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Abstract
We study the phenomenon of parametric amplification in the context of time-periodic dielectric slabs. These structures show particular promise inasmuch as they are capable of very large amplifications when illuminated by an electromagnetic wave of half the modulation frequency. Successive studies have corroborated this finding but none have yet been able to ascertain the nature of amplification in such devices. On top of that, some studies have raised speculations regarding the instability of a time-periodic slab which are off the mark. The problem lies in the poor understanding (or lack thereof) of the mathematical devices necessary to tackle such problems. We successfully carry out the tasks by tapping into the rich mathematical theory of Hill’s equation. Specifically, we make use of the Folquet’s theorem in its complete form which brings to light novel physical phenomena that the more prevalent simplified form fails to account for. Also, useful mathematical concepts such as coexistence are employed which to the best of our knowledge have not yet been applied in the field of time-varying optics. Our analytical method proves an effective means of assessing the amplifier’s performance, e.g., estimating how long it takes for the device to reach steady state. We further delineate the link between amplification and instability and correct the misconceptions surrounding the subject by presenting a rigorous analysis of the instability problem in such structures.
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1. Introduction
Time-varying electromagnetic structures have been an attractive research topic in the field of microwaves and optics. Especially in the past decade, they have held the interest of many a researcher and found various applications such as realization of nonmagnetic nonreciprocal devices [1–3], parametric amplification [4–7], frequency conversion [8,9], frequency combs [10,11], photonic topological insulators [12,13] and time-varying metasurfaces [14,15].
One of the earliest studies to have been conducted in connection with time-varying electromagnetic structures, concerned parametric amplification [16]. It was noted back then that an electromagnetic wave travelling through a dielectric medium would experience a frequency shift along with an increase in total energy density, if the medium underwent a modulation of permittivity or permeability. The excess energy was then attributed to the modulating source doing work upon the electromagnetic field. In regard to time-periodic media this observation corresponds to the existence of certain propagating modes (with purely imaginary propagation constant), the amplitudes of which increase exponentially in time [17–19].
Having treated unbounded (isolated) temporally periodic media, practicality demanded that open structures be attended to as well. The simplest of these would be a temporally modulated dielectric slab surrounded by free space [4,5]. The problem of normal incidence upon this structure was addressed and it turned out that an incident wave with half the modulation frequency could occasion considerable amplification, which come about by operating the device near the points of singularity (infinite transmission and reflection). Singularities show up at certain points while altering the slab’s characteristics such as thickness, modulation frequency, modulation strength, etc. Their emergence suggests that a time-periodic slab supports parametric resonances of infinite quality factor [6].
As yet, no effort has been directed towards explaining the nature of infinite transmission and reflection that emerges in the treatment of a time-periodic dielectric slab. Since amplification in such a device is closely connected to singularity, knowledge of the system’s behaviour at these points is much needed. Furthermore, owing to the obscurity of the actual mechanism by which the parametric resonance occurs, flawed conclusions have been formerly drawn regarding the instability of such structures [4–7]. For instance, by gradually increasing (starting from zero) the modulation frequency of a time-periodic slab, while holding all the other parameters constant, we come upon the first occurrence of singularity in normal incidence. Some hold the view [4,7], albeit loosely, that from this point onwards the structure is unequivocally unstable. Others [5,6] insist that such points constitute mere resonances and tackle this subject with disregard for instability. These perceptions have their basis solely in speculations and are not backed by rigorous analyses.
In this paper, we set out to carefully examine the phenomenon of parametric amplification and the nature of singularity in the context of a time-periodic dielectric slab. We rigorously demonstrate that a time-periodic slab can sustain spontaneous unstable (exponentially growing in time) outward radiation and see that this happens in certain bounded instability intervals, the boundaries of which coincide with the points of singularity. The task is carried out in the following fashion (outlined in Fig. 1): In Section 2, we study wave propagation in time-periodic media with the aid of the well established theory of Hill’s equation. We put these findings to use in Section 3 to examine the nature of singularity emerging in the problem of scattering by a time-periodic slab. In Section 4, we address the misapprehensions surrounding the instability of a time-periodic slab by presenting a rigorous analytical method and also by providing full-wave simulation results. Lastly, concluding remarks are contained in Section 5.
