**2.2 Specific properties of surface plasmons**

<sup>ϵ</sup>*r*ð Þ¼ *<sup>ω</sup>* <sup>1</sup> � *<sup>ω</sup>*<sup>2</sup>

and hydrodynamic velocity (*v r*ð Þ¼� , *t* ∇*ψ*ð Þ *r*, *t* ) [23].

*dt <sup>ψ</sup>*ð Þ¼ *<sup>r</sup>*, *<sup>t</sup>*

*d*

8 >>>>>><

>>>>>>:

*βk ωp*

*S r*,*r*

*d*

∇2

*<sup>ω</sup>*<sup>2</sup> <sup>¼</sup> <sup>1</sup>

<sup>0</sup> ð Þ¼ ; *<sup>ω</sup>* <sup>X</sup>

dependent density functional theory is used [23].

**62**

<sup>2</sup> *<sup>ω</sup>*<sup>2</sup> *<sup>p</sup>* <sup>þ</sup> *<sup>β</sup>*<sup>2</sup>

1 2

*G n*½ �¼ <sup>3</sup>

∇2

*d*

*d*

Eq. (8) for the first time [23].

values of wavenumber.

*Nanoplasmonics*

*p <sup>ω</sup>*<sup>2</sup> <sup>þ</sup> *<sup>i</sup>γ ω*ð Þ*<sup>ω</sup>* � *<sup>β</sup>k*<sup>2</sup> , *<sup>ω</sup>*<sup>2</sup>

where "*β*" is the Landau nonlocal parameter which becomes important for large

More precise treatment of surface plasmons can be done using the hydrodynamic model which includes solving Bloch equations, i.e., continuity, and Bernoulli and Poisson's equations simultaneously. According to Eqs. (4–6), one can describe collective oscillations of electrons in an arbitrary system using electron density (n)

j j <sup>∇</sup>*ψ*ð Þ *<sup>r</sup>*, *<sup>t</sup>* <sup>2</sup> <sup>þ</sup>

<sup>10</sup> <sup>3</sup>*π*<sup>2</sup> � �<sup>2</sup> 3 ½ � *n r*ð Þ , *<sup>t</sup>* <sup>5</sup> 3

In the general form, Bloch equations are nonlinear and quite difficult to solve.

However, using the perturbation theory, one can find linearized equations of Eq. (7) which helped Ritchie and his team to find plasmon dispersion equation in

> *n r*ð Þ¼ , *t n*0ð Þþ *r n*1ð Þþ *r*, *t* … *ψ*ð Þ¼ *r*, *t* 0 þ *ψ*1ð Þþ *r*, *t* …

*dt <sup>ψ</sup>*1ð Þ¼ *<sup>r</sup>*, *<sup>t</sup>* ½ � *<sup>β</sup>*ð Þ*<sup>r</sup>*

<<<sup>1</sup> ! *<sup>ω</sup>* <sup>¼</sup> *<sup>ω</sup><sup>p</sup>*

*n*

*ϕ*1ð Þ¼ *r*, *t* 4*πn*1ð Þ *r*, *t*

*dt <sup>n</sup>*1ð Þ¼ *<sup>r</sup>*, *<sup>t</sup>* <sup>∇</sup>*: <sup>n</sup>*0ð Þ*<sup>r</sup>* <sup>∇</sup>*ψ*<sup>1</sup> ½ � <sup>ð</sup>*r*, *<sup>t</sup>*<sup>Þ</sup>

*<sup>k</sup>*<sup>2</sup> <sup>þ</sup> *<sup>β</sup><sup>k</sup>*

ffiffi 2 <sup>p</sup> <sup>þ</sup> *<sup>β</sup><sup>k</sup>* 2

The most accurate model for dealing with surface plasmons in atomic scales is solving Schrodinger's equation and calculating dynamical structure factor in Eq. (9) which is related to the oscillations of particle density in a many-particle system [23]:

where the first two terms are elements of the operator "*ρ*(*r*)-*n*0(*r*)" relating the ground state "*ψ0*" with energy "*E0*" and "*δ*" is the Dirac function, "*n*0(*r*)" represents

Using this model one can precisely calculate electron density profile in a manyelectron system like a metal. However, solving the required equations is not easy, and most often approximations like random phase approximation or time-

ground state density of particles, and "*ρ*(*r*)" is the particle density operator.

