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A Green function method to study thin diffraction gratings

Daniel A. Travo,1, ∗ Rodrigo A. Muniz,1 M. Liscidini,2 and J. E. Sipe

Department of Physics, University of Toronto, Toronto, Ontario M5S 1A7, Canada

Department of Physics, University of Pavia, Pavia I-27100, Italy

The anomalous features in diffraction patterns first observed by Wood over a century ago have been the subject of many investigations, both experimental and theoretical. The sharp, narrow structures - and the large resonances with which they are sometimes associated - arise in numerous studies in optics and photonics. In this paper we present an analytical method to study diffracted fields of optically thin gratings that highlights the nonanalyticities associated with the anomalies. Using this approach we can immediately derive diffracted fields

for any polarization in a compact notation. While our equations are approximate, they fully respect energy conservation in the electromagnetic field, and describe the large exchanges of energy between incident and diffracted fields that can arise even for thin gratings.

I. INTRODUCTION tivity in the region of these anomalies, the simplest pertur-

bation theories are not sufficient to describe them; it is es- Over a century ago, Wood observed anomalies in the an- sential to consider the full interaction between diffracted and gular dependence of light reflected from a metal sheet , and specularly reflected beams. Over the years a wide range of since then there have been many studies of these anomalies approaches have arisen to treat such systems. These include and their applications in optics and photonics, particularly as guided-mode techniques such as coupled mode theory16–1 ,

they arise in the reflection from surfaces on which a grating transfer matrix approaches , and a variety of robust numeri- is intentionally deposited. Maystre has recently presented a cal techniques based on finite element and RCWA (scattering detailed review of the history of research in this field. Gen- matrix) methods20–2 . erally, two mechanisms have been identified as sources of the In this paper we present a semi-analytic method for the

anomalies: The first is a transition between propagation and treatment of thin gratings, with advantages that are not all evanescence in one of the diffracted orders , and the second present in earlier work. We consider the grating structure is the excitation of a leaky mode within the grating region . shown in Fig. 1(a) in this first communication. Based on Some ambiguity exists in the literature concerning the distinc- a Green function formalism that treats the scattered light

tion between these mechanisms and the terms used to refer to in terms of its s− and p−polarized components, the method them. In this paper, we follow the convention that anomalies leads to an immediate identification of the features in the scat- related to a transition in a diffracted order are referred to as tering equations that describe the anomalies, and allows for Rayleigh anomalies, and those associated with the resonant the easy inclusion of effects of surface waves of the substrate

excitation of a leaky mode in the grating region are referred to as well as leaky modes in the grating region. Light with any as Wood anomalies. Maradudin has recently shown that the plane of incidence, any polarization, and at any incident an- resonant excitation of any surface wave in a substrate below gle is treated, and anisotropy in the response of the material the grating, by scattering from the grating, can lead to anoma- in the grating region is included; the substrate can consist of

lies in the reflectance as well; we also refer to these as Wood an arbitrary set of layers with uniaxial optical properties. The anomalies. description of the reflected and diffracted light is necessar- Rayleigh anomalies are square root-like, sharp, narrow ily approximate, since we simplify our equations based on peaks that arise in the irradiance of both the specularly re- the grating being thin, but it is nonetheless completely robust

flected and diffracted beams. Wood anomalies are associated with respect to energy conservation: In the absence of any with extraordinary increases in the specular reflectance6–8 , absorption in the material media, at whatever number the in- and have seen wide use in applications where gratings serve clusion of diffracted and evanescent fields in the calculation is as filters9–1 , modulators12,1 , and sensors14,1 . Because even truncated, the approximate equations respect conservation of

very thin gratings can lead to very large effects on the reflec- energy in the electromagnetic field, despite large exchanges of energy between diffracted and reflected fields. For simple incidence configurations, and if only a few diffracted orders ∗ [email protected] are important, the set of equations to be solved is small and

the physics easily identified. This is an important advantage, since gratings are now being used to access resonances for enhanced sensing applications25,2 and in novel 2D materials such as graphene27–3 . Our simple but robust treatment of the optics of the grating should allow for such work to focus on the physics of the medium being probed.

The outline of the paper is as follows. In section II we first

treat the simpler, symmetric grating structure shown in Fig. 1(b). In the limit of a thin grating, we show how the scatter- (a) ing equations lead naturally to the assignment of a uniform dielectric tensor for a layer associated with the grating region; see Fig. 1(c). The scattering by the grating can be best under- stood as occurring with this as part of the background optical response, and it is the waveguide modes of this nominal layer

that become the resonances associated with the Wood anoma- lies, discussed alongside Rayleigh anomalies in section III and identified in the resulting scattering equations in section IV. (b) In section IV A we build a scattering matrix for the problem.

This can of course be done in many ways, but we adopt an ap-

proach that leads to a proof that the equations respect energy conservation, and allows for an easy generalization to include an arbitrary layered substrate (Fig. 1(a)). These equations are separated by polarization and simplified in section IV B for a simple configuration chosen as an example. In section IV C a two wave-vector model is used to derive analytic expressions (c) for the scattered fields alongside a discussion of their poles

that signal the Wood anomalies. We discuss how the Wood anomalies associated with the waveguide modes of the grating region (Fig. 1(c)) are modified – or disappear – in the presence of the substrate in section IV D and present, as a sample calcu- lation, results for a simple silicon grating atop a glass substrate and confirm the validity of our approximate treatment by com- parison with convergent, numerically exact calculations. Our conclusions are presented in section V. Some of the details of

the derivations, a discussion of waveguide dispersion, and our (d) proof of energy conservation are relegated to appendices. FIG. 1. (Color online) The general structure studied and discussed in this paper (not to scale). (a) A thin grating placed on top of a mul- II. THIN GRATINGS tilayer structure with a substrate with relative dielectric constant εQ .

The grating parameters are the same as in (b), and the relative dielec-

tric constant of the cladding is ε . (b) An isolated grating with rela-

We begin by considering a grating in the region −D/2 < z <

tive dielectric tensor εg suspended in a medium with relative dielec- D/2, with the rest of space taken to be filled with an isotropic tric constant ε . (c) An effective dielectric slab with relative dielectric dielectric; see Fig. 1(b). In the presence of a field Ein (r, t) tensor εlayer that characterizes the average properties of the grating incident on the grating region we write the total field as relation. (d) The corresponding effective planar structure consisting

of the effective dielectric slab and the multilayer structure.

E (r, t) = Ein (r, t) + E sc (r, t) , (1)

where E sc (r, t) is the scattered field. We take all such time dependent fields F (r, t) to be stationary, F (r, t) = F (r) e−iωt + c.c.,

and we assume the refractive index of the surrounding dielec- The scattered field contribution to (3) is determined by √ tric, n = ε , is real at frequency ω. Denoting by R the Z D/2 projection in the xy plane of a position vector r = R + zẑ, E sc (κ; z) = dz G κ; z − z · P κ; z ,

(8)

−D/2

we take ê to be the unit vector in the xy plane that identifies the direction in which the susceptibility varies, and write the where the Green function is (possibly complex) spatially dependent (tensor) susceptibility in the grating region as χ (ζ), where ζ = ê · R. G κ; z − z = g κ; z − z − δ z − z ẑ ẑ,


(9) 0 ε

It will be convenient to Fourier transform our field ampli-

tudes F (r) only in the xy plane, with iω̃2 Z dκ iκ·R 0 F (r) = F (R; z) = e F (κ; z) , g κ; z − z = θ z − z eiw (z−z ) (ŝŝ + p̂1+ p̂1+ )

(2) (2π)2 20 w where κ has x and y components, so for example iω̃2 0

+ θ z − z e−iw (z−z ) (ŝŝ + p̂1− p̂1− ) ,
20 w

E(R; z) = Ein (R; z) + E sc (R; z), (3)

and where E(κ; z) = Ein (κ; z) + E sc (κ; z) . ŝ = κ̂ × ẑ, (10)

In the usual example of an incident plane wave, the incident

κẑ ∓ w κ̂ field will be characterized by a single κin , and there will be p̂1± = , scattered fields characterized by κin + mK, where m ranges ω̃n over all positive and negative integers, and identify the s− and p−polarized field components of the radi- p 2π ated fields. Here w ≡ ω̃2 ε − κ , where κ = |κ|; to ensure

