1. INTRODUCTION

The scattering matrix is an important object in the electromagnetic theory of gratings. It gives the response of a grating to a set of incident monochromatic plane waves of a common wavelength. The full scattering matrix is an infinite-square matrix, whose columns correspond to incident plane waves and rows to diffracted plane waves. Once the wave vector projection of any one of the incident plane waves in the grating plane is determined, so are those of all incident and diffracted plane waves owing to the periodicity of the grating. A finite section of the full scattering matrix restricted to all propagating incident and diffracted orders fully describes the grating’s far-field properties. It is referred to as the propagating-order scattering matrix or simply $S$ matrix in this paper.

Uretsky [1] was the first to introduce the $S$ matrix into grating theory. He studied perfectly conducting gratings in non-conical mounting and showed that the matrix of reflection coefficients of the grating is unitary and has symmetry properties reminiscent of the quantum-mechanical $S$ matrix (in grating theory, the term “conical” describes the grating mounting in which the plane of incidence differs from the cross-sectional plane that is perpendicular to the grating lines; therefore, using it infers the grating under study is one-dimensionally periodic). Nevière and Vincent [2] were the first, in their study of reciprocity theorem of gratings in 1976, to use $S$ matrix for dielectric transmission gratings in non-conical mounting. Three years later, Vincent and Nevière [3] extended their work in [2] to transmission gratings in conical mountings and cast their results in $S$-matrix form. Maystre et al. [4,5] and Popov et al. [6] applied the symmetry and analytic properties of the $S$ matrix to study anomalies, energy absorption, and perfect blazing of gratings. The gratings in these three studies are in non-conical mounting and can be categorized as two-order gratings, i.e., gratings only allowing either the 0th and ${-}{1}$st orders in reflection or the 0th orders in reflection and transmission; therefore, the $S$ matrices are ${2} \times {2}$. In 2002, Fehrembach et al. [7] extended the work of [6] in a phenomenological study of filtering by crossed transmission gratings. The gratings in [7] are still two-order, allowing the 0th orders in reflection and transmission, but the polarization coupling phenomenon makes the size of the $S$ matrix ${4} \times {4}$. In 2015, Alaridhee et al. [8] extended the work of [7] to treat perfectly conducting slanted annular aperture arrays, a special type of crossed grating that has space reversal symmetry. The wavelength-to-period ratio of [8] still only allows the two 0th orders propagating.

The general formulation and global and elemental properties of the $S$ matrix in non-conical mounting are well-documented in the literature (see, e.g., [9]). Here the global properties refer to those that belong to the $S$ matrix as a whole, which are primarily the unitarity of $S$ matrix for lossless gratings and the relationship between the $S$ matrices of an arbitrary grating for two sets of mutually reciprocal incident plane waves, and the elemental properties refer to those that belong to individual $S$ matrix elements. The status for conically mounted gratings is surprisingly different: there is a general lack of information on the above aspects of the $S$ matrix in the existing literature (some content of the present paper was presented orally at the EOS Topical Meeting on Diffractive Optics, 16–19 September 2019, Jena, Germany). The work of Vincent and Nevière in [3] is, to the best of my knowledge, the only one that touches the topic, but the treatment there is sketchy and incomplete. That a plane wave diffracted by a conically mounted grating is in general elliptically polarized adds a new dimension to the $S$ matrix and makes the $S$-matrix story more interesting than its non-conical counterpart. In [3] an important vector dot-product reciprocity relation for gratings in conical mounting is derived, but the rich information contained in it is not fully exploited. The authors derived an $S$-matrix expression of reciprocity theorem, but they did not pay due attention to the formation of the column vectors that the $S$ matrix operates on and generates. I will show in Section 3 a symmetric global $S$-matrix expression of the reciprocity theorem is achieved only when the column vectors are properly formulated. Furthermore, only a reciprocity theorem for the efficiencies of natural (unpolarized) light incidences are given in [3]. To date, there is still no reciprocity theorem that relates both the diffraction efficiencies and the states of polarization of two reciprocally incident plane waves.

Publications on the $S$ matrix of crossed gratings besides [7,8,10] are virtually non-existent. References [7,8] only consider the 0th-order transmission gratings. In [10], the relationship between $S$ matrices of two reciprocal incidence cases is derived for numerical testing purpose. Their $S$-matrix elements are not normalized; therefore, the derived reciprocity relation does not have the matrix transpose symmetry. Using the $S$ matrix is the best way to display the reciprocity theorem of gratings. Historically, discussions of the $S$ matrix have often included a discussion of $S$-matrix expression of the reciprocity theorem, but the converse is not true. A dot-product reciprocity relation for crossed gratings, similar to that derived for conically mounted gratings in [3], is mentioned in research papers [10–12] and reference books [9,13]; however, except in [10] the dot-product relation is not converted to an $S$-matrix relation and all of [9–13] trace the detailed derivation of the dot-product relation, apparently in the context of infinitely conducting inductive grids, to an internal report.

The aim of this paper is to present a systematic and formal treatment of the $S$ matrix of conically mounted and crossed gratings. It includes three parts: the formulation of the general $S$ matrix, the global properties of the $S$ matrix as a whole (henceforth global $S$ matrix), and the properties of ${2} \times {2}$ block matrices (henceforth elemental matrices). The analytical and resonant properties of the elemental matrices fall outside its coverage. The first two parts may be known or easily predictable to an expert of grating theory; they are included because they are not yet available in the literature. The third part contains original contributions including the definition of elemental $S$ matrix, a discussion of the eigensolutions of the elemental $S$ matrix and those of its Hermitian product, the extremal property of diffraction efficiency, the sum rules of diffraction efficiencies, and most importantly the vectorial reciprocity theorem of gratings.

2. NOTATION SYSTEM AND REPRESENTATIONS OF POLARIZATION

A. Notations

For the sake of saving space, we only explicitly consider diffraction by a crossed (two-dimensionally periodic) grating. Diffraction by a conically mounted grating is a special case for which one of the two periodicities vanishes. Let the $z$ axis of a rectangular Cartesian coordinate system $Oxyz$ be normal to the grating plane, and its $x$ axis be along one of the two principal periodic directions. Define an oblique Cartesian coordinate system $O{x^1}{x^2}{x^3}$ such that its ${x^1}$ and ${x^3}$ axes coincide with the $x$ and $z$ axes, respectively, and its ${x^2}$ axis is in the $Oxy$ plane and at an angle $\zeta$ to the $y$ axis. The set of reciprocal space (or contravariant) basis vectors associated with $O{x^1}{x^2}{x^3}$ that is the most natural for expressing the incident and diffracted wave vectors is

The incident and diffracted wave vectors of a crossed grating are given by

 

Fig. 1. Wave vector diagram of a crossed grating in two reciprocal incidences.

