A New Theory of Wood’s Anomalies on Optical Gratings

A. Hessel; A. A. Oliner

doi:10.1364/AO.4.001275

I. Introduction

A. General Introduction

Wood’s anomalies are an effect observed in the spectrum of light resolved by optical diffraction gratings; they manifest themselves as rapid variations in the intensity of the various diffracted spectral orders in certain narrow frequency bands. They were first discovered by Wood[1] in 1902 in experiments on reflection gratings, and were termed anomalies because the effects could not be explained by ordinary grating theory. Since that time these effects constituted a topic of many experimental and theoretical investigations, the principal ones being summarized in Sec. I-B. A physical discussion of the effects themselves is included under Sec. II.

As indicated in Sec. I-B, the existing theories explain some of the observed effects, but fail with regard to a number of other significant experimental observations. For example, these theories cannot explain the P anomalies investigated thoroughly by Palmer,[2] or the reluctance of two anamolies to coincide, as reported by Stewart and Gallaway.[3] In addition, existing theories are satisfactory only for certain classes of structures, such as arrays of small conducting semicylinders on a plane or metallic gratings with shallow grooves. They are inadequate for metallic gratings with deep grooves, or other structures such as periodically grooved, or otherwise modulated, dielectric layers.

The new theory presented here is based on a guided wave approach, in which all multiple-scattering effects are implicitly accounted for. It employs a surface reactance boundary condition which takes into account the standing waves in the grooves, and is, therefore, valid for a very wide class of structures, including gratings with deep grooves and modulated dielectric layers. The theory is intimately associated with the complex guided waves supportable by the grating, and presents a physically satisfying explanation of the anomalies, even with regard to their shapes and line-widths. It is also more general than the previous theories, and it explains in a direct fashion the existence of the P anomalies and the effect reported by Stewart and Gallaway.[3] The theory may also be used to predict the presence of anomalies in situations in which they have not as yet been detected experimentally; the significance of this statement lies in the fact that these predictions could not be made on the basis of the previous, multiplescattering theories.

The historical summary in Sec. I-B outlines in context some of the previously proposed theories and their limitations, and points out which experimental observations remain to be explained.

Section II then briefly describes the nature of the Wood anomalies, and presents a physical discussion of the major features of the new theory. It is pointed out that the anomalies may be of two types, one associated with the Rayleigh wavelengths and the other related to a resonance effect. These two types of anomalies may occur separately and independently, or they may almost coincide, as one finds in most optical reflection gratings. The Rayleigh wavelength type is well known. The resonance type is the one which is shown to be related to the guided complex waves supportable by the grating. An analogy with a multi-mode resonant cavity is presented to explain the origin of the resonance anomaly, and its relation to these guided waves. A physical discussion of the implications of this new approach then follows, together with an explanation of the P anomalies.

The terminology employed here may differ somewhat from that used by others. The modal components of the total scattered field are here called propagating and nonpropagating (evanescent) spectral orders, even though that term customarily implies only the propagating waves. The evanescent waves are sometimes incorrectly called surface waves; surface waves are not properly part of a plane-wave field but represent a form of guided wave along the grating which by itself satisfies all boundary conditions. The concept of such guided waves plays a key role in the present theory, and, since our discussions will involve both the modal guided waves and the modal scattered waves, we have tried not to use the term mode in order to avoid confusion. Another possible, but inapplicable, term is space harmonic, which is used for the periodic components of a guided wave. The choice of the term spectral order for both the propagating and nonpropagating scattered waves seems, therefore, the best compromise, and should not cause any confusion.

In Sec. III, the mathematical details of the theory are presented. A plane which is characterized by a periodic surface reactance along one coordinate is chosen as an appropriate model for mathematical analysis. This surface reactance is not only a function of frequency and angle of incidence of the plane wave, but is also assumed to be an analytic function of a parameter which is called the depth of modulation and which is a measure of the groove depth. The field is represented in the form of a Floquet-type expansion and is then subjected to the surface reactance boundary condition. As a consequence, an inhomogeneous infinite set of linear equations results for the infinite number of amplitudes of the various spectral orders. The amplitudes of these spectral orders are then found by inverting the system of equations. These amplitudes then undergo rapid variations in the neighborhood of a zero of the infinite determinant. Such a zero, which corresponds to a pole in the amplitudes, occurs near to that value of frequency for which the phase velocity of a spectral order of the incident plane wave along the surface becomes equal to that of the surface wave which would be present in the absence of modulation. Because of the introduction of periodicity, this surface wave becomes in actuality a leaky (complex) wave and the resulting pole becomes complex. With increase in the depth of modulation the imaginary part of the complex pole in general increases, thus decreasing the peak of the resonance, and increasing its width. The treatment includes a discussion of the specific shape of the anomalies, including the locations of the maxima and minima, and the Q’s or line-widths. Other implications of the theory are considered, and the class of structures for which this analysis is valid is also discussed.

A simple graphical method which yields the number of anomalies occurring and their location in frequency and incidence angle has also been devised, but it will be presented in a subsequent paper. The method makes use of a construction in the k vs β plane, where k(= 2π/λ) is the free-space wavenumber and β is the phase constant of the guided wave which is supportable by the grating. A plot of this type is also termed a Brillouin diagram or a dispersion curve plot. The graphical method informs us readily as to whether the anomaly is of the Rayleigh or the resonance type, or a combination thereof, and also indicates where double anomalies will occur. It is also shown to be of value in explaining the stop-band effect described by Stewart and Gallaway.[3]

The material contained in Sec. III is intended to be general in order to stress the basic characteristics of this new approach to the explanation of Wood’s anomalies. As an illustrative example of the general theory described herein, the problem of scattering by a sinusoidally modulated, planar nondispersive reactance surface has been solved rigorously, and a discussion of this solution appears in Sec. IV. The solution is valid for all values of modulation, periodicity, frequency, and angles of incidence. The amplitudes of all of the spectral orders are determined explicitly, and the Wood’s anomaly effects are demonstrated clearly for a variety of cases. This section also contains a detailed theoretical discussion on the shape of the anomalies and the Q of the resonant spectral order, and presents numerical data in agreement with these theoretical considerations. The specific behavior at the Rayleigh wavelengths is also included, but most of the anomalous effects discussed are removed from these wavelengths. Numerical data in graphical form are presented to illustrate many of the various points which emerge as contributions from the new theory.

B. Historical Summary

Good historical introductions have been presented in papers by Twersky,[4] Millar,[5] and Stewart and Gallaway,[3] and a comprehensive summary, with an extensive bibliography, was compiled by Twersky.[6] However, in order to make clear the historical continuity of the present new theory, and because reference must be made in context to the various unexplained effects which this new theory clarifies, it is desirable that a partial historical summary be included here which contains a different stress from the previous ones.

In 1902, Wood[1] discovered the presence of unexpected narrow bright and dark bands in the spectrum of an optical reflection grating illuminated by a light source with a spectral intensity distribution which was only slowly varying. He noted, furthermore, that these bands could be weakened or abolished completely in some cases, but not in others, by simply rubbing the tops of the gratings (we understand this now to be a groove depth effect, rather than an edge effect). In addition, he found that the occurrence of these bands was dependent on the polarization of the incident light. The bands were present only for S polarization, when the electric vector was perpendicular to the rulings of the gratings. Since these effects could not be explained by means of ordinary grating theory, Wood termed them “anomalies”.

For a number of years, further experiments on gratings by Ingersoll,[7] Strong,[8] Wood himself,[9] and others seemed to bear out these basic observations. The behavior of the bright and dark bands as a function of wavelength or incidence angle was rather involved, but the polarization dependence was clear: anomalies were observed when the incident electric field was perpendicular to the grooves of the grating (S anomalies, E-mode incidence), but when the electric field was parallel to the gratings (P anomalies, H-mode incidence) no anomalous effects were found.

The first theoretical treatment of these anomalies is due to Rayleigh[10] in 1907. His “dynamical theory of the grating” was based on an expansion of the scattered electromagnetic field in terms of outgoing waves only. With this assumption, he finds that the scattered field is singular at wavelengths for which one of the spectral orders emerges from the grating at the grazing angle. He then observed that these wavelengths, which have come to be called the Rayleigh wavelengths λ_R, correspond to the Wood anomalies. Furthermore, these singularities appear only when the electric field is polarized perpendicular to the rulings, and thus account for the S anomalies; for P polarization, his theory predicts a regular behavior near λ_R. Thus, Rayleigh’s theory correctly predicted the major features observed experimentally at that time: the wavelengths at which the S anomalies occurred, and the absence of P anomalies.

One of the limitations in Rayleigh’s theory is that it indicates a singularity at the Rayleigh wavelength, and, therefore, does not yield the shape of the bands associated with the anomaly. In an attempt to overcome this difficulty, Fano[11] assumed a grating consisting of lossy dielectric material, but the coefficients of his expansion in the vicinity of λ_R were still too large to be useful. Artmann[12] also employed Rayleigh’s outgoing wave assumption, but derived a convergent approximate representation for the field near the Rayleigh wavelength. His expansion, however, exhibits only the maxima, but not the minima, characteristic of the Wood anomalies.

For a period of some years, therefore, the theory was essentially in agreement with the basic experimental observations, even though many of the finer points, such as the detailed shape of the anomalies, could not be predicted. Wood’s later papers,[9],[13] however, suggest that P anomalies can sometimes be observed. More recently, Palmer[2],[14],[15] very clearly demonstrated that P anomalies do exist under appropriate circumstances. In fact, Palmer attributes the appearance of P anomalies to the use, both by Wood and himself, of deeply ruled gratings. He also points out that in other experiments,[7],[8] for which P anomalies were absent, gratings with shallow grooves were used. Hence, anomalies of both the S and P type are obtainable, but P anomalies are found only on gratings with deep grooves.

The theories which state that P anomalies are forbidden must therefore be reconsidered. Such a careful reexamination was conducted by Lippmann[16] and Lippmann and Oppenheim.[17] They analyzed the implications of Rayleigh’s initial assumption of including outgoing waves only, and concluded that this approximation is valid for shallow grooves only. Since this approximation is basic to all of the theories mentioned above, they are all valid for shallow grooves only. Since P anomalies are observed only on gratings with deep grooves, no inconsistency is present, but these theories are, however, clearly incomplete.

In recent years, with the developing sophistication in treatments of electromagnetic scattering from various obstacles, another approach to the explanation of Wood anomalies was adopted. This approach is based on a multiple-scattering point of view. In most of the treatments, the total scattered field is expressed in terms of the multiple-scattering amplitude of one scatterer within the grating, and this amplitude is, in turn, given in terms of the presumed known scattering amplitude of the same scatterer when isolated. The expression for the multiple-scattering amplitude specifies completely all the coupling effects between the various scatterers. Such analyses have been applied by a number of writers to a variety of periodic arrays of scatterers; for appropriate sets of parameters, Wood’s anomalies are obtained. Some of the authors with whom such solutions are associated are Karp and Radlow,[18] Millar[5],[19] (integral equation approach), and Twersky.[4],[6],[20]–[24] Of these, the most extensive investigations are those due to Twersky, who not only examined the scattering properties of gratings in general terms[4],[21] but was also concerned specifically with the occurrence of Wood’s anomalies.[20],[23],[24] He has presented several different scattering approaches to the problem, including an earlier orders of scattering treatment[20] and a later sum-integral equation method,[4],[24] has considered both finite and infinite gratings, and includes results for certain specific classes of scatterers, such as semicircular bosses on a plane,[24] elliptical semicylinders on a plane,[22] etc. While the existence of maxima in the anomalies had already been demonstrated by Artmann,[12] Twersky was the first to discuss the minima.

