1. Introduction

The resonance phenomenon in optical structures drew much attention in the past decade [1–3]. Resonant photonic structures found numerous application as optical filters, sensors, laser resonators, nonlinear optical devices, (magneto-)optical modulators and switches [1]. Optical properties of such structures are usually characterized by the resonance lineshape representing a dependence of the scattering amplitude on the light frequency. The coupled-mode theory provides an intuitive approach for deriving the resonance lineshapes.

Different coupled-mode theories were developed to describe various optical structures including coupled waveguides and fibers [4–6], Bragg reflectors in waveguides [4, 5], coupled resonators [4, 5], waveguide-coupled resonators [7–9], gratings [1, 8, 10], individual scatterers [11–13]. The referenced papers use either the spatial formulation of the coupled-mode theory [4–6] or the temporal one [4, 5, 7–13]. Spatial theories describe spatial evolution of the propagating mode, while temporal coupled-mode theories deal with temporal evolution of optical field. A generalization of these two approaches is the time-dependent coupled-mode theory [14–16] or the spatiotemporal coupled-mode theory [17]. This theory, considering both spatial and temporal evolution of optical fields, was used to investigate parallel waveguides [14,16] and various photonic-crystal devices [15].

In the current paper we develop a new formulation of the spatiotemporal coupled-mode theory to describe resonant light scattering by guided-mode resonant gratings. The paper is organized as follows. Following the Introduction, in Section 2 we derive partial differential equations that describe both temporal and spatial field evolution inside the grating. In Section 3 we use these equations to obtain approximation of the transmission and reflection coefficients of the structure as a function of both frequency and in-plane wave vector. In Section 4 we compare the results of the coupled-mode theory with the rigorous computations. Besides, we show that the proposed approach describes a number of important optical effects: mode coupling, emergence of a photonic band gap, emergence of even and odd modes at normal incidence of light.

2. Spatiotemporal coupled-mode theory

Let us consider an optical pulse diffraction by a guided-mode resonant grating (Fig. 1). We assume that the structure and the incident field are invariant along the y-axis. For the sake of simplicity we consider the symmetric grating with yOz symmetry plane. We also assume that the grating period is subwavelength, i.e. the grating supports only zeroth diffraction orders (reflected and transmitted). However, inside the waveguide layer several diffraction orders exist.

 

Fig. 1 Diffraction of a pulse on the guided-mode resonant grating (waveguide with a grating on its top) supporting two quasiguided modes.

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We start our analysis from replacing a grating with a slab waveguide (with an effective refractive index). In this case the field inside a waveguide is a superposition of two uncoupled counter-propagating modes. We suppose that at the angular frequency ω₀ the mode wave number is k_x0. Moreover, we suppose the following linear dispersion law for this mode:

Now let us consider a mode wave packet (pulse). Its field distribution can be written as

Note that the first part of this equation $(\frac{\partial u}{\partial t}=-v_g\frac{\partial u}{\partial x})$ is a simple unidirectional wave equation for a wave packet envelope P. While the last term in Eq. (2) takes account of the phase of the wave packet.

Let us consider two wave packets with opposite directions [see Fig. 2(a)]:

 

Fig. 2 (a) Two counter-propagating modes in a slab waveguide. (b) Modes excitation by ±m-th diffraction orders. (c) Mode leakage into −m-th diffraction order and mode coupling by means of −2m-th diffraction order.

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Now let us add a grating on the top of the waveguide (Fig. 1). Moreover, we assume that grating period d satisfies the mode excitation condition

For the incident pulse central frequency ω₀ this condition provides matching of the propagation constants of the mode and of the ±m-th diffraction orders. In this case a number of new effects will arise. First, due to the diffraction on the grating grooves, at every moment of time an α₀-fraction of the mode will scatter away from the waveguide. Second, an extra energy can be introduced into the waveguide from the incident field. Third, a coupling between two counter-propagating modes will take place (i.e. energy from a mode will be transferred to the counter-propagating one). All these effects are sketched in Figs. 1 and 2.

