Temporal optical activity and chiral time-interfaces [Invited]

Shixiong Yin; Shixiong Yin; Yao-Ting Wang; Andrea Alù; Andrea Alù; Andrea Alù

doi:10.1364/OE.480199

1. Introduction

Time modulation has been recently harnessed as a new degree of freedom in metamaterials and photonics, leveraging non-trivial light-matter interactions emerging in the presence of time-varying optical properties of materials [1–3]. A variety of intriguing wave phenomena have been unveiled through periodic temporal modulations, such as nonreciprocity [4–6], surface-wave control [7,8] and parametric wave-mixing [9–11]. When the timescale of the change in optical properties becomes comparable to the period of the wave, the involved signals can experience at a temporal interface, dual to a spatial boundary. Such time-interfaces — at which the property of the entire medium change abruptly in time and uniformly in space [12,13] — have attracted broad interest from various wave-related communities. Unlike the scenarios involving periodic modulations, numerous promising applications have been emerging that leverage time-interfaces, due to their intrinsically broadband nature, including time-reversal [14], frequency conversion [15,16], as well as impedance matching and transformations [17–20] and absorption of short pulses [21]. While most of the work on time interfaces has been focused on lossless, isotropic and nondispersive media, even more exotic scattering features arise at time-interfaces when considering frequency dispersion [22–26], loss and gain [27,28], anisotropy [29–33], and nonreciprocity [34].

Here, we explore wave scattering at a time-interface in the presence of electromagnetic chirality. A chiral medium features circular birefringence, in which the left-handed circularly polarized (LCP) light and right-handed circularly polarized (RCP) light experience different refractive indices [35]. As a result, a linearly polarized (LP) wave traveling through a chiral medium is known to undergo polarization rotation in space, which is known as optical activity and obeys time-reversal symmetry [36]. A dual phenomenon in time may be envisioned at a time-interface in which optical chirality is switched on. If a host non-chiral medium supporting a LP wave is suddenly switched in time to become chiral, the wave is expected to split into a set of LCP and RCP waves traveling with different phase velocities, and thus experience polarization rotation as time advances. Different from the spatial scenario, here momentum is conserved, while the frequencies of the time-reflected and time-refracted waves are different. In this article, we formally investigate such circularly birefringent scattering phenomenon at chiral time-interfaces, and showcase the temporal analog of optical activity. Dual to its spatial counterpart, temporal optical activity is time-reciprocal, in contrast with temporal Faraday rotation studied in [34]. We also discuss the role of material dispersion at chiral time-interfaces, and derive the temporal boundary conditions in this scenario. Recent investigations on chiral time-interfaces have unveiled their role for spin-dependent temporal scattering [37] and, combined with the present work, they may provide a designer perspective for time-metamaterials with added functionalities.

2. Non-dispersive chiral time-interfaces

We consider a monochromatic plane wave of frequency ${\omega _{\textrm {inc}}}$ propagating in a non-dispersive, isotropic and homogeneous dielectric along the $+ z$ direction (with a ${e^{ - i{\omega _{\textrm{inc}}}t}}$ notation). We assume that the host medium features a refractive index ${n_1} = \sqrt {{\varepsilon _1}{\mu _1}} $ and wave impedance ${\eta _1} = \sqrt {{\mu _0}{\mu _1}/({\varepsilon _1}{\varepsilon _0})} $, where ${\varepsilon _0}$ (${\varepsilon _1}$) and ${\mu _0}$ (${\mu _1}$) are the vacuum (relative) permittivity and permeability, respectively. At $t = 0$ the entire medium is assumed to abruptly acquire electromagnetic chirality, characterized by a real-valued and nondispersive chirality coefficient $\kappa $, and establishing a chiral time-interface [37]. Experimentally, this may be implemented by suddenly turning a degree of magneto-electric coupling in each unit cell of a metamaterial using time-varying media [2] or suitably engineered transmission-line metamaterials [38]. The chirality at $t > 0$ can be modeled with the constitutive relations [35]