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When formulating the reciprocity theorem of crossed gratings, we consider two mutually reciprocal cases. In Case (a), an incident plane wave with $\textbf{k}_{\textit{mn}}^{\text{i }(p)}$ generates a diffracted plane wave with $\textbf{k}_{\textit{jl}}^{\text{d}(q)}$. In Case (b), an incident plane wave with

We use $(\textit{m,n;p}) \to (\textit{j,l;q})$ and $(\textit{m,n;p})^\prime \leftarrow (\textit{j,l;q})^\prime$ to denote the diffraction processes in Case (a) and Case (b), respectively. The left pointing arrow goes with the primed quantities. In both cases, the triple-index group at the arrow tail side labels the incident order and that at the arrow head side labels the diffracted order. When there is no need to distinguish the incident and diffracted orders, the triple-index group ($\textit{m,n;p}$) is referred to as a propagating order. Furthermore, when there is no need to indicate the medium, the double-index group ($\textit{m,n}$) is used instead.

B. Representations of Polarization

The polarization of the incident and diffracted monochromatic plane waves in the far field are in general elliptically polarized. It takes four real parameters to fully define a polarization ellipse, which is in a plane perpendicular to the wave vector. In geometric language, a polarization ellipse is characterized by its major axis orientation angle $\psi$ and its ellipticity angle $\chi = {\pm}{\rm arcsin}(b/a)$, where $a$ and $b$ are the lengths of the major and minor semiaxes, respectively, and the ${\pm}\;{\rm signs}$ correspond to two senses of rotation. The other two parameters, the size of the ellipse (may be measured as ${a^2} + {b^2}$) and the phase of rotation (at time $t = {t_0}$), are less important. The geometric characterization is intuitive but not convenient for mathematical manipulation. In our discussion of $S$ matrices, we will use some two-dimensional complex linear-space representations.

1. Vectorial Representation

In this representation, the electric field of a plane wave is a complex vector that has this general expression:

2. Optical Representation

The most familiar polarization representation to an optics person and useful in the context of planar guided-wave optics and plasmonics is to decompose the electric field vector into its ${ p}$ and ${ s}$ components:

3. Electromagnetic Representation

In numerical solutions of grating problems, one usually first solves Maxwell’s equations in some electromagnetic components and then from which derives the other field components. The polarization of a plane wave in the free space can be uniquely determined with two properly chosen field components. In what follows for definiteness we sometimes choose the third components. Then, the orthogonal basis is (${E_z} = {1}$, 0) and (0, ${H_z} = {1}$), and the polarization is given by ${\boldsymbol x} = {({E_z}, {H_z})^{\text{T}}}$, which we will refer to as the standard electromagnetic representation. Other bases are possible. When a plane wave propagates normal to the grating plane, the ${({E_z}, {H_z})^{\text{T}}}$ representation is inapplicable and must be replaced with, for example, ${({E_2}, {H_2})^{\text{T}}}$ or ${({E_1}, {E_2})^{\text{T}}}$. Note that the electromagnetic representations describe the polarization indirectly via Maxwell’s equations. They are in physical spaces different from that of the optical representations. The transformation from electromagnetic to optical or vice versa requires knowing the wave vector $\textbf{k}$.

4. Generic Linear-Space Representation

Since the optical and electromagnetic polarization spaces are isomorphic, we may talk about a generic two-dimensional complex linear-space representation, in which the polarization can be expressed as ${\boldsymbol x} = {({x_1},\;{x_2})^{\text{T}}}$, with the optical and electromagnetic representations being special cases. To work with the $S$ matrices, we assume that all polarization representations use an orthogonal basis, which may be complex. When associated with linear-space characterization of polarization the term representation has the connotation of the physical space and the underlying basis. Two polarizations ${\boldsymbol x}$ and ${\boldsymbol y}$ are in different representations when either they are in different physical spaces or they use different bases. The generic representation suppresses these differences. To further discuss the generic representation and to prepare for presenting the $S$-matrix properties, we need to normalize the polarization vectors first.

C. Normalization of Field Vectors

We normalize the plane waves such that the absolute value squared of the electric field vector is equal to the absolute value of the $z$ component of the Poynting vector, up to a medium and diffraction-order independent multiplier. We use a tilde to denote the normalized field quantities. Then,

For describing polarization, all representations defined in the previous subsection are equivalent to one another. The following their common properties will be used in this paper:

3. FORMULATION AND GLOBAL PROPERTIES OF THE S MATRIX

A. Formulation of the S Matrix

Let ${\cal I}$ and ${\cal D}$ be column vectors of the normalized and regularized incident and diffracted field amplitudes, respectively, that we have prepared in the previous section. Based on the linearity of Maxwell’s equations and the assumption that the grating is composed of only linear materials, without any derivation we can write

The further writing of the column vectors and $S$ matrix at the element level needs to bring in their representations. In vectorial representation, the elements of ${\cal I}$ and ${\cal D}$ are two-dimensional vectors $\tilde{\textbf E}_{\textit{mn}}^{{\text{i}}(p)}$ and $ \tilde{\textbf E}_{\textit{jl}}^{\text{d}(q)}$, respectively, and an element of $\mathbb{S}$ is a two-dimensional second-order tensor ${\stackrel{\leftrightarrow}{\textbf{S}}}_{jl;mn}^{(q;p)}$ that describes the diffraction process $(\textit{m,n;p}) \to (\textit{j,l;q})$. Henceforth, for convenience we will refer to ${\stackrel{\leftrightarrow}{\textbf{S}}}_{jl;mn}^{(q;p)}$ as an elemental $S$ tensor, although tensor is not a mathematically correct term here as it becomes apparent in the next two paragraphs. With these notations, we may write

In a linear-space representation, the elements of ${\cal I}$ and ${\cal D}$ are ${2} \times {1}$ column vectors $ \tilde{\boldsymbol{x}}_{\textit{mn}}^{\text{i}(p)}$ and $ \tilde{\boldsymbol{x}}_{\textit{jl}}^{\text{d}(q)}$, respectively, and at the $(jl, mn)$ position of $\mathbb{S}$ is a ${2} \times {2}$ elemental $S$ matrix $S_{jl;m n}^{(q;p)}$. In component form,

Similar to Eq. (19), this equation can also be interpreted in two ways. Then, at the matrix level, Eq. (21) may be viewed as an expression of the $2N$-square $\mathbb{S}$ matrix when the indices $(j,\!l{;}{\sigma _\nu},q)$ and $(m,\!n;{\tau _\upsilon},\!p)$ run through their full ranges $(\nu ,\;\upsilon = {1},\;{2})$. The elemental matrix $S_{jl;m n}^{(q;p)}$ relates the two vectors $ \tilde{\boldsymbol{x}}_{\textit{mn}}^{\text{i}(p)}$ and $ \tilde{\boldsymbol{x}}_{\textit{jl}}^{\text{d}(q)}$, both in an orthogonal bases, in different polarization planes; therefore, by definition it is a creature of mixed row and column representations. Furthermore, different elemental $S$ matrices in a global $S$ matrix obviously use different pairs of representations (bases and possibly physical spaces).