The multiple-scattering approach has afforded us with the best theory to date of these anomalies. It can satisfactorily predict the location and shape of the anomalies in the case of certain specific forms of the basic scatterer in the grating. The results, however, have been limited to gratings of relatively shallow groove depth because of the restriction to scatterers of small ka, where a is the radius, say, of a semicircular cylinder on a plane. These multiple-scattering results have not, therefore, exhibited the P anomalies found experimentally by Palmer.[2],[14],[15]

A recent and clearly written paper by Stewart and Gallaway[3] summarizes much of the basic experimental information available on Wood anomalies, and presents some new experimental data. Included in this new data is an observation regarding double anomalies which had not been previously reported. These writers plot the occurrence of the zero-order anomalies on an incidence angle vs wavelength plane. The loci of anomalies then form sets of straight lines; when these lines cross, double anomalies should be expected. The authors found, however, that instead of crossing the loci of anomalies “appear to repeal each other, exchange identities, and then separate”. Experimental data are presented in verification of this statement. Stewart and Gallaway[3] also comment that this “previously unreported property of Wood anomalies is disclosed as a challenge to our theoretical colleagues who may wish to explain it”. (Although this effect can indeed be readily explained by the new theory presented herein—as related to a stop band in the dispersion plot of the guided wave which influences the resonance anomalies—it can be discussed most easily in terms of the graphical procedure which is to be presented in a subsequent paper. For this reason, an explanation of the effect will not be included here.)

The new theoretical approach which is given in the present report is phrased in such a way as to yield new insight into the character of Wood’s anomalies. It is not an extension of the multiple-scattering approach it is a new guided wave viewpoint which formally may be viewed as an extension of the earlier work of Rayleigh and Artmann, but which contains a number of new elements. It readily explains both the occurrence of P anomalies under appropriate conditions, and the new effect described by Stewart and Gallaway.

II. Physical Discussion of the New Theory

A. Two Types of Anomalous Behavior

Let us first review the basic effect which is observed with the Wood anomaly. In this connection, let us assume that light is incident at angle θ with respect to the normal on a periodic reflection grating of period d, as shown in Fig. 1. In addition to the reflected wave, specified by n = 0, let us assume the presence of several diffracted spectral orders. Next, we assume that the source of light possesses a spectral intensity distribution which is continuous (no line spectra) and is only slowly varying with wavelength. As the wavelength is varied, the n = 0 beam remains still while the others move. If the amplitude (or intensity) of the n = −1 (say) beam is measured as the wavelength is varied (and the beam is tracked), one might expect, as Wood did, that the amplitude vs wavelength variation of the n = −1 beam would be a slow one. Instead, Wood found rapid variations in intensity in narrow wavelength ranges, as shown in Fig. 2. Since such variations did not follow from ordinary grating theory, Wood called them anomalous.

Rayleigh’s early theory[10] yielded singularities in the scattered field at wavelengths (Rayleigh wavelengths) which correspond, for S polarization, to the entry of a new spectral order. He also pointed out that the occurrence of such singularities corresponded to the appearance of the Wood anomalies. As a new spectral order appears, one could expect a rearrangement in the amplitudes of the other diffracted orders. One might note, however, that for P polarization for shallow grooves the electric field is parallel to the metallic surface so that the radiation field of a new spectral order is essentially short-circuited as it enters at grazing angle. Hence, the Rayleigh singularities (or, alternatively, this type of anomaly) do not occur for P polarization for shallow grooves.

When the intensity variations of a Wood anomaly are examined carefully, it is seen that the appearance of a new spectral order is in itself not sufficient for a description of the form of variation. There is, in addition, a resonance effect present (which Artmann’s theory, for example, could not explain, but which Twersky’s multiple-scattering theory described). However, for some structures these resonance effects can occur for wavelengths far removed from the Rayleigh wavelength. It is, therefore, important to recognize that two types of anomalous effects occur:

(1) a rapid variation in the amplitudes of the diffracted spectral orders, corresponding to the onset or disappearance of a particular spectral order, and
(2) a resonance type of behavior in these amplitudes.

Depending on the type of periodic structure, the anomalies of type (1) and (2) may occur separately or almost superimposed. Examples of structures which fit into these two classes are shown in Fig. 3(a) and (b). In the latter, the anomalous effects occur separately, while in the former, the anomalies exhibit their classical form: a superposition of the two types. The distinction between the structures of Fig. 3(a) and (b) may be made conveniently in terms of their surface reactance. For the structure of Fig. 3(a), the surface reactance becomes zero (a conducting plane) when the periodic modulation is removed; for the other structures (b) this reactance remains finite. When the surface reactance value in the absence of modulation is small, the anomalous effects (1) and (2) occur very close to each other, as in the usual (classical) anomalies. The resulting intensity variation is therefore rather involved. It would be easier to consider structures for which these effects are separated, so that the effects may individually be understood more clearly. The theoretical analysis in Sec. III, therefore, assumes a general form for the surface reactance in order to permit such a separation.

Effect (1), related to the Rayleigh wavelength, is well understood and requires no further discussion. Effect (2), the resonance effect, is the one which has heretofore been incompletely appreciated and which is shown here to be related to the leaky (complex) waves[25]–[27] supportable by the grating. Section II-B presents a qualitative discussion of this resonance effect.

B. The Forced Resonance Nature of the Resonance Anomalies

In order to clarify the nature of the resonance anomalies, the anomaly of type (2) referred to above, let us consider a simple electromagnetic analogy involving a closed waveguide. Suppose that this waveguide, shown in Fig. 4, is operated under multi-mode conditions, i.e., the dimensions and frequency are such that more than one mode can propagate in the guide. Under these conditions, if the dominant mode is incident (traveling downward in Fig. 4) on the termination at plane T, the geometrical discontinuity at T will reflect the dominant mode and will excite higher modes, some of which can propagate in this multimode guide. The proportion of energy in the various propagating modes depends on the nature of the termination. Now, if the amplitude of incident wave is essentially flat with frequency, we should expect that the amplitude of any given higher propagating mode would also vary little with frequency, unless the termination itself contained a strongly frequency-dependent element. The termination in Fig. 4 below the line T contains just such an element—a resonant cavity. When the frequency of the incident mode passes very near to a frequency for which this associated cavity is resonant, a violent rearrangement of amplitudes will occur, and we can expect a resonance form of response to appear.

The free resonance of this cavity corresponds to a complex resonance wavenumber, the imaginary portion being due to loss of power from the cavity proper into the connecting waveguide through the coupling hole (plus loss due to dissipation in the cavity walls, if any). However, the propagating waveguide modes possess real wavenumbers, and they, therefore, cannot excite the true free resonance of the cavity. As is usual in resonance phenomena, however, the cavity produces a strong reaction back on these propagating modes in the form of a forced resonance whenever the (real) wavenumber of any of these modes approaches very close to the (complex) free resonance wavenumber of the cavity. Alternatively, we could say that this forced resonance occurs whenever the (real) frequency of the modes is very near to the (complex) frequency associated with the cavity free resonance.

The analogy to the case of the diffraction grating is a strong one, as we shall see. A plane wave traveling at an angle to the normal to the diffraction grating, as in Fig. 1, may be interpreted as an E mode or an H mode traveling parallel to this normal (perpendicular to the grating), depending on the polarization of the plane wave. Thus, the dominant and higher propagating modes traveling vertically in the waveguide of Fig. 4 correspond precisely to the incident, reflected, and diffracted waves associated with the diffraction grating in Fig. 1. The termination at plane T corresponds, of course, to the top surface of the diffraction grating. To pursue the analogy further, one must examine the nature of the terminations. What in the diffraction grating corresponds to the resonant cavity hidden below the plane T in Fig. 4?

We must recall that the condition for a guided wave to exist along a structure, open or closed, is that a transverse resonance must be satisfied. For a guided wave to be present along the diffraction grating, the conditions must, therefore, be proper for a resonance in the plane perpendicular to it. This transverse resonance is, furthermore, a free resonance. If the period of the diffraction grating is small, no diffracted spectral orders will be excited upon plane-wave incidence. Correspondingly, a guided wave along the grating, if excited by an appropriate source, would be a surface (purely bound) wave. When the period of the diffraction grating is increased sufficiently so that one or more diffracted spectral orders will be created on plane-wave incidence, the grating can no longer support a purely bound guided wave, but only a leaky wave with one or more radiating space harmonics. This leaky (complex) wave possesses a complex wavenumber. The resonant cavity hidden below the plane T in Fig. 4, therefore, corresponds precisely to the hidden transverse resonance which would produce a leaky wave along the grating. It is also to be noted that the wavenumbers in both cases are complex, while those of the incident waves and modes are all real.

The resonance form of the Wood anomaly is thus seen to occur when the frequency is such that either the reflected beam or one of the diffracted beams appears at the same angle as would be taken by the leaky wave which is supportable by the grating. Consider the situation shown schematically in Fig. 5, in which the fields of the diffracted plane wave in (a) are strikingly similar to those of the leaky wave in (b). As indicated above, the fields are not identical since the plane waves possess real wavenumbers while the wavenumber of the leaky wave is complex. For this reason, the scattered field cannot excite the leaky wave, but a forced resonance occurs which produces a rearrangement of the amplitudes of the various beams present in Fig. 5(a). The numbering of the various beams follows from certain details in the theory of Sec. III. However, n = 0 corresponds to the reflected plane wave, while n′ = 0 represents the basic slow guided wave along the grating. The other n′ values represent radiating space harmonics.

It is of interest to note that for this example the n = 1 nonpropagating spectral order of the plane-wave couples to the (dominant) n′ = 0 nonpropagating space harmonic of the leaky wave. The n = 1 spectral order actually exhibits a very sharply peaked increase in amplitude during the resonance. It is shown in Sec. III that this is the only spectral order to possess this form of amplitude response. The others exhibit a maximum on one side of the resonance and a minimum on the other, as one finds experimentally for the propagating spectral orders.

The relation discussed in the foregoing between the resonance type of anomaly and the capability of the grating to support a leaky wave affords a ready explanation of the P anomalies. It was found experimentally[2],[14],[15] that P anomalies occur for gratings with deep grooves, but not shallow grooves. If we examine the conditions under which H-mode (corresponding to P polarization) guided waves can be supported by a corrugated surface, we find that the groove depth l must satisfy approximately the following stipulation:

\frac{n λ_{g}}{2} > l > (2 n - 1) \frac{λ_{g}}{4}, n = 1, 2, . . . .

This requirement on the groove depth is equivalent to specifying a capacitive surface for the H-mode guided waves, in contrast to the inductive surface needed for the E-mode guided waves (corresponding to S polarization). Shallow grooves (l < λ_g/4) produce an inductive surface. Hence, the P anomalies will not appear when the grating possesses shallow grooves, but only when the groove depth satisfies the above relation.

III. Scattering by Periodically Modulated Reactance Surfaces

As a general model of a scattering structure with periodic properties, we choose a plane which possesses a periodically modulated surface reactance. Such a planar surface is an idealization of a large variety of scattering structures, including metallic reflection gratings with either shallow or deep grooves, dielectriccoated metallic gratings, periodically grooved dielectric layers, etc. The variation in the surface reactance as a function of frequency, angle of incidence, position on the surface, etc., takes into account the specific characteristics of the structure to be represented. The distinct virtue of the surface reactance approach is that the scattered field can be represented above the reactance plane without approximation by outgoing waves alone, since the standing waves which are present within the grating grooves are actually accounted for by an appropriate choice of surface reactance variation. This procedure thus retains the simplicity, enjoyed by Rayleigh[10] and Artmann,[12] of working with outgoing waves only, but does not restrict the solution to gratings with shallow grooves.

The periodically modulated reactance plane under consideration is assumed to extend infinitely in the x and y directions, with period d in the x direction only, as shown in Fig. 6. A plane wave with its magnetic field parallel to the y axis is incident on this plane at angle θ with respect to the normal to the plane. The magnetic field of this incident wave of S polarization is, therefore,

H_{i} (x, z) = H e^{i k_{s} x} e^{- i_{κ_{0} z}},

where

κ_{0} = {(k^{2} - {k_{s}}^{2})}^{1 / 2},

and the wavenumbers k, k_s, and κ₀ are the free-space wavenumber and its components along the surface and normal to it, respectively, as seen from Fig. 6. The relation between k_s and the angle θ of incidence is seen to be

k_{s} = k sin θ .

We also note that the time dependence exp(-iwt) is suppressed in (1), and that the scattered field must be independent of y.

