Analysis of dispersive and dissipative media with optical resonances

Frederico Dias Nunes; Ben-Hur Viana Borges; John Weiner

doi:10.1364/OE.20.015679

1. Introduction

Advances in nanofabrication technology realized in the past few years have opened the door to sensitive, rugged, miniaturized devices for signal processing, communication, medical treatment, as well as environmental and biological sensing. All these applications push conventional electronics beyond intrinsic limits, for they require high processing speeds, small circuit sizes, and low power dissipation. These challenges are hard to overcome with conventional bulky optical devices (limited in size by the Abbe's diffraction limit). Fortunately, with the advent of plasmonics, new technologies with unprecedented small sizes have become available, paving the way to large scale integrated plasmonic devices. These devices are based on the propagation of surface plasmon-polariton waves (SPPs) that correspond to the coupling between an electromagnetic wave and electron charge density at the interface between a metal and a dielectric. The excitation of SPP waves in structures such as subwavelength optical cavities, nano-slits, or nano-holes have enabled new technological solutions that overcome the well-known limitations of conventional electronics and optics. More recently, Sun et al. [1] have demonstrated the first two-dimensional optical lumped nanocircuit at mid-infrared wavelengths, demonstrating the reality of the concept of “metatronics,” a term coined by Engheta [2].

Despite all these exciting new practical possibilities, old-fashioned problems such as signal degradation due to dispersion or attenuation still remain to be tackled when operating at frequencies close to a spectral resonance of the device medium. Around the resonant regions, properties of propagating SPP waves such as group and energy velocities, or stored and dissipated energy require additional considerations. The conventional mathematical description of the electric energy density associated with a propagating electromagnetic field applies Brillouin’s [3] or Landau’s [4] model, and both authors calculate the energy density using the common expression $(E \cdot d D / dt)$ together with the medium permittivity. Brillouin considers a modulated electric field, while Landau describes an electric field via the Fourier transform. In both cases, absorption effects are neglected and the material permittivity is assumed to be a slowly varying function of the frequency. This assumption leads to a very simple and well-known expression for the average stored energy density $〈 μ_{s} 〉$ :

〈 u_{s} (t) 〉 = \frac{ε_{0} {| E_{0} |}^{2}}{4} [ε^{'} (ω) + ω \frac{d ε^{'} (ω)}{d ω}],

where

ε^{'}

is the real part of the relative permittivity,

ε_{0}

the free-space permittivity, ω the angular frequency, and

E_{0}

the electric field amplitude.

A different approach to this problem was suggested by Loudon [5], in which a generic material was modeled as a collection of damped harmonic oscillators, all with the same resonant frequency, interacting with a monochromatic field. The averaged stored energy density and dissipated power obtained for this case are given by, respectively:

〈 u_{s} 〉 = \frac{1}{4} ε_{0} {| E_{0} |}^{2} ((ε^{'} - 1) + \frac{2 ε^{″} ω}{γ})

〈 W_{d} 〉 = \frac{1}{2} (ε^{″} ε_{0} ω) {| E_{0} |}^{2},

where ε″ is the imaginary part of the relative permittivity and γ is the collision frequency. Loudon has shown that when a material spectral resonance is present, his model does not agree with Brillouin’s result. Nonetheless, both approaches converge for two specific situations, namely, when γ vanishes [5] or when the medium permittivity can be described by a Drude model, as shown by Nunes et al. [6].

Differently from Loudon’s treatment, the present work focuses on the analysis of electromagnetic field interaction with a real medium whose permittivity may be described as a combination of Drude and Lorentz oscillators, each with its proper oscillator characteristics. Our approach is also similar to the earlier work of Oughstun and Shen [7]. The present analysis emphasizes Al, but it can be readily extended to any metal. We discuss many important quantities such as group and energy velocity, and stored and dissipated energy. Magnetic properties can be straightforwardly incorporated by assuming the magnetic permeability to be described by a set of resonances with Lorentzian lineshape. This strategy can be useful for the analysis of complex media, such as metamaterials. The properties of materials exhibiting single Lorentzian resonances in the permeability have been analyzed by Ruppin [8].