2. Wave propagation in unbounded time-periodic media
We start off by investigating propagating modes in temporally periodic media. We take the dielectric medium in question to be characterized as $\varepsilon _r(t)=\sum _{m=-\infty }^{+\infty } \epsilon _{m} e^{im\Omega t}$ with period $T\equiv 2\pi /\Omega$. From Maxwell’s equations
Since Eq. (2) is a linear differential equation with real coefficients, the complex conjugate of the above solution is also a solution to Eq. (2) and so it suffices to find a solution of the above form and then take its real part to obtain the real (physical) field. To simplify matters, we confine our analysis to the complex exponential forms throughout this work. By substituting Eq. (3) in Eq. (2) we get
which according to the Floquet’s theorem admits a solution of the formEquation (5) along with Eq. (4) provides us with an eigenvalue problem which can be solved for the wavenumber $k$ granted that the frequency $\omega$ of the mode is specified:
For demonstration purposes, we take up a hypothetical case where $\varepsilon _r(t)=2+1 \cos \Omega t$, we duly truncate the infinite-dimensional problem of Eq. (6) and solve the resulting matrix eigenvalue problem in the Brillouin zone $0<\omega /\Omega <1$; the result of which is presented in Fig. 2. Certain momentum gaps ($k$ gaps) can be discerned from an inspection of Fig. 2, hinting at the existence of propagating modes (with real $k$) whose frequencies $\omega$ are not real. To proceed with identifying these gaps, we first note that $\Psi (t)=\varepsilon _r(t)\Xi (t)$ transforms Eq. (4) into the Hill’s equation [20,21]:
which over the last century, has enjoyed extensive analysis at the hands of many illustrious mathematicians [21]. In Appendix A, we seek to acquaint the reader with the theory of Hill’s equations, insofar as is necessary to the progression of our analysis. Here is a summary of the key points raised in the Appendix.Although a band diagram like that of Fig. 2 is quite comprehensive in its description of time-harmonic propagating modes in a time-periodic medium, it leaves out some important details which, as far as parametric amplification is concerned, are indispensable. The solutions that we have obtained here, in quite the same way as the previous studies [22,23], have their origin in the simplified statement of Floquet’s theorem which is pervalent in engineering circles. According to this simplified form, the solutions to Eq. (7) are as follows:
where $p_1(t)$ and $p_2(t)$ are periodic with period $T$. At the band edges however, which happen to be at the center of attention for the purpose of parametric amplification, Eq. (8) fails to provide an exhaustive set of solutions. For those $k$s that lie on the band edges, Flouqet’s theorem speaks of a different form of solutions [21] to Eq. (7)These are exclusive to the cases $\omega = \Omega$ ($\rho =1$) or $\omega = \Omega /2$ ($\rho =-1$). The constant $\alpha$ can be zero which constitutes an instance of coexistence, i.e., there are two periodic solutions of Eq. (7) with period $T$ ($\omega =\Omega$) or $2T$ ($\omega =\Omega /2$). The reach of the Floquet’s theorem extends even further [21], inasmuch as it determines the nature of the modes within the momentum gaps. Inside the gaps, the solutions are of the same form specified in Eq. (8) but $\omega$ is complex, which corresponds to exponentially increasing and decreasing modes. Moreover, within a certain gap, only the imaginary part of $\omega$ varies and its real part is fixed at a value of either $\Omega /2$ or $\Omega$.
Floquet’s theorem alone provides all that information about the modes of a time-periodic medium, but that is as far as it gets. Another question that comes to mind in this context is: how exactly is the linearly growing solution of Eq. (9) related to the momentum gaps and under what circumstances does coexistence ($\alpha =0$) happen? These questions along with many others were answered in a treatise of time-periodic systems by the celebrated Liapounoff [24]. His Oscillation Theorem establishes that coexistence happens if and only if the edges of the neighbouring bands come together, i.e., the momentum gap is closed. There is no linearly growing solution associated with such band edges, nor is there a solution for nearby $k$s that grows exponentially in time since the gap is closed. In other words, amplification and instability in time-periodic media occur only when momentum gaps exist, i.e., when coexistence does not happen. The problem of coexistence in the theory of Hill’s equation has been the subject of intensive research over the past century [21]. For instance, Winkler and Magnus have worked out [25] that for a sinusoidal modulation, only the gaps characterized by $Re\{\omega \}=\Omega /2$ exist. In the case of the band diagram of Fig. 2, which corresponds to a sinusoidal $\varepsilon _r(t)$, there is no gap on the line $\omega =\Omega$ and all the gaps on the line $\omega = \Omega /2$ are open. This explains why the previous studies working on sinusoidal modulation [5,6], have encountered amplification only at $\omega =(2n+1)\Omega /2$.