The total field is derivable from the single nonvanishing y component of magnetic field, and it satisfies at the grating plane z = 0 the periodic impedance boundary condition

Z^{s} (x) = - \frac{E_{x} (x, 0)}{H_{y} (x, 0)} .

The surface impedance Z^s(x) is a periodic reactance function which we shall assume to be representable within a given period d by the Fourier series expansion

Z^{s} (x) = ∑_{ν = - \infty}^{\infty} {Z_{ν}}^{s} e^{i (2 π / d) ν x},

with

{Z^{s}}_{- ν} = - {Z^{s *}}_{ν} .

Representation (5) for the surface impedance is valid for a large number of structures, such as grooved gratings made of metal or dielectric, but may not be valid for others, such as a zero-thickness metal strip grating, where the strips lie in the xy plane. If expansion (5) is valid, boundary condition (4) takes into account the standing waves within the grooves so that the scattered field can be represented for z ≥ 0 in terms of outgoing waves only.

We shall also postulate that: (a) to ensure that the fields remain finite, all coefficients Z_ν^s are bounded. They are also analytic functions of the groove depth or the modulation parameter as either of these quantities approaches zero, and of k_s and k [save for isolated branch point singularities, which occur at κ_n = 0, where κ_n is defined by (11), and which correspond physically to the location of the Rayleigh wavelengths], (b) as the groove depth or the modulation parameter approaches zero, so that the periodic structure approaches a uniform one, all Z_ν^s with υ ≠ 0 tend to zero, (c) in this limit in which the periodic grating becomes a uniform structure Z₀^s may also approach zero or it may remain finite and become equal to an average surface impedance value Z^s₀₀, (d) if Z^s₀₀ ≠ 0 a surface wave can be supported which propagates along the z = 0 plane in the x direction. The properties of this surface wave then depend upon the value of the average surface impedance Z^s₀₀. The propagation characteristics are determined by the transverse resonance, or dispersion, relation

{Z^{s}}_{00} + \frac{κ}{ω ∊} = 0,

where

κ = {(k^{2} - {k_{s}}^{2})}^{1 / 2} = i {({k_{s}}^{2} - k^{2})}^{1 / 2} = i α, α > 0 .

The magnetic field of this E-mode surface wave is then given by

H_{s w} = H_{0} e^{i k_{s} x} e^{- α z}, z \geq 0,

so that the field is purely bound to the grating surface. One realizes, of course, that such a surface wave cannot be excited by an incident plane wave, since the k_s for a surface wave must be greater than k while that for the plane wave at any real incidence angle is less than k.

Returning to the case of plane-wave incidence on the periodic surface, the total magnetic field H_y(x, z) satisfies the scalar-wave equation

(\nabla^{2} + k^{2}) H_{y} = 0, z > 0,

together with the variable impedance boundary condition (4) at z = 0, and the radiation condition for its scattered portion H_s as z → ∞. Due to periodicity, H_s(x, z) can be represented in the form

H_{s} (x, z) = ∑_{n = - \infty}^{\infty} I_{n} (k_{s}) e^{i [k_{s} + (2 π n / d)] x} e^{i κ_{n} z}, for z \geq 0, n = 0, \pm 1, \pm 2, . . .,

where I_n are the amplitudes of the various spectral orders, k_s is determined by the incident wave, and

κ_{n} = {[k^{2} - {(k_{s} + \frac{2 π n}{d})}^{2}]}^{1 / 2} .

For k² > [k_s + (2πn/d)]², κ_n is positive real, and

κ_{n} = i {[{(k_{s} + \frac{2 π n}{d})}^{2} - k^{2}]}^{1 / 2}, for k^{2} < {(k_{s} + \frac{2 π n}{d})}^{2} .

With the help of Maxwell’s equations, (4) may be written in terms of magnetic field only, as

{[Z^{s} (x) H_{y} (x, z) + \frac{1}{i ω ∊} \frac{\partial H_{y} (x, z)}{\partial z}]}_{z = 0} = 0 .

When expansions (10) and (5) are inserted together with (1) into (12), the latter becomes

∑_{n = - \infty}^{\infty} \frac{κ_{n}}{ω ∊} {\bar{\bar{I}}}_{n} e^{i (2 π / d) n x} - 2 H \frac{κ_{0}}{ω ∊} + ∑_{m = - \infty}^{\infty} ∑_{ν = - \infty}^{\infty} {Z_{ν}}^{s} Î_{m} e^{i (2 π / d) (ν + m) x} = 0,

where we have written

\begin{array}{l} Î_{0} & = I_{0} + H, \\ Î_{m} & = I_{m}, m \neq 0 . \end{array}

We may rewrite (13) as

∑_{n = - \infty}^{\infty} ∑_{m = - \infty}^{\infty} (Z_{n - m}^{s} + \frac{κ_{n}}{ω ∊} δ_{n m}) Î_{m} e^{i (2 π / d) n x} = \frac{2 κ_{0}}{ω ∊} H .

Using the orthogonality properties of the exponential over the period d, one then obtains an infinite set of simultaneous linear equations for the scattered field amplitudes Î_m as

∑_{m = - \infty}^{\infty} ({Z_{n - m}}^{s} + \frac{κ_{n}}{ω ∊} δ_{n m}) Î_{m} = \frac{2 κ_{0}}{ω ∊} H δ_{n 0},

with

\begin{array}{l} δ_{n m} & = \begin{matrix} 0, m \neq n, \\ 1, m = n, \end{matrix} \\ n & = 0, \pm 1, \pm 2, . . ., \end{array}

and Î_m defined by (14). Equations (15) may also be put into the matrix form:

[\begin{array}{l} ⋱ & ⋮ & ⋰ \\ Z_{0}^{s} + \frac{κ_{2}}{ω ∊} & Z_{1}^{s} & Z_{2}^{s} & Z_{3}^{s} & Z_{4}^{s} \\ Z_{- 1}^{s} & Z_{0}^{s} + \frac{κ_{1}}{ω ∊} & Z_{1}^{s} & Z_{2}^{s} & Z_{3}^{s} \\ . . . & Z_{- 2}^{s} & Z_{- 1}^{s} & Z_{0}^{s} + \frac{κ_{0}}{ω ∊} & Z_{1}^{s} & Z_{2}^{s} & \dots \\ Z_{- 3}^{s} & Z_{- 2}^{s} & Z_{- 1}^{s} & Z_{0}^{s} + \frac{κ_{- 1}}{ω ∊} & Z_{1}^{s} \\ Z_{- 4}^{s} & Z_{- 3}^{s} & Z_{- 2}^{s} & Z_{- 1}^{s} & Z_{0}^{s} + \frac{κ_{- 2}}{ω ∊} \\ ⋰ & ⋮ & ⋱ \end{array}] [\begin{matrix} ⋮ \\ Î_{2} \\ Î_{1} \\ Î_{0} \\ Î_{- 1} \\ Î_{- 2} \\ ⋮ \end{matrix}] = [\begin{matrix} ⋮ \\ 0 \\ 0 \\ \frac{2 κ_{0}}{ω ∊} H \\ 0 \\ 0 \\ ⋮ \end{matrix}] .

The set of Equations (15) may be expressed in the compact form

(Z) Î = V .

The amplitudes Î_n of the propagating or nonpropagating spectral orders may be obtained from

Î_{n} = \frac{Δ_{n}}{Δ},

where Δ is the determinant of (Z) in (17), and Δ_n is the formal infinite determinant which is produced by replacing the nth column in Δ by the column vector V of (17). The specific forms taken by Δ and Δ_n may be seen from an inspection of (16). Although neither determinant exists without the inclusion of appropriate convergence factors, we know physically that their ratio must exist.

In seeking the occurrence of the Wood anomalies, we recall that they manifest themselves as rapid changes in the amplitudes Î_n, or, more strictly, in the amplitudes I_n, which are related to Î_n by (14) and which are defined by (10). These anomalies are commonly assumed to be associated with the Rayleigh wavelengths only, but, as we show below, they are of two different types and one type may occur at wavelengths far removed from the Rayleigh wavelengths, if the structure is appropriate. The two types are associated, respectively, with the following conditions:

(a) D_{n} = {Z_{0}}^{s} + \frac{κ_{n}}{ω ∊} = 0,

(b) κ_{n} = {[k^{2} - {(k_{s} + \frac{2 π n}{d})}^{2}]}^{1 / 2} = 0 .

Condition (a) is satisfied in the neighborhood of a resonance form of anomaly, while condition (b) represents the branch point singularities occurring at the Rayleigh wavelengths. The influence on the Î_n terms due to condition (b) arises in view of the infinite derivative associated with the square root; this effect will be considered later. The recognition that anomalies of the resonance type may occur quite independently of the Rayleigh wavelength is one of the contributions of this new theoretical approach, and, for this reason, more attention will be paid to the resonance anomalies, and they will be considered first.

The appearance of resonance anomalies is associated with the existence of solutions of the homogeneous equations, and, therefore, with the vanishing of determinant Δ, defined in (18). Such solutions correspond to guided waves that may be supported by the structure if an appropriate excitation were furnished. The effect of such guided waves on plane-wave scattering was discussed qualitatively in Sec. II-B. For a given, specific structure, one can readily determine whether or not such guided waves can be established and, if so, for what sets of parameter values. From such knowledge one can then predict the location and Q (as we shall see below) of the resonance anomalies. It is difficult, however, to prove the existence of roots of Δ for a Fourier series representation of an arbitrary periodic surface impedance [Eq. (5)]. However, one can show under certain assumptions, such existence in the limit of zerogroove depth or modulation amplitude, and then analytically continue such solutions into the range of the periodic modulation, at least for small values of groove depth or modulation amplitude.

Let us note first that all of the diagonal terms of the matrix (Z) in (16) are of the form

{Z^{s}}_{0} + \frac{κ_{n}}{ω ∊},

so that the relation to condition (a) will soon be evident. We then recall that, in the limit of groove depth or modulation parameter approaching zero, the impedance component Z₀^s approaches the average surface reactance value Z₀₀^s. Hence, in this limit of vanishing periodicity, the matrix (Z) becomes diagonal with a typical term

{Z^{s}}_{00} + \frac{κ_{n}}{ω ∊},

which, in view of (7), will vanish for an appropriate combination of parameters. The vanishing of term (21) implies a surface wave field

H_{s w} (x, z) = H_{0} e^{i k_{s} x} e^{i κ_{n} z} = H_{0} e^{i k_{s} x} exp (- i ω ∊ {Z^{s}}_{00} z),

which is independent of n. Hence, the vanishing of term (21) expresses the same resonance condition for any value of n.

One readily verifies that the usual convergence difficulties characteristic of a resonance region are obtained when a solution in a Neumann series form is attempted. If one writes for the matrix (Z) of (17)

(Z) = (D + Z^{'}),

where

{(D)}_{i j} = ({Z_{0}}^{s} + \frac{κ_{i}}{ω ∊}) δ_{i j},

and premultiplies (17) by the inverse of the diagonal matrix D, one has

(1 + D^{- 1} Z^{'}) Î = {(D)}^{- 1} V,

which shows that the bound of (D⁻¹Z′) near condition (19) is very large, if not infinite, so that the Neumann series approach fails there. One notes, however, that if (Z′) is bounded and if Z^s₀₀ ≠ 0 the convergence difficulties no longer appear at the Rayleigh wavelength, and the field can be calculated there by means of a Neumann series if Z^s₀₀ is not too small.

We wish next to demonstrate that resonant solutions of the field do exist, and that the reason for these convergence difficulties lies in the presence of a complex pole of Î_n in the neighborhood of condition (19). We proceed by recognizing that such resonant solutions are given by the vanishing of determinant Δ(k,k_s,l), as defined in (18), where l signifies either the groove depth or a modulation parameter. Then, because of the assumed analyticity of Z_ν^s in l near l = 0, we note that Δ(k, k_s,l), being the limit of an infinite sequence of analytic functions, is itself an analytic function. Also, since the off-diagonal terms Z_ν^s of matrix (Z) vanish as l approaches zero, it follows that Δ(k,k_s,0) is simply the product of all of the diagonal terms. If proper non-vanishing convergence factors f_ν are introduced, we may write

Δ (k, k_{s}, 0) = \frac{∏_{ν = - \infty}^{\infty} ({Z^{s}}_{00} + \frac{κ_{ν}}{ω ∊})}{∏_{ν = - \infty}^{\infty} f_{ν}} .