This rest of this paper is organized as follows. Section 2 describes the mathematical framework developed to describe the interaction between the electromagnetic field and matter. Section 3 applies the proposed model to Al, and describes the effect of spectral resonances on group and energy velocities, and stored and dissipated energies. Section 4, in turn, solves an Al-based metal-dielectric-metal waveguide, and investigates how the above quantities are influenced by material spectral resonances. Finally, Section 5 summarizes the conclusions.

2. Interaction between electromagnetic field and matter

In this section we present the formalism that describes the interaction of harmonic electromagnetic fields with matter. We start with the Poynting theorem [9], which reads (for non-magnetic material):

- \nabla \cdot S = ε_{0} E \cdot \frac{d E}{d t} + μ_{0} H \cdot \frac{d H}{d t} + E \cdot \frac{d P}{d t} .

The first two terms on the right in (4) correspond to the stored electric and magnetic field power density, respectively. The third term,

E \cdot d P / d t

, corresponds to power density relative to the interaction between field and matter, and requires the description of the medium optical properties. To calculate this third term let us first obtain the polarization vector

P

using the well-known definition [9],

P = \sum_{j = 1}^{M} ρ_{j} p_{j} = \sum_{j = 1}^{M} ρ_{j} (- e r_{j})

where M is the number of vibrational modes, ρ_j is the density of oscillators belonging to mode j with an electric dipole moment p_j. Here, the medium is considered as a collection of oscillator groups, where each group accounts for one characteristic oscillatory process. The electric field is directly obtained from the equation of motion for any dipole j, which is given by:

m_{e} {\ddot{r}}_{j} + m_{e} γ {\dot{r}}_{j} + m_{e} ω_{0} {\dot{r}}_{j} = - e E

where m_e and e are the electron mass and charge, respectively, γ_j is the collision frequency, and ω_0j is the restoring frequency. The dot(s) on top of r stands for time derivative(s). The electric field then reads

E = \frac{- m_{e}}{e} ({\ddot{r}}_{j} + γ_{j} {\dot{r}}_{j} + ω_{0 j}^{2} r_{j}) .

Now, dot multiplying E by the time derivative of P yields

E \cdot \frac{d P}{dt} = \sum_{j = 1}^{M} (m_{e} ρ_{j}) ({\ddot{r}}_{j} + γ_{j} {\dot{r}}_{j} + ω_{0 j}^{2} r_{j}) \cdot {\dot{r}}_{j} .

However, it can be shown that:

({\ddot{r}}_{j} \cdot {\dot{r}}_{j} + ω_{0 j}^{2} r_{j} \cdot {\dot{r}}_{j}) = \frac{1}{2} \frac{\partial}{\partial t} ({\dot{r}}_{j} \cdot {\dot{r}}_{j} + ω_{0 j}^{2} r_{j} \cdot r_{j}) .

Hence, after substituting Eq. (9) into Eq. (8), we have

E \cdot \frac{d P}{d t} = {\frac{\partial}{\partial t} \sum_{j = 1}^{M} ρ_{j} (\frac{1}{2} m_{e} ({\dot{r}}_{j} \cdot {\dot{r}}_{j}) + \frac{1}{2} m_{e} ω_{0 j}^{2} (r_{j} \cdot r_{j})) + \sum_{j = 1}^{M} m_{e} ρ_{j}_{j} γ_{j} ({\dot{r}}_{j} \cdot {\dot{r}}_{j})} .

We can further simplify Eq. (10) by means of the following definitions:

K_{j} = \frac{1}{2} m_{e} ({\dot{r}}_{j} \cdot {\dot{r}}_{j}) a n d U_{j} = \frac{1}{2} m_{e} ω_{0 j}^{2} (r_{j} \cdot r_{j}) .