Based on the polarization property (ii) stated in Subsection 2.C, an elemental $S$ matrix multiplied from the left by a unitary matrix ${V^{\dagger}}$, or from the right by a unitary matrix $U$, or simultaneously, is still an elemental $S$ matrix of the same diffraction process. Therefore, the elemental $S$ matrices are determined only to an equivalence class, which we denote by $C({\stackrel{\leftrightarrow}{\textbf{S}}})$. Different $S$ matrices $A$ and $B$ in $C({\stackrel{\leftrightarrow}{\textbf{S}}})$ are related to each other by a matrix bi-unitary equivalence relation $A = {V^{\dagger}}\!BU$. The term bi-unitary equivalence is used here to distinguish it from unitary equivalence that refers to the special case of $V = U$.

B. Formulation of the S Matrix

The $\mathbb{S}$ matrix as a whole has a few interesting symmetry properties. The foremost is the reciprocity symmetry that is available only for gratings composed of reciprocal media. The constitutive relations for a bianisotropic medium can be written as

On both sides of Eq. (25), the two electric field vectors forming the dot product are associated with plane waves of opposite propagation directions. By using Eq. (16) and taking the normalization and regularization Eq. (14) into account, the linear-space version of Eq. (25) is obtained:

The transpose symmetry in Eqs. (30) and (31) and the elegant expression of the reciprocity theorem in Eq. (32) are achieved thanks to the regularization of polarization column vectors in Eq. (14). Without the regularization, the plus signs between the two terms after row–column vector multiplications on both sides of Eq. (29) would become minus signs, and in front of the off-diagonal elements of the matrix on the right-hand side of Eq. (31) would appear a negative sign, thus breaking the transpose symmetry and making Eq. (30) invalid. This feature is seen in Eq. (7) of [10] and also in Eq. (4.15) of [16]. Equation (22) of [3] for conically mounted gratings parallels the present Eq. (31), but Eq. (21) of [3] suggests that no regularization is made there. In [7] the transpose symmetry is retained by using an inconsistent ($\hat{\boldsymbol p},\hat{\boldsymbol s}$) polarization reference system. The triplets ($\hat{\boldsymbol p},\hat{\boldsymbol s},\textbf{k}$) for their upward and downward propagating waves are left-handed and right-handed, respectively. While this approach may be okay with the optical representation, it is unacceptable with the electromagnetic representation because it is universally accepted that ($\textbf{E},\textbf{H},\textbf{k}$) for a right-handed triplet. Therefore, the regularization adopted in Eq. (14) seems to be the most reasonable.

The global reciprocity symmetry of the $S$ matrix exhibited in Eq. (32) is valid for any grating geometry, any wavelength-to-period ratios, any angles of incidence, and any grating materials provided they are all reciprocal. As noted by Fehrembach et al. [7], when a crossed grating possess the ${C_2}$ symmetry, $\mathbb{S}^\prime = \mathbb{S}$; therefore, it follows from Eq. (32) that $\mathbb{S}^{\text{T}}= \mathbb{S}$, i.e., $\mathbb{S}$ is a symmetric matrix. Except for the ${C_2}$ symmetry requirement, this result has the same range of validity as Eq. (32). The reader is reminded that the ${C_2}$ symmetry requires the coordinate origin of the grating plane be placed at the symmetry center of the grating. Other symmetries can be derived, but they impose further restrictions on the grating and the angles of incidence. A discussion in this direction is beyond the scope of this paper.

Another global property of the $S$ matrix is its unitarity for a grating composed of lossless media. A medium is lossless if its constitutive parameters in Eq. (23) satisfy these conditions [15]:

C. Numerical Obtainment of the S Matrix

Once a numerical code for normal grating simulation is available, getting numerical values of the $S$ matrix is easy. In a typical grating problem, only one incident plane wave exists in only one outermost medium. Suppose the incident medium is the ${+}{1}$ medium. By the time the work for getting the grating’s response to this single incident wave is done, the two entire block matrices ${R^{(+ 1)}}$ and $T{^{(- 1, + 1)}}$ in Eq. (18) are already available. If the grating is of reflection type, ${T^{(- 1, + 1)}}$ does not exist but ${R^{(+ 1)}}$ always does. For a transmission grating, the other two blocks of Eq. (18) still have to be calculated, but it is unnecessary to launch another normal grating calculation for incidence from the bottom medium. The missing pieces can be obtained at relatively little extra cost in the process of solving the normal grating problem for the upper medium incidence, if the full $S$-matrix recursion is used, instead of the half- or quarter-$S$-matrix recursion [17]. What the numerical code produces so far is not the final object. Only after restricting to propagating orders and normalizing and regularizing in accordance with Eq. (14) does one get the needed $\mathbb{S}$.

4. PROPERTIES OF ELEMENTAL S MATRICES

In this section, since we focus on the elemental $S$ matrices, we drop all diffraction order and medium identifying superscripts and subscripts but keep in mind that an arbitrary but fixed diffraction process $(\textit{m,n;p}) \to (\textit{j,l;q})$, possibly along with its reciprocal process $(\textit{m,n;p})^\prime \leftarrow (\textit{j,l;q})^\prime$, is under discussion. Then the expression of the elemental $S$ matrix is simplified to be

A. Physical Meanings of $S$-Matrix Elements and Elemental $S$ Matrices

The physical meaning of an $S$-matrix element for a non-conically mounted grating is straightforward. In this case, only one integer is needed to label the diffraction order. The TE and TM directions are customarily taken as the basis directions and the matrix in Eq. (36) is diagonal. Suppose the incident plane wave is TE- (TM)-polarized with a unit amplitude and zero phase. Then the diagonal element of $S$ at the TE (TM) position and its absolute value squared give directly the complex amplitude and diffraction efficiency of the diffraction order, respectively.