*<sup>k</sup>*<sup>2</sup> h i <sup>q</sup>

<sup>2</sup> *n*1ð Þ *r*, *t n*0ð Þ*r*

*<sup>p</sup>* <sup>¼</sup> *ne*<sup>2</sup>

*dt n r*ð Þ¼ , *<sup>t</sup>* <sup>∇</sup>*n r*ð Þ¼ , *<sup>t</sup>* <sup>∇</sup>*:*½*n r*ð Þ , *<sup>t</sup>* <sup>∇</sup>*ψ*ð � *<sup>r</sup>*, *<sup>t</sup>*<sup>Þ</sup> (4)

þ *ϕ*ð Þ *r*, *t*

*ϕ*ð Þ¼ *r*, *t* 4*πn r*ð Þ , *t* (6)

þ *ϕ*ð Þ *r*, *t*

*δ*^*ρ*0*<sup>n</sup>*ð Þ *r*<sup>1</sup> *δ*^*ρ<sup>n</sup>*0ð Þ *r*<sup>2</sup> *δ ω*ð Þ � *En* þ *E*<sup>0</sup> (9)

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2*ω*<sup>2</sup> *<sup>p</sup>* <sup>þ</sup> *<sup>β</sup>*<sup>2</sup>

*δG n*½ � *δn*

<sup>ϵ</sup>0*<sup>m</sup>* , *<sup>γ</sup>* <sup>¼</sup> <sup>1</sup>

*<sup>τ</sup>* (3)

(5)

(7)

(8)

Various applications of plasmonic technology in development of nanoscale devices and systems are all based on the same fundamental properties of plasmons. These specific properties include field confinement, enhancement of local density of optical states, and ultrawide bandwidth and fast response [11].

Confinement of electromagnetic fields in scales much smaller than the wavelength is the most crucial property of surface plasmon modes and can be defined in both parallel and orthogonal planes. Due to the high rate of loss, propagation length of the surface plasma waves in any direction is inversely related to the imaginary part of the wavenumber "1/Im(ksp)." This length for good plasmonic metals like gold and silver is limited to a few microns and is considered as the upper limit of confinement [11]. The lower limit of confinement is exerted by Fourier transform properties with considering a monochromatic field with frequency "ω" and wavenumber "k = ω/c" in the vacuum with far from any surface. It can be concluded that in the "x" direction "ΔxΔα ≥ 2π" in which α is the x component of the wavenumber. Therefore, the lower limit of field confinement is "2π/αmax = λ" which is also known as the diffraction limit. However, for surface plasmons, the wavenumber according to the dispersion relation (see **Figure 1**) can be much higher than "ω/c" which implies that surface plasmon modes can be confined in extremely tiny dimensions (much smaller than the wavelength) [11].

Enhancement of local density of optical states (LDOS) for surface plasmons can be investigated both near the metal surface and in a metallic nanoresonator. In a metallic nanoresonator, this effect which is also known as Purcell effect or enhancement of spontaneous emission is the vital property of plasmonic nanolasers. Purcell factor (*Fp*) is defined by the ratio of decay rate due to the spontaneous emission in a cavity over the decay rate in the free space. It can be calculated by Fermi's golden rule in a two-level atomic system and expressed by Eq. (10) [24].

$$\begin{split} \frac{\Gamma\_{\rm cav}}{m\_{1}\Gamma\_{0}} &= \frac{3}{4\pi^{2}} \left(\frac{\lambda\_{\rm em}}{n\_{1}}\right)^{3} \frac{\text{Q}}{V\_{\text{eff}}} \frac{\left|\hat{u} \cdot \vec{f}\left(\vec{r}\right)\right|^{2}}{1 + 4Q^{2} \left(\frac{\alpha\_{\rm em} - \alpha\_{\rm e}}{w\_{\text{e}}}\right)^{2}} = F\_{p} \frac{\left|\hat{u} \cdot \vec{f}\left(\vec{r}\right)\right|^{2}}{1 + 4Q^{2} \left(\frac{\alpha\_{\rm em} - \alpha\_{\rm e}}{w\_{\text{e}}}\right)^{2}} \\ F\_{p} &= \frac{3}{4\pi^{2}} \left(\frac{\lambda\_{\rm em}}{n\_{1}}\right)^{3} \frac{\text{Q}}{V\_{\text{eff}}} \end{split} \tag{10}$$

In which "Γ*cav*" is the decay rate in the cavity, "Γ0" is the decay rate in the free space, "*n*1" is the refractive index of the propagation medium, "*λem*" and "*ωem*" are the emission wavelength of the medium, "ωc" is the cavity resonance frequency, "*Q*" is the quality factor of the cavity, "*Veff*" is the effective mode volume of the propagating mode in the cavity, and the dot product of the nominator corresponds to the mismatch between directions of transition dipole and the field.

