Frequency conversion in time-varying graphene microribbon arrays

Mohammadreza Salehi; Pegah Rahmatian; Mohammad Memarian; Khashayar Mehrany

doi:10.1364/OE.467479

1. Introduction

Metasurfaces have brought about a revolution in the field of Electromagentics, in that they provided us with extraordinary and curiously versatile ways of controlling light by means of spatially structuring materials [1–3]. They consist of sub-wavelength scatterers and resonators densely packed in a two-dimensional array; organized in a manner so as to customize the overall response of the metasurface by locally altering the amplitude, phase, and polarization state of light waves. Numerous applications have been proposed for metasurfaces, such as flat lenses [4,5], holograms [6–9], polarization converters [10–12], anomalous deflectors [13–16], and photonic spin Hall effects [17,18] to name a few.

These advances occasioned the introduction of a generalized law of reflection and refraction which accounts for the spatially varying phase response of metasurfaces [3]. This in turn paved the way for the creation of ultrathin, flat optical components capable of manipulating light in ways that were not accessible before. With all this at hand, it was only a matter of time before methods of controlling light by structuring metasurfaces in time domain appeared; inasmuch as space and time play similar roles in Maxwell’s equations [19]. Although the first examples of "active metasurfaces" were simply static metasurfaces with additional features to adjust the optical response with an external stimulus [20], soon, attempts turned towards dynamic manipulation of light with fast temporal as well as spatial modulation of material properties owing to its potential for unlocking new physical phenomena [19].

The coming of tunable and ultrathin van der Waals materials [21], such as graphene, and thin-film semiconductors such as indium tin oxide [22], played a major role in the surfacing of new possibilities to make proper use of temporal modulation in the form of time-varying metasurfaces. Graphene has already shown promise in allowing us to exploit temporal modulation as a means to effectively control light waves. As an example, in [23] the temporal counterpart of the Wood anomaly has been proposed which proved superior in coupling propagating waves to surface waves since it manages to replace spatial with temporal modulation and thereby omit the need for subwavelength structuring of material which is known to give rise to problems with regard to cost-effectiveness, reversibility, and also losses.

Having already racked up various achievements in the field of electromagnetics, time-varying systems are being extensively studied in connection with microwaves and optics, suggesting that they are likely to play a determinant role in the future of devices. From them, several novel approaches to classical electromagnetic problems have originated, namely parametric amplification [24,25], non-reciprocity without using magnets [26–28], frequency conversion [29], frequency combs [30], and indeed time-varying metasurfaces [31–33].

In an endeavor to utilize time-varying metasurfaces, a structure comprising temporally modulated graphene microribbon array (GMRAs) has been developed for inducing a change of frequency into incoming light waves [34]. The whole exercise is predicated on extending the generalized Snell’s law to include time variations by relaxing the constraints on the conservation of energy [35]. The method presented in [34] for achieving frequency conversion bears a likeness to that used in the case of a spatial metasurface; in that the objective is to devise a time-gradient phase variation which, similar to its spatial analogue, is expected to produce the desired change in wavelength. Although the method used, roughly complies with prescribed instructions for designing metasurfaces, it leaves out some important details. Furthermore, no attempts whatsoever has been made to validate the results obtained by the static analysis. This indeed calls for an investigation to assess the validity of the claim and to clarify the underlying principles for frequency conversion in time-varying metasurfaces.

To this end, in this paper, we set out to carefully examine the phenomenon of frequency conversion in the context of temporally modulated GMRAs. We discuss the shortcomings that accompany such devices when it comes to manipulating light in time domain and we seek to alleviate these limitations by utilizing multilayered structures. The task is carried out in the following fashion: In Section 2, we present the structure in question and provide solutions for the problem of normal incidence upon it. In Section 3, we proceed by introducing a scheme, comprising a temporal phase gradient, to exploit the results of the previous section to reach a desired frequency change in the incident wave. In Section 4, we examine the usefulness of the proposed scheme by rigorously analyzing the problem with the aid of spatiotemporal Fourier modal method (FMM) and also by providing full-wave simulation results. Lastly, concluding remarks are contained in Section 5. A time dependence of the form $e^{-i\omega t}$ is assumed throughout the article.

