## Abstract

The Rayleigh hypothesis and the related method of diffraction analysis are revisited. It is shown that the Rayleigh method can be applied to deep grating modeling without numerical problems and that it gives any desired accuracy whatever the groove depth. This proves the validity of the Rayleigh hypothesis and rehabilitates the Rayleigh method.

©2009 Optical Society of America

## 1. Introduction

The Rayleigh hypothesis (RH) was first formulated by Lord Rayleigh [1] and then applied to the theory of diffraction gratings [2]. If one considers the reflection of a plane wave from the plane interface between two homogeneous media, three plane waves only exist: the incident wave, the reflected outgoing wave, and the transmitted (refracted) wave. Considering the scattering from a sinusoidally undulated interface, Rayleigh looked for a solution in a similar form, assuming that the field above and under the grating interface only consists of outgoing waves with spatially constant amplitudes.

Whereas such assumption is undoubtedly true in the half-spaces adjacent to the grating region, it is clearly questionable inside the grating region. About half a century ago it was stated that the Rayleigh method (RM) is incorrect [3], [4], because the diffracted field in the neighbourhood of the scattering surface should actually consist of both outgoing and incoming waves. This more intuitive than properly founded point of view has been generally accepted by the diffraction community for more than 50 years. A significant number of theoreticians have disproved the RH and even given precise limits of its validity [5]–[10]. There has been a broad consensus that the RM gives wrong results when applied to deep gratings. It was however not realized that the numerical problems arising in the implementation of the RM might be due to specific properties of improperly conditioned diffraction matrices and not to the inconsistency of the RH itself as we are going to show hereafter. The critics of the RH, following Rayleigh himself, used a Fourier series expansion to match the fields at the periodic (usually sinusoidal) interface. The truncation of an infinite system of equations led them to numeric instabilities.

The consensus about the incapability of the RH to represent the physical reality was however not unanimous. Despite the overwhelming numerical evidence of the RM failure to correctly calculate gratings of depth larger than a small fraction of the period, some voices left the question open [11]–[13], and few authors wondered about the unexpected agreement between their exact results and those given by the RM [14]–[16]. Amazingly, the RM even had a short golden age in the seventies at the beginning of the integrated optics era where waveguide gratings were considered as possible coupling means between free space, fibers or lasers and planar waveguides [17]–[18]. Since then, the interest has shifted towards resonant grating filters where the association of a planar waveguide and a corrugation gives rise to a family of novel free space wave filters with high selectivity and high efficiency [19]. The limitations of the RM to small corrugation depth were no limitations for such resonant corrugation structures since the concentration of the modal field in the grating region gives high diffraction strength with a shallow corrugation. Under such conditions the RM reveals to be a powerful modeling tool and a physically meaningful analytical representation for the design of free space resonant diffractive optical elements [20].

The author of the present paper has long been emphasizing the physically enlightening power of the RM [21]–[23] whose main analytical results and teachings are reported in a special issue of the General Physics Institute (Moscow) [24]. Lately, when applying the exact true-mode method [25] in the transformed space of the C-method [26] and finding out analytically that the eigenmodes are identical to the Rayleigh orders, it became evident for the author that the RH can only be true. There remained to bring the numerical proof of this belief which is what the present paper does. The analytical derivation and its consequences will be reported elsewhere.

The proof of the validity of the RH is made here at the very level it has been claimed invalid so far: through a numerical experiment. The presentation of the actual potential of the RH is organized in three steps. In the next two sections the electromagnetic problem is formulated together with the version of the RM which will be used hereafter. Then, a convergent and exact far field solution is demonstrated. Finally, a new numerical scheme is presented giving a convergent and correct near field solution.

A numerical experiment delivers numbers and tables of numbers. There are a lot of them in the present paper as we decided to sacrifice the aesthetics of its appearance to allow the reader to repeat the experiment and verify our statements on the validity of the RH and the relevance of the RM.

## 2. Diffraction problem formulation

The considered structure is represented in Fig. 1. It is composed of a substrate half-space of permittivity ε_{1}, a superstrate halfspace of permittivity ε_{2}, and a corrugated zone in-between. The corrugation is expressed as a periodical surface undulation of period *d*:

Function *ζ*(*x*) is continuous and piecewise differentiable. It can be represented by a Fourier series:

where *K*=2π/*d* is the corrugation wavenumber.

