Temporal analog of Bragg gratings

Stefanos Fr. Koufidis; Theodoros T. Koutserimpas; Martin W. McCall

doi:10.1364/OL.499359

Although the concept of time-varying media has been established for more than six decades, with Morgenthaler’s seminal publication [1] and subsequent works (see, e.g., [2]) having indeed covered several theoretical aspects, the development of artificial meta-media in recent years has renewed the interest of the scientific community in this idea. From gain in time-dependent media [3] to $\mathcal {PT}$-symmetric parametric amplifiers [4] and time-domain Fabry–Pérot etalons [5], the degree of freedom that time offers provides a prominent platform for achieving exotic light behavior. The usual treatment for examining such media involves the Floquet–Bloch theory, yielding analytic solutions that involve cumbersome hypergeometric functions.

The well-known coupled-wave theory approach was recently extended in [6] to media with a periodically time-modulated permittivity, and the resulting expressions were found to be in a remarkable agreement with the analytic solutions, albeit much simpler. Inspired by that work, in this Letter, for the first time to the authors’ knowledge, we combine such an approach with the Möbius transformation method [7] to investigate the temporal analog of Bragg gratings in different time-modulation scenarios. Moreover, our work contributes to the theoretical foundations of reverse-engineering problems [8,9].

For an instantaneously responding medium whose permittivity is a function of time but not of space, Maxwell’s macroscopic postulates are supplemented by the constitutive relations [10]

(1)$${\bf D} = \epsilon\left(t\right){\bf E} \quad {\rm and} \quad {\bf B}=\mu{\bf H} ,$$

where ${\bf E}$, ${\bf B}$ are the complex-valued phasors of the primitive electromagnetic fields and ${\bf D}$, ${\bf H}$ those of their associated stimulated excitation fields. Here, $\epsilon \left (t\right )=\epsilon _r\left (t\right )\epsilon _0$ and $\mu =\mu _r\mu _0$, with $\epsilon _r$, $\mu _r$ being the relative permittivity and permeability, respectively, and $\epsilon _0$, $\mu _0$ the corresponding free-space parameters. The relative permittivity is presumed to be a periodic function of time, $\epsilon _r\left (t+T_p\right )=\epsilon _r\left (t\right )$, with a temporal period $T_p>0$ and a temporal modulation frequency $\Omega = {2\pi }/{T_p}$, and the medium is deemed non-magnetic, i.e., $\mu _r=1$.

Combining the curl of Faraday’s law with Ampére–Maxwell’s equation and the constitutive relations of Eq. (1), we obtain

(2)$$\nabla^2{\bf D}=\epsilon\left(t\right)\mu\frac{\partial^2 {\bf D}}{\partial t^2} .$$

For a plane wave propagating along the $z$-direction, being polarized along, say, the $x$-axis, due to the separability of Maxwell’s equations [11], its electric excitation field can be decomposed as ${\bf D}\left (z,t;k\right )=D_s\left (z;k\right )D_t\left (t;k\right )\hat {\bf x}$. Here, $D_s$ and $D_t$ are the spatial and temporal parts of the field, respectively, and $\hat {\bf x}$ is the unit vector associated with the $x$-axis. Therefore, upon substitution of $D_s D_t$ into Eq. (2), via separation of the variables, we are led to two equations. In particular, for the spatial part of the field, we have the Helmholtz wave equation (cf. Eq. (6a) of [2]):

(3a)$$\frac{{\rm d}^2D_s\left(z\right)}{{\rm d} z^2}+k^2D_s\left(z\right)=0 ,$$

where $-k^2$ is the separation constant, with $k$ corresponding to the wavenumber within the medium. In the absence of sources, the solutions to Eq. (3a) are $D_s\left (z;k\right )=D_{s}^+e^{ikz}+D_{s}^-e^{-ikz}$, where $D_{s}^+$ and $D_{s}^-$ are $z$-independent (complex) amplitudes determined by the boundary conditions. Regarding the temporal part, we derive the Hill equation (cf. Eq. (6b) of [2]):

(3b)$$\frac{{\rm d}^2 D_t\left(t\right)}{{\rm d} t^2}+k^2\theta\left(t\right) D_t\left(t\right)=0 ,$$

where we have defined the auxiliary variable $\theta \left (t\right )=\epsilon ^{-1}\left (t\right )\mu ^{-1}$.