The physical interpretation of an individual $S$-matrix element for a conically mounted or crossed grating is less straightforward and actually not very useful, because in general the $S$ matrix in Eq. (36) is not diagonal. Although formally one can understand, for example, ${S^{({\sigma _1};\,{\tau _2})}}$ as the complex amplitude of the first polarization component of the diffracted order due to an incident order with only the second polarization component non-zero, $| {S^{({\sigma _1};\,{\tau _2})}}{|^2}$ gives only one of the two contributions to the diffraction efficiency of the diffraction process. Therefore, for a conically mounted or crossed grating it is the elemental $S$ matrix, not its individual elements, that completely describes the diffraction process. The elemental $S$ matrix or $S$ tensor is the basic functional unit that acts as a transformer turning the incident polarization into the diffracted polarization with energy efficiency:

B. Elemental S Matrix

As indicated at the end of Subsection 3.A, the rows and columns of the elemental $S$ matrix use different representations and they are associated with polarizations in different planes. This character of $S$ limits its usefulness. For example, the eigenvalue problem ${\rm S} \tilde{\boldsymbol{x}} = \lambda \tilde{\boldsymbol{x}}$, although mathematically well-defined, is not physically meaningful because in general $S \tilde{\boldsymbol{x}}$ and $\lambda \tilde{\boldsymbol{x}}$, being in two different polarization planes, cannot be equal in physical space. In this sense, the equation ${ S} \tilde{\boldsymbol{x}} = \lambda \tilde{\boldsymbol{x}}$ is only formal. In addition, given a diffraction process, the eigensolution of $S$ depends on the representations chosen for the incident wave and diffracted order; therefore, the eigenvalue problem does not have the invariance that an equation describing a physical process should possess. Because $S$ is in general non-Hermitian, its two eigenvalues ${\lambda _1}$ and ${\lambda _2}$ may be complex and the eigenvectors ${ \tilde{\boldsymbol{x}}_1}$ and ${ \tilde{\boldsymbol{x}}_2}$ are in general complex and non-orthogonal. By definition Eq. (37), $|\lambda _a|^2 = \eta ({ \tilde{\boldsymbol{x}}_a})$, $a = {1}$, 2.

As an application of the formal eigenvalue problem, we consider two mutually reciprocal Cases (a) and (b) that are represented by elemental matrices $S$ and $S^\prime$. Let $({\lambda_i, \tilde{\boldsymbol{x}}_i})$ and $({\lambda ^\prime _i},{ \tilde{\boldsymbol y}_i})$, $i = {1}$, 2, be the two eigensolutions of $S$ and $S^\prime$, respectively:

C. Elemental ${{S}^{\dagger}}{S}$ Matrix

The product ${S^{\boldsymbol\dagger}}S$ is not a strange object in grating theory, but in most cases it appears as a part of the unitarity expression of the global $S$ matrix for a lossless grating, as in Eq. (35). Outside this context, studies of properties of ${S^{\dagger}}S$ are rare. In [4], ${S^{\dagger}}S$ is used to study anomalies and absorption of two-order metallic gratings in non-conical mountings. The ${2} \times {2}$ $S$ matrix there has its off-diagonal elements go across two diffraction orders, thus it is not an elemental $S$ matrix defined in this paper. In [7,8], Hermitian products of elemental $S$ matrices for crossed gratings are studied briefly. In both these works only the 0th-order diffractions are considered and the standard optical representation (except for the handedness switch mentioned in Subsection 3.B) is used. I have not seen other studies of elemental ${S^{\dagger}}S$ in the literature.

It is beneficial to study the eigenvalue problem ${{\rm S}^{\dagger}}S \tilde{\boldsymbol{x}} = \rho \tilde{\boldsymbol{x}}$. We first remark that this problem is defined with respect to the column representation of $S$, independent of its row representation. Given an $S$ in some row representation, a unitary matrix $U$ makes $US$ an $S$ matrix in a different row representation; however, because ${(US)^{\dagger}}(US) = {S^{\dagger}}S$, the problem is unchanged. This problem is also invariant with respect to a change of the column representation of $S$. Indeed, from ${S^{\dagger}}S \tilde{\boldsymbol{x}}= \rho \tilde{\boldsymbol{x}}$ one can easily derive ${{\overset{\frown}{S}}^{\boldsymbol{\dagger }}}\overset{\frown}{S}\tilde{y} = \rho \tilde{y}$ with $ \tilde{\boldsymbol y}= V\tilde{\boldsymbol{x}}$ and ${{\overset{\frown}{S}}} = VSV^\dagger$, where $V$ is a unitary matrix. Using the terminology defined at the end of Subsection 3.A, we say that the solutions of the eigenvalue problem ${S}^{\dagger} S{\tilde{\textbf x}}= \rho \tilde{\textbf x}$ are shared by the whole bi-unitarily equivalent class $C({\stackrel{\leftrightarrow}{\textbf{S}}})$.

From the composition of ${S^{\dagger}}S$ it follows immediately that the two eigenvalues ${\rho _1}$ and ${\rho _2}$ are real and non-negative and the eigenvectors ${{\boldsymbol u}_1}$ and ${{\boldsymbol u}_2}$ are mutually orthogonal, ${\boldsymbol u}_1^{\boldsymbol \dagger}{{\boldsymbol u}_2}= 0$. We will assume ${\rho _1} \le {\rho _2}$. The right-hand side of Eq. (37) is in the form of a Rayleigh–Ritz ratio. From a theorem in matrix theory ([18], p. 176),

The results stated in the preceding paragraph are of course also valid for ${S^{\prime {\dagger}}}S^\prime$. It is easy to verify that if ($\rho$, $ \tilde{\boldsymbol{x}}$) is an eigensolution of ${S^{\dagger}}S$, then ($\rho$, $\overline {S \tilde{\boldsymbol{x}}}$) is an eigensolution of ${S^{\prime {\dagger}}}S^\prime$. Therefore, ${S^{\dagger}}S$ and ${S^{\prime \dagger}}S^{\prime}$ share the same eigenvalues; however, they are in different representations.

D. Relationship between $S$ and ${S^{\boldsymbol\dagger}}S$

As we have seen in the preceding two subsections, although the elemental $S$ matrix is the basic object that transforms the incident polarization into a diffracted polarization, its Hermitian self-product has better physical and mathematical properties. In general, the eigenvectors of $S$ are not related to those of ${S^{\dagger}}S$ and $|\;{\lambda _{a}}{|^2} \ne {\rho _a}$; however, it follows from $\rm{det}({S^{\dagger}}S) = \rm{det}({S^{\dagger}})\;\rm{det}(S)$ that $|{\lambda _1}{\lambda _2}{|^2} = {\rho _1}{\rho _2}$. In physical terms, the geometric mean of the efficiencies of the two eigenpolarizations of $S$ and that of ${S^{\dagger}}S$ are the same.