The first two effects will modify Eqs. (3) in the following way:

The third effect caused by the grating, coupling between modes, can be taken into account by introducing two coupling coefficients, c₁ and c₂:

The energy leaked out of the grating forms the resonant contribution to the reflected and transmitted fields. The non-resonant part of the scattered field is the field transmitted/reflected directly through the structure without the mode excitation (see Fig. 1). Hence the transmitted and reflected field can be defined as

The exponential terms here take account of the fact that the modes are scattered out of the waveguide by means of ∓m-th diffraction orders [see Fig. 2(c)].

Equations (5)–(7) define the spatiotemporal coupled-mode theory for guided-mode resonant gratings.

3. Transmission and reflection coefficients

In what follows we will analyze Eqs. (5)–(7) to obtain the analytical expressions for the reflection and transmission coefficients of the grating. To do this, we need to solve Eq. (5) with respect to the total scattered field (7). We first rewrite Eq. (5) in terms of functions $\tilde{u}=u{e}^{-\mathrm{i}{k}_{x0}x}$ and $\tilde{v}=v{e}^{\mathrm{i}{k}_{x0}x}$:

Finally, we eliminate $\frac{\partial w}{\partial x}$ from Eq. (9) and substitute it into Eq. (10) differentiated with respect to x. This leads us to the following partial differential equation for s:

Equations (6) and (11) relate the incident and the scattered fields. These equation are formulated in spatiotemporal domain. Let us rewrite these equations in the frequency domain. To do this, we define the incident field spectrum as the Fourier transform of the incident field:

The reflected and transmitted field spectra, F_R and F_T, are defined in a similar way. Besides, we define the scattered field spectrum S as the Fourier transform of s(x,t).

By applying the Fourier transform to Eq. (11) we obtain the following relation between the incident and scattered field spectra:

Fourier transform of Eq. (6) gives

Let us consider the incident field to be a monochromatic plane wave with angular frequency ω and in-plane wave vector k_x. In this case the complex transmission coefficient is the ratio of the transmitted and the incident field spectra [T(k_x,ω) = F_T(k_x,ω)/F(k_x,ω)]. A similar expression is valid for the reflection coefficient. According to Eqs. (13) and (12), the transmission coefficient T and the reflection coefficient R are given by.

By substituting Eq. (15) into Eq. (14) we obtain the following form of the reflection and transmission coefficients:

Equation (16) can be considered as a generalization of the Fano lineshape. Indeed, at fixed ω the transmission and reflection coefficients take the form of two-resonance Fano lineshape [18] a + b/(k_x − k_p) − b/(k_x + k_p). At fixed k_x Eq. (16) can be rewritten in a similar two-resonance form [8]: a + b/(ω −ω₁) + c/(ω −ω₂). Note that at k_x = 0 (at normal incidence) the transmission coefficient has only one resonance with frequency ω_p1. The physical origin of this effect is that the second mode with frequency ω_p2 cannot be excited due to the symmetry conditions [10]. At normal incidence this mode has an odd (antisymmetric) field distribution with respect to the plane yOz and cannot be excited by the symmetric incident plane wave. Note that the second resonance with frequency ω_p1 corresponds to the even (symmetric) mode that can be excited.

4. Numerical validation

To check the validity of the proposed theory we considered guided-mode resonant filter (Fig. 1) with the following parameters: grating period d = 320nm, grating fill-factor a = 70/d, grating height h₁ = 50nm, waveguide layer height h₂ = 200nm, superstrate refractive index n_s = 1, grating and waveguide layer refractive index n₂ = 1.7, substrate refractive index n₁ = 1.5. The presented parameters are typical for guided modes resonant filters. This filter was designed to exhibit pronounced peaks in the visible spectrum range for TE-polarized light.

