E-Wave Interaction with the One-Dimensional Photonic Crystal with Weak Conductive and Transparent Materials

Abstract

The interaction of electromagnetic E-waves with one-dimensional photonic crystals consisting of weak conductive and transparent layers is studied theoretically. If the thicknesses of both the conductive and transparent layers are of the order of skin depth, then the domains of transmission and reflection appear. However, if the thickness of the conductive layers is much less than the skin depth but much more than the Debye screening radius, the resonant behavior of the optical power coefficients appears at a frequency close to the plasma frequency.

1. Introduction

Owing to the rapid development of transparent electronic coatings used in displays, there is currently a great demand to create materials with a controlled, thin bandwidth of radiation transmission or reflection. Such promising materials are one-dimensional (1D) photonic crystals [1,2,3,4,5,6,7]. These 1D photonic crystals are periodic structures consisting of identical alternating thin, flat layers of two different transparent substances. The alternation of layers results in the appearance of allowed and forbidden frequency areas of radiation transmission and reflection. One of the most well-known examples of 1D photonic crystal is the cholesteric liquid crystal and its effect of selective reflection [8,9,10,11]. As to the structure investigated, its potential materials and experimental realization have been presented in papers [12,13,14,15,16]. The application of such materials and structures involves electromagnetic beam steering in a range from UV to MIR and THz, optical tweezers, sub-microsecond electro-optical switching in photonic crystal cells and fibers, and other devices.

Typically, the optical thickness of a layer is equal to a quarter of the wavelength of the radiation used, which allows for exploitation of 1D photonic crystal as a narrow-band filter. For the convenience of controlling such a filter, it is necessary to use a weakly conductive matter as the substance in one of the layers. Weak conductors, i.e., semi-metals or semiconductors, are substances with a sufficiently low concentration of conduction electrons, usually in a range of concentration of atoms of 10${}^{-5}$–10${}^{-2}$. The gas of conduction electrons in weak conductors is practically non-degenerate; therefore, the performance of these substances significantly depends on changes in surrounding parameters such as field strength, temperature, pressure, etc. [17].

However, at frequencies of the order of the plasma frequency of conduction electrons of ordinary metal conductors or weak conductors, the longitudinal plasma oscillations between thin layers of these conductors influence the interaction of electromagnetic radiation with these layers [18,19,20,21,22,23]. Such influence is observed in the form of resonant peaks of reflection, transmission and absorption power coefficients. The longitudinal plasma oscillations appear only in the case of electromagnetic E-waves when the E-vector of the wave lies in the incidence plane. Investigations of longitudinal plasma oscillations were performed well in the case of a single conductive layer; however, the influence of these plasma oscillations has not been taken into account in cases of weak conductive multilayer structures. Here, in a physical model of 1D photonic crystal with conductive layers, one should use such scale parameters as the skin depth as well as the Debye screening radius.

Hence, the purpose of this article is to numerically investigate the longitudinal plasma oscillations of conduction electrons in the conductive layers of 1D photonic crystal described and their influence on its interaction with electromagnetic E-waves. We also find the conditions of the longitudinal plasma oscillations’ contribution to E-wave interaction with such photonic crystals.

2. Materials and Methods

Let us consider 1D photonic crystal made of N similar weak conductor flat layers (M) of the thickness $d_1$ separated between by $N-1$ similar transparent dielectric flat layers of the thickness $d_2$ having the dielectric permittivity of ${\epsilon }_2$. We suppose that the photonic crystal is localized between two transparent media having the dielectric permittivities of ${\epsilon }_1$ and ${\epsilon }_3$ (see Figure 1). We also assume that the weak conductive and transparent substances are the uniform and isotropic media, and that the dielectric permittivities are the positive constant values. As we will see in the text, the photonic crystal has a forbidden band gap that is determined by its geometry and material properties. As to the structure investigated, its potential materials and experimental realization similar to the photonic crystal systems are presented in [2,3,5,8,9].

