Coupled-mode theory and Fano resonances in guided-mode resonant gratings: the conical diffraction mounting

Dmitry A. Bykov; Leonid L. Doskolovich; Victor A. Soifer

doi:10.1364/OE.25.001151

1. Introduction

In the past decades much attention has been paid to studying transmission and reflection resonances in periodic optical structures. An important example of such structures is guided-mode resonant grating (GMRG). Possible applications of GMRGs include optical filters, polarizers, reflectors, and sensors [1,2].

A distinctive feature of resonant gratings is pronounced peaks in the transmission and reflection spectra. The shape of these peaks (the dependence of the transmission or reflection coefficient on the light’s frequency) is referred to as the line shape of the resonance. A good approximation of the resonance line shape is given by the Fano formula [3], which describes the scattering amplitude as the sum of resonant and non-resonant terms. When the non-resonant scattering is negligible, the line shape of the resonance takes symmetrical Lorentzian form. Structures with such resonances are used as optical filters [1].

A general theoretical description of the resonance line shape is of great interest. It allows to describe theoretically a class of feasible filtering operations (both in spectral and angular domains) implemented by GMRGs. The most known resonance approximations are based on the Fano formula [3] and represent the transmission and reflection coefficients by rational functions of incident light’s frequency ω [4, 5] or in-plane wave vector component k_x [6]. More general resonant approximations (ω−k_x Fano line shapes) consider the scattering amplitudes as functions of both ω and k_x [7–13]. In particular, in [7,8], the authors propose an approach to derive the ω−k_x transmission (reflection) coefficient of the GMRG (weak grating on the top of a waveguide layer) by representing the field inside the waveguide layer as two plane waves, which are coupled with the incident, transmitted, and reflected plane waves by the grating located on the waveguide layer. In papers [9, 10] the authors propose an alternative approach to derive the ω−k_x spectrum approximation for extraordinary-optical-transmission gratings. In [11], the authors exploit the theory of functions of several complex variables to derive simple approximations of the ω−k_x transmission spectrum. Later in paper [12], this approach is generalized by taking into account the mode coupling phenomenon at k_x = 0. In paper [13], the authors obtain similar approximations, using a spatiotemporal formulation of 1D coupled-mode theory.

The considered ω−k_x Fano line shapes [7–13] do not describe a number of important resonant effects arising in conical diffraction mounting when the wave vector of incident light has generally two non-zero in-plane components (k_x and k_y). In particular, in papers [14–16] it was shown that GMRG angular tolerances (the resonance width with respect to the angle of incidence) in planar diffraction geometry and in conical mounting differ by an order of magnitude. This effect was studied numerically and is still not rigorously described. Moreover, in the case of conical mounting, polarization conversion results in another optical effect of cross-polarization mode excitation. In this regard, a derivation of general ω−k_x−k_y resonant approximations for GMRGs is of great importance. Let us note that ω−k_x−k_y resonant approximations are currently proposed only for multilayer structures [17,18]. However, these approximations cannot be used to describe the spectra of GMRGs.

In this paper, we propose a two-dimensional formulation of the spatiotemporal coupled-mode theory (CMT) for GMRGs. We apply this theory to derive simple analytical approximations for the ω−k_x−k_y reflection and transmission spectra of the GMRG. We show that the obtained approximations can be used to investigate the symmetry of the quasiguided modes of the grating in the case of conical mounting. In particular, the theory describes the effect of cross-polarization mode excitation. Besides, we use the proposed approximations to derive simple estimations for angular tolerances of GMRGs.

The paper is organized as follows. Following the Introduction, Section 2 contains a general formulation of the 2D time-dependent coupled-mode theory (or spatiotemporal coupled-mode theory) which neglects polarization effects. In Section 3, we reformulate the theory to take light’s polarization into account. We use the proposed theory to derive analytical approximations of the ω−k_x−k_y reflection and transmission spectra of the grating. In Section 4, we analyze the obtained approximations and compare them to the rigorously calculated spectra.

2. 2D coupled-mode theory

In this section we consider an optical pulse (spatiotemporal wave packet) that is normally incident on a 1D guided-mode resonant grating (Fig. 1). The grating is periodic in the x-direction with period d and is uniform in the y-direction. The incident pulse is defined by its field distribution f(x, y, t) at the grating’s upper interface (at z = 0). The pulse central wavelength λ₀ is assumed to be higher than the grating’s period d (i. e. the grating is subwavelength: λ₀ > d), so that the reflected and transmitted pulses are formed by the zeroth diffraction orders.

Fig. 1 Pulse diffraction by guided-mode resonant grating

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2.1. Pulse in a slab waveguide

Let us start the analysis with replacing the GMRG with a slab waveguide. Suppose that in a vicinity of the angular frequency ω₀ = 2πc/λ₀ the mode supported by the waveguide has the following linear dispersion law:

k - k_{1} = (ω - ω_{0}) / v_{g} .

Here v_g is the mode group velocity;

k = \sqrt{k_{x}^{2} + k_{y}^{2}}

is the mode wave number; k₁ is the wave number of the mode with angular frequency ω₀.

Now let us consider a pulse, which propagates inside the waveguide. We will assume that the pulse central frequency is ω₀, and it propagates in the positive-x direction, i. e. the pulse central wave numbers are (k_x, k_y) = (k₁, 0). Assuming that the waveguide is thin, compared to the spatial extent of the pulse, we can represent the field inside the waveguide as the following superposition of the modes with different wave numbers:

u (x, y, t) = \iint G (k_{x}, k_{y}) e^{i (k_{x} x + k_{y} y - ω t)} d k_{x} d k_{y},

where G is the pulse spectrum. Note that k_x, k_y, and ω in Eq. (2) are related by Eq. (1).

According to Eqs. (1) and (2), the pulse field u(x, y, t) obeys the following PDE:

v_{g}^{2} [\frac{\partial^{2} u}{\partial x^{2}} + \frac{\partial^{2} u}{\partial y^{2}}] = \frac{\partial^{2} u}{\partial t^{2}} - 2 i (v_{g} k_{1} - ω_{0}) \frac{\partial u}{\partial t} - {(v_{g} k_{1} - ω_{0})}^{2} u .

The validity of this equation can be verified by means of direct substitution. Note that Eq. (3) is a form of a wave equation which takes into account the linear dispersion law (1).