In ordinary spatially modulated dielectric media, such as Bragg gratings [12] or structurally chiral media [13], and in the context of coupled-wave theory, an incident wave with an amplitude $A^+$ generates a wave counter-propagating in space with an amplitude $A^-$. Dual to this, we may express the temporal part of the electric excitation field as [6]

(4)$$D_t\left(t\right)=D_t^+\left(t\right)e^{i\omega_{0} t}+D_t^-\left(t\right)e^{{-}i\omega_{0} t}\, ,$$

where negative time is interpreted as $+\omega _{0} t=-\left (-\omega _{0}\right )t$, i.e., as the generation of a wave with angular frequency $-\omega _{0}$, with $\omega _{0}$ being the unperturbed angular frequency. Unlike the static case, energy is not conserved, but momentum is (see, e.g., [11,14]). For weak modulation amplitudes, we may retain only the first two terms of the the Fourier expansion of $\theta$ and write

(5)$$\theta\left(t\right)=\sum_{n=0}^{+\infty}{\theta_n\cos{\left(n\Omega t\right)}}\approx\bar{\theta}+\delta\theta\frac{e^{i\Omega t}+e^{{-}i\Omega t}}{2},$$

where $\bar {\theta }={c_0^2}/{\epsilon _{\rm eff}}$ and $\delta \theta \approx -c_0^2\delta \epsilon /\bar {\epsilon }_{r}^{2}$. Here, $\epsilon _{\rm eff}={T}/{\int _{0}^{T} {{\rm d}\tau }/{\epsilon _{r}(\tau)}}$, $\bar {\epsilon }_{r}$ and $\delta \epsilon$ are the mean value and the modulation depth of the relative permittivity, respectively, and $c_0={1}/{\left (\epsilon _0\mu _0\right )^{1/2}}$.

Substituting the ansatz of Eq. (4) into Eq. (3b), if the time-dependence of the amplitudes is sufficiently mild, we can apply the slowly varying envelope approximation. This is reasonable provided that we are not concerned with pulse propagation, where, e.g., material dispersion is important (see Section IV of [15]). Phase-matching potentially synchronous terms yields

(6)$$\frac{{\rm d}}{{\rm d} t}\left(\begin{matrix}D_t^+\\ {{D}}_t^-\\\end{matrix}\right)=\left(\begin{matrix}0 & -i\kappa e^{{-}i2\delta\omega t}\\i\kappa e^{i2\delta\omega t} & 0\\\end{matrix}\right)\left(\begin{matrix}D_t^+\\{{D}}_t^-\\\end{matrix}\right),$$

where we have identified the key parameters

\delta\omega=\omega_{0}-\frac{\Omega}{2}\quad {\rm and} \quad \kappa=\frac{k^2}{4\omega_{0}}\delta\theta\triangleq\frac{\omega_{0}}{4}\frac{\delta\theta}{\bar{\theta}}\, ,

which express the detuning from the unperturbed frequency and the coupling coefficient, respectively (cf. Eq. (7) of [16]).

Although Eq. (6) deviates from the previously derived Eqs. (7) and (8) of [6], it is straightforward to show that via the transformation $\tilde {D_t}^{\pm }=D_{t}^{\pm }e^{\pm i\delta \omega t}$, the characteristic matrix ${\bf M}$ on the right-hand side of Eq. (6) is rendered constant and equivalent to that of [6]. Around the Bragg resonance $\omega _0\approx {\Omega }/{2}$, we have $\delta \omega \approx 0$, and by differentiating both rows of the transformed system, we find its eigenvalues. These lead to the dispersion relation

(7)$$\omega=\frac{\Omega}{2}+{\rm sign}\left(\delta\omega\right)\left[\left(\sqrt{\bar{\theta}}k-\frac{\Omega}{2}\right)^2-\left(\frac{k\delta\theta}{4c_0^2\sqrt{\bar{\theta}}}\right)^2\right]^{1/2} ,$$

where we emphasize the subtlety that the choice of sign in front of the square root is dictated by the necessity that $\omega \rightarrow 0$ as $k\rightarrow 0$ and that $\omega \rightarrow \sqrt {\bar {\theta }}k$ as $k\rightarrow +\infty$. Upon careful observation of Eq. (7), around ${\Omega }/{2}$, $\omega$ acquires an imaginary part in the regime $|\delta \omega |<|\kappa |$, where parametric amplification occurs (cf. ${\bf E} e^{-i\left (kz-\omega t\right )}$ for $\omega =\omega ^{\prime }-i\omega ^{\prime \prime }$ rather than $k=k^{\prime }-i k^{\prime \prime }$). Such energy-momentum dispersion is illustrated as a Brillouin diagram in Fig. 1, and should be compared with the dispersion of a uniform Bragg grating (seen, e.g., in Fig. 3 of [7]).