2. Graphene microribbon array

Graphene seems to be an ideal platform for realizing time-varying metasurfaces, since by applying a bias voltage and thereby tuning its Fermi level (electrically induced carrier doping), its conductivity can be temporally controlled [36]. It also permits fast modulation rates, having shown a response time of approximately 2.2 ps [37–39]. Graphene can be modeled as a 2D conductive film with a surface conductivity $\sigma _{s}$ given by the Kubo formula [40], which at sufficiently low frequencies ($\hbar \omega \ll 2E_{F}$), given that the condition $kBT \ll E_{F}$ holds, reduces to the Drude form:

(1)$$\sigma_{s}(\omega) = \frac{e^2 E_{F}}{\pi \hbar^2}\;\frac{i}{\omega+i\tau^{{-}1}}\ .$$

where $e$ is the electron charge, $\hbar$ is the reduced Planck constant, $E_{F}$ is the Fermi energy, $\omega$ is the angular frequency, and $\tau$ is the relaxation time which is given by $\mu E_{F}/ev_{F}^2$ with $v_{F}$ being the Fermi velocity and $\mu$ the electron mobility.

As is costumary with spatially gradient metasurfaces, we are mainly concerned with the phase change resulting from alteration of a control parameter (here $E_{F}$). In order to achieve an appreciable amount of phase change, GMRAs were put to use in [34]; the reason being the fact that these devices make for efficient excitation of the localized surface plasmons at terahertz and infrared wavelengths [41]. Figure 1(a) shows a schematic illustration of a single-layer GMRA terminating in a perfect electric conductor (PEC), as proposed in [34]. Also shown, in Fig. 1(b), is a generic multilayered graphene microribbon array (MGMRA), which stands as our own proposal for efficient frequency conversion.

Fig. 1. Schematic drawing of (a) the structure proposed in [34] and (b) a generic multi-layer graphene microribbon array terminating in a PEC. The structures are periodic in $x$ with period $L$. The relative electric permittivty of region $n$ and the width of the graphene ribbons on top of it are designated $\varepsilon _n$ and $w_n$ respectively.

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Polyimide is chosen herein as the substrate due to its suitability at Terahertz range. It has a small refractive index, low absorption, and its thickness can be precisely adjusted within the range of a few micrometers via spin-coating [42]. On top of that, it is relatively easy to grow and transfer graphene onto a polyimide substrate [43], where an abundance of lithographic techniques can be used for micro- and nano-scale patterning of graphene [44]. A structure much similar to the MGMRA of Fig. 1(b) is demonstrated in [45], where multiple layers of aluminum microribbon arrays were embedded in polyimide via repeated stages of layer-by-layer spin-coating, metal deposition, photolithography, and thermal fusion of successive layers of polyimide. Similar processes may be used for fabricating MGMRAs on account of their compatibility with graphene. Also, time-modulation of graphene ribbons can be achieved by applying a time-varying voltage via conductive contacts between the graphene layers and the PEC reflector [46].

We start off the analysis of GMRAs by attending to the single-layer structure of [34]. The ribbon width and the periodicity of the structure are chosen to be 42.5 and 50 µm; the refractive index of the spacer (polyimide) is set to $n_1 = 1.7$ and its thickness is $z_1=10$ µm. Also, $\mu = 5 \text { m}^2/(\text {V} \cdot \text {s})$. We consider the problem of normal incidence where a TM polarized plane wave (magnetic field along the y direction) with a free-space wavelength of $\lambda _{0} = 290$ µm impinges upon the structure. Following [47], we solve the problem by using FMM; the result of which is presented in Fig. 2. Amplitude and phase of the reflected wave is evaluated for a range of $E_{F}$ and a resonant behavior can be easily discerned from an inspection of Fig. 2. Varying $E_{F}$ in the vicinity of the point of resonance can yield a substantial phase change which although it does not amount to $2\pi$, according to [34], it is just enough to prompt a frequency shift. This assertion, as we will see, is inaccurate.

Fig. 2. (a) Amplitude of reflected wave versus Fermi energy for the problem of TM polarized normal incidence ($\lambda _0 = 290$ µm) upon the structure of Fig. 1 with $L = 50$ µm, $w_1 = 42.5$ µm, $z_1 = 10$ µm, and $\varepsilon _1= 1.7^2$. (b) Phase of the reflected wave for the same problem.

