic losses in metals, these modes propagate just few microns, in opposition to dielectric waveguides where light can

These structures are known as plasmonic waveguides, and their operation principle is based on SPP mode propagation. These surface waves are the result of collective oscillations of the conductive electrons at a metal-dielectric interface induced by the electric field of an electromagnetic wave. For a system invariant in the ^*y* direction, SPP modes can only be excited if the electric field oscillates in the *xz* plane. Hence, only TM polarized electromagnetic fields couple to SPP modes (for

Different combinations of insulator (I) and metallic (M) materials can be used to

As plasmonic waveguides allow light propagation beyond the diffraction limit, these structures have been used for the development of integrated nanophotonic devices for optical signal transportation, optical communications, biosensing and

From the previous waveguiding configurations, it is natural to think that modes propagating through a dielectric waveguide can be coupled to a plasmonic waveguide. This kind of structures is named hybrid photonic-plasmonic waveguide, or

The structure depicted in **Figure 3a** can be considered as a hybrid plasmonic waveguide, but more complex multilayered systems can be designed to propagate more than one mode in these structures. For instance, in **Figure 4** are presented two

*Schematic representation of plasmonic waveguides for (a) IIM, (b) IMI and (c) MIM configurations and SPP*

*modes profiles. For IMI and MIM waveguides, symmetric and antisymmetric modes are excited.*

TE polarized waves the electric field only oscillates along the ^*y* direction).

define a plasmonic waveguide. In **Figure 3** are represented IIM, IMI and MIM plasmonic waveguide structures as well as the amplitude distribution of the outof-plane electromagnetic field (*Hy* component) of the SPP modes. For the IIM structure, there is only one SPP mode at the interface between the metal (*εm*) and first dielectric (*ε<sup>d</sup>*<sup>1</sup> ). For both IMI and MIM configurations, two SPP modes can be excited. They result from in-phase and out-of-phase coupling of SPP at the first and second dielectric-metal interfaces, and they are known as symmetric and

$$\frac{1}{\nu\_{\rm II}} \frac{\partial \nu\_{\rm II}(\mathbf{x}, \mathbf{z}, \boldsymbol{\alpha})}{\partial \mathbf{z}} \bigg|\_{\mathbf{z} = -d} = \frac{1}{\nu\_{\rm III}} \frac{\partial \nu\_{\rm III}(\mathbf{x}, \mathbf{z}, \boldsymbol{\alpha})}{\partial \mathbf{z}} \bigg|\_{\mathbf{z} = d},\tag{6}$$

with *ν<sup>m</sup>* ¼ 1 for TE polarized electromagnetic fields and *ν<sup>m</sup>* ¼ *εm*ð Þ *ω* for TM polarized fields. Substituting Eq. (1) in Eqs. (3–6), and considering that *AI* ¼ *BIII* ¼ 0 because both, *I* and *III* are semi-infinite media and no back-reflections from the boundaries are present, it is obtained a two coupled equation system that can be represented in a matrix way of the form

$$
\begin{bmatrix}
\frac{\alpha\_I}{\nu\_I} - \frac{a\_{II}}{\nu\_{II}} & \frac{a\_I}{\nu\_I} + \frac{a\_{II}}{\nu\_{II}} \\
 e^{-ia\_{II}d} \left(\frac{a\_I}{\nu\_I} + \frac{a\_{III}}{\nu\_{III}}\right) & -e^{-ia\_{II}d} \left(\frac{a\_I}{\nu\_I} - \frac{a\_{III}}{\nu\_{III}}\right)
\end{bmatrix}
\begin{bmatrix} A\_{II} \\ B\_{II} \end{bmatrix} = \begin{bmatrix} \mathbf{0} \\ \mathbf{0} \end{bmatrix}.\tag{7}
$$

By equating to zero the determinant of the matrix it is possible to obtain the non-trivial solutions of this eigenmode equation system, resulting in the dispersion relation of a three-layered media

$$\frac{\left(\frac{a\overline{\mu}}{\nu\_{\rm II}} - \frac{a\overline{\iota}}{\nu\_{\rm I}}\right)\left(\frac{a\overline{\iota}}{\nu\_{\rm II}} - \frac{a\overline{\iota}\overline{\iota}}{\nu\_{\rm III}}\right)}{\left(\frac{a\overline{\iota}\_{\rm II}}{\nu\_{\rm II}} + \frac{a\overline{\iota}}{\nu\_{\rm I}}\right)\left(\frac{a\overline{\iota}}{\nu\_{\rm II}} + \frac{a\overline{\iota}\overline{\iota}}{\nu\_{\rm III}}\right)} = e^{i2d\alpha\underline{\iota}}.\tag{8}$$

We must notice that Eq. (8) is a transcendental function with no analytical solution, thus, numerical methods should be employed to solve it.

When solving this dispersion relation, it is obtained the mode propagation constant, *β*, that depends on the optical frequency or wavelength of light and determines how the amplitude and phase of light varies along the *x* direction. In the same way as wavenumber can be related to the refractive index of a homogeneous medium, the propagation constant can be regarded as the wavenumber (spatial frequency) of light propagating through an effective medium composed by the inhomogeneous three-layered structure. The propagation constant is then related to the so-called effective index through the relationship

$$
\beta = \frac{2\pi}{\lambda} n\_{\text{eff}},
\tag{9}
$$

being *λ* the wavelength of light in vacuum. We must notice that the effective index is only defined for a mode of the waveguide and it should not be understood as a material property. We can say then that each mode of the waveguide will "see" different effective media.

As the refractive index of a dielectric medium, as well as the dielectric constant, is a real number equal or greater than the unit (*n*≥1) the modes in a dielectric waveguide are diffraction limited: if the thickness of the waveguide, *d*, is smaller than *λ=* 2*neff* � �, the solutions for the dispersion Eq. (8) will lead to evanescent waves, meaning that no modes can be propagated below this limit.