We used the Fourier modal method [19, 20] to simulate the diffraction of a plane wave on the considered structure. Figure 3(a) shows the transmission coefficient as a function of both ω and k_x. Let us note that due to the grating symmetry, T(ω,+k_x) = T(ω, −k_x). For the sake of comparison, the simulated spectrum is presented on the left-hand side of the figure (at k_x < 0) while the approximating spectrum obtained from Eq. (16) is presented on the right (at k_x > 0). Figure 3(b) presents the simulated and approximating transmission spectra at different angles of incidence. Circles in Fig. 3(b) presents the transmission coefficient obtained from Eq. (16) with k_x = (ω/c)n_s sinθ.

 

Fig. 3 (a) Transmission coefficient of the grating, |T|², vs. incident light’s k_x and ω: rigorous simulation (left part, at k_x < 0) and analytical formula (16) (right part, at k_x > 0). The dashed lines specify the light lines for θ = 0.2°,1°. (b) Transmission coefficient of the grating, |T|², for different angles of incidence (θ = 0°,0.2°,1°): rigorous simulation (solid lines) and analytical formula (16) (circles).

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We used the following parameters in Eq. (16): v_g = 1.768 10¹⁷ nms⁻¹, ω₀ = 3.751×10¹⁵ s⁻¹, α₀ +c₁ = 8.694×10¹¹ s⁻¹, c₂ = −8.694×10¹¹+6.968×10¹² is⁻¹,qγ₁ = (−2.023+8.181i)×10¹¹,qγ₂ = (−5.408−3.706i)×10¹¹, $\tilde{r}=-0.1620-0.1923\mathrm{i}$, $\tilde{t}=-0.2827-0.9257\mathrm{i}$. To estimate these parameters we used the following approach. First, we calculated ω_p1 and ω_p2 as the pole of the scattering matrix at k_x = 0 using the scattering matrix approach [21]. Then, we calculated ω_r and ω_t as the zeros of the reflection and transmission coefficients at k_x = 0 using the Fourier modal method [19,20]. Using the calculated parameters we obtained c₂, ω₀, α₀ +c₁, $q{\gamma }_1{\tilde{t}}^{-1}$, $q{\gamma }_2{\tilde{r}}^{-1}$. Finally, the remaining parameters (v_g, $\tilde{r}$, $\tilde{t}$) were estimated using the conjugate gradient method aimed at minimizing the root-mean-square deviation between the rigorously computed transmission spectrum [left part of Fig. 3(a)] and approximate relation (16).

Figure 3 shows pronounced transmission minimum governed by the quasi-guided mode excitation. According to Fig. 3(a), the dispersion law of the mode resembles a hyperbola. The latter’s equation can be obtained from analytical Eq. (16) by equating to zero its denominator. As we discussed in the previous section, in the case of oblique incidence there are two resonances at fixed angle of incidence. However, in the case of normal incidence only one resonance is present in Fig. 3(a) and Fig. 3(b). The second resonance corresponds to the antisymmetric mode and cannot be excited by a normally incident plane wave.

According to Fig. 3, the spatiotemporal coupled-mode theory provides a good approximation of the transmission spectrum at different angles of incidence. For the considered frequency range and |θ| < 0.2° the approximation error |T₁ − T₂|² is less than 0.017; when |θ| < 1° and |θ| < 2° the error does not exceed 0.018 and 0.035, respectively.

5. Conclusion

We presented the spatiotemporal coupled-mode theory for guided-mode resonant grating. The theory describes resonant light scattering using two coupled partial differential equations (unidirectional wave equations for dispersive medium). By solving these equations we obtained analytical approximation of the transmission and refection spectra of the grating. This approximation takes into account both frequency and the in-plane wave vector of the incident light. The obtained formulae are in good agreement with rigorous simulations, describing such grating-theory phenomena as mode coupling, emerging of “dark” and “bright” resonances at normal incidence of light. The theory can be generalized to the case of conical mount, as well as to the case of crossed grating. In the latter case four coupled modes can be excited simultaneously.

The presented analytical expressions for the reflection and transmission coefficients can be used for the design of a wide range of photonic devices, such as filters, lasers, sensors. Besides, the developed theory provides theoretical description of the general class of spatiotemporal transformations of optical fields that can be implemented by resonant periodic structures.