Let us simplify Eq. (3) using slowly varying envelope approximation (SVEA) [19]. Under this approximation the pulse envelope

\bar{u} = u \cdot e^{- i k_{1} x + i ω_{0} t}

is a slowly varying function of the arguments ω and k_x. Hence, we can neglect the second-order derivatives of ū with respect to t and x. Therefore, the second-order derivatives of the u-function can be approximated as

\frac{\partial^{2} u}{\partial t^{2}} \approx - 2 i ω_{0} \frac{\partial u}{\partial t} + ω_{0}^{2} u, \frac{\partial^{2} u}{\partial x^{2}} \approx 2 i k_{1} \frac{\partial u}{\partial x} + k_{1}^{2} u .

Using these approximations in Eq. (3), we obtain a unidirectional paraxial wave equation that describes field evolution inside the waveguide:

\frac{\partial u}{\partial t} = - v_{g} \frac{\partial u}{\partial x} + \frac{i v_{g}}{2 k_{1}} \frac{\partial^{2} u}{\partial y^{2}} + i (v_{g} k_{1} - ω_{0}) u .

By replacing v_g → −v_g, k₁ → −k₁ we can obtain a similar equation for the pulse v propagating in the negative-x direction:

\frac{\partial v}{\partial t} = v_{g} \frac{\partial v}{\partial x} + \frac{i v_{g}}{2 k_{1}} \frac{\partial^{2} v}{\partial y^{2}} + i (v_{g} k_{1} - ω_{0}) v .

2.2. Coupled-mode equations

The previous subsection describes light propagation inside the waveguide. In this subsection we add a thin grating on top of the waveguide. The period of the grating is chosen according to the following mode excitation condition:

\frac{2 π}{d} m = k_{1} .

In the case of normally incident wave with frequency ω₀ this condition ensures that the ±m-th diffraction orders have the same in-plane wave numbers, as the modes supported by the waveguide.

The grating substantially modifies the optical properties of the waveguide [13], which is no longer described by Eqs. (4) and (5). Firstly, the modes u and v can be excited by the incident pulse f. Secondly, due to reciprocity, a part of the mode energy will scatter away from the waveguide and contribute to the reflected and transmitted fields. Finally, the coupling between the u-mode and v-mode will take place [13]. All these effects can be taken into account [13,20] by adding coupling terms to the right-hand side of Eqs. (4) and (5):

{\begin{cases} \frac{\partial u}{\partial t} = - v_{g} \frac{\partial u}{\partial x} + \frac{i v_{g}}{2 k_{1}} \frac{\partial^{2} u}{\partial y^{2}} + i (v_{g} k_{1} - ω_{0}) u - c_{1} u + c_{2} e^{2 i k_{1} x} v + c_{e} e^{i k_{1} x} f \\ \frac{\partial v}{\partial t} = v_{g} \frac{\partial v}{\partial x} + \frac{i v_{g}}{2 k_{1}} \frac{\partial^{2} v}{\partial y^{2}} + i (v_{g} k_{1} - ω_{0}) v - c_{1} v + c_{2} e^{- 2 i k_{1} x} u + c_{e} e^{- i k_{1} x} f . \end{cases}

Here c_e, c₁, and c₂ are coupling coefficients: c_e describes modes excitation by the incident pulse; c₁ describes mode decay due to diffraction by the grating; c₂ describes the coupling between the u and v modes. The exponential terms e^±ik₁x in system (6) takes account of the phase change due to scattering into the ±m-th diffraction order [13,20].

To write a full set of coupled-mode equations we need to define the reflected and transmitted field. According to the Fano resonance theory, the scattered field is a sum of the non-resonantly scattered field and the field scattered out of the waveguide:

{\begin{cases} f_{R} = r_{0} f + (c_{r} e^{- i k_{1} x} u + c_{r} e^{i k_{1} x} v), \\ f_{T} = t_{0} f + (c_{t} e^{- i k_{1} x} u + c_{t} e^{i k_{1} x} v) . \end{cases}

Here f_R (x, y, t) and f_T (x, y, t) are field distributions of the reflected and transmitted pulses at the structure’s top and bottom interfaces; r₀ and t₀ are non-resonant reflection and transmission coefficients; c_r and c_t are coupling coefficients that describe the leakage of the modes into the 0-th reflected and transmitted diffraction orders.

Equations (6) and (7) can be simplified by expressing them in terms of the modes’ spatial envelopes ũ = u · e^−ik₁x and ṽ = v · e^ik₁x:

{\begin{cases} \frac{\partial \tilde{u}}{\partial t} = - v_{g} \frac{\partial \tilde{u}}{\partial x} + \frac{i v_{g}}{2 k_{1}} \frac{\partial^{2} \tilde{u}}{\partial y^{2}} - (c_{1} + i ω_{0}) \tilde{u} + c_{2} \tilde{v} + c_{e} f, \\ \frac{\partial \tilde{v}}{\partial t} = v_{g} \frac{\partial \tilde{v}}{\partial x} + \frac{i v_{g}}{2 k_{1}} \frac{\partial^{2} \tilde{v}}{\partial y^{2}} - (c_{1} + i ω_{0}) \tilde{v} + c_{2} \tilde{u} + c_{e} f, \\ f_{R} = r_{0} f + c_{r} (\tilde{u} + \tilde{v}), \\ f_{T} = t_{0} f + c_{t} (\tilde{u} + \tilde{v}) . \end{cases}

Note that all coupling coefficient here are generally complex except for c₁, which is real [13].

We will solve Eq. (8) in the Fourier domain with the Fourier transform defined as follows:

ℱ u = \int \iint u (x, y, t) e^{- i (k_{x} x + k_{y} y - ω t)} d k_{x} d k_{y} d ω .

By applying the Fourier transform to Eq. (8), we obtain the following system of linear equations:

{\begin{array}{l} - i ω U & = - v_{g} i k_{x} U - \frac{i v_{g}}{2 k_{1}} k_{y}^{2} U - (c_{1} + i ω_{0}) U + c_{2} V + c_{e} F, \\ - i ω V & = v_{g} i k_{x} V - \frac{i v_{g}}{2 k_{1}} k_{y}^{2} V - (c_{1} + i ω_{0}) V + c_{2} U + c_{e} F, \\ F_{R} & = r_{0} F + c_{r} (U + V), \\ F_{T} & = t_{0} F + c_{t} (U + V), \end{array}

where U = ℱ ũ, V = ℱ ṽ, F = ℱ f, F_R = ℱ f_R, F_T = ℱ f_T.