Let the plane monochromatic E-wave (the E-vector which is parallel to the plane of incidence $OXZ$), having the cyclic frequency $\omega$, falls on 1D photonic crystal from the medium with the ${\epsilon }_1$ under the incidence angle $\theta$ (Figure 1). Then, the power coefficients reflectance R, transmittance $T_r$ and absorptance A are evaluated by the equations [7,24]:

$$R={\left|\frac{m_{21}}{m_{22}}\right|}^2,T_r={\left|m_{11}-\frac{m_{12}{m}_{21}}{m_{22}}\right|}^2\mathrm{Re}\left(\sqrt{\frac{{\epsilon }_3}{{\epsilon }_1}}\frac{\cos {\theta }^{\prime }}{\cos \theta }\right),A=1-R-T_r.$$

(1)

Here, $m_{jk}$ ($j,k=1,2$) are the elements of the transfer matrix for the radiation transmitting through 1D photonic crystal. The matrix can be shown as

$$\left(\begin{array}{cc}{m}_{11}& m_{12}\\ m_{21}& m_{22}\end{array}\right)={\mathbf{M}}_{3out}^{-1}{\left({\mathbf{M}}_{2in}{\mathbf{M}}_{2pr}{\mathbf{M}}_{2out}^{-1}\right)}^{N-1}{\mathbf{M}}_{1in}.$$

(2)

In the equation, ${\mathbf{M}}_{\alpha in}$ and ${\mathbf{M}}_{\alpha out}$ are, respectively, the matrix of the E-wave input into the conductive layer from the transparent medium with the permittivity of ${\epsilon }_{\alpha }$, and the matrix of the E-wave output from the conductive layer into the transparent medium with ${\epsilon }_{\alpha }$ ($\alpha$ = 1, 2, 3). However, ${\mathbf{M}}_{2pr}$ is the matrix of E-wave propagation in the transparent layer with the permittivity of ${\epsilon }_2$. The matrices have the form of

$$\begin{array}{ccc}& & {\mathbf{M}}_{\alpha in}=\left(\begin{array}{cc}\cos {\theta }_{\alpha }-Z_E^{\left(1\right)}\sqrt{{\epsilon }_{\alpha }}& -\cos {\theta }_{\alpha }-Z_E^{\left(1\right)}\sqrt{{\epsilon }_{\alpha }}\\ -\cos {\theta }_{\alpha }+Z_E^{\left(2\right)}\sqrt{{\epsilon }_{\alpha }}& \cos {\theta }_{\alpha }+Z_E^{\left(2\right)}\sqrt{{\epsilon }_{\alpha }}\end{array}\right),\hfill \end{array}$$

(3)

$$\begin{array}{ccc}& & {\mathbf{M}}_{\alpha out}=\left(\begin{array}{cc}\cos {\theta }_{\alpha }+Z_E^{\left(1\right)}\sqrt{{\epsilon }_{\alpha }}& -\cos {\theta }_{\alpha }+Z_E^{\left(1\right)}\sqrt{{\epsilon }_{\alpha }}\\ \cos {\theta }_{\alpha }+Z_E^{\left(2\right)}\sqrt{{\epsilon }_{\alpha }}& -\cos {\theta }_{\alpha }+Z_E^{\left(2\right)}\sqrt{{\epsilon }_{\alpha }}\end{array}\right),\hfill \end{array}$$

(4)

$$\begin{array}{ccc}& & {\mathbf{M}}_{2pr}=\left(\begin{array}{cc}\exp \left(\mathrm{i}{k}_{2z}{d}_2\right)& 0\\ 0& \exp (-\mathrm{i}{k}_{2z}{d}_2)\end{array}\right).\hfill \end{array}$$

(5)

Here, ${\theta }_1=\theta$, ${\theta }_3={\theta }^{\prime }$ is the refraction angle into the medium with ${\epsilon }_3$, and ${\theta }_2$ is the refraction angle into the transparent layers with ${\epsilon }_2$. These angles obey the refraction law:

$$\sqrt{{\epsilon }_1}\sin \theta =\sqrt{{\epsilon }_2}\sin {\theta }_2=\sqrt{{\epsilon }_3}\sin {\theta }^{\prime }.$$

(6)

Further, in Equations (3) and (4), $Z_E^{\left(j\right)}$ is the dimensionless surface impedance of the E-wave on the border of the conductive layer:

$$Z_E^{\left(j\right)}={\left.\frac{1}{Z_0}\frac{E_x^{\left(j\right)}}{H_y^{\left(j\right)}}\right|}_{z=0}(j=1,2),$$

where $Z_0$ is the vacuum wave impedance (in Ohm) and j denotes the mode number of the E-wave inside the conductive layer. When one assumes the specular reflection of conduction electrons from the borders of the layer, the impedance can be shown as [18,25]

$$Z_E^{\left(j\right)}=\frac{2\mathrm{i}c\omega }{d_1}\sum _n\frac{1}{k_n^2}\left(\frac{k_x^2}{{\omega }^2{\epsilon }_l(\omega ,k_n)}+\frac{{(\pi n/d_1)}^2}{{\omega }^2{\epsilon }_{tr}(\omega ,k_n)-{\left(c{k}_n\right)}^2}\right).$$