3. Singularities in the problem of incidence upon a time-periodic dielectric slab
Our investigation henceforth concerns the time-periodic dielectric slab of Fig. 1(b). This structure is similar in all aspects to that of the previous studies [4,5], except it terminates on one side in a PEC reflector. It can be considered equivalent to the previous structures considering that they were chiefly studied in a symmetric configuration. It also has the advantage of making for a less tedious analysis. For an ample discussion of the suitable materials and the means to realize time-varying permittivity, we refer the reader to [26]. The problem of time-periodic slab under normal incidence has been treated by the method of modal expansion [4,5] and points of infinite transmission and reflection, i.e. singularity, have been discerned. Still, no thought was given to solving the Maxwell’s equations at the point of singularity, where the previously used method fails to do so. Moreover, the momentous problem of ascertaining instability and identifying its nature remains. Therefore we recapitulate the method in Appendix B, review its results in what follows and further along, seek to surmount its shortcomings. Specifically, we make use of the coupled-wave formalism (otherwise known as RCWA) [27] which is more convenient than and mathematically equivalent to the method of modal expansion.
Fig. 1. (a) Schematic drawing of a time-periodic unbounded medium with relative permittivty $\varepsilon _r(t)$ whose propagating modes are studied in Section 2. (b) A time-periodic slab terminating in a PEC. The structure has a thickness of $L$ and is unbounded in the $x$ and $y$ directions. The solution to the problem of normal incidence upon this slab is presented in Section 3 and the points of singularity are thoroughly analysed. (c) The radiative modes of slab are investigated in Section 4 and instability is rigorously demonstrated.
Fig. 2. The band diagram of a time-periodic dielectric medium with relative permittivity specified as $\varepsilon _r(t) = 2+1\cos \Omega t$.
Upon a time-periodic grounded slab of thickness $L$ with $\varepsilon _r(t)=\sum _{m=-\infty }^{+\infty } \epsilon _{m} e^{im\Omega t}$, a normally incident plane wave of the form $\exp i(\omega _0 t + k_{0}z)$ produces a series of reflected waves of the form $\sum _{m=-\infty }^{+\infty } R_{m} \exp {i(\omega _m t-k_{z,m}z)}$ where $\omega _m=\omega _0+m\Omega$. The reflection coefficients $R_m$ can be obtained via a matrix equation $\mathbf {M}\mathbf {R}=\mathbf {b}$, the derivation of which is detailed in Appendix B. The previous studies noted that a sinusoidally modulated slab is capable of arbitrarily large amplifications when illuminated by a plane wave of frequency $\omega _0=\Omega /2+n\Omega$ [5]. For a slab with fixed characteristics whose modulation frequency $\Omega$ is allowed to vary, such amplifications are achieved in the vicinity of certain discrete $\Omega$s where the coefficient matrix $\mathbf {M}$ becomes singular, i.e., $\det (\mathbf {M})=0$ [6]. The closer the device’s operating point to a singularity, the larger the amplification gets and it does so without bound. Although the previous studies make it clear that large amplifications are at hand with a time-periodic slab, from a practical standpoint one should also inquire as to the nature of amplification in such devices; for example, how long does it take for such large amplifications to take effect? We note that amplification here is closely connected with singularity and thus, in order to answer the above question, we proceed with an examination of the solution to the problem of incidence at the point of singularity.
The fact that singularities in a sinusoidally modulated slab are reported to only occur when $\omega _0 = \Omega /2+n\Omega$ [5], recalls the rather extraordinary form of solutions to the Hill’s equation on the edge of the momentum gaps (one periodic and the other linearly growing), which in the case of sinusoidal modulation happen to reside exclusively at $\omega =\Omega /2$. This can be utilized to develop a novel method of solution based on modal expansion that complements the previously used method when $\omega _0 = \Omega /2+n\Omega$. In this case, the electric field in the slab may be expressed in terms of modes as $E_y(z,t)=\sum _{n} e^{i\beta _nz}\Xi _n(t)$, where according to Eq. (9)
In the above equation $\omega _m = \Omega /2+m\Omega$, $\chi _{nm}, \xi _{nm}$ represent the Fourier series coefficients of $p_1(t), p_2(t)$ in Eq. (9), and $a_n, b_n$ are the mode coefficients. From Eq. (10) and $E_y(z,t)=\sum _{n} e^{i\beta _nz}\Xi _n(t)$ it follows that
We therefore arrive at the following coupled-wave form:
This form of solution includes linearly growing terms $tQ_m(z)$ which are absent in the standard RCWA (Appendix B). Since our method is intended to complement the standard RCWA, the two solutions must be the same whenever the standard RCWA is applicable, i.e., in the absence of singularity. In other words, the linearly growing terms $tQ_m(z)$ must be identically zero when $\mathbf {M}$ is invertible. It is only in the event of singularity that the method is expected to produce different results.