K= ê, (4) proper radiation conditions, the square root is made unique by

a √ √ √ where a is the fundamental period of the grating. For taking Im Z ≥ 0 , and taking Re Z ≥ 0 if Im Z = 0. |κin + mK| > ω̃n , where ω̃ ≡ ω/c, the scattered fields are Of the two terms on the right hand side of (9), the second evanescent, confined to the neighborhood of the grating re- will typically lead to the larger contribution for our thin grat-

gion; for |κin + mK| < ω̃n the scattering leads to diffracted ings of interest, and it can be dealt with explicitly. If we define fields that can carry energy away from the grating region. a modified field,

We write Z D/2

Emod (κ; z) =Ein (κ; z) + g(κ; z − z ) · P(κ; z )dz , (11) χ (ζ) = χ + χadd (ζ) , (5) −D/2

where in dyadic form we have χ = (ε − 1) (x̂x̂ + ŷ ŷ + ẑ ẑ) , E (R; z) =Emod (R; z) − ẑ ẑ · P(R; z), 0 ε is just the susceptibility that would be present were the and we can write the expression (7) for the polarization as

background dielectric extended into the grating region, and

χadd (ζ) is an additional contribution that is responsible for P (R; z) =0 χmod (ζ) · Emod (R; z) , (12) the ζ dependence of the susceptibility in the grating region. We assume that one of the principal axes of χadd (ζ) is the z where axis, and we choose the x and y axes to coincide with the other χmod (ζ) =x̂x̂χmod xx (ζ) + ŷ ŷχyy (ζ) + ẑ ẑχzz (ζ) , (13) principal axes, mod mod

χadd (ζ) = x̂x̂χadd xx (ζ) + ŷ ŷχyy add (ζ) + ẑ ẑχzz add (ζ) . (6) with

We let P(r) denote the polarization in the grating region, χmod

xx (ζ) ≡ χadd xx (ζ) , above and beyond that which would result if the grating re- χyy (ζ) ≡ χyy (ζ) , mod add gion consisted solely of the background isotropic dielectric χzz (ζ) medium. Then χzz (ζ) ≡ add . mod 1 + χadd (ζ) /ε zz

P (R; z) = 0 χadd (ζ) · E (R; z) , (7)

From (11) we see that as D → 0 we typically have

for − D/2 < z < D/2. Emod (R; z) → Ein (R; z), and so in that limit χmod (ζ) can be

understood as an effective local susceptibility relating the (ex- Our equations (15,16) can then be written as cess) polarization to the incident f ield, rather than to the field

P (κ) = 0 hχmod i · Emod (κ) + Pv (κ) , (18)

in the grating region itself. So far we have made no approximations, and an exact de- Emod (κ) = Ein (κ) + g (κ) · P (κ) , scription of the scattering could proceed by numerically solv- where ing (11,12) for any specified Ein (κ; z). Instead, we develop an approximate description of the scattering based on the condi- Pv (R) =0 χv (ζ) · Emod (R) (19) tion that the thickness D of the grating region is much less is the only contribution from the variation of the effective sus-

than the wavelength of light, ω̃n D  1, and as well that the ceptibility with ζ. If we define χlayer according to variation in z of the scattered fields over the grating region D E D E is negligible for κ of interest, |w | D  1. This leads to the χlayer xx ≡ χmod xx = χadd xx ,

D E D E

ansatz that a number of fields can be taken as independent of χyy yy yy layer ≡ χmod = χadd , (20) z within the grating region, χlayer zz

D E *

χadd zz + ≡ χzz = ,

F (κ; z) → F (κ) (14) 1 + χlayer /ε

zz mod 1 + χzz add /ε

for − D/2 < z < D/2, where the last equation is to be solved for χzz layer , we can iden- tify χlayer as the effective (excess) susceptibility of the thin and for such fields we write layer that would lead to the optical response of the grating Z dκ iκ·R region were the variation χv (ζ) in the effective susceptibility F(R) = e F (κ) . (2π)2 ignored; this is the scenario sketched in Fig. 1(c). For if we

Naturally for the incident field we simply take Ein (κ) = would return to (7,8), take χadd (ζ) → χlayer and repeat the Ein (κ; 0), while to determine P (κ) self-consistently we ap- derivation and approximations leading to (18), we would re- proximate the field Emod (κ; z) as uniform over the grating re- cover precisely those equations with Pv (κ) absent, and with gion by taking Emod (κ; z) → Emod (κ), where the components of hχmod i replaced by the components of

Z D/2 Z D/2 χlayer according to (20). Note that if we write the full relative

1 dielectric tensor in the grating region as ε (ζ) ≡ ε + χadd (ζ),

Emod (κ) =Ein (κ) + dz g κ; z − z · P (κ) .
dz

D −D/2 −D/2 and the full relative dielectric tensor associated with χlayer as

In the limit |w | D  1 this leads to εlayer ≡ ε + χlayer , where ε = ε (x̂x̂ + ŷ ŷ + ẑ ẑ), we have

Emod (κ) =Ein (κ) + g (κ) · P (κ) , (15) εlayer

xx = hε xx i , εyy layer = hε i , yy (21) where * + 1 1 iω̃2 D " # = zz . g (κ) = ŝŝ + (p̂1+ p̂1+ + p̂1− p̂1− ) , εzz layer ε 20 w 2

We return to the equations (18), and can now understand

iω̃2 D iw D iκ D = ŝŝ + κ̂κ̂ + ẑ ẑ, them as describing the scattering due to a variation in the ef- 20 w 20 ε 20 ε w fective excess susceptibility, χv (ζ), in the presence of a uni- and in this limit the equation (12) reduces to form background dielectric tensor εlayer in the grating region. P (R) =0 χmod (ζ) · Emod (R) . (16) Below we will construct an expression for Pv (κ), and then

these equations can be solved consistently for P (κ). Once that Within these approximations the fields in the grating region is done we can construct the scattered fields above the grating are determined by the solution of (15,16). region (z > D/2) and below the grating region (z < −D/2). At this point it is useful to separate out the spatial average We denote these by E+sc (κ; z) and E−sc (κ; z) respectively, and of our various quantities. In particular for χmod (ζ) we have they follow immediately from the general expression (8) for

1 a/2

Z hχmod i = χmod (ζ) dζ, a −a/2 E±sc (κ; z) = e±iw z E±sc (κ) , (22) and we put where

χv (ζ) ≡χmod (ζ) − hχmod i . (17) E±sc (κ) =G± (κ) · P (κ) , (23)

and the non-analyticity associated with the passing of a diffracted order into evanescence appears as well in the expressions for iω̃2 D G± (κ) = (ŝŝ + p̂1± p̂1± ) , (24) the amplitudes of the other diffracted orders, and in those of 20 w the specularly reflected and transmitted fields. These are the and we have again assumed |w | D  1. Now the incident Rayleigh anomalies. field satisfies the Maxwell equations with a uniform relative Another non-analyticity implicit in these equations can be

dielectric constant ε , and so everywhere in space it is of the revealed by inserting the second of (18) into the first and for- form mally solving for P (κ), Ein (κ; z) =E+in (κ; z) + E−in (κ; z), (25) P (κ) = (I − 0 hχmod i · g (κ))−1 (28) where · [0 hχmod i · Ein (κ) + Pv (κ)] ,

E±in (κ; z) =e±iw z E±in (κ) , (26) where I = x̂x̂+ŷ ŷ+ẑ ẑ is the unit dyadic. The expression (28) is valid as long as (I − 0 hχmod i · g (κ))−1 has no divergent (c f. (22)). Then given any κ , for z > D/2 we label the full up- components, and this holds as long as the determinant of a ward propagating (or evanescent) fields as E+out (κ) exp(iw z), matrix representing (I − 0 hχmod i · g (κ)) does not vanish. In while for z < −D/2 we label the full downward propagating the special case where εlayer = εyy

layer ≡ εlayer (recall (20,21)), xx k

(or evanescent) fields as E−out exp(−iw z); we clearly have that matrix can be easily written out in the (ŝ, κ̂, ẑ) basis, since x̂x̂ + ŷ ŷ = ŝŝ + κ̂κ̂; we find

E±out (κ) =E±in (κ) + E±sc (κ) . (27)

det (I − 0 hχmod i · g (κ)) (29)