So far, we have assumed that the two eigenvectors ${ \tilde{\boldsymbol{x}}_1}$ and ${ \tilde{\boldsymbol{x}}_2}$ of $S$ are not mutually orthogonal, which is true in general, but it is possible that $ \tilde{\boldsymbol{x}}_1^{\boldsymbol \dagger} { \tilde{\boldsymbol{x}}_2}$= 0. This happens when $S$ is a normal matrix. A square matrix $A$ is said to be normal if $A{A^{\dagger}} = {A^{\dagger}}A$. Unitary matrices and Hermitian matrices are normal, but a normal matrix is not necessarily unitary or Hermitian. For the ${2} \times {2}$ elemental $S$ matrix, the two statements $ \tilde{\boldsymbol{x}}_1^{\boldsymbol \dagger} { \tilde{\boldsymbol{x}}_2}$= 0 and $S$ is normal is equivalent to each other ([18], p. 101). It can be easily shown that if $S$ is normal, then $S$, ${S^{\dagger}}$, and ${S^{\dagger}}S$ share the same eigenvectors that correspond to eigenvalues $\lambda$, $\bar \lambda$, and $|\lambda {|^2}$, respectively.

E. Vectorial Reciprocity Theorem

The expressions of the reciprocity theorem stated in Section 3 are general but formal and perhaps too mathematical. It is highly desirable to derive a reciprocity theorem stated in physical terms that a researcher who cares less about mathematics can understand. So far, besides the dot-product reciprocity relations like Eq. (15) and the $S$-matrix-element relation like Eq. (22), both of [3], the only grating reciprocity theorem is about the diffraction efficiency of natural light [3,9]. The fact that in non-conical mounting the reciprocity can be expressed in terms of the vector diffraction amplitudes directly without using vector product, such as in Eq. (11) of [2] and Eq. (2.87) of [9], makes one want to have a similar result for conically mounted and crossed gratings.

A question can be asked naturally: Supposing a plane wave of arbitrary polarization $\tilde{\boldsymbol{x}}$ is incident on a grating at an arbitrary angle and the diffraction efficiency of a diffraction order is $\eta$ in Case (a), can we get the same diffraction efficiency and a polarization $\tilde{\boldsymbol x^\prime}$ that correlates to $\tilde{\boldsymbol{x}}$ in the reciprocal Case (b)? If we keep the polarization part of the question in Case (a), the answer is negative—consider the reflection of a circularly polarized incident plane wave by an air–glass interface at the Brewster angle. If we drop altogether the polarization contents of the question, the answer is affirmative, but the solution is not unique. Since ${S^{\dagger}}S$ and ${S^{\prime \dagger}}S^{\prime}$ have the same eigenvalues, an incident polarization,

The answer to the above unaltered question is given by the following theorem:

Theorem 1 (vectorial reciprocity theorem). Given an (original) incident polarization $ \tilde{\boldsymbol{x}}$ that generates a diffracted polarization $S \tilde{\boldsymbol{x}}$ with diffraction efficiency $\rho$. The diffracted wave generated by the incident wave with polarization $\overline {S \tilde{\boldsymbol{x}}}$ in the reciprocal diffraction case has the same diffraction efficiency $\rho$ and the polarization ${ {\bar {\tilde{\boldsymbol{x}}}}}$, if and only if ${S^{\dagger}}S\tilde{\boldsymbol{x}}= \rho \tilde{\boldsymbol{x}}$.

Proof of Sufficiency. Taking complex conjugate of ${S^{\dagger}}S$$ \tilde{\boldsymbol{x}}$$= \;\rho$$ \tilde{\boldsymbol{x}}$ leads to

Proof of Necessity. Any original incident polarization can be written as $\tilde{\boldsymbol{x}}=a_{1} \boldsymbol{u}_{1}+a_{2} \boldsymbol{u}_{2}$, where ${{\boldsymbol u}_1}$ and ${{\boldsymbol u}_2}$ are the two principal polarizations of ${S^{\dagger}}S$. Suppose ${S^{\dagger}}S \tilde{\boldsymbol{x}} \ne \;\rho \tilde{\boldsymbol{x}}$, then ${a_1}{a_2} \ne{0}$. We may assume ${\rho _1} \ne{\rho _2}$ because otherwise any $ \tilde{\boldsymbol{x}}$ can be a principal polarization. Then,

Theorem 1 is a general vectorial reciprocity theorem for conically mounted and crossed gratings. We restate it in physical terms. Given a pair of mutually reciprocal diffraction cases, suppose in Case (a) the (original) incident polarization is along one of the two principal directions. If in Case (b) the incident polarization is chosen to be the time-reversed image of the diffracted polarization of Case (a), then the two cases have the same diffraction efficiency, and the diffracted polarization of Case (b) is the time-reversed image of the original incident polarization of Case (a). If either or both incident polarizations are not as specified, the time reversal relationship between the incident polarization of Case (a) and that of Case (b) does not hold.

F. Sum Rules of Diffraction Efficiencies

Many researchers are familiar with this grating property, which we call the sum rule of non-conical diffraction efficiencies: the diffraction efficiency of an incident plane wave is given by a weighted average of TE and TM efficiencies, the weights being the powers in TE and TM polarizations. Is there a similar sum rule for diffraction efficiencies of conically mounted or crossed gratings, and if yes, how is it formulated? To answer this question, we offer this theorem:

Theorem 2. Given any diffraction process represented by the elemental matrix $S$ for which ${\rho _1} \ne {\rho _2}$, the diffraction efficiency $\eta ( \tilde{\boldsymbol{x}}$) of an arbitrary incident polarization $\tilde{\boldsymbol{x}}=a_{1} \boldsymbol{u}_{1}+a_{2} \boldsymbol{u}_{2}$ is given by

When ${{\boldsymbol u}_1}$ and ${{\boldsymbol u}_2}$ are not as specified, Eq. (45) may still hold but it can only happen accidentally for some special combinations of ${a_1}$ and ${a_2}$. Note that the required ${{\boldsymbol u}_1}$ and ${{\boldsymbol u}_2}$ are eigenvectors of ${S^{\dagger}}S$, not those of $S$. The proof of the theorem is left to the reader. Evidently, the classical sum rule in the non-conical case is a special case of Eq. (45). To distinguish this general sum rule from another one to be discussed below, we call Theorem 2 the first sum rule of grating diffraction efficiencies (or decomposition of grating diffraction efficiency).

