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Investigating non-reciprocity in time-periodic media using a perturbative approach

Amirhosein Sotoodehfar, Somayeh Boshgazi, Mohammad Memarian, Behzad Rejaei, and Khashayar Mehrany

Open Access Open Access

Abstract

Lorentz famous theorem leads to clear reciprocity conditions for linear, time-invariant media based on their constitutive parameters. By contrast, reciprocity conditions for linear time-varying media are not fully explored. In this paper, we investigate whether, and how a structure containing a time-periodic medium can be truly identified as reciprocal or not. To that end, a necessary and sufficient condition is derived which requires both the constitutive parameters and the electromagnetic fields inside the dynamic structure. As solving for the fields for such problems is challenging, a perturbative approach is proposed which expresses the aforementioned non-reciprocity condition in terms of the electromagnetic fields and the Green’s functions of the unperturbed static problem and is particularly applicable for the case of structures with weak time modulation. Reciprocity of two famous canonical time-varying structures are then studied using the proposed approach and their reciprocity/non-reciprocity is investigated. In the case of one-dimensional propagation in a static medium with two point-wise modulations, our proposed theory clearly explains the often observed maximization of non-reciprocity when the modulation phase difference between the two points is 90 degrees. In order to validate the perturbative approach, analytical and Finite-Difference Time-Domain (FDTD) methods are employed. Then, solutions are compared and considerable agreement between them is observed.

© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement

1. Introduction

Non-reciprocity is an interesting yet elusive concept in electromagnetics and physics. Many works in literature have been dedicated to topic of reciprocity in electromagnetics [1], which almost exclusively had analyzed linear time-invariant media, up until a decade ago. From a practical perspective, non-reciprocal devices such as circulators and isolators are essential components in various applications, e.g. in full-duplex communication systems which allow us to separate, and send and receive data simultaneously, or to protect lasers from undesired probable reflections. Non-reciprocal devices have been traditionally realized using magneto-optical materials like ferrites which require an external DC magnetic field as bias [2,3]. This magnetic bias makes the structure bulky and not-integratable which is challenging for non-reciprocal nanophotonics devices [4]. Another method to achieve non-reciprocal response is utilizing non-linear effects [5,6].This approach is also challenging due to the weakness of nonlinear effects and the requirement of high intensity electric fields [7,8].

Over the past decade, research on linear "time-varying" (TV) structures has grown rapidly, and one major result is the demonstration of non-reciprocal behavior without the need for magnets or non-linearity, and by appropriate spatio-temporal modulation of the permittivity of the structure [9–15]. According to recent research, if the modulation profile does not have a generalized time-reversal symmetry, the media shows non-reciprocal behavior [16]. This property is used in many structures but it is not proved rigorously using Lorentz reciprocity condition. In many structures, a linear momentum bias is applied to the structure using an appropriate travelling wave modulation profile for the permittivity of the structure. In this manner, the incoming wave can be transmitted properly in one direction while it scatters in the opposite direction, and thereby the structure becomes non-reciprocal [9,17,18]. In such structures, a large area of the structure should be spatio-temporally modulated in such a way that a uni-directional linear momentum is exerted upon the incoming wave. This is hard to realize on account of imperfections which makes the applied linear momentum partially bi-directional. Thus, cases where modulation is restricted to only two or few areas are more desirable [18–20]. Interestingly, noticeable non-reciprocity in such structures is achieved whenever the phase difference between the two points has a $\pi /2$ phase difference. This begs the question why a certain phase difference maximizes non-reciprocity. To show the non-reciprocal response in these structures, numerical methods like FDTD simulations are typically used and in some cases analytical methods such as Bloch-Floquet decomposition of electromagnetic fields are available [15]. Nevertheless, analytical solutions are not available for general time-modulated structures which makes it hard to delve into the problem and possibly find the proper modulation to achieve noticeable non-reciprocity.

In this paper, we investigate the reciprocity of linear, dispersive, in-homogenous, space-time modulated isotropic media. According to the Lorentz reciprocity theorem, in reciprocal media, the fields produced by a current source at an observation point, are the same compared with the case when the current source and observation point are interchanged [21]. In order to investigate the reciprocity/non-reciprocity of space-time modulated media, we start from a generalized version of the Lorentz reciprocity theorem which is then used to derive a sufficient condition for determining reciprocity, based on the constitutive parameters of the medium, be it static or dynamic. These conditions may be viewed as generalization of the results reported in [21] to dispersive TV media. The formalism developed, however, allows us to go beyond the derivation of sufficient reciprocity conditions; it provides a means to evaluate the degree of non-reciprocity of a system when the aforementioned conditions on the medium are not met. Also, Even if these conditions are violated, reciprocity may still be preserved depending on the geometry of the system and the type of excitation. Thus, a necessary and sufficient condition for reciprocity is derived which involves the electromagnetic fields inside the time-varying media in addition to the constitutive parameters. However, the calculation of the electromagnetic fields inside time-varying media is not an easy task for general time-modulated structures. In many practical scenarios, the dynamically modulated components of permittivity and permeability are small. This observation motivated us to propose a perturbative approach that provides accurate solutions based on the fields of the underlying static medium. Using perturbative theory, the aforementioned necessary and sufficient condition for reciprocity can be written in terms of the electromagnetic fields of the static problem. Therefore, the proposed approach can be used for investigation of many time-varying structures that do not have an analytical solution, but their static counterparts can be analytically solved. Moreover, inspired by the integrals derived from Lorentz reciprocity, we define a metric for the amount of non-reciprocity. This can provide an effective method for increasing isolation, and for example we can show that when the phase difference of modulation between two modulated parts is $\pi /2$, the non-reciprocity factor will be maximized.

For demonstration purposes, the famous one-dimensional propagation inside a slab having a travelling-wave modulation profile is first considered. Since analytical results can be found for this canonical problem, it is a good validation example as well. We solve this problem in 3 ways, using the analytical method, the FDTD, and our proposed perturbative approach. The perturbative approach is used to calculate the amount of non-reciprocity. The analytical approach uses Bloch-Floquet analysis to calculate the scattered electromagnetic fields from the time-varing structure. Moreover, the analytical and perturbative approaches are then both validated using an in-house FDTD simulator. In a second example, the travelling-wave modulation is removed and instead, only two point-like modulations in a distance create time variations in the structure. Our investigation using the perturbative approach which utilizes the fields of the static structure shows promising and insightful results which are also validated using FDTD simulations.

This paper is organized as follows. The reciprocity condition using the Lorentz reciprocity theorem is extracted in Sec. 2. The perturbative approach for calculating the electromagnetic fields inside the time-varying structure is presented in Sec. 3. In Sec. 4, the necessary and sufficient condition for reciprocity is calculated for the two mentioned structures. Finally, the simulation results of these two examples are presented in Sec. 5 using the perturbative approach and are validated using analytical and FDTD simulations.

2. Reciprocity in space-time dependent systems

The constitutive relations in a causal, local, linear, and isotropic time-modulated medium are [22]

$$\begin{aligned} & \mathscr{B} (\mathbf{r}, t) = \int_{-\infty}^{t} \tilde{\mu}(\mathbf{r}, t; t-t') \mathscr{H} (\mathbf{r}, t') d t^\prime, \\ & \mathscr{D} (\mathbf{r}, t) = \int_{-\infty}^{t} \tilde{\epsilon}(\mathbf{r}, t; t-t') \mathscr{E} (\mathbf{r}, t') d t^\prime, \end{aligned}$$
according to which, the magnetic and electric flux densities at an arbitrary time $t$ are functions of all the previous values of magnetic and electric fields, respectively. This function, however, is itself dependent on the observation time $t$, in contrast to static structures. We assume a medium with a time-periodic behavior in which the kernels of the integrals have the property $\tilde {\epsilon }(\mathbf {r}, t; \tau ) = \tilde {\epsilon }(\mathbf {r}, t + \frac {2\pi }{\Omega }; \tau ), \tilde {\mu }(\mathbf {r}, t; \tau ) = \tilde {\mu }(\mathbf {r}, t + \frac {2\pi }{\Omega }; \tau )$, where $\Omega$ is the angular frequency of modulation. Due to the periodicity, these functions can be expanded in Fourier series as
$$\tilde{\epsilon}(\mathbf{r}, t; \tau) = \sum_{\textrm{n}} e^{j\textrm{n}\Omega t} \tilde{\epsilon}_{\textrm{n}} (\mathbf{r}, \tau), \quad \tilde{\mu}(\mathbf{r}, t; \tau) = \sum_{\textrm{n}} e^{j\textrm{n}\Omega t} \tilde{\mu}_{\textrm{n}} (\mathbf{r}, \tau).$$