Therefore, the final expression for the third term of Eq. (4) becomes,

E \cdot \frac{d P}{d t} = {\frac{\partial}{\partial t} \sum_{j = 1}^{M} ρ_{j} (K_{j} + U_{j}) + \sum_{j = 1}^{M} ρ_{j} γ_{j} ({\dot{r}}_{j} \cdot {\dot{r}}_{j})} .

From Eq. (12) it becomes clear that the power of field and matter interaction has two main contributions, namely the stored energy density (given in terms of the dipoles kinetic K_j and potential U_j energies), and the power dissipated as a result of the interaction (given by the second summation).

Substituting Eq. (12) back into Eq. (4), the Poynting vector theorem then becomes:

- \nabla \cdot S = \frac{\partial}{\partial t} [\frac{1}{2} ε_{o} (E \cdot E) + \frac{1}{2} μ_{o} (H \cdot H) + u_{s} (t)] + W_{d} (t),

with the following definitions:

u_{s} (t) = \sum_{j = 1}^{M} ρ_{j} (K_{j} + U_{j}) and W_{d} (t) = \sum_{j = 1}^{M} ρ_{j} γ_{j} ({\dot{r}}_{j} \cdot {\dot{r}}_{j})

where u_s is the total stored energy in matter and W_d is the total power dissipated in the field-matter interaction. The sum of all terms inside the squared bracket in Eq. (13) gives the total energy density at any point in space where field and matter are present and interacting.

After highlighting each contribution of the field and matter interaction, we now write the energy density and dissipated power in the usual form using optical properties of matter such as permittivity. We start with the expression for r_j which is obtained as the solution of Eq. (6),

r_{j} = [\frac{- e / m_{e}}{(ω_{0 j}^{2} - ω^{2}) - i (γ_{j} ω)}] E .

Therefore, the electric dipole moment p_j_, will be:

p_{j} = - e r_{j} = [\frac{e^{2} / m_{0} ε_{0}}{(ω_{0 j}^{2} - ω^{2}) - i (γ_{j} ω)}] ε_{0} E,

where for convenience we have multiplied Eq. (16) by ε₀/ε₀. According to Eq. (5), the sum of all electric dipole moments per unit volume for all j groups gives the polarization of matter,

P = \sum_{j = 1}^{M} \frac{ρ_{j} e^{2} / m_{e} ε_{0}}{(ω^{2} - ω_{0 j}^{2}) - i (γ_{j} ω)} = χ_{e} (ω) ε_{0} E,

where χ_e(ω) is the susceptibility of the media given by

χ_{e} (ω) = \sum_{j = 1}^{M} \frac{ρ_{j} e^{2} / m_{e} ε_{0}}{(ω_{0 j}^{2} - ω^{2}) - i (γ_{j} ω)} .

From Eq. (18) we can easily obtain the expression for the permittivity ε using the general definition ε = (1 + χ_e), which becomes,

ε = ε^{'} +i ε^{″} = [1 + \sum_{j = 1}^{M} \frac{f_{j} ω_{p}^{2}}{(ω_{0 j}^{2} - ω^{2}) - i (γ_{j} ω)}]

where the plasma frequency ω_p given by:

ω_{p}^{2} = (\frac{ρ e^{2}}{m_{e} ε_{0}})

with ρ as the density of the total oscillators in the medium. Equation (19) can be split into real and imaginary parts of the dielectric function, as given below:

ε^{'} = 1 + \sum_{j=1}^{M} \frac{f_{j} ω_{p}^{2} (ω^{2} - ω_{0 j}^{2})}{{(ω_{0 j}^{2} - ω^{2})}^{2} + {(γ_{j} ω)}^{2}}

and

ε^{″} = \sum_{j = 1}^{M} \frac{f_{j} ω_{p}^{2} (γ_{j} ω)}{{(ω_{0 j}^{2} - ω^{2})}^{2} + {(γ_{j} ω)}^{2}}