Since in this limit of vanishing periodicity the z = 0 plane can support a surface wave if Z₀₀^s ≠ 0, one of the terms in (22) of the form (21) will vanish for an appropriate combination of k_s and k. Therefore, the equation

Δ (k, k_{s}, 0) = 0

will possess a solution k⁰_s for a given value of k⁰, where the zero superscript signifies the value in the limit of vanishing periodicity. Since Δ(k,k_s,l) is analytic in k_s and l near l = 0, we have that the determinantal equation

Δ (k^{0}, k_{s}, l) = 0

will, in general, possess, by the implicit function theorem, for not too large values of l, a unique solution k_s located near k_s⁰.

If Z₀₀^s in (22) were equal to zero, the vanishing of form (21) would imply that Δ(k,k_s,0) would still possess a zero, but Δ(k,k_s,l) would no longer be analytic near l = 0 because of the branch point singularity at κ_n = 0. In this case, one can perform a uniformizing transformation by setting

t = {[k^{2} - {(k_{s} + \frac{2 π n}{d})}^{2}]}^{1 / 2} .

With this substitution,

Δ (k, k_{s}, l) = \hat{Δ} (k, t, l)

becomes an analytic function of t near t = 0, and, since

\hat{Δ} (k^{0}, t^{0}, 0) = 0

possesses roots, we are assured, via the implicit function theorem, of the existence of solutions of

\hat{Δ} (k^{0}, t, l) = 0

if l is not too large.

We must next recognize that these roots of Δ(k,k_s,l) are complex, and, therefore, represent complex poles in the field solution. It has been shown[25] that when kd is such that a plane wave incident upon a grating excites one or more propagating spectral orders in addition to the reflected wave, the propagation wavenumber k_sp (where the subscript p signifies pole) of a guided wave supportable by the same grating at the same frequency must be complex. The guided wave wavenumber can no longer be purely real, as at lower frequencies or closer periodic spacings, because under these conditions one of the space harmonics of the guided wave becomes radiating, and what was a purely bound surface wave now becomes a leaky (complex) wave.

A consequence of the complex nature of this guided wave, and, correspondingly, the complex roots k_sp of (23) or (24), is that the spectral-order amplitudes Î_n, in the light of (18), can never become infinite due to plane-wave excitation since the values of (k_s + 2πn/d) due to such excitation are always real. However, when the (k_s + 2πn/d) value of the plane-wave spectral order becomes approximately equal to the real part of the complex root k_sp, the denominator Δ in (18) becomes small (although not zero) and varies rapidly so that the value of Î_n becomes correspondingly large and rapidly varying. The resulting behavior is a resonant one, and corresponds to what we have termed the resonance anomaly. Since the vanishing of form (21) and the condition (19) are very similar to each other, at least for small values of modulation, we see that satisfaction of condition (19) yields the neighborhood in which a resonance anomaly can be expected.

Let us next examine a bit more closely the form of the variation in the various Î_n’s in the neighborhood of a resonance anomaly. We assume that the determinant Δ has a zero

k_{s p} = β + i α,

and that for a particular angle of incidence θ, which corresponds to a given value of k_s from (3), the nth spectral order has the value

k_{s} + \frac{2 π n}{d} ≃ β .

We can then expand Δ in a Taylor series around k_s = k_sp. Retaining only the linear term, we have for the nth spectral order amplitude

Î_{n} = I_{n} = \frac{Δ_{n}}{Δ} = \frac{Δ_{n} (k_{s})}{(k_{s} - β - i α) g (k_{s})} .

In the neighborhood of k_s = k_sp, the ratio Δ_n(k_s)/g(k_s) can be considered essentially constant (say equal to C_n), because the term D_n which becomes equal to zero in view of (19) has been eliminated from Δn. Hence, (27) may be written as

I_{n} ≃ \frac{C_{n}}{k_{s} - β - i α}

or

| I_{n} | ≃ \frac{| C_{n} |}{\sqrt{{(k_{s} - β)}^{2} + α^{2}}} .

Expression (29) is seen to result in a standard resonance behavior for the magnitude of I_n, which is seen from (10) to be the complex amplitude of the magnetic field of the nth spectral order. Since this spectral order is the one which couples to the complex guided wave which would be supported by the grating if it were excited, let us call it the resonant spectral order. The additional effect of the resonance is to redistribute the scattered energy among the various other spectral orders, and, in particular, among the observable propagating ones. It should be recognized, of course, that the resonant spectral order is always evanescent (nonpropagating).

The Q factor of this resonance follows directly from (29), and is seen to be simply

Q = \frac{β}{2 α} .

If the complex guided wave which could be supported by the grating would leak energy very slowly along its length, α would be small and the Q would be high. If the grating consists of a modulated dielectric slab, for example, the Q would increase when the modulation amplitude is decreased.

An entirely different behavior is exhibited by the magnitudes of the other spectral orders. Let us consider, for example, I_n₋₁. The determinant Δ_n₋₁ is obtained in the usual fashion by replacing the column in Δ containing D_n₋₁ by the vector V. In this case, however, the term D_n, which is the one set equal to zero in view of (19), is retained in Δ_n₋₁. Hence, Δ_n₋₁ possesses a zero near to the value of k_s given by (26); however, this value of k_s is in general complex and differs from k_sp. In order to obtain an approximate expression for I_n₋₁, we can expand Δ_n₋₁ in a Taylor series about k_s = β₁ + iα₁, the new root under discussion. Determinant Δ is treated as before. We, therefore, find for the (n − 1)th spectral order amplitude

Î_{n - 1} = I_{n - 1} = \frac{Δ_{n - 1}}{Δ} ≃ C_{n - 1} \frac{(k_{s} - β_{1} - i α_{1})}{(k_{s} - β - i α)}

or

| I_{n - 1} | ≃ C_{n - 1} {[\frac{{(k_{s} - β_{1})}^{2} + {α_{1}}^{2}}{{(k_{s} - β)}^{2} + α^{2}}]}^{1 / 2} .

Expression (31) contains the product of terms involving a simple complex pole and a first-order complex zero, with the pole and zero near to each other but not at identical locations. Viewed in this way, it is easy to see that (32) represents a function which has a maximum and a minimum in its proximity. Which occurs first depends on the relative values of β and β₁. Form (32) is indeed the one which is found experimentally for the propagating spectral orders. The occurrence of both maxima and minima has been noted by Twersky, and later by Millar.

From the meaning of β₁, we may write

\frac{β_{1}}{k} = sin θ_{min},

where θ_min is angle corresponding to the minimum in the plot of |I_n₋₁|vs θ. If α₁ = 0, as can occur when only a few propagating spectral orders are present (examples are shown in Sec. IV), then θ_min represents a null in the plot. The ratio β/k does not, however, yield the maximum in this plot, since β corresponds to a complex pole. The location of this maximum follows readily from (32), by taking

\frac{\partial {| I_{n - 1} |}^{2}}{\partial k_{s}} = 0

and solving for k_s. One finds readily that

(k_{s} - β_{1}) (k_{s} - β) (β_{1} - β) = (k_{s} - β) {α_{1}}^{2} - (k_{s} - β_{1}) α^{2} .

When α₁ = 0, we see that k_s − β₁ = 0 affirms the fact that we have a null at the minimum, and we also find the following simple relation for the location of the maximum:

\frac{k_{s_{max}}}{k} = sin θ_{max} = \frac{β}{k} + \frac{{(\frac{α}{k})}^{2}}{\frac{β}{k} - \frac{β_{1}}{k}} .

When α₁≠0, we see from (34) that in general k_s_max is obtained from the solution of a quadratic.

Whether the minimum or the maximum intensity of a given propagating spectral order at a resonance anomaly occurs nearer to the normal to the grating depends on whether β is greater than or less than β₁, respectively. It is difficult to prove in general terms which behavior will result, but we conjecture that it depends upon whether the resonant (evanescent) spectral order couples to a guided wave which propagates in the same direction or in the opposite one. Justification for this assertion is given in connection with the example treated in detail in Sec. IV.

So far, we have discussed the properties of the resonance type of anomaly. We consider next the second type, given by condition (20), which is associated with the Rayleigh wavelength. The wavenumber κ_n in the direction normal to the grating bears a square root relation to k and k_s [as seen from (20)], and when the nth spectral order either just emerges or just reenters at grazing angle, κ_n becomes zero. Those terms containing the derivative of κ_n thus may become divergent, in view of the square root form for κ_n. We are actually concerned with the variation of spectral order amplitude I_n with angle or frequency. If dependence on angle is considered, we may write

\frac{\partial | I_{n} |}{\partial k_{s}} = \frac{\partial}{\partial k_{s}} | \frac{Δ_{n}}{Δ} | = | \frac{Δ \frac{\partial Δ_{n}}{\partial k_{s}} - Δ_{n} \frac{\partial Δ}{\partial k_{s}}}{Δ^{2}} | .

Since some of the terms in both Δ and Δ_n contain κ_n, certain of the terms in the derivatives of these determinants may contain the vanishing square root in their denominators. Hence, it is entirely possible that the curve of |I_n| vs θ will have an infinite slope at the Rayleigh wavelength, although we do not seem to be able to determine whether or not such behavior is assured. In Sec. IV, we demonstrate in a particular case that an infinite slope does indeed occur.

Let us now consider the type of grating on which most measurements of Wood’s anomalies have been performed, namely, a metallic grating with shallow grooves. Corresponding to S polarization, for which strong anomalies have been reported, the grating can support a guided wave, and condition (19) yields the neighborhood in which a resonance anomaly can be expected. For such a grating, however, the value of Z₀^S is near to zero, in view of the shallow grooves and the fact that a short-circuiting plane is obtained if the groove depth is reduced to zero. As a result, we note that conditions (19) and (20) are not much different from each other, and that they become equal when the groove depth is made vanishingly small. Therefore, for such a grating the resonance anomaly occurs near to the Rayleigh wavelength, and the resonance effect may be enhanced by the infinite slope effect, if present, resulting from condition (20). The Wood’s anomalies which are customarily observed are therefore a superposition of these two effects. As a result, the variation with angle may be somewhat more complicated in form but it is generally also more pronounced.

If the plane wave is incident with P, rather than S, polarization, the equations are set up formally in the same fashion, but the diagonal terms in (16) now have the form

D_{n} \equiv {Z^{s}}_{0} + \frac{ω μ}{κ_{n}} .

It is again possible to obtain anomalies of the resonance type for values of Z₀^s for which (37) vanishes. This stipulation is analogous to condition (19) for S polarization. Since, under these conditions, κ_n is imaginary (evanescent), Z₀^s must be inductive for S polarization, but capacitive for P polarization. For ordinary metallic gratings with shallow grooves, Z₀^s is close to zero and inductive. It is, therefore, clear that resonance anomalies will not appear for P polarization on such structures.

A dual form of expansion is possible which utilizes a phrasing in terms of the surface admittance Y^s(x), rather than the surface impedance Z^s(x). This dual formulation results in a matrix relation dual to (16) in which the diagonal terms are

{Y^{s}}_{0} + \frac{κ_{n}}{ω μ},

and for which the neighborhood for resonance anomalies is given by the vanishing of (38). For surface admittance values Y₀^s which are small but capacitive, resonance anomalies may be observed near the Rayleigh wavelengths, and such anomalies should be rather pronounced. Gratings with grooves which are slightly greater than λ_g/4 or 3λ_g/4 in depth (the phase shift introduced at the interface modifies this value somewhat) furnish an example of a structure which exhibits such a surface admittance.