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3. Time-gradient phase variation

Let us consider a hypothetical reflecting structure with an electromagnetic response characterized by the function $\Gamma (V)$ representing the reflection coefficient as a function of a control parameter $V$. Now assume that $V$ is to be modulated in time with the frequency of modulation being considerably lower than that of the incident light. Given the situation outlined above, it is only logical to apply a quasi-static analysis to the problem; which is to say that the solution is, approximately:

(2)$$\mathbf{E}_{ref} = \mathbf{E}_{0} \Gamma (V(t)) e^{{-}i(\omega t-k z)} ,$$

where $\mathbf {E}_{0}$ and $\mathbf {E}_{ref}$ represent the incident and the reflected wave respectively.

We now proceed by inquiring into the specific qualities that the structure should possess in order to be able to produce frequency conversion. Assume:

(3)$$\Gamma (V(t)) = A(t) e^{i\phi(t)} ,$$

where $A(t)$ and $\phi (t)$ denote the amplitude and phase response of the device as a function of time. Provided that $A(t)$ is a constant and that $\phi (t)$ changes linearly during the course of a period, covering a total of exactly $2\pi$ rad, it is easy to see that, since the phase function can be unwrapped to form a simple line, a change in frequency follows:

(4)$$\begin{aligned} A(t)&=A_0\rightarrow\mathbf{E}_{ref} = A_0\mathbf{E}_{0} e^{{-}i[\omega t-\phi (t)-k z]}, \\ \phi(t) &={-}\Omega t \rightarrow \mathbf{E}_{ref} = A_0\mathbf{E}_{0} e^{{-}i[(\omega+\Omega) t-k z]}. \end{aligned}$$

To illustrate this point, we take as an example a case where an amplitude modulated Gaussian pulse is fed to the structure, purposing a raise in carrier frequency. Figure 3(a) gives a taste of what a quasi-static solution to this problem might look like when all is taken care of, that is the amplitude $A(t)$ and the phase $\phi (t)$ abide by the above instructions. If, however, the phase coverage is anything short of $2\pi$, discontinuities will emerge in the output signal; as is shown in Fig. 3(b) which pertains to the case of $\pi$ phase coverage and constant amplitude. Apart from distorting the signal, discontinuities are liable to introduce unwanted frequency components which, in a practical setting, translates into adjacent channel interference.

Fig. 3. A typical quasi-static solution when the input is a Gaussian pulse, $A(t)$ is constant, and $\phi (t)$ progresses linearly, covering (a) $2\pi$ and (b) $\pi$ radians throughout the course of a period.

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These principles are also applicable to the case of a GMRA. Essentially, achieving frequency conversion comes down to whether the structure in question can dynamically modulate the reflection phase over a range of $2\pi$ while maintaining a constant amplitude. Single-layer GMRAs fall short of the criteria. However in contrast to what we have shown here, [34] claims that the above requirements are not necessary when it comes to time-varying metasurfaces; although it does not present any evidence to substantiate the claim. In order to further clarify the subject, we carry out a rigorous analytical investigation of the matter in the next section.

4. Time-varying graphene microribbon array

For the problem of time-varying graphene, we are confronted with a lack of an established model concerning the interaction of light and graphene under modulation. Yet, we can seek to circumvent such a difficulty by resorting to perturbative approaches, as is done in [23]. To this end, we start from the Drude equation:

(5)$$\frac {d J_{x}}{d t} + \gamma J_{x}(t) = W_D E_x(x,t) ,$$

with $J_{x}$ being the surface current density, $W_D=e^2E_F/\pi \hbar ^2$ the Drude weight, $\gamma = 1/\tau$ the loss rate, and $E_x$ the in-plane electric field, and by introducing a time-varying Fermi energy, we arrive at:

(6)$$\frac {d J_{x}}{d t} + \frac{\gamma_0}{\xi(t)} J_{x}(t) = \xi(t)W_{D,0}(x) E_x(x,t) ,$$

where $W_{D,0}$ and $\gamma _0$ pertain to a Fermi level of $E_{F,0} = 1 \text { eV}$ and $\xi (t)$ constitutes the dimensionless modulation term. Also, $W_{D,0}(x)$ represents the conductivity profile of an array of graphene ribbons; it equals $e^2E_{F,0}/\pi \hbar ^2$ wherever graphene exists and vanishes elsewhere.

