Method of eigenvectors for numerical studies of multilayer gratings 

V.I. Erofeev

e-mail Yerofeyev@iae.nsk.su 
Institute of Automation & Electrometry, Academy of Sciences, RUSSIA 

N.V. Kovalenko

e-mail koval@sunsr.inp.nsk.su 
Budker Institute of Nuclear Physics, Academy of Sciences, RUSSIA

A new method for numerical study of ideal multilayer gratings (MG) with
rectangle grooves is developed. In this method electric field inside MG
is regarded as series of special solutions to Helmholts equation -- some
"natural harmonics." The latter solutions are based on eigenvectors of
transition matrix for etched bilayer. This matrix describes
transformation of field vector -- a vector composed of electric field
Fourier components and their derivatives, -- from the bottom surface of
bilayer to its top surface. The natural harmonics are the convenient
substitution to plane waves. Half of the harmonics correspond to
eigenvalues with absolute value smaller than unity , the other half --
to those with modulus greater than unity . The natural harmonics of the
first type carry energy from the top of multilayer towards the substrate
surface, those of the second type-- in opposite direction. These two
types of natural harmonics are treated different way in iterative
procedure in tended to reveal picture of diffraction. Approximation of
the lo w est order to diffracted field satisfies b oundary conditions at
the top of MG in assumption of infinite width of the grating. For such a
grating field consists only of natural oscillations of the first type
that are dying in depth of multilayer. Through boundary conditions at
the bottom of multilayer their amplitudes yeld amplitudes of natural
harmonics of the second type. Matching electric field above the grating
to this the second type natural harmonics, one obtains correction to
refracted field and some correction to incident field. The latter
correction should be compensated since it distorts real picture of
diffraction. Successive iterations results in its compensation. This
method works sufficiently well in many extreme situations where its
ancestors fails. It can be applied even to study performance of MG in
hard X-ra ys with grazing angles of radiation incidence. Some other
applications of the method are under consideration.