Before solving for these fields, we identify how the Rayleigh

iω̃ D k " #" # and Wood anomalies are captured in their calculation.   iw D k   = 1− ε − ε 1 − εlayer − ε 2w layer 2ε iκ D εlayer − ε   ⊥  , 

III. RAYLEIGH AND WOOD ANOMALIES × 1 −

Returning to the expression (24) for G± (κ), and writing where we have put ε⊥layer ≡ εzz layer . For reasonable εlayer the k

p̂1± in terms of κ̂ and ẑ, we see that in this basis of real unit middle bracketed term in (29) cannot vanish, since by assump- vectors there are terms in G± (κ) proportional to w , and terms tion |w | D  1; thus the determinant in (29) can vanish only proportional to 1/w These are both non-analytic in κ, since if one of the following conditions is met: w is purely real for κ < ω̃n , purely imaginary for κ > ω̃n , and vanishes at κ = ω̃n ; 1/w thus diverges at κ = ω̃n . The iω̃2 D  k 

1− εlayer − ε = 0, (30) transition from real to imaginary w can arise as the angle of 2w iκ D εlayer − ε ⊥ incidence is varied, and κ is associated with a diffracted order 1− = 0. that becomes evanescent in the background dielectric as κ first 2w ε⊥layer approaches, and then exceeds, ω̃n . Of course, although the

In Appendix B we show that the first of (30) is the dispersion

G± (κ) diverge as w → 0, the E±sc (κ) do not; the same non- relation for the fundamental s-polarized mode, and the sec- analyticity as w → 0 appears in g (κ), since ond for the fundamental p-polarized mode, of a thin enough 1 + planar uniaxial waveguide with relative dielectric tensor sus- g (κ) = G (κ) + G− (κ) ,

2 ceptibility (x̂x̂ + ŷ ŷ)εklayer + ẑ ẑε⊥layer , bounded above and be-

and once the expression for Pv (κ) is included the self- low by a uniform isotropic dielectric with dielectric constant consistent solution of the set of equations (18) leads to finite ε ; recall that in the limit of a thin enough planar waveg- fields everywhere at all κ, as we show in detail below. This uide at most one waveguide mode of each polarization exists. is enforced by the coupling among the different diffracted and Around the values of κ where they vanish, the left-hand-sides

evanescent orders, and by the coupling between each of them of (30) can be written as proportional to (κ − κS ) and (κ − κP ) to the specularly reflected and transmitted fields; the source of respectively, where at frequency ω the s− and p−polarized these couplings is of course the grating that is itself respon- waveguide modes have wave numbers κS and κP respectively; sible for the existence of the diffracted and evanescent orders if there is no absorption, κS and κP are real. Thus the non-

themselves. Another consequence of these couplings is that analyticities of (I − 0 hχmod i · g (κ))−1 are poles, on the real κ

axis if there is no absorption, associated with the waveguide where m ranges over the integers and K is given by (4); here modes of the “effective waveguide” established by the average 1 a/2 −imKζ Z optical response in the grating region. χv[m] = e χv (ζ) dζ, a −a/2

Despite these divergences, the solution of (28) for P (κ) is

with K = |K|. Note that by virtue of the definition (17) of again always finite. The waveguide modes exist for κ > ω̃n , χv (ζ) we have χv = 0. Since Pν (R) is the response (19) to “beyond the light line,” and no physical field incident from in- Edr (R) due to χv (ζ), we seek a solution for our fields of the finity can be described by nonzero Ein (κ) for κ in the range of form the divergences. Of course, by coupling through the grating, X Pv (κ) can acquire κ components for κ at wave numbers near F (κ) = (2π)2 δ (κ − κin − mK) F (κin + mK) ,

or at the waveguide modes if the angle of incidence of the inci- m

dent field is properly chosen, as we see in detail below. How- where ever, a grating that allows Pv (κ) to acquire those κ compo- X

F (R) = F (κm ) ei(κin +mK)·R , (31)

nents from the incident field will also couple part of any field m that Pv (κ) generates back to the wave vector of the incident and here and henceforth we put field, thus modifying the effective incident field driving Pv (κ) and ameliorating the response; the effective waveguide pole is κm ≡κin + mK. moved off the real κ axis, as we illustrate in an example later. Here κin characterizes the incident field, but we actually allow

Another consequence is that the resonant structure associated the incident field Ein (R) to be of the general form (31), with with one of the evanescent orders being close to an effective Ein (κm ) nonzero for m , 0; in later sections we will consider waveguide mode will lead, through coupling by the grating, to a grating above a substrate, and terms with m , 0 will arise resonant structures in other diffracted and evanescent orders, from reflection of scattered light off the substrate. Using the

and in the specularly reflected and transmitted fields. These expansion (31) in (19) we have are the Wood anomalies. X Thus within the approximation of a thin grating region Pν (κm ) =0 χv[m−m ] · Emod (κm ), (32) m even a schematic discussion as presented above can identify for example; equations for the Fourier components of other

Rayleigh and Wood anomalies with non-analyticities in the

quantities will be given below. The set of these equations can response of the grating structure to an incident field: Rayleigh be organized as matrix equations in many ways; below we anomalies are associated with square root divergences as a present one approach that is both useful for calculations, and diffracted order becomes evanescent, and Wood anomalies allows for an easy proof of energy conservation even when the are associated with pole divergences as an evanescent order

number of Fourier components is truncated. approaches an effective waveguide mode of the grating re- gion. Full calculations within this approximation presented below will confirm this connection, and show that our equa- A. S-matrix equations tions, while approximate, exhibit exact energy conservation. As well, since for thin grating regions the dispersion relations To complete a calculation we approximate sums over m by of the effective waveguide modes lie close to the light line, a restriction to |m| ≤ N, where the threshold integer N includes

we can expect a complicated response because the resonances at least all diffracted, propagating orders. For each field F(R) associated with the anomalies, considered independently, lie we then introduce F¯ , a column of columns close to each other. This is considered in some examples pre-  ¯   F (κN )  sented in section IV.  ¯ 

F = 

¯ ..  , (33)   .  

IV. COUPLED WAVE VECTOR EQUATIONS F¯ (κ(−N) )

We now turn to the solution for the fields in the presence of where each F¯ (κm ) is a column with the three Cartesian com- a grating χ (ζ) of the form (5), where since χv (ζ) is taken as ponents of F (κm ),   periodic with period a, we can expand it in a Fourier series  x̂ · F (κm )  F¯ (κm ) =  ŷ · F (κm )  ,  

X (34)

χv (ζ) = χv[m] eimK·R ,   m ẑ · F (κm )

resenting the grating is not block diagonal, but is given by

χ̄v;N(N−1) · · · χ̄ν;N(−N)     0̄  χ̄v;(N−1)(N) · · · χ̄v;(N−1)(−N)    0̄ χ̄v =   .. .. .. ..  , (36) . . . .    

χ̄v;(−N)(N) χ̄v;(−N)(N−1) · · · 0̄ 

where   x̂ · χv[m−m ] · x̂ 0 0 

FIG. 2. Plane containing the set of axes (x̂, ŷ) and basis vectors

χ̄v;mm =   ,   (ŝm , κ̂m ). 0 ŷ · χv[m−m ] · ŷ 0   0 0 ẑ · χv[m−m ] · ẑ

and so the full column F¯ has 3(2N + 1) elements. For the and the diagonal elements of χ̄v vanish because χv = 0. In tensors we introduce (2N+1)×(2N+1) matrices with elements this notation the equations (32) for the Pν (κm ) can be written that are themselves 3 × 3 matrices; thus in each of these there in full matrix form as are 3(2N + 1) × 3(2N + 1) elements in all. We put

P̄v =0 χ̄v Ēmod ,    ḡNN 0̄ ··· 0̄  and combining this with the matrix form of (18) we find    0̄ ḡ(N−1)(N−1) · · · 0̄  ḡ =  . . . .  ,  .. .. .. ..  P̄ = 0 (χ̄o + χ̄v )Ēmod ,  