The parameter f_j = ρ_j/ρ is the oscillator strength obeying the normalization

\sum_{j = 1}^{M} f_{j} = 1

, and is interpreted as a weight quantifying the influence of each vibrational process upon the total permittivity. Assuming a harmonic electromagnetic field of the form E(t) = E_oe^-^iωt, r_j will be complex and the following expansions can be obtained:

{\dot{r}}_{j}^{2} = ω^{2} r_{j}^{2} and {\dot{r}}_{j}^{2} + ω_{j}^{2} r_{j}^{2} = (ω_{}^{2} + ω_{j}^{2}) r_{j}^{2} .

By substituting Eq. (23) into Eq. (11) and the result in the expression for u_s(t) in Eq. (14), one finally obtains the averaged stored energy density during the interaction between electromagnetic field and matter, which reads:

〈 u_{s} (t) 〉 = 〈 \sum_{j = 1}^{M} ρ_{j} (K_{j} + U_{j}) 〉 = \frac{1}{4} ε_{0} {| E |}^{2} [(ε^{'} - 1) + \sum_{j} \frac{2 ω}{γ_{j}} {ε^{″}}_{j}] .

The dissipated power, in its turn, is obtained from (16) as follows,

〈 W_{d} (t) 〉 = 〈 \sum_{j = 1}^{M} m n_{j} γ_{j} {\dot{r}}_{j}^{2} 〉 = \frac{1}{2} ω ε^{″} ε_{0} {| E |}^{2},

where ε″ accounts for the sum of all imaginary parts of the permittivity contributions ε_j. An algebraic manipulation of the sum on the right hand side of Eq. (24) shows that this term can be recast as a weighted mean value of ε_j″ if (1/γ_j) is used as the weighting coefficient. By doing this, Eq. (24) becomes

〈 u_{s} (t) 〉 = \frac{1}{4} ε_{0} {| E |}^{2} [(ε^{'} - 1) + \frac{2 ω {〈 {ε^{″}}_{j} 〉}_{γ_{j}}}{Γ}] .

In Eq. (26) the following definitions were used:

{〈 ε^{″} 〉}_{γ_{j}} = \frac{\sum_{j} γ_{j}^{- 1} {ε^{″}}_{j}}{\sum_{j} γ_{j}^{- 1}} Γ = \sum_{j} γ_{j}^{}

Therefore, Eq. (26) is a generalization of the expression obtained by Loudon [see Eq. (2)] for a medium exhibiting one resonant line. Two obvious differences between Eqs. (26) and (2) are observed. The first difference is the imaginary part of the permittivity ε″ found in Loudon’s equation which is now replaced in the multi-Lorentz approach Eq. (26) by the weighted mean value of the different terms ε_j″ corresponding to each resonant process. Each one can be related to a different inter-band transition [10]. The second difference is the collision frequency γ found in Loudon, Eq. (2), which is replaced hereby Γ given in Eq. (27), and whose value is determined by all the processes corresponding to each resonance.

It may seem strange at first glance that the imaginary part of the medium permittivity appears in the expression for the stored energy. However, the influence of each dissipative process upon the stored energy is inversely dependent on the collision frequency. Consequently, the larger the value of γ_j, the smaller the contribution of ε_j″ to the total weighted mean value 〈ε″〉. Higher values of γ_j correspond to higher number of collisions and, therefore, dissipation should lead to smaller contributions to the dispersion and storage energies.