In a dielectric microcavity despite the large quality factor, large mode volume results in infinitesimal Purcell factors, but nanoscale metallic resonators (the building block of a plasmonic nanolaser) provide a very small equivalent mode volume expressed by Eq. (11) which results in a large Purcell factor which is crucial for the nanolaser operation. Moreover, since the emission rate is proportional to the LDOS, the higher Purcell factor means the higher local density of optical states [24].

$$V\_{\rm eff} = \frac{\int \varepsilon \left(\overrightarrow{r}\right) \left|\overrightarrow{E} \left(\overrightarrow{r}\right)\right|^2 dr}{\text{Max}\left[\varepsilon \left(\overrightarrow{r}\right) \left|\overrightarrow{E} \left(\overrightarrow{r}\right)\right|^2\right]}\tag{11}$$

where "*ε*(*r*)" is the permittivity as a function of position inside the resonator volume in which the integral is calculated and "*E*(*r*)" is the electrical field related to the propagating mode. However, this condition for a plasmonic nanocavity may not be satisfied. Therefore, the electromagnetic energy density of a dispersive and dissipative medium should be used in Eq. (11). A dispersive lossless medium like the dielectric side of the interface (12) can provide a good estimation and a lossy medium like the metal side of the interface (13) should be used [25].

$$
\langle \mu \rangle = \frac{1}{2} \varepsilon\_0 \left| E \left( \overrightarrow{r} \right) \right|^2 \left( \varepsilon'\_r(r, \alpha) + \alpha \frac{\partial \varepsilon'\_r(r, \alpha)}{\partial \alpha} \right) \tag{12}
$$

Surface scattering due to roughness of the surface (*γsc*) is inversely related to the mean free path due to surface roughness (a) and directly proportional to the Fermi velocity of electrons [26]. Fermi velocity is the maximum velocity an electron can achieve due to Fermi-Dirac distribution. For instant, for gold Fermi velocity is

The third type of losses in plasmonic devices is because of a process called Landau's damping [26]. Landau damping arises from the electrons oscillating with the velocity equal to the phase velocity of the surface plasmon mode. During acceleration, these electrons absorb energy from the plasmon mode. In other words, a quanta (plasmon) of the surface plasma wave is annihilated, and an electron-hole pair is generated in this procedure. Simply put, Landau damping is the plasmonelectron interaction mechanism. Landau damping rate (*γL*) which is significant for the large wavenumber values (see **Figure 2**) can be estimated by Eq. (16) [26].

*:* exp �*k*<sup>2</sup>

Plasmonic nanolasers are considered to be quantum nanogenerators of surface plasmons. Similar to lasers, these devices also operate based on both particle and wave properties of the electromagnetic waves. Therefore, finding a valid quantization approach for surface plasmon modes is necessary for explaining operation

The first attempts for finding a quantized description of plasmons are done by Bohm and Pines in the 1950s, and their works lead to the Pines model. In the Pines model, metal is considered to be a free electron gas material and electrons share long-range correlations in their positions in the form of collective oscillations in the whole system [27]. Pines model describes a quantized model of these collective

*D* 2*k*<sup>2</sup> !

, *kD* <sup>¼</sup> *<sup>ω</sup><sup>p</sup> vF*

(16)

*<sup>γ</sup><sup>L</sup>* <sup>≈</sup>*ωp: kD*

*Nanoscale Plasmon Sources: Physical Principles and Novel Structures*

*DOI: http://dx.doi.org/10.5772/intechopen.90842*

**2.4 Surface plasmons in quantum mechanical picture**

*k* � �<sup>3</sup>

about 2.5 � <sup>10</sup><sup>6</sup> m/s.

principles of the nanolasers.

**Figure 2.**

**65**

*Landau damping effect on the total damping rate [26].*

where "*ε*´(*r*)" is the real part of the permittivity.

$$
\langle u \rangle = \frac{1}{4} \varepsilon\_0 \left| E \left( \overrightarrow{r} \right) \right|^2 \left( \varepsilon'\_r(r, \alpha) + \frac{2 \varepsilon''\_r(r, \alpha) \alpha}{\gamma} \right) \tag{13}
$$

where "*ε*˝(*r*)" is the real part of the permittivity and "*γ*" is the loss rate introduced in Drude's model.