(1)$$\left[ {\begin{array}{c} {{\varepsilon_0}^{ - 1}\mathbf{D}}\\ {c\mathbf{B}} \end{array}} \right] = \left[ {\begin{array}{cc} {{\varepsilon_r}}&{i\kappa }\\ { - i\kappa }&{{\mu_r}} \end{array}} \right]\left[ {\begin{array}{c} \mathbf{E}\\ {{\eta_0}\mathbf{H}} \end{array}} \right], $$

relating the electric displacement $\mathbf{D}$ and magnetic induction $\mathbf{B}$ to the electric $\mathbf{E}$ and magnetic $\mathbf{H}$ fields. All field quantities are assumed to generally have non-zero x- any y-components. $c = 1/\sqrt {{\varepsilon _0}{\mu _0}} $ is the speed of light and ${\eta _0} = \sqrt {{\mu _0}/{\varepsilon _0}} $ is the wave impedance in free space. After the time-interface, the chiral medium described by Eq. (1) supports two circularly polarized eigenstates ${\psi _ \pm } = {[1,\textrm{ } \pm i]^T}/\sqrt 2 $ (normalized Jones vectors), as illustrated in Fig. 1(a) by the red and blue circular arrows with opposite handedness and different phase velocities, associated with the circular bi-refringence of the material. The associated effective refractive indices are ${n_ \pm } = n \pm \kappa $ where $n = \sqrt {{\varepsilon _r}{\mu _r}} $. For an incident wave [star symbol in Fig. 1(b)] undergoing through such a chiral time-interface at $t = 0$, the wavenumber ${k_{\textrm{inc}}}$ is conserved due to spatial symmetry, as illustrated by the vertical bar at $k/{k_{\textrm{inc}}} = 1$ in the dispersion diagram in Fig. 1(b). As a result, the circular birefringence is manifested in frequency: the incident monochromatic light is now scattered onto multiple frequencies ${\omega _ \pm } = c{k_{\textrm{inc}}}/{n_ \pm }$, dual to the spatial birefringence where the frequency is unaltered yet different wavenumbers are excited. Negative frequencies $- {\omega _ \pm }$ are also generally excited, corresponding to time-reflected waves possessing the same handedness when defined for the propagation direction flipped in space.

Fig. 1. (a) A time-interface between a dielectric and a chiral medium. An incoming wave experiences a refractive index ${n_1}$ before the time-interface at $t = 0$. It then splits into two circularly polarized waves that experience distinct effective refractive indices ${n_ + }$ and ${n_ - }$. (b) Dispersion diagram of a dielectric (dashed line) and a chiral medium with circular birefringence (red and blue curves). The vertical bar in cyan represents the scattering at the chiral time-interface in (a).

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To determine the temporal scattering coefficients, dual to the scenario of spatial circular birefringence, we first project the total fields at $t = {0^ - }$ into the eigenstates by applying the wavefield postulates [35] ${\mathbf{E}_ \pm } = (\mathbf{E} \pm i{\eta _2}\mathbf{H}) \propto {\psi _ \pm }$ and ${\mathbf{H}_ \pm } = \mathbf{H} \pm \mathbf{E}/(i{\eta _2})$ with ${\eta _2} = {\eta _0}\sqrt {{\mu _r}/{\varepsilon _r}} $. Now the decoupled LCP ($+ $) and RCP ($- $) waves effectively propagate in media with effective relative permittivity ${\varepsilon _ \pm } = {\varepsilon _r}(1 \pm \kappa /n)$ and permeability ${\mu _ \pm } = {\mu _r}(1 \pm \kappa /n)$, yet with the same wave impedance ${\eta _ \pm } = {\eta _0}\sqrt {{\mu _ \pm }/{\varepsilon _ \pm }} = {\eta _2}$. The associated ${\mathbf{D}_ \pm } = {\varepsilon _0}{\varepsilon _ \pm }{\mathbf{E}_ \pm }$ and ${\mathbf{B}_ \pm } = {\mu _0}{\mu _ \pm }{\mathbf{H}_ \pm }$ across the time-interface must be continuous, as required by the temporal boundary conditions [12] and the orthogonality between the two eigenmodes, which implies

(2)$$\begin{array}{c} 1 = {{\rm T}_{D, \pm }} + {\Gamma _{D, \pm }},\\ {\eta _1} = {\eta _2}({{{\rm T}_{D, \pm }} - {\Gamma _{D, \pm }}} ), \end{array}$$

where ${{\rm T}_{D, \pm }}$ and ${\Gamma _{D, \pm }}$ are the temporal transmission and reflection coefficients defined with respect to the amplitude of the electric displacement fields ${\mathbf{D}_ \pm }$. We then obtain