Let a plane monochromatic wave be incident from the upper side of the grating with *x* and *z* wave vector components *k*
^{x}
_{0} and *k ^{z}*

_{20}(the temporal term exp(-

*jωt*) is omitted for sake of brevity). A 1D grating under non-conical incidence is considered. The optogeometrical parameters and the fields do not depend on the

*y*coordinate. Therefore, the TE and TM problems can be considered independently. The electric field of TE waves as well as the magnetic field of TM waves is directed along the y axis. Therefore, the corresponding

*y*field component characterizes all fields completely. The resulting formulae are different for different polarizations but the analysis is similar. The transverse field of the incident wave is

**F**

_{inc}=

**ŷ**exp(

*jk*

^{x}_{0}

*x*-

*jk*

^{z}_{20}z), where

*F*can either be the electric

*E*, or the magnetic

*H*field. Then, in the regions above the grating,

*z*>max

*ζ*(

*x*), and below the grating,

*z*<min

*ζ*(

*x*), the scattered field is represented by the sum of diffracted plane and evanescent waves:

where *a*
_{1m} and *a*
_{2m} are the constant amplitudes of the waves diffracted into the lower and upper media, respectively,

where subscripts 1 and 2 denote the media below and above the corrugated interface. In the case of a complex wavenumber *k _{p}*, projections

*k*are chosen according to the rule:

^{z}_{pm}Thus, solving the diffraction problem amounts to determining all the unknown complex diffracted field amplitudes *a*
_{1m} and *a*
_{2m}.

## 3. Rayleigh method

Under the Rayleigh hypothesis the scattered field is assumed to be in the form of outgoing waves even in the grating region:

The transverse field is continuous at the periodic interface *z*=*ζ*(*x*). This gives the first infinite set of equations on the unknown amplitudes *a*
_{1m} and *a*
_{2m} :

All functions in equation (9) are continuous along x and can be represented by a product of some periodic function by factor exp(*jk ^{x}*

_{0}

*x*). This allows for the development of both sides of Eq. (9) into series ∑∞

*q*=-∞

*F*exp(

_{yq}*jk*). To determine coefficients

^{x}_{q}x*F*, we first multiply both sides of Eq. (9) by exp(-

_{yq}*jk*), then integrate the product over one grating period. Finally, we get the first infinite set of linear equations for unknown amplitudes

^{x}_{q}x*a*

_{1m}and

*a*

_{2m}:

where *I*
^{p±}
_{qm} represent the integrals:

In the case of a sinusoidal groove profile *ζ*(*x*)=σsin *Kx*, for example, the integrals of Eq. (11) are represented by Bessel functions:

The second boundary condition is obtained by applying the continuity of the other tangent field component which is found from Maxwell’s equations as the product

where *χ* means permeability µ for the TE polarization and permittivity ε for the TM polarization, *∂/∂n* means the derivative in the direction normal to the grating surface. Taking the derivatives yields the second equation:

$$=\sum _{m=-\infty}^{\infty}\genfrac{}{}{0.1ex}{}{-j{k}_{1m}^{z}-j{k}_{m}^{x}{\zeta}^{\prime}\left(x\right)}{{\chi}_{1}\sqrt{{1+\left[{\zeta}^{\prime}\left(x\right)\right]}^{2}}}{a}_{1m}\mathrm{exp}\left[j{k}_{m}^{x}x-j{k}_{1m}^{z}\zeta \left(x\right)\right]$$

We multiply both sides of the latter equation by factor

$j{\chi}_{1}{\chi}_{2}\sqrt{1+{\left[{\zeta}^{\prime}\left(x\right)\right]}^{2}\mathrm{exp}\left(-j{k}_{q}^{x}x\right)}$

then integrate them over one grating period. Finally, integrating by parts, we get the second infinite set of linear equations for amplitudes *a*
_{1m} and *a*
_{2m} :

Note that the particular case *k ^{z}_{pm}*=0 does not lead to any difficulty since the corresponding coefficient writes:

Thus, in the RM the diffraction problem is reduced to resolving the infinite system of linear Eqs. (10) and (15). The numerical implementation of this system can be made by truncating it to 2*M* equations for 2*M* unknown amplitudes of *M* diffraction orders *a*
_{1m} and *a*
_{2m}.