By assuming a monochromatic current source, the response of a linear and time-periodic system can be expanded as a series of harmonics, as follows

$$\begin{aligned}\mathbf{E}(\mathbf{r}, t) &= \sum_{\textrm{m}} \mathbf{E}_{\textrm{m}}(\mathbf{r}) e^{j\omega_{\textrm{m}} t} + c.c., \\ & \mathbf{H}(\mathbf{r}, t) = \sum_{\textrm{m}} \mathbf{H}_{\textrm{m}}(\mathbf{r}) e^{j\omega_{\textrm{m}} t} + c.c., \end{aligned}$$
where $\omega _{\textrm{m}} = \omega _0 + \textrm{m} \Omega$ and $\omega _0$ is the source frequency. By substituting Eqs. (1) and (3) into Maxwell’s equations, and using the harmonic balance approach, Faraday’s and Ampere’s laws can be expressed as
$$\begin{aligned} & \frac{1}{j\omega_{\textrm{m}}}\nabla\times \mathbf{E}_{\textrm{m}}(\mathbf{r}) ={-}\sum_{\textrm{n}}\mu_{\textrm{m-n}}(\mathbf{r}, \omega_{\textrm{n}})\mathbf{H}_{\textrm{n}}(\mathbf{r}), \\ & \frac{1}{j\omega_{\textrm{m}}}\nabla\times \mathbf{H}_{\textrm{m}}(\mathbf{r}) = \sum_{\textrm{n}}\epsilon_{\textrm{m-n}}(\mathbf{r}, \omega_{\textrm{n}})\mathbf{E}_{\textrm{n}}(\mathbf{r}) + \frac{1}{j\omega_{\textrm{m}}}\mathbf{J}(\mathbf{r})\delta_{\textrm{m0}}, \end{aligned}$$
where $\epsilon _{\textrm{n}}(\mathbf{r}, \omega )$ and $\mu _{\textrm{n}}(\mathbf{r}, \omega )$ are the Fourier transforms of each Fourier component defined as $\epsilon _{\textrm{n}}(\mathbf{r}, \omega ) = \int _{0}^{\infty }\tilde {\epsilon }_{\textrm{n}}(\mathbf{r}, \tau )e^{-j\omega \tau }d\tau$ and so on. Also, $\delta _{\textrm{m0}}$, Kronecker delta, is equal to $1$ when $\textrm{m} =0$ and $0$ otherwise. Note that this formulation automatically takes dispersion into account as $\epsilon _{\textrm{n}}$ and $\mu _{\textrm{n}}$ are functions of $\omega$. This is needed when dealing with realistic electromagnetic structures. In time-varying media whose electromagnetic response may be considered instantaneous, however, this dependence on frequency may be neglected.

In order to obtain the reciprocity condition, we consider two monochromatic current sources $\mathbf{J}^{\textrm{a}}$ and $\mathbf{J}^{\textrm{b}}$ with the same angular frequency $\omega _0$ as Fig. 1. Then, we use the generalized Lorentz reciprocity theorem which is as follows [21]

$$\frac{1}{j\omega_{\textrm{0}}}\int \mathbf{E}_{\textrm{0}}^{\textrm{a}}\cdot\mathbf{J}^{\textrm{b}} dV^b = \frac{1}{j\omega_{\textrm{0}}}\int \mathbf{E}_{\textrm{0}}^{\textrm{b}}\cdot\mathbf{J}^{\textrm{a}} dV^a ,$$
where $\mathbf{E}_{\textrm{0}}^{\textrm{a},\textrm{b}}$ is the electric field harmonic at the source frequency.

For the sake of simplicity, let us drop the notation for the dependency of fields on $\mathbf{r}$, and let $\epsilon _{\textrm{m}\hbox{-}\textrm{n}}^{\textrm{n}} = \epsilon _{\textrm{m}\hbox{-}\textrm{n}}(\mathbf{r}, \omega _{\textrm{n}}), \mu _{\textrm{m}\hbox{-}\textrm{n}}^{\textrm{n}} = \mu _{\textrm{m}\hbox{-}\textrm{n}}(\mathbf{r}, \omega _{\textrm{n}})$. As described in Appendix A, starting from Maxwell’s equations for both cases "a" and "b", one can obtain the following equality

$$\begin{aligned} \frac{1}{j\omega_{\textrm{0}}}\int \big[ & \mathbf{E}_{\textrm{0}}^{\textrm{a}}\cdot\mathbf{J}^{\textrm{b}} - \mathbf{E}_{\textrm{0}}^{\textrm{b}}\cdot\mathbf{J}^{\textrm{a}}\big] dV = \\ & -\int\sum_{\textrm{m,n}} \big(\mu_{\textrm{m-n}}^{\textrm{n}} - \mu_{\textrm{n-m}}^{\textrm{m}}\big)\mathbf{H}_{\textrm{m}}^{\textrm{b}}\cdot\mathbf{H}_{\textrm{n}}^{\textrm{a}} d V - \int\sum_{\textrm{m,n}} \big(\epsilon_{\textrm{m-n}}^{\textrm{n}} - \epsilon_{\textrm{n-m}}^{\textrm{m}}\big)\mathbf{E}_{\textrm{m}}^{\textrm{a}}\cdot\mathbf{E}_{\textrm{n}}^{\textrm{b}} d V . \end{aligned}$$

 figure: Fig. 1.

Fig. 1. The electromagnetic interaction between two current sources $\mathbf{J}^{\textrm{a}}$ and $\mathbf{J}^{\textrm{b}}$ in a space-time dependent medium.

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Equation (6) will be our starting point for evaluating the reciprocity of the structures under study. For Lorentz reciprocity condition (Eq. (5)) to hold, the integral on the left-hand side of Eq. (6) must vanish. Otherwise, the value of this integral represents a measure for non-reciprocity which will be utilized in Sec. 4. This value is demonstrating the isolation at source frequency which is a degree of non-reciprocity in problems with a monochromatic current source [13]. A large magnitude for this integral implies a "less reciprocal" system. It readily follows from (6) that a sufficient condition for reciprocity is when each term inside the summations is identically zero, i.e.

$$\epsilon_{\textrm{m-n}}^{\textrm{n}} = \epsilon_{\textrm{n-m}}^{\textrm{m}}, \mu_{\textrm{m-n}}^{\textrm{n}} = \mu_{\textrm{n-m}}^{\textrm{m}},$$
which only involves the medium and not the geometry of the structure nor the fields or type of excitation. This sufficient condition means that if Eq. (7) is met, the electromagnetic structure at hand is definitely reciprocal. However, if Eq. (7) does not hold, the structure may or may not still show reciprocal behavior since the integral of a non-zero integrand can still be zero. To reach the final verdict, one needs to calculate the complete right-hand side of Eq. (6), and thus the volume integrals and thus the fields.

A well-known example of a time-varying structure that is reciprocal, is a non-magnetic in-homogeneous non-dispersive sinusoidal time-modulated medium [21]. One can readily check the sufficient reciprocity conditions, Eq. (7), holds for this example, as $\epsilon _{\pm 1} = \epsilon _{\mp 1}$.

It must be mentioned that sufficient conditions for reciprocity of a non-dispersive medium which has an instantaneous response, have been discussed in [16,21]. Equation (7) can be easily reduced to conditions in [21].

As the reciprocity condition (7) is a sufficient condition, if it does not hold, one can not know whether the structure is still reciprocal or not, or to what degree is it non-reciprocal. We need to calculate the electromagnetic fields and the integrals in Eq. (6) which is not an easy task for arbitrary time-periodic structures. However, in most practical applications the time-varying components of permittivity and permeability are small compared to their static parts. This motivates a perturbative approach as outlined below.

3. Perturbative approach

Many optical time-varying structures have relatively weak temporal modulation strengths. Thus the possibility of obtaining field solutions in such structures using a perturbative approach must be explored. This will then help us to properly investigate the non-reciprocity of such media. Let us confine ourselves to a non-dispersive non-magnetic time-periodic medium, for which Eq. (4) can be written as

$$\begin{aligned} & \nabla\times\mathbf{E}_{\textrm{m}}(\mathbf{r}) ={-}j\omega_{\textrm{m}}\mu_0\mathbf{H}_{\textrm{m}}(\mathbf{r}), \\ & \nabla\times\mathbf{H}_{\textrm{m}}(\mathbf{r}) = j\omega_{\textrm{m}}\epsilon_0(\mathbf{r}) \mathbf{E}_{\textrm{m}}(\mathbf{r}) + \mathbf{J}_{\textrm{m}}^{\textrm{ e q}}(\mathbf{r}) + \mathbf{J}(\mathbf{r})\delta_{\textrm{m0}}, \end{aligned}$$
where
$$\mathbf{J}_{\textrm{m}}^{\textrm{eq}}(\mathbf{r}) = j\omega_{\textrm{m}} \sum_{\mathrm{n(\ne m)}} \epsilon_{\textrm{m-n}}(\mathbf{r}) \mathbf{E}_{\textrm{n}}(\mathbf{r}).$$

Let us assume that a static medium with the permittivity $\epsilon _0(\mathbf{r})$ (and permeability $\mu _0$) has a corresponding dyadic Green’s function $\bar {\bar {\mathbf{G}}}(\mathbf{r}, \mathbf{r}'; \omega )$. Equations (8) and (9) will then yield the integral equation

$$\mathbf{E}_{\textrm{m}}(\mathbf{r}) - \omega_{\textrm{m}}^2 \mu_0 \sum_{\mathrm{n(\ne m)}}\int\bar{\bar{\mathbf{G}}}(\mathbf{r}, \mathbf{r}'; \omega_{\textrm{m}}) \cdot \epsilon_{\textrm{m-n}}(\mathbf{r}') \mathbf{E}_{\textrm{n}}(\mathbf{r}') dV' = \delta_{\textrm{m0}}\mathbf{E}^0(\mathbf{r}),$$
where
$$\mathbf{E}^0(\mathbf{r}) ={-}j\omega_0\mu_0 \int\bar{\bar{\mathbf{G}}}(\mathbf{r}, \mathbf{r}'; \omega_{\textrm{0}}) \cdot \mathbf{J}(\mathbf{r}') dV'$$
is the field produced by the current source in the static structure. Assuming that the modulation strength of different harmonics of permittivity is much smaller than the static permittivity, that is $|\epsilon _{\mathrm{m(\ne 0)}} (\mathbf{r})| \ll |\epsilon _0(\mathbf{r})|$, the perturbative approach can be used as follows.