One can easily solve Eq. (9) with respect to the ratio of the transmitted and incident pulses spectra. This ratio gives the transmission coefficient of the grating [13]:

T = \frac{F_{T}}{F} = t_{0} \frac{v_{g}^{2} k_{x}^{2} - (ω - ω_{z t} - η k_{y}^{2}) (ω - ω_{p 2} - η k_{y}^{2})}{v_{g}^{2} k_{x}^{2} - (ω - ω_{p 1} - η k_{y}^{2}) (ω - ω_{p 2} - η k_{y}^{2})},

where η = v_g/(2k₁), ω_p1 = ω₀ −i(c₁ − c₂), ω_p2 = ω₀ −i(c₁ + c₂),

ω_{zr} = ω_{0} - i (c_{1} - c_{2} + 2 c_{e} c_{r} r_{0}^{- 1})

. A similar equation holds for the reflection coefficient.

If we equate the denominator of Eq. (10) to zero, we obtain the dispersion law of the mode. Indeed, when

v_{g}^{2} k_{x}^{2} = (ω - ω_{p 1} - η k_{y}^{2}) (ω - ω_{p 2} - η k_{y}^{2})

solution to Eq. (9) exists without incident field (at F = 0). This solution describes an eigenmode of the grating. The mode’s field distribution inside the structure is defined by the functions ũ and ṽ. According to Eq. (11), at k_x = k_y = 0 the theory predicts two modes with complex eigenfrequencies ω_p1 and ω_p2. According to Eq. (9), when F = 0, k_x = k_y = 0, and ω = ω_p1 the mode’s field is symmetric: ũ = ṽ; this mode is called an even mode. At the same time, for mode with eigenfrequency ω = ω_p2 we obtain ũ = −ṽ; this mode is called an odd mode. The excitation of even and odd modes is discussed in subsection 4.2.

3. Polarization in coupled-mode theory

In the previous section, the coupled-mode equations were obtained without taking light’s polarization into account: the incident light, the excited modes, and the considered diffraction orders were assumed to have the same polarization. However, in the case of conical mount, gratings transform the polarization of the incident light. In particular, the transmitted and reflected diffraction orders at k_y ≠ 0 contain both TE and TM components. Moreover, at k_y ≠ 0 the incident wave can excite cross-polarized modes. In the current section, we generalize the coupled-mode equations of Section 2 to take these effects into account. For simplicity we will assume that the incident wave is TE-polarized (the case of TM-polarized incident plane wave can be considered in a similar way).

3.1. Coupling operators

The coupled-mode equations obtained in Section 2 depend on the coupling coefficients c_e, c₁, c₂, c_r, and c_t. These coefficients describe the light scattering by the grating (e. g., the coefficient c_e describes modes excitation by the incident wave).

In the previous section, we assumed that the coupling coefficients are numbers. This means that the shape of the pulse does not change due to diffraction by the grating. Generally, this is not the case: the shape of the pulse may undergo substantial transformations [21]. To take these transformations into account we need to replace coupling coefficients c with coupling operators ĉ. The same replacement should be carried out for the non-resonant scattering coefficients r₀ and t₀. In this case, the system of coupled-mode equations (Eq. (8)) takes the following general form:

{\begin{cases} \frac{\partial \tilde{u}}{\partial t} = - v_{g} \frac{\partial \tilde{u}}{\partial x} + \frac{i v_{g}}{2 k_{1}} \frac{\partial^{2} \tilde{u}}{\partial y^{2}} - i ω_{0} \tilde{u} - {\hat{c}}_{1} \tilde{u} + {\hat{c}}_{2} \tilde{v} + {\hat{c}}_{e 1} f, \\ \frac{\partial \tilde{v}}{\partial t} = v_{g} \frac{\partial \tilde{v}}{\partial x} + \frac{i v_{g}}{2 k_{1}} \frac{\partial^{2} \tilde{v}}{\partial y^{2}} - i ω_{0} \tilde{v} - {\hat{c}}_{1} \tilde{v} + {\hat{c}}_{2} \tilde{u} + {\hat{c}}_{e 2} f, \\ f_{R}^{TE} = {\hat{r}}_{0}^{TE} f + {\hat{c}}_{r 1}^{TE} \tilde{u} + {\hat{c}}_{r 2}^{TE} \tilde{v}, \\ f_{T}^{TE} = {\hat{t}}_{0}^{TE} f + {\hat{c}}_{t 1}^{TE} \tilde{u} + {\hat{c}}_{t 2}^{TE} \tilde{v}, \\ f_{R}^{TM} = {\hat{r}}_{0}^{TM} f + {\hat{c}}_{r 1}^{TM} \tilde{u} + {\hat{c}}_{r 2}^{TM} \tilde{v}, \\ f_{T}^{TM} = {\hat{t}}_{0}^{TM} f + {\hat{c}}_{t 1}^{TM} \tilde{u} + c_{t 2}^{TM} \tilde{v} . \end{cases}

Here we introduced different coupling operators (e. g., ĉ_e1 and ĉ_e2) for u-mode and v-mode. Besides, comparing to system (8), we introduced two cross-polarized scattering channels:

f_{R}^{TM}

and

f_{T}^{TM}

.

Let us investigate the properties of the coupling operators. Firstly, if the structure contains no nonlinear materials, the coupling operators are linear: if incident field f₁ excites a mode with field distribution ĉ_e f₁ then incident field α f₁ + β f₂ excites a mode with field distribution ĉ_e(α f₁ + β f₂) = α · ĉ_e f₁ + β · ĉ_e f₂. Secondly, coupling operators are both time- and space-invariant, since the optical properties of the structure does not depend on x, y, or t. Hence, coupling operators are multiplier operators, i. e. they can be interpreted as multiplication by some transfer function in the Fourier domain:

ℱ ({\hat{c}}_{e} f) = C_{e} \cdot F .

Here F = ℱ f is the spectrum of the incident field; C_e is the transfer function (TF or the multiplier) of the coupling operator.