## 4. Far field calculation

Most works in the scientific literature on the RM base their assessment of the latter on far field calculations, i.e., on the calculation of the efficiency of the diffraction orders. In the present section the RM is applied to calculate the diffraction of a plane wave from a sinusoidal grating. The grating period is *d*=1µm, the wavelength λ=0.6328 µm. The incidence is from the air side (*n*
_{2}=1) under angle θ=arcsin(1/3). Two types of substrates are considered: a dielectric (*n*
_{1}=2.5) and a lossless metal (*n*
_{1}=0+i·5). For economy of place, we present the modeling results of the TE wave diffraction on a dielectric grating and of the TM wave diffraction on a metal grating only. The first case allows for possible comparison with any rigorous slicing techniques, the second case is the most critical since there are few possible reference techniques.

The Bessel functions in Eq. (12) were calculated by their Maclaurin series [27]:

Such technique is known as leading to potential loss of accuracy for large values of argument x. In the considered RM implementation, however, the numerical limitations have another cause and the argument of the Bessel functions never reach a dangerous level. The truncated equation system was solved by the Gauss elimination procedure [28] using an ordinary personal computer.

First, relatively shallow gratings were analyzed as a benchmark: *h*=2σ=0.15µm. The results are summarized in Tables 1 and 2. One can conclude that in all cases the results converge fast with the number of orders *M*. The energy balance corresponds well to the accuracy reached in the calculation of diffraction efficiencies. Increasing the number *M* enables the grating calculation with any desired accuracy. Such behavior is not surprising since it is recognized that the RM leads to reliable results if the groove depth is below *K*σ<0.448 [6]

Quite different results are obtained when the groove depth increases (Tables 3-5). Unlike in the shallow grating case, the accuracy reached in the calculation of diffraction efficiencies does not increase monotonically with the number of orders: there is an order number M which provides the highest accuracy. Increasing M beyond this optimum value leads to a fast loss of accuracy. Such behavior of the RM also is known and was considered for a long time as a decisive disproof of the validity of the method itself as well as of the underlying hypothesis. Nevertheless, this behavior is typical of a convergence problem. The existence of a finite order number M giving the best accuracy is very often the sign of numerical instabilities due to a limited computer precision.

To check on the relevance of such assumption, the same calculation was repeated with the doubled processor precision which corresponds to 128 bits in the mantissa representation. The results are given in Tables 6 to 8. They show a drastic difference in comparison with those obtained with a simple processor precision (Tables 3 to 5). All the calculated efficiencies keep on converging with increasing number *M*. The Rayleigh method still gives results within 1e-15 accuracy even when the grating grooves become deeper and exceed twice the well-known limit *K*σ<0.448 [6].

The clue for these astonishingly accurate results is well hidden and explains why it has not been suspected for so long: evanescent waves which correspond to high diffraction orders exhibit extremely rapid exponential spatial decrease. The resulting S matrix of a deep grating is not well conditioned. It can contain, for example, eigenvalues of order of 1e-30 and even less. This explains why any numerical treatment of such matrix has to be made with increased processor precision.

In the RM implementation of the present paper the intermediate matrices are first calculated analytically and, at the latest stage, they are transformed to the diffraction S matrix of the grating. Such transformation includes implicitly a matrix inversion procedure which can lead to numerical instabilities because of specific properties of the S matrix and an insufficient processor precision.

Thus, the accuracy problems in applying the RM are only caused by the limited precision of the processor. Even the doubled processor precision fails to establish perfect convergence in the case of *h*=2σ=0.6 µm, *K*σ=1.885 (Table 8). Nevertheless, increasing the processor precision improves the convergence and finally leads to exact results whatever the groove depth and the prescribed accuracy.