IV. Example: Scattering by a Sinusoidally Modulated Reactance Plane

The general theoretical treatment in Sec. III shows that Wood’s anomalies are actually of two distinct types, a resonance type and a form which appears at the Rayleigh wavelengths due to the emergence or reentry of another propagating spectral order. It is also shown that while these two types are often almost merged together (e.g., in metallic reflection gratings with shallow grooves), they need not be, and that when the average surface reactance Z₀₀^s in the limit of vanishing periodicity is unequal to zero, the resonance form of anomaly generally occurs away from the Rayleigh wavelengths. Various other statements were made in general terms regarding the existence of complex poles in the field solution, and the location and shape, including the Q, of resonance anomalies. In the present section, a rigorous solution is given for a special case, a sinusoidal form of periodic modulation, and the results obtained are shown to be in agreement with the general theoretical predictions. The results of numerical calculations are presented in graphical form for a variety of different situations, including different numbers of propagating spectral orders, the behavior of both the resonant and the propagating spectral orders, and an illustration of a double anomaly.

A. Formal Solution

We assume the presence of a periodically modulated reactance sheet of the form shown in Fig. 6, and we employ the notation used in Sec. III. However, for convenience, and in order to permit a rigorous solution to be obtained, we specialize the general surface impedance function of (5) to one sinusoidal in form. The surface impedance variation for the structure under investigation may be written as

Z^{s} (x) = Z_{s} (1 + M cos \frac{2 π x}{d}),

where Z_s = −iX_s, representative of an inductive reactance, M is the modulation index, and d is the period along the x direction on the surface. For simplicity, we also specify that Z_s is a constant, independent of frequency. Comparison with (5) indicates that the parameters of (39) are related to the Fourier series expansion (5) by

\begin{array}{c} {Z^{s}}_{0} & = Z_{s} = {Z^{s}}_{00} \\ {Z^{s}}_{1} & = {Z^{s}}_{- 1} = \frac{Z_{s} M}{2} \\ {Z^{s}}_{ν} & = 0, ν \neq 0, \pm 1 . \end{array}

A plane wave of S polarization is again assumed to be incident at angle θ on this surface, as shown in Fig. 6, with the magnetic field of this incident wave given by (1). Following the general approach outlined in Sec. III, we find for the matrix form corresponding to (16):

[\begin{array}{c} ⋱ & ⋮ & ⋰ \\ Z_{s} + \frac{κ_{2}}{ω ∊} & \frac{Z_{s} M}{2} & 0 & 0 & 0 \\ \frac{Z_{s} M}{2} & Z_{s} + \frac{κ_{1}}{ω ∊} & \frac{Z_{s} M}{2} & 0 & 0 \\ . . . & 0 & \frac{Z_{s} M}{2} & Z_{s} + \frac{κ_{0}}{ω ∊} & \frac{Z_{s} M}{2} & 0 & \dots \\ 0 & 0 & \frac{Z_{s} M}{2} & Z_{s} + \frac{κ_{- 1}}{ω ∊} & \frac{Z_{s} M}{2} \\ 0 & 0 & 0 & \frac{Z_{s} M}{2} & Z_{s} + \frac{κ_{- 2}}{ω ∊} \\ ⋰ & ⋮ & ⋱ \end{array}] [\begin{matrix} ⋮ \\ Î_{2} \\ Î_{1} \\ Î_{0} \\ Î_{- 1} \\ Î_{- 2} \\ ⋮ \end{matrix}] = [\begin{matrix} ⋮ \\ 0 \\ 0 \\ \frac{2 κ_{0}}{ω ∊} H \\ 0 \\ 0 \\ ⋮ \end{matrix}]

The terms Î_n are the modified diffracted amplitudes, related to I_n by (14); I_n is defined in terms of the magnetic field by (10). Because of the sinusoidal form of the surface impedance variation, the equations in (41) are seen to reflect only nearest neighbor interaction, and to be in the form of a three-term recursion relation except for the n = 0 equation, which involves the driving term. These equations may be rewritten as

Î_{n + 1} + {\bar{D}}_{n} Î_{n} + Î_{n - 1} = 0, n = 1, 2, . . . and n = - 1, - 2, . . .

Î_{1} + {\bar{D}}_{0} Î_{0} + Î_{- 1} = V_{0},

with

\begin{array}{c} {\bar{D}}_{n} & = \frac{2}{M} {1 + \frac{1}{{Z_{s}}^{'}} {[1 - {(\frac{k_{s}}{k} + \frac{2 π n}{k d})}^{2}]}^{1 / 2}} \\ = \frac{2}{M} {1 + \frac{1}{{Z_{s}}^{'}} {[1 - {(sin θ + \frac{n λ}{d})}^{2}]}^{1 / 2}} \end{array}

V_{0} = \frac{4}{M} \frac{1}{{Z_{s}}^{'}} {[1 - {(\frac{k_{s}}{k})}^{2}]}^{1 / 2} = \frac{4}{M} \frac{cos θ}{{Z_{s}}^{'}},

where

{Z_{s}}^{'} = \frac{Z_{s}}{\sqrt{μ / ∊}} .

The solution of the inhomogeneous infinite set of linear Equations (42) and (43) is obtained by first solving the homogeneous equations (42) and then substituting these results into the only inhomogeneous Equation (43). From Eq. (42) with positive n, one can derive the recursion formula

\frac{Î_{n}}{Î_{n + 1}} = - \frac{1}{{\bar{D}}_{n} + \frac{Î_{n + 1}}{Î_{n}}}, n = 1, 2, . . .,

which, when iterated, results in the ratio of two consecutive amplitudes in the following continued fraction forms:

\frac{Î_{n}}{Î_{n - 1}} = - \frac{1 |}{{\bar{D}}_{n}} - \frac{1 |}{| {\bar{D}}_{n + 1}} - \frac{1 |}{| {\bar{D}}_{n + 2}} - . . . = A_{n}, n = 1, 2, . . .,

which for n = 1 becomes

\frac{Î_{1}}{Î_{0}} = - \frac{1 |}{{\bar{D}}_{1}} - \frac{1 |}{| {\bar{D}}_{2}} - \frac{1 |}{| {\bar{D}}_{3}} - . . . = A_{1} .

Similarly, one has from Eq. (42) for negative n

\frac{Î_{n}}{Î_{n + 1}} = - \frac{1 |}{{\bar{D}}_{n}} - \frac{1 |}{| {\bar{D}}_{n - 1}} - \frac{1 |}{| {\bar{D}}_{n - 2}} - . . . = B_{n}, n = - 1, - 2, . . .,

and

\frac{Î_{- 1}}{Î_{0}} = - \frac{1 |}{{\bar{D}}_{- 1}} - \frac{1 |}{| {\bar{D}}_{- 2}} - . . . = B_{- 1} .

If

| {\bar{D}}_{n} | > 2

for all sufficiently large values of n, these continued fractions may be shown[28] to converge absolutely and uniformly. This sufficiency condition is satisfied here, as seen from (44), for all values of kd and M. One also verifies by direct substitution that (48) and (50) satisfy (42) for positive and negative n, respectively. By substituting (48) and (50) into (43), one finds:

Î_{0} = \frac{V_{0}}{{\bar{D}}_{0} + A_{1} + B_{- 1}},

Î_{1} = Î_{0} A_{1}

Î_{n} = Î_{0} ∏_{ν = 1}^{n} A_{ν}, n = 1, 2, . . .

Î_{- 1} = Î_{0} B_{- 1}

Î_{n} = Î_{0} ∏_{ν = - 1}^{n} B_{ν}, n = - 1, - 2, . . .

Thus, all of the complex spectral-order amplitudes Î_n, or I_n using (14), can be found. From (54) and (56), it is evident that Î_n ∼ 1/n!, as n → ∞, and, therefore, the series expansion (10) of the scattered field converges absolutely and uniformly in x and z.

After substituting (3) in (52) and (56), one finds upon rearrangement

\frac{Î_{0}}{H} = \frac{(2 / {Z_{s}}^{'}) cos θ}{1 + \frac{1}{{Z_{s}}^{'}} cos θ - \frac{M^{2}}{4} (\frac{1 |}{D_{1}} - \frac{M^{2} / 4 |}{| D_{2}} - . . . + \frac{1 |}{D_{- 1}} - \frac{M^{2} / 4 |}{| D_{- 2}} - . . .)}

or

\frac{I_{0}}{H} = \frac{\frac{cos θ}{{Z_{s}}^{'}} - 1 + \frac{M^{2}}{4} [\frac{1 |}{D_{1}} - \frac{M^{2} / 4 |}{| D_{2}} - . . . + \frac{1 |}{D_{- 1}} - \frac{M^{2} / 4 |}{| D_{- 2}} - . . .]}{\frac{cos θ}{{Z_{s}}^{'}} + 1 - \frac{M^{2}}{4} [\frac{1 |}{D_{1}} - \frac{M^{2} / 4 |}{| D_{2}} - . . . + \frac{1 |}{D_{- 1}} - \frac{M^{2} / 4 |}{| D_{- 2}} - . . .]}

Î_{1} = I_{1} = - [\frac{M / 2 |}{D_{1}} - \frac{M^{2} / 4 |}{| D_{2}} - . . .] Î_{0}

Î_{- 1} = I_{- 1} = - (\frac{M / 2 |}{D_{- 1}} - \frac{M^{2} / 4 |}{| D_{- 2}} - . . .) Î_{0},

while

\frac{I_{n}}{I_{n - 1}} = - \frac{M / 2 |}{D_{n}} - \frac{M^{2} / 4 |}{| D_{n + 1}} - . . ., n = 1, 2, . . .

and

\frac{I_{n}}{I_{n + 1}} = - \frac{M / 2 |}{D_{n}} - \frac{M^{2} / 4 |}{| D_{n - 1}} - . . ., n = - 1, - 2, . . .,

with

D_{n} = 1 + \frac{1}{{Z_{s}}^{'}} {[1 - {(sin θ + \frac{n λ}{d})}^{2}]}^{1 / 2} = \frac{M}{2} {\bar{D}}_{n} .

Relation (63) for D_n differs from the D_n used in (19) by the factor 1/Z_s. Expressions (57) to (62) present explicit solutions for the complex amplitudes of all of the spectral orders, whether propagating or not, in a simple and rapidly convergent continued fraction form. The parameters are seen to be, in addition to the usual free-space wavelength λ, angle of incidence θ and period d, the average surface impedance value Z_s′ normalized to the impedance of free space, and the modulation index M.

Relations (57) to (62) may alternatively be obtained by following the general method outlined in Sec. III. Let Δ represent the system determinant of matrix Eqs. (41) after removing the factor Z_s from each term. This Δ differs from the Δ used in Sec. III by this constant factor, and is written as

Δ = | \begin{array}{c} ⋱ & ⋮ & ⋰ \\ D_{2} & \frac{M}{2} & 0 & 0 & 0 \\ \frac{M}{2} & D_{1} & \frac{M}{2} & 0 & 0 \\ . . . & 0 & \frac{M}{2} & D_{0} & \frac{M}{2} & 0 & . . . \\ 0 & 0 & \frac{M}{2} & D_{- 1} & \frac{M}{2} \\ 0 & 0 & 0 & \frac{M}{2} & D_{- 2} \\ ⋰ & ⋮ & ⋱ \end{array} |,

where the D_n are those defined by (63). Let us calculate the expression for I₋₁ using this procedure. We then have

\frac{I_{- 1} = | \begin{array}{c} ⋱ & ⋰ \\ D_{2} & \frac{M}{2} & 0 & 0 & 0 & 0 \\ \frac{M}{2} & D_{1} & \frac{M}{2} & 0 & 0 & 0 \\ 0 & \frac{M}{2} & D_{0} & \bar{V} & 0 & 0 \\ 0 & 0 & \frac{M}{2} & 0 & \frac{M}{2} & 0 \\ 0 & 0 & 0 & 0 & D_{- 2} & \frac{M}{2} \\ 0 & 0 & 0 & 0 & \frac{M}{2} & D_{- 3} \\ ⋰ & ⋱ \end{array} |,}{Δ}

where

\bar{V} = \frac{2 κ_{0}}{ω ∊} \frac{H}{Z_{s}} .