In terms of coupling operators’ TFs, the Fourier transform of system (12) is

{\begin{cases} - i ω U = - v_{g} i k_{x} U - \frac{i v_{g}}{2 k_{1}} k_{y}^{2} U - (i ω_{0} + C_{1}) U + C_{2} V + C_{e 1} F, \\ - i ω V = v_{g} i k_{x} V - \frac{i v_{g}}{2 k_{1}} k_{y}^{2} V - (i ω_{0} + C_{1}) V + C_{2} U + C_{e 2} F, \\ F_{T}^{TE} = T_{0}^{TE} F + C_{t 1}^{TE} U + C_{t 2}^{TE} V, \\ F_{R}^{TE} = R_{0}^{TE} F + C_{r 1}^{TE} U + C_{r 2}^{TE} V, \\ F_{T}^{TM} = T_{0}^{TM} F + C_{t 1}^{TM} U + C_{t 2}^{TM} V \\ F_{R}^{TM} = R_{0}^{TM} F + C_{r 1}^{TM} U + C_{r 2}^{TM} V, \end{cases}

where the uppercase letters C, R, and T denote the transfer functions of the corresponding coupling operators ĉ, r̂, and t̂ (with the corresponding indices). As it is shown in Fig. 2, each TF corresponds to the scattering of light into a particular diffraction order.

Fig. 2 TFs describing coupling between the modes and the incident, reflected, and transmitted fields

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Since the spectrum of the incident pulse is centered at the point (k_x, k_y, ω) = (0, 0, ω₀) we can approximate each TF with its Taylor-series expansion in the vicinity of this point. The simplest approximation is the lowest-degree term of Taylor series. Due to symmetry considerations, the lowest-degree term is not always a constant. In the following subsections, we will find the approximations for all TFs, using the symmetry conditions (23)–(26) obtained in the Appendix.

3.2. Same-polarization mode excitation

Let us consider the form of the coupling operators for the case when the mode has the same polarization as the incident wave. The latter, as noted above, is assumed to be TE-polarized.

First, we consider TFs C_e1(k_x, k_y, ω) and C_e2(k_x, k_y, ω), which describe mode excitation by means of the incident light scattering into the grating’s ±m-th diffraction orders (see Fig. 2). According to Eq. (23), C_e1(k_x, k_y, ω) = C_e2(−k_x, k_y, ω). Assuming c_e = C_e1(0, 0, ω₀) ≠ 0, we can approximate each TF with the zeroth term of its Taylor series:

C_{e 1} (k_{x}, k_{y}, ω) \approx C_{e 2} (k_{x}, k_{y}, ω) \approx c_{e} .

Similar approximations can be used for the following TFs: $C_{r 1}^{TE}$ , $C_{r 2}^{TE}$ , $C_{t 1}^{TE}$ , $C_{t 2}^{TE}$ , T^TE, R^TE.

Now let us consider the cross-polarization coupling operator with the TF $C_{t 1}^{TM}$ , which corresponds to the scattering of the waveguide mode with in-plane wave vector (k_x + 2πm/d, k_y) to the cross-polarized −m-th diffraction order (see Fig. 2). According to Eq. (26), the considered TF is an odd function of k_y: $C_{t 1}^{TM} (k_{x}, k_{y}, ω) = - C_{t 1}^{TM} (k_{x}, - k_{y}, ω)$ . Hence, $C_{t 1}^{TM} (0, 0, ω_{0}) = 0$ and the lowest-degree non-zero Taylor term is the following:

C_{t 1}^{TM} (k_{x}, k_{y}, ω) \approx k_{y} c_{t}^{TM},

where

c_{t}^{TM} = {\partial C_{t 1}^{TM} / \partial k_{y} |}_{k_{x} = k_{y} = 0, ω = ω_{0}}

. Let us note that this TF corresponds to the following coupling operator:

{\hat{c}}_{t 1}^{TM} \approx - i c_{t}^{TM} \frac{\partial}{\partial y} .

Similarly, the TF $C_{t 2}^{TM}$ corresponds to the +m-th cross-polarized diffraction order for the incident wave with in-plane wave vector (k_x − 2πm/d, k_y). According to Eq. (24), $C_{t 1}^{TM} (k_{x}, k_{y}, ω) = - C_{t 2}^{TM} (- k_{x}, k_{y}, ω)$ . Hence, in the vicinity of (k_x, k_y, ω) = (0, 0, ω₀) the following approximate equation holds:

C_{t 2}^{TM} (k_{x}, k_{y}, ω) \approx - k_{y} c_{t}^{TM} .

Similar approximations are valid for $C_{r 1}^{TM}$ and $C_{r 2}^{TM}$ :

C_{r 1}^{TM} (k_{x}, k_{y}, ω) \approx - C_{r 2}^{TM} (k_{x}, k_{y}, ω) \approx k_{y} c_{r}^{TM} .

The non-resonant cross-polarization scattering operators with the TFs denoted as $R_{0}^{TM}$ and $T_{0}^{TM}$ have a different form. According to Eqs. (24) and (26), $R_{0}^{TM}$ is an odd function with respect to k_x and k_y: $R_{0}^{TM} (k_{x}, k_{y}, ω) = - R_{0}^{TM} (- k_{x}, k_{y}, ω) = - R_{0}^{TM} (k_{x}, - k_{y}, ω)$ . Hence, its Taylor approximation is

R_{0}^{TM} (k_{x}, k_{y}, ω) \approx k_{x} k_{y} r_{0}^{TM},

where

r_{0}^{TM} = {\partial^{2} R_{0}^{TM} / (\partial k_{x} \partial k_{y}) |}_{k_{x} = k_{y} = 0, ω = ω_{0}}

. A similar equation holds for

T_{0}^{TM}

:

T_{0}^{TM} (k_{x}, k_{y}, ω) \approx k_{x} k_{y} t_{0}^{TM} .