The following expressions represent the electromagnetic fields within the slab (denoted by the subscript slab) and the region above (denoted by the subscript up):
If we set $\mathbf {Q}(z)$ equal to zero, Eq. (14) reduces to
The solution to Eq. (14) is obtained in a similar fashion, only this time we set $\mathbf {S}(0) =\mathbf {Q}(0)=\mathbf {0}$
The above $4\times 2$ coefficient matrix results from omitting those elements of $\exp {(\widetilde {\mathbf {C}}L)}$ that are multiplied by $\mathbf {S}(0)$ and $\mathbf {Q}(0)$. It is easy to show that within the exponential of $\widetilde {\mathbf {C}}$, which is a triangular block matrix, the diagonal blocks equal the exponential of the individual blocks that lie on the diagonal of $\widetilde {\mathbf {C}}$; hence the specific form of the $4\times 2$ coefficient matrix above. On a further note, the diagonal blocks of $\widetilde {\mathbf {C}}$ are equal to $\mathbf {C}$ and thus $\mathbf {P}_{1}$ and $\mathbf {P}_{2}$ in Eq. (17) are the same as the ones in Eq. (16).
Next we invoke the electric and magnetic field boundary conditions which yield (details in Appendix C)
In Eq. (18) when $\mathbf {M}$ is invertible, $\mathbf {Q'}(0)$ is necessarily zero and so we are in effect left with Eq. (19), meaning that the solution is identical to that of the standard RCWA. This is indeed what we expected. When $\mathbf {M}$ is singular, however, the coefficient matrix $\widetilde {\mathbf {M}}$ is also singular—being triangular with $\mathbf {M}$ on the diagonal—in which case $\mathbf {Q'}(0)$ can be a nonzero vector lying in the null space of $\mathbf {M}$ provided, of course, that Eq. (18a) admits of a solution. To ascertain whether or not this is the case, we need to locate the point of singularity and then solve the singular system of equations with the help of suitable devices. But in doing so we are confronted with a challenge. Although the whereabouts of singularity can be determined, the exact location of the point where $\det (\mathbf {M})=0$ cannot be represented by a terminating decimal. So with a finite number of decimal places we can only get close to the point of singularity, in which case the coefficient matrix is still invertible irrespective of how close to the point we are. This in turn means $\mathbf {Q'}(0)=\mathbf {0}$ and thereby Eq. (18) is effectively reduced to Eq. (19). To circumvent this difficulty we compute the singular value decomposition (SVD) of $\widetilde {\mathbf {M}}$, we treat its smallest singular value as zero and thereby arrive at an approximation of $\widetilde {\mathbf {M}}$ which is precisely singular. Then, by employing the Moore-Penrose inverse, we proceed to obtain the minimum-norm least-squares solution to the singular system of equations $\widetilde {\mathbf {M}}\mathbf {x}=\mathbf {b}$ and subsequently check whether it actually solves the equation by evaluating the relative error $R.E. = ||\widetilde {\mathbf {M}}\mathbf {x}-\mathbf {b}||/||\mathbf {b}||$ and seeing if it vanishes. The method is explained in greater detail in Appendix D.
Here, we seek to further clarify the process by means of a numerical example. We consider a slab of thickness $L = 10 \hspace {1mm} \text {cm}$, modulated in the same manner as [5] with $\varepsilon _r(t) = 5.25+0.087 \cos \Omega t$ and illuminated by a plane wave of frequency $\omega _0=\Omega /2$. This particular choice of permittivity pertains to lithium niobate which allows for reasonable modulation strengths and also modulation frequencies as high as a few tens of GHz [28,29]. The problem of normal incidence for different modulation frequencies is solved by the ordinary coupled-wave analysis (Appendix B) and the results are plotted in Fig. 3. Figure 3(a) shows the first instances of large reflection as modulation frequency progresses. A finer resolution of the $\Omega$-axis prompts even larger reflections. As illustrated by Fig. 3(b), at these points, the phase of $\det (\mathbf {M})$ undergoes an abrupt jump of magnitude $\pi$, implying that $\det (\mathbf {M})$ crosses zero. What is more, this feature is exclusive to $\omega _0 = \Omega /2$ (or more accurately $\omega _0 = \Omega /2+n\Omega$) which accords with our findings in Section 2, where we demonstrated that for a sinusoidal modulation, a time-periodic medium exhibits parametric amplification, or instability for that matter, only of the type characterized by $Re\{\omega \} = \Omega /2$.