Developing the numerator determinant along its n = − 1 column, we obtain

(67)

We now apply to both numerator and denominator the Laplace development along the partitions indicated in (67):

\frac{I_{- 1}}{\bar{V}} = \frac{- \frac{M}{2} | D_{- 2} D_{- 3} . . . | \cdot | D_{1} D_{2} . . . |}{Δ},

where |D_nD_n₊₁…| or |D_nD_n₋₁…| denote the principal minors, of

| \begin{array}{c} D_{n} \frac{M}{2} & . . . \\ \frac{M}{2} D_{n + 1} \\ ⋱ \end{array} | and | \begin{array}{c} D_{n} \frac{M}{2} & . . . \\ \frac{M}{2} D_{n - 1} \\ ⋱ \end{array} | .

Determinant Δ in (64) is developed similarly along the partition just above the n = − 2 row. One then obtains

\begin{array}{c} \frac{I_{- 1}}{\bar{V}} & = - \frac{\frac{M}{2} | D_{- 2} D_{- 3} . . . | \cdot | D_{1} D_{2} . . . |}{| D_{- 2} D_{- 3} . . . | | D_{- 1} D_{0} . . . | - \frac{M^{2}}{4} | D_{- 3} D_{- 4} . . . | | D_{0} D_{1} . . . |} \\ = \frac{- \frac{M}{2} \frac{| D_{- 2} D_{- 3} . . . |}{| D_{- 3} D_{- 4} . . . |} \cdot \frac{| D_{1} D_{2} . . . |}{| D_{0} D_{1} . . . |}}{\frac{| D_{- 2} D_{- 3} . . . |}{| D_{- 3} D_{- 4} . . . |} \cdot \frac{| D_{- 1} D_{0} . . . |}{| D_{0} D_{1} . . . |} - \frac{M^{2}}{4}} \end{array} .

Upon inspection, one observes that these ratios of determinants are identical with the continued fractions of expressions (57)–(62); for example,

\begin{array}{c} \frac{| D_{- 2} D_{- 3} . . . |}{| D_{- 3} D_{- 4} . . . |} & = \frac{D_{2} | D_{- 3} D_{- 4} . . . | - \frac{M^{2}}{4} | D_{- 4} D_{- 5} . . . |}{| D_{- 3} D_{- 4} . . . |} \\ = D_{- 2} - \frac{M^{2}}{4} \frac{| D_{- 4} D_{- 5} . . . |}{| D_{- 3} D_{- 4} . . . |} = D_{- 2} - \frac{M^{2}}{4} \frac{| D_{- 4} D_{- 5} . . . |}{D_{- 3} | D_{- 4} D_{- 5} . . . | - \frac{M^{2}}{4} | D_{- 5} D_{- 6} . . . |} \\ = D_{- 2} - \frac{M^{2} / 4 |}{D_{- 3}} - \frac{M^{2} / 4 |}{| D_{- 4}} - . . . \end{array}

Similarly, one finds

\frac{| D_{1} D_{2} . . . |}{| D_{0} D_{1} . . . |} = \frac{1 |}{D_{0}} - \frac{M^{2} / 4 |}{| D_{1}} - . . .

and

\frac{| D_{- 1} D_{0} . . . |}{| D_{0} D_{1} . . . |} = D_{- 1} - \frac{M^{2} / 4 |}{D_{0}} - \frac{M^{2} / 4 |}{| D_{1}} - . . .

Expression (68) thus becomes:

\frac{I_{- 1}}{\bar{V}} - \frac{- \frac{M}{2} (D_{- 2} - \frac{M^{2} / 4 |}{D_{- 3}} - \frac{M^{2} / 4 |}{| D_{- 4}} - . . .) (\frac{1 |}{D_{0}} - \frac{M^{2} / 4 |}{| D_{1}} - \frac{M^{2} / 4 |}{| D_{2}} - . . .)}{(D_{- 2} - \frac{M^{2} / 4 |}{D_{- 3}} - . . .) (D_{- 1} - \frac{M^{2} / 4 |}{D_{0}} - \frac{M^{2} / 4 |}{| D_{1}} - . . .) - \frac{M^{2}}{4}} .

After further rearrangements, we obtain

\frac{I_{- 1}}{H} = \frac{- \frac{M}{2} (\frac{1 |}{D_{- 1}} - \frac{M^{2} / 4 |}{| D_{- 2}} - . . .) \frac{2 κ_{0}}{ω ∊ Z_{a}}}{D_{0} - \frac{M^{2} / 4 |}{| D_{1}} - \frac{M^{2} / 4 |}{| D_{2}} - . . . - \frac{M^{2} / 4 |}{D_{1}} - \frac{M^{2} / 4 |}{D_{- 2}} - . . .}

which is actually identical to (60) when (57) is used for Î₀/H. Results identical to those of (57) through (62) are thus obtainable by means of the above-indicated direct expansion procedure.

Another rearrangement alters (69) into the following form:

\frac{I_{- 1}}{H} = \frac{- \frac{M}{2} (D_{- 2} - \frac{M^{2} / 4 |}{| D_{- 3}} - . . .) (\frac{1 |}{D_{0}} - \frac{M^{2} / 4 |}{| D_{1}} - . . .) (\frac{1 |}{D_{- 1}} - \frac{M^{2} / 4 |}{| D_{0}} - . . .) \frac{2 κ_{0}}{ω ∊ Z_{s}}}{(D_{- 2} - \frac{M^{2} / 4 |}{| D_{- 3}} - . . . - \frac{M^{2} / 4 |}{D_{- 1}} - \frac{M^{2} / 4 |}{| D_{0}} - \frac{M^{2} / 4 |}{| D_{1}} - . . .)}

Form (70) is valuable if the n = − 2 spectral order is the resonant one. We then see clearly that the positions of the pole and the minimum are slightly different from each other. We will return to this aspect later.

B. Some Theoretical Relations for the Detailed Behavior of the Anomalies

From the formal solution obtained above one can deduce information regarding the detailed nature of the Wood anomalies. It is too messy to attempt a solution valid for any spectral order n, so that we will restrict our considerations to n = −1, for which the relations are slightly simpler. In addition, we shall assume that only two spectral orders are propagating: the n = 0 and the n = − 1; the remainder are evanescent. We will first treat the resonance anomalies; we shall determine the location of the minima, the complex poles and the maxima, and then demonstrate the factors which determine whether the maximum or the minimum lies closer in angle to the normal. After that, consideration is given to the anomalous effect associated with the Rayleigh wavelength, and a case is presented for which an infinite slope is obtained at the Rayleigh wavelength in the variation of |I₋₁| with angle.

Suppose that we examine the behavior of the intensity of the n = −1 spectral order as a function of angle, when the resonant spectral order is n = −2. We may then use to advantage form (70) for I₋₁. This form permits us to estimate rapidly the positions of the minimum and the maximum of the resonance anomaly as a function of the physical parameters. From (70), we see that, since D₋₂ = 0 prescribes the neighborhood of the anomaly, the position of the minimum is given approximately by

D_{- 2} - \frac{M^{2} / 4}{D_{- 3}} = 0 .

Because the n = −2, −3, … spectral orders are all evanescent, the corresponding D_n values are real, so that (71) represents a null rather than simply a minimum. The complex pole of the resonance is given by the vanishing of the complete denominator of (70), but a reasonable approximation is afforded, for M not too large, by

D_{- 2} - \frac{M^{2} / 4}{D_{- 3}} - \frac{M^{2} / 4}{D_{- 1}} = 0 .

For additional accuracy the terms in D₀ and D₋₄ could be included. From Eqs. (71) and (72) one obtains the quantities β, β₁, β and α (α₁ = 0, of course, in this case), discussed in Sec. III. The maximum in the curve of intensity vs angle is then found from the simple relation (35).

We wish next to learn whether the maximum or the minimum is encountered first when scanning from the normal, say. Toward this end, we seek to determine the relative locations in angle of the null and the pole without explicitly solving (71) and (72). Since both the null and the pole occur close to the condition D₋₂ = 0, we shall expand around the wavenumber k_s⁰ corresponding to this condition. Since

D_{- 2} = 1 - \frac{1}{{X_{s}}^{'}} {[{(\frac{k_{s}}{k} - \frac{4 π}{k d})}^{2} - 1]}^{1 / 2},

the condition D₋₂ = 0 yields

\frac{{k_{s}}^{0}}{k} = - {(1 + {X_{s}}^{' 2})}^{1 / 2} + \frac{4 π}{k d},

where the negative sign before the square root must be chosen since k_s⁰/k must be less than unity. In expanding the square root of (73) about k_s⁰, and writing

k_{s} = {k_{s}}^{0} + δ,

one finds

{[{(\frac{{k_{s}}^{0}}{k} - \frac{4 π}{k d})}^{2} + \frac{2 δ}{k} (\frac{{k_{s}}^{0}}{k} - \frac{4 π}{k d}) - 1]}^{1 / 2} ≃ {X_{s}}^{'} + \frac{\frac{δ}{k} (\frac{{k_{s}}^{0}}{k} - \frac{4 π}{k d})}{{X_{s}}^{'}}

using (74). Thus, we have

D_{- 2} ({k_{s}}^{0} + δ) ≃ \frac{δ}{k} \frac{{(1 + {X_{s}}^{' 2})}^{1 / 2}}{{X_{s}}^{' 2}} .

Expression (76) can be used in a perturbation solution of (71) for the location of the null. Noting that D₋₃ is slowly varying, and writing δ_m as the appropriate angular shift δ corresponding to the minimum, we have

\frac{δ_{m}}{k} = \frac{\frac{M^{2}}{4} \frac{{X_{s}}^{' 2}}{{(1 + {X_{s}}^{' 2})}^{1 / 2}}}{1 - \frac{1}{{X_{s}}^{'}} {[{(\sqrt{1 + {X_{s}}^{' 2}} + \frac{λ}{d})}^{2} - 1]}^{1 / 2}}

Proceeding in the same manner for the location of the pole, by using (72) and letting δ_p represent the shift δcorresponding to the pole, we find

\begin{array}{c} \frac{δ_{p}}{k} \frac{\sqrt{1 + {X_{s}}^{' 2}}}{{X_{s}}^{' 2}} & = \frac{M^{2}}{4} {\frac{1}{1 - \frac{1}{{X_{s}}^{'}} {[{(\sqrt{1 + {X_{s}}^{' 2} +} \frac{λ}{d})}^{2} - 1]}^{1 / 2}}} \\ + \frac{1}{1 - \frac{i}{{X_{s}}^{'}} {[1 - {(\sqrt{1 + {X_{s}}^{' 2}} - \frac{λ}{d})}^{2}]}^{1 / 2}}} \end{array} .

Relations (77) and (78) may be used for the rough computation of the angular locations of the null and the pole, respectively, but we are concerned with their relative positions. However, δ_p is complex, so that the real part of this difference must be taken. Hence, we obtain

\begin{array}{c} Re [\frac{δ_{p}}{k} - \frac{δ_{m}}{k}] & = \frac{(M^{2} / 4) {X_{s}}^{' 2}}{{(1 + {X_{s}}^{' 2})}^{1 / 2}} \times \frac{1}{1 + \frac{1}{{X_{s}}^{' 2}} {1 - {[{(1 + {X_{s}}^{' 2})}^{1 / 2} - \frac{λ}{d}]}^{2}}} \\ = \frac{{X_{s}}^{' 4}}{{(1 + {X_{s}}^{' 2})}^{1 / 2}} \times \frac{M^{2} / 4}{(\frac{λ}{d}) [2 {(1 + {X_{s}}^{' 2})}^{1 / 2} - \frac{λ}{d}]} \end{array} .

Since we are considering the range in which two or more propagating spectral orders appear, λ/2d < 1, so that δ_m is seen to be less than the real part of δ_p for this case. In view of definition (75) for δ, and the relation (3) between k_s and angle θ, result (79) tells us that the null lies closer to the normal than does the pole, and, therefore, than the maximum, which must lie on the other side of the pole.

Suppose now that the resonant spectral order was not n = −2, as above, but rather n = 1. The difference in sign of the spectral order is equivalent to coupling to a guided wave which travels along the surface in the opposite direction. As we shall see, this distinction between the two cases causes a reversal as regards the relative angular locations of the minimum and the maximum.