By applying the perturbation theory, the amplitude of harmonic ${m}$ can be expanded to the first order using the electric field of the static structure, as

$$\begin{aligned} \mathbf{E}_{\textrm{m}}(\mathbf{r}) & = \delta_{\textrm{m0}} \mathbf{E}^0 (\mathbf{r}) + \omega_{\textrm{m}}^2 \mu_0 \sum_{\mathrm{n(\ne m)}}\int\bar{\bar{\mathbf{G}}}(\mathbf{r}, \mathbf{r}'; \omega_{\textrm{m}}) \cdot \epsilon_{\textrm{m-n}}(\mathbf{r}') \delta_{\textrm{n0}} \mathbf{E}^0(\mathbf{r}') dV' \\ & = \delta_{\textrm{m0}} \mathbf{E}^0 (\mathbf{r}) + (1-\delta_{\textrm{m0}})\omega_{\textrm{m}}^2\mu_0\int\bar{\bar{\mathbf{G}}}(\mathbf{r}, \mathbf{r}'; \omega_{\textrm{m}}) \cdot \epsilon_{\textrm{m}}(\mathbf{r}') \mathbf{E}^0(\mathbf{r}') dV'. \end{aligned}$$

In order to evaluate the reciprocity by using the perturbative approach, the integrals on the right-hand side of Eq. (6) need to be calculated. Since the medium is non-magnetic, evaluating the volume integral on the electric fields is needed and will suffice. By substituting Eq. (12) for the two cases "a" and "b" into Eq. (6) and performing some algebraic manipulations, one obtains

$$\begin{aligned} \int \sum_{\textrm{n, m}}(\epsilon_{\textrm{m-n}} - \epsilon_{\textrm{n-m}}) \mathbf{E}_{\textrm{m}}^a & \cdot \mathbf{E}_{\textrm{n}}^b d V = \mu_0 \sum_{\textrm{n>0}} \int \int \big[\epsilon_{\textrm{n}}^*(\mathbf{r}) \epsilon_{\textrm{n}}(\mathbf{r}') - \epsilon_{\textrm{n}}^*(\mathbf{r}')\epsilon_{\textrm{n}}(\mathbf{r})\big] \\ & \mathbf{E}^{0,\textrm{a}}(\mathbf{r}) \cdot\big[\omega_{\textrm{n}}^2 \bar{\bar{\mathbf{G}}}(\mathbf{r}, \mathbf{r}'; \omega_{\textrm{n}}) - \omega_{\textrm{-n}}^2\bar{\bar{\mathbf{G}}}(\mathbf{r}, \mathbf{r}'; \omega_{\textrm{-n}})\big] \cdot \mathbf{E}^{ 0,\textrm{b}}(\mathbf{r}')d V^\prime d V, \end{aligned}$$
where the details of calculations are presented in Appendix B. Notice that this equation enables us to evaluate the reciprocity of a linear time-periodic medium by using the electric fields and the Green’s function of the unperturbed (static) medium. Since calculating the electric fields and the Green’s function of a static structure is a much easier task than its dynamic counterpart, Eq. (13) simplifies the procedure to check if a structure is reciprocal or not. For the cases where the permeability also varies with time, a similar equation for the integral on magnetic fields in Eq. (6) can be achieved. Thus, one can simply apply Lorentz theorem, and check reciprocity to see if the right-hand or left-hand side of the Eq. (6) is zero.

4. Examples

In this section, two non-dispersive time-periodic modulated structures are presented and the integral in the Lorentz reciprocity theorem is calculated through the perturbative approach and validated with two other methods.

4.1 Travelling-wave modulated slab

As the first example, consider a non-dispersive non-magnetic 1D slab in free space with the travelling-wave modulation, as shown in Fig. 2. The problem is invariant in the $x$ and $y$ directions. The dielectric permittivity of the entire problem which has an instantaneous response is as follows

$$\tilde{\epsilon}(\mathbf{r}, t; \tau) = \big[\epsilon_0\epsilon_{\textrm{r}} + \Delta\epsilon \cos (qz - \Omega t)\big] \delta(\tau^+),$$
where $q$ is the wave number of the modulation, and consequently, the speed of modulation is $v_{\textrm{m}} = \frac {\Omega }{q}$. Evaluating Eq. (7) shows that this form of modulation does not satisfy the sufficient condition (i.e., $\epsilon _{\pm 1} \neq \epsilon _{\mp 1}$), and therefore, we need to consider the volume integral of Eq. (6) to check if the structure is reciprocal. Using the perturbative approach and substituting Eq. (14) in Eq. (13), results in
$$\begin{aligned} \int \big[\mathbf{E}_0^{\textrm{a}}\cdot\mathbf{J}^{\textrm{b}} - \mathbf{E}_0^{\textrm{b}}\cdot\mathbf{J}^{\textrm{a}}\big] & dV ={-}j\omega_0 \mu_0 \big(\frac{\Delta \epsilon}{2}\big)^2 \int\limits_{z = 0}^{z=\textrm{L}} \int\limits_{z' = 0}^{z'=\textrm{L}} \big[e^{jq(z-z')} - e^{{-}jq(z-z')}\big] \\ & \mathbf{E}^{0,\textrm{a}}(z)\cdot \big[\omega_1^2 \bar{\bar{\mathbf{G}}}(z, z'; \omega_1) - \omega_{{-}1}^2 \bar{\bar{\mathbf{G}}}(z, z'; \omega_{{-}1})\big] \cdot \mathbf{E}^{0,\textrm{b}}(z') dz dz'. \end{aligned}$$

 figure: Fig. 2.

Fig. 2. Demonstration of a non-dispersive slab with travelling-wave modulation which is surrounded by free space. Two current sources $\mathbf{J}^{\textrm{a}}$ and $\mathbf{J}^{\textrm{b}}$ excite this structure in cases "a" and "b", respectively. This structure is invariant under transformation in the x-y plane.

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In order to calculate this expression, we need to find the electric fields related to the static slab and its relevant Green’s function. The procedure of solving the static slab is well-known in the literature and can be found in Appendix C.

4.2 Two point-like time-modulated structure

Generally, a non-dispersive point-like time-dependent modulation located at $\mathbf {r}_\mathrm {p}$ in free space can be expressed as

$$\tilde{\epsilon}(\mathbf{r},t;\tau) = \sum _{\textrm{n}} \epsilon _{\textrm{n}} (\mathbf{r}) e^{j \textrm{n}\Omega t} \delta(\tau ^+),$$
where,
$$\epsilon_{\textrm{n}}(\mathbf{r}) = \sum_{\textrm{p}=1}^{\textrm{P}_{\textrm{n}}}\epsilon_{\textrm{n}}^{(\textrm{p})} \delta(\mathbf{r} - \mathbf{r}_{\textrm{p}}),$$
in which $\textrm{P}_{\textrm{n}}$ is the number of the perturbed points in space. By substituting this form of permittivity in Eq. (13), we obtain
$$\begin{aligned} \int \big[\mathbf{E}_0^{\textrm{a}}\cdot\mathbf{J}^{\textrm{b}} - \mathbf{E}_0^{\textrm{b}}\cdot\mathbf{J}^{\textrm{a}}\big] & dV = \mu_0 \sum_{\textrm{n}>0}\sum_{\textrm{p}}\sum_{\textrm{q}}\Big[\big(\epsilon_{\textrm{n}}^{(\textrm{p})}\big)^* \epsilon_{\textrm{n}}^{\rm(q)} - \big(\epsilon_{\textrm{n}}^{(\textrm{q})}\big)^* \epsilon_{\textrm{n}}^{(\textrm{p})}\Big]\times \\ & \mathbf{E}^{0,\textrm{a}}(\mathbf{r}_{\textrm{p}}) \cdot \Big[\omega_{\textrm{n}}^2 \bar{\bar{\mathbf{G}}}(\mathbf{r}_{\textrm{p}}, \mathbf{r}_{\textrm{q}}; \omega_{\textrm{n}}) - \omega_{-\textrm{n}}^2\bar{\bar{\mathbf{G}}}(\mathbf{r}_{\textrm{p}}, \mathbf{r}_{\textrm{q}}; \omega_{\textrm{n}})\Big]\cdot \mathbf{E}^{0,\textrm{b}}(\mathbf{r}_{\textrm{q}}). \end{aligned}$$