The third specific property of surface plasmons is their ultrawide bandwidths and fast response. In plasmonic nanoresonators due to considerable loss levels, quality factor is limited and in many cases between 10 and 100. Low-quality factor despite its negative effect on the Purcell factor provides an ultrawide bandwidth of several terahertz. This wide bandwidth resulting in fast time response has applications in generating femtosecond pulses in nanoscale dimensions and ultra-wideband nanoantennas [11].

#### **2.3 Plasmon loss mechanisms**

Inherent lossy nature of plasmon propagations requires more attention to the loss mechanisms both for using the loss as a beneficial application like biomarkers and biosensors [13] and minimizing its unwanted effects like plasmonic nanolasers.

Surface plasmons decay because of several elastic or inelastic loss channels, for instance, scattering because of other electrons, phonons, or crystal defects and so on. We will categorize them into three groups. The first is bulk decay rate or (*γb*) which can be expressed by Eq. (14) [26].

$$
\gamma\_b = \chi\_{\epsilon - \epsilon} + \chi\_{\epsilon - phomon} + \chi\_{\epsilon - defect} + \dots \tag{14}
$$

which the first term is due to electron-electron scattering as can be derived by Eq. (15) and the second term is due to electron-phonon scattering mechanism. Furthermore, the third term is the electron-defect decay rate. It should be noticed that for metals in the room temperature, electron-phonon decay rate is about 10<sup>14</sup> Hz and increases with the temperature. But, the other two factors remain constant with the temperature and exist even in the absolute zero [26].

$$\gamma\_{\varepsilon-\varepsilon} \approx 10^{15} \cdot \left(\frac{\hbar o}{E\_F}\right)^2 \text{ Hz} \tag{15}$$

As can be seen from Eq. (15), the electron-electron scattering has a direct relation with the frequency and in the visible frequencies is in the same order of magnitude as the first term [26].

In the plasmonic structures with dimensions about few nanometers, i.e., shorter than the mean free path of electrons, the second type of decay should be considered. *Nanoscale Plasmon Sources: Physical Principles and Novel Structures DOI: http://dx.doi.org/10.5772/intechopen.90842*

where "*ε*(*r*)" is the permittivity as a function of position inside the resonator volume in which the integral is calculated and "*E*(*r*)" is the electrical field related to the propagating mode. However, this condition for a plasmonic nanocavity may not be satisfied. Therefore, the electromagnetic energy density of a dispersive and dissipative medium should be used in Eq. (11). A dispersive lossless medium like the dielectric side of the interface (12) can provide a good estimation and a lossy

medium like the metal side of the interface (13) should be used [25].

 2 *ε*0

 2 *ε*0

where "*ε*˝(*r*)" is the real part of the permittivity and "*γ*" is the loss rate intro-

The third specific property of surface plasmons is their ultrawide bandwidths and fast response. In plasmonic nanoresonators due to considerable loss levels, quality factor is limited and in many cases between 10 and 100. Low-quality factor despite its negative effect on the Purcell factor provides an ultrawide bandwidth of several terahertz. This wide bandwidth resulting in fast time response has applications in generating femtosecond pulses in nanoscale dimensions and ultra-wideband

Inherent lossy nature of plasmon propagations requires more attention to the loss mechanisms both for using the loss as a beneficial application like biomarkers and biosensors [13] and minimizing its unwanted effects like plasmonic nanolasers. Surface plasmons decay because of several elastic or inelastic loss channels, for instance, scattering because of other electrons, phonons, or crystal defects and so on. We will categorize them into three groups. The first is bulk decay rate or (*γb*)

which the first term is due to electron-electron scattering as can be derived by Eq. (15) and the second term is due to electron-phonon scattering mechanism. Furthermore, the third term is the electron-defect decay rate. It should be noticed that for metals in the room temperature, electron-phonon decay rate is about 10<sup>14</sup> Hz and increases with the temperature. But, the other two factors remain

constant with the temperature and exist even in the absolute zero [26].