(7)

In the equation, c is the speed of light, and the summation by n at $j=1$ is performed over all odd $n=\pm 1,\pm 3,\pm 5,\dots$ but at $j=2$ is performed over all even $n=0,\pm 2,\pm 4,\dots$. In Equations (5) and (7), the quantities are shown as

$$k_n=\sqrt{{\left(\frac{\pi n}{d_1}\right)}^2+k_x^2},k_x=\frac{\omega }{c}\sqrt{{\epsilon }_1}\sin \theta ,k_{2z}=\frac{\omega }{c}\sqrt{{\epsilon }_2}\cos {\theta }_2.$$

(8)

Finally, in (7), ${\epsilon }_l(\omega ,k)$ and ${\epsilon }_{tr}(\omega ,k)$ are, respectively, the longitudinal and the transverse dielectric permittivities of the conduction electron plasma in the weak conductive substance. These permittivities were evaluated using the quantum kinetic Mermin approach, where both collective movement and quantum wave properties of the plasma electrons are taken in account, and looks as follows [26,27] (see also [28]):

$$\begin{array}{ccc}& & {\epsilon }_l^{\left(qu\right)}(\omega ,k)=1-\frac{2}{Q^2}\frac{(\Omega +\mathrm{i}\gamma )G(\Omega +\mathrm{i}\gamma ,Q)G(0,Q)}{\Omega G(0,Q)+\mathrm{i}\gamma G(\Omega +\mathrm{i}\gamma ,Q)},\hfill \end{array}$$

(9)

$$\begin{array}{ccc}& & {\epsilon }_{tr}^{\left(qu\right)}(\omega ,k)=1-\frac{1}{{\Omega }^2}\left(1+\frac{\Omega G(\Omega +\mathrm{i}\gamma ,Q)+\mathrm{i}\gamma G(0,Q)}{\Omega +\mathrm{i}\gamma }\right).\hfill \end{array}$$

(10)

In these equations, the function can be shown as

$$G(\Omega +\mathrm{i}\gamma ,Q)=\frac{Q^2}{\sqrt{\pi }}\underset{0}{\overset{+\infty }{\int }}\frac{[({\Omega }_{+}+\mathrm{i}\gamma )({\Omega }_{-}+\mathrm{i}\gamma )+{\left(Q\xi \right)}^2]\exp (-{\xi }^2)}{[{({\Omega }_{+}+\mathrm{i}\gamma )}^2-{\left(Q\xi \right)}^2][{({\Omega }_{-}+\mathrm{i}\gamma )}^2-{\left(Q\xi \right)}^2]}\mathrm{d}\xi ,$$

(11)

where the dimensionless quantities are

$$\Omega =\frac{\omega }{{\omega }_p},\gamma =\frac{\nu }{{\omega }_p},Q=\frac{v_{T}k}{{\omega }_p},{\Omega }_{\pm }=\Omega \pm \frac{\hslash {\omega }_p}{2m_e{v}_T^2}{Q}^2.$$

(12)

Here, ${\omega }_p$ is the plasma frequency, $\nu$ is the frequency of the conduction electron collisions in the plasma, $m_e$ is the effective mass of the conduction electrons, ℏ is the Planck constant, and $v_T$ is the heat velocity of the conduction electrons of the non-degenerate electron plasma:

$$v_T=\sqrt{\frac{2k_{B}T}{m_e}},$$

(13)

where T is the absolute temperature of the plasma electrons and $k_B$ is the Boltzmann constant.

The dielectric permittivities (9) and (10) in the long-wave limit $k\to 0$ go over to the permittivity of the classical electron gas in the Drude–Lorentz approach [18,19,29]:

$${\epsilon }_l^{\left(DL\right)}\left(\omega \right)={\epsilon }_{tr}^{\left(DL\right)}\left(\omega \right)=1-\frac{1}{\Omega (\Omega +\mathrm{i}\gamma )}.$$

(14)

In this approach, one neglects both the kinetic and quantum properties of the conduction electrons.

Further, to evaluate the plasma frequency ${\omega }_p$ and the frequency of conduction electron collisions $\nu$, we use the equations [29]:

$${\omega }_p=\sqrt{\frac{e^2{n}_0}{{\epsilon }_0{m}_e}},\nu =\frac{e^2{n}_0{\rho }_0}{m_e}.$$

(15)

Here, e is the elementary charge, $n_0$ is the concentration of the conduction electrons, ${\rho }_0$ is the static specific resistance of a conductor, and ${\epsilon }_0$ is the SI electric constant.