As a second example, let us assume only two perturbation points, i.e. $\textrm{P}_{\textrm{n}} = 2$ and $\epsilon _{\textrm{n}} = 0$ for $|\textrm{n}|>1$, as shown in Fig. 3. Using the symmetry of the Green’s function for the static structure, and by expressing $\epsilon _1^{(1)}, \epsilon _1^{(2)}$ in terms of magnitude and phase $\big (\epsilon _{\textrm{n}}(\mathbf{r}) = |\epsilon _{\textrm{n}}(\mathbf{r})|e^{-j\Theta _{\textrm{n}}(\mathbf{r})}\big )$, Eq. (18) can be extracted as

$$\begin{aligned} & \int \big[\mathbf{E}_0^{\textrm{a}}\cdot\mathbf{J}^{\textrm{b}} - \mathbf{E}_0^{\textrm{b}}\cdot\mathbf{J}^{\textrm{a}}\big] dV = \\ & j2\mu_0 |\epsilon_1^{(1)}||\epsilon_1^{(2)}|\sin(\Theta_1^{(1)} - \Theta_1^{(2)}) \times\bigg[ \mathbf{E}^{0,\textrm{a}}(\mathbf{r}_1) \cdot \Big[\omega_1^2 \bar{\bar{\mathbf{G}}}(\mathbf{r}_1, \mathbf{r}_2; \omega_1) - \omega_{{-}1}^2\bar{\bar{\mathbf{G}}}(\mathbf{r}_1, \mathbf{r}_2; \omega_{{-}1})\Big] \cdot\mathbf{E}^{0, \textrm{b}}(\mathbf{r}_2) \\ & \quad \quad \quad \quad \quad \quad \quad \quad - \mathbf{E}^{0, \textrm{a}}(\mathbf{r}_1) \cdot \Big[\omega_1^2 \bar{\bar{\mathbf{G}}}(\mathbf{r}_1, \mathbf{r}_2; \omega_1) - \omega_{{-}1}^2\bar{\bar{\mathbf{G}}}(\mathbf{r}_1, \mathbf{r}_2; \omega_{{-}1})\Big] \cdot\mathbf{E}^{0, \textrm{b}}(\mathbf{r}_2)\bigg]. \end{aligned}$$

Equation (19) shows why in two-part modulated structures, a phase difference of 90 degrees yields the highest degree of non-reciprocity [20], as the $\sin (\Theta _1^{(1)} - \Theta _1^{(2)})$ term is maximum only for the cases of $\Theta _1^{(1)} - \Theta _1^{(2)} = \pi /2+\textrm{n}\pi$. Also, this expression shows that if the two points are modulated with the same phase, the structure is reciprocal. This result can also be seen using the reciprocity condition (7) if both of the phases $\Theta _1^{(1)}$ and $\Theta _1^{(2)}$ are equal to zero but for non-zero phases the reciprocity condition (7) is inconclusive and calculation of electromagnetic fields inside the structure is essential. This emphasizes that our investigation for the full reciprocity condition including the fields is of vital importance as the same structure can be reciprocal or not, while always violating the sufficient condition.

 figure: Fig. 3.

Fig. 3. Demonstration of structure with point-like modulation. Two distributive points’ permittivity oscillates in time with frequency of $\Omega$ and phase different $\phi$.

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5. Results

In this section, we illustrate the results of the perturbative approach for the two examples discussed in the previous section. In order to validate the results obtained by the perturbative approach, one can use the FDTD method to find the electric and magnetic fields numerically. After obtaining electric and magnetic fields numerically, integrals in Eq. (6) can also be solved numerically. For this purpose, Helmholtz equation

$$\nabla\times\nabla\times\mathscr{E}(\mathbf{r}, t) ={-}\mu _0 \frac{\partial^2}{\partial t^2} \mathscr{D}(\mathbf{r}, t) - \mu _0 \frac{\partial}{\partial t}\mathscr{J}(\mathbf{r}, t)$$
could be discretized in both space and time. Note that since we have considered a non-dispersive medium, we have $\mathscr {D}(\mathbf{r}, t) = \epsilon (\mathbf{r}, t) \mathscr {E}(\mathbf{r}, t)$ in Eq. (20).

Moreover, for the travelling-wave modulated slab, the electromagnetic fields can be extracted analytically using the harmonic balance approach based on [13,15] which has been illustrated, additionally.

To compare the results of the perturbative approach, a normalized expression is defined based on the left-hand side of Eq. (6) which is $\Delta =\vert S_{12}-S_{21} \vert$ as

$$\Delta =\vert \int\big[\mathbf{E}_0^{\textrm{a}}\cdot \mathbf{J}^{\textrm{b}} - \mathbf{E}_0^{\textrm{b}} \cdot \mathbf{J}^{\textrm{a}}\big]dV / \Big[-\frac{\eta_0}{2}J_0^{\textrm{a}}J_0^{\textrm{b}}e^{{-}jk_0(z_{\textrm{b}} - z_{\textrm{a}})}\Big] \vert,$$
which is a metric to show the amount of non-reciprocity. In the following subsections, this expression is calculated using the mentioned approaches for two examples presented before.

5.1 Travelling wave modulated slab

Consider the first example presented in Subsec. 4.1 which is a 1D slab with a travelling-wave modulation profile. Here, we assume a free space slab, $\epsilon _{\textrm{r}} = 1$, with thickness (thickness of modulation on free space) of $d = 1$ mm. The source frequency is $f_0 = 0.5 \mathrm {THz}$. The speed of modulation is $v_{\textrm{m}} = 0.06 c_0$, where $c_0$ is speed of light in free space. Also, the strength of the modulation is $\Delta \epsilon = 0.1 \epsilon _0$, which is relatively large with respect to the practical values used in [9]. This structure shows non-reciprocal response and is used in the literature [9,10].

A modulation with low strength was the primary assumption of the perturbative approach. Therefore, it is worth computing the value of $\Delta$ as a function of $\Delta \epsilon$. Figure 4 illustrates $\Delta$ w.r.t. $\Delta \epsilon$ for three different values of modulation frequency, $\Omega$. The results are calculated using three approaches, perturbative theory, FDTD, and analytical solution. This figure verifies that the result of the perturbation approach is in very good agreement with analytical and FDTD solutions. It is worth noting that higher modulation strengths up to $\Delta \epsilon / \epsilon _0 =0.5$ for the example of Fig. 4(b), still results in excellent agreement between the perturbative approach and the analytical results.

 figure: Fig. 4.

Fig. 4. $\Delta$ vs. strength of modulation for different $\Omega$. For 3 cases (a) $\Omega = 10$ GHz, (b) $\Omega = 20$ GHz and (c) $\Omega = 50$ GHz the results of three different methods have been demonstrated.

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In order to evaluate the effect of changing $\epsilon _{\textrm{r}}$ on the results, let us sweep $\Delta$ over the excitation frequency, $f_0$ for three different values of $\epsilon _{\textrm{r}}$. To do so, we assume $d = 1$ mm, $v_{\textrm{m}} = 0.06 c_0$, $\Omega = 20$ GHz and $\Delta \epsilon / \epsilon _0 = 5\times 10^{-3}$. The results are illustrated in Fig. 5. It can be seen that in some frequencies the travelling-wave modulation results in a reciprocal behavior while Eq. (7) is not capable to show this. As is shown in Fig. 5(a) and (b), there exists considerable agreement between perturbative and analytical solutions. Also, Fig. 5(c) shows that the results of the perturbative approach follow the general trend but has deviations compared to the other approaches around resonance points. This is due to the fact that the perturbation theory loses accuracy in presence of resonances. As the intensity of the associated fields are increased for higher values of $\epsilon _r$, the accuracy in these points decreases.

 figure: Fig. 5.

Fig. 5. $\Delta$ vs. excitation frequency for different $\epsilon _{\textrm{r}}$. For 3 cases (a) $\epsilon _{\textrm{r}} = 1$, (b) $\epsilon _{\textrm{r}} = 3$ and (c) $\epsilon _{\textrm{r}} = 5$ the results of three different methods have been demonstrated.

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 figure: Fig. 6.

Fig. 6. (a),(b),(c). $\Delta$ vs. excitation frequency for different $\phi$. For 3 cases (a) $\phi = 5^{\circ}$, (b) $\phi = 45^{\circ}$ and (c) $\phi = 90^{\circ}$ the results of two different methods, perturbation theory and FDTD solution, have been demonstrated. Based on the perturbative solution, we can state that these diagrams are approximately different by a factor of $\sin {\phi }$.(d). $\Delta$ vs. $\phi$ for excitation frequency $f_0 = 400$GHz.

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5.2 Two point-like time-modulated structure

A 1D structure consisting of two time-modulated points presented in Subsec. 4.2 is now considered. Note that the modulation is in the form of Eq. (17) but Delta function is a one dimensional function in the form of $\delta (z-z_{\mathrm {p}})$. Consider the 1D configuration, as shown in Fig. 3, where two point-like time-dependent modulations are placed with distance $d = 1$ mm of each other. The strength and frequency of modulation for each point are $\Delta \epsilon = 0.1\epsilon _0$ and $\Omega = 20$ GHz, respectively, and there is a modulation phase difference $\phi$ between the two points. This modulated structure can be considered as two time-varying slabs which their thickness much smaller than their distance and wavelength of the incident wave. It should be noted that this structure reduces to free space in the static case.

In order to validate the accuracy of the perturbation approach and investigate the reciprocity of the structure, the parameter $\Delta$ has been calculated over the excitation frequency $f_0$ for three different phase differences, $\phi$ as shown in Fig. 6 (a)–(c). The perturbation and FDTD results are illustrated in Fig. 6 and an acceptable overlap between the two results can be seen. It should be noted that in some excitation frequencies, the structure shows reciprocal behaviour even when the phase difference between the two points is equal to 90 degrees. In these frequencies, the reciprocity condition (7) does not meet yet the integral calculations of Eq. (19) reveals the reciprocity of the structure at those frequencies. Also, the parameter $\Delta$ is calculated in one of the incident frequencies in the range of $\phi = [0^\circ,90^\circ ]$ which shows that the non-reciprocity of the structure is maximized at $\phi = 90^\circ$ as expected.

