Warping space to test the validity
of Rayleigh’s hypothesis [ ]
d1 d2 d
d1 d2 d x
(a) Saw-tooth grating as an example of periodically modulated interface. (b) The grating transformed in a curvilinear
The dominant opinion is that it is an approximation whose
validity limit has even been quantified at h/d < 0.1426 (h being
the corrugation depth and d its period). Against common sense
and pretended mathematical proofs, I had the curiosity to test
the Rayleigh hypothesis numerically. We found that it is exactly
true up to a corrugation depth of at least 15 times the claimed
validity limit. After revisiting the rationale of this century-old
hypothesis, we believe that its rehabilitation will do more than
just pay tribute to Lord Rayleigh’s intuition. It will reopen an
unexplored buried path for electromagnetic theory.
A touch of physics and a few simple formulae will permit us
to grasp the plausibility of the validity of Rayleigh’s hypothesis. Let a plane monochromatic wave incident on a periodically modulated interface z=a(x)=a(x+d ). At the limit of zero
modulation amplitude, the diffraction product is a single plane
wave, reflected in the upper medium and refracted in the lower
medium whose amplitude is simply given by the Fresnel reflection and transmission coefficients.
Non-zero corrugation depth generates higher-order diffracted waves whose wave vector projection on the grating
plane k x m are:
2p kxm=kx0 +—m. d
Only a finite number of these waves propagate into the two
adjacent media, since kxm exceeds the wavenumber in each
medium as from a finite order m, beyond which their field
decays exponentially with the distance from the grating. In
accordance with the Floquet theorem (1883), the basis of the
diffracted waves is complete. This means that the solution in
the region above and under the corrugation is a superposition
of reflected and transmitted diffracted waves.
Solving the diffraction problem amounts to finding the
complex amplitude of the diffracted waves and to determining the field in the corrugation. Rayleigh assumed that, within
the corrugation, the solution in the incident and transmission
media is a superposition of reflected and transmitted diffracted
waves, respectively. This allowed him to obtain a simple analytical solution to the problem of a shallow corrugation. However,
it left open the question of its validity for larger depth.
In a modern interpretation, the problem can be formulated
as follows: Are there periodic functions a(x) different from
a(x)= 0 , for which the Rayleigh hypothesis exactly holds?
Revisiting this question today is important; the perspective
of obtaining exact analytic solutions in diffraction problems
involving a surface corrugation is very attractive for 2-D periodically microstructured surfaces as well as, beyond these, for
non-periodic microoptical elements.
Validity of the hypothesis
Let us warp space and express the above corrugation a(x) in
another, curvilinear coordinate system
x=x,h=y,z=z2a(x). ( 2)
Such transformation is the key idea in the C method, which
is known to be rigorous in diffraction theory. In the new
coordinate system, the corrugated interface between the two
media has become a plane, z = 0, and both dielectric permittivity and magnetic permeability have become periodic tensors
of coordinate x.
An important property of transformation ( 2) is that it
preserves the translational symmetry of the problem along
coordinate z. This permits us to represent the solution of the
diffraction problem by a superposition of Bloch waves (i.e., the
grating eigenmodes propagating up and down along coordinate z). The correctness of this modal representation is a
consequence of the completeness of the modal basis.
The combination of coordinate transformation ( 1) with the
modal representation leads to the astonishingly simple equation for the modal field Ym(x) in an analytical form:
9 2 Y9m (x) Y9m (x) 3 2222 2jbma9(x) 4 132222 2jbma9(x) 4 1 k 2 2b 2 m = 0 ( 3)
Ym (x) Ym (x)
where bm is the propagation constant of the mode of index m
along coordinate z. The general solution of equation ( 3) is eas-
ily found by direct integration in the form