(3)$${{\rm T}_{D, \pm }} = \frac{{{\eta _2} + {\eta _1}}}{{2{\eta _2}}}\textrm{ and }{\Gamma _{D, \pm }} = \frac{{{\eta _2} - {\eta _1}}}{{2{\eta _2}}}, $$

which are identical to the temporal scattering coefficients at an achiral time-interface [2], and independent of the polarization, because the chiral medium is reciprocal and both LCP and RCP light waves see the same impedance ${\eta _2}$. The scattering coefficients for the electric fields ${\mathbf{E}_ \pm }$, however, are different for the two modes as a result of birefringence:

(4)$${{\rm T}_ \pm } = \frac{1}{2}\left( {\frac{{{\varepsilon_1}}}{{{\varepsilon_ \pm }}} + \frac{{{n_1}}}{{{n_ \pm }}}} \right)\textrm{ and }{\Gamma _ \pm } = \frac{1}{2}\left( {\frac{{{\varepsilon_1}}}{{{\varepsilon_ \pm }}} - \frac{{{n_1}}}{{{n_ \pm }}}} \right). $$

For an arbitrarily polarized incident wave ${\mathbf{E}_{\textrm{inc}}} = {a_ + }{\psi _ + } + {a_ - }{\psi _ - }$, the energy balance across this chiral time-interface is simply the sum of the one for the two orthogonal polarizations ${\psi _ \pm }$:

(5)$$\frac{{U(t = {0^ + })}}{{U(t = {0^ - })}} = \frac{{{{|{{a_ + }} |}^2}}}{{{{|{{\mathbf{E}_{\textrm{inc}}}} |}^2}}}\frac{{{\varepsilon _1}}}{{{\varepsilon _ + }}} + \frac{{{{|{{a_ - }} |}^2}}}{{{{|{{\mathbf{E}_{\textrm{inc}}}} |}^2}}}\frac{{{\varepsilon _1}}}{{{\varepsilon _ - }}}. $$

3. Temporal optical activity

3.1 Reciprocal polarization rotation through a chiral temporal slab

Chiral media act as polarization rotators through a phenomenon known as optical activity. Here we show that a temporal chiral slab, defined by two consecutive time-interfaces in which the entire medium becomes chiral with $\kappa \ne 0$ for a finite time interval of duration $\Delta t$, can also rotate a LP wave, dual to chiral slab in space. For simplicity, we assume ${\varepsilon _r} = {\varepsilon _1}$ and ${\mu _r} = {\mu _1}$ across all time instants, while the electromagnetic chirality is turned on abruptly at $t = 0$ and then suddenly turned off at $t = \Delta t$, forming a chiral temporal slab of length $t = 0\sim \Delta t$. In this scenario, the wave impedance is matched for both LCP and RCP individually at both time-interfaces [37]. Hence, the temporal reflection coefficients are zero and only positive frequencies are excited, with transmission coefficients

(6)$${{\rm T}_ \pm } = \frac{n}{{n \pm \kappa }}\textrm{ and }{{\rm T}_ \pm }^\prime = \frac{{n \pm \kappa }}{n}, $$

where ${{\rm T}_ \pm }^\prime $ is the one at the second time-interface as the chirality is switched off.

We consider a LP wave propagating along $+ z$ direction (left-incidence) with polarization forming an angle $\theta$ with the x-axis. The corresponding Jones vector can be written as

(7)$${\mathbf{E}_{l,\textrm{inc}}} = \left[ {\begin{array}{c} {\cos \theta }\\ {\textrm{sin}\theta } \end{array}} \right] = \frac{1}{{\sqrt 2 }}({{e^{ - i\theta }}{\psi_ + } + {e^{i\theta }}{\psi_ - }} ). $$

Due to orthogonality, the LCP (RCP) component of the incident wave conserve the same handedness at the time interface, while changing its frequency ${\omega _ \pm }$ inside the temporal slab because of the change in effective refractive index. The total transmission coefficient of such a reflection-less chiral temporal slab is ${{\rm T}_ \pm }^\prime {e^{ - i{\omega _ \pm }\Delta t}}{{\rm T}_ \pm } = {e^{ - i{\omega _ \pm }\Delta t}}$. Thus, the output Jones vector at $t = \Delta {t^ + }$ is given by

