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\centerline{\large\bf Echelette-diffraction grating in the short-wave 
approximation}
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\centerline{\it Victor V. Zalipaev}
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\centerline{St.-Petersburg Institute of Cinema and Television,}
\centerline{Pravda street 13, St.-Petersburg, 191126, Russia.}
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One of the most important aspects of diffraction grating theory is the 
scattering by echelette (a grating with saw-tooth boundary), a particular 
interest of which arises owing to the possibility to concentrate the energy 
of reflected field for one diffraction order (corresponding reflected 
efficiency is just under unit) for both polarizations if any conditions are 
observed. In this case we need to employ  the high diffraction orders of 
reflected field that leads to the situation where the period of grating is 
far greater than the wave length (the case of high-frequency approximation). 
As a result, the numerical methods face difficulties to calculate 
characteristics of reflected field since the order of system of complex 
linear equations becomes very high and calculations are time consuming.
 Under these conditions it is natural to employ the asymptotic methods of 
geometric theory of diffraction (GTD) to investigate the problem of 
short-wave scattering by a diffraction grating. 

 The following time-harmonic diffraction problem is considered. The periodic 
saw-tooth surface $S$ is an echelette-diffraction grating with right angle of a
 wedge which is illuminated by a plane electromagnetic wave. It is 
electrically perfectly conductive grating. The problem is two-dimensional. 
The total wave field $u(x,y)$ satisfies above $S$ the Helmholtz 
equation $\Delta u+k^2u=0$ ($k=2\pi/\lambda$ is the wave number),  and 
Dirichlet's  or Neumann's boundary conditions on $S$  corresponding to the  
case of TE or TM polarization. The diffracted field can be written for 
$y\ge0$ in the form of a series of diffracted plane waves outgoing
 in  the upper half-plane (the Rayleigh series) 
$$u_d=\sum_{n=-\infty}^{+\infty}R_n\exp\,\{ik(x\cos\varphi_n+y\sin\varphi_n)\},$$
with $$\cos\varphi_n=2\pi n/kl+\cos\varphi_0,$$
 where $R_n$ are called the Rayleigh coefficients, $\varphi_n$
is the diffraction angle, $0\le\varphi_n\le\pi$ and $\varphi_0$ is the grazing angle
of incident plane wave. We 
consider a special case. The wave length is assumed to be small compared with
 the period of a grating, namely, $kl\gg1$, i.e., we deal with the
 high-frequency approximation.

In the papers [1,2] the author has developed a new ray-optical approach to 
the scattering by an echelette-grating in the short-wave approximation. The 
asymptotic solution of the problem was constructed by means of the method of 
summation of multiple diffracted fields (MDF) scattered forward and backward 
by wedges of grating. The sum of the series of MDF for the diffracted field 
in each term of the short-wave asymptotics was found with the aid of the 
solution of the Wiener-Hopf type integral equation. The uniform asymptotic 
expansion for the homogeneous plane waves Rayleigh coefficients takes the 
form 
$$R_n=R_n^{(0)}+R_n^{(1)}/p+O(p^{-2}), p=(kl)^{1/2}\to\infty,$$
 where expressions for $R_n^{(0)}, R_n^{(1)}$ are derived from the asymptotic 
expansion of the wave field above grating (see [1,2]).

 In the report we are going to present new optical properties of 
echelette-grating in the short-wave approximation ($\lambda\to0$, 
$kl\to\infty$) related with the possibility to concentrate the energy of
 reflected field for one diffraction order and which could be derived on the 
basis of the ray-optical approach.

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\centerline{\bf References}
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\noindent[1] Zalipaev V.V. Calculation of the wave field scattered by an 
echelette-grating in the short-wave approximation. International Symposium 
on Electromagnetic Theory. URSI. St.-Peters\-burg, Russia, 1995.

\noindent[2] Zalipaev V.V. Calculation of the wave field scattered by an 
echelette-grating in the short-wave approximation. TE case. Radio Science, 
to be published.

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