6. Conclusion

In this paper, we seek to find whether and to what degree, do we truly achieve reciprocity/non-reciprocity in time-varying electromagnetic systems. Starting from Maxwell’s equations and constitutive relations for a causal, local, linear, and isotropic time-periodic medium, Lorentz reciprocity theorem is invoked, leading to a definition for reciprocity/non-reciprocity in such systems. Initially, a sufficient condition is extracted for reciprocity, solely based on the Fourier components of the dynamic constitutive parameters. As this strict sufficient condition is not met in many scenarios, it is shown that further calculations are needed to see if a system is truly reciprocal or not, necessitates the solution to the fields inside the dynamic problem space, in addition to the constitutive parameters. The calculations can also serve as a metric for the degree of non-reciprocity in a TV system. Due to difficulty in finding the fields for general TV problems, a perturbative approach was proposed that is valid for weak time modulations, showing promising results, that were also validated. The benefit of this approach is that the Lorentz condition of reciprocity for time-periodic structures can be evaluated using Green’s function and the electromagnetic fields of the static structure instead, essentially turning off the time variations in the calculations. Two famous and canonical problems were studied: 1) travelling wave modulation and 2) point-like modulation. It is shown that a great agreement is observed between the perturbative approach and FDTD simulations, as well as analytic solutions, were available.

Appendix A. Derivation of Eq. (6)

Maxwell’s equations in form of Eq. (4), for cases "a" and "b" can be expressed as

$$\begin{aligned} & \frac{1}{j\omega_{\textrm{m}}}\nabla\times \mathbf{E}_{\textrm{m}}^{\textrm{a,b}} ={-}\sum_{\textrm{n}}\mu_{\textrm{m-n}}^{\textrm{n}}\mathbf{H}_{\textrm{n}}^{\textrm{a,b}}, \\ & \frac{1}{j\omega_{\textrm{m}}}\nabla\times \mathbf{H}_{\textrm{m}}^{\textrm{a,b}} = \sum_{\textrm{n}}\epsilon_{\textrm{m-n}}^{\textrm{n}}\mathbf{E}_{\textrm{n}}^{\textrm{a,b}} +\frac{1}{j\omega_{\textrm{m}}}\mathbf{J}^{\textrm{a, b}}\delta_{\textrm{m0}}. \end{aligned}$$

Multiplying the so-called counterpart field to each equation yields

$$\begin{aligned} & \frac{1}{j\omega_{\textrm{m}}}\mathbf{H}_{\textrm{m}}^{\textrm{b}}\cdot\nabla\times \mathbf{E}_{\textrm{m}}^{\textrm{a}} ={-}\sum_{\textrm{n}}\mu_{\textrm{m-n}}^{\textrm{n}}\mathbf{H}_{\textrm{m}}^{\textrm{b}}\cdot\mathbf{H}_{\textrm{n}}^{\textrm{a}}, \\ & \frac{1}{j\omega_{\textrm{m}}}\mathbf{E}_{\textrm{m}}^{\textrm{b}}\cdot\nabla\times \mathbf{H}_{\textrm{m}}^{\textrm{a}} = \sum_{\textrm{n}}\epsilon_{\textrm{m-n}}^{\textrm{n}}\mathbf{E}_{\textrm{m}}^{\textrm{b}}\cdot\mathbf{E}_{\textrm{n}}^{\textrm{a}} +\frac{1}{j\omega_{\textrm{m}}}\mathbf{E}_{\textrm{m}}^{\textrm{b}}\cdot\mathbf{J}^{\textrm{a}}\delta_{\textrm{m0}}, \\ & \frac{1}{j\omega_{\textrm{m}}}\mathbf{H}_{\textrm{m}}^{\textrm{a}}\cdot\nabla\times \mathbf{E}_{\textrm{m}}^{\textrm{b}} ={-}\sum_{\textrm{n}}\mu_{\textrm{m-n}}^{\textrm{n}}\mathbf{H}_{\textrm{m}}^{\textrm{a}}\cdot\mathbf{H}_{\textrm{n}}^{\textrm{b}}, \\ & \frac{1}{j\omega_{\textrm{m}}}\mathbf{E}_{\textrm{m}}^{\textrm{a}}\cdot\nabla\times \mathbf{H}_{\textrm{m}}^{\textrm{b}} = \sum_{\textrm{n}}\epsilon_{\textrm{m-n}}^{\textrm{n}}\mathbf{E}_{\textrm{m}}^{\textrm{a}}\cdot\mathbf{E}_{\textrm{n}}^{\textrm{b}} +\frac{1}{j\omega_{\textrm{m}}}\mathbf{E}_{\textrm{m}}^{\textrm{a}}\cdot\mathbf{J}^{\textrm{b}}\delta_{\textrm{m0}}. \end{aligned}$$

Basic manipulations yield

$$\begin{aligned} & -\frac{1}{j\omega_{\textrm{m}}}\nabla\cdot\big[\mathbf{E}_{\textrm{m}}^{\textrm{a}}\times\mathbf{H}_{\textrm{m}}^{\textrm{b}}\big] = \sum_{\textrm{n}}\mu_{\textrm{m-n}}^{\textrm{n}}\mathbf{H}_{\textrm{m}}^{\textrm{b}}\cdot\mathbf{H}_{\textrm{n}}^{\textrm{a}} + \sum_{\textrm{n}}\epsilon_{\textrm{m-n}}^{\textrm{n}}\mathbf{E}_{\textrm{m}}^{\textrm{a}}\cdot\mathbf{E}_{\textrm{n}}^{\textrm{b}} + \frac{1}{j\omega_{\textrm{m}}} \mathbf{E}_{\textrm{m}}^{\textrm{a}}\cdot\mathbf{J}^{\textrm{b}}\delta_{\textrm{m0}}, \\ & -\frac{1}{j\omega_{\textrm{m}}}\nabla\cdot\big[\mathbf{E}_{\textrm{m}}^{\textrm{b}}\times\mathbf{H}_{\textrm{m}}^{\textrm{a}}\big] = \sum_{\textrm{n}}\mu_{\textrm{m-n}}^{\textrm{n}}\mathbf{H}_{\textrm{m}}^{\textrm{a}}\cdot\mathbf{H}_{\textrm{n}}^{\textrm{b}} + \sum_{\textrm{n}}\epsilon_{\textrm{m-n}}^{\textrm{n}}\mathbf{E}_{\textrm{m}}^{\textrm{b}}\cdot\mathbf{E}_{\textrm{n}}^{\textrm{a}} + \frac{1}{j\omega_{\textrm{m}}} \mathbf{E}_{\textrm{m}}^{\textrm{b}}\cdot\mathbf{J}^{\textrm{a}}\delta_{\textrm{m0}}. \\ \end{aligned}$$

Subtracting these two equations and applying summation on $\textrm{m}$ results in

$$\begin{aligned} \sum_{\textrm{m}} \frac{1}{j\omega_{\textrm{m}}}\nabla\cdot\big[\mathbf{E}_{\textrm{m}}^{\textrm{a}}\times\mathbf{H}_{\textrm{m}}^{\textrm{b}} & - \mathbf{E}_{\textrm{m}}^{\textrm{b}}\times\mathbf{H}_{\textrm{m}}^{\textrm{a}}\big] ={-}\sum_{\textrm{m,n}} \mu_{\textrm{m-n}}^{\textrm{n}}\big[\mathbf{H}_{\textrm{m}}^{\textrm{b}}\cdot\mathbf{H}_{\textrm{n}}^{\textrm{a}} - \mathbf{H}_{\textrm{m}}^{\textrm{a}}\cdot\mathbf{H}_{\textrm{n}}^{\textrm{b}}\big] \\ & -\sum_{\textrm{m,n}} \epsilon_{\textrm{m-n}}^{\textrm{n}}\big[\mathbf{E}_{\textrm{m}}^{\textrm{a}}\cdot\mathbf{E}_{\textrm{n}}^{\textrm{b}} - \mathbf{E}_{\textrm{m}}^{\textrm{b}}\cdot\mathbf{E}_{\textrm{n}}^{\textrm{a}}\big] -\frac{1}{j\omega_{\textrm{0}}}\big[\mathbf{E}_{\textrm{0}}^{\textrm{a}}\cdot\mathbf{J}^{\textrm{b}} - \mathbf{E}_{\textrm{0}}^{\textrm{b}}\cdot\mathbf{J}^{\textrm{a}}\big]. \end{aligned}$$

If we take volume integral on all the space, the above result may be written as

$$\begin{aligned} \frac{1}{j\omega_{\textrm{0}}}\int \big[\mathbf{E}_{\textrm{0}}^{\textrm{a}}\cdot\mathbf{J}^{\textrm{b}} - \mathbf{E}_{\textrm{0}}^{\textrm{b}}\cdot\mathbf{J}^{\textrm{a}}\big] dV = & -\int\sum_{\textrm{m,n}} \big(\mu_{\textrm{m-n}}^{\textrm{n}} - \mu_{\textrm{n-m}}^{\textrm{m}}\big)\mathbf{H}_{\textrm{m}}^{\textrm{b}}\cdot\mathbf{H}_{\textrm{n}}^{\textrm{a}} dV \\ & - \int\sum_{\textrm{m,n}} \big(\epsilon_{\textrm{m-n}}^{\textrm{n}} - \epsilon_{\textrm{n-m}}^{\textrm{m}}\big)\mathbf{E}_{\textrm{m}}^{\textrm{a}}\cdot\mathbf{E}_{\textrm{n}}^{\textrm{b}} dV. \end{aligned}$$

Notice that the volume integral on the left part of the Eq. (25) reduces to a surface integral by using Divergence theorem, and vanishes due to the fact that fields become zero at infinity.