(8)$$\begin{aligned} {\mathbf{E}_{r,\textrm{out}}} &= {e^{ - i{\omega _ + }\Delta t}}\frac{{{e^{ - i\theta }}}}{{\sqrt 2 }}{\psi _ + } + {e^{ - i{\omega _ - }\Delta t}}\frac{{{e^{i\theta }}}}{{\sqrt 2 }}{\psi _ - }\\ &= {e^{ - i\bar{\phi }}}\left[ {\begin{array}{c} {\cos ({\theta - \Delta \phi } )}\\ {\sin ({\theta - \Delta \phi } )} \end{array}} \right], \end{aligned}$$

where $\bar{\phi } = ({\omega _ - } + {\omega _ + })\Delta t/2$ and $\Delta \phi = ({\omega _ - } - {\omega _ + })\Delta t/2$. This result suggests that the polarization of the incident wave is rotated by the angle $\Delta \phi $, which is half of the difference of phase accumulation between the two modes during the temporal slab.

In [34] we studied the related problem of a time-interface involving a magnetically biased medium, which also supports the emergence of chiral birefringence and polarization rotation. A crucial difference between the two scenarios is that at a chiral time interface reciprocity holds, as expected due to the absence of broken time-reversal symmetry. To verify this statement, we consider an incident wave polarized at the same angle $(\theta - \Delta \phi )$ as in Eq. (8), yet propagating along the $- z$ direction following ${e^{i{\omega _{\textrm{inc}}}t}}{e^{i{k_{\textrm{inc}}}z}}$. We can similarly decompose the incident wave into the two orthogonal polarizations as

(9)$${\mathbf{E}_{r,\textrm{inc}}} = \left[ {\begin{array}{c} {\cos ({\theta - \Delta \phi } )}\\ {\textrm{sin}({\theta - \Delta \phi } )} \end{array}} \right] = \frac{{{e^{i(\theta - \Delta \phi )}}}}{{\sqrt 2 }}\psi _ + ^\ast{+} \frac{{{e^{ - i(\theta - \Delta \phi )}}}}{{\sqrt 2 }}\psi _ - ^\ast . $$

Note that LCP and RCP modes are now denoted by conjugated Jones vectors $\psi _ \pm ^\ast{=} {\psi _ \mp }$ due to the opposite propagation direction. The output polarization at $t = \Delta {t^ + }$ undergoing the same chiral temporal slab becomes

(10)$$\begin{aligned} {\mathbf{E}_{l,\textrm{out}}} &= {e^{i{\omega _ + }\Delta t}}\frac{{{e^{i(\theta - \Delta \phi )}}}}{{\sqrt 2 }}{\psi _ - } + {e^{i{\omega _ - }\Delta t}}\frac{{{e^{ - i(\theta - \Delta \phi )}}}}{{\sqrt 2 }}{\psi _ + }\\ &= {e^{i\bar{\phi }}}\left[ {\begin{array}{c} {\cos \theta }\\ {\sin \theta } \end{array}} \right], \end{aligned}$$

which rotates back to the polarization angle $\theta $ in the previous scenario, confirming the temporal reciprocity of the chiral slab. By choosing $\theta = \{ 0,\pi /2\} $ in Eqs. (7), (8), and then $\theta = \{ \Delta \phi ,\Delta \phi + \pi /2\} $ in Eqs. (9), (10), we can then define the temporal transmission matrix relating ${\mathbf{E}_{r,\textrm{out}}} = {\mathbf{T}_{l \to r}}{\mathbf{E}_{l,\textrm{out}}}$ and ${\mathbf{E}_{l,\textrm{out}}} = {\mathbf{T}_{r \to l}}{\mathbf{E}_{r,\textrm{out}}}$, which reads

(11)$${\mathbf{T}_{l \to r}} = {e^{ - i\bar{\phi }}}R({\Delta \phi } )\textrm{ and }{\mathbf{T}_{r \to l}} = {e^{i\bar{\phi }}}R({ - \Delta \phi } ), $$

where $R(\Delta \phi ) = [\cos \Delta \phi ,\textrm{ } - \sin \Delta \phi ;\textrm{ }\cos \Delta \phi ,\textrm{ }\sin \Delta \phi ]$ is the rotation matrix. We can verify that ${\mathbf{T}_{l \to r}} = \mathbf{T}_{r \to l}^\dagger$, with $\dagger$ denoting conjugate transpose, which obeys the reciprocity condition at time-interfaces defined in Ref. [34].