For the last pair of TFs, C₁ and C₂, we will use the following Taylor approximations:

\begin{array}{l} C_{1} (k_{x}, k_{y}, ω) \approx c_{1}, \\ C_{2} (k_{x}, k_{y}, ω) \approx c_{2} . \end{array}

Using the obtained approximations one can solve system (13) to find complex reflection and transmission coefficients. The transmission coefficients $T^{same} = F_{T}^{TE} / F$ and $T^{cross} = F_{T}^{TM} / F$ are the following:

T^{same} = t_{0}^{same} \frac{v_{g}^{2} k_{x}^{2} - (ω - ω_{zt} - η_{zt} k_{y}^{2}) (ω - ω_{p 2} - η_{p 2} k_{y}^{2})}{v_{g}^{2} k_{x}^{2} - (ω - ω_{p 1} - η_{p 1} k_{y}^{2}) (ω - ω_{p 2} - η_{p 2} k_{y}^{2})},

T^{cross} = k_{x} k_{y} [t_{0}^{cross} + \frac{b}{v_{g}^{2} k_{x}^{2} - (ω - ω_{p 1} - η_{p 1} k_{y}^{2}) (ω - ω_{p 2} - η_{p 2} k_{y}^{2})}],

where

\begin{array}{l} t_{0}^{same} & = t_{0}^{TE}, t_{0}^{cross} = t_{0}^{TM}, \\ b & = - 2 i c_{e} c_{t}^{TM} v_{g}, \\ ω_{p 1} & = ω_{0} - i (c_{1} - c_{2}), \\ ω_{p 2} & = ω_{0} - i (c_{1} + c_{2}), \\ ω_{zt} & = ω_{p 1} - 2 i c_{e} c_{t}^{TE} / t_{0}^{TE}, \\ η_{p 1} & = η_{p 2} = η_{zt} = v_{g} / (2 k_{1}) . \end{array}

The equations describing the TE- and TM-polarized reflection coefficients have a similar form to Eqs. (15) and (16).

Let us obtain a more general form of Eqs. (15) and (16). To do this we add higher-order terms into the Taylor approximations of C₁ and C₂. According to Eq. (25), we introduce $O (k_{y}^{2})$ terms:

\begin{array}{l} C_{1} (k_{x}, k_{y}, ω) \approx c_{1} + c_{1}^{'} k_{y}^{2}, \\ C_{2} (k_{x}, k_{y}, ω) \approx c_{2} + c_{2}^{'} k_{y}^{2} . \end{array}

By solving system (13), we obtain Eqs. (15) and (16), where η_p2 and η_p1 = η_zt are different complex numbers:

\begin{array}{l} η_{p 1} = v_{g} / (2 k_{1}) - i (c_{1}^{'} - c_{2}^{'}), \\ η_{p 2} = v_{g} / (2 k_{1}) - i (c_{1}^{'} + c_{2}^{'}) . \end{array}

Using the same approach for TFs C_e1, C_e2,

C_{t 1}^{TE}

,

C_{t 2}^{TE}

,

C_{r 1}^{TE}

, and

C_{r 2}^{TE}

, one can show that η_p1 and η_zt are generally not equal as well.

3.3. Cross-polarization mode excitation

In the case when the mode’s polarization differs from that of the incident field, the TFs have a different form. The TFs C_e1(k_x, k_y, ω) and C_e2(k_x, k_y, ω) correspond to cross-polarization scattering. According to Eqs. (24) and (26), the TFs can be approximated as

C_{e 1} (k_{x}, k_{y}, ω) \approx - C_{e 2} (k_{x}, k_{y}, ω) \approx k_{y} c_{e} .

Similar approximations hold for

C_{t 1}^{TE}

and

C_{t 2}^{TE}

:

C_{t 1}^{TE} (k_{x}, k_{y}, ω) \approx - C_{t 2}^{TE} (k_{x}, k_{y}, ω) \approx k_{y} c_{t}^{TE} .

In contrast, the TFs

C_{t 1}^{TM}

and

C_{t 2}^{TM}

correspond to the light scattering without polarization con version. Hence,

C_{t 1}^{TM} (k_{x}, k_{y}, ω) \approx C_{t 2}^{TM} (k_{x}, k_{y}, ω) \approx c_{t}^{TM} .

The other TFs have the same form as in Subsection 3.2.

By substituting the obtained approximations into Eq. (13) we can find analytical expressions for the complex reflection and transmission coefficients. The form of the cross-polarized transmission coefficient is the same as in Eq. (16), while the TE-polarized transmission coefficient has a different form:

T^{same} = t_{0}^{same} \frac{v_{g}^{2} k_{x}^{2} - (ω - ω_{p 1} - η_{p 1} k_{y}^{2}) (ω - ω_{p 2} - η_{zt} k_{y}^{2})}{v_{g}^{2} k_{x}^{2} - (ω - ω_{p 1} - η_{p 1} k_{y}^{2}) (ω - ω_{p 2} - η_{zt} k_{y}^{2})} .

Here

η_{zt} = η_{p 2} - 2 i c_{e} c_{t}^{TE} / t_{0}^{TE}

, while the other parameters are defined after Eq. (16). A similar approximation can be obtained for the reflection coefficient.

4. Numerical validation and discussion

In this section, in order to verify the proposed theoretical model, we compare its predictions with the simulations performed using rigorous coupled-wave analysis (RCWA) [22]. Being developed for guided-mode grating shown in Fig. 1, the proposed model is also valid for photonic crystal slabs without waveguide layer (h_wg = 0). As a matter of example, we consider the structure with the following parameters: h_wg = 0, h_gr = 700 nm, w = 200 nm, d = 1000 nm. The dielectric permittivities of the grating and of the surrounding medium are ε_gr = 2 and ε_sub = 1. The optical properties of the structure are studied in the following frequency range: ω ∈ [1.4, 1.7] × 10¹⁵ s⁻¹.

4.1. Simulation results

In the considered spectral range the structure supports TE- and TM-polarized modes. For each mode we constructed the transmission spectrum approximations given by Eqs. (15) and (18). The parameters of the approximations, presented in Tables 1 and 2, were calculated as follows. First, we estimated the parameters of the denominator of Eq. (15) using the scattering matrix approach [23, 24]. The mode’s eigenfrequencies ω_p1, ω_p2 were calculated as the poles of the scattering matrix at k_x = k_y = 0. The mode’s group velocity v_g, was estimated by calculating the scattering matrix pole at k_x ≠ 0. A similar calculations at k_y ≠ 0 allowed us to estimate the values of η_p1 and η_p2. Then, we estimated the value of ω_z by numerically solving the following equation: T (ω_z) = 0 at k_x = k_y = 0. By solving the same equation at k_y ≠ 0 we estimated the value of η_zt. Finally, we calculated the value of the non-resonant transmission coefficient $t_{0}^{same}$ by equating the rigorously calculated transmission coefficient to the approximated one at the point k_x = k_y = 0, ω = 1.57 × 10¹⁵ s⁻¹. Let us note that some of the parameters in Tables 1 and 2 are real, since the considered structure is lossless.