*<sup>γ</sup><sup>e</sup>*�*<sup>e</sup>* <sup>≈</sup>10<sup>15</sup>*:* <sup>ℏ</sup>*<sup>ω</sup>*

As can be seen from Eq. (15), the electron-electron scattering has a direct relation with the frequency and in the visible frequencies is in the same order of

*EF* <sup>2</sup>

In the plasmonic structures with dimensions about few nanometers, i.e., shorter than the mean free path of electrons, the second type of decay should be considered.

*<sup>r</sup>*ð Þþ *<sup>r</sup>*, *<sup>ω</sup> <sup>ω</sup>∂ε*<sup>0</sup>

*<sup>r</sup>*ð Þþ *<sup>r</sup>*, *<sup>ω</sup>* <sup>2</sup>*ε*<sup>00</sup>

*γ<sup>b</sup>* ¼ *γ<sup>e</sup>*�*<sup>e</sup>* þ *γ<sup>e</sup>*�*phonon* þ *γ<sup>e</sup>*�*defect* þ … (14)

Hz (15)

*<sup>r</sup>*ð Þ *r*, *ω ∂ω*

*<sup>r</sup>* ð Þ *r*, *ω ω γ*

(12)

(13)

*<sup>ε</sup>*<sup>0</sup> *E r*! 

<sup>4</sup> *<sup>ε</sup>*<sup>0</sup> *E r*! 

h i *<sup>u</sup>* <sup>¼</sup> <sup>1</sup> 2

h i *<sup>u</sup>* <sup>¼</sup> <sup>1</sup>

duced in Drude's model.

*Nanoplasmonics*

nanoantennas [11].

**2.3 Plasmon loss mechanisms**

which can be expressed by Eq. (14) [26].

magnitude as the first term [26].

**64**

where "*ε*´(*r*)" is the real part of the permittivity.

Surface scattering due to roughness of the surface (*γsc*) is inversely related to the mean free path due to surface roughness (a) and directly proportional to the Fermi velocity of electrons [26]. Fermi velocity is the maximum velocity an electron can achieve due to Fermi-Dirac distribution. For instant, for gold Fermi velocity is about 2.5 � <sup>10</sup><sup>6</sup> m/s.

The third type of losses in plasmonic devices is because of a process called Landau's damping [26]. Landau damping arises from the electrons oscillating with the velocity equal to the phase velocity of the surface plasmon mode. During acceleration, these electrons absorb energy from the plasmon mode. In other words, a quanta (plasmon) of the surface plasma wave is annihilated, and an electron-hole pair is generated in this procedure. Simply put, Landau damping is the plasmonelectron interaction mechanism. Landau damping rate (*γL*) which is significant for the large wavenumber values (see **Figure 2**) can be estimated by Eq. (16) [26].

$$\gamma\_L \approx o\_p. \left(\frac{k\_D}{k}\right)^3. \exp\left(\frac{-k\_D^2}{2k^2}\right), k\_D = \frac{o\_p}{v\_F} \tag{16}$$

#### **2.4 Surface plasmons in quantum mechanical picture**

Plasmonic nanolasers are considered to be quantum nanogenerators of surface plasmons. Similar to lasers, these devices also operate based on both particle and wave properties of the electromagnetic waves. Therefore, finding a valid quantization approach for surface plasmon modes is necessary for explaining operation principles of the nanolasers.

The first attempts for finding a quantized description of plasmons are done by Bohm and Pines in the 1950s, and their works lead to the Pines model. In the Pines model, metal is considered to be a free electron gas material and electrons share long-range correlations in their positions in the form of collective oscillations in the whole system [27]. Pines model describes a quantized model of these collective

**Figure 2.** *Landau damping effect on the total damping rate [26].*

oscillations which have both wave and particle properties, and the corresponding quanta (plasmon) is a boson [27].

Moreover, in PCQED loss and dispersion are critical factors and are necessary for correct modeling. Finding precise quantum mechanical models for these phenomena in plasmonic nanocavities still needs more research. However, we can use the photon/plasmon analogy and developed methods and tools of the photons like the density of state matrix and decay channels for estimating quantum mechanical

Modeling phenomena like quantum fluctuations, spectral narrowing,

*HAF* <sup>¼</sup> *<sup>i</sup>*ℏ*g a*†

by Eq. (18) in which "*σjx*" and "*σjy*" are Pauli matrices [28].