3.2 Numerical example

In analogy to the definition of spatial rotatory power of a chiral medium, we define the temporal rotatory power measuring the rotation angle per unit time

(12)$$\rho = \frac{{\Delta \phi }}{{\Delta t}} = \frac{\pi }{{{T_{\textrm{inc}}}}}({\hat{n}_ -^{ - 1} - \hat{n}_ +^{ - 1}} ), $$

where ${T_{\textrm{inc}}} = 2\pi /{\omega _{\textrm{inc}}}$ is the oscillation period of the incident wave and ${\hat{n}_ \pm } = {n_ \pm }/n$. We are now ready to explore polarization rotation by tailoring the rotatory power and the duration of the chiral temporal slab. As a numerical demonstration, we assume ${\varepsilon _1} = {\varepsilon _r} = {\mu _1} = {\mu _r}$ and $\kappa /n = 0.5$ with negnigible frequency dispersion (${\omega _{\textrm{inc}}},{\omega _ \pm }$ are all far from the material resonances). The normalized effective indices of the chiral medium are ${\hat{n}_ + } = 1.5$ for LCP waves and ${\hat{n}_ - } = 0.5$ for RCP, exhibiting temporal rotatory power $\rho = 4\pi /(3{T_{\textrm{inc}}})$. If we want to convert an x-polarized ($\theta = 0$) incident light to $45^\circ $-polarization (with respect to $+ x$-axis) through such a chiral slab in the time domain, the required duration $\Delta t = 3(4m - 1)/16{T_{\textrm{inc}}}$, where $m = 1,2,3\ldots $, corresponding to the rotation angle $\Delta \phi = m\pi - \pi /4$. In the following FDTD simulations, we choose the minimum duration $\Delta t = 9/16{T_{\textrm{inc}}}$ to form a temporal $45^\circ $-polarizer, as shaded by the violet boxes in Fig. 2. For a left-incident x-polarized plane wave ${\mathbf{E}_{l,\textrm{inc}}} = {[0,\textrm{ }1]^T}$, as illustrated in the blue plane in Fig. 2(a), its polarization undergoes a rotation $\Delta \phi = 135^\circ $ after the chiral temporal slab, and becomes $45^\circ $-polarized (yellow plane) with Jones vector ${\mathbf{E}_{r,\textrm{out}}} \propto {[1,\textrm{ }1]^T}/\sqrt 2$, as predicted by Eq. (8). To validate the reciprocity of temporal optical activity, we then launch a $45^\circ $-polarized wave ${\mathbf{E}_{r,\textrm{inc}}} = {[1,\textrm{ }1]^T}$ propagating along the $- z$-direction. The simulated temporal evolution of the electric field is plotted in Fig. 2(b), showing that the output wave indeed goes back to the x-polarization after the same temporal slab, as predicted by Eq. (10).

Fig. 2. FDTD simulations for temporal optical activity. (a) Simulated time evolution of a monochromatic plane wave propagating along $+ z$ and undergoing a chiral temporal slab (violet box). The light evolves in time from x-polarization to $45^\circ $ with respect to the x-axis. (b) A $45^\circ $-polarized light propagating along $- z$ direction is converted back to x-polarization after the same chiral temporal slab as in (a).

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4. Chiral time-interfaces with Lorentz dispersion

So far our discussions have been assuming negligible dispersion for the involved chiral media, which is a good approximation when all frequencies are far from the material resonances [22,37] and spatial dispersion is also negligible [39]. However, in pursuit of extreme wave phenomena determined by strong chirality, e.g., chiral nihility and negative refraction [40–42], dispersion plays an important role in regulating electromagnetic energy and imposing causality. In this section, we investigate wave scattering at chiral time-interfaces in the presence of material dispersion.