Fig. 3. (a) Amplitude of $R_0$ versus modulation frequency for the problem of normal incidence with $\omega _0 = \Omega /2$ upon a time-periodic slab with relative permittivity $\varepsilon _r(t) = 5.25 + 0.087 \cos \Omega t$ and thickness $L=10$ cm. (b) Phase of the determinant of the coefficient matrix $\mathbf {M}$ for the same problem.
In the following, we confirm that the standard RCWA is not sufficient for the task of ascertaining the solution at the point of singularity. To this end, we find the least-squares solution to the singular system of equations, following the method detailed in Appendix D. We examine the first occurrence of singularity in Fig. 3, which is determined up to 13 decimal places of accuracy as $\Omega _1 = 266.9082012348271 \times 10^{9} \mathrm {rad/s}$. We compute the SVD of $\mathbf {M}$ for different accuracies of $\Omega _1$ ranging from 3 to 13 decimal places, and confirm that only one singular value approaches zero as the accuracy is increased. This is illustrated in Fig. 4(a). We then calculate the least-squares solution and evaluate the relative error $R.E.$. It is evident from Fig. 4(b) that the error does not vanish and hence, the singular equation does not have a solution.
Fig. 4. (a) The two smallest singular values of matrix $\mathbf {M}$ versus different accuracies of $\Omega _1 = 266.9082012348271 \times 10^{9} \mathrm {rad/s}$ for the problem of Fig. 3. (b) The relative error of the least-squares solution to Eq. (19a) versus different accuracies of $\Omega _1$.
Now we demonstrate that the augmented equations presented herein, do indeed allow us to fulfil the task we set ourselves. Once again, we attend to the first occurrence of singularity in Fig. 3 and carry on as before. Figure 5(a) shows that the matrix $\widetilde {\mathbf {M}}$ has only one singular value sufficiently close to zero. We therefore proceed as described in Appendix D to calculate the Moore-Penrose inverse of $\widetilde {\mathbf {M}}$, by means of which we find the minimum-norm least-squares solution to Eq. (18a). A cosinusoidal input of frequency $\Omega /2$ is assumed which is equivalent to $\mathbf {E}$ having only two components $E_0=E_{-1}=0.5$. The relative error is depicted in Fig. 5(b). Since $R.E.$ is vanishingly small, we come to the conclusion that the singular equation does indeed admit of a solution. To determine the rate of linear growth we refer to $R'_m$, displayed in Fig. 5(c). Also presented in Fig. 5 is a plot of the significant component of $\mathbf {R'}$ ($R'_0$) versus different accuracies of $\Omega _1$, showing clearly that the solution has converged. The results are verified numerically by Finite Element Time Domain (FETD) method via COMSOL Multiphysics v5.6. For two frequencies near $\Omega _1$, the solutions pertaining to the input signal of Fig. 6(a) are computed and compared against the analytical solution for $\Omega _1$ which suggests a linear growth in the form of $6.6\times 10^8 t\cos \frac {\Omega _1 t}{2}$. It is evident that as $\Omega$ approaches $\Omega _1$, the solution draws closer to the analytical solution at the point of singularity. Our method thus proves a valuable asset by means of which one can estimate the amount of time that it takes for a certain amplification to take effect.
Fig. 5. (a) The two smallest singular values of matrix $\widetilde {\mathbf {M}}$ versus different accuracies of $\Omega _1 = 266.9082012348271 \times 10^{9} \mathrm {rad/s}$ for the problem of Fig. 3. (b) the relative error of the least-squares solution to Eq. (18) versus different accuracies of $\Omega _1$ with $\mathbf {E}$ specified as having only two components $E_0=E_{-1}=0.5$. (c) The minimum-norm least-squares solution $\mathbf {R'}$ to Eq. (18). (d) The amplitude of the significant component of $\mathbf {R'}$ ($R'_0$) versus different accuracies of $\Omega _1$.
