Electromagnetic wave diffraction by semi-infinite discrete structures

The theory of wave diffraction by semi infinite set of obstacles is important for the electromagnetic field analysis in bounded photonic band-gap structures. Methods for solving the problems of wave diffraction by semi-infinite discrete structures have been developed much worse than the theory of wave diffraction by semi-infinite structures that have Wiener-Hopf geometry.

The spectral operator *R* of reflection by a semi-infinite
periodic structure is most essential in the developed theory. This operator
assigns to an incident field with some known space spectrum the reflected field
generated by diffraction.

A semi-infinite system of partly transmitted layers
is the one with the infinite number of identical obstacles. The neighbour
obstacles are equally spaced at the range *h*
along the *Oz* axis. This discrete
semi-infinite structure isn’t a periodic one but can be transformed to self by
space displacement *h* to the positive
direction along the *Oz* axis and
reducing one edge structure element. Such specific geometry can be used
efficiently for solving the problem of wave diffraction by semi-infinite system
of partly transmitted layers.

In the concept of this method it isn’t important what kind of obstacle would be a partially transmitted layer. For example it may be a plane-parallel dielectric slab, metal screen, or diffraction array. It is important that solution was known for the problem of wave diffraction by a single obstacle, i.e. operators were known for both the transmission and reflection by a single layer. Therefore further we will consider the semi-infinite system of the periodic strip gratings in the dielectric half-space. However all the below operator equations will be correct for any other kind of obstacles.

According to the Floquet theorem, the scattering
electromagnetic field over and under a single grating can be represented by a
series of partial waves. The vectors of space partial wave amplitudes in the
region _{} and _{} satisfy to equations

_{}

where _{} are the operators of
transmission by the boundary _{}:

_{}, _{};

_{} are the operators of
reflection by the boundary _{}:

_{};

_{} and _{} are the propagation
constants of the *n*-th space partial
wave along the *Oz* axis in the free
space and in the dielectric, accordingly; *R*
is the operator of reflection by a semi-infinite system of strip gratings in
the dielectric half-space with permittivity _{}; _{} is the transform
operator of space partial wave vector by the forward displacement _{} along the *Oz* axis.

Let us analyse the field in the region _{} to determine the
operator _{} in the above system
of equations. In this region the equations for the vectors of space harmonic
amplitudes are

_{}

Here _{}and _{}are the transmission and reflection operators of the single
strip grating in the dielectric [L.N. Litvinenko and S.L. Prosvirnin, *Spectral scattering operators in the
problems of wave diffraction by plane screens.* – Kiev: Naukova dumka, 1984.
– 240 p. (in Russian)]; _{} is the transform
operator of amplitude vector of space partial waves by the field displacement *h* along the *Oz* axis. We can evaluate *B*,
then *C* and *b*, and obtain the operator equation

_{} (1)

to determine the unknown matrix operator *R*, *I *is the identity
operator.

Equation (1) is the non-linear operator equation. The generalized Newton method can be used for solving this equation.

The electromagnetic field is the sum of incident wave and space partial waves in the free half-space over the semi-infinite structure. It can be written as the following expression

_{}

Complex amplitudes of reflection space partial waves could be found as

_{}

Thus, the solution of the problem is reduced to the solving of
operator equation (1) relative to *R*
by an iterate method.

Some numerical results respect to the electromagnetic wave diffraction by semi-infinite periodical structures of dielectric slabs and strip gratings may be found in the papers referenced below.

A field incident on a semi-infinite periodic
structure excites the eigenfield of the corresponding infinite periodic
structure. The operator *R* can express
the transmission operator, which permits finding the vector of spectral
amplitudes of the excited eigenfield. The set of eigenwaves can be analyzed and
the expression for finding the propagation constants for these waves can be
obtained using the known operator *R*.
Furthermore, if the eigenwave in a semi-infinite structure is propagating
towards the free-space boundary, the reflection and transmission operators for
such field can also be expressed by *R*.
The operator method, taking the interaction of waves reflected by the
free-space boundaries into account, has been used in obtaining the reflection
and transmission operators for a periodic structure with finite number of
screens. Operator *R* can express these
operators also. Thus the knowledge of operator *R* allows obtaining a completely phenomenological characterization
of the electromagnetic properties of an infinite periodic screen sequence, as
well as of its semi-infinite or finite-layered parts.

References

[1] Litvinenko L.N., Reznik I.I., Litvinenko D.L., Wave diffraction
by semi-infinite periodical structure // *Dokladi AN Ukraine*, 1991, no. 6.

[2] Litvinenko L.N., Pogarsky S.A., Prosvirnin S.L., Wave diffraction by semi-infinite periodical structures, *Journal of Infrared and Millimeter Waves*, 1996, v.17, ïî.5, pp.897-903.

[3] Litvinenko D.L., Litvinenko L.N., Prosvirnin S.L., Reznik I.I., Wave diffraction by semi-infinite system of partially transmitted layers. - *Proc. of the VI Inter. conf. on Mathem. Methods in Electromag.
Theory*, 1996, Lviv, Ukraine.- pp. 96 - 99.

[4] Ëèòâèíåíêî Ë.Í., Ïðîñâèðíèí Ñ.Ë., Àíàëèç äèôðàêöèè âîëí íà ïîñëåäîâàòåëüíîñòè èäåíòè÷íûõ ëåíòî÷íûõ ðåøåòîê. Ìíîãîâîëíîâûé ðåæèì, *Ðàäèîôèçèêà è ðàäèîàñòðîíîìèÿ*, 1999, ò.4, ¹3, 276-286 __[PDF-1.14M]__.

[5] Lytvynenko Leonid, Prosvirnin Sergey, Schunemann Klaus, Wave scattering by semi-infinite structure Programme poster presentations, International Union of Radio Science XXVII-th General Assembly, Maastricht, Aug. 17-24, 2002, report B2P.12, paper 911.

[6] Lytvynenko L. M., Prosvirnin S. L., Schuenemann K., Wave diffraction by periodic multilayered structures, *Radio Physics and Radio Astronomy,* 2005, vol. 10, Special issue, pp. S186-S201 __ [PDF-246K] __.