Tables 9 and 10 present the diffraction efficiencies in deep dielectric gratings calculated by the RM up to groove depth *h*=2σ=2 µm, *K*σ=6.283 (to be compared with the pretended validity limit of *K*σ=0.448), with an accuracy better than 1e-15. They confirm the relevance of the RM for deep grating calculation and, most importantly, establish the validity of the RH well beyond its pretended limitations.

The data in Tables 9-10 show how strong is the diffraction and the corresponding diffraction order amplitudes in deep gratings. To author knowledge no other method exists which is capable to provide a solution to such diffraction problems at the same level of accuracy. This illustrates and reveals that the RM has a strong potential for even becoming an exact modeling tool for deep gratings.

## 5. Numerical experiment on the near field calculation

It is in the capability of the RH to accurately represent the field in the grating region that the skepticism of the scientific community has been the most pronounced. How can a hypothesis stating that the field is composed of outgoing waves represent the near field in deep corrugations? Even for relatively small grating depths where the diffraction efficiencies can be found with a good accuracy, the RM failed to give reasonable values of the near field [29]. Thus, the rehabilitation of the RH naturally calls for a demonstration of the RM ability to exactly calculate the diffraction near field. As we will show hereafter, the RM gives also, and against all expectations, an accurate solution for the near field.

The numerical experiment considered for such demonstration is still the diffraction of a plane wave from a sinusoidal grating. All the parameters are the same as in the previous section. The near-field calculation starts with resolving the equation systems (10) and (15) to deliver the diffraction order amplitudes *a*
_{1m} and *a*
_{2m}. Then, in the regions above the grating, *z*≥max *ζ*(*x*), and below, *z*≤min *ζ*(*x*), the scattered field is calculated by sum (3). Such approach reveals however to be numerically unstable when applied to calculate the fields in the grating region min *ζ*(*x*)<*z*<max *ζ*(*x*). The reason of such instability lies in the fast exponential growth of high-order evanescent diffraction orders. One can conclude that although the diffracted near field can be represented by a superposition of outgoing plane and evanescent waves, such representation is not the best from a physical point of view. Alternative electromagnetic solutions, as for instance that using grating modes in the modal method [25], could be better suited for the near field representation. An alternative approach will therefore be later set up which is more in agreement with the Fourier representation of the field being an essential part of the RM.

In fact, resolving Eq. (10) ensures the equality between the periodic series components of the fields above and below the grating surface. This means that the RM basically deals with field harmonics rather than with the fields. Therefore, when calculating the field on the sinusoidal grating surface it is safe and more correct to first calculate the field harmonics as given by the expressions at the left-hand and the right-hand of Eq. (9):

Then, the field is found by the sum of harmonics:

The summation on *q* in formula (19) is formally over all the harmonics taken into account. In practice, higher order harmonics are not well conditioned and taking them into account leads to numerical instabilities. Therefore, we made the summation in formula (19) over N central harmonics. The results of numerical modeling of the near field in a deep sinusoidal grating *h*=2σ=1µm, *K*σ=3.142 are shown in Tables 11 and 12. The transverse y-components of the electric field are calculated directly at the grating interface. The best results are obtained when the number of central harmonics N taken for summation in formula (19) is slightly less than half the total number of orders M.

The convergence with the total number M of diffraction orders taken into account is clearly established by comparison of the data of Table 11 with those of Table 12. No numerical instability was encountered during this calculation. All the fields converge well to their exact value when the total number M of considered diffraction orders increases.

There is no rigorous justification why skipping half of the M diffraction orders used for the exact calculation of the far field gives the most accurate values of the near field. Such selection was also advantageously used without justification to prevent the numerical instabilities in the RCWA applied to metal gratings [30]. The rationale for such practice lies in that the diffracted waves of large order have zero amplitude at infinity but are needed for the exact calculation of the propagating orders in amplitude and phase whereas their presence in the near field plays an important role, in particular those which arise from the very truncation process. The elimination of the latter permits to have access to those orders only which have a physical meaning.

Figure 2 represents examples of the near field calculated at the sinusoidal grating interface. No quantitative accuracy assessment can be drawn from this graph. However it gives a vivid illustration of how complicated the field can be in the corrugation region at large depths and how well can the RM account for it.