Fig. 6. (a) The input electric field (having a frequency of $\Omega /2$) for the numerical FETD simulations. (b) and (c) The total electric field for the specified $\Omega$s compared against the analytical solution of Fig. 5 pertaining to $\Omega _1 = 266.9082012348271 \times 10^{9} \mathrm {rad/s}$. $T = 2\pi /\Omega$.
4. Instability in time-periodic dielectric slabs
Now that we have solved the problem of normal incidence upon a time-periodic slab at the point of singularity, it is time we discussed the underlying physics of the singularity itself, since it is certain to have a bearing on the applicability of such structures. For a system that is not driven ($\mathbf {E}=\mathbf {0}$), it follows from $\mathbf {M}\mathbf {S'}(0)=\mathbf {0}$ that in the case of singularity, $\mathbf {S'}(0)$ will be nonzero given nonzero initial conditions. Thus $\mathbf {S'}(0)$ and thereupon $\mathbf {R}$ can be uniquely determined which in turn speaks of the existence of a self-sustained outward radiation with $\omega _0=\Omega /2$.
The sudden emergence of singularities in a time-periodic slab at a critical point $\Omega _c$ suggests that the phenomenon of parametric amplification associated with time-periodic media, which we identified in Section 2, provides means to counter the losses in the slab that ensue from radiating into the free space. Assuming all the other parameters are constant, the modulation frequency determines the rate at which energy increases in the slab. This amplification may, at a certain critical modulation frequency, balance out the radiative losses and thereby make for outward radiation with constant amplitude. It might further be the case that passing this critical frequency would bring about instability in the system which in turn would thwart all attempts at using the device as an amplifier, because the amplified signal would then be completely submerged in an exponentially growing natural mode. So we are obliged to ask ourselves the following question: exactly how does the state of our system evolve as the modulation frequency varies in a neighbourhood of $\Omega _c$.
Since the instability intervals (momentum gaps) in a sinusoidally modulated dielectric medium are all characterized by $Re\{\omega \}=\Omega /2$, we postulate that the unstable modes of a sinusoidally modulated slab possess the same property and we test our hypothesis as follows. For a given modulation frequency $\Omega$ near $\Omega _c$ we seek electromagnetic fields of the same form as that of the standard RCWA (Appendix B) but with $\omega _m=\Omega /2 - i\omega _i+m\Omega$ , that satisfy Maxwell’s equations when there is no incident wave. To this end we let $\omega _i$ vary in a certain range, we compute $\det (\mathbf {M})$ in Eq. (19a) for each $\omega _i$ and look for the value for which $\det (\mathbf {M})=0$. But let us not get ahead of ourselves; we first need to discuss the physical implications of introducing complex $\omega _m$ into the analysis. We rewrite the fields in the upper space as follows:
We dismiss the above sinusoidal terms since they alternate between positive and negative values and hence do not contribute to the overall flow of energy. The constant terms however are of great significance. On the grounds that a physical mode of slab is one that radiates outwards, the constant terms provide us with the condition
which we can use to determine $k_{z,m}$ uniquely. Now we can proceed with our approach. We analyze a numerical example and this time we opt for a stronger modulation, $\varepsilon _r(t) = 5.25 + 1 \cos \Omega t$. The problem of normal incidence upon a slab of thickness $L = 1$ cm is solved and the result is shown in Fig. 7. The aforementioned approach is applied to this problem and the results are verified numerically by Finite Element Time Domain method. The numerical simulations are carried out with very short Gaussian pulses as incident waves, which is roughly equivalent to specifying initial conditions and zeroing out input. It appears that in certain bounded intervals of $\Omega$ ($\Omega _1<\Omega <\Omega _2$), the time-periodic slab generates considerable amplification, overcoming radiative losses and giving rise to instability. Beyond the confines of such intervals however, $\omega _i$ is negative, meaning that the radiative losses prevail and thus the structure is stable. This observation contravenes the notion that the time-periodic slab is unstable for all $\Omega >\Omega _1$ [4,7].More importantly, parametric resonances with infinite quality factor (singularities) are not mere theoretical conceptions that are dispensed with by the slightest addition of material losses, as some would have it [6]. Singularities indicate the existence of instability intervals, within which the slab’s amplification exceeds radiative losses by some margin. Introducing material losses into the slab does not abruptly eliminate an instability interval but rather gradually narrows it. This goes on until the radiative and material losses have overpowered the amplification, in which case the instability interval will vanish. This is confirmed by Fig. 8 which results from allowing for a conduction current $\sigma \mathbf {E}$ in the Maxwell’s equations. To accommodate the resistance loss in our analysis, we have only to rewrite the matrix $\mathbf {A}$ in Eq. (15) as follows:
The rest of the treatment remains unchanged.