Table 1. Parameters of the modes

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Table 2. Parameters of the resonances

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Figure 3 shows the transmission spectra |T|² of the structure. The left part of each subfigure (k_x < 0, k_y < 0) was calculated rigorously using the RCWA [22], while the right part (k_x > 0, k_y > 0) was calculated using the obtained approximations. To take into account both TE- and TM- polarized modes in Figs. 3(c) and 3(d), we used the product of approximations given by Eqs. (15) and (18). Figures 3(a) and 3(b) correspond to the case of planar diffraction (k_y = 0), while Figs. 3(c) and 3(d) correspond to the conical mount at k_x = 0. Figures 3(a) and 3(c) give the transmission spectrum for TM-polarized incident wave, while Figs. 3(b) and 3(d) correspond to the TE polarization. Note that according to Eqs. (24) and (26), in the considered cases (k_x = 0 or k_y = 0) the cross-polarization transmission coefficient is equal to zero.

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As follows from Fig. 3, the developed theory gives a visually good approximation of the transmission spectrum. The calculated relative root-mean-square deviation between the spectra shown in Fig. 3 does not exceed 0.068 which confirms a good agreement between the simulation results and the proposed model.

4.2. Mode symmetry

Each spectrum in Fig. 3 contains pronounced resonant minimum areas that forms two branches. These branches correspond to the excitation of the structure’s eigenmode. At k_x = k_y = 0 the TM-polarized modes are denoted as TM₁ and TM₂; similarly, the TE-polarized modes are denoted as TE₁ and TE₂. In what follows, we will use the same notations to refer to the whole branch.

According to Figs. 3(a) and 3(b), in the case of planar diffraction (k_y = 0) the TE mode is excited by the TE wave, while TM modes are excited by the TM wave. Thus, no cross-polarized mode excitation takes place. This fact is described by Eq. (18), since at k_y = 0 the fraction in Eq. (18) vanishes.

In the case of normal incidence (k_x = k_y = 0) the TM-polarized incident wave does not excite the mode TM₂. Similarly, TE-polarized normally incident wave does not excite the mode TE₂. This is due to the fact that these are the odd modes which, due to symmetry, cannot be excited by the normally incident plane wave [5]. This fact agrees with Eq. (15): at k_x = k_y = 0 the odd mode’s term ( $ω - ω_{p 2} - η_{p 2} k_{y}^{2}$ )vanishes from the equation.

According to Figs. 3(c) and 3(d), in the case of conical diffraction the cross-polarized mode excitation takes place. However, at k_x = 0 the symmetry still imposes some restrictions: only the odd cross-polarized modes are excited: TE₂ mode is excited by the TM-polarized plane wave; TM₂ mode is excited by the TE-polarized plane wave. Equation (18) takes this fact into account in the following way: at k_x = 0 the term ( $ω - ω_{p 1} - η_{p 1} k_{y}^{2}$ ), which corresponds to the even cross-polarized mode, vanishes.

4.3. Angular tolerances

According to Fig. 3(a), the frequency ω of the minimum changes with increasing |k_x|: while for the TM₁ branch the frequency decreases, for the TM₂-branch the frequency increases. Moreover, for both TM₁ and TE₂ branches in Fig. 3(c) the frequency of the minimum point increases with increasing |k_y|. The shape of the two branches in Figs. 3(a) and 3(b) resembles a hyperbola. In contrast, the two branches in Figs. 3(c) and 3(d) resemble a pair of parabolae.

According to Fig. 3, the frequency of the minimum varies slower with varying k_y rather than k_x (note that the scales of the k_x and k_y axes differ by an order of magnitude). The dependence of minimum’s frequency on k_x and k_y is referred to as the angular tolerance of the GMRG. The angular tolerance is governed by the dispersion law of the mode, Eq. (15), which was obtained by equating to zero the denominator of the obtained approximations. According to Eq. (15), the mode frequency ω at k_x = 0 is given by

ω = ω_{p 1} + η_{p 1} k_{y}^{2} .

The frequency of the mode at k_y = 0 can be approximately calculated as

ω = ω_{p 1} + \frac{v_{g}^{2}}{ω_{p 1} - ω_{p 2}} k_{x}^{2} + O (k_{x}^{4}) .

Therefore, in the case of conical diffraction the angular tolerance is defined by the value of η_p1 that depends on the group velocity of the mode v_g, the period of the grating d, and the number m of the diffraction order, which excites the mode. In contrast, in the case of planar diffraction the angular tolerance depends on the squared group velocity of the mode

v_{g}^{2}

and the mode coupling coefficient c₂ = i(ω_p2 − ω_p1)/2.

Let us estimate the angular tolerances of the TM-polarized mode TM₁ using Eqs. (19) and (20). According to Table 1, the angular tolerance with respect to k_y is defined by |Re η_p1| = 1.56 × 10¹⁹ nm²/s. The angular tolerance with respect to k_x is defined by $| Re v_{g}^{2} / (ω_{p 1} - ω_{p 2}) | = 5.53 \times 10^{20} {nm}^{2} / s$ , which is 35 times larger than |Re η_p1|. Therefore, the difference in k_x- and k_y-angular tolerances, which is seen in Fig. 3, is explained by the proposed model.

5. Conclusion

The optical properties of 1D guided-mode resonant grating in conical mount can be described using two coupled paraxial unidirectional wave equations. This approach allows to describe Fano resonances in the case of conical mounting. Simple expressions for the line shape of the resonance are obtained in a form of transmission and reflection coefficients approximations which depend on light’s frequency and two in-plane wavenumbers. The proposed theory takes into account symmetry and polarization effects. In particular, the theory provides mode excitation conditions and angular tolerances of guided-mode resonant gratings. The approximations are in good agreement with simulation results obtained using rigorous coupled-wave analysis.

The proposed approach for 1D gratings can be generalized to the case of crossed (2D) gratings. In crossed gratings the modes can be simultaneously excited by four diffraction orders, which results in a different form of the ω−k_x−k_y resonance line shape.

Appendix: Symmetry properties of gratings

In this appendix we derive symmetry properties of guided-mode resonant gratings. The latter are considered to be symmetric with respect to both xz and yz planes. Assuming the incident wave has the wave vector (k_x, k_y, k_z), we will be interested in the complex amplitude of the m-th diffraction order, which has the wave vector (k_x + 2πm/d, k_y, k′_z). The amplitude of the diffraction order with the same polarization as the incident wave is denoted as $A_{m}^{same} (k_{x}, k_{y})$ . A similar notation is used for the amplitude of the cross-polarized diffraction order: $A_{m}^{cross} (k_{x}, k_{y})$ .