## 6. Discussion and concluding remarks

The present paper shows that the very much questioned Rayleigh hypothesis is true. It is true to the extent that it is valid up to a grating depth as large as fifteen times the depth which has been considered as its validity limit and up to the point where it is more accurate than any of the known exact methods taken as a reference. The paper reveals that, very regrettably, this fact has long been overlooked for a common reason of a limited computer precision.

To overcome this limitation the author has written a special numerical library. This private library allows to perform calculations with any desired precision on a standard personal computer. This is not a remarkable achievement but to author knowledge there exist no easily accessible analogue on the market. Most of accessible tools are designed to calculate big integer numbers.

The library is based on techniques developed lately for arbitrary precision computing [31]. The simplest and most straightforward algorithms are used for the arithmetic operations. Those are written in Assembler since they intensively use the carry flag of the processor. All other functions are written in C++ and are compatible with codes performing standard precision calculations. The calculations with high precision take more time than with standard precision. For example, on a computer with processor AMD Turion 64 X2 1,60 GHz, the standard precision calculation (double precision, 52 mantissa bits) of diffraction efficiencies involving *M*=45 orders is performed in 0.10 s, whereas the same calculation with (2×64 bits) precision takes 1.25 s; (5×64 bits) precision requires 2.87 s, and (25×64 bits) precision 33.15 s, respectively. The needed computer memory increases proportionally to the desired precision.

All the results of sinusoidal grating modeling confirm the excellent convergence and self-consistency of the RM. We performed a benchmark for the dielectric grating of Section 4 by comparing the results given by the RM for the diffraction efficiencies with those obtained by the RCWA method with slices [32]. The results are presented in Table 13. The achieved agreement is within 1e-5; it is limited by the slow convergence of the RCWA with the number of orders and slices. Table 14 presents a benchmark for the metal grating of Section 4 comparing the results given by the RM with those obtained by the C method [26]. The achieved agreement is better than 1e-5.

In the case of dielectric grating, the near-field calculated by the RM coincides with the values obtained by RCWA within 1e-4. Whereas it is not too committing to admit that the far field is composed of outgoing waves it is much less evident to accept it for the near field in deep gratings. This is yet what the near-field calculation with the increased processor precision shows.

The present rehabilitation of the RH concretely shows that the once relegated hypothesis and related RM have now become a reference method. Furthermore, beyond its vivid physical representation power, it provides the potential of expressing diffraction problems analytically a long way towards the solution. These possibilities have been discarded for half century; it is now time to catch up at an accelerated pace and to explore and exploit the horizons opened by the RH.

It is worth mentioning here that the RH was first used in the problem of light scattering on 3D objects [1]. Similarly to the diffraction problem, there is no consensus so far on the validity of the RH because of numerical instabilities in its applications [33]. Very probably, the cause of instabilities will be identified similarly and increasing the processor precision will result in the demonstration of the RH validity for the 3D scattering problem.

The domains which are bound to benefit from this new vision and related tools extend well beyond the boundaries of optical sciences and are all phenomena governed by the Helmholtz operator from acoustics [34] to particle scattering [35].

Arguing that the need for higher precision computing is a deterring hurdle hindering the RH from deploying its potential in electromagnetic theory and preventing the RM from becoming a practical exact modeling tool would not be relevant. The potential of the RH encompasses yet unexplored possibilities to treat electromagnetic problems analytically as we will show in further publications. The interest of these horizons by far exceeds the temporary software problem of increasing the number of digits which a processor can crunch. This nevertheless refers to an important strategic issue in numerical modeling that is presently under debate [35].

## Acknowledgements

The author is deeply grateful to Vladimir A. Sychugov, Institute of General Physics Moscow, for having led him in the early days of integrated optics into the field of waveguide grating coupling that they have explored together by a consistent resort to the physically evocative Rayleigh hypothesis under its Fourier-Kiselev implementation. Jean Chandezon is respectfully acknowledged for his profound vision of the fundamentals of electromagnetism and for his critical analysis which has motivated the writing up of the paper. The author is thankful to Olivier Parriaux for his long interest in the Rayleigh method as well as for his intense help in the preparation of the manuscript.

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