The following result can be called the second sum rule of grating diffraction efficiency:

Theorem 3. The sum of diffraction efficiencies in any two mutually orthogonal polarizations is independent of the choice of orthogonal polarizations. In mathematical terms,

This theorem can be proven in many ways. Among them the simplest, although not the most elementary, is to use the double equalities

G. Other Properties of Elemental $S$ Matrices

The four complex elements of an elemental $S$ matrix are not completely arbitrary and mutually independent. Although equality restrictions are hard to find, many inequality restrictions exist. For example, $|{S_{i1}}{|^2} + |{S_{i2}}{|^2}\; \le \;{1}$, $|{S_{1j}}{|^2} + |{S_{2j}}{|^2}\; \le \;{1}$, ${0}\; \le \;{\rho _1}\; \le \;{\rho _2}\; \le \;{1}$, and ${0}\; \le \;{\rm |}\lambda {{\rm |}^2}\; \le \;{1}$, where $\lambda$ is an eigenvalue of $S$. These restrictions are not necessarily independent. A complete study is beyond the scope of this paper.

As the final note of this section, I mention that the elemental $S$ matrix can be singular, i.e., have a zero eigenvalue. It follows from inequality Eq. (40) that when this happens the incident plane wave must be in a principal polarization. The most obvious example is the Brewster reflection at a planar medium interface for the $p$-polarized light. A non-trivial example is the total absorption by a metallic grating [19]. Theoretically, nothing prevents both eigenvalues of $S$ from being zero, which means $S = {0}$. This extreme case has been studied by Popov et al. [20] and Bonod et al. [21]. In the latter three references, excitation of surface plasmons occurs.

5. CONCLUSION

I have presented a systematic and formal treatment of the global and elemental properties of the propagating-order $S$ matrices of conically mounted and crossed gratings. The most general $S$ matrix of gratings is formulated, including its clear and symmetry-preserving indexing system, polarization representation, and normalization and regularization. General mathematical proofs of the well-known global properties (reciprocity and unitarity) previously not available in the literature are given in Appendices A and B. The distinctive contribution of this work, contained in Section 4, is its exposition of the properties of the elemental $S$ and ${S^{\dagger}}S$ matrices. The key results of the exposition are the two sum rules of diffraction efficiencies and the reciprocity theorem that directly relates the vectorial amplitudes of the incident and diffracted plane waves.

APPENDIX A: DERIVATION OF DOT-PRODUCT RECIPROCITY RELATION

For simplicity and without loss of generality we assume the grating has one finite-thickness layer (Fig. 2). Assuming conditions Eq. (24) are met, from Maxwell’s equations we have

(A1)$$\nabla \cdot \left({\textbf{E} \times \textbf{H}^\prime - \textbf{E}^\prime \times \textbf{H}} \right) = 0,$$

where ($\textbf{E},\textbf{H}$) and (${\textbf{E}^\prime},{\textbf{H}^\prime}$) are the total monochromatic electromagnetic fields satisfying Maxwell’s equations and boundary conditions in Cases (a) and (b), respectively. Consider the coordinate-surface-aligned parallelepiped in Fig. 2, whose cross-section parallel to the ${x^1}{x^2}$ plane is a unit cell of the crossed grating and whose top and bottom faces ${{A}^{{\pm}}}{{B}^{{\pm}}}{{C}^{{\pm}}}{{D}^{{\pm}}}$ are located above and below the top and bottom grating surfaces, respectively. We denote the faces ${{A}^{{\pm}}}{{B}^{{\pm}}}{{C}^{{\pm}}}{{D}^{{\pm}}}$ by ${\Sigma ^ +}$ and ${\Sigma ^ -}$, and the two curved surfaces with corners ${{A}_1}{{B}_1}{{C}_1}{{D}_1}$ and ${{A}_2}{{B}_2}{{C}_2}{{D}_2}$ by ${\Sigma _1}$ and ${\Sigma _2}$, respectively. We also denote the subvolumes between ${\Sigma ^ -}$ and ${\Sigma _1}$, ${\Sigma _1}$ and ${\Sigma _2}$, and ${\Sigma _2}$ and ${\Sigma ^ +}$ by ${\mathcal{V}^ -}$, ${\mathcal{V}^0}$, and ${\mathcal{V}^ +}$, respectively.

 

Fig. 2. Integration volume for proving the reciprocity relation and the unitarity of the $\mathbb{S}$ matrix.

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Next, we integrate Eq. (A1) over the three subvolumes and apply the divergence theorem of vector calculus to convert the volume integrals to surface integrals. All surface integrals except the two over ${\Sigma ^{{\pm}}}$ vanish pair-wise due to either the periodicity or the continuity (across the surfaces ${\Sigma _1}$ and ${\Sigma _2}$) of the integrands; therefore,

(A2)$$\int_{{\Sigma ^ +}} \!{({\textbf{E} \times \textbf{H}^\prime - \textbf{E}^\prime \times \textbf{H}} )\, \cdot \hat{\boldsymbol z}\,}\, {\rm d} s = \int_{{\Sigma ^{ -}}} \!{({\textbf{E} \times \textbf{H}^\prime - \textbf{E}^\prime \times \textbf{H}} ) \cdot \hat{\boldsymbol z}}\, {\rm d} s.$$

The total fields are the sums of the incident and diffracted fields:

(A3)$$\begin{split}(\textbf{E}, \textbf{H}) &= ({\textbf{E}^{\text{i}}} + {\textbf{E}^{\text{d}}},{\textbf{H}^{\text{i}}} + {\textbf{H}^{\text{d}}}),\\ (\textbf{E}^\prime, \textbf{H}^\prime) &= ({\textbf{E}^{\prime{\text{i}}}} + {\textbf{E}^{\prime{\text{d}}}},{\textbf{H}^{\prime{\text{i}}}} + {\textbf{H}^{\prime{\text{d}}}}).\end{split}$$

Since ${\Sigma ^{{\pm}}}$ are outside the corrugated region, we can substitute the diffracted fields in Eq. (A2) with their Rayleigh expansions. Assume that the incident wave vector in Case (a) is given by Eq. (2) and that in Case (b) by Eq. (8). Then, for example,

(A4)$${\textbf{E}^{\text{i}}} = \textbf{E}_{m n}^{\text{i}\, (p)}\exp (\,{i}\,\textbf{k}_{m n}^{\text{i}\, (p)} \cdot \textbf{r}),$$

(A5)$${\textbf{E}^{\text{d}\, (p)}} = \sum\limits_{j,l} {\textbf{E}_{j l}^{\text{d}\, (p)}\exp ({i}\,\textbf{k}_{\textit{jl}}^{\text{d}\, (p)} \cdot \textbf{r})}.$$