3. Results

For the numerical study of E-wave interactions with 1D photonic crystals consisting of weak conductor and transparent dielectric slabs, we used graphite as a weak conductor. The graphite material was chosen due to the fact that it is a semi-metal, and its electrophysical properties are well-presented in the literature [30,31]. Other examples of weak conductivity materials (semiconductors, ZrO${}_2$, InSb, and polymers) are presented in [2,3,5]. These papers are devoted to the experimental parameters of the systems under actions of different natural physical impacts (optical, mechanical, electrical, etc.) without their detailed theoretical descriptions.

Owing to the anisotropic properties of graphite, we used the following average values [31]:

$$\begin{array}{ccc}& & {\rho }_0=4·{10}^{-7}\mathrm{Ohm}·\mathrm{m},\hfill \\ & & n_0=2·{10}^{25}{\mathrm{m}}^{-3},\hfill \\ & & m_e=9·{10}^{-31}\mathrm{kg}.\hfill \end{array}$$

Using these data and Equation (15), we evaluated the plasma frequency and the frequency of electron collisions for graphite:

$${\omega }_p=2.54·{10}^{14}{\mathrm{s}}^{-1},\nu =2.273·{10}^{11}\mathrm{Hz}.$$

As the transparent medium with ${\epsilon }_1$, we used air or vacuum when ${\epsilon }_1=1$. The transparent substance for both layers with ${\epsilon }_2$ and the medium with ${\epsilon }_3$ selected by us was quartz with ${\epsilon }_2={\epsilon }_3=2$.

Numerical calculations of the power coefficients R, $T_r$ and A performed by Equations (1)–(14) show that for the number of graphite layers $N>4$ at the thickness $d_1\sim d_2\sim$ 1–3 $\mathsf{\mu }$m in the frequency range $\omega \sim$ (0.25–2.5)${\omega }_p$, one observes the reflection and transmission frequency domains for the considered 1D photonic crystal (Figure 2). Here, the results obtained in the framework of the quantum electron plasma with permittivities (9) and (10) coincide with the data evaluated in the case of the classical electron gas having the permittivity (14). Similar results were observed in experimental studies of 1D photonic crystals [2,3,5,12,13,14,15,16] as well as in the case of the cholesteric liquid crystals as 1D photonic crystals [8,9,10,11].

Let us consider a case where the graphite layer thickness is $d_1\sim$ 25–150 nm and the thickness of the quartz layers $d_2$ is unchanged. For the frequencies $\omega >{\omega }_p$ close ${\omega }_p$, one observes the difference of the power coefficients R, $T_r$ and A evaluated in the case of the quantum electron plasma permittivities (9) and (10) in comparison with these coefficients in the framework of the classical electron gas permittivity (14) (Figure 3). Such discrepancy is displayed by the resonant behavior of the power coefficients in the case of the quantum electron plasma. Here, a decrease in the graphite layer thickness $d_1$ results in an increase in the distance between the resonant frequency peaks (Figure 4).

Further, an increase in graphite layer number N results in a sharper behavior of the power coefficients close the resonant frequencies (Figure 5). Finally, one observes temperature dependence of the quantum electron plasma power coefficients in the vicinity of the resonant frequencies (Figure 6). These resonant frequencies grow with an increase in temperature.

4. Discussion

We have selected the frequencies $\omega \sim {\omega }_p$, since one observes at these frequencies the peculiarities of E-wave interaction with the electron plasma [18,21,22,23]. Note that the considered radiation frequencies $\omega \sim$ (0.25–2.5) ${\omega }_p$ lie in the infrared light domain in the case of graphite. In the case of a weak conductor and layer thickness $d_1\sim d_2\sim c/{\omega }_p$, quartz permittivity ${\epsilon }_{\alpha }\sim 1$, and graphite permittivity (14), the frequency domains of the power coefficients appear, caused by the interference of the waves. The waves are either reflected many times by the layers of 1D photonic crystal or penetrate through them [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16]. For graphite, the value $c/{\omega }_p=1.18\mathsf{\mu }$m and, hence, the frequency domains of the studied 1D photonic crystal appear at the layer thicknesses $d_1\sim d_2\sim$ 1–3 $\mathsf{\mu }$m (see Figure 2).

Note that the quantity $c/{\omega }_p$ is the skin depth of the conductor which characterizes the penetration depth of an alternating electromagnetic field into a plasma [19]. For the conductive layer thickness $d_1$, a leading (at small n) contribution of the value $k_n$ by (8) into the permittivities (9) and (10) of the quantum electron plasma, taking into account (11) and (12), has the order ${\displaystyle \frac{\pi n{v}_T}{c}\ll 1}$. Therefore, the influence of the quantum permittivities (9) and (10) on the surface impedance (7) is almost the same as that for the classical permittivity (14) [19]. That is why longitudinal plasma oscillations have no visible contribution to E-wave interactions with photonic crystals, and one observes a coincidence of the results for quantum electron plasma and for classical electron gas (see [7,23]).

However, when the thickness $d_1$ of the weak conductor layers satisfies the condition $v_T/{\omega }_p\ll d_1\ll c/{\omega }_p$, the longitudinal plasma oscillations in the conductive layers between their borders contribute to the power coefficients (1) in the form of resonant peaks (see Figure 3, Figure 4, Figure 5 and Figure 6) [18,21,22,23]. The resonant frequencies of the peaks ${\omega }_{res}$ are localized near the plasma frequency ${\omega }_p$, exceed it, and are determined by the equation [18,23]

$$\mathrm{Re}{\epsilon }_l({\omega }_{res},k_n)=0.$$

(16)

In the case of classical electron gas, the resonant peaks are absent, as can be seen from Equations (14) and (16).

It follows from (16) and Equation (9) for the longitudinal permittivity of quantum electron plasma with use of the (11) and (12), that a distance between neighbouring frequencies of the resonant peaks has the order [18,21,22]

$$\Delta {\omega }_{res}\sim \frac{\pi v_T}{d_1}.$$

(17)

Hence, one sees an increase in the distance between the resonant frequencies at the point of decrease in weak conductor layer thickness $d_1$ (Figure 4). From (17) and condition $d_1\gg v_T/{\omega }_p$, one gets $\Delta {\omega }_{res}\ll {\omega }_p$ (see Figure 3, Figure 4, Figure 5 and Figure 6). Using (13), one gets for graphite, at temperature $T=294$ K, the value $v_T/{\omega }_p=0.374$ nm. Therefore, the thickness range $d_1\sim$ 50–150 nm satisfies the $v_T/{\omega }_p\ll d_1\ll c/{\omega }_p$ condition. It is also worth noting that the quantity $v_T/{\omega }_p$ is just the Debye screening radius [29]; hence, one can study the conduction electrons inside the weak conductive layers having the thickness $d_1\gg v_T/{\omega }_p$ using the plasma technique.

Further, with the use of (13) and (17), one can see a qualitative increase in resonant frequencies with an increase in temperature T (Figure 6). Finally, the resonant peaks of the power coefficients (1) at the growth of the conductive layer number N become more contrasted (Figure 5) owing to multiple wave reflections from and penetrations through 1D photonic crystal [1,2,3,4,5,6,7,8,9].

5. Conclusions

We have numerically studied an electromagnetic E-wave interaction with a one-dimensional photonic crystal consisting of weak conductive and transparent matter. The weak conductor we selected was a graphite substance, and the investigated transparent matter was quartz. The studied frequencies of the E-waves were of the order of the graphite plasma frequency and lay in the infrared domain.

Our studies have shown that if the thickness of both the conductive and transparent layers is of the order of skin depth, then frequency domains of the reflected and transmitted radiation appear. Such domains are caused by interference with the radiation at its multiple reflection points from the layers and from its penetration into other layers. Here, one notices an absence of the influence of longitudinal plasma oscillations on the E-wave’s interaction with the photonic crystal.

However, in the case of a conductive layer thickness much less than skin depth but much larger than the Debye screening radius, there is a resonant behavior of the power coefficients in the form of resonant peaks at frequencies slightly exceeding the plasma frequency. This behavior is caused by the influence of longitudinal plasma oscillations in the conductive layers between their borders. It becomes more contrasted with an increase in the number of conductive layers.

Here, the distance between the resonant peaks increases with a decrease in conductive layer thickness. In addition, there is a dependence of the resonant frequencies on temperature when these frequencies increase with increasing temperature.

The results obtained for the one-dimensional photonic crystals can be experimentally tested in samples with perfect geometry and structure. Such analogs are cholesteric liquid crystals. They have selective reflection spectra with well-defined, wide band gaps and narrow satellite peaks as in Figure 2. For other systems, the satellite peaks are usually smoothed, owing to the imperfectness of the sample geometry and structure or non-monochromaticity of the probe beam.

Potential application of such materials and structures includes electromagnetic beam steering in the range from UV to MIR and THz, optical tweezers, sub-microsecond electro-optical switching in photonic crystal cells and fibers, and other devices.