Fig. 7. (a) Amplitude of $R_0$ versus modulation frequency for the problem of normal incidence with $\omega _0 = \Omega /2$ upon a time-periodic slab with relative permittivity $\varepsilon _r(t) = 5.25 + 1 \cos \Omega t$ and thickness $L=1$ cm. (b) Plot of $1/|\det (\mathbf {M})|$ versus $\Omega$ and $\omega _i$ showing the evolution of the slab’s mode as modulation frequency varies in a neighbourhood of $\Omega _c$. (c), (d), and (e) Numerical solution to the above problem when the slab is illuminated by a short Gaussian pulse for the specified modulation frequencies.
Fig. 8. (a), (b), and (c) Plot of $1/|\det (\mathbf {M})|$ versus $\Omega$ and $\omega _i$ for a problem similar to that of Fig. 7, differing only in terms of the electrical conductivity specified in the slab.
Our findings shed light on the nature of parametric amplification in a time-periodic slab, which is as follows. While adjusting the frequency of modulation, certain intervals are reached where the time-periodic slab has exponentially growing natural modes. Within these intervals, the existence of a growing mode renders the device impractical for parametric amplification. At the boundaries of the instability intervals, an incident wave of frequency $\omega =(2n+1)\Omega /2$ coincides with the modes, resulting in a reflection that grows linearly in time (refer to Section 3). Approaching these boundaries occasions sinusoidal reflections of arbitrarily large amplitudes. To ensure that the device reaches steady state, ergo functioning as a viable parametric amplifier, one should see to it that the points of singularity are approached from outside the instability intervals.
5. Conclusion
We have looked into the problem of parametric amplification and instability as concerns time-periodic dielectric media, particularly, temporally modulated dielectric slab. We have presented an analytical method based on rigorous coupled wave analysis to resolve the problem of incidence upon a time-periodic slab at the points of singularity, which proved an efficient way to assess the performance of such amplifiers. We have also demonstrated the connection between amplification and instability and made corrections to previous misconceptions regarding the nature of instability in time-periodic slabs.
Appendix A: Theory of Hill’s equation
We had best begin by giving a detailed account of Floquet’s theorem [21]. For a piecewise continuous $W(t)$ periodic with period $T$, Eq. (7) has two linearly independent continuously differentiable solutions $\overline {\Psi }_1(t)$ and $\overline {\Psi }_2(t)$, referred to as normalized solutions and uniquely determined by the conditions
The Hill’s equation can then be characterized with the aid of normalized solutions by the following characteristic equation:
Now, if the roots $\rho _1$ and $\rho _2$ of the characteristic equation differ from one another, Eq. (7) has two linearly independent solutions of the form
where the characteristic exponent (or natural frequency) $\omega$ is defined—up to an integral multiple of $\Omega$—by $e^{i\omega T}=\rho _1=\rho _2^{-1}$ and $p_1(t)$ and $p_2(t)$ are periodic with period $T$. When $\rho _1=\rho _2=\rho$, the solutions obey which can easily be shown to correspond toThis special case happens when $\overline {\Psi }_1(T)+\overline {\Psi }_2'(T)=\pm 2$, yielding $\rho =\pm 1$. The constant $\alpha$ can be zero which constitutes an instance of coexistence, i.e., there are two periodic solutions of Eq. (7) with period $T$ ($\rho =1$ and $\omega =\Omega$) or $2T$ ($\rho =-1$ and $\omega =\Omega /2$).