Appendix B. Derivation of Eq. (13)

Consider the integral

$$\int\sum_{\textrm{n, m}}\big(\epsilon_{\textrm{m-n}}^{\textrm{n}} - \epsilon_{\textrm{n-m}}^{\textrm{m}}\big)\mathbf{E}_{\textrm{m}}^{\textrm{a}} \cdot \mathbf{E}_{\textrm{n}}^{\textrm{b}} dV = \int\sum_{\textrm{n, m}}\big(\epsilon_{\textrm{m-n}} - \epsilon_{\textrm{n-m}}\big)\mathbf{E}_{\textrm{m}}^{\textrm{a}} \cdot \mathbf{E}_{\textrm{n}}^{\textrm{b}} dV ,$$
in which we substitute Eq. (12). The lowest order term in the reciprocity integral Eq. (27) is
$$\begin{aligned} \int \int & \sum_{\textrm{n, m}} (\epsilon_{\textrm{m-n}}(\mathbf{r}) - \epsilon_{\textrm{n-m}}(\mathbf{r})) \epsilon_{\textrm{n}}(\mathbf{r}') \delta_{\textrm{m0}} (1-\delta_{\textrm{n0}}) \omega_{\textrm{n}}^2\mu_0 \mathbf{E}^{0, a}(\mathbf{r}) \cdot \bar{\bar{\mathbf{G}}}(\mathbf{r}, \mathbf{r}'; \omega_{\textrm{n}})\cdot \mathbf{E}^{0, b}(\mathbf{r}') dVdV'\\ +\int \int & \sum_{\textrm{n, m}} (\epsilon_{\textrm{m-n}}(\mathbf{r}) - \epsilon_{\textrm{n-m}}(\mathbf{r})) \epsilon_{\textrm{m}}(\mathbf{r}') \delta_{\textrm{n0}} (1-\delta_{\textrm{m0}}) \omega_{\textrm{m}}^2\mu_0 \mathbf{E}^{0, b}(\mathbf{r}) \cdot \bar{\bar{\mathbf{G}}}(\mathbf{r}, \mathbf{r}'; \omega_{\textrm{m}})\cdot \mathbf{E}^{0, a}(\mathbf{r}') dVdV'\\ = \int \int & \sum_{\textrm{n}} (\epsilon_{\textrm{-n}}(\mathbf{r}) - \epsilon_{\textrm{n}}(\mathbf{r})) \epsilon_{\textrm{n}}(\mathbf{r}') (1-\delta_{\textrm{n0}}) \omega_{\textrm{n}}^2\mu_0 \mathbf{E}^{0, a}(\mathbf{r}) \cdot \bar{\bar{\mathbf{G}}}(\mathbf{r}, \mathbf{r}'; \omega_{\textrm{n}})\cdot \mathbf{E}^{0, b}(\mathbf{r}') dVdV'\\ -\int \int & \sum_{\textrm{n}} (\epsilon_{\textrm{-n}}(\mathbf{r}) - \epsilon_{\textrm{n}}(\mathbf{r})) \epsilon_{\textrm{n}}(\mathbf{r}') (1-\delta_{\textrm{n0}}) \omega_{\textrm{n}}^2\mu_0 \mathbf{E}^{0, b}(\mathbf{r}) \cdot \bar{\bar{\mathbf{G}}}(\mathbf{r}, \mathbf{r}'; \omega_{\textrm{n}})\cdot \mathbf{E}^{0, a}(\mathbf{r}') dVdV'. \end{aligned}$$

Note that

$$\bar{\bar{\mathbf{G}}}^{\textrm{T}} (\mathbf{r}, \mathbf{r}'; \omega_{\textrm{n}}) = \bar{\bar{\mathbf{G}}}(\mathbf{r}', \mathbf{r}; \omega_{\textrm{n}}).$$

As a result

$$\begin{aligned} \int \sum_{\textrm{n, m}}( & \epsilon_{\textrm{m-n}}(\mathbf{r}) - \epsilon_{\textrm{n-m}}(\mathbf{r})) \mathbf{E}_{\textrm{m}}^a \cdot \mathbf{E}_{\textrm{n}}^b dV = \sum_{\mathrm{n(\ne0)}} \omega_{\textrm{n}}^2 \mu_0 \times \\ & \int \int \big[\epsilon_{\textrm{-n}}(\mathbf{r})\epsilon_{\textrm{n}}(\mathbf{r}') - \epsilon_{\textrm{-n}}(\mathbf{r}')\epsilon_{\textrm{n}}(\mathbf{r})\big] \mathbf{E}^{0, a}(\mathbf{r}) \cdot \bar{\bar{\mathbf{G}}}(\mathbf{r}, \mathbf{r}'m; \omega_{\textrm{n}}) \cdot \mathbf{E}^{0, b}(\mathbf{r}') dV' dV. \end{aligned}$$

Note that, since $\epsilon _{\textrm{}\hbox{-}\textrm{n}}(\mathbf{r}) = \epsilon _{\textrm{n}}^*(\mathbf{r})$, we have

$$\begin{aligned} \int \sum_{\textrm{n, m}}(\epsilon_{\textrm{m-n}} - \epsilon_{\textrm{n-m}}) \mathbf{E}_{\textrm{m}}^a & \cdot \mathbf{E}_{\textrm{n}}^b d V = \mu_0 \sum_{\textrm{n>0}} \int \int \big[\epsilon_{\textrm{n}}^*(\mathbf{r}) \epsilon_{\textrm{n}}(\mathbf{r}') - \epsilon_{\textrm{n}}^*(\mathbf{r}')\epsilon_{\textrm{n}}(\mathbf{r})\big] \mathbf{E}^{0,a}(\mathbf{r}) \cdot\\ & \big[\omega_{\textrm{n}}^2 \bar{\bar{\mathbf{G}}}(\mathbf{r}, \mathbf{r}'; \omega_{\textrm{n}}) - \omega_{\textrm{-n}}^2\bar{\bar{\mathbf{G}}}(\mathbf{r}, \mathbf{r}'; \omega_{\textrm{-n}})\big] \cdot \mathbf{E}^{0, b}(\mathbf{r}')d V^\prime d V. \end{aligned}$$

Appendix C. Calculation of Green’s function and the electromagnetic fields in a static slab

For the fields inside the slab, Fig. 7, we have (Note that $0\le z, z' \le \textrm{L}$)

$$\begin{aligned} & \mathbf{E}^{0, \textrm{a}}(z) ={-}\frac{\eta_1 J_0^{\textrm{a}}}{2}\big(A^{\textrm{a}} e^{{-}jk_1 |z-z_{\textrm{a}}|} + B^{\textrm{a}} e^{jk_1 |z-z_{\textrm{a}}|}\big)\hat{x}, \\ & \mathbf{E}^{0, \textrm{b}}(z') ={-}\frac{\eta_1 J_0^{\textrm{b}}}{2}\big(A^{\textrm{b}} e^{{-}jk_1 |z'-z_{\textrm{b}}|} + B^{\textrm{b}} e^{jk_1 |z'-z_{\textrm{b}}|}\big)\hat{x}. \end{aligned}$$

The coefficients $A^{\textrm{a}}$, $A^{\textrm{b}}$, $B^{\textrm{a}}$, and $B^{\textrm{b}}$ can be found by applying the Boundary conditions. Also $\eta _1 = \sqrt {\mu _0 / \epsilon _0 \epsilon _{\textrm{r}}}$ and $k_1 = \omega _0 \sqrt {\mu _0 \epsilon _0 \epsilon _{\textrm{r}}}$ are impedance and wave number of the static slab. Since $0\le z' \le \textrm{L}$, we need to find the Green function related to a current sheet source (representing the delta function) placed inside the slab. Regarding Fig. 7, depending on to which the parameter $z$ belongs to, the Green’s function can be written as

$$\begin{aligned} & (1): \hat{x} \cdot \bar{\bar{\mathbf{G}}}(z, z'; \omega) \cdot \hat{x} = T_1 \frac{e^{{-}jk_0 |z-z'|}}{j2\omega/c_0 }, \\ & (4): \hat{x} \cdot \bar{\bar{\mathbf{G}}}(z, z'; \omega) \cdot \hat{x} = T_4 \frac{e^{{-}jk_0 |z-z'|}}{j2\omega/c_0 }, \\ & (2): \hat{x} \cdot \bar{\bar{\mathbf{G}}}(z, z'; \omega) \cdot \hat{x} = A^0 \frac{e^{{-}jk_1 |z-z'|}}{j2\omega/c_1} + B^0 \frac{e^{jk_1 |z-z'|}}{j2\omega/c_1}, \\ & (3): \hat{x} \cdot \bar{\bar{\mathbf{G}}}(z, z'; \omega) \cdot \hat{x} = C^0 \frac{e^{{-}jk_1 |z-z'|}}{j2\omega/c_1} + D^0 \frac{e^{jk_1 |z-z'|}}{j2\omega/c_1}. \end{aligned}$$