Given a periodic modulation, the problem of solving Maxwell’s equations along with Eq. (6) is best addressed by invoking Floquet-Bloch theorem. The details of the treatment is available in the Appendix. We consider the same structure described in Section 2 but now modulated, according to Fig. 4(a), with a modulation frequency of 10 GHz. This specific modulation pattern is proposed in [34] and we intend to ascertain whether or not it really holds promise. Results coming from analytical treatment of the problem as well as numerical simulation (Finite Element Time Domain via COMSOL Multiphysics v5.6) are plotted in Fig. 4(b). The results appertain to the electric field and in the case of FMM, they belong to the zeroth diffraction order. The numerical simulations are carried out with incident waves in the form of Gaussian pulses.

Fig. 4. (a) The temporal modulation pattern proposed in [34]. (b) Analytical and numerical solution to the problem of modulated GMRA. Same values are retained for those parameters specified in Fig. 2. the modulation frequency is $\Omega = 2\pi \times 10$ GHz.

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From observation of Fig. 2(b) it follows that upon varying $E_F$ across the specified range, a total phase change of $\Delta \phi \approx 3$ rad is collected, which according to [34], implies a change in angular frequency equal to $\Delta \omega = \Delta \phi / T$ where $T$ is the modulation period; here in particular, a frequency shift equal to about half the modulation frequency. This is clearly not consistent with the results presented here. However, the results obtained from the quasi-static analysis (Section 3) correspond closely to those obtained from FMM.

Now that we have demonstrated the practicality of the quasi-static analysis, we emphasize once again that attaining perfect frequency shift (necessarily an integer multiple of modulation frequency) consists in simultaneously acquiring a reflection phase spanning $2\pi$ radians (or an integer multiple of it for larger frequency shifts) and a constant reflection amplitude in a certain interval of $E_F$. Single-layer GMRAs cannot satisfy this requirement; the phase change hardly reaches $2\pi$ and if—by employing a modulation pattern stretching over an unfavorably vast range of $E_F$—it gets close to $2\pi$, it will always be accompanied by a sharp drop in amplitude which violates the latter part of the above condition. Hence, we shift our focus to MGMRAs and see if they prevail in that respect.

The idea is that by supplementing the aforementioned structure with additional layers of GMRA, one can elicit an electromagnetic response comprising multiple resonances and hence an enhanced phase variation. One also has to take care that the dips in reflection amplitude are aligned and that they are in close proximity to one another so that the amplitude response undergoes minimal undulation in the Fermi energy range of interest. Let us elucidate the matter with some examples.

We begin by considering a two-layered structure with the following characteristics: $w_1 = 0.88 L$, $w_2 = 0.8 L$, $z_1 = 7$ µm, $z_2 = 14$ µm, and $n_1 = n_2 = 1.7$. The other parameters remain unchanged. For simplicity, we assume that all graphene layers share the same Fermi energy. The solution to the problem of TM polarized normal incidence upon this structure is displayed in Fig. 5 (we have used [47]). As it happens, the reflection phase change extends over a range well over $2\pi$. Moreover, it is possible to modulate the device in a way that the reflection amplitude remains fairly constant. One such modulation pattern and its corresponding amplitude and phase profile (amplitude and phase of $\Gamma (E_F(t))$) is depicted in Fig. 6. Note that the phase profile should be strictly linear (refer to section 3) which necessitates $E_F(t)$ being devised based on the device’s phase response (Fig. 5(b)). The time-varying problem is solved using FMM and the results for 1 GHz and 10 GHz modulation frequencies is presented in Fig. 6. In the case of 1 GHz modulation frequency, 65% of input power is converted into the desired output (conversion efficiency). Also, the total output power amounts to 66% of that of the input which suggests that the output signal is subject to very little distortion. With 10 GHz modulation frequency, these figures get to be 63% and 68% respectively.

Fig. 5. (a) Amplitude of reflected wave versus Fermi energy for the problem of TM polarized normal incidence ($\lambda _0 = 290$ µm) upon an MGMRA with the following characteristics: $L = 50$ µm, $w_1 = 0.88 L$, $w_2 = 0.8 L$, $z_1 = 7$ µm, $z_2 = 14$ µm, and $n_1 = n_2 = 1.7$. (b) Phase of the reflected wave for the same problem.