The magnetic field vectors have similar expressions, so do the field vectors in Case (b)—one only needs to attach a prime to the field and wave vector symbols. Note that given a pair of reciprocal cases, the incident waves in Cases (a) and (b) are independently non-zero in only one of the two media ${\pm}\;{1}$. Substituting Eqs. (A3)–(A5) into Eq. (A2) gives

(A6)$$({I_{+ 1}} + J_{+ 1}^{\text{i}} + J_{+ 1}^{\text{d}}) = ({I_{- 1}} + J_{- 1}^{\text{i}} + J_{- 1}^{\text{d}}),$$

with

(A7a)$${I_{\pm 1}} = \int_{{\Sigma ^ \pm}} {({{\textbf{E}^{\text{i}}} \times {{\textbf{H}^\prime}^{\text{d}}} + {\textbf{E}^{\text{d}}} \times {{\textbf{H}^\prime}^{\text{i}}} - {{\textbf{E}^\prime}^{\text{i}}} \times {\textbf{H}^{\text{d}}} - {{\textbf{E}^\prime}^{\text{d}}} \times {\textbf{H}^{\text{i}}}} ) \cdot \hat{\boldsymbol z}\,{\rm d} s},$$

(A7b)$$J_{\pm 1}^{\text{i}} = \int_{{\Sigma ^ \pm}} {({{\textbf{E}^{\text{i}}} \times {{\textbf{H}^\prime}^{\text{i}}} - {{\textbf{E}^\prime}^{\text{i}}} \times {\textbf{H}^{\text{i}}}} ) \cdot \hat{\boldsymbol z}\,{\rm d} s},$$

(A7c)$$J_{\pm 1}^{\text{d}} = \int_{{\Sigma ^ \pm}} {({{\textbf{E}^{\text{d}}} \times {{\textbf{H}^\prime}^{\text{d}}} - {{\textbf{E}^\prime}^{\text{d}}} \times {\textbf{H}^{\text{d}}}} ) \cdot \hat{\boldsymbol z}\,{\rm d} s} .$$

The evaluations of the above three integrals are elementary, although tedious. It turns out that $J_{\pm 1}^{\text{i}} = J_{\pm 1}^{\,\text{d}} = 0$. For ${I_{{\pm}1}}$, we need to distinguish two scenarios. In scenario 1, the incident plane waves in both Cases (a) and (b) are in medium $p$. Then ${I_{- p}} = {0}$ due to the absence of incident waves in medium ${-}p$, and only evaluating ${I_p}$ is needed, which leads to the reciprocity relation linking two reciprocal cases in reflection. In scenario 2, the incident plane wave of Case (a) is in medium $p$ and that in Case (b) is in medium ${-}p$. Then, the second and third terms in ${I_p}$ and the first and fourth terms in ${I_{- p}}$ are zero due to the absence of incident waves. The evaluations of the non-zero terms give the reciprocity relation linking two reciprocal cases in transmission. Combining the two results together, we get the general reciprocity relation of crossed gratings valid for both reflection and transmission:

(A8)$$\frac{{\gamma _{m n}^{(p)}}}{{{\mu ^{(p)}}}}(\textbf{E}_{m n}^{\text{i} (p)} \cdot \textbf{E}_{m n}^{\prime\text{d}\, (p)}) = \frac{{\gamma _{j l}^{(q)}}}{{ {\mu ^{(q)}}}} (\textbf{E}_{j l}^{\prime\text{i}\, (q)} \cdot \textbf{E}_{j l}^{\text{d}\, (q)}),\quad p,q = \pm 1.$$

APPENDIX B: PROOF OF UNITARITY

The proof of unitarity follows a path similar to that of deriving the reciprocity relation. Let Cases (a) and (b) here be the same as Case (a) in Appendix A, with incident wave vectors $\textbf{k}_{m n}^{\text{i} (p)}$ and $\textbf{k}_{j l}^{\text{i} (q)}$, respectively. The two incident orders ($\textit{m,n;p}$) and ($\textit{j,l;q}$) are arbitrary but fixed, and they may be the same or different; when they are different, they must belong to the same family (sharing the same ${\alpha _0}$ and ${\beta _0}$). From the complex Poynting’s theorem, in a lossless and source-free region,

(B1)$$\nabla \cdot ({\textbf{E}_{a}} \times \;{{ {\bar {\textbf H}}}_{b}} + \;{{ {\bar {\textbf E}}}_{b}} \times {\textbf{H}_{a}}) = 0,$$

where (${\textbf{E}_a},{\textbf{H}_a}$) and (${\textbf{E}_b},{\textbf{H}_b}$) are two independent monochromatic solutions of Maxwell’s equations subject to the same boundary conditions and the radiation condition. Integrating Eq. (B1) over the volume of the parallelepiped in Fig. 2 and using the same reasoning as that in Appendix A, we get

(B2)$$\begin{split}&\int_{{\Sigma ^ +}} {({{\textbf{E}_{a}} \times {{{ {\bar {\textbf H}}}}_{b}} + {{{ {\bar {\textbf E}}}}_{b}} \times {\textbf{H}_{a}}} ) \cdot \hat{\boldsymbol z}\,} \,{\rm d} s \\&\quad= \int_{{\Sigma ^{ -}}} {({{\textbf{E}_{a}} \times {{{{\bar {\textbf H}}}}_{b}} + {{{{\bar {\textbf E}}}}_{b}} \times {\textbf{H}_{a}}} ) \cdot \hat{\boldsymbol z}}\, {\rm d} s.\end{split}$$

Writing the total fields as a sum of incident and diffracted fields in Cases (a) and (b) and substituting the results into Eq. (B2) leads to an equation that is formally the same as Eq. (A6), but now

(B3a)$${I_{\pm 1}} = \int_{{\Sigma ^ \pm}} {\left({\textbf{E}_{a}^{\text{i}} \times \overline {\textbf{H}_{b}^{\text{d}}} + \textbf{E}_{a}^{\text{d}} \times \overline {\textbf{H}_{b}^{\text{i}}} + \overline {\textbf{E}_{b}^{\text{i}}} \times \textbf{H}_{a}^{\text{d}} + \overline {\textbf{E}_{b}^{\text{d}}} \times \textbf{H}_{a}^{\text{i}}} \right)\, \cdot \hat{\boldsymbol z}\,{\rm d} s},$$

(B3b)$$J_{\pm 1}^{\,\text{i}} = \;\int_{{\Sigma ^ \pm}} {\left({\textbf{E}_{a}^{\text{i}} \times \overline {\textbf{H}_{b}^{\text{i}}} + \overline {\textbf{E}_{b}^{\text{i}}} \times \textbf{H}_{a}^{\text{i}}} \right)\, \cdot \hat{\boldsymbol z}\,{\rm d} s} ,$$