*<sup>J</sup><sup>α</sup>* <sup>¼</sup> <sup>X</sup> *N*

> *γ*↑ 2

<sup>2</sup>*<sup>σ</sup> <sup>j</sup>*�*ρσ <sup>j</sup>*<sup>þ</sup> � <sup>1</sup>

X *N*

2*σ <sup>j</sup>*þ*ρσ <sup>j</sup>*� þ

*Jz<sup>ρ</sup>* � <sup>1</sup> 2

equation describing the system by collective atomic operators [28].

*j*¼1

2

!

*j*¼1

coherency, threshold behavior, and precise dynamic analysis of plasmon nanolasers need an atomic-scale quantum mechanical model. However, in order to find closedform equations, several simplifications are necessary, and thus this model just provides a theoretical tool for investigating fundamental properties of plasmonic

To do so, consider an N-atom system in a low-quality factor nanocavity in which the decay rate of the cavity (κ) is the fastest decay rate of this system. This condition is called the "bad cavity assumption" [28]. Therefore, resonator mode can be adiabatically eliminated, and the system state is totally determined by "N" active atoms. Considering two energy levels for each atom which are coupled to a cavity with resonance frequency (ω) and plasmon lifetime (1/2κ), one can describe the interactions between the atoms and field by Tavis-Cumming Hamiltonian of

*J*� � *aJ*<sup>þ</sup>

In which "*g*" is the coupling factor which is identical for all of the atoms, "*a*" and

*<sup>σ</sup>j<sup>α</sup>*, *<sup>α</sup>* <sup>¼</sup> f g *<sup>x</sup>*, *<sup>y</sup>*, *<sup>z</sup>* , *<sup>σ</sup> <sup>j</sup>*� <sup>¼</sup> *<sup>σ</sup>jx* � *<sup>σ</sup>jy*

*ρJz* � *Nρ*

X *N*

2*σjzρσjz* � *Nρ* !

*j*¼1

þ *γp* 2

� �" and its Fourier transform "*P v* <sup>~</sup> , *<sup>v</sup>* <sup>∗</sup> ð Þ , *<sup>m</sup>* " as the

" are annihilation and creation operators of plasmons, respectively, and "*Jα*" is the operator of collective atomic oscillations in the "*α*" direction and can be defined

Using atomic density operator "*ρ*" in a quantum system with state vector "*ψ*" and by considering the Hamiltonian of Eq. (17), Schrodinger's equation leads us to the dynamic equation of Eq. (19) in which "*γ*↑" is the pumping rate and "*γ*↓" is the spontaneous emission rate and "*γp*" is the dephasing rate of oscillating atoms [28]. The last term describes interaction of active atoms through the cavity mode [28].

> 1 2 *Jzρ* þ 1 2

!

*ρJz* � *Nρ*

atomic polarization operator, we can conclude Eq. (20) as a closed-form dynamic

After several mathematic manipulations on Eq. (19) and by defining

� � (17)

<sup>2</sup> (18)

þ *g*2

*<sup>κ</sup>* ð Þ <sup>2</sup>*J*�*<sup>ρ</sup>*

(19)

behavior of plasmonic nanocavities [14].

*DOI: http://dx.doi.org/10.5772/intechopen.90842*

nanolasers.

Eq. (17) [28].

"*a*†

*ρ*\_ ¼ �*i*

**67**

1 2

<sup>þ</sup> *<sup>γ</sup>*<sup>↓</sup> 2

*ω Jz* ½ �þ , *ρ*

X *N*

*j*¼1

"*XN <sup>ξ</sup>*, *<sup>ξ</sup>* <sup>∗</sup> ð Þ¼ , *<sup>η</sup> tr <sup>ρ</sup>ei<sup>ξ</sup>* <sup>∗</sup> *<sup>J</sup>*<sup>þ</sup> *<sup>e</sup><sup>i</sup>ηJz <sup>e</sup><sup>i</sup>ξJ*�

**3.2 Quantum mechanical atomic-scale model**

*Nanoscale Plasmon Sources: Physical Principles and Novel Structures*

Polariton is a joint state of light and matter introduced by Hopefield for providing a quantum model for the polarization field describing the response of matter to light [30]. Based on Hopfield's model, Ritchie and Elson proposed the first quantized description of surface plasma waves called Surface Plasmon Polariton or SPP. However, Hopefield's model did not consider the scattering and loss in the metal and effects of valance electrons and later Huttner and Barnett propose a model based on the Hopefield model including dispersion and loss, and recently a macroscopic quantization model based on Green's functions has also been published [27].