Let us again examine the behavior of the intensity of the n = −1 propagating spectral order. We proceed exactly as we did above, but it is more convenient to first rearrange the continued fraction form (70) into

\frac{I_{- 1}}{H} = \frac{- \frac{M}{2} (\frac{1 |}{D_{- 1}} - \frac{M^{2} / 4 |}{| D_{- 2}} - . . .) (D_{1} - \frac{M^{2} / 4 |}{D_{2}} - . . .) (\frac{1 |}{D_{0}} - \frac{M^{2} / 4 |}{| D_{- 1}} - . . .) \frac{2 κ_{0}}{ω ∊ Z_{s}}}{(D_{1} - \frac{M^{2} / 4 |}{D_{2}} - . . . - \frac{M^{2} / 4 |}{D_{0}} - . . .)} .

Since the neighborhood of the anomaly is now given by D₁ = 0, we see from (80) that the approximate positions of the minimum and the pole are given, respectively, by

D_{1} - \frac{M^{2} / 4}{D_{2}} = 0

and

D_{1} - \frac{M^{2} / 4}{D_{2}} - \frac{M^{2} / 4}{D_{0}} = 0 .

Since D₁ and D₂ are real, but D₀ is complex, we again see that the pole is complex and that the minimum is a null. For the perturbation treatment, we take

D_{1} = 1 - \frac{1}{{X_{s}}^{'}} {[(\frac{k_{s}}{k} + \frac{2 π^{2}}{k d}) - 1]}^{1 / 2},

so that the condition D₁ = 0 yields

\frac{{k_{s}}^{0}}{k} = \sqrt{1 + {X_{s}}^{' 2}} - \frac{2 π}{k d},

where now the positive sign must be chosen, and

D_{1} ({k_{s}}^{0} + δ) = - \frac{δ}{k} \frac{\sqrt{1 + X^{' 2}}}{{X_{s}}^{' 2}},

using (75). Proceeding exactly as above for the n = −2 resonant spectral order, we find, on use of (81) and (82), together with (85),

Re [\frac{δ_{p}}{k} - \frac{δ_{m}}{k}] = - \frac{X^{' 4}}{\sqrt{1 + {X_{s}}^{' 2}}} \frac{M^{2} / 4}{(\frac{λ}{d}) (2 \sqrt{1 + {X_{s}}^{' 2}} - \frac{λ}{d})},

which is identical to (79) except for a negative sign. Hence, for this resonance anomaly the maximum, rather than the minimum, lies closer to the normal. We later show numerical data computed from the rigorous expression (60) which verify the above results.

Let us next consider the anomalous effect associated with the Rayleigh wavelength when a resonance anomaly is not nearby. We consider the amplitude behavior of the I₀ spectral order (reflected wave), at the angle of incidence for which the n = −1 spectral order is at grazing angle. In view of the vanishing of κ₋₁ at this angle, we wish to determine whether or not the slope in the curve of |I₀| vs θ remains finite there.

We consider first the derivative

\frac{\partial I_{0}}{\partial k_{s}} = \frac{\partial Î_{0}}{\partial k_{s}} = {Î_{0}}^{'} = \frac{F {V_{0}}^{'} - V_{0} [{\bar{D}}_{0}^{'} + {A_{1}}^{'} + {B_{- 1}}^{'}]}{F^{2}}

on use of (14) and (52), where we have set

F = {\bar{D}}_{0} + A_{1} + B_{- 1} .

The square root (of κ₋₁) which goes to zero at the Rayleigh wavelength occurs in B₋₁ but not in

{\bar{D}}_{0}

or A₁. Hence, since the terms of greatest significance contain the reciprocal of this square root, we must examine B₋₁′. If we write B₋₁ = −1/G, we see from (51) that

{B_{- 1}}^{'} = \frac{1}{G^{2}} G^{'} = {B_{- 1}}^{2} ({\bar{D}}_{- 1} - . . .) .

But, from (44) we have that

\begin{array}{l} {\bar{D}}_{- 1}^{'} & = \frac{\partial}{\partial k_{s}} (\frac{2}{M} {1 + \frac{i}{{X_{s}}^{'}} [1 - {(\frac{k_{s}}{k} - \frac{2 π}{k d})}^{2}]}^{1 / 2}) \\ = - \frac{i}{{X_{s}}^{'}} \frac{2}{M k} \frac{(\frac{k_{s}}{k} - \frac{2 π}{k d})}{\sqrt{1 - {(\frac{k_{s}}{k} - \frac{2 π}{k d})}^{2}}} \end{array},

so that (89) becomes, approximately,

{B_{- 1}}^{'} = \frac{i}{{X_{s}}^{'}} \frac{2 {B_{- 1}}^{2}}{M k} \frac{(\frac{2 π}{k d} - \frac{k_{s}}{k})}{\sqrt{1 - {(\frac{k_{s}}{k} - \frac{2 π}{k d})}^{2}}} .

Since B₋₁′ is the only term in (87) which contains the reciprocal of the pertinent square root, we may write for (87)

\frac{\partial I_{0}}{\partial k_{s}} = {Î_{0}}^{'} ≃ - \frac{V_{0} {B_{- 1}}^{'}}{F^{2}} = - Î_{0} \frac{{B_{- 1}}^{'}}{F},

in view of (52) and (88).

Rather than (92), however, we actually wish to examine

\begin{array}{l} \frac{\partial {| I_{0} |}^{2}}{\partial k_{s}} & = 2 Re [I_{0} * \frac{\partial I_{0}}{\partial k_{s}}] \\ = 2 Re [- I_{0} * Î_{0} \frac{{B_{- 1}}^{'} F *}{{| F |}^{2}}] \end{array}

from (92). On use of (14) and (91), (93) becomes

\frac{\partial {| I_{0} |}^{2}}{\partial k_{s}} = - \frac{2 i {B_{- 1}}^{'}}{{| F |}^{2}} I m [F * ({| I_{0} |}^{2} + H I_{0} *)] .

Since F and I₀ are complex, the coefficient of B₋₁′ does not vanish. Hence, the reciprocal square root multiplies a nonvanishing factor, and the derivative is indeed infinite. An infinite slope in the curve of |I₀| as a function of incidence angle is therefore to be expected. This prediction is borne out in the numerical data presented below.

The foregoing analysis has shown that an infinite slope appears in the curve for |I₀| when the n = −1 spectral order is at grazing angle. By means of a few additional steps we can determine the behavior of |I₋₁| under the same conditions, i.e., when it itself is at the grazing angle. Since I₋₁ is given by (55), we have

\frac{\partial I_{- 1}}{\partial k_{s}} = {B_{- 1}}^{'} Î_{0} + B_{- 1} {Î_{0}}^{'} .

But B₋₁′ and I₀′ have already been obtained as (91) and (92), respectively, so that (95) becomes

\frac{\partial I_{- 1}}{\partial k_{s}} = \frac{{B_{- 1}}^{'} Î_{0}}{F} ({\bar{D}}_{0} + A_{1}) .

However, we want

\begin{array}{l} \frac{\partial {| I_{- 1} |}^{2}}{\partial k_{s}} & = 2 Re [I_{- 1} * \frac{\partial I_{- 1}}{\partial k_{s}}] \\ = 2 {| Î_{0} |}^{2} \frac{{| B_{- 1} |}^{2}}{{| F |}^{2}} Re [{B_{- 1}}^{'} {\bar{D}}_{0}], \end{array}

on use of (88), and after dropping those terms which have no real part. The term B₋₁′ is seen from (91) to be positive imaginary, and

{\bar{D}}_{0}

is complex with a positive imaginary part. Hence, (97) is divergent with a negative sign. This infinite slope in the behavior of |I₋₁| at the Rayleigh wavelength is commented upon in the next section.

Double anomalies are always a feature of interest. For metallic gratings with shallow grooves, it is customary to view double anomalies as occurring when two spectral orders emerge at grazing angle at essentially the same frequency or incidence angle, with the two waves emerging in opposite directions. Since, as is shown above, resonance anomalies need not occur at the Rayleigh wavelengths, double resonance anomalies need not, and, in general, will not be associated with anything happening at the grazing angle. The condition for a double anomaly is simply that

D_{n} = 0

and

D_{m} = 0, n \neq m

are satisfied essentially simultaneously [see the discussion associated with condition (19)]. In order for this to occur, n and m must be of opposite sign, implying that the two resonances each couple to a guided wave traveling along the grating in opposite directions. We cannot without further investigation say anything about the shape of the double resonance; the discussion above concerning the shape of a single isolated resonance is no longer applicable.

C. Discussion of Numerical Results

Numerical calculations have been made for a variety of examples, and the results are presented below in graphical form. Most of the data apply to the case for which two propagating spectral orders are present, but the behavior for only a single propagating spectral order and for four of them are also considered.

1. One Propagating Spectral Order (Reflected Wave Only)

We do not ordinarily expect to find anomalous scattering effects when only a reflected wave is present. Since the grating is a perfect reflector, the magnitude of the reflection coefficient remains equal to unity, and any anomalous effect can manifest itself only in the phase of the reflected wave. If the grating can support a slow wave, however, it is possible for it to sustain a leaky (complex) wave even though the dimensions are such that no diffracted waves are produced upon plane wave incidence. Under these conditions, a resonance anomaly can be produced even though the plane wave scattering results in only a reflected wave.

The current reflection coefficient Γ_c for this wave is given directly by (58), or, on use of (63), (49), and (51), by

Γ_{c} = \frac{I_{0}}{H} = \frac{\frac{i cos θ}{{X_{s}}^{'}} - [1 + \frac{M}{2} (A_{1} + B_{- 1})]}{\frac{i cos θ}{{X_{s}}^{'}} + [1 + \frac{M}{2} (A_{1} + B_{- 1})]} .

When no diffracted waves are present, A₁ and B₋₁ are both real, so that |Γ_c| = 1 as expected. We may define an equivalent input reactance X_eq′ to the grating in view of this complete reflection, and we know from elementary transmission line theory that

Γ_{c} = \frac{1 + i {X_{eq}}^{'}}{1 - i {X_{eq}}^{'}} .

Comparison between (99) and (100) yields

{X_{eq}}^{'} = \frac{{X_{s}}^{'}}{cos θ} [1 + \frac{M}{2} (A_{1} + B_{- 1})] .

In the limit of zero modulation, i.e., a smooth surface with surface reactance X_s′, the equivalent input reactance becomes

{X_{eq}}^{'} = \frac{{X_{s}}^{'}}{cos θ},

in agreement with direct considerations. The data can thus be plotted either in the form of X_eq′ or as the phase of I₀/H. Either form will exhibit the anomalous behavior, if present.

The behavior of X_eq′ as a function of the incidence angle θ is plotted in Fig. 7 for three different values of λ/d, with M and X_s′ maintained constant. For the largest value of λ/d, an anomaly does not appear, and the curve follows the 1/cosθ behavior of (102), valid for a smooth surface. For the other two λ/d values, anomalous behavior is manifested in an unmistakable resonance response. The phase behavior at these resonance anomalies is shown in Fig. 8 to result in a rapid change in phase equal to 2π. For either method of plotting the data, the manifestation of the anomaly is very sharp and severe.

It is to be recalled that a resonance anomaly occurs in the vicinity of D_n = 0. A simple calculation shows that n = −1 is the relevant evanescent spectral order in this case. Using (63), we find that D₋₁ = 0 reduces to

sin θ = \frac{λ}{d} - {(1 + {X_{s}}^{' 2})}^{1 / 2},

so that one can readily check that the angles at which the sharp changes occur are given very closely by (103).

The influence of the modulation index M [see relation (39)] on the resonances is shown in Figs. 9 and 10. As one would expect, the location of the anomalies is unchanged, but the resonances are less sharp when the surface is more strongly modulated.

For the case of metallic gratings with shallow grooves, X_s′ is very small. If we set X_s′ = 0 in (103), we then obtain the condition for a Rayleigh wavelength for which the n = −1 spectral order just emerges at grazing angle. The resonance anomaly discussed above then occurs approximately at the Rayleigh wavelength, in accord with observations.

