CC BY
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y^K 517.968.519.6.
Nesvit K.V.
PROPAGATION AND DIFFRACTION OF TM WAVE ON PERIODIC IMPEDANCE GRATING
ABOVE A FLAT SCREEN REFLECTOR
Boundary-value problems in integral equation form for the stationary wave equations lead to problems of diffraction on plane-parallel structures from the theory of electromagnetic waves. They have been studied in the monograph [1] and in papers [2] - [4], [9], [10]. Problems of modeling the interaction of electromagnetic (EM) radiation with pre-fractal [8] structures, among which are the limited diffraction gratings with the flat screen reflector [1], [9], have been actual to the present day.
The overall aim of this work is to build a discrete mathematical model of the diffraction problem of a plane monochromatic TM wave on periodic impedance pre-Cantor strips, placed above a flat screen reflector. The discrete mathematical model is based on hypersingular integral equations (HSIE) of the 1st kind and on the Fredholm integral equations of 2nd kind. It allows one to perform numerical simulations of this diffraction problem with the help of an efficient discrete singularities method (DSM).
Diffraction problem of TM mode
We consider the diffraction problem of an H-polarized plane wave on periodic impedance pre-Cantor strips, located above a flat screen reflector. In this case the components of electric and magnetic fields are
(Hx ,0,0), (o, Ey, Ez ). Propagation direction of a plane wave is given by the direction of wave vector k , where k = sfj,<a (Fig. 1).
Strips«"» /X // z / y/ SS
" ArAr,
r Reflector ^^
Fig. 1. Schematic of the considered diffraction structure. Strips(N) = j(x,y,z)g y g Stz = o}
st 2N} = +u u (s
r=-œ q =1
aN + 2l ■ r, bN + 2l ■ r )
(1)
For convenience let us switch to dimensionless coordinates and 2j period:
^ = ^y, C=-z, * = -k, d h, K= l-k= *aN, PN = *bN,
l l j l J q l q q lq
i ