Significant amplification of electromagnetic waves in disordered dispersive temporal multilayered structures with randomness in plasma frequency or dielectric permittivity

Abstract

In this paper, two disordered temporal multilayered structures are proposed based on dispersive Lorentzian materials for amplification of electromagnetic waves in the frequency range of [25 GHz, 38 GHz] without using gain media. The first structure is composed of nondispersive dielectric and dispersive Lorentzian slabs alternatively. The electric permittivity of dispersive slabs is the same while the electric permittivity of nondispersive slabs changes randomly after the same time intervals. The averages of transmission and reflection coefficients of this structure are calculated versus the frequency of incident wave by employing the 4 × 4 temporal transfer matrix. This structure demonstrates the significant amplification of electromagnetic waves incident onto the structure for both forward and backward waves in the final medium at two different generated frequencies. It is shown that the increase of disorder level in the designed temporal structure leads to the enhancement of wave amplification. To indicate the effect of dispersion on the amount of amplification, the transmission and reflection coefficients are calculated for a disordered multilayered temporal structure composed of only nondispersive slabs. Our results confirm that the wave amplification in this nondispersive structure is so weaker compared to the structure containing dispersive materials. The effects of average of electric permittivity of nondispersive slabs are also investigated on the amount of wave amplification in the first proposed temporal disordered multilayered structure. The second disordered temporal structure proposed in this paper consists of spatially homogeneous dispersive Lorentzian materials whose plasma frequency changes abruptly and randomly after equal time intervals. This structure is also able to amplify the electromagnetic propagating through it efficiently. With increasing the disorder level and number of slabs existing in this structure, stronger amplification of incident wave can be observed. However, the temporal width of slabs should be selected suitably in order to achieve significant amplification.

Introduction

The interesting idea of changing the electromagnetic properties of materials with time and investigating the interaction of electromagnetic waves with these time varying media dates back to 1958. Frederic. R. Morgenthaler in this year tried to answer to the question, what will happen to electromagnetic waves when they pass through a dielectric medium whose propagation constants change with time? By using Maxwell’s equations, he indicated that when the wave passes through a temporal boundary, the momentum of wave is conserved, while its frequency changes¹. After reporting these results, which had never been presented before, a lot of attention was attracted to this issue. Therefore, with the progress of different aspects of science, extensive research has been carried out continuously on time varying media. Many of these researches are devoted to changing the frequency of waves, which is one of the unique features of these media. For example, the sudden creation of a plasma slab in free space and the simultaneous existence of spatial and temporal boundaries showed that the frequency shift of electromagnetic radiation occurred only in passing through the temporal boundary². In another work, the frequency shift of surface plasmons was observed during the propagation in a graphene sheet in which the carrier density varies with time³. Miyamaru et al. experimentally demonstrated the frequency-shift dynamics for a THz wave propagating through a metal-semiconductor waveguide by controlling the structural dispersion⁴. So far, some optical effects were observed upon passing the EM waves through temporal boundaries such as temporal aiming⁵, temporal Brewster angle⁶, inverse prism⁷, antireflection temporal coating⁸, polarization conversion⁹, nonreciprocity and Faraday rotation¹⁰, and Goos-Hunchen effect¹¹. In addition, the possibility of amplification and attenuation of waves passing through time varying media has been investigated in some works^12,13.

The phenomena observed in time varying media and temporal boundaries are very diverse, however, they can be placed in two general categories: 1- phenomena and effects that are specific to these media and 2- phenomena that they were inspired by their spatial counterparts. In the second case, we can mention the temporal multilayered structure, which is the subject of our discussion in this paper. In spatial multilayered structures, the refractive index changes with space. Then, inspired by the same definition, but with the difference that the refractive index changes with time, researchers introduced temporal multilayered structures. In order to know these structures and their applications, various researches have been carried out. Changing the refractive index periodically with time can lead to the creation of photonic time crystal (PTC)¹⁴. In 2009, a group examined the propagation of EM waves through a PTC created by changing the dielectric function of the medium periodically with time in a sinusoidal profile¹⁵. It was shown in this work that the dispersion relationship of a PTC has a form similar to the band structure in photonic space crystals (PSCs), but with some interesting differences. For example, in PTCs, unlike PSCs, the dispersion relation is periodic in the frequency. Therefore, there are momentum bands and momentum gaps in their band structure instead of frequency bands and frequency gaps in PSCs. In another study conducted in 2021, the researchers investigated in detail a PTC obtained by changing the permittivity and permeability with time periodically in a square profile¹⁶. In this paper, the authors achieved the same results as those in ref¹⁵ by obtaining the dispersion relation and plotting the band structure. But there are some differences between the band structure in ref^15,16 due to the different profiles of temporal variation employed in these two references. Another important difference between PTCs and PSCs is that it is possible to amplify the EM waves in the gap of PTCs unlike PSCs. For example, in a research, it has been shown that by controlling the momentum of the incident pulse, the intensity of the pulse passing through the momentum gaps of a PTC can be enhanced¹⁷. It was also shown in ref¹⁸ that when the wave propagated in the momentum gap, it was amplified. However, when it propagated in the momentum band, no amplification occurred.

In 2021, Ramaccia et al. introduced a transfer matrix method to obtain the scattering coefficients for an electromagnetic wave propagating through a temporal multilayered structure consisting of isotropic and nondispersive materials¹⁹. They studied four different classes of these multilayered systems with four layers between two semi-infinite media. For the case of equally travel-distance multilayered device with ${L_s}=\lambda /2$ in which the refractive indices of four layers were arbitrary, there was a wide frequency range around the central frequency with optical transparency. However, for the case of equally travel-distance multilayered device with ${L_s}=\lambda /2$ composed of alternating high/low refractive index materials, the forward and backward coefficients became unity and zero at the central frequency. Conversely, for the case of equally travel-distance multilayered device with ${L_s}=\lambda /4$ made of alternating high/low refractive index layers, strong amplification of both forward and backward waves was observed. In another work, Koutserimpas has employed the coupled wave theory to demonstrate the parametric amplification in the momentum gap for a wave propagating through time periodic optical media²⁰. Comparison of the results of coupled wave theory with finite difference time domain (FDTD) method in this paper confirms the ability of coupled wave theory in the modeling of first and second momentum gaps. Furthermore, Gaxiola-Luna et al. have obtained the band structure of a photonic time crystal with square modulation of permittivity¹³. They found that the band structure is periodic in wave number. For certain values of modulation strength and period, frequencies of the modes within the wave number band gap have the same imaginary parts. When these modes are excited simultaneously, the strong amplification of electromagnetic waves is achieved.

Recently, Sharabi et al. have examined the dynamic of a Gaussian light pulse propagating through a disordered PTC using FDTD method²¹. They indicated that the group velocity of pulse exponentially decreased when the temporal disorder started and eventually the wave propagation completely stopped. However, the amplitude and energy of pulse showed an exponential growth in contrast to Anderson localization in spatially disordered one-dimensional structures. In addition, the group velocity of modes on band edges exhibited the strongest declaration and their amplitude displayed the greatest enhancement. In this structure, it was assumed that two alternating segments were nondispersive dielectric materials. The effects of temporal dispersion on the wave propagation in time varying media have been explored in different papers^22,23. Feng et al. proposed an antireflection temporal coating (ATC) by using a nonmagnetic dielectric material with the Lorentzian dispersion temporally sandwiched between two nondispersive dielectric media²⁴. They found that the time duration of this dispersive ATC is shorter than that of a nondispersive ATC. Furthermore, depending on the value of plasma frequency of the dispersive material, there is a certain value for the time duration at which the reflection is eliminated which means that the tunability of time duration with the plasma frequency is possible. Unlike to nondispersive ATCs, the transmission in the dispersive ATCs can be made tunable with changing the plasma frequency and in some cases, it can be enhanced. In this paper, we aim to examine a new type of temporal multilayered structure containing Lorentzian dispersive materials. The first structure considered in this paper is composed of alternative Lorentzian dispersive and nondispersive temporal slabs. To introduce randomness in this structure, it is assumed that the electric permittivity of the nondispersive dielectric material changes randomly after special time intervals. Using the transfer matrix method, the transmission and reflection coefficients for this structure are calculated versus the frequency of incident wave. Our results show that strong amplification of electromagnetic waves can be achieved in the disordered structure compared to the periodic one. The effect of disorder level and dispersion on the amplification are also investigated. Then, we propose another temporal multilayered structure consisting of only Lorentzian dispersive materials in which the plasma frequency of slabs experience an abrupt random change. This structure is also able to amplify the incident waves strongly. Finally, the effect of number of slabs existing in the structure is investigated on the improvement of amplification.

The remaining parts of this paper are organized as follows. In Section “Theoretical method and the designed structure”, the transfer matrix method used in this paper for calculation of transmission and reflection coefficients of a plane wave incident onto a temporal multilayered structure composed of Lorentzian dispersive and nondispersive materials is described. In Section “Results and discussion”, the proposed structures are presented and the corresponding results and discussion are also given. Finally, the paper is finished with some conclusion in Section “Conclusion”.

Theoretical method and the designed structure

In this paper, we first consider a temporal multilayered structure consisting of nondispersive dielectric and lossless dispersive slabs alternatively. In addition, all slabs are taken to be nonmagnetic. The proposed structure is schematically shown in Fig. 1.

Fig. 1

A schematic representation of the temporal multilayered structure consisting of Lorentzian dispersive and nondispersive temporal slabs alternatively between two temporally semi-infinite media with permittivity of $\:{\varepsilon\:}_{a}$ and $\:{\varepsilon\:}_{L,final}$.

It is assumed that both dispersive and nondispersive slabs have the same temporal width $\:\varDelta\:t$. To introduce the disorder in the structure, the relative electric permittivities of nondispersive slabs $\varepsilon _{d}$ are taken to be random numbers obtained from the relation $\varepsilon _{d} = \varepsilon _{{av}} \left( {1 + q} \right)$ where q is a random number distributed uniformly in the range [-Q, Q] and $\varepsilon _{{av}}$ is the average of $\varepsilon _{d}.$ Q is named the disorder level or disorder strength. To generate random permittivities, we write a MATLAB code using the rand command. In the structure, we assume that all dispersive slabs have a Lorentzian dispersion described by the following relation:

$$\:{\varepsilon\:}_{L,m}\left(\omega\:\right)={\varepsilon\:}_{\infty\:,m}+\frac{{\omega\:}_{p,m}^{2}}{{\omega\:}_{0,m}^{2}-{\omega\:}^{2}}\:$$

(1)

where $\:{\omega\:}_{p,m}$ and $\:{\omega\:}_{0,m}$ are the plasma and resonance frequency of the mth dispersive layer, respectively. $\:{\varepsilon\:}_{\infty\:,m}$ in Eq. (1) denotes the relative permittivity at frequency $\:\omega\:$ tending to infinity. It should be noted that the relative permittivity of materials such as fiber-filled composites and FR-4 (fiberglass—epoxy) dielectric^25,26,27 is described by Eq. (1). To calculate the transmission and reflection coefficients of a plane wave incident onto the temporal structure displayed in Fig. 1, we use the temporal transfer matrix method. Therefore, we first obtain the temporal transfer matrix for relating the field amplitudes of forward and backward waves in the mth dispersive medium to ones in the (m-1)th dispersive medium after an abrupt change in the permittivity of the (m-1)th medium. The first point to be considered is the change in the frequency of waves crossing the temporal boundary. Due to the conservation of the momentum at the temporal boundaries, it is possible to obtain the frequency changes of the EM waves in passing the temporal boundaries between two dispersive media. For example, for the temporal boundary at $\:t={t}_{m}$, the wave number k, the angular frequency $\:\omega\:$ and the refractive index n obey the following relation:

$$\:{k}_{m-1}(t<{t}_{m})={k}_{m}(t>{t}_{m})$$

(2)

$$\:{n}_{m-1}\left({\omega\:}_{i}\right){\omega\:}_{i}={n}_{m}\left({\omega\:}_{f}\right){\omega\:}_{f}$$

It is well known that when an optical medium with a plane wave propagating through it, is abruptly converted to a Lorentzian dispersive medium at the time $\:t={t}_{1}$, four new modes with frequencies $\:{\pm\:\omega\:}_{1\:}\:and\:{\pm\:\omega\:}_{2\:}$ are created in the new medium for $\:t>{t}_{1}$²². The positive sign means propagation in the direction of the incident wave while the negative sign means propagation in the opposite direction of the incident wave. By imposing the temporal continuity of electric and magnetic fields, electric displacement and the first-time derivative of polarization at the time interface $\:t={t}_{m}$, one can derive the temporal matching matrix $\:{MM}_{m}$ as follows:

$$\:{MM}_{m}=\frac{1}{2({\varepsilon\:}_{1,m}-{\varepsilon\:}_{2,m})}\left[\begin{array}{cc}\begin{array}{cc}{{A}_{m}}^{+}&\:{{A}_{m}}^{-}\\\:{{A}_{m}}^{-}&\:{{A}_{m}}^{+}\end{array}&\:\begin{array}{cc}{{B}_{m}}^{+}&\:{{B}_{m}}^{-}\\\:{{B}_{m}}^{-}&\:{{B}_{m}}^{+}\end{array}\\\:\begin{array}{cc}{{C}_{m}}^{+}&\:{{C}_{m}}^{-}\\\:{{C}_{m}}^{-}&\:{{C}_{m}}^{+}\end{array}&\:\begin{array}{cc}{{D}_{m}}^{+}&\:{{D}_{m}}^{-}\\\:{{D}_{m}}^{-}&\:{{D}_{m}}^{+}\end{array}\end{array}\right]\:$$

(3)

where the matrix elements of $\:{MM}_{m}$ are defined as follows:

$$\:{A}_{m}^{\pm\:}=\left({\varepsilon\:}_{1,\left(m-1\right)}-{\varepsilon\:}_{2,m}\right)\left(1\pm\:\sqrt{\frac{{\varepsilon\:}_{1,m}}{{\varepsilon\:}_{1,\left(m-1\right)}}}\right),\:\:\:\:\:\:\:{B}_{m}^{\pm\:}=\left({\varepsilon\:}_{2,\left(m-1\right)}-{\varepsilon\:}_{2,m}\right)\left(1\pm\:\sqrt{\frac{{\varepsilon\:}_{1,m}}{{\varepsilon\:}_{2,\left(m-1\right)}}}\right)\:$$

(4a)

$$\:{C}_{m}^{\pm\:}=\left({\varepsilon\:}_{1,m}-{\varepsilon\:}_{1,\left(m-1\right)}\right)\left(1\pm\:\sqrt{\frac{{\varepsilon\:}_{2,m}}{{\varepsilon\:}_{1,\left(m-1\right)}}}\right),\:\:\:\:\:\:{D}_{m}^{\pm\:}=\left({\varepsilon\:}_{1,m}-{\varepsilon\:}_{2,\left(m-1\right)}\right)\left(1\pm\:\sqrt{\frac{{\varepsilon\:}_{2,m}}{{\varepsilon\:}_{2,\left(m-1\right)}}}\right)\:$$

(4b)

where$\:\:{\varepsilon\:}_{i,j}$ is the electric permittivity of the jth dispersive Lorentzian slab in the structure at frequency of $\:{\omega\:}_{i}$ which is calculated from Eq. (1). Therefore, the field amplitudes at before and after the temporal boundary at $\:t={t}_{m}$ can be related as follows:

$$\left( {\begin{array}{*{20}c} {f_{{1,m}} } \\ {b_{{1,m}} } \\ {f_{{2,m}} } \\ {b_{{2,m}} } \\ \end{array} } \right) = MM_{m} \left( {\begin{array}{*{20}c} {f_{{1,m - 1}} } \\ {b_{{1,m - 1}} } \\ {f_{{2,m - 1}} } \\ {b_{{2,m - 1}} } \\ \end{array} } \right)$$

(5)

Now, we consider a plane wave propagating in an initial dielectric nondispersive medium at $\:t<{t}_{1}\:$with electric permivity of $\:{\varepsilon\:}_{a}$. If this medium changes abruptly to a dispersive Lorentzian medium at $\:t={t}_{1}$, the matching transfer matrix connecting the field amplitudes around both sides of the temporal boundary can be written as:

$$\:{MM}_{1}=\frac{1}{2({\varepsilon\:}_{\text{1,1}}-{\varepsilon\:}_{\text{2,1}})}\left[\begin{array}{cc}\begin{array}{cc}{a}^{+}&\:0\\\:{a}^{-}&\:0\end{array}&\:\begin{array}{cc}0&\:0\\\:0&\:0\end{array}\\\:\begin{array}{cc}{c}^{+}&\:0\\\:{c}^{-}&\:0\end{array}&\:\begin{array}{cc}0&\:0\\\:0&\:0\end{array}\end{array}\right]\:$$

(6)

where the matrix elements in $\:{MM}_{1}$ are given as follows:

$$\:{a}^{\pm\:}=\left({\varepsilon\:}_{a}-{\varepsilon\:}_{\text{2,1}}\right)\left(1\pm\:\sqrt{\frac{{\varepsilon\:}_{\text{1,1}}}{{\varepsilon\:}_{a}}}\right)\:\:$$

(7a)

$$\:{c}^{\pm\:}=\left({\varepsilon\:}_{\text{1,1}}-{\varepsilon\:}_{a}\right)\left(1\pm\:\sqrt{\frac{{\varepsilon\:}_{\text{2,1}}}{{\varepsilon\:}_{a}}}\right)\:$$

(7b)

where$\:\:{\varepsilon\:}_{\text{1,1}}$ and$\:\:{\varepsilon\:}_{\text{2,1}}$ are the electric permittivity of the first dispersive Lorentzian slab in the structure, at frequencies of $\:{\omega\:}_{1}$ and $\:{\omega\:}_{2}$, respectively.

If the time duration of the mth dispersive slab in the structure is $\:{\varDelta\:t}_{m}$, the connection between the field amplitudes at after the mth interface and before the (m + 1)th interface can be established by the temporal delay matrix $\:{DM}_{m}$:

$$\:{DM}_{m}=\left[\begin{array}{c}\begin{array}{c}\begin{array}{ccc}{e}^{+i{\omega\:}_{1,m}{\varDelta\:t}_{m}}&\:0&\:\begin{array}{cc}0&\:0\end{array}\end{array}\\\:\begin{array}{ccc}0&\:{e}^{-i{\omega\:}_{1,m}{\varDelta\:t}_{m}}&\:\begin{array}{cc}0&\:0\end{array}\end{array}\end{array}\\\:\begin{array}{c}\begin{array}{ccc}0&\:0&\:\begin{array}{cc}{e}^{+i{\omega\:}_{2,m}{\varDelta\:t}_{m}}&\:0\end{array}\end{array}\\\:\begin{array}{ccc}0&\:0&\:\begin{array}{cc}0&\:{e}^{-i{\omega\:}_{2,m}{\varDelta\:t}_{m}}\end{array}\end{array}\end{array}\end{array}\right]\:$$

(8)

To obtain the temporal matching and delay matrices for a dielectric medium in the structure, it is sufficient to take $\:{\varepsilon\:}_{\infty\:,m}={\varepsilon\:}_{d}$ in Eq. (1), $\:{\omega\:}_{0,m}\to\:\infty\:$, $\:{\omega\:}_{2,m}\to\:\infty\:$ and $\:{\varepsilon\:(\omega\:}_{2,m})=0,\:{\varepsilon\:(\omega\:}_{1,m})={\varepsilon\:}_{d}$.

Using the temporal matching and delay matrices introduced above one can express the forward and backward propagating waves in the final medium in terms of the incident field in the initial medium as follows:

$$\left[ {\begin{array}{*{20}l} {f_{{1,\:final}} } \hfill \\ {b_{{1,\:final}} } \hfill \\ {f_{{2,\:final}} } \hfill \\ {b_{{2,\:final}} } \hfill \\ \end{array} } \right] = \left( {\prod\nolimits_{{m = 1}}^{M} {MM_{{m + 1}} DM_{m} } } \right)MM_{1} \left[ {\begin{array}{*{20}c} {f_{{initial}} } \\ 0 \\ 0 \\ 0 \\ \end{array} } \right]$$

(9)

If the initial medium is a nondispersive dielectric material with permittivity of $\:{\varepsilon\:}_{a}$, four frequencies in the mth dispersive slab material are $\:\pm\:{\omega\:}_{1,m}$ and $\:\pm\:{\omega\:}_{2,m}$ where we can obtain them from the conservation of momentum ${\omega _i}\sqrt {{\varepsilon _a}} ={\omega _m}\sqrt {{\varepsilon _{\infty ,m}}+\frac{{\omega _{{p,m}}^{2}}}{{\omega _{{0,m}}^{2} - \omega _{m}^{2}}}}$and they are expressed as follows:

$${\omega _{1,m}}=\sqrt {\frac{{{K_m}+\sqrt {K_{m}^{2} - 4\omega _{{0,m}}^{2}\omega _{i}^{2}{\varepsilon _a}{\varepsilon _{\infty ,m}}} }}{{2{\varepsilon _{\infty ,m}}}}} ,\,\,\,\,{\omega _{2,m}}=\sqrt {\frac{{{K_m} - \sqrt {K_{m}^{2} - 4\omega _{{0,m}}^{2}\omega _{i}^{2}{\varepsilon _a}{\varepsilon _{\infty ,m}}} }}{{2{\varepsilon _{\infty ,m}}}}}$$

(10)

where ${K_m}=\omega _{{p,m}}^{2}+\omega _{{0,m}}^{2}{\varepsilon _{\infty ,m}}+\omega _{i}^{2}{\varepsilon _a}$; $\:{\omega\:}_{p,m}$, $\:{\omega\:}_{\infty\:,m}$ and $\:{\omega\:}_{0,m}$ are the plasma, infinite and resonance frequency of the mth Lorentzian dispersive slab in the structure, respectively, and $\:{\omega\:}_{i}$ is the frequency of incident wave in the initial medium. In the same way, when a plane wave with frequency $\:{\omega\:}_{-}$ propagates through a Lorentzian medium in which the plasma frequency changes abruptly from $\:{\omega\:}_{p-}$ to $\:{\omega\:}_{p+}$, from the momentum conservation $\:{\omega\:}_{-}\sqrt{{\varepsilon\:}_{\infty\:}+\frac{{\omega\:}_{p-}^{2}}{{\omega\:}_{0}^{2}-{\omega\:}_{-}^{2}}}={\omega\:}_{+}\sqrt{{\varepsilon\:}_{\infty\:}+\frac{{\omega\:}_{p+}^{2}}{{\omega\:}_{0}^{2}-{\omega\:}_{+}^{2}}}$ we can derive four new modes with frequencies $\:\pm\:{\omega\:}_{1,+}$ and $\:\pm\:{\omega\:}_{2,+}$ which are defined as follows:

$$\begin{gathered} {\omega _{1,+}}=\sqrt {\frac{{kk+\sqrt {k{k^2} - 4{\varepsilon _\infty }\omega _{0}^{2}\omega _{ - }^{2}\left( {\omega _{ - }^{2} - \omega _{0}^{2}} \right)\left( {{\varepsilon _\infty }\omega _{ - }^{2} - {\varepsilon _\infty }\omega _{0}^{2} - \omega _{{p - }}^{2}} \right)} }}{{2{\varepsilon _\infty }\left( {\omega _{ - }^{2} - \omega _{0}^{2}} \right)}}} \hfill \\ {\omega _{2,+}}=\sqrt {\frac{{kk - \sqrt {k{k^2} - 4{\varepsilon _\infty }\omega _{0}^{2}\omega _{ - }^{2}\left( {\omega _{ - }^{2} - \omega _{0}^{2}} \right)\left( {{\varepsilon _\infty }\omega _{ - }^{2} - {\varepsilon _\infty }\omega _{0}^{2} - \omega _{{p - }}^{2}} \right)} }}{{2{\varepsilon _\infty }\left( {\omega _{ - }^{2} - \omega _{0}^{2}} \right)}}} \hfill \\ \end{gathered}$$

(11)

where $kk=\omega _{ - }^{2}\left( {{\varepsilon _\infty }\omega _{ - }^{2}+\omega _{{p+}}^{2} - \omega _{{p - }}^{2}} \right) - \omega _{0}^{2}\left( {{\varepsilon _\infty }\omega _{0}^{2}+\omega _{{p+}}^{2}} \right)$.

In the next section we use the method mentioned above to calculate the transmission and reflection coefficients of the proposed disordered temporal multilayered structure.

Results and discussion

We first consider the structure shown in Fig. 1. It is assumed that the initial spatially homogeneous medium is air. It is also supposed that air is abruptly changes to a Lorentzian dispersive medium such as FR-4 (fiberglass—epoxy) dielectric with $\:{\varepsilon\:}_{\infty\:}=4.096,\:$ $\:{\omega\:}_{p}=1.1237\times\:{10}^{11}\frac{rad}{s}$ and $\:{\omega\:}_{0}=2.4819\times\:{10}^{11}\frac{rad}{s}$at t₁ = 0²⁶. It should be noted that the Lorentzian permittivity in ref²⁶, is ${\varepsilon _\infty }+\frac{{{A_k}\omega _{{0k}}^{2}}}{{\omega _{{0k}}^{2} - {\omega ^2}+j\omega 2{\delta _k}}}$while in our paper it is described by Eq. (1) in which the absorption is neglected. In ref²⁶, ${A_k}\omega _{{0k}}^{2}=\left( {{\varepsilon _{sk}} - {\varepsilon _\infty }} \right)\omega _{{0k}}^{2}=\omega _{p}^{2},$ therefore, for the values of $\:{\varepsilon\:}_{s}=4.301,\:{\varepsilon\:}_{\infty\:}=4.096\:$ and $\:{\omega\:}_{0}=2.4819\times\:{10}^{11}\frac{rad}{s}$, we can obtain $\:{\omega\:}_{p}=\sqrt{{\varepsilon\:}_{s}-{\epsilon}_{\infty\:}}{\omega\:}_{0}=1.1237\times\:{10}^{11}\:rad/s$. It is also worth mentioning that the value of imaginary part of permittivity for the frequency range 25 to 38 GHz used here is lower than 0.05 according to Fig. 2b in ref²⁶ which is significantly smaller than the real part of permittivity which is about 4.1 (Fig. 2a in ref²⁶). Therefore, we neglect the loss effect. However, taking into account the effect of absorption in time varying structures with Lorentzian dispersion is under consideration and the corresponding results will be reported in the near future.

Fig. 2

A schematic of multilayered structure composed of two nondispersive dielectric slabs alternatively. The permittivity of one of slabs is constant as $\:{\varepsilon\:}_{d}=4.8$ and the other changes randomly. The temporal width of all slabs is the same as $\:\varDelta\:t=0.2\:ns.$

The temporal width of the first Lorentzian dispersive medium is taken to be $\:\varDelta\:t=0.2\:ns.$ At $\:t=\varDelta\:t$, the first dispersive medium is converted to a nondispersive dielectric medium whose relative electric permittivity is a random number obtained from the relation given in the Section “Theoretical method and the designed structure”. The average of relative permittivity and disorder level are taken to be ${\varepsilon _{av}}=3.5$ and Q$\:=$0.9, respectively. The temporal width of the first nondispersive dielectric slab is also $\:\varDelta\:t=0.2\:ns.$ This process is repeated 30 times. Then at $\:t=6$ ns the nondispersive medium becomes a dispersive one. The relative permittivity of other dispersive slabs in the temporal multilayered structure is the same as the first one. But the relative permittivity of dielectric slabs is chosen randomly. Using the transfer matrix given in Eq. (11), the average of transmission and reflection coefficients for two frequencies in the final dispersive medium are calculated versus the incident frequency and the corresponding results are displayed in Fig. 3 for the frequency range [25 GHz, 33 GHz]. To obtain the results in Fig. 3, ensemble averaging is performed over $\:2000$ realizations with the same disorder level. It should be noted that the values of amplitude of forward and backward waves in the final medium correspond to transmission and reflection coefficients because the amplitude of incident wave is taken to be unity. Figure 3a shows the average of absolute value of transmission coefficients ($\:\left|{f}_{1,final}\right|\:and\:\left|{f}_{2,final}\right|$). One can see that $\:{f}_{2,final}$ has smaller value in all frequencies compared to $\:{f}_{1,final}$. Furthermore, around the frequency of 28.43 GHz and 28.91 GHz both $\:{f}_{1,final}$ and $\:{f}_{2,final}$, respectively experience a significant amplification. Figure 3b shows the average of reflection coefficients $\:\left|{b}_{1,final}\right|\:and\:\left|{b}_{2,final}\right|$ at two generated frequencies versus the incident frequency. It should be noted that the transmission coefficients $\:\left|{f}_{1,final}\right|\:and\:\left|{f}_{2,final}\right|$ are the amplitudes of forward waves at frequencies of $\:{\omega\:}_{1,final}$ and $\:{\omega\:}_{2,final}$, respectively. However, the reflection coefficients of $\:\left|{b}_{1,final}\right|\:and\:\left|{b}_{2,final}\right|$ are the amplitudes of backward waves at frequencies of $\:-{\omega\:}_{1,final}$ and $\:-{\omega\:}_{2,final},$ respectively. The frequencies of $\:{\omega\:}_{1,final}$ and $\:{\omega\:}_{2,final}$ in the final medium are calculated using Eq. (10). As shown in Fig. 3b, the value of $\:{b}_{1,final}$ is greater than $\:{b}_{2,final}$ in all incident frequencies. Comparison of Fig. 3a,b indicates that there is no significant difference between the reflection and transmission coefficients. It should be noted that the amplitude of electric field of incident wave is unity while the values of transmission and reflection coefficients are larger than one. Therefore, the amplitudes of forward and backward waves in the final medium indicate a maximum amplification by $\:1560$ times due to the abrupt change of permittivity in the disordered dispersive structure displayed in Fig. 1. When electromagnetic waves passe through a spatially homogenous medium with disordered time interfaces, copropagating and counterpropagating waves are created due to time reflections and transmissions at different interfaces. The constructive interference between these multiple waves leads to the amplification of incident waves. It should be noted that there are many methods to change the permittivity of dielectric materials. For example, the permittivity of porous dielectric materials can be changed by varying the porosity. In ref²⁸, it was demonstrated that electrochemical etching of bulk silicon crystalline with different current densities leads to samples with different porosities as well as different permittivities in the range 1 to 40 GHz. To obtain random permittivities, it is sufficient to employ random current densities in the electrochemical etching.

Fig. 3

(a) Average of absolute value of transmission coefficients ($\:\left|{f}_{1,final}\right|\:and\:\left|{f}_{2,final}\right|$) and (b) reflection coefficients $\:\left(\left|{b}_{1,final}\right|\:and\:\left|{b}_{2,final}\right|\right)\:$versus the frequency of incident wave onto the structure displayed in Fig. 1.

To understand the effect of disorder level on the reflection and transmission coefficients, in Fig. 4a–d the absolute values of coefficients $\:\left|{f}_{1,final}\right|,\:\left|{f}_{2,final}\right|,\:\left|{b}_{1,final}\right|\:and\:\left|{b}_{2,final}\right|$ are plotted as a function of incident frequency for different disorder levels Q$\:=$0, 0.35, 0.7 and 0.95. It is clearly seen that the values of all coefficients increase with increase of disorder level. This effect results from the enhancement of multiple scattering due to the increase of disorder level. In other words, electromagnetic waves propagating through a spatially invariant medium with disordered temporal boundaries experiences stronger scattering compared to one passing through a spatially homogenous medium with periodic temporal boundaries. As a result, the probability of constructive interference between waves generated at time boundaries increases with disorder level leading to more amplification. Therefore, disordered temporal multilayered structures are good platforms for amplification of electromagnetic waves compared to periodic ones without using gain media.

Fig. 4

The average of absolute value of transmission and reflection coefficients versus the incident frequency at different disorder levels Q = 0, 0.35, 0.70 and 0.95. Figure (a–d) correspond to $\:\left|{f}_{1,final}\right|,\:\left|{f}_{2,final}\right|,\:\left|{b}_{1,final}\right|\:and\:\left|{b}_{2,final}\right|$, respectively.

To demonstrate the effect of dispersion on the wave propagation in the temporal multilayered structure, we consider another structure like one shown in Fig. 1 but all dispersive slabs are replaced with nondispersive dielectric media. This structure is schematically shown in Fig. 2. In this figure, $\:{\varepsilon\:}_{d,i}\left(i=1,\:2,\:3,\:\dots\:,m\right)$ are random numbers selected in the same way as in Fig. 1. Because the relative permittivity of dispersive medium in the structure of Fig. 1 is in the range $\:4.4<{\varepsilon\:}_{L}\left({\omega\:}_{1}\right)<4.8$ and $\:3.5<{\varepsilon\:}_{L}\left({\omega\:}_{2}\right)<3.8$ for the frequency range [25 GHz, 33 GHz] and the average of permittivity of dielectric slabs is 3.5, we choose the relative permittivity of similar slabs in Fig. 2 to be $\:{\varepsilon\:}_{d}=4.8$ and ${\varepsilon _{av}}=3$0.5. Such selections for the parameters cause the average of permittivity contrast in Fig. 2 to be equal to the maximum of average of permittivity contrast in Fig. 1. Because the final medium in this figure is a nondispersive material, there exist only one forward wave and only one backward wave in the final medium in contrast to the structure in Fig. 1.

In the calculations, the disorder level is taken to be Q$\:=$0.9 and the other parameters are the same as those in Fig. 1. The corresponding results are displayed in Fig. 5 and are compared with the results of structure shown in Fig. 1 with disorder level of Q$\:=$0.9. It is clearly seen in this figure that when we use nondispersive materials instead of dispersive ones in the disordered temporal multilayered structure, we observe no significant amplification. In the temporal disordered structure based on dispersive materials the maximum output amplitude of forward and backward waves increases by $\:1560$ times at $\:f=28.43$ GHz. However, for the structure containing only nondispersive dielectric materials, the maximum output amplitude of forward and backward waves increases by 6.7 times at $\:30.12$ GHz. Therefore, disordered structures based on dispersive materials make it possible the strong enhancement of intensity of electromagnetic waves incident on them. It should be noted that when a medium abruptly changes to a dispersive medium two new frequencies are generated while an abrupt change of the medium to a nondispersive one leads to generation of only one new frequency. It seems that this effect causes more amplification to occur in the disordered temporal multilayered structure in the presence of dispersive slabs. Since a greater number of transmitted and reflected waves are generated at temporal interfaces in Fig. 1 due to the dispersion effect, the probability of constructive interference between copropagating and counterpropagating waves, increases resulting in more amplification in comparison with the structure in Fig. 2.

Fig. 5

The average of absolute value of transmission coefficients $\:\left|{f}_{1,final}\right|$ (a) and reflection coefficients $\:\left|{b}_{1,final}\right|$ (b) versus the incident frequency corresponding to the structure displayed in Fig. 3 (blue solid-line curve) and the structure represented in Fig. 1 (red dotted-line curve), respectively.

Now we examine how the value of average dielectric permittivity of dielectric slabs in the structure of Fig. 1 affects the amount of amplification. To this end, in Fig. 6 we plot the average of absolute values of transmission and reflection coefficients as a function of incident frequencies for different values of ${\varepsilon _{av}}=3.5,\,3.0\,and\,\,2.5$. As shown in this figure, when the value of $\:{\varepsilon\:}_{av}$ decreases, all coefficients $\:\left|{f}_{1,final}\right|,\:\left|{f}_{2,final}\right|,\:\left|{b}_{1,final}\right|\:and\:\left|{b}_{2,final}\right|$ increase at most frequencies. Furthermore, the maximum of amplification factor in both transmission and reflection increases with decreasing the value of $\:{\varepsilon\:}_{av}$. This effect is due to the enhancement of refractive index contrast between nondispersive dielectric and dispersive materials with decrease of $\:{\varepsilon\:}_{av}$ resulting in the raise of multiple scattering.

Fig. 6

The average of absolute values of transmission and reflection coefficients versus the incident frequency at different averages of permittivity of dielectric slabs $\:{\varepsilon\:}_{av}=3.5,\:3.0\:and\:2.5$. Figure (a–d) correspond to $\:\left|{f}_{1,final}\right|,\:\left|{f}_{2,final}\right|,\:\left|{b}_{1,final}\right|\:and\:\left|{b}_{2,final}\right|$, respectively.

Now, we consider a temporal disordered structure consisting of dispersive slabs only whose plasma frequencies change randomly after the same intervals $\:\varDelta\:t$ while their resonance frequency is the same as $\:{\omega\:}_{0}=2.4819\times\:{10}^{11}\frac{rad}{s}$ corresponding to FR-4 Lorentzian material²⁶. It is assumed that all slabs have the same temporal width of $\:\varDelta\:t=1\:ns$. The schematic representation of this disordered temporal dispersive multilayered structure is shown in Fig. 7. Thus, in this structure all slabs are dispersive materials with random plasma frequencies.

Fig. 7

A schematic of multilayered structure composed of dispersive slabs only. The plasma frequencies of slabs change randomly after the same time intervals $\:\varDelta\:\text{t}$ while their resonance frequencies are the same.

The value of plasma frequency of the (m-1)th slab in the structure of Fig. 7$\omega _{{p,m - 1}}$changes abruptly to $\omega _{{p,m}}$ which is a random number obeying from the relation $\omega _{{p,m}} = \omega _{{p,av}} \left( {1 + q} \right)$ where $\:q$ is a random number distributed uniformly in the range [-Q, Q] and $\omega _{{p,av}}$ is the average of plasma frequency. Q is the disorder level as in the first structure in Fig. 1. The temporal profile of plasma frequency changes is shown in Fig. 8.

Fig. 8

The temporal profile of plasma frequency which changes randomly in the structure displayed in Fig. 7.

In the calculations, the average of plasma frequency is chosen as $\:{\omega\:}_{p}=1.1238\times\:{10}^{11}\frac{rad}{s}$ corresponding to FR-4 Lorentzian material²⁶. In addition, the number of slabs is taken to be 40 and the ensemble averaging is performed over $\:2000$ different random realizations with the same disorder level. It is assumed that the first slab is a Lorentzian material whose plasma frequency is equal to the average plasma frequency while its resonance frequency and infinite permittivity are the same as other slabs. For calculation of frequency of new generated modes in the temporal multilayered structure one can use Eq. (11). For the structure in Fig. 7, we calculate all transmission and reflection coefficients versus incident frequency for different disorder levels Q$\:=$0.1, 0.4, 0.7 and 0.9 and show the corresponding results in Fig. 9 in the frequency range [31 GHz, 38 GHz]. It is clearly seen in this figure that the values of average of reflection and transmission coefficients at all incident frequencies increase with increasing the disorder level. Therefore, a Lorentzian dispersive medium whose plasma frequency changes randomly after equal time intervals $\:\varDelta\:t=1\:ns$ can be a good candidate for amplification of electromagnetic waves in the frequency range [31 GHz, 38 GHz]. This effect, results from the enhancement of multiple scattering due to the increases of disorder strength. To change the plasma frequency of Lorentzian materials experimentally, the number density of polarizable atoms can be altered under illumination by short optical pulses²⁹ or electrical gating³⁰. In other words, for random variation of plasma frequencies in these materials, the intensity of laser pulses or voltage gating should be changed randomly.

Fig. 9

The average of absolute value of transmission and reflection coefficients versus the incident frequency at different disorder levels Q = 0.1, 0.4, 0.7 and 0.9 corresponding to the structure displayed in Fig. 7. Figure (a–d) correspond to $\:\left|{f}_{1,final}\right|,\:\left|{f}_{2,final}\right|,\:\left|{b}_{1,final}\right|\:and\:\left|{b}_{2,final}\right|$, respectively.

In order to understand how the number of slabs in the structure affects the amplification of electromagnetic waves propagating through the structure, we calculate the transmission and reflection coefficients for different slab numbers $\:N=10$, $\:20$, $\:30$ and $\:40$ existing in the temporal structure represented in Fig. 7. In the calculations, the disorder level is selected to be Q$\:=$0.9 and the other parameters are the same as those in Fig. 9. The corresponding results are shown in Fig. 10. One can see in this figure that the increase of number of slabs in the disordered temporal structure leads to the enhancement of the amplification of the incident wave. The reason for this behavior is that the increase of number of slabs results in the increase of number of temporal boundaries. Therefore, the number of reflected and transmitted waves at the boundaries increases leading to the enhancement of multiple scattering and probability of constructive interference. As a result, in the cases we tend to observe more amplification of the beams incident onto a disordered temporal structure, it is sufficient to use a greater number of slabs in the structure.

Fig. 10

The average of absolute value of transmission and reflection coefficients versus the incident frequency at different numbers of slabs N = 10, 20, 30 and 40 corresponding to the structure displayed in Fig. 7. Figure (a–d) correspond to $\:\left|{f}_{1,final}\right|,\:\left|{f}_{2,final}\right|,\:\left|{b}_{1,final}\right|\:and\:\left|{b}_{2,final}\right|$, respectively.

Finally, we examine the effect of temporal width of dispersive slabs on the amplification of electromagnetic waves incident onto the structure displayed in Fig. 7. To this end, the transmission and reflection coefficients for different temporal widths of slabs $\:\varDelta\:t=1,\:0.1,\:0.01\:and\:0.001\:ns$ are calculated and the corresponding results are shown in Fig. 11. In the calculations, the number of layers is taken to be $\:N=40$. The other parameters are the same as those in Fig. 10.

Fig. 11

The average of absolute value of transmission and reflection coefficients versus the incident frequency at different temporal widths of slabs $\:\varDelta\:t=1,\:0.1,\:0.01\:$and 0.001 ns corresponding to the structure displayed in Fig. 7. Figure (a–d) correspond to $\:\left|{\text{f}}_{1,\text{f}\text{i}\text{n}\text{a}\text{l}}\right|,\:\left|{f}_{2,final}\right|,\:\left|{b}_{1,final}\right|\:and\:\left|{b}_{2,final}\right|$, respectively.

It is clearly seen in Fig. 11 that by decreasing $\:\varDelta\:t$ from 1 ns to 0.1 ns, the transmission and reflection coefficients increase in most frequencies. However, when $\:\varDelta\:t$ decreases from 0.1 ns to 0.001 ns, the transmission and reflection coefficients decrease in all frequencies. The reason for this behavior is that the temporal phase shift $\:\phi\:=2\pi\:f\varDelta\:t$ within each slab becomes so small when $\:\varDelta\:t$ is very short. In this case, the difference between consecutive temporal phase shifts is not so large to make strong multiple scattering leading to a giant amplification. Consequently, to obtain strong amplification of incident wave in the proposed structure, the temporal width of slabs should be chosen properly. In addition, in Fig. 11, there are some local maxima in the curves of transmission and reflection coefficients for the case $\:\varDelta\:t=1\:ns$. It seems that the interference effect resulted from the enough phase accumulation in each temporal layer is responsible for this behavior such that this phase accumulation weakens with decrease of layer thickness $\:\varDelta\:t$. Therefore, local maxima disappear for the cases $\:\varDelta\:t<1\:ns$ in Fig. 11.

Consequently, the disordered temporal multilayered structures proposed in this paper based on Lorentzian dispersive materials are promising candidates for amplification of electromagnetic waves in Ka-band without using gain media and stimulated emission. Hence, the experimental implementation of the proposed structure seems interesting and finds many applications in different area.

Conclusion

We have proposed two temporal multilayered structures based on Lorentzian dispersive materials to amplify the electromagnetic waves in the Ka-band. In the first structure which composed of dispersive and nondispersive slabs alternatively, the permittivity of dispersive slabs is taken to be the same while the permittivities of nondispersive slabs are random numbers changing abruptly after equal time intervals. In this structure, the strong amplification of incident wave has been observed such that the amount of amplification increases with increase of the disorder level. Furthermore, decrease of average of permittivity of nondispersive dielectric slabs leads to more wave amplification. Comparison of transmission and reflection coefficients of this disordered temporal structure with the disordered temporal structure without Lorentzian dispersive slabs demonstrates that the presence of dispersion is essential for observation of significant amplification. The second disordered temporal multilayered structure proposed in this paper consists of spatially homogeneous Lorentzian dispersive slabs whose plasma frequencies change abruptly and randomly after equal time intervals. In this structure, the amount of amplification also increases with increase of disorder level as well as the number of temporal slabs forming the structure. In addition, it has been shown that the temporal width of dispersive slabs in the second structure should be selected properly in order to witness strong wave amplification.