We assume a single-resonance Lorentz dispersion for both permittivity and permeability of the chiral medium

(13)$${\varepsilon _r}(\omega ) = 1 + \frac{{\omega _e^2}}{{\omega _0^2 - {\omega ^2} - i\gamma \omega }}\textrm{ and }{\mu _r}(\omega ) = 1 + \frac{{\omega _m^2}}{{\omega _0^2 - {\omega ^2} - i\gamma \omega }}, $$

where ${\omega _0}$ is the resonant frequency, $\gamma $ is the collision rate and ${\omega _{e,m}}$ are the plasma frequencies for the electric and magnetic response, respectively. The chirality parameter is assumed to follow the Condon model, where the same dominant resonant frequency as in Eq. (13) is assumed [35],

(14)$$\kappa (\omega ) = \frac{{\tau \omega _0^2\omega }}{{\omega _0^2 - {\omega ^2} - i\gamma \omega }}, $$

where $\tau $ is the characteristic time constant describing the magnitude of chirality. Upon a time-interface at $t = 0$ when the entire medium is switched to this dispersive chiral medium described by Eqs. (13) and (14), an incident wave in the general form ${\mathbf{E}_{\textrm{inc}}} = {a_ + }{\psi _ + } + {a_ - }{\psi _ - }$ excites two decoupled circular polarizations, similar to our formulation in Sec. 2, following $\mathbf{E}({t > 0} )= {\mathbf{E}_ + }({t > 0} )+ {\mathbf{E}_ - }({t > 0} )$. For an incident wavenumber ${k_{\textrm{inc}}}$, which is conserved across the time-interface, the associated eigenfrequencies ${\omega _ \pm }$ are given by

(15)$$\frac{{c{k_{\textrm{inc}}}}}{{{\omega _ \pm }}} = \sqrt {1 + \frac{{\omega _e^2}}{{\omega _0^2 - \omega _ \pm ^2 - i\gamma {\omega _ \pm }}}} \sqrt {1 + \frac{{\omega _m^2}}{{\omega _0^2 - \omega _ \pm ^2 - i\gamma {\omega _ \pm }}}} \pm \frac{{\tau \omega _0^2{\omega _ \pm }}}{{\omega _0^2 - \omega _ \pm ^2 - i\gamma {\omega _ \pm }}}. $$

Due to the Lorentz dispersion, three different frequencies for each polarization are compatible with the incoming momentum: $\omega _ \pm ^{(j)}\textrm{, }j = 1,2,3$. The decomposed electric field after this dispersive chiral time-interface can then be written as

(16)$${\mathbf{E}_ \pm }({t > 0} )= {a_ \pm }{\psi _ \pm }\sum\limits_{j = 1}^3 {\{{{\rm T}_ \pm^{(j)}{e^{ - i\omega_ \pm^{(j)}t}} + \Gamma _ \pm^{(j)}{e^{i{{[\omega_ \pm^{(j)}]}^\ast }t}}} \}}, $$

where ${\rm T}_ \pm ^{(j)}$ and $\Gamma _ \pm ^{(j)}$ are the transmission and reflection coefficients describing the amplitude of each excited mode normalized to the incident field component in the same polarization. To determine these scattering coefficients, additional temporal boundary conditions are required, due to the temporal nonlocality (material dispersion) [43]. These boundary conditions are determined by the microscopic structure of the time interface. For instance, if we assume that the time-switched chirality is achieved by suddenly increasing the number of microscopic oscillators, as considered in [22] and [25], the additional boundary conditions demand the continuity of the decomposed ${\mathbf{E}_ \pm }$, ${\mathbf{H}_ \pm }$, the electric and magnetic polarization current density ${\mathbf{J}_{e, \pm }} ={-} i{\omega _ \pm }({\mathbf{D}_ \pm } - {\varepsilon _0}{\mathbf{E}_ \pm })$, ${\mathbf{J}_{m, \pm }} ={-} i{\omega _ \pm }({\mathbf{B}_ \pm }/{\mu _0} - {\mathbf{H}_ \pm })$. Together with the continuity of ${\mathbf{D}_ \pm }$ and ${\mathbf{B}_ \pm }$, we can solve for the temporal scattering coefficients for both polarizations using