3. Numerical results for Aluminum

Next, we apply the model proposed to the analysis of Al whose permittivity is described by Drude-Lorentz functions. The procedure consists in fitting the experimental data for Al permittivity [10] using the multi-resonant model as given in Eqs. (21) and (22). The expansions for the real and imaginary parts of ε are given by,

ε^{'} (ω) = 1 - \frac{f_{1} ω^{2} ω_{p}^{2}}{ω^{4} + {(γ_{1} ω)}^{2}} + \frac{f_{2} (ω_{02}^{2} - ω^{2}) ω_{p}^{2}}{{(ω_{02}^{2} - ω^{2})}^{2} + {(γ_{2} ω)}^{2}} + \frac{f_{3} (ω_{03}^{2} - ω^{2}) ω_{p}^{2}}{{(ω_{03}^{2} - ω^{2})}^{2} + {(γ_{3} ω)}^{2}}

ε^{″} (ω) = \frac{f_{1} γ_{1} ω ω_{p}^{2}}{ω^{4} + {(γ_{1} ω)}^{2}} + \frac{f_{2} γ_{2} ω ω_{p}^{2}}{{(ω_{02}^{2} - ω^{2})}^{2} + {(γ_{2} ω)}^{2}} + \frac{f_{3} γ_{3} ω ω_{p}^{2}}{{(ω_{03}^{2} - ω^{2})}^{2} + {(γ_{3} ω)}^{2}},

with the adopted fitting parameters f_j, ω_p, ω_0j, γ_j listed in Table 1 . The first contribution in Eq. (28), as well as in Eq. (29), corresponds to the damped Drude model which is the fundamental process contributing to the complex permittivity. Basically, it corresponds to the sea of quasi-free electrons in the metal, and as such their natural resonance frequency ω₀₁ is null. The other contributions correspond to interband transitions that may occur in the medium when the photons corresponding to the electromagnetic wave have enough energy to excite these transitions.

Table 1. Model Parameters for Al Permittivity

View Table

Figure 1 shows the spectral dependence of both real and imaginary parts of Al permittivity from the experimental data of [10] (solid circles) and the fitting curves (solid lines) obtained with Eqs. (28) and (29) using the parameters listed in Table 1 with ω_p = 2.27215 × 10¹⁶ rad/s. For the sake of comparison, the Drude model alone is also shown in this figure. Observe that this model does not accurately follow the data points around material resonances (2.825 × 10¹⁵ rad/s). However, Eqs. (28) and (29) are in good agreement with the experimental data. Once all the essential quantities have been defined, we can proceed to the calculation of Al refractive index using the definition

η = \sqrt{ε} =n+i κ

with ε as the medium permittivity (ε = ε′ + ιε″).

Fig. 1 Frequency dependence of the real (left panel) and imaginary (right panel) parts of Al permittivity. Experimental data from Smith et al. [9] (circles), and their fitting curves (solid lines) using Eq. (28) for left panel and Eq. (29) for right panel. Dashed curves represent the Drude model contribution alone, shown here for the sake of comparison.

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Here n and κ are the real and imaginary parts of the refractive index, both obtained from Eqs. (28) and (29), respectively. Therefore, η is a complex quantity as well, and is calculated taking into account all the physical processes that define the value of Al permittivity. Both real and imaginary parts of the complex refractive index are shown in the left and right panels, respectively, in Fig. 2 .

Fig. 2 Real (left panel) and imaginary (right panel) parts of the refractive index of aluminum calculated with multi-Lorentz approach. Dashed curves show the same quantity when only the Drude contribution is taken into account.

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Observe that the real part n presents a pronounced resonant behavior, but less so for the imaginary part κ. For the sake of comparison, the same figure shows n taking into account only the Drude contribution to point out the influence caused by inter-band transitions upon the refractive index. It is easily seen that the inter-band transitions cause a strong effect on the Al refractive index, changing its value along a broad frequency range.

The imaginary part κ, on the other hand, presents a smaller perturbation caused by inter-band transitions. But for consistency, we use the complete model for both real and imaginary parts of η. Now we calculate the group index using the definition [9]:

N_{g} = Re (η + ω \frac{d η}{d ω}) .