2. Two Propagating Spectral Orders (Reflected Wave Plus One Diffracted Wave)

Numerical data are presented below for a variety of cases in which only the n = 0 and then n = − 1 spectral orders are propagating. Let us first view Figs. 11 and 12, which demonstrate the amplitude behavior of the n = 0 and n = −1 spectral orders as a function of incidence angle. For angles of incidence less than 30° the n = −1 spectral order is nonpropagating; θ = 30°, therefore, corresponds to a Rayleigh wavelength. For angles greater than 30° we observe two resonance anomalies, both removed from the Rayleigh wavelength and each separated from the other. When the |I₋₁| curve has a maximum, the |I₀| curve possesses a minimum; this long-observed (and well-understood) feature corresponds to power conservation across any plane parallel to the grating plane, and for S polarization takes the form

\frac{κ_{0}}{k} {| I_{0} |}^{2} + \frac{κ_{- 1}}{k} {| I_{- 1} |}^{2} = \frac{κ_{0}}{k} {| H |}^{2} .

One notes that the shape of the resonances in Fig. 12 for |I₋₁|, for example, corresponds to a sharp maximum followed by a rapid fall and then a null, or vice versa. In fact, in the first anomaly the maximum precedes the minimum, while in the second anomaly the reverse is true. All of this behavior has in fact been explained in the preceding theoretical sections. The general shape has been discussed in Sec. III, in which it is pointed out that a complex pole and a minimum, which may be a null as in the present case, occur very close together, and that the general shape is given by (32), with the curve’s minimum and maximum obtainable from (33) and (35).

Before being able to examine the behavior in Fig. 12 in greater depth, one must know which evanescent spectral order is the resonant one for each of the two anomalies. A quick numerical calculation, employing relations (73), (74), (83), and (84), verifies that for the first anomaly, i.e., the one corresponding to smaller angles (nearer to the normal), the n = 1 is the resonant spectral order, while the n = −2 spectral order is the resonant one for the second anomaly. We may then refer to the discussion in Sec. IV-B, in which it is pointed out that for the first anomaly the locations of the null and the pole are given approximately by (81) and (82), respectively, and that one should expect the maximum of this resonance to lie closer to the normal. This observation is indeed borne out by Fig. 12. The discussion which pertains to the second anomaly shows that approximate calculations for the null and pole locations are given by (71) and (72), respectively, and that the null is expected in this case to lie closer to the normal, as is found in Fig. 12.

Returning to Fig. 11, we note that the curve exhibits an infinite slope at the Rayleigh wavelength. While we have not been able to determine whether or not this feature is true generally, the discussion in Sec. IV-B pertinent to this particular case [Eq. (94) and preceding] does predict the occurrence of an infinite slope. An infinite slope, with a negative sign, is also predicted for the |I₋₁| curve at the Rayleigh wavelength, but such behavior is not pronounced in Fig. 12 and must occur rather suddenly over a very short angular range.

It is interesting also to note the phase behavior associated with these anomalies, as shown in Fig. 13. At each of the resonances, the phase of I₋₁ jumps by π, and does so in a sequence which corresponds to the reversal between the roles of maximum and minimum observed in Fig. 12.

Figures 14–16 correspond to another set of parameters. Only a single resonance anomaly appears now, also far removed from the Rayleigh wavelength. The variation of intensity (power) as a function of incidence angle (actually sinθ) is shown in Fig. 14 for both the n = 0 and n = −1 spectral orders. All of the other spectral orders are evanescent. The amplitude variation of the n = 1 spectral order is shown in Fig. 15, and it is seen to be the resonant one. In accordance with the fact that the resonant spectral order is n = 1, the power plot in Fig. 14 for the n = −1 spectral order exhibits a maximum located closer to the normal than the minimum, in agreement with Fig. 12. It is also seen that the sum of the powers in the n = 0 and n = −1 spectral orders is a constant, in accordance with (104).

The form of the variation in Fig. 15 for |I₁| is seen to be rather different from that shown by |I₋₁|, a propagating spectral order. The true resonance shape shown in Fig. 15 is in agreement with (29) and its associated discussion. It is shown there that the resonance corresponds to the condition for a leaky wave which is supportable by the grating (but not excited by the plane wave), and that the Q of this resonance is given simply by (30), where the β and α are the phase and attenuation constants of the above-mentioned leaky wave.

For the parameters associated with Fig. 15, the values of β and α were computed and found to be

\frac{β}{k} = 1.203, \frac{α}{k} = 0.00688,

and the associated Q becomes equal to 87.49, using (30). This Q is referred to in Fig. 15 as Q_p, the subscript signifying pole. The value of the Q computed directly from the curve in terms of its half-power points is denoted as Q_c, the subscript meaning curve, and it is found to be 87.21, which is in excellent agreement.

The value of β in (105) is equal to the value that would be possessed by the basic slow-wave space harmonic of the leaky wave if it were present on the grating. This value of β differs from the corresponding k_s value which is associated with the angle of incidence via (3), and is, in fact, related to it by β = k_s + 2π/d. The discrepancy between the β and k_s values is reflected in the two sets of abscissa values given in Fig. 15. The upper set corresponds to the k_s and the lower to the β in (105).

From Fig. 14, one can ascertain immediately that the null of the anomaly is located at sinθ = 0.278. This value of sinθ is equal to β₁/k, where β₁ is discussed in Sec. III in connection with (32) and (33). From this knowledge of β₁ and that of α and β (actually k_s = β − 2π/d) from (105) one can readily compute the location of the maximum of the curve, using (35). From these values one obtains that the maximum is located at sinθ = 0.2635, in exact agreement with an expanded version (not shown here) of the curve of P₋₁ in Fig. 14.

From this knowledge of β,α, and β₁, one can do more than locate the maximum of the curve; using (32), one should be able to obtain a close approximation to the whole curve in the neighborhood of the anomaly. Using the numbers given above for α, β, and β₁ in (32), a calculation was made of |I₋₁| for a number of values of sinθ, and the results are presented in Fig. 16 as individual crosses on the rigorous curve of |I₋₁| computed using (60). As seen, the agreement is remarkably good in the region of the anomaly, but deviation sets in as one departs from it. One concludes, therefore, that the anomaly is indeed due to the resonance effect ascribed to it in the present theory.

Figures 17 and 18 correspond to parameters which are identical to those of Figs. 14–16, except for the value of the modulation index M. The previous set applies to M = 0.4, while these now hold for M = 0.8, a considerably deeper modulation. Upon examination of the curve for the resonant spectral order n = 1, shown in Fig. 17, we find, as expected, that the Q for the resonance is much lower, being about 20 rather than 80, as before. The Q value was again computed from the β and α values, and the agreement with the value obtained directly via the curve is again excellent. The behavior for the propagating spectral orders is shown in Fig. 18. It is qualitatively similar to that found for the smaller modulation, but the anomalous region is wider and the power interchanges greater.

Figures 19–22 illustrate effects we have not encountered as yet in this section. Figure 19 shows the variation of the n = −1 (propagating) spectral order as a function of λ/d. Near to λ/d equal to 0.975 we see a double anomaly, composed of two almost overlapping resonance anomalies and located far from the Rayleigh wavelength (λ/d = 1.500). The resonant spectral orders corresponding to this double anomaly are given by n = 2 and n = −3.

The other effect to be noted on Fig. 19 is the presence of a pronounced single anomaly located near to the Rayleigh wavelength, which is at λ/d = 1.500. The associated resonant spectral order is that due to n = −2, and the reason for the enhanced effect is its proximity to the Rayleigh wavelength. Thus, two separate anomalous effects enhance each other, and this observation serves to explain why the resonance effects on metallic gratings with shallow grooves, which necessarily occur near the Rayleigh wavelengths, are so pronounced.

Figure 20 presents the corresponding information for the power in the (n = 0) reflected wave, and verifies the power conservation relation (104). The phase information presented in Fig. 21 for the n = −1 spectral order is interesting. Again, a phase jump of π is associated with the resonances, and the behavior at the double anomaly clearly indicates that two separate resonances are involved.

The last figure in this group pertaining to only two propagating spectral orders is concerned with an indication of the convergence properties of the continued fraction expansion. The calculations referred to above have included 13 spectral orders (or modes), ranging from n = −6 to n = 6. In many cases, sufficient accuracy is obtainable on use of a smaller number of these modes. Figure 22 shows a comparison of results for a typical resonance anomaly; the usual calculation employing 13 modes is shown by the solid line, while the dashed line represents the result found by using only 5 modes (n = −1, 0, 1, 2, and 3). The agreement is seen to be rather good. However, caution must be exerted with regard to which modes are most pertinent, since they change with the circumstances.

3. Four Propagating Spectral Orders

Numerical data have been obtained for one set of parameters for which four spectral orders are propagating, the orders corresponding to n = 0, −1, +1, and −2. The associated resonant spectral order is n = −3, and its response curve as a function of λ/d is given in Fig. 23. As seen, the usual sharply peaked resonance shape is obtained. The corresponding variations in the power in the n = −2 and n = −1 spectral orders, as a function of λ/d, are shown in Figs. 24 and 25, respectively. In both cases, the expected shape is obtained, with clearly defined maxima and minima. For the n = −2 case a null is found at the minimum, since the factor that defines the minimum is purely real; the other propagating spectral orders do not have a null, however. It is also seen that the maximum for n = −2 corresponds to the minimum for n = −1, and vice versa.

The remaining propagating waves, the n = 0 and n =1, exhibit only a very slight reaction to the resonance. For the n = 0 spectral order, this effect is understandable in view of the low powers present in the n = −2 and n = −1 waves, as seen from Figs. 24 and 25. In the case of the n = 1 spectral order, the effect is weakly felt because of the sinusoidal nature of the modulation. As one notes in the derivation at the beginning of Sec. IV, for example, in (42) and (43), only nearest-neighbor coupling is present. As a result, as seen from relations (57)–(62), if one wishes the behavior to first order only in modulation index M, only the two nearest spectral order neighbors need be considered, while to order M² the four nearest are involved, etc. Hence, because of the special example chosen here the anomalous effects are more localized to neighboring spectral orders than one would expect to find in general.

The research described herein was performed under contract with the Air Force Cambridge Research Laboratories, Office of Aerospace Research, Bedford, Massachusetts.

Figures

Fig. 1 Reflected and diffracted spectral orders produced by plane-wave incidence on a reflection grating.

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Fig. 2 A spectrogram exhibiting Wood’s anomalies (reproduced from Wood[13]).

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Fig. 3 Classes of reflection gratings which exhibit Wood’s anomalies.

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Fig. 4 A multimode waveguide terminated by a resonant cavity—an analogy to a diffraction grating.

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Fig. 5 Example of correspondence between the fields of a diffracted plane wave and a leaky wave: (a) plane-wave case; (b) leaky-wave case.

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Fig. 6 An S-polarized plane wave incident on a plane with a periodically varying surface reactance.

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Fig. 7 Equivalent surface reactance vs angle of incidence for a single propagating spectral order (reflected wave only). (The parameter is λ/d.)

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Fig. 8 Phase of I₀ vs angle of incidence for a single propagating spectral order. (The parameter is λ/d.)

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Fig. 9 Equivalent surface reactance vs angle of incidence for a single propagating spectral order. (The parameter is M.)

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Fig. 10 Phase of I₀ vs angle of incidence for a single propagating spectral order. (The parameter is M.)

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Fig. 11 Relative magnitude of the n = 0 spectral order vs angle of incidence for two propagating orders.

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Fig. 12 Relative magnitude of the n = − 1 spectral order vs angle of incidence for two propagating orders.

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Fig. 13 Phase of the n = − 1 spectral order vs angle of incidence for two propagating orders.

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Fig. 14 Relative powers in the n = −1 and n = 0 propagating spectral orders vs sinθ (θ; = angle of incidence).

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Fig. 15 Relative amplitude of the n = 1 resonant sp ectral order vs sinθ (θ = angle of incidence).

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Fig. 16 Comparison of calculations for the magnitude of the n = − 1 spectral order for a resonance anomaly: solid curve computed from exact solution, crosses obtained from Eq. (32) using only the knowledge of the positions of the null and the complex pole.

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