(17)$$\left[ {\begin{array}{ccc} {Y_ \pm^{(1)}}&{Y_ \pm^{(2)}}&{Y_ \pm^{(3)}}\\ {\Lambda _ \pm^{(1)}Y_ \pm^{(1)}}&{\Lambda _ \pm^{(2)}Y_ \pm^{(2)}}&{\Lambda _ \pm^{(3)}Y_ \pm^{(3)}}\\ {({\Lambda _ \pm^{(1)} - {I_2}} )Z_ \pm^{(1)}}&{({\Lambda _ \pm^{(2)} - {I_2}} )Z_ \pm^{(2)}}&{({\Lambda _ \pm^{(3)} - {I_2}} )Z_ \pm^{(3)}} \end{array}} \right]\left[ {\begin{array}{c} {{\rm T}_ \pm^{(1)}}\\ {\Gamma _ \pm^{(1)}}\\ {{\rm T}_ \pm^{(2)}}\\ {\Gamma _ \pm^{(2)}}\\ {{\rm T}_ \pm^{(3)}}\\ {\Gamma _ \pm^{(3)}} \end{array}} \right] = {\left[ {\begin{array}{c} 1\\ {{\eta^{ - 1}}}\\ \varepsilon \\ {\mu {\eta^{ - 1}}}\\ {(\varepsilon - 1)\omega }\\ {(\mu - 1)\omega {\eta^{ - 1}}} \end{array}} \right]_{\textrm{inc}}}, $$

where $Y_ \pm ^{(j)} = [1,\textrm{ }1;\textrm{ }1/\eta _ \pm ^{(j)},\textrm{ } - 1/{(\eta _ \pm ^{(j)})^\ast }]$, $Z_ \pm ^{(j)} = [\omega _ \pm ^{(j)},\textrm{ }\omega _ \pm ^{(j)};\textrm{ }\omega _ \pm ^{(j)}/\eta _ \pm ^{(j)},\textrm{ (}\omega _ \pm ^{(j)}/\eta _ \pm ^{(j)}{)^\ast }]$, $\Lambda _ \pm ^{(j)} = [\varepsilon _ \pm ^{(j)},\textrm{ }0;\textrm{ }0,\textrm{ }\mu _ \pm ^{(j)}]$, and ${I_2}$ is a 2-by-2 identity matrix. $\varepsilon _ \pm ^{(j)}$ and $\mu _ \pm ^{(j)}$ are the effective relative permittivity and permeability evaluated at the frequency $\omega _ \pm ^{(j)}$, which leads to the effective wave impedance $\eta _ \pm ^{(j)} = {\eta _0}\sqrt {\mu _ \pm ^{(j)}/\varepsilon _ \pm ^{(j)}}$ seen by each eigenmode. The column vector on the right-hand side of Eq. (17) involves the corresponding electromagnetic parameters of the medium just before the time-interface, which are evaluated at frequency ${\omega _{\textrm{inc}}}$.

As a basic example, we assume a chiral time-interface formed by switching the entire medium from free space to a chiral medium with Lorentz parameters ${\omega _0} = 1.5{\omega _{\textrm{inc}}}$, ${\omega _e} = 2{\omega _{\textrm{inc}}}$, ${\omega _m} = 1.5{\omega _{\textrm{inc}}}$, $\gamma = 0$, and a chirality characteristic time constant $\tau = 0.64/{\omega _0}$. The dispersion diagram of the chiral medium is shown in Fig. 3(a), in which the red (blue) curves correspond to the eigen-polarization ${\psi _ + }$ (${\psi _ - }$). In this scenario, we solve for the eigenfrequencies after the time-interface using Eq. (15): ${\omega _ + }/{\omega _{\textrm{inc}}} \approx \{ - 1.647,0.414,3.301\}$ and ${\omega _ - }/{\omega _{\textrm{inc}}} \approx \{ - 2.544,0.492,1.797\}$. The first solutions $\omega _ \pm ^{(1)}$ are both negative, yet they fall on branches with positive slopes, as labeled in Fig. 3(a). This implies that the associated time-refracted waves propagate with negative phase velocities but positive group velocities, dual to negative refraction ($k < 0,\textrm{ }\partial \omega /\partial k > 0$) at a spatial interface involving chiral media [39–41]. Meanwhile, the associated time-reflected waves travel with positive phase velocities by taking $- \omega _ \pm ^{(1)} > 0$, while delivering power flow backward in space.