Fig. 1. Brillouin diagram for the dispersion of a medium in which the permittivity is a periodic functions of time, as per Eq. (7). The permittivity’s modulation depth is taken as $\delta \epsilon =0.1$.

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Let us now express the general solution to Eq. (6) as

{\bf D}_t\left(t\right) = {\bf T}\left(t\right)\cdot{\bf D}_t\left(t_0\right), \quad t\geq t_0,

where ${\bf D}_t=\left (D_t^+, D_t^-\right )^T$, with $T$ denoting transpose, and ${\bf T}$ is a transfer matrix relating the amplitudes of the electric excitation field at some time $t$ to those at an initial time, say, $t_0=0$. It follows immediately that ${\bf T}$ solves the equation

(8)$$\frac{{\rm d} {\bf T}\left(t\right)}{{\rm d} t}={\bf M}\left(t\right)\cdot{\bf T}\left(t\right)\,$$

and that ${\bf T}\left (0\right )=\mathbb {I}$, where $\mathbb {I}$ is the $2\times 2$ identity. Since ${\bf M}$ is traceless, Liouville’s formula implies that ${\rm det}\left [{\bf T}\left (t\right )\right ]=1\, , \forall \ t\geq 0$. Furthermore, for real parameters, ${\bf M}$ is also Hermitian; hence ${{\bf T}}^{\dagger }\cdot {\bf J}\cdot {{\bf T}}={\bf J}$, where ${\bf J}={\rm diag}\left (1, -1\right )$ and the dagger denotes Hermitian conjugation. Consequently, $I_0=|D_t^+|^2-|D_t^-|^2$ is constant, i.e., flux is conserved. Combining both the aforementioned conditions, we have that ${\bf T}\in SU\left (1,1\right )$, with the special unitary group having a well-known representation.

Without any assumption on whether $\delta \omega$ is time-dependent or not, we introduce a Möbius transformation $w:\mathbb {C}^2\rightarrow \mathbb {C}^2$ as

{\bf T}=\left(\begin{matrix}T_{11} & T_{12}\\{{ T}}_{21} & T_{22}\\\end{matrix}\right)\rightarrow w\left(w_0\right)=\frac{T_{11}\left(t\right)w_0+T_{12}\left(t\right)}{T_{21}\left(t\right)w_0+T_{22}\left(t\right)},

whereby considering Eq. (8), the variable $w$ evolves as

(9)$$\frac{{\rm d} w\left(t\right)}{{\rm d} t}={-}i\kappa e^{i2\delta\omega t}w^2\left(t\right)-i\kappa e^{{-}i2\delta\omega t},$$

which is a complex Riccati equation (cf. Eq. (5) of [8]).

As implied by the $SU\left (1,1\right )$ symmetry, restricting the transformation’s action to the unit circle $|w_0|=1$, by setting $w=e^{i\psi }$, $\psi \in \mathbb {R}$, Eq. (9) reduces to a first-order nonlinear differential equation of a single real variable, encoding all information

(10)$$\frac{{\rm d} \psi}{{\rm d} t}={-}2\kappa \cos{\left(\psi+2\delta\omega t\right)} .$$

We note that if we were to include an additional phase term $\phi _0$ in the permittivity’s modulation profile, the argument of Eq. (10) would then become $\psi +2\delta \omega t-\phi _0$, thus describing, for instance, even functions when $\phi _0={\pi }/{2}$. The Möbius method is indeed successful for arbitrarily chosen phase profiles [7].

For $\psi ^\prime \rightarrow \psi +2\delta \omega t$, it is readily seen that Eq. (10) is the Adler equation of coupled-oscillator theory (cf. Eq. (21) of [17] for $\lambda _{\rm Adl}={\delta \omega }/{\kappa }$), with its stability condition $|\lambda _{\rm Adl}|\leq 1$ corroborating the abovementioned parametric amplification regime. If $\delta \omega$ is constant, Eq. (10) can be solved analytically, admitting solutions that can be cast in terms of a transfer matrix as