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Fig. 6. (a) A proper modulation pattern for the structure of Fig. 5 and its corresponding (b) amplitude and (c) phase profile. Solution to the specified time-varying problem with modulation frequencies equal to (d) $\Omega = 2\pi \times 1$ GHz and (e) $\Omega = 2\pi \times 10$ GHz. The result indicates a frequency shift of value +1.

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Now assume a three-layered structure with $w_1 = 0.95 L$, $w_2 = 0.85 L$, $w_3 = 0.75 L$, $z_1 = 5$ µm, $z_2 = 10$ µm, $z_3 = 15$ µm, and $n_1 = n_2 = n_3 = 1.7$. Figure 7 shows the solution to the static problem. This time, the phase change exceeds $4\pi$ which renders the circumstances most opportune to introduce a scheme where the frequency shift is twice the modulation frequency. The proper modulation pattern and the solution to the time-varying problem is given in Fig. 8. The conversion efficiencies are 66% for 1 GHz and 61% for 10 GHz modulation frequencies with the total output power being 67% and 68% of the input power respectively.

Fig. 7. (a) Amplitude of reflected wave versus Fermi energy for the problem of TM polarized normal incidence ($\lambda _0 = 290$ µm) upon an MGMRA with the following characteristics: $L = 50$ µm, $w_1 = 0.95 L$, $w_2 = 0.85 L$, $w_3 = 0.75 L$, $z_1 = 5$ µm, $z_2 = 10$ µm, $z_3 = 15$ µm, and $n_1 = n_2 = n_3 = 1.7$. (b) Phase of the reflected wave for the same problem.

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Fig. 8. (a) A proper modulation pattern for the structure of Fig. 7 and its corresponding (b) amplitude and (c) phase profile. Solution to the specified time-varying problem with modulation frequencies equal to (d) $\Omega = 2\pi \times 1$ GHz and (e) $\Omega = 2\pi \times 10$ GHz. The result indicates a frequency shift of value +2.

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To conclude this section we will briefly touch upon some considerations concerning the design of MGMRAs. Through speculation, we have come to realize that positioning the graphene layers at integer multiples of a certain distance from the PEC and adjusting the ribbons’ width from the top down is a rather convincing design procedure. To be more precise, we first remove all but the topmost graphene and determine its specifications to acquire a certain resonant response. we then add a layer of graphene underneath and adjust its width to come by two adjacent resonances. The same goes for the rest of the graphene layers if more resonances are to be effected.

5. Conclusion

We have looked into the problem of frequency conversion as concerns time-varying metasurfaces, particularly, temporally modulated GMRAs. We have presented a quasi-static model that can predict the structure’s behavior when the modulation frequency is considerably smaller than that of the incident wave. We have also developed a time-varying model based on Drude equation and rigorously solved the electromagnetic problem using spatiotemporal FMM. As for the effectiveness of GMRAs, we have learned that single layer structures fall short in this regard; only with multilayer construction do GMRAs provide occasion for efficient wavelength conversion.

Appendix

The following is a recursive algorithm suitable for analytical treatment of time-varying MGMRAs. First, we note that since $\xi (t)$ varies periodically, it can be expanded in a Fourier series as:

(7)$$\xi (t) = \sum_{p={-}\infty}^{+\infty} \xi_{p} e^{{-}ip\Omega t},$$

where $\Omega \equiv 2\pi /T$. According to Floquet theorem, we express the electromagnetic fields in region $l$ (with relative permittivity $\varepsilon _{l}$) as:

(8a)$$H_{y}^{(l)} (x,z,t) = \sum_{m} \left[I_{m}^{(l)}(x,z) + R_{m}^{(l)}(x,z)\right] e^{{-}i\omega_{m} t},$$

(8b)$$E_{x}^{(l)} (x,z,t) ={-}i \sum_{m}\left[\frac{{\partial I_{m}^{(l)}}/{\partial z} }{\omega_{m} \varepsilon_{0} \varepsilon_{l}} + \frac{{\partial R_{m}^{(l)}}/{\partial z}}{\omega_{m} \varepsilon_{0} \varepsilon_{l}}\right] e^{{-}i\omega_{m} t}.$$