The result for N_g is shown in Fig. 3 as a function of the angular frequency. As can be seen, N_g ranges from negative to positive values depending on the frequency. Moreover, positive values of N_g can be smaller than unity (suggesting superluminal behavior if a group velocity is defined as v_g = c/N_g). Superluminal group velocity raises questions about causality and instabilities, which are not within the scope of this work [11–15]. As well known the group velocity is often physically interpreted as the velocity with which energy propagates in a medium, or as the velocity of a light pulse. These effects are connected with the anomalous dispersion region of the medium. However, even in this region, the energy velocity is always subluminal as will be discussed below.

Fig. 3 Group index for jAl calculated with the multiresonance model. Painted area highlights the spectral range Δω where N_g > 1.0. Dashed curve shows group velocity when only Drude model contribution is considered.

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The energy velocity is defined in terms of the energy flux and energy density of a light field propagating in a medium.

v_{E} = \frac{〈 S 〉}{〈 \frac{1}{2} ε_{o} E \cdot E^{*} + \frac{1}{2} μ_{o} H \cdot H^{*} + u_{s} 〉} .

In Eq. (32), 〈S〉 is the optical-cycle averaged energy flux and the denominator is the cycle-averaged total stored energy. The quantity 〈S〉 is directly obtained from the Poynting vector with

〈 S 〉 = (1 / 2) Re (E \times H^{*})

. In the denominator the average energy density 〈u_s〉 is obtained from Eq. (26).

The spectral dependence of 〈u_s〉, and the dissipated power obtained with Eq. (25), are shown in Fig. 4 . Both quantities clearly show the influence of the interband transition effects, where high levels of dissipation and stored energy in the matter can be observed. The peak values for stored and dissipated energy occur, respectively, at 2.352 × 10¹⁵ rad/s and 2.392 × 10¹⁵ rad/s. Next, we can calculate the energy velocity, using Eq. (32). Once the energy velocity has been calculated, we can now use it to obtain a very useful parameter known as energy index, defined as N_E = c/v_E [4]. Using Eqs. (26) and (32) we end up with the following expression for N_E:

N_{E} = [n + \frac{ω {〈 {ε^{″}}_{j} 〉}_{γ_{j}}}{n Γ}] .

This result is a general expression which also agrees with the one obtained in [7, Eq. (25)].

Fig. 4 Stored energy density (left) and dissipated power (right) for interaction of electromagnetic radiation in Al calculated with multi-Lorentz approach and harmonic field. Quantities are plotted in units of (ε₀∣E₀∣²).

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It should be noticed in Eq. (33) that the energy index (and of course the energy velocity) depends on dissipative processes through 〈ε_j_″〉_{γ j} and Γ. The energy index N_E as a function of frequency is shown in Fig. 5 (solid line). For the sake of comparison, we also show in this figure the same parameter calculated using only the Drude contribution to the material model (dashed line). Algebraic manipulations show that N_E for the limit γ_j→0 for all j, corresponding to the case of lossless medium. Observe the large difference between the energy indices calculated with both material models. In addition, these indices are larger than one within the frequency range, resulting in energy velocities smaller than the velocity of light in free space.

Fig. 5 Energy index (using Eq. (31)) calculated with multi-Lorentz approach and when only Drude contribution is taken into account.

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4. SPP propagation in a metal-dielectric-metal waveguide

In this section we investigate the SPP propagation in the metal-dielectric-metal (MDM) waveguide shown in Fig. 6 using the extended Drude-Lorentz material model.

Fig. 6 Metal-Dielectric-Metal waveguide.

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For convenience, we provide below the analytical solution of the Helmholtz equation for transverse magnetic (TM) polarized waves,

\frac{\partial^{2} H_{y}}{\partial z^{2}} - (ε ε_{0} k_{0}^{2} - β^{2}) H_{y} = 0

complemented with the following set of equations,

E_{x} = i \frac{1}{ω ε ε_{0}} \frac{\partial H_{y}}{\partial z} E_{z} = \frac{β}{ω ε ε_{0}} H_{y} .