The necessary means for identifying the momentum gaps of Fig. 2 is now at hand. We define the discriminant of Hill’s equation as
where $\overline {\Psi }(t, k)$ denotes the normalized solution of Eq. (7) for the specified $k$ and differentiation is carried out with respect to $t$. Since $W(t)$ in Eq. (7) is a real function for real values of $k$, the normalized solutions are bound to be real-valued when propagating modes are of interest, rendering Eq. (25) an equation with real coefficients. The momentum gaps can therefore be ascribed to $|\Delta (k)|> 2$; inasmuch as $|\Delta (k)|< 2$ entails the roots of Eq. (25) lying on the unit circle and consequently $\omega$ being real, which by definition cannot be the case in a momentum gap. Furthermore, $|\Delta (k)|> 2$ implies that the roots of Eq. (25) are real, an thus $\omega$ is complex. In those gaps where $\Delta (k)>2$, the roots are positive and hence $Re\{\omega \}=\Omega$. In the rest of the gaps where $\Delta (k)<-2$, the roots are negative and $Re\{\omega \}=\Omega /2$.To gain further insight into the nature of these momentum gaps and to ascertain their connection with one another and also with the problem of coexistence, we invoke the following theorem due to Liapounoff (Oscillation Theorem) [21,24]. We leave out the insignificant details, such as transforming Eq. (7) into the standard form by applying Liouville’s transformation, and get straight to the point. To Eq. (7), with positive $\varepsilon _r(t)$, there belong two monotonically increasing infinite sequence of real numbers,
and such that Eq. (7) has a solution of period $T$ ($\omega =\Omega$) if and only if $k=k_n, n\ge 0$, and a solution of period $2T$ ($\omega =\Omega /2$) if and only if $k=k'_n, n \ge 1$. In other words, these numbers are the roots of equation $|\Delta (k)|=2$. The $k_n$ and $k'_n$ satisfy the inequalitiesThe solutions of Eq. (7) for real $k$, are unstable only in the following closed intervals ($n \ge 1$):
Also, $k_n = k_{n+1}$ and $k'_n=k'_{n+1}$ are the only cases where coexistence happens ($\alpha =0$ in Eq. (28)).
Having dealt with the generalities of time-periodic systems, the next step would be to delve into the specifics of our modulation of choice; sinusoidal modulation. Upon multiplying both sides of Eq. (7) by $\varepsilon _r(t)$ and dividing by the mean of $\varepsilon _r(t)$, we arrive at the following form:
which falls under the category of Ince’s equations:Appendix B: Rigorous coupled-wave analysis
The wave equation inside the slab is
The following expressions represent the electromagnetic fields within the slab (denoted by the subscript slab) and the region above (denoted by the subscript up):
Now, since $z=0$ marks the position of PEC, vanishing of the electric field requires $S_m(0)=0$ for all $m$, hence $\mathbf {S}(0)=\mathbf {0}$. Thus, the fields’ components at the slab’s boundary ($z=L$) can be expressed as
By invoking electric and magnetic field boundary conditions we get
In order to reduce the complexity of solving these equations, we rewrite them as
We then put that in matrix form and use Eq. (48) to come by
Appendix C: Boundary conditions
We invoke the electric and magnetic field boundary conditions (at $z=L$) each of which produce two equations; one originates from equating the coefficients of the linearly growing terms in Eq. (13) at the opposite sides of the boundary and the other from equating those of the sinusoidal terms
We can rewrite the above equations as
The first two equations yield
Finally, by applying Eq. (17) to Eqs. (54) and (55) we arrive at Eq. (18).
Appendix D: Least-squares solution to a singular system of equations
It is well-known that when a system of linear equations
is singular, it either has no solution or has an infinite number of solutions. One way to find out which is the case, would be to obtain the least-squares solution to Eq. (56) and see if the relative error defined by vanishes. Here, we use Moore-Penrose inverse [30] to solve the least-squares problem. The Moore-Penrose inverse is the unique matrix $\mathbf {M}^+$ satisfyingWe further use the singular value decomposition [31] to determine $\mathbf {M}^+$ [32]. SVD of an square matrix $\mathbf {M}$ is a factorization of the form $\mathbf {M}=\mathbf {U}\mathbf {\Sigma }\mathbf {V}^*$ where $\mathbf {U}$ and $\mathbf {V}$ are square unitary matrices ($\mathbf {U}\mathbf {U}^*=\mathbf {V}\mathbf {V}^*=\mathbf {I}$) and $\mathbf {\Sigma }$ is a diagonal matrix whose elements $\sigma _{ii}$ are called the singular values of matrix $\mathbf {M}$. It follows from the definition of SVD, that a singular matrix has to have one or more zero singular values. In the specific case of our problem, $\mathbf {M}$ is numerically confirmed to have only one singular value sufficiently close to zero. We treat that singular value as zero and thereby obtain the following representation of $\mathbf {M}$:
It is the readily seen that
since it satisfies the conditions of Eq. (58). We then use Eq. (59) to come by the minimum-norm least-squares solution.Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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