(11)$$\left(\begin{matrix}D_{t}^+\\ {{ D}}_{t}^-\end{matrix}\right)_{t}=\left(\begin{matrix}e^{- i\delta\omega \left(t-t_0\right)}p^+ & e^{- i\delta\omega \left(t+t_0\right)}q^+\\{{ e}}^{ i\delta\omega \left(t+t_0\right)}q^- & e^{ i\delta\omega \left(t-t_0\right)}p^-\end{matrix}\right)\left(\begin{matrix}D_{t}^+\\{{ D}}_{t}^-\end{matrix}\right)_{t_0},$$

where $p^\pm =\cosh {\left [\Delta \left (t-t_0\right )\right ]}\pm i\left ({ \delta \omega }/{\Delta }\right )\sinh {\left [\Delta \left (t-t_0\right )\right ]}$ and $q^\pm =\pm \left ({ i\kappa }/{\Delta }\right )\sinh {\left [\Delta \left (t-t_0\right )\right ]}$, with $\Delta ^2=\kappa ^2-\delta \omega ^2$. In our analysis, however, we shall follow the less restrictive method outlined in Section IIIC of [7], but with the local conductance at $t_0$ being now infinite. Numerical integration of Eq. (10) leads to the complete identification of both the amplitude and phase of all elements of the transfer matrix ${\bf T}$ via Eq. (16) of [7].

We may now consider a geometry similar to that depicted in Fig. 1 of [18] but with a periodic modulation of the refractive index of the temporal slab, in lieu of the rectangular function considered in [18]. Specifically, we will examine the scenario in which the relative permittivity of a regular dielectric medium, characterized by $\epsilon$, $\mu$, starts at $t_0=0$ to vary as per Eq. (5) up until a moment $t$, after which it returns to its pre-modulation values. The temporal slab’s time duration is then $\Delta t = t-t_0$, and by applying the obvious initial condition $D_{t}^{-}\left (0\right )=0$ [19], we may define the intensity remittances as

(12)$$R = \left|T_{21}\right|^2\quad {\rm and} \quad T = \left|T_{11}\right|^2 .$$

Hitherto, the mapping of the temporal coupled-wave theory to its spatial counterpart was “one-to-one,” with the crucial difference being the imposition of a different initial condition that resulted in the transfer matrix parameters becoming scattering coefficients. Therefore, henceforth, we shall refer to the intensity reflectance $R$ and transmittance $T$ of Eq. (12) as the backward (respectively, forward) reflectance for impedance-matched media (see [19]). However, the latter condition requires additional modulation of the medium for $t\in \left (-\infty,0^-\right )\cup \left (t^+,+\infty \right )$. Nevertheless, we can employ the standard calculation from thin-film theory, as prescribed in [20]. In fact, setting the time-averaged refractive index of the temporal slab to $\bar {n}=\left (\epsilon _0\mu _0\right )^{1/2}$, and via Eqs. (20) to (26) of [20], we can match the fields at both interfaces as ${\bf D}_t^{\pm }\left (0^{-},t^{-}\right )={\bf D}_t^{\pm }\left (0,t\right )$ [21]. However, for small values of $\delta \epsilon$, the temporal slab approximately achieves impedance- (and index-) matching with the “surrounding” medium, and hence the expressions of Eq. (12) can be safely regarded as the total intensity remittances. Such an approximation will not, of course, be valid if we set $\delta \epsilon =0$ and abruptly change, at $t=t_0$, the relative permittivity of the temporal medium while keeping it constant for times outside the duration of the slab (see, e.g., [5,18]).

The electromagnetic response of the considered medium is illustrated in Fig. 2 for a time duration $\Delta t=20T_p$, long enough so that the amplification process is enacted (cf. Fig. 5 of [19]). Apparently, for $\Delta t \in \left (t_0, t_0+2T_p\right )$, the duration is not sufficient for the always-in-phase waves inside the temporal medium to extract the required energy, pumped by an idler, and get amplified. This is consistent with the condition of [4], where the modulation frequency is required to be twice of that associated with the incident lightwave signal. The behavior of the photonic bandgap resembles that of regular Bragg gratings, i.e., increasing the modulation depth augments its width. However, by contrast to these, increasing the duration does not force the resonance to become steeper, as the temporal slab acts as an active medium rather than as a platform in which successive reflections from an alternating refractive index build-up coherently in phase. Additionally, as proven in [22], Fig. 2 shows the energy pseudo-conservation relation $T-R=1$, which is valid independently of the modulation profile [8], being similar to that of $\mathcal {PT}$-symmetric systems [23] and time-periodic systems [4].

Fig. 2. Optical response of a finite “time-slab,” with a permittivity modulation profile as seen in the inset. The modulation depth is that of Fig. 1, and the slab’s time duration is set to $\Delta t=20T_p$.