0̄ 0̄ · · · ḡ(−N)(−N)  Ēmod = Ēin + ḡP̄,

a block diagonal matrix where 0̄ indicates a 3 × 3 matrix of with a formal solution zeros, and the 3 × 3 matrices ḡmm are given by h i−1 P̄ =0 (χ̄o + χ̄v ) 1̄3 − 0 ḡ(χ̄o + χ̄v ) Ēin ,    (x̂ · g (κm ) · x̂) (x̂ · g (κm ) · ŷ) (x̂ · g (κm ) · ẑ)  where 1̄ j denotes the j(2N + 1) × j(2N + 1) unit matrix, with j

ḡmm =  (ŷ · g (κm ) · x̂) (ŷ · g (κm ) · ŷ) (ŷ · g (κm ) · ẑ)  .     an integer. Introducing columns Ē±sc to describe the scattered (ẑ · g (κm ) · x̂) (ẑ · g (κm ) · ŷ) (ẑ · g (κm ) · ẑ) fields, from (23) we then have h i−1 For example, let the associated polarization vectors associ- Ē±sc =0Ḡ± (χ̄o + χ̄v ) 1̄3 − 0 ḡ(χ̄o + χ̄v ) Ēin .

ated with κm be ŝm and p̂m± , such that

Separating out the upward propagating (or evanescent) contri-

ŝm = κ̂m × ẑ, (35) butions of the incident field from the corresponding downward κm ẑ ∓ w (κm ) κ̂m propagations (see (26)), we have Ēin = Ē+in + Ē−in , and introduc- p̂1±,m = , ing columns for the full outward propagating (or evanescent) ω̃n fields for z > D/2 and z < −D/2 (see (27)), with new columns with κ̂m = κm / |κm | and w (κm ) = ω̃2 ε − κm (compare

p defined as indicated we can write (10)). If we then let φm indicate the rotation in the xy plane i−1  Ē+out = 0Ḡ+ (χ̄o + χ̄v ) 1̄3 − 0 ḡ(χ̄o + χ̄v ) Ē+in + Ē−in + Ē+in , h  between the sets of unit orthogonal vectors (x̂, ŷ) and (ŝm , κ̂m ) (see Fig. 2), we have (37)

i−1  Ē+in + Ē−in + Ē−in . h 

Ē−out = 0Ḡ− (χ̄o + χ̄v ) 1̄3 − 0 ḡ(χ̄o + χ̄v )

 1 − ω̃κ2mε sin φm 2ω̃κm ε sin 2φm   iω̃ D   κm ḡmm = sin 2φm 1 − ω̃κ2mε cos φm  . 

0 The sub-columns of Ē±in , Ē±in (κm ), contain the three Carte-

20 w (κm )  2ω̃ ε  2 κm  0 0 ω̃2 ε sian components of E±in (κm ) (recall (34)). However, these are not independent, since for any κm there are only s− and Block diagonal matrices Ḡ± and χ̄o are defined similarly, p−polarized components,

D yy Eχ̄o;mm

whereD the Eblocks E χ̄o are all identical, χ̄o;mm = of E±in (κm ) =ŝm E±in;s (κm ) + p̂1±,m E±in;p (κm ) , D diag( χmod , χmod , χmod ). The matrix of matrices χ̄v rep- xx zz

with two independent amplitudes E±in;s (κm ) and E±in;p (κm ). As where such we can write  Ē±out (κN )       ±   x̂ · E±in (κm )   Ēout (κ(N−1) )  Ē±out =  ..  , Ē±in (κm ) =  ŷ · E±in (κm )  ,

   .      ẑ · E±in (κm ) 

Ē±out (κ(−N) )

= σ̄±in (κm ) Ē±in (κm ) , is a 2(2N + 1) column for each example (±) once the elements where of  ±  
  Eout;s (κm )   (x̂ · ŝm ) x̂ · p̂1±,m  Ēout m =  ± ± (κ )   σ̄±in (κm ) =  (ŷ · ŝm ) ŷ · p̂1±,m  


(38) Eout;p (κm ) 
 (ẑ · ŝm ) ẑ · p̂1±,m are written out, and

 σ̄out (κN )  ±  is a 3 × 2 matrix for each of the (+) and (−) examples, and 0̄ ··· 0̄  σ̄out (κ(N−1) ) · · · ±      0̄ 0̄   E± (κm )  σ̄out = 

± . . . .  , (44)

Ēin (κm ) =  ±

±  in;s  (39)  .. .. .. .. 

Ein;p (κm )   

0̄ 0̄ · · · σ̄±out (κ(−N) )  is a column of two elements. Constructing the full column for all κm components of Ē±in we have (recall (33)) where the 0̄ denote 2 × 3 matrices with all their elements van- ishing, and

Ē±in =σ̄±in Ē±in , (40)  

 (x̂ · ŝm ) (ŷ · ŝm ) (ẑ · ŝm )  σ̄out (κm ) =  ±


 . (45) where x̂ · p̂1±,m ŷ · p̂1±,m ẑ · p̂1±,m

 Ē±in (κN )    Using (40,43) in (37), we can write  ±   Ēin (κ(N−1) )  + −

Ēin = 

±  , Ē+out = T̄g Ein + R̄g Ein , .. (41) . Ē−out = R̄g Ē+in + T̄g Ē−in ,    

Ē±in (κ(−N) )

 where R̄g and T̄g are 2(2N + 1) × 2(2N + 1) matrices, which for each of the (+) and (−) examples is a column with  i−1  2(2N + 1) elements, once all the Ē±in (κm ) are written out. Fur- h T̄g = σ̄±out 1̄3 + 0Ḡ± (χ̄o + χ̄v ) 1̄3 − 0 ḡ (χ̄o + χ̄v ) σ̄±in , ther, h i−1

R̄g = σ̄±out 0Ḡ± (χ̄o + χ̄v ) 1̄3 − 0 ḡ (χ̄o + χ̄v ) σ̄∓in .  σ̄in (κN )  ± 0̄ ··· 0̄   (46)  0̄ σ̄in (κ(N−1) ) · · · ± 0̄  Since we consider the same dielectric above and below the σ̄in =  ±  ,   . . . . . . .

. (42) grating, the transmission and reflection properties are the  . . . .    same whether light is incident from above or below; thus the 0̄ 0̄ · · · σ̄±in (κ(−N) ) expressions (46) are the same whether the + or − matrices on which for each of the examples is a 3(2N + 1) × 2(2N + 1) the right-hand-side of the equations are used in their evalua-

matrix, once all the elements of the σ̄±in (κm ) are written out; tion. here 0̄ are 3 × 2 matrices with all elements vanishing. Finally, combining the two columns Ē+out and Ē−out , each with Similarly, for each Ē±out (κm ) in Ē±out there will be only s− 2(2N + 1) elements, to form one column with 4(2N + 1) ele- and p−polarized components, ments, and likewise for Ē+in and Ē−in , we can form a 4(2N + 1) ×

4(2N + 1) scattering matrix S , E±out (κm ) =ŝm E±out;s (κm ) + p̂1±,m E±out;p (κm ) ,    T̄g R̄g 

S ≡   , (47)

which we can immediately see will be identified by the R̄g T̄g G± (κm ) (see (24)) that appear in Ḡ± . Nonetheless, we can so that formally extract those amplitudes E±out;s,p (κm ) by writing  +   +   Ēout   Ē   −  =S  in  . (48)

Ē±out =σ̄±out Ē±out , (43) Ēout Ē−in

and similarly for the form of the expressions (45) for σ̄±out (κm ), which are the transpose of the σ̄±in (κm ). When these are assembled into σ̄±in and σ̄±out in (42,44) and the results used in (46) for R̄g and T̄g , we find that because of the high symme- try of the problem each of these 2(2N +1)×2(2N +1) matrices can be reorganized into two (2N + 1) × (2N + 1) matrices, one relevant for s-polarized light and one for p-polarized light.

For each polarization the relevant matrices can then be com-

bined into a 2(2N + 1) × 2(2N + 1) scattering matrix, and in place of (48) we have two sets of equations,  +   +   Ēout,α   Ē   −  =Sα  in,α  , (49)

Ēout,α Ē−in,α

where α = s, p, each Ē+in,α is a (2N + 1) element column,  −   Ein,α (κN )   −   Ein,α (κN−1 ) 

Ēin,α = 

− ..  , (50)   .  

