There has been a long-standing debate in the optics community as to whether
Rayleigh’s hypothesis is true or simply an approximation for shallow corrugations.
The 6 sign in ( 4) corresponds to the wave propagating
along and opposite the x axis.
Despite the rather complicated form of the tensor functions representing the magnetic and dielectric constants in
the warped space, their ratio remains constant and does not
depend on the profile function a(x). Therefore, the impedance
√— m/e remains constant everywhere in the modulated structure.
This property is important for understanding the behavior of the modal solution at the vertical interfaces, where the
profile function derivative a9(x) is discontinuous. One usually
assumes the existence of two independent solutions at each side
of the discontinuity and matches them. But constant impedance leads to the absence of any reflection at discontinuities.
Therefore, the solution with the “plus” sign in ( 4) is never
coupled with the solution corresponding to the “minus” sign;
they represent two independent modes in the grating region.
This property widens the scope of the analysis to any piecewise
differentiable function a(x).
Besides, the modal solutions satisfy the periodicity condition
Y6m (x+d )= Y6m(x)exp( jk x0 d ) ( 5)
specified by the incident wave. Substituting solution ( 4) in
expression ( 5) and taking into account definition ( 1), we get
the modal propagation constants bm
bm = 6√k 2 21k x m 22 . ( 6)
The field of each mode π corresponds to one of the
solutions ( 4):
Ym (x, z) = exp 3 jkx m x + jbma(x) 6 jbmz 4 . ( 7)
The 6 sign in equations ( 6) and ( 7) corresponds to the
mode propagating upward and downwards along coordinate
z. Since the interface between the media coincides with the
coordinate plane in the transformed system, all the modes in
the lower medium are directed downwards, and all the modes
in the upper medium are directed upwards (except for those
modes that represent the incident wave). This is the key to the
Rayleigh hypothesis: All modes involved propagate outward
relative to the interface plane or are evanescent.
The transformation back to the original (Cartesian) coordi-
nate system reveals that the field of each mode
Diffracted field in sinusoidal grooves calculated under
the Rayleigh hypothesis.
corresponds exactly to one of the orders diffracted by the corrugation grating without coupling between them. Therefore,
the exact solution in the lower medium everywhere in the corrugated region is expressed by a superposition of transmitted
Similarly, the exact solution in the upper medium everywhere, including in the corrugation region, contains only the
superposition of reflected diffracted waves. As a consequence,
the Rayleigh hypothesis is valid in the case of the considered profile a(x). Whereas we showed numerically that the
Rayleigh hypothesis was true, the above development, which
resorts to the grating mode representation in the warped
space, confirms it fundamentally.
The figure above shows field mapping within the corrugation of a grating 15 times deeper than the once-believed
validity limit of the Rayleigh hypothesis under 20 degree
TE incidence. It illustrates how little evident it is that such
a complex picture might be created by few diffracted orders.