To formulate the symmetry conditions we have to define the polarizations for the case of conical mount (k_x ≠ 0, k_y ≠ 0). In the case of planar diffraction (k_y = 0), the TM- (TE-) polarized wave is defined in such a way that its magnetic (electric) field is perpendicular to the plane of incidence xz. In this paper we say that the plane wave is TM-polarized if

[\begin{matrix} E_{x} \\ E_{y} \\ E_{z} \\ H_{x} \\ H_{y} \\ H_{z} \end{matrix}] = [\begin{matrix} k_{0} k_{z} \\ 0 \\ - k_{0} k_{x} \\ - k_{x} k_{y} \\ k_{x}^{2} + k_{z}^{2} \\ - k_{y} k_{z} \end{matrix}] \cdot exp {i (k_{x} x + k_{y} y + k_{z} z - ω t)} .

The TE-polarized plane wave is defined as follows:

[\begin{matrix} E_{x} \\ E_{y} \\ E_{z} \\ H_{x} \\ H_{y} \\ H_{z} \end{matrix}] = [\begin{matrix} 0 \\ k_{0} k_{z} \\ - k_{0} k_{y} \\ - (k_{y}^{2} + k_{z}^{2}) \\ k_{x} k_{y} \\ k_{x} k_{z} \end{matrix}] \cdot exp {i (k_{x} x + k_{y} y + k_{z} z - ω t)} .

In the case of planar diffraction (k_y = 0) the above defined waves are reduced to conventional TE- and TM-polarized waves.

Let us assume, that we know the solution of Maxwell’s equations when the incident wave is TE-polarized. In this case, $A_{m}^{same} (k_{x}, k_{y})$ and $A_{m}^{cross} (k_{x}, k_{y})$ denote the complex amplitudes of the TE and TM components of the m-th transmitted diffraction order. Now let us “reflect” the solution (i. e. the field distribution) with respect to the structure’s symmetry plane yz. After the “reflection”, the incident wave will have the wave vector (−k_x, k_y, k_z), while the considered diffraction orders will have the wave vector (−k_x − 2πm/d, k_y, k′_z), which correspond to the −m-th diffraction order. Since E⃗ is a vector while H⃗ is a pseudovector, one can deduce from Eq. (22) that the amplitude of the (TE-polarized) incident wave will not change due to the “reflection”. Similarly, the considered TE-polarized diffraction order will have the amplitude $A_{m}^{same} (k_{x}, k_{y})$ , while the cross-polarized (TM-polarized) diffraction order will change the sign due to the “reflection”: according to Eq. (21), the “reflected” wave’s amplitude will be $- A_{m}^{cross} (k_{x}, k_{y})$ .

Since the “reflected” solution satisfies the Maxwell’s equations, we can equate the amplitudes of the diffraction orders before and after the “reflection”. This allows us to obtain the following symmetry conditions:

A_{m}^{same} (k_{x}, k_{y}) = A_{- m}^{same} (- k_{x}, k_{y}),

A_{m}^{cross} (k_{x}, k_{y}) = - A_{- m}^{cross} (- k_{x}, k_{y}) .

These conditions are valid for both reflected and transmitted diffraction orders. Besides, one can consider the case of TM-polarized incident wave in a similar manner to show that Eqs. (23) and (24) hold for both TM- and TE-polarized incident wave.

A similar analysis of the structure’s xz symmetry plane results in the following conditions:

A_{m}^{same} (k_{x}, k_{y}) = A_{m}^{same} (k_{x}, - k_{y}),

A_{m}^{cross} (k_{x}, k_{y}) = - A_{m}^{cross} (k_{x}, - k_{y}) .

Funding

The work was funded by the Russian Science Foundation grant 14-19-00796 (the results pertaining to the polarization-dependent CMT and numerical simulations presented in Sections 3 and 4), Russian Foundation for Basic Research grant 16-29-11683 (the results pertaining to the CMT without polarization effects presented in Section 2), and Russian Science Foundation grant 14-31-00014 (the results pertaining to analysis of angular tolerances and symmetry properties of GMRGs presented in Section 4).

References and links

1. S. S. Wang and R. Magnusson, “Theory and applications of guided-mode resonance filters,” Appl. Opt. 32, 2606–2613 (1993). [CrossRef] [PubMed]

2. W. Zhou, D. Zhao, Y.-C. Shuai, H. Yang, S. Chuwongin, A. Chadha, J.-H. Seo, K. X. Wang, V. Liu, Z. Ma, and S. Fan, “Progress in 2D photonic crystal Fano resonance photonics,” Prog. Quantum Electron. 38, 1–74 (2014). [CrossRef]

3. U. Fano, “Effects of configuration interaction on intensities and phase shifts,” Phys. Rev. 124, 1866–1878 (1961). [CrossRef]

4. S. Fan, W. Suh, and J. D. Joannopoulos, “Temporal coupled-mode theory for the Fano resonance in optical resonators,” J. Opt. Soc. Am. A 20, 569–572 (2003). [CrossRef]

5. N. A. Gippius, S. G. Tikhodeev, and T. Ishihara, “Optical properties of photonic crystal slabs with an asymmetrical unit cell,” Phys. Rev. B 72, 045138 (2005). [CrossRef]

6. M. Nevière, E. Popov, and R. Reinisch, “Electromagnetic resonances in linear and nonlinear optics: phenomenological study of grating behavior through the poles and zeros of the scattering operator,” J. Opt. Soc. Am. A 12, 513–523 (1995). [CrossRef]

7. A. Sharon, S. Glasberg, D. Rosenblatt, and A. A. Friesem, “Metal-based resonant grating waveguide structures,” J. Opt. Soc. Am. A 14, 588–595 (1997). [CrossRef]

8. D. Rosenblatt, A. Sharon, and A. A. Friesem, “Resonant grating waveguide structures,” IEEE J. Quantum Electron. 33, 2038–2059 (1997). [CrossRef]

9. H. Liu and P. Lalanne, “Microscopic theory of the extraordinary optical transmission,” Nature 452, 728–731 (2008). [CrossRef] [PubMed]

10. H. Liu and P. Lalanne, “Comprehensive microscopic model of the extraordinary optical transmission,” J. Opt. Soc. Am. A 27, 2542–2550 (2010). [CrossRef]