Fig. 3. Dispersive chiral time-interfaces. (a) Dispersion diagrams of a chiral medium with Lorentz dispersion. The curves in red and blue correspond to the two orthogonal circular polarizations, respectively. The frequencies $\omega _ \pm ^{(j)}$ labeled on the right denote the solutions of the dispersion equation with positive group velocities. (b) The evolution of normalized electric fields upon the chiral time-interface with Lorentz dispersion.

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Substituting the eigenfrequencies into Eq. (17), we obtain all the temporal scattering coefficients. The calculated time-variation of the total electric fields is plotted with solid curves in Fig. 3(b) for an incident electric field ${\mathbf{E}_{\textrm{inc}}} = {[1,\textrm{ }0]^T}$. To verify the theoretical results, we perform direct numerical calculations by recasting Eqs. (13) and (14) into time-dependent ordinary differential equations:

(18)$$\left[ {\frac{{{d^2}}}{{d{t^2}}} + \gamma (t)\frac{d}{{dt}} + \omega_0^2(t)} \right]\left[ {\begin{array}{c} \mathbf{P}\\ {{\mu_0}\mathbf{M}} \end{array}} \right] = \left[ {\begin{array}{cc} {{\varepsilon_0}\omega_e^2(t)}&{ - \frac{{\tau (t)\omega_0^2(t)}}{c}\frac{d}{{dt}}}\\ {\frac{{\tau (t)\omega_0^2(t)}}{c}\frac{d}{{dt}}}&{{\mu_0}\omega_m^2(t)} \end{array}} \right]\left[ {\begin{array}{c} \mathbf{E}\\ \mathbf{H} \end{array}} \right], $$

where $\mathbf{P} = \mathbf{D} - {\varepsilon _0}\mathbf{E}$ and ${\mu _0}\mathbf{M} = \mathbf{B} - {\mu _0}\mathbf{H}$. Combining Eq. (18) with Maxwell’s curl equations and realizing that $\nabla \to i\hat{z}{k_{\textrm{inc}}}$, we can solve for the electric fields using the Runge-Kutta method, as shown by the circles in Fig. 3(b). The results agree well with our theoretical predictions and confirm our expectations on the excitation of multiple frequencies and the polarization rotation of the wave.

5. Conclusion

In summary, we have investigated wave scattering at time-interfaces involving chiral media. We started from the simple assumption of absence of material dispersion, showing that the wavefield decomposition in chiral media also applies to the temporal scattering problem. By projecting the total fields into the two orthogonal polarizations (LCP and RCP), we demonstrated the temporal analogue of optical activity, in which a chiral temporal slab can rotate the polarization of a linearly polarized incidence light in a reciprocal manner. We then considered the dispersive scenario by assuming a single-resonance Lorentz model for chiral media, and solved the temporal scattering problem at a chiral time-interface in the presence of material dispersion, demonstrating scattering into multiple frequencies and nontrivial phenomena, such as the occurrence of temporal negative refraction at such interfaces. All theoretical results have been validated with numerical calculations.

Our results not only contribute another building block — electromagnetic chirality — in the nascent field of time-interfaces and temporal scattering, but can also facilitate future explorations on time-metamaterials and Floquet physics with polarization/spin degree of freedom [37]. More exotic wave phenomena, such as temporal circular dichroism, chiral nihility, negative refraction, and chiral momentum-gap in photonic time crystals, can be investigated based on this formalism.

Funding

DEVCOM Army Research Laboratory; Air Force Office of Scientific Research; Simons Foundation.

Acknowledgments

The authors thank helpful discussions with Dr. Huanan Li, M. H. Mostafa, Dr. M. S. Mirmoosa, and Dr. S. A. Tretyakov.

Disclosures

The authors declare no conflicts of interest.

Data availability

No data were generated or analyzed in the presented research.

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Temporal optical activity and chiral time-interfaces [Invited]

Abstract

1. Introduction

2. Non-dispersive chiral time-interfaces

3. Temporal optical activity

3.1 Reciprocal polarization rotation through a chiral temporal slab

3.2 Numerical example

4. Chiral time-interfaces with Lorentz dispersion

5. Conclusion

Funding

Acknowledgments

Disclosures

Data availability

References

Data availability

Cited By

Figures (3)

Equations (18)

Optics Express