Here, $\omega _{0}$ is the angular frequency of the incident wave, $\omega _{m} = \omega _{0} + m\Omega$, $\varepsilon _{0}$ is the electric permittivity of vacuum, and $I^{(l)}(x,z)$ together with $R^{(l)}(x,z)$ respectively symbolize the spatial distribution of the wave incident upon and reflected from the array of graphene ribbons beneath region $l$. Equation (8b) is derived by applying Ampere-Maxwell’s Law to Eq. (8a). Further expansion can be carried out due to periodicity with respect to $x$:

(9a)$$I_{m}^{(l)} (x,z,t) = \sum_{n} I_{mn}^{(l)} e^{{-}ik_{z,mn}^{(l)} (z-z_{l-1})} e^{ik_{x,n} x},$$

(9b)$$R_{m}^{(l)} (x,z,t) = \sum_{n} R_{mn}^{(l)} e^{ik_{z,mn}^{(l)} (z-z_{l-1})} e^{ik_{x,n} x},$$

in which, $k_{x,n} = 2\pi n/L$ and:

(10)$$k_{z,mn}^{(l)} = \begin{cases} \text{sgn} (\omega_m)\sqrt{\varepsilon_{l}\omega_{m}^{2}/c^{2}-k_{x,n}^2}, & \text{if } \varepsilon_{l}\omega_{m}^{2}/c^{2} \geq{k_{x,n}} \\ i \sqrt{\varepsilon_{l}\omega_{m}^{2}/c^{2}-k_{x,n}^2}, & \text{otherwise} \end{cases}.$$

The above choice of $k_{z,mn}^{(l)}$ serves to ensure that the boundary condition at $z \to +\infty$ is satisfied. The surface current density $J_{x}$ is as follows:

(11)$$J_{x} (x,t) = \sum_{m} J_m(x) e^{{-}i\omega_{m} t},$$

By substituting Eq. (7)–(9) into Eq. (6) we get:

(12)$$\{[{-}i\omega_n\xi_{m-n}]+[\gamma_0\delta_{mn}]\} [J_m] = W_{D,0}(x)[\xi_{m-n}]^2[E_m],$$

where the bracketed entries denote a matrix and $E_m$ is given by:

(13)$$E_m ={-}i\left[\frac{{\partial I_{m}^{(l+1)}}/{\partial z} }{\omega_{m} \varepsilon_{0} \varepsilon_{l+1}} + \frac{{\partial R_{m}^{(l+1)}}/{\partial z}}{\omega_{m} \varepsilon_{0} \varepsilon_{l+1}}\right]_{z=z_l},$$

We solve Eq. (12) for $[J_m]$ and thus arrive at the following compact form:

(14)$$\mathbf{J}(x) = W_{D,0}(x)\mathbf{W}\mathbf{E}(x),$$

Note that in Eq. (14), $W_{D,0}(x)$ and $\textbf {E}(x)$ are periodic functions with a simultaneous discontinuity located where the graphene’s edge resides; knowing that their product ($\textbf {J}(x)$) is continuous, according to [48], we are required to use inverse rule in expanding Eq. (14) and not Laurent’s rule. This, as much as it seems straightforward, has a major obstacle lying in its way; after all, $W_{D,0}(x)$ has no reciprocal since, being representative of the conductivity of GMRA, it vanishes in certain portions of a period.

We proceed by considering the boundary condition concerning magnetic fields. Following the argument presented in [47], we surmount the above difficulty by formulating the boundary condition as follows:

(15)$$H_y^{(l)} (x, z=z_l-h/2, t) - H_y^{(l+1)}(x, z=z_l+h/2, t) = \varepsilon_0\varepsilon_{eff}h\frac{\partial E_x}{\partial t}+J_x(x, t),$$

where $\varepsilon _{eff} = (\varepsilon _l+\varepsilon _{l+1})/2$, $E_x$ is the in-plane electric field in the vicinity of the interface ($z=z_l$). Also $h\ll w_l$, which legitimizes the simplifying assumption that $E_x$ is constant in $z_l^-<z<z_l^+$ (a value of $h=w_l/100$ avails in most cases). Equation (15) yields:

(16)$$H_{y,m}^{(l)} (x, z=z_l^{-}) - H_{y,m}^{(l+1)}(x, z=z_l^{+}) ={-}i\omega_{m}\varepsilon_0\varepsilon_{eff}h E_m(x) + W_{D,0}(x)\sum_{p} W_{mp} E_p(x),$$

Where $W_{mp}$ denotes the elements of $W$ (refer to Eq. (14)). In matrix form, Eq. (16) becomes:

(17)$$\mathbf{H}_{y}^{(l)} (x, z=z_l^{-}) - \mathbf{H}_{y}^{(l+1)}(x, z=z_l^{+}) = [\mathbf{D}+W_{D,0}(x)\mathbf{W}] \mathbf{E}(x).$$

According to inverse rule, we should seek to expand:

(18)$$[\mathbf{D}+W_{D,0}(x)\mathbf{W}]^{{-}1}\mathbf{H}_{y}^{(diff)} (x) = \mathbf{E}(x).$$

for no $x$ does $\mathbf {D}+W_{D,0}(x)\mathbf {W}$ equal zero and thus it poses no immediate problem, although we cannot declare the matrix assured of invertibility. The matrix $\mathbf {D}+W_{D,0}(x)\mathbf {W}$ is a two-level step-periodic function of $x$, and so is its inverse; hence, it can easily be expressed in terms of a Fourier series:

(19)$$[\mathbf{D}+W_{D,0}(x)\mathbf{W}]^{{-}1} = \sum_{q} \boldsymbol{\rho}_q e^{iqKx}; \hspace{1em} K = 2\pi/L.$$

Also, by virtue of Floquet theorem, we have:

(20a)$$\mathbf{H}_{y}^{(diff)} (x) = \sum_{n} \mathbf{H}_{n}^{(diff)} e^{ik_{x,n} x},$$

(20b)$$\mathbf{E} (x) = \sum_{n} \mathbf{E}_{n} e^{ik_{x,n} x},$$

which combined with Eq. (19) and upon being substituted in Eq. (18) yields:

(21)$$\sum_{q} \boldsymbol{\rho}_{n-q} \mathbf{H}_{q}^{(diff)} =\mathbf{E}_{n}.$$

By rewriting in terms of $I_{mn}$ and $R_{mn}$, Eq. (21) along with the equation that follows from electric field boundary condition, provides us with a system of linear equations that, given they are properly combined with those of the other layers, can be solved for the desired variable. We begin by defining the following block matrices:

(22a)$$\mathbf{R}^{(l)} = \begin{bmatrix} \cdots & [R_{m,-1}^{(l)}] & [R_{m,0}^{(l)}] & [R_{m, 1}^{(l)}] & \cdots \end{bmatrix}^T, \hspace{1em} \mathbf{I}^{(l)} \text{ Likewise},$$

(22b)$$\mathbf{P} = [\boldsymbol{\rho}_{m-n}], \hspace{1em} \mathbf{Z}^{(l)} = \text{diag} \left(\begin{bmatrix} \cdots & \left[ Z_{m,-1}^{(l)} \right] & \left[ Z_{m,0}^{(l)} \right] & \left[ Z_{m,1}^{(l)} \right] & \cdots \end{bmatrix}^T\right),$$

(22c)$$\mathbf{E}_{l} = \text{diag} \left(\begin{bmatrix} \cdots & \left[ e^{ik_{z,m,-1}^{(l)}(z_{l}-z_{l-l})} \right] & \left[ e^{ik_{z,m,0}^{(l)}(z_{l}-z_{l-l})} \right] & \left[ e^{ik_{z,m,1}^{(l)}(z_{l}-z_{l-l})} \right] & \cdots \end{bmatrix}^T\right),$$

(22d)$$\mathbf{e}_{l} = \text{diag} \left(\begin{bmatrix} \cdots & \left[ e^{ik_{z,m,-1}^{(l)}h/2} \right] & \left[ e^{ik_{z,m,0}^{(l)}h/2} \right] & \left[ e^{ik_{z,m,1}^{(l)}h/2} \right] & \cdots \end{bmatrix}^T\right),$$

where $Z_{mn}^{(l)} = {{k_{z,mn}^{(l)}}/\omega _{m} \varepsilon _{0} \varepsilon _{l}}$. By invoking electric and magnetic field boundary conditions we get:

(23a)$$\mathbf{Z}^{(l)} \left(-\mathbf{E}_l^{{-}1}\mathbf{I}^{(l)}+\mathbf{E}_l\mathbf{R}^{(l)}\right) = \mathbf{Z}^{(l+1)} \left(-\mathbf{I}^{(l+1)}+\mathbf{R}^{(l+1)}\right),$$

(23b)$$\left( \mathbf{E}_l^{{-}1}\mathbf{e}_l\mathbf{I}^{(l)} + \mathbf{E}_l\mathbf{e}_l^{{-}1}\mathbf{R}^{(l)} \right) - \left( \mathbf{e}_{l+1}^{{-}1}\mathbf{I}^{(l+1)} + \mathbf{e}_{l+1}\mathbf{R}^{(l+1)} \right) = \mathbf{P}^{{-}1} \mathbf{Z}^{(l+1)} \left(-\mathbf{I}^{(l+1)}+\mathbf{R}^{(l+1)}\right).$$

All that remains to conclude the analysis, is to employ a recursive algorithm to account for the entirety of the layers. To see to it that the implementation is numerically stable, we opt for the $S$-matrix formulation [49]; whereby we restate Eq. (23) in the following matrix form:

(24)$$\begin{bmatrix} \mathbf{Z}^{(l+1)} & \mathbf{Z}^{(l)}\mathbf{E}_l^{{-}l} \\ \mathbf{P}^{{-}1}\mathbf{Z}^{(l+1)}+\mathbf{e}_{l+1} & -\mathbf{E}_{l}^{{-}1}\mathbf{e}_{l} \end{bmatrix} \begin{bmatrix} \mathbf{R}^{(l+1)} \\ \mathbf{I}^{(l)} \end{bmatrix} = \begin{bmatrix} \mathbf{Z}^{(l)}\mathbf{E}_{l} & \mathbf{Z}^{(l+1)} \\ \mathbf{E}_{l}\mathbf{e}_{l}^{{-}1} & \mathbf{P}^{{-}1}\mathbf{Z}^{(l+1)}-\mathbf{e}_{l+1}^{{-}1} \end{bmatrix} \begin{bmatrix} \mathbf{R}^{(l)} \\ \mathbf{I}^{(l+1)} \end{bmatrix},$$

in other words:

(25)$$\begin{aligned} & \begin{bmatrix} \mathbf{R}^{(l+1)} \\ \mathbf{I}^{(l)} \end{bmatrix} =\\ &\begin{bmatrix} \mathbf{I} & \mathbf{0} \\ \mathbf{0} & \mathbf{E}_{l} \end{bmatrix} \begin{bmatrix} \mathbf{Z}^{(l+1)} & \mathbf{Z}^{(l)}\\ \mathbf{P}^{{-}1}\mathbf{Z}^{(l+1)}+\mathbf{e}_{l+1} & -\mathbf{e}_{l} \end{bmatrix}^{{-}1} \begin{bmatrix} \mathbf{Z}^{(l)} & \mathbf{Z}^{(l+1)} \\ \mathbf{e}_{l}^{{-}1} & \mathbf{P}^{{-}1}\mathbf{Z}^{(l+1)}-\mathbf{e}_{l+1}^{{-}1} \end{bmatrix} \begin{bmatrix} \mathbf{E}_{l} & \mathbf{0} \\ \mathbf{0} & \mathbf{I} \end{bmatrix} \begin{bmatrix} \mathbf{R}^{(l)} \\ \mathbf{I}^{(l+1)} \end{bmatrix}. \end{aligned}$$

Note that $\mathbf {e}_l^{-1}$ and $\mathbf {e}_{l+1}^{-1}$ constitute the only exponentially growing terms (refer to Eqs. (22d) and (10)) in Eq. (25) which could give rise to numerical instability [49]. However, this is not a matter of concern here, since $h$ is very small.

The rest of the treatment proceeds along the same lines as in [49] and is hence not included here.

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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Frequency conversion in time-varying graphene microribbon arrays

Abstract

1. Introduction

2. Graphene microribbon array

3. Time-gradient phase variation

4. Time-varying graphene microribbon array

5. Conclusion

Appendix

Disclosures

Data availability

References

Data availability

Cited By

Figures (8)

Equations (32)

Optics Express