The solution of Eq. (32) in all three layers for the (fundamental) symmetric mode is given by,

H_{y} (z) = A {\begin{cases} \cos h (k_{d} w / 2) e^{i β x} e^{- k_{m} (z - w / 2)} (w / 2) < z < \infty \\ \cos h (k_{d} z) e^{i β x} (- w / 2) \leq z \leq (w / 2) \\ \cos h (k_{d} w / 2) e^{i β x} e^{- k_{m} (z + w / 2)} - \infty < z < (w / 2) \end{cases}

where A is an amplitude constant, assumed here as A = 1 without loss of generality. Also,

k_{d, m} = \sqrt{β^{2} - ε_{d, m} k_{0}^{2}} .

After applying the boundary conditions for electric and magnetic fields we end up with the following transcendental equation for the symmetric mode of this waveguide system,

\tan h (\frac{k_{d} w}{2}) = - (\frac{ε_{d} k_{m}}{ε_{m} k_{d}}),

which is used to obtain the complex propagation constant β.

We begin by calculating the frequency dependence of the effective index (n_eff = β/k₀), as shown in Fig. 7 , for waveguide channel w = 50nm. This channel width supports a mode that strongly interacts with the metallic walls, therefore reinforcing the effect of the metal spectral resonance. The real and imaginary parts of the effective index are shown in Fig. 7. With these values obtained, we next calculate the group index and group velocity for this waveguide.

Fig. 7 Real and imaginary parts of the effective index n_eff as a function of angular frequency.

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The group index calculated for the MDM waveguide is shown in Fig. 8 . The effect of Al spectral resonance is observed around 2.295 × 10¹⁵ rad/s as a result of the anomalous dispersion. Figure 9 shows the energy density (left panel) and average dissipated power (right panel) as a function of z (waveguide transverse direction) for ω = 2.285 × 10¹⁵ rad/s (the Al spectral resonance) and ε_d = 2.25, for both multi-Lorentz and Drude approaches for the sake of comparison. Observe that in both cases, the values of stored energy density and dissipated power (solid lines in both figures) significantly differ from their respective values obtained with Drude model (dashed lines) around the metal interface. Also observe in Fig. 9 that no dissipation is found in the dielectric channel because its imaginary part is null (lossless), and all dissipation occurs within metallic region.

Fig. 8 Group index N_g as a function of angular frequency for an Al-based MDM waveguide with w = 50 nm. The Al material model used is the Drude-Multi-Lorentz approach.

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Fig. 9 Average total stored energy density (left panel) and averaged dissipated power (right panel) obtained for the fundamental mode of the MDM waveguide at the Al spectral resonance (ω = 2.285 × 10¹⁵ rad/s) at x₀ = 0.

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Finally, we calculate the energy velocity and energy index for this waveguide. The energy velocity in this case requires a generalized definition (directly obtained from the standard definition in Eq. (31). The definition is,

v_{E} = \frac{\int_{- D / 2}^{D / 2} d y \int_{- \infty}^{\infty} 〈 S_{x} (x_{0}, z) 〉 d z}{\int_{- D / 2}^{D / 2} d y \int_{- \infty}^{\infty} 〈 \frac{1}{2} ε_{o} E \cdot E + \frac{1}{2} μ_{o} H \cdot H + u_{s} (x_{0}, z) 〉 d z} = \frac{P_{d} + P_{m}}{U_{e m} + U_{d} + U_{m}},

where D is an arbitrary length along y with the terms P_d, P_m, U_d, U_m given by,

P_{m} = 2 \int_{w / 2}^{\infty} 〈 S_{m z} 〉 d z = \frac{β}{2 {k^{'}}_{m} ω ε_{d} (ω) ε_{0}} {| A |}^{2}

P_{d} = 2 \int_{0}^{w / 2} 〈 S_{d z} 〉 d z = \frac{β}{4 ω ε_{d} (ω) ε_{0}} [\frac{\sin h ({k^{'}}_{d} w)}{{k^{'}}_{d}} + \frac{\sin ({k^{″}}_{d} w)}{{k^{″}}_{d}}] {| A |}^{2}