(B3c)$$J_{\pm 1}^{\,\text{d}} = \int_{{\Sigma ^ \pm}} {\left({\textbf{E}_{a}^{\text{d}} \times \overline {\textbf{H}_{b}^{\text{d}}} + \overline {\textbf{E}_{b}^{\text{d}}} \times \textbf{H}_{a}^{\text{d}}} \right)\, \cdot \hat{\boldsymbol z}\,{\rm d} s} .$$

Substituting the Rayleigh expansions of diffracted fields in medium $t$, $t = {\pm}\;{1}$,

(B4)$$\textbf{E}_{a}^{\text{i}\, (p)} = \textbf{E}_{m n}^{\text{i}\, (p)}\exp ({i}\,\textbf{k}_{m n}^{\text{i}\, (p)} \cdot \textbf{r}),$$

(B5)$$\textbf{E}_{a}^{\text{d}\, (t)} = \sum\limits_{r,s} {\textbf{E}_{r s;m n}^{\text{d} (t;p)}\exp (\,{i}\,\textbf{k}_{r\,s}^{\text{d}\, (t)} \cdot \textbf{r})} ,$$

for example, into Eqs. (B3a) and (B3c) and carrying out elementary evaluations of the integrals, we get ${I_{{\pm}1}} = {0}$,

(B6a)$$J_t^{\text{i}} = - \frac{{2A}}{{{k_0} {\mu ^{( t)}}}} |\textbf{E}_{m n}^{\text{i} (t)}{|^2}\,t\,\gamma _{m n}^{(t)}\,{\delta _{\textit{mj}}} {\delta _{\textit{nl}}} {\delta _{\textit{pt}}}{\delta _{\textit{qt}}},$$

(B6b)$$J_t^{\text{d}} = \frac{{2A}}{{{k_0} {\mu ^{( t)}}}} \sum\limits_{(r,s) \in \;{P^{(t)}}} {(\textbf{E}_{rs;\,m n}^{\text{d} (t;p)} \cdot \overline {\textbf{E}_{r\,s;j l}^{\text{d} (t;q)}}) t \gamma _{r s}^{(t)}} ,$$

where $A$ is the area of ${\Sigma ^{{\pm}}}$. Substituting ${I_{{\pm}1}} = {0}$ and Eq. (B6) into Eq. (A6) and using the electric field normalization defined in Eq. (14), we obtain

(B7)$$\sum\limits_{(r,s)\; \in \;{P^{(t)}},t\, = \,\, \pm \,1\,} {\overline { \tilde{\boldsymbol E}_{r s;jl}^{\text{d} (t;q)}} \cdot \tilde{\boldsymbol E}_{r s;m n}^{\text{d} (t;p)}} = \overline { \tilde{\boldsymbol E}_{\textit{jl}}^{\text{i} ( q)}} \cdot \tilde{\boldsymbol E}_{\textit{mn}}^{\text{i} ( p)}{\delta _{\textit{mj}}} {\delta _{\textit{nl}}}\,{\delta _{\textit{pq}}}.$$

After we use the $S$ tensor defined in Subsection 3.A to write

(B8)$$\tilde{\textbf{E}}_{rs;mn}^{\text{d}(t;p)}={\stackrel{\leftrightarrow}{\textbf{S}}}_{rs;mn}^{(t;p)}\,\cdot \tilde{\textbf{E}}_{mn}^{\text{i}(p)}, \quad \tilde{\textbf{E}}_{rs;jl}^{\text{d}(t;q)}= {\stackrel{\leftrightarrow}{\textbf{S}}}_{rs;jl}^{(t;q)}\cdot \tilde{\textbf{E}}_{jl}^{\text{i}(q)},$$

Eq. (B7) becomes

(B9)$$\begin{split}&\overline{\tilde{\boldsymbol{E}}_{m n}^{\text{i}(p)}} \cdot\left(\sum_{(r, s) \in P^{(t)}, t=\pm 1} {\stackrel{\leftrightarrow}{\textbf{S}}}_{{\textit{r s ; m n}}}^{(t ; p)\dagger} \cdot {\stackrel{\leftrightarrow}{\textbf{S}}}_{{\textit{r s: m n}}}^{(t ; p)}-{\stackrel{\leftrightarrow}{\textbf{I}}}\right) \cdot \tilde{\boldsymbol{E}}_{m n}^{\text{i}(p)}=0,\\& \quad{\rm if }\;(j, l)=(m, n) \quad {\rm and }\quad q=p; \\ &\overline{\tilde{\boldsymbol{E}}_{j l}^{\text{i}(q)}} \cdot\left(\sum_{(r, s) \in P^{(t)}, t=\pm 1} {\stackrel{\leftrightarrow}{\textbf{S}}}_{r s; j l}^{(t, q)} \cdot {\stackrel{\leftrightarrow}{\textbf{S}}}_{r s; m n}^{(t ; p)}\right) \cdot \tilde{\boldsymbol{E}}_{m n}^{\text{i}(p)}=0, \quad{\rm else }.\end{split}$$

In Eq. (B9), ${\stackrel{\leftrightarrow}{\textbf{I}}}$ is the two-dimensional identity tensor in polarization space and $\stackrel{\leftrightarrow}{\textbf{A}}^{\dagger}=\overline{\stackrel{\leftrightarrow}{\textbf{A}^{\rm{T}}}}$, where $\stackrel{\leftrightarrow}{\textbf{A}^{\rm{T}}}$ is defined in Subsection 3.B. Since $ \tilde{\textbf E}_{\textit{jl}}^{\text{i} (q)}$ and $ \tilde{\textbf E}_{m n}^{\text{i} (p)}$ are arbitrary complex vectors, the contents inside the two pairs of parentheses must be zero; therefore,

(B10)$$\sum\limits_{(r,s)\ \in \ {{P}^{(t)}},t\,=\,\,\pm \,1\,}{{\stackrel{\leftrightarrow}{\boldsymbol{S}}}{{_{rs;jl}^{(t;q)}}^{\dagger }}\,\cdot {\overset{\leftrightarrow}{\boldsymbol{S}}}_{rs;mn}^{(t;p)}}= {\overset{\leftrightarrow}{\boldsymbol{I}}} {{\delta }_{mj}}{{\delta }_{nl}}{{\delta }_{pq}}.$$

This is the expression of the unitarity of the $S$ matrix of crossed gratings in terms of elemental $S$ tensors.

Disclosures

The author declares no conflicts of interest.

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