Having the Green function, we can write the form of fields in each region

$$\begin{aligned} & (1): \mathbf{E}_1 (z) ={-}\frac{\eta_0 T_1}{2}J_0 e^{{-}jk_0 |z-z'|} \hat{x}, \\ & (1): \mathbf{H}_1 (z) ={-}\frac{J_0 T_1}{2}Je^{{-}jk_0 |z-z'|} \hat{y}, \\ & (2): \mathbf{E}_2(z) = \bigg[-\frac{\eta_1 A^0}{2}J_0 e^{{-}jk_1 |z-z'|} - \frac{\eta_1 B^0}{2}J_0 e^{jk_1 |z-z'|}\bigg] \hat{x}, \\ & (2):\mathbf{H}_2(z) = \bigg[-\frac{A^0 J_0}{2} e^{{-}jk_1 |z-z'|} + \frac{B^0 J_0}{2} e^{jk_1 |z-z'|}\bigg] \hat{y}, \\ & (3): \mathbf{E}_3(z) = \bigg[-\frac{\eta_1 C^0}{2}J_0 e^{{-}jk_1 |z-z'|} - \frac{\eta_1 D^0}{2}J_0 e^{jk_1 |z-z'|}\bigg] \hat{x}, \\ & (3):\mathbf{H}_3(z) = \bigg[-\frac{C^0 J_0}{2} e^{{-}jk_1 |z-z'|} + \frac{C^0 J_0}{2} e^{jk_1 |z-z'|}\bigg] \hat{y}, \\ & (4): \mathbf{E}_4 (z) ={-}\frac{\eta_0 T_4}{2}J_0 e^{{-}jk_0 |z-z'|} \hat{x}, \\ & (4): \mathbf{H}_4 (z) ={-}\frac{J_0 T_4}{2}Je^{{-}jk_0 |z-z'|} \hat{y}. \end{aligned}$$

Finding the coefficients $A^0$, $B^0$, $C^0$, $D^0$, $T_1$, and $T_4$ requires applying the following boundary conditions

$$\begin{aligned} & (1): \mathbf{E}_1 = \mathbf{E}_2 , \quad \mathbf{H}_1 = \mathbf{H}_2 \\ & (2): \mathbf{E}_2 = \mathbf{E}_3 , \quad \big[\mathbf{H}_2 - \mathbf{H}_3\big].\hat{y} = J_0 \\ & (3): \mathbf{E}_3 = \mathbf{E}_4 , \quad \mathbf{H}_3 = \mathbf{H}_4 . \end{aligned}$$

 figure: Fig. 7.

Fig. 7. Calculating Green’s function and electric fields of the static structure.

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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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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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Figures (7)
Fig. 1.
Fig. 1. The electromagnetic interaction between two current sources $\mathbf{J}^{\textrm{a}}$ and $\mathbf{J}^{\textrm{b}}$ in a space-time dependent medium.

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Fig. 2.
Fig. 2. Demonstration of a non-dispersive slab with travelling-wave modulation which is surrounded by free space. Two current sources $\mathbf{J}^{\textrm{a}}$ and $\mathbf{J}^{\textrm{b}}$ excite this structure in cases "a" and "b", respectively. This structure is invariant under transformation in the x-y plane.

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Fig. 3.
Fig. 3. Demonstration of structure with point-like modulation. Two distributive points’ permittivity oscillates in time with frequency of $\Omega$ and phase different $\phi$ .

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Fig. 4.
Fig. 4. $\Delta$ vs. strength of modulation for different $\Omega$ . For 3 cases (a) $\Omega = 10$ GHz, (b) $\Omega = 20$ GHz and (c) $\Omega = 50$ GHz the results of three different methods have been demonstrated.

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Fig. 5.
Fig. 5. $\Delta$ vs. excitation frequency for different $\epsilon _{\textrm{r}}$ . For 3 cases (a) $\epsilon _{\textrm{r}} = 1$ , (b) $\epsilon _{\textrm{r}} = 3$ and (c) $\epsilon _{\textrm{r}} = 5$ the results of three different methods have been demonstrated.

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Fig. 6.
Fig. 6. (a),(b),(c). $\Delta$ vs. excitation frequency for different $\phi$ . For 3 cases (a) $\phi = 5^{\circ}$ , (b) $\phi = 45^{\circ}$ and (c) $\phi = 90^{\circ}$ the results of two different methods, perturbation theory and FDTD solution, have been demonstrated. Based on the perturbative solution, we can state that these diagrams are approximately different by a factor of $\sin {\phi }$ .(d). $\Delta$ vs. $\phi$ for excitation frequency $f_0 = 400$ GHz.

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Fig. 7.
Fig. 7. Calculating Green’s function and electric fields of the static structure.

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Equations (35)