E−in,α (κ−N )

(compare (39,41)), and likewise for Ē+in,α and Ē±out,α , and where   FIG. 3. (Color online) A simple 1D grating configuration with the  Tg,α Rg,α  grating oriented such that ê = ŷ. Incident, reflected, transmitted, Sα ≡   ,

Rg,α Tg,α

and diffracted rays are shown by black (thick) lines, and labeled by the notation used to indicate their field amplitudes; the projection of with the wave vectors on the xy plane are shown by red (thin) lines. #−1 iω̃2 D −1 k "

Tg,s = β̄ 1̄1 − w̄1 χ̄ β̄, (51)

With the equations in this form, a proof of energy conservation Rg,s = Tg,s − 1̄1 , is possible, and is presented in Appendix C. That proof, and and the equations from which it was derived, hold for any orienta- " #−1 " #−1 tion of the grating direction ê in the xy plane and any plane of iD ⊥ iD k Tg,p = 1̄1 − κ̄χ̄ κ̄ + β̄ 1̄1 − χ̄ w̄1 β̄ − 1̄1 , (52)

incidence. 2ε 2ε " #−1 " #−1 iD ⊥ iD k Rg,p = 1̄1 − κ̄χ̄ κ̄ − β̄ 1̄1 − χ̄ w̄1 β̄. 2ε 2ε

B. A simple configuration

Here each of Tg,s , Rg,s , Tg,p , and Rg,p is a (2N + 1) × (2N + 1) matrix. The matrices β̄, κ̄, and w̄1 are di-

In this subsection we simplify the equations above for a

agonal matrices of the same dimension, β̄ = diag(κ̂N · common scenario of interest: We take the grating susceptibil- ŷ, κ̂N−1 · ŷ, ..., κ̂−N · ŷ), κ̄ = diag(|κN | , |κN−1 | , ..., |κ−N |), ity to be uniaxial, χaddxx (ζ) = χyyadd (ζ) (recall (5,6)), choose and w̄1 = diag(w (κN ), w (κN−1 )..., w (κ−N )), with w (κ) = ê = ŷ, and assume the plane of incidence contains ẑ and q ê = ŷ, as illustrated in Fig. 3; for the isolated grating treated ω̃2 n − κ . Finally, χ̄k and χ̄⊥ are (2N +1)×(2N +1) matrices

D E

above and in this section, we have ε = ε . The wave vec- with (mm ) elements χ̄kmm = δmm χmod xx + x̂ · χv[m−m ] · x̂ =

D yy E D E

tors κm that are relevant here are then either in the ŷ or −ŷ δmm χmod + ŷ · χv[m−m ] · ŷ and χ̄⊥mm = δmm χzz mod +
direction, so κ̂m · ŷ = sign(κ̂m · ŷ) = ŝm · x̂ ; the form of the
ẑ · χv[m−m ] · ẑ . We note that the relation between T s and R s expressions (38) for the σ̄±in (κm ) simplifies to is simple because the reference vectors ŝm for the fields are all   the same or differ simply by a minus sign; while that between

 (κ̂m · ŷ) 0 T p and R p is more complicated because, even for a particular  1 (κm ) σ̄±in (κm ) =  ∓ wω̃n  ,   0 (κ̂ · ŷ)  m κm , the z components of p̂1+,m and p̂1−,m are identical, but the  κm  0 ω̃n y components differ by a sign (see (35)).

C. An example and upward and downward diffracted fields

E±out,p (κ−1 )

The expressions (51,52) for the reflection and transmission (56)

E+in,p (κ )

matrices, and indeed the more general expressions (46), can iκ κ−1 D ⊥ ! ! be used to calculate specular reflection and transmission, and −1 iw (κ )D k = Uz−1 χ̄ ± Uκ χ̄(−1)0 , diffraction, for the choice of any number 2N + 1 of wave vec- 2w (κ−1 )ε (−1)0 2ε tors κm in the calculation. However, in certain circumstances where further approximations are possible. For example, if the grat- ! !

iw (κ )D k iw (κ−1 )D k ing period a (see Fig. 1b) is small enough, then for at least Uκ = 1 − χ̄0 1 − χ̄0 2ε 2ε some angles of incidence there will be only one propagating w (κ )w (κ−1 )D k diffracted order (m = −1) in addition to the specularly re- + χ̄0(−1) χ̄k(−1)0 , 4ε flected and transmitted fields (see Fig. 3, again with ε = ε ). iκ D 2

   A choice of 2N + 1 = 3 could be adopted, but since the field iκ−1 D

Uz = 1 − χ̄  1 −

⊥ χ̄  ⊥     2w (κ )ε 0 2w (κ−1 )ε 0    associated with κ is evanescent we can neglect that field and still respect energy conservation in a lossless structure if we κ κ−1 D + χ̄⊥ χ̄⊥ . keep only the fields at κ and κ−1 , simply neglecting the fields 4ε w (κ )w (κ−1 ) 0(−1) (−1)0 at κ . If we do this, and consider the simple excitation sce-

Of course, the diffracted fields only appear for κ−1 < ω̃n , and

nario presented above, each of the Tg,α and Rg,α is a 2 × 2 the expressions above are to be used only in that range. The matrix, and the resulting equations for the specularly reflected more complicated form of the results for p−polarized light and transmitted fields, and the diffracted fields, can be solved arises because of the two components (κ̂ and ẑ) of the light easily. We refer to this as the “two wave vector model.” Con- that arise, as opposed to the simpler results for s−polarization

sidering an incident field from z = −∞, for s−polarization we where there is only one component (ŝ). find specularly transmitted and reflected fields

As an example, we consider a grating with thickness D =

E+out,s (κ ) iω̃2 D k

! = Us 1 − −1 χ̄ , (53) 2 nm consisting of an isotropic medium with refractive in- E+in,s (κ ) 2w (κ−1 ) 0 dex 3.5 embedded in vacuum; we take the grating period to E−out,s (κ ) E+out,s (κ ) be a = 1.2 µm, with a fill fraction of one-half (d/a = 0.5), = + − 1, E+in,s (κ ) Ein,s (κ ) and consider incident s−polarized light at a wavelength of and upward and downward diffracted fields that are equal in 1.5 µm. In Fig. 4 we plot the relative irradiance of the ra-

amplitude, diated electric fields in this system,

E±out,s (κ−1 ) −iω̃2 D k 2

=U −1

χ̄ , E+out,s (κm ) n cos θ (κm )

E+in,s (κ ) s

2w (κ−1 ) (−1)0 (54) I+s (κm ) = 2 , (57)

E+in,s (κ ) nQ cos θ

where 2

E−out,s (κm ) cos θQ (κm )

iω̃ D k iω̃ D k I−s (κm ) = , ! !

Us = 1 − χ̄ 1− χ̄ 2

E+in,s (κ ) cos θ

2w (κ ) 0 2w (κ−1 ) 0 ω̃4 D for κm < ω̃n , and I±s (κm ) = 0 for κm > ω̃n where the + χ̄k χ̄k , 4w (κ )w (κ−1 ) 0(−1) (−1)0 fields are evanescent. Here θ is the angle of incidence, cos θ = while for p−polarization we find specularly transmitted and wQ (κ ) /ω̃nQ , cos θ (κm ) = w (κm ) /ω̃n , and cos θQ (κm ) = reflected fields wQ (κm ) /ω̃nQ . In green (dashed lines) we show the predic-

E+out,p (κ ) tions of the specularly and diffracted reflectance and trans-

(55) E+in,p (κ ) mission as a function of incident angle θ (see Fig. 3) for the  2 iκ−1 D  iw (κ−1 )D k ! two wave-vector model (53,54); we plot in blue (dash-dot) the = Uz 1 − −1  χ̄0  + Uκ−1 1 − ⊥ χ̄0 − 1,   2w (κ−1 )ε 2ε predictions of the full thin grating model (49) with 2N +1 = 7; E−out,p (κ ) and we plot in red (solid) the predictions of a full numerical

calculation using the approach of Whittaker and Culshaw21,2 ,

E+in,p (κ )

which can be considered exact. We see that even our sim-  2  ! iκ−1 D iw (κ−1 )D k ple analytic two-wave vector model (53,54) gives a very good = Uz−1 1 − χ̄⊥0  − Uκ−1 1 − χ̄0 ,   2w (κ−1 )ε 2ε approximation of the diffracted and specularly reflected and

a shift in the real component of the pole. To verify this, we restrict ourselves to excitation with κ · ŷ > 0 and expand our analytic expressions for the specular component of the electric field in (53) and (55) about κ = κ̌, with κ̌ defined by the expression 2π κ̌ − = −κWG , a where κWG is the magnitude of the wave vector satisfying the approximate dispersion relations of the isolated waveg- uide mode given by (30); expressions for κWG for s− and p- polarization are given by (B2) and (B3) in Appendix B. For κ

in this region κ−1 · ŷ < 0, and κ−1 = −(2π/a − κ ) = −κ−1 ŷ is close to the wave vector of a waveguide mode propagating in the −ŷ direction, κ−1 ≈ κ̌0−1 ≡ −(2π/a − κ̌)ŷ = −κWG ŷ.

Since κWG > ω̃n , w (κWG ) is purely imaginary; we put

q = −iw (κWG ), use superscripts s and p on κ̌, κWG , and q FIG. 4. (Color online) Comparison between the numerically exact to indicate the appropriate polarization, and also use w1s and relative irradiances (red solid curves), the irradiances predicted by w1p as short-hand for w (κ̌ s ) and w (κ̌ p ) respectively. Looking the full thin grating model with 2N + 1 = 7 (blue dash-dot curves) at the transmitted specular field, for κ in the neighborhood and the irradiances predicted from the two wave-vector model (green of κ̌ŷ we find the expressions (53,55) can be written approxi-

dashed curves). The calculation is for a 2 nm thick grating with a = mately as 1.2 µm, d = 0.6 µm, and a refractive index of ng = 3.5 suspended in vacuum and subject to an s-polarized field incident at angle θ at E+out,s (κ ) κ−1 − κWG ≈ ηs , (59) a wavelength of 1.5 µm. The vertical, dotted, black line marks the E+in,s (κ ) κ−1 − (κRs + iκIs )

Rayleigh anomaly. E+out,p (κ ) κ−1 − κWG

≈ ηp + C,

E+in,p (κ ) κ−1 − (κRp + iκIp )

transmitted fields, and the calculation with (2N + 1) = 7 wave where κRs,p ≡ κWG s,p + κδs,p , with vectors is essentially exact. Similar good agreement between the approximate calculations and the numerically exact calcu- !2 χ̄k χ̄k (q s )2 ω̃2 D 0(−1) (−1)0 lation is found for p-polarized light. κδs = − s 2 ,

κWG 2w1s  1 + ω̃2wDs χ̄k The vertical, dotted, black lines in Fig. 4 identify the on- 1 set of diffraction, and thus the angle at which the Rayleigh ω̃ D χ̄0(−1) χ̄(−1)0 s 2 2 ! k k (q ) anomalies appear in the specularly reflected and transmitted κIs = 2 , χ̄k κWG s 2w1s 

1 + ω̃2wDs χk fields. At lower angles is the Wood anomaly: The peak in the specularly reflected intensity, and the dip in the specu- 1 larly transmitted intensity, arise from a pole in the response ηs = ω̃ D k , s χ̄0 1 − i2w functions of the structure; the pole is associated with the “ef- 1

fective waveguide” discussed in section III. Returning to the and full response equations (46), we see that the poles of the full structure are given by χ̄⊥0(−1) χ̄⊥(−1)0 !2 p (κ̌ p )2 D κWG κδp = − 2 # , 2ε w1p  2 "  2 p h i κWG / (q p )2 − 2 1 + (κ̌ p ) Dp χ̄⊥

det 1̄3 − 0 ḡ (χ̄o + χ̄v ) = 0, (58) 2ε w 0 1 1

p 2 p κWG χ̄⊥0(−1) χ̄⊥(−1)0 ! 1 (κ̌ ) D (compare (29,30)). Poles here are off the real κ axis; the K , κIp = 2 # , χ̄⊥0 2ε w1p κ p 2 / (q p )2 − 2  "  2 p 0 components of the grating provide coupling into and out of WG 1 + (κ̌ ) Dp χ̄⊥ 2ε w 0 the uniform waveguide, with a dispersion relation identified 1

approximately by the expressions (30), giving the position of ηp = p )2 D , 1 − i(κ̌ p χ̄ ⊥ the pole in the κ plane an imaginary contribution, as well as 2ε w 0

and where substrate, with relative dielectric constant εQ , extends, takes the form qp D k ! 1 TQ − R1Q T1Q −1 RQ R1Q T1Q −1  



C= 1+ χ̄ − 1 M1Q = 

Uκ 2ε 0  . (62)

− T1Q −1 RQ T1Q −1

is negligible for sufficiently thin gratings. We do not plot (59), This is a 4 × 4 matrix, but as long as the layered materials are but note that in the region of the dip of the specular transmis- sion for both s- and p-polarized light the pole expansion gives isotropic or uniaxial  it will  be composed of 2 × 2 block ma- trices Ti j = diag Tisj , Tipj where Tis,p j is the Fresnel coefficient an extremely good fit to the more exact expressions (53,55) for the transmitted s− or p−polarized fields from εi to ε j and

in the two-wave-vector model, as well of course to the results

Ri j is similarly defined for their reflected counterparts . We

(51,52) of the (2N +1)-wave-vector model and to the exact nu- can immediately extend this to a transfer matrix M̄1Q of the merical results with which the two-wave-vector model agrees layered structure involving all our (2N + 1) κm of interest by well. The inclusion of the imaginary parts κIs,p of the pole writing positions are obviously essential in achieving this, but the in- clusion of the shifts κδs,p in the real part of the pole positions are as well and should not be neglected; both are second order T̄Q − R̄1Q T̄1Q −1 R̄Q R̄1Q T̄1Q −1 

      in the grating coupling amplitudes. M̄1Q =    −1  −1  , (63)

− T̄1Q R̄Q T̄1Q

h i D. Including a substrate a 4(2N + 1) × 4(2N + 1) matrix where T̄1Q =   mm p diag T1Q (κm ) , T1Q (κm ) and other terms are similarly de- s

Returning to our general scattering treatment (48) of an iso- fined. lated grating, we can move to a transfer matrix treatment by We can now construct a transfer matrix for the full structure solving for the upward and downward propagating (or evanes- shown in Fig. 1(a) by imagining an infinitesimal layer of ma- cent) field amplitudes above the grating (Ē+out and Ē−in ) in terms terial with relative dielectric constant ε inserted between the

of the upward and downward propagating (or evanescent) field bottom of the grating structure and the top of the highest layer amplitudes below the grating (Ē+in and Ē−out ), in the multilayer structure below. Then the transfer matrix relating the upward and downward propagating (or evanes-  +   Ēout   +   Ē  cent) field amplitudes just above the grating to the upward and  −  = M̄g  −in  , (60) downward propagating (or evanescent) field amplitudes at the

Ēin Ēout

largest z in the substrate is given by where

 T̄g − R̄g T̄g −1 R̄g R̄g T̄g −1        M̄g =   −1  −1  (61) M̄01Q = M̄g M̄1 (D/2) M̄1Q , − T̄g R̄g T̄g where has 4(2N + 1) × 4(2N + 1) elements, as does the scattering  +  matrix (47). Returning to our general structure of Fig. 1(a), L̄ 0̄ 

M̄1 (D/2) =   

we can now combine the transfer matrix (61) of the grat- 0̄ L̄− ing region with the transfer matrix of the multilayer below to form a transfer matrix for the whole structure, in terms is composed of 2 (2N + 1) × 2 (2N + 1) block matrices with of which the optical properties of the structure can be cal- 1 0

L̄± (κm ) = e±iw (κm )D/2   and propagates the fields from

culated. To do this, consider first light characterized by a 0 1 the center of the grating at z = 0 to the position of the sub- single κ in the presence of the multilayer structure of Fig. strate at z = −D/2. Through simple algebra we can write the 1(a), but without the presence of the grating. The transfer ma- elements of M̄01Q as trix of the multilayer structure relating upward- and downward propagating (or evanescent) amplitudes of light just above the

multilayer in the medium with relative dielectric constant ε T̄0 − R̄0 T̄0 −1 R̄0     −1 

R̄01Q T̄01Q 

(at z = (−D/2)+ ) to upward- and downward propagating (or M̄1Q = 

0 Q

 1Q 1Q

−1 Q

 −1  , (64) 0 0 evanescent) amplitudes of light at the largest z to which the − T̄1Q R̄Q T̄01Q

FIG. 5. (Color online) Comparison between relative irradiance calculations that are numerically exact (red solid curves) and those predicted by the full thin grating model with 2N + 1 = 7 (blue dash-dot curves) around a Rayleigh anomaly. The calculation is for a 2 nm thick grating with a = 1.8 µm, d = 0.7 µm, and a refractive index of ng = 3.5. The system has a substrate with index nQ = 1.44, a vacuum cladding, and is subject to an s-polarized incident field from the substrate at angle θ and with a vacuum wavelength of 1.5 µm.

where constant ε which also serves as a cladding, then resides on   2 −1 a semi-infinite substrate of dielectric constant εQ , which we T̄01Q = T̄1Q L̄+ 1̄2 − R̄g R̄1Q L̄+ T̄g , now relabel ε (see Fig. 3). Taking ε to be real and positive  2   2 −1  −1 there are no modes associated with the substrate, and so the T̄0Q = T̄g R̄1Q L̄+ 1̄2 − R̄g R̄1Q L̄+ R̄1Q T̄Q L̄− ,

only effect of the substrate will be to modify the modes iden-  2   2 −1 R̄01Q = R̄g + T̄g R̄1Q L̄+ 1̄2 − R̄g R̄1Q L̄+ T̄g , (65) tified by the poles of the isolated grating structure (see Fig.   2 −1  1(b)).

R̄0Q = R̄Q + T̄1Q L̄+ 1̄2 − R̄g R̄1Q L̄+ R̄g T̄Q L̄+

Insight into the nature of this modification can be gleaned

are easily identified as the transmission and reflection matrices from recalling the simplest picture of the grating region as an of the entire structure (compare (61,63)). effective anisotropic slab (see Fig. 1(c)). In a symmetric envi- The analytic structure of the new Fresnel matrices (65) is ronment both s− and p−polarized waveguide modes exist, but inherited from that of the isolated grating (46) and from the for an environment with ε , ε (see Fig. 3) the modes will

R̄i j and T̄i j of the multilayer below it. Besides the poles of not survive if the asymmetry is large enough. In such a situ- T̄g and R̄g signaling the waveguide modes in the isolated grat- ation we expect the Wood anomalies will vanish, although of ing structure, we can in general expect poles in R̄i j and T̄i j course the Rayleigh anomalies will remain. We demonstrate signaling the presence of waveguide modes in the multilayer. how our thin grating model describes this situation by con-

The positions of the poles in the new Fresnel matrices (65) sidering a grating with D = 2 nm, with a dielectric constant will exhibit the interaction between these excitations, and we εg = (3.5)2 appropriate for silicon, a period of a = 1.8 µm, will turn to a general analysis of the new excitations in a later and a fill fraction d/a = 0.4 in vacuum (ε = 1), located publication. Here we focus on a first application our thin grat- above a substrate of fused silica (ε = (1.44)2 ) and subject

ing model in the presence of a substrate, and on some of the to s−polarized light from below at a vacuum wavelength of qualitative features that arise from the interaction. Thus we λ = 1.5 µm. For the resulting εlayer we would require an consider the simplest multilayer structure possible, taking the ε > (1.38)2 for a waveguide mode to be contained within substrate with relative dielectric constant εQ to extend up to the guiding layer, so the asymmetry here is too great to al-

z = −D/2. The grating, suspended in a medium of dielectric low for Wood anomalies, and only Rayleigh anomalies should

FIG. 6. (Color online) Comparison of relative irradiance calculations that are numerically exact (red solid curves) and those predicted by the full thin grating model (blue dash dot curves). The calculation is for a 2 nm thick grating with a = 1.8 µm, d = 0.7 µm, and a refractive index of ng = 3.5. The system has a substrate with index nQ = 1.44, a cladding with index n = 1.42, and is subject to an s−polarized field, incident from the substrate at angle θ with a vacuum wavelength 1.5 µm.

survive. Assuming ε > ε , this can be confirmed by solving culation using (48) with 2N + 1 = 7, while in red (solid) √ (B1) for ε with κ = ω̃ ε . In agreement with this simple we give the results from a full numerical calculation using argument, the reflected, transmitted, and diffracted light in- the approach of Whittaker and Culshaw21,2 . For s−polarized tensities exhibits only cusp-like Rayleigh anomalies, as seen light we focus on the region around the Wood anomaly; note

in Fig. 5. In blue (dashed-dot) we plot a calculation with that with the field incident from the substrate, which has a (2N + 1) = 7 wave vectors using (48), while in red (solid) higher index than the cladding, the forward diffracted fields we plot the exact result found numerically from the approach become evanescent before the backward diffracted fields. For of Whittaker and Culshaw21,2 . There is excellent qualitative p−polarized light we plot the response for all incident angles;

and good quantitative agreement between the results of the the absence of a Wood anomaly leaves somewhat unremark- thin grating model and the exact result, especially consider- able results for specularly reflected and transmitted light, but √ ing that a parameter 2π εg D/λ , which should obviously be yields several noteworthy features in the diffracted compo- small for our thin grating approximations (14) to be valid, is nents, which can propagate up to m = −3. Rayleigh anoma-

here about 0.35. There is only a significant relative correc- lies when the m = +1, −2, and −3 diffracted orders transi- tion in the diffracted intensities at κ−1 , where the diffracted tion between evanescence and propagation lead to Rayleigh intensities themselves are very small. anomalies that appear as non-analyticities in the m = −1 and m = −2 beams. Additionally, for angles of incidence beyond

In order for this grating to exhibit a Wood anomaly the mis-

that which would yield total internal reflection were the grat- match between ε and ε must be decreased. To move into ing absent, the specular reflectance does not remain at unity. this regime, we raise ε to (1.42)2 , and keep all other pa-

A small dip in the specular reflectance follows the Rayleigh

rameters the same; for this value the simple argument used anomaly associated with this transition, which is compensated above predicts a waveguide mode for s−polarized light, but by an increase in the irradiance of the remaining diffracted not for p−polarized light. In accord with this, the calcu- components. They display peaks over this range, which fi- lated reflected, transmitted, and diffracted light irradiances nally drop to zero as the incidence approaches grazing. We shown in Figs. 6 and 7 exhibit Wood and Rayleigh anoma-

note excellent agreement between the thin grating results (48) lies for s−polarized light, but only Rayleigh anomalies for and the exact calculation throughout the plots in Figs. 6 and p−polarized light. Again in blue (dashed-dot) we plot a cal-

FIG. 7. (Color online) Comparison of relative irradiance calculations that are numerically exact (red solid curves) and those predicted by the full thin grating model (blue dash dot curves). The calculation is for a 2 nm thick grating with a = 1.8 µm, d = 0.7 µm, and a refractive index of ng = 3.5. The system has a substrate with index nQ = 1.44, a cladding with index n = 1.42, and is subject to a p−polarized field, incident from the substrate at angle θ with a vacuum wavelength 1.5 µm. The insert in the graph of I−p (κ+1 ) shows the detail around θ = 23.5o .

7, with the largest relative disagreements appearing only when see how the anomalies arise from the structure of the equa- the intensities involved are very small. tions that describe the specular and diffracted fields, with Although not shown, we note that if the asymmetry be- square root singularities associated with Rayleigh anomalies tween cladding and substrate is decreased further so that a and poles with Wood anomalies; the poles are linked to ef-

p-polarized Wood anomaly appears, we observe a small shift fective waveguide modes of the grating region that are easily between its location as predicted by (48) and the full numer- identified in the thin grating limit. This helps in understand- ical results, which does not occur for the s−polarized Wood ing the optical response even if a set of coupled wave vec- anomaly shown in Fig. 6. For both s- and p−polarized Wood tor equations must be solved for the specular and diffracted

anomalies, the disagreements with the full numerical calcula- fields. Yet, where only a few wave vectors are important, an- tions increase as the dimensionless optical thickness param- alytic expressions can be given directly for the specular and eters D̃ s and D̃ p , given by (B4) and (B5) in Appendix B, diffracted fields. Comparison with full numerical solutions approach unity. In that Appendix we show that this signals of a 1D grating response confirms that our approximate solu-

the breakdown of our approximate treatment of the waveguide tions is in excellent agreement with the exact response, even modes in the effective anisotropic slab. near the anomalies.

V. CONCLUSIONS

In this work we have presented a treatment for the optical We expct that the development of approximate yet accurate response of thin gratings. Although approximate, it nonethe- treatments of thin gratings, such as the one presented here, less respects energy conservation exactly, even if there are will play an important role in enabling their use as probes of large exchanges of energy between specular and diffracted optical systems. The calculations can be made more easily

fields, and between specularly transmitted and reflected fields. than full numerical treatments, and the physics can be identi- These large exchanges are associated with Rayleigh and Wood fied in the reasonably simple sets of equations that are used in anomalies. Our Green function approach makes it easy to calculations.

ACKNOWLEDGMENTS The exact solution for the waveguide modes of this system

The authors thank the Natural Sciences and Engineering

h − qp Research Council of Canada (NSERC) for partial funding of cot (hD) = , (B1) h (q + p) this work including the award of an Alexander Graham Bell

Canada graduate scholarship to D. A. Travo. Additionally, we where

thank Matteo Galli, Sharon Weiss, and Daniele Aurelio for v u  εlayer   t k  useful conversations throughout this work.  h =  b  ω̃2 εblayer − κ , εlayer v  εblayer u t

Appendix A: Fourier components for a rectangular grating 

q = aq ω̃2 εblayer − ε − k h , εlayer

In this section we evaluate χv[m] for a one dimensional rect- v

 εblayer u t angular grating, composed of isotropic dielectric materials,  p = a p ω̃ εlayer − ε − k h ,

2 b

such as the one presented in Fig. 1(b). Recalling (17), we εlayer have and where εblayer = εklayer and aq = a p = 1 for s−polarization,

1 a/2 −imKζ

Z χv[m] = e (χmod (ζ) − hχmod i) dζ, (A1) and εblayer = ε⊥layer , aq = εklayer /ε and a p = εklayer /ε for a −a/2 p−polarization. Taking ε = ε , approximate dispersion re- where from (13) and our definition of ε (ζ) ≡ ε + χadd (ζ), lations can be determined directly from (30); solving those we can write equations yields

χkmod (ζ) = εk (ζ) − ε , κ 1 = 1 + D̃2s , (B2) ω̃2 ε 4 ε ! χ⊥mod (ζ) = ε 1 − . ε⊥ (ζ) for s−polarization, and

Note that for the system considered, we have κ 2

= , (B3)

ω̃2 ε q 1 + 1 − D̃ p  εg |ζ| ≤ d/2   ε (ζ) = ε (ζ) =  ⊥ k  ε |ζ| > d/2   for p−polarization , where

 εlayer  k within a single period of our grating.D Additionally, we note  

D̃ s = ω̃n 

E that hχmod i, found from (20), yields χmod = εlayer − ε and k k − 1 D, (B4) 

D E   ε

χ⊥mod = ε 1 − ε /ε⊥layer where εklayer and ε⊥layer are found   ε D̃ p = ω̃n 1 − ⊥  D.   from (21). At this point we can evaluate (A1) to find (B5) εlayer !  k  mKd χv[m] = εlayer − ε sinc k , The approximate dispersion relations can be shown to agree   with the lowest order solution of the exact relations to first

ε  ! mKd χ⊥v[m] = ε 1 − ⊥  sinc .  order in the grating thickness. The exact and approximate dis- εlayer 2 persion relations are shown in Fig. 8 over a range of waveg- uide thicknesses. For the purposes of the calculation we set

Appendix B: Waveguide Modes εklayer = 6.1 and ε⊥layer = 3.03, with ε = (1.42)2 , as used in

Fig. 6 with the absence of any diffraction. Here we compare the exact and approximate dispersion re- For both polarizations we begin to see significant deviations lations of the waveguide modes of a thin uniaxial slab. For in Fig. 8 as D̃ j → 1, where j = s, p. The s-polarized case has ẑ perpendicular to the slab we take the relative dielectric ten- a relative deviation of 1.5% at D̃ s = 1, which corresponds to sor to be εlayer = εklayer (x̂x̂ + ŷ ŷ) + ε⊥layer ẑẑ, with the cladding a thickness of 8 nm, while the p-polarized case has a devia-

and substrate of the slab taken to be isotropic media respec- tion of 27% at D̃ p = 1. The significantly larger deviation in tively characterized by relative dielectric constants ε and ε . the p-polarized results can be attributed to two factors. The

Appendix C: Energy Conservation

Here we confirm that our approximate treatment of diffrac-

tion and scattering across the grating satisfies energy conser- vation exactly in the limit of no absorption. To do this, we start with the difference between the total irradiance of the outgoing and incident fields

P + 2 2

 ∆I = 2c0 n Ēout (κm ) + Ē−out (κm ) W (κm ) cos θ (κm ) m  

P + 2 2

−2c0 n Ēin (κm ) + Ē−in (κm ) W (κm ) cos θ (κm ) , m (a) s-polarization

(C1)

where W (κm ) ≡ 1 for propagating fields, and W (κm ) ≡ 0 for evanescent fields, such that (C1) considers the difference in the incoming and outgoing energy from the grating via prop- agating fields; also cos θ (κm ) = w (κm ) /ω̃n . Denoting by θ̄ the diagonal matrix with elements θ̄mm = 2c0 n cos θ (κm ) we can write the difference in irradiance as i  W̄ 0̄   θ̄ 0̄   W̄ 0̄   Ē+      

∆I = Ē+∗

Ē−∗         out 

out out 0̄ W̄ 0̄ θ̄ 0̄ W̄ Ē−out i  W̄ 0̄   θ̄ 0̄   W̄ 0̄   Ē+       +∗ h −∗ − Ēin Ēin         in  , 0̄ W̄ 0̄ θ̄       0̄ W̄ Ē−in

(C2)

where 0̄ is a 2 (2N + 1) × 2 (2N + 1) matrix of zeros, and W̄ is a 2 (2N + 1) × 2 (2N + 1) block diagonal matrix with diagonal elements   (b) p-polarization 1 0 W̄ (κm ) = W (κm )   .

FIG. 8. (Color online) Comparison of exact (solid red curves)

0 1 and approximate (blue dash-dot curves) waveguide mode disper- After recalling (47), (C2) becomes sion calculations over a range of slab thicknesses. The calcula-  θ̄ 0̄   θ̄ 0̄     0   0  i  W̄ 0̄   tion is for a thin uniaxial slab characterized by εklayer = 6.1 and ∆I = Ē+∗ h in Ēin

−∗     S †

S −

0̄ θ̄0 0̄ θ̄0        ε⊥layer = 3.03, suspended in a medium with index 1.42, subject to ei- 0̄ W̄     

ther an s−polarized (a) or p−polarized incident field with a vacuum (C3) wavelength of 1.5 µm. The thickness parameters, D̃ s and D̃ p are   +   W̄ 0̄   Ēin  give by (B4) and (B5) respectively. ·     , 0̄ W̄ Ē−in

where θ̄0 = W̄θ̄W̄ and we have made use of the fact that W̄2 = first is that for D̃ p > 1 the square root in the denominator W̄. of (B2) becomes imaginary, giving a firm cut-off for its valid

For convenience we introduce the matrix

comparison to the exact solution, and the second is due to the  −1 fact that εklayer > ε⊥layer for the cases considered in this pa- C̄± = 0Ḡ± χ̄tot 1̄3 − 0 ḡχ̄tot , (C4) per. While the chosen cut-off for the s-polarized calculation where χ̄tot = χ̄o + χ̄v . Using (C4), we can write our scattering was found to be 8 nm, the breakdown of the p-polarized case matrix from (47) as occurs at 5 nm, a thickness well beyond our underlying as-  +  + +   +

 σ̄  C̄ C̄   σ̄   sumption that w D  1. To provide a better comparison to 0̄   0̄  S =  out −  1̄6 +  − −   in −  , (C5) the s−polarized case, we note that (B3) has a relative devia- 0̄ σ̄out C̄ C̄ 0̄ σ̄in tion of approximately 1.5% at D̃ p = 0.7 which corresponds to and can then reduce (C3) to

Ē+∗ W̄σ̄+out Ē−∗

h i ∆I = in in W̄σ̄−out  +

 Θ̄ 0̄   C̄+ C̄+   C̄+† C̄−† 