U_{e m} = \int_{- D / 2}^{D / 2} d y \int_{- \infty}^{\infty} 〈 \frac{1}{2} ε_{o} E \cdot E + \frac{1}{2} μ_{o} H \cdot H 〉 d z

U_{m} = 2 \int_{w / 2}^{\infty} 〈 u_{m} 〉 d z = (\frac{1}{4 {k^{'}}_{m}}) {μ_{0} + [(ε^{'} - 1) + \sum_{j} \frac{2 ω}{γ_{j}} {ε^{″}}_{j}] \frac{({| β |}^{2} + {| k_{m} |}^{2})}{ω^{2} {| ε_{m} |}^{2} ε_{0}}} {| A |}^{2}

\begin{matrix} U_{d} = 2 \int_{0}^{w / 2} 〈 u_{d} 〉 d z = \frac{w}{8} {[\frac{{(| k_{m} | w)}^{2}}{2 ω^{2} ε_{d} ε_{0}}] + [μ_{o} + \frac{(β^{2} + {| k_{m} |}^{2})}{ω^{2} ε_{d} ε_{0}}] [\frac{\sin h ({k^{'}}_{d} w)}{{k^{'}}_{d} w}] + \\ [μ_{o} + \frac{({| β |}^{2} - {| k_{m} |}^{2})}{ω^{2} ε_{d} ε_{0}}] [\frac{\sin ({k^{″}}_{d} w)}{{k^{″}}_{d} w}]} {| A |}^{2} \end{matrix}

The energy index is then obtained as before. This quantity gives the velocity with which the energy flows along the waveguide. The energy index for the MDM waveguide is shown in Fig. 10 . Observe that the energy index values are above those obtained for the waveguide effective index n_eff (see Fig. 7), resulting in a subluminal energy velocity.

Fig. 10 Energy index for the fundamental mode of a MDM (Al-Glass-Al) waveguide with channel width w = 50 nm.

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Finally, we would like to emphasize that the present approach makes use of an accurate representation of the material permittivity and, as a consequence, the analysis carried out here can be interpreted as a much closer representation of what would be expected from the experimental characterization of these structures. Moreover, this statement can be extended to any metal (with spectral resonances or not), as long as its dielectric parameters are accurately fitted with a Drude-Lorentz approach

5. Conclusions

In this paper we revisited the problem of optical field interaction with dispersive and dissipative media using an improved approach to describe this interaction when spectral resonances are present. This approach is applied to a medium whose optical behavior is described in terms of Drude-Lorentz oscillators, allowing for more accurate calculations of quantities such as group velocity, stored energy density, and energy velocity, regardless the medium adopted. The medium adopted here was aluminum (a medium with several optical resonances), and for which we have presented the details of the multiresonance model. In addition, this metal plays and important role for nanoplasmonic applications, and we presented some useful results for an Al-based MDM slot waveguide. The model suggested here can be successfully applied to any medium that can be described by Drude-Lorentz approach.

Acknowledgments

The authors would like to acknowledge the financial support from FAPESP and CNPq. The authors are grateful to Masud Mansuripur for helpful discussions.

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J	f_j	ω_0j (rad/s)	${\bar{γ}}_{j}$ (rad/s)
1	0.80	0	2.12636 × 10¹⁴
2	0.08	2.37987 × 10¹⁵	5.93456 × 10¹⁴
3	0.12	3.04042 × 10¹⁵	1.85737 × 10¹⁵

Analysis of dispersive and dissipative media with optical resonances

Abstract

1. Introduction

2. Interaction between electromagnetic field and matter

3. Numerical results for Aluminum

4. SPP propagation in a metal-dielectric-metal waveguide

5. Conclusions

Acknowledgments

References and links

Cited By

Figures (10)

Tables (1)

Equations (44)

Optics Express