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B ( r , t ) = ∫ − ∞ t μ ~ ( r , t ; t − t ′ ) H ( r , t ′ ) d t ′ , D ( r , t ) = ∫ − ∞ t ϵ ~ ( r , t ; t − t ′ ) E ( r , t ′ ) d t ′ ,
ϵ ~ ( r , t ; τ ) = ∑ n e j n Ω t ϵ ~ n ( r , τ ) , μ ~ ( r , t ; τ ) = ∑ n e j n Ω t μ ~ n ( r , τ ) .
E ( r , t ) = ∑ m E m ( r ) e j ω m t + c . c . , H ( r , t ) = ∑ m H m ( r ) e j ω m t + c . c . ,
1 j ω m ∇ × E m ( r ) = − ∑ n μ m-n ( r , ω n ) H n ( r ) , 1 j ω m ∇ × H m ( r ) = ∑ n ϵ m-n ( r , ω n ) E n ( r ) + 1 j ω m J ( r ) δ m0 ,
1 j ω 0 ∫ E 0 a ⋅ J b d V b = 1 j ω 0 ∫ E 0 b ⋅ J a d V a ,
1 j ω 0 ∫ [ E 0 a ⋅ J b − E 0 b ⋅ J a ] d V = − ∫ ∑ m,n ( μ m-n n − μ n-m m ) H m b ⋅ H n a d V − ∫ ∑ m,n ( ϵ m-n n − ϵ n-m m ) E m a ⋅ E n b d V .
ϵ m-n n = ϵ n-m m , μ m-n n = μ n-m m ,
∇ × E m ( r ) = − j ω m μ 0 H m ( r ) , ∇ × H m ( r ) = j ω m ϵ 0 ( r ) E m ( r ) + J m  e q ( r ) + J ( r ) δ m0 ,
J m eq ( r ) = j ω m ∑ n ( ≠ m ) ϵ m-n ( r ) E n ( r ) .
E m ( r ) − ω m 2 μ 0 ∑ n ( ≠ m ) ∫ G ¯ ¯ ( r , r ′ ; ω m ) ⋅ ϵ m-n ( r ′ ) E n ( r ′ ) d V ′ = δ m0 E 0 ( r ) ,
E 0 ( r ) = − j ω 0 μ 0 ∫ G ¯ ¯ ( r , r ′ ; ω 0 ) ⋅ J ( r ′ ) d V ′
E m ( r ) = δ m0 E 0 ( r ) + ω m 2 μ 0 ∑ n ( ≠ m ) ∫ G ¯ ¯ ( r , r ′ ; ω m ) ⋅ ϵ m-n ( r ′ ) δ n0 E 0 ( r ′ ) d V ′ = δ m0 E 0 ( r ) + ( 1 − δ m0 ) ω m 2 μ 0 ∫ G ¯ ¯ ( r , r ′ ; ω m ) ⋅ ϵ m ( r ′ ) E 0 ( r ′ ) d V ′ .
∫ ∑ n, m ( ϵ m-n − ϵ n-m ) E m a ⋅ E n b d V = μ 0 ∑ n>0 ∫ ∫ [ ϵ n ∗ ( r ) ϵ n ( r ′ ) − ϵ n ∗ ( r ′ ) ϵ n ( r ) ] E 0 , a ( r ) ⋅ [ ω n 2 G ¯ ¯ ( r , r ′ ; ω n ) − ω -n 2 G ¯ ¯ ( r , r ′ ; ω -n ) ] ⋅ E 0 , b ( r ′ ) d V ′ d V ,
ϵ ~ ( r , t ; τ ) = [ ϵ 0 ϵ r + Δ ϵ cos ⁡ ( q z − Ω t ) ] δ ( τ + ) ,
∫ [ E 0 a ⋅ J b − E 0 b ⋅ J a ] d V = − j ω 0 μ 0 ( Δ ϵ 2 ) 2 ∫ z = 0 z = L ∫ z ′ = 0 z ′ = L [ e j q ( z − z ′ ) − e − j q ( z − z ′ ) ] E 0 , a ( z ) ⋅ [ ω 1 2 G ¯ ¯ ( z , z ′ ; ω 1 ) − ω − 1 2 G ¯ ¯ ( z , z ′ ; ω − 1 ) ] ⋅ E 0 , b ( z ′ ) d z d z ′ .
ϵ ~ ( r , t ; τ ) = ∑ n ϵ n ( r ) e j n Ω t δ ( τ + ) ,
ϵ n ( r ) = ∑ p = 1 P n ϵ n ( p ) δ ( r − r p ) ,
∫ [ E 0 a ⋅ J b − E 0 b ⋅ J a ] d V = μ 0 ∑ n > 0 ∑ p ∑ q [ ( ϵ n ( p ) ) ∗ ϵ n ( q ) − ( ϵ n ( q ) ) ∗ ϵ n ( p ) ] × E 0 , a ( r p ) ⋅ [ ω n 2 G ¯ ¯ ( r p , r q ; ω n ) − ω − n 2 G ¯ ¯ ( r p , r q ; ω n ) ] ⋅ E 0 , b ( r q ) .
∫ [ E 0 a ⋅ J b − E 0 b ⋅ J a ] d V = j 2 μ 0 | ϵ 1 ( 1 ) | | ϵ 1 ( 2 ) | sin ⁡ ( Θ 1 ( 1 ) − Θ 1 ( 2 ) ) × [ E 0 , a ( r 1 ) ⋅ [ ω 1 2 G ¯ ¯ ( r 1 , r 2 ; ω 1 ) − ω − 1 2 G ¯ ¯ ( r 1 , r 2 ; ω − 1 ) ] ⋅ E 0 , b ( r 2 ) − E 0 , a ( r 1 ) ⋅ [ ω 1 2 G ¯ ¯ ( r 1 , r 2 ; ω 1 ) − ω − 1 2 G ¯ ¯ ( r 1 , r 2 ; ω − 1 ) ] ⋅ E 0 , b ( r 2 ) ] .
∇ × ∇ × E ( r , t ) = − μ 0 ∂ 2 ∂ t 2 D ( r , t ) − μ 0 ∂ ∂ t J ( r , t )
Δ = | ∫ [ E 0 a ⋅ J b − E 0 b ⋅ J a ] d V / [ − η 0 2 J 0 a J 0 b e − j k 0 ( z b − z a ) ] | ,
1 j ω m ∇ × E m a,b = − ∑ n μ m-n n H n a,b , 1 j ω m ∇ × H m a,b = ∑ n ϵ m-n n E n a,b + 1 j ω m J a, b δ m0 .
1 j ω m H m b ⋅ ∇ × E m a = − ∑ n μ m-n n H m b ⋅ H n a , 1 j ω m E m b ⋅ ∇ × H m a = ∑ n ϵ m-n n E m b ⋅ E n a + 1 j ω m E m b ⋅ J a δ m0 , 1 j ω m H m a ⋅ ∇ × E m b = − ∑ n μ m-n n H m a ⋅ H n b , 1 j ω m E m a ⋅ ∇ × H m b = ∑ n ϵ m-n n E m a ⋅ E n b + 1 j ω m E m a ⋅ J b δ m0 .
− 1 j ω m ∇ ⋅ [ E m a × H m b ] = ∑ n μ m-n n H m b ⋅ H n a + ∑ n ϵ m-n n E m a ⋅ E n b + 1 j ω m E m a ⋅ J b δ m0 , − 1 j ω m ∇ ⋅ [ E m b × H m a ] = ∑ n μ m-n n H m a ⋅ H n b + ∑ n ϵ m-n n E m b ⋅ E n a + 1 j ω m E m b ⋅ J a δ m0 .
∑ m 1 j ω m ∇ ⋅ [ E m a × H m b − E m b × H m a ] = − ∑ m,n μ m-n n [ H m b ⋅ H n a − H m a ⋅ H n b ] − ∑ m,n ϵ m-n n [ E m a ⋅ E n b − E m b ⋅ E n a ] − 1 j ω 0 [ E 0 a ⋅ J b − E 0 b ⋅ J a ] .
1 j ω 0 ∫ [ E 0 a ⋅ J b − E 0 b ⋅ J a ] d V = − ∫ ∑ m,n ( μ m-n n − μ n-m m ) H m b ⋅ H n a d V − ∫ ∑ m,n ( ϵ m-n n − ϵ n-m m ) E m a ⋅ E n b d V .
∫ ∑ n, m ( ϵ m-n n − ϵ n-m m ) E m a ⋅ E n b d V = ∫ ∑ n, m ( ϵ m-n − ϵ n-m ) E m a ⋅ E n b d V ,
∫ ∫ ∑ n, m ( ϵ m-n ( r ) − ϵ n-m ( r ) ) ϵ n ( r ′ ) δ m0 ( 1 − δ n0 ) ω n 2 μ 0 E 0 , a ( r ) ⋅ G ¯ ¯ ( r , r ′ ; ω n ) ⋅ E 0 , b ( r ′ ) d V d V ′ + ∫ ∫ ∑ n, m ( ϵ m-n ( r ) − ϵ n-m ( r ) ) ϵ m ( r ′ ) δ n0 ( 1 − δ m0 ) ω m 2 μ 0 E 0 , b ( r ) ⋅ G ¯ ¯ ( r , r ′ ; ω m ) ⋅ E 0 , a ( r ′ ) d V d V ′ = ∫ ∫ ∑ n ( ϵ -n ( r ) − ϵ n ( r ) ) ϵ n ( r ′ ) ( 1 − δ n0 ) ω n 2 μ 0 E 0 , a ( r ) ⋅ G ¯ ¯ ( r , r ′ ; ω n ) ⋅ E 0 , b ( r ′ ) d V d V ′ − ∫ ∫ ∑ n ( ϵ -n ( r ) − ϵ n ( r ) ) ϵ n ( r ′ ) ( 1 − δ n0 ) ω n 2 μ 0 E 0 , b ( r ) ⋅ G ¯ ¯ ( r , r ′ ; ω n ) ⋅ E 0 , a ( r ′ ) d V d V ′ .
G ¯ ¯ T ( r , r ′ ; ω n ) = G ¯ ¯ ( r ′ , r ; ω n ) .
∫ ∑ n, m ( ϵ m-n ( r ) − ϵ n-m ( r ) ) E m a ⋅ E n b d V = ∑ n ( ≠ 0 ) ω n 2 μ 0 × ∫ ∫ [ ϵ -n ( r ) ϵ n ( r ′ ) − ϵ -n ( r ′ ) ϵ n ( r ) ] E 0 , a ( r ) ⋅ G ¯ ¯ ( r , r ′ m ; ω n ) ⋅ E 0 , b ( r ′ ) d V ′ d V .
∫ ∑ n, m ( ϵ m-n − ϵ n-m ) E m a ⋅ E n b d V = μ 0 ∑ n>0 ∫ ∫ [ ϵ n ∗ ( r ) ϵ n ( r ′ ) − ϵ n ∗ ( r ′ ) ϵ n ( r ) ] E 0 , a ( r ) ⋅ [ ω n 2 G ¯ ¯ ( r , r ′ ; ω n ) − ω -n 2 G ¯ ¯ ( r , r ′ ; ω -n ) ] ⋅ E 0 , b ( r ′ ) d V ′ d V .
E 0 , a ( z ) = − η 1 J 0 a 2 ( A a e − j k 1 | z − z a | + B a e j k 1 | z − z a | ) x ^ , E 0 , b ( z ′ ) = − η 1 J 0 b 2 ( A b e − j k 1 | z ′ − z b | + B b e j k 1 | z ′ − z b | ) x ^ .
( 1 ) : x ^ ⋅ G ¯ ¯ ( z , z ′ ; ω ) ⋅ x ^ = T 1 e − j k 0 | z − z ′ | j 2 ω / c 0 , ( 4 ) : x ^ ⋅ G ¯ ¯ ( z , z ′ ; ω ) ⋅ x ^ = T 4 e − j k 0 | z − z ′ | j 2 ω / c 0 , ( 2 ) : x ^ ⋅ G ¯ ¯ ( z , z ′ ; ω ) ⋅ x ^ = A 0 e − j k 1 | z − z ′ | j 2 ω / c 1 + B 0 e j k 1 | z − z ′ | j 2 ω / c 1 , ( 3 ) : x ^ ⋅ G ¯ ¯ ( z , z ′ ; ω ) ⋅ x ^ = C 0 e − j k 1 | z − z ′ | j 2 ω / c 1 + D 0 e j k 1 | z − z ′ | j 2 ω / c 1 .
( 1 ) : E 1 ( z ) = − η 0 T 1 2 J 0 e − j k 0 | z − z ′ | x ^ , ( 1 ) : H 1 ( z ) = − J 0 T 1 2 J e − j k 0 | z − z ′ | y ^ , ( 2 ) : E 2 ( z ) = [ − η 1 A 0 2 J 0 e − j k 1 | z − z ′ | − η 1 B 0 2 J 0 e j k 1 | z − z ′ | ] x ^ , ( 2 ) : H 2 ( z ) = [ − A 0 J 0 2 e − j k 1 | z − z ′ | + B 0 J 0 2 e j k 1 | z − z ′ | ] y ^ , ( 3 ) : E 3 ( z ) = [ − η 1 C 0 2 J 0 e − j k 1 | z − z ′ | − η 1 D 0 2 J 0 e j k 1 | z − z ′ | ] x ^ , ( 3 ) : H 3 ( z ) = [ − C 0 J 0 2 e − j k 1 | z − z ′ | + C 0 J 0 2 e j k 1 | z − z ′ | ] y ^ , ( 4 ) : E 4 ( z ) = − η 0 T 4 2 J 0 e − j k 0 | z − z ′ | x ^ , ( 4 ) : H 4 ( z ) = − J 0 T 4 2 J e − j k 0 | z − z ′ | y ^ .
( 1 ) : E 1 = E 2 , H 1 = H 2 ( 2 ) : E 2 = E 3 , [ H 2 − H 3 ] . y ^ = J 0 ( 3 ) : E 3 = E 4 , H 3 = H 4 .
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