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Having not confined our analysis to constant detuning/coupling coefficient, the Möbius method provides access to more sophisticated modulation scenarios, e.g., chirping and apodization. Regarding the former, for, say, linear chirping, we may alter the modulation frequency to $\Omega \left (t\right ) =\Omega _0+{\rm d}\Omega$, where $\Omega _0\approx 2\omega _0$ and ${\rm d}\Omega =\left (2F/\Delta t^2\right )t$, with $F$ being the chirping coefficient. Then, upon substitution of $\delta \omega \left (t\right )=\omega _0-{\Omega \left (t\right )}/{2}$ into Eq. (10), numerical integration leads to the spectra seen in Fig. 3, plotted as functions of the normalized detuning, for various values of $F$. Manifestly, the farther away we move from the central period $T_p$, the more the effective time duration is reduced, thus giving the waves less time to get amplified. Nonetheless, if abundant amplification is not the sole concern and broader bandwidths are required, linear chirping provides amplification over a wider wavelength range than linear detuning.

Fig. 3. Forward reflectances of the “time-slab” in the presence of linear chirping. The rest of the scenario is that of Fig. 2.

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Although linear chirping increases the amplification bandwidth, it also generates significant side-lobes to the optical spectrum. An effective method to suppress them is to consider apodization. This requires the coupling coefficient to become time-dependent, which appears to be quite an ambitious task from an implementation point of view. We contend that it is not, as it is natural for the idler, modulating the permittivity, to bequeath its spatial and temporal characteristics to $\epsilon _r\left (t\right )$. For a Gaussian idler beam, the coupling coefficient will be $\kappa \left (t\right )=\kappa e^{\left [-\left ({4}/{\Delta t}\right )^2\left (t-{\Delta t}/{2}\right )^2\right ]}$. Such a coupling profile may be combined with linear chirping, but even alone, linear apodization yields almost linear phase response. The optical spectrum is seen in Fig. 4, wherein it is obvious that the gradual reduction of the modes’ coupling toward the two time-edges of the temporal medium reduces its amplifying abilities. Still, we can compensate such an amplification decrement by extending $\Delta t$.

Fig. 4. Forward reflectances of a linearly chirped and linearly apodized “time-slab.” Here, $\Delta t=40T_p$ and $\delta \epsilon =0.1$.

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Finally, we may examine a scheme in which the detuning parameter is discontinuous. In fact, we will consider the case where at some moment $t_{d}\in \left (0,\Delta t\right )$, the period of the relative permittivity slightly changes, possibly due to a synchronization failure. Then, for $t\in \left (0,t_{d}\right )$, we will have a temporal period $T_{1}$, whereas for $t\in \left (t_{d},t\right )$, a temporal period $T_{2}$. We will further assume that $T_{1}$, $T_{2}$ are slightly different, satisfying the condition ${1}/{T_1}-{1}/{T_2}={1}/{\Delta t}$, i.e., the temporal analog of the spectral hole condition of Eq. (20) in [12]. As seen in Fig. 5, by cascading two matrices as per Eq. (11), with slightly different detunings, a spectral hole also occurs here, around which the phase discontinuity leads to destructive interference [19]. Such a phenomenon is typically used in sensing applications and in Mach–Zehnder add-drop multiplexers. Our proposed “time-domain” version may prove valuable for measuring the accuracy of the permittivity’s modulation, in time-directional couplers or for real-time optical modulation.

Fig. 5. Optical response of two cascaded “time-slabs” of equal time durations $\Delta t=20T_p$ and slightly dissimilar periods: ${1}/{T_1}-{1}/{T_2}={1}/{\Delta t}$. The situation is equivalent to having one slab of $\Delta t=40T_p$ with a ${\pi }/{2}$ phase delineation at $t_d=20T_p$.

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In this communication, we synthesized coupled-wave theory with the Möbius transformation method to investigate the dispersion and optical response of a finite “time-slab” in which the permittivity is a periodic function of time. We demonstrated the fundamental differences between temporal and spatial Bragg gratings and provided insights into various scenarios of time modulation. A direction that should be further pursued is the incorporation of material dispersion through the treatment of [24] and the exploration of reverse-engineering algorithms, such as layer-peeling, for the reconstruction of tailor-made pulses.

Funding

Bodossaki Foundation; Schweizerischer Nationalfonds zur Förderung der Wissenschaftlichen Forschung (203176).

Disclosures

The authors declare no conflicts of interest.

Data availability

The research data of the presented work are generated by the analysis that is derived and the direct application of the associated equations.

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Temporal analog of Bragg gratings

Abstract

Funding

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Data availability

REFERENCES

Data availability

Cited By

Figures (5)

Equations (16)

Optics Letters

Stefanos Fr. Koufidis	https://orcid.org/0000-0002-2888-7809
Theodoros T. Koutserimpas	https://orcid.org/0000-0003-1288-781X
Martin W. McCall	https://orcid.org/0000-0003-0643-7169