11. S. P. Shipman and S. Venakides, “Resonant transmission near nonrobust periodic slab modes,” Phys. Rev. E 71, 026611 (2005). [CrossRef]

12. D. A. Bykov and L. L. Doskolovich, “ω − k_x Fano line shape in photonic crystal slabs,” Phys. Rev. A 92, 013845 (2015). [CrossRef]

13. D. A. Bykov and L. L. Doskolovich, “Spatiotemporal coupled-mode theory of guided-mode resonant gratings,” Opt. Express 23, 19234–19241 (2015). [CrossRef] [PubMed]

14. E. Sakat, G. Vincent, P. Ghenuche, N. Bardou, C. Dupuis, S. Collin, F. Pardo, R. Haïdar, and J.-L. Pelouard, “Freestanding guided-mode resonance band-pass filters: from 1D to 2D structures,” Opt. Express 20, 13082–13090 (2012). [CrossRef] [PubMed]

15. D. W. Peters, R. R. Boye, and S. A. Kemme, “Angular sensitivity of guided mode resonant filters in classical and conical mounts,” Proc. SPIE 8633, 86330W (2013). [CrossRef]

16. Y. H. Ko, M. Niraula, and R. Magnusson, “Divergence-tolerant resonant bandpass filters,” Opt. Lett. 41, 3305–3308 (2016). [CrossRef] [PubMed]

17. S. P. Shipman and A. T. Welters, “Resonant electromagnetic scattering in anisotropic layered media,” J. Math. Phys. 54, 103511 (2013). [CrossRef]

18. N. V. Golovastikov, D. A. Bykov, L. L. Doskolovich, and V. A. Soifer, “Analytical description of 3D optical pulse diffraction by a phase-shifted Bragg grating,” Opt. Express 24, 18828 (2016). [CrossRef] [PubMed]

19. F. Träger, Springer Handbook of Lasers and Optics (Springer, 2007). [CrossRef]

20. H. A. Haus, Waves and Fields in Optoelectronics (Prentice-Hall, 1984).

21. N. V. Golovastikov, D. A. Bykov, L. L. Doskolovich, and V. A. Soifer, “Spatiotemporal optical pulse transformation by a resonant diffraction grating,” J. Experimental Theoretical Phys. 121, 785–792 (2015). [CrossRef]

22. M. G. Moharam, E. B. Grann, D. A. Pommet, and T. K. Gaylord, “Formulation for stable and efficient implementation of the rigorous coupled-wave analysis of binary gratings,” J. Opt. Soc. Am. A 12, 1068–1076 (1995). [CrossRef]

23. S. G. Tikhodeev, A. L. Yablonskii, E. A. Muljarov, N. A. Gippius, and T. Ishihara, “Quasiguided modes and optical properties of photonic crystal slabs,” Phys. Rev. B 66, 045102 (2002). [CrossRef]

24. D. A. Bykov and L. L. Doskolovich, “Numerical methods for calculating poles of the scattering matrix with applications in grating theory,” J. Lightwave Technol. 31, 793–801 (2013). [CrossRef]

	Mode polarization

	TM	TE
ω_p1, s⁻¹	1.5361 × 10¹⁵ − 3.1952 × 10¹²i	1.5963 × 10¹⁵ − 3.5938 × 10¹²i
ω_p2, s⁻¹	1.6222 × 10¹⁵	1.4705 × 10¹⁵
η_p1, nm²/s	1.5615 × 10¹⁹ + 7.0941 × 10¹⁶i	1.4951 × 10¹⁹ + 7.1408 × 10¹⁶i
η_p2, nm²/s	1.8035 × 10¹⁹ − 9.9797 × 10¹⁶i	2.0849 × 10¹⁹ − 2.5641 × 10¹⁷i
v_g, nm/s	2.1834 × 10¹⁷	2.1915 × 10¹⁷

			Mode polarization

			TM	TE
Incident wave polarization	TM	$t_{0}^{same}$	0.1241 − 0.9572i
		ω_zt, s⁻¹	1.5975 × 10¹⁵	1.5543 × 10¹⁵
		η_zt, nm²/s	1.5584 × 10¹⁹	2.0948 × 10¹⁹

	TE	$t_{0}^{same}$	0.2509 − 0.9196i
		ω_zt, s⁻¹	1.6942 × 10¹⁵	1.6572 × 10¹⁵
		η_zt, nm²/s	1.7998 × 10¹⁹	1.4908 × 10¹⁹

	Mode polarization

	TM	TE
ω_p1, s⁻¹	1.5361 × 10¹⁵ − 3.1952 × 10¹²i	1.5963 × 10¹⁵ − 3.5938 × 10¹²i
ω_p2, s⁻¹	1.6222 × 10¹⁵	1.4705 × 10¹⁵
η_p1, nm²/s	1.5615 × 10¹⁹ + 7.0941 × 10¹⁶i	1.4951 × 10¹⁹ + 7.1408 × 10¹⁶i
η_p2, nm²/s	1.8035 × 10¹⁹ − 9.9797 × 10¹⁶i	2.0849 × 10¹⁹ − 2.5641 × 10¹⁷i
v_g, nm/s	2.1834 × 10¹⁷	2.1915 × 10¹⁷

			Mode polarization

			TM	TE
Incident wave polarization	TM	$t_{0}^{same}$	0.1241 − 0.9572i
		ω_zt, s⁻¹	1.5975 × 10¹⁵	1.5543 × 10¹⁵
		η_zt, nm²/s	1.5584 × 10¹⁹	2.0948 × 10¹⁹

	TE	$t_{0}^{same}$	0.2509 − 0.9196i
		ω_zt, s⁻¹	1.6942 × 10¹⁵	1.6572 × 10¹⁵
		η_zt, nm²/s	1.7998 × 10¹⁹	1.4908 × 10¹⁹

Coupled-mode theory and Fano resonances in guided-mode resonant gratings: the conical diffraction mounting

Abstract

1. Introduction

2. 2D coupled-mode theory

2.1. Pulse in a slab waveguide

2.2. Coupled-mode equations

3. Polarization in coupled-mode theory

3.1. Coupling operators

3.2. Same-polarization mode excitation

3.3. Cross-polarization mode excitation

4. Numerical validation and discussion

4.1. Simulation results

4.2. Mode symmetry

4.3. Angular tolerances

5. Conclusion

Appendix: Symmetry properties of gratings

Funding

References and links

Cited By

Figures (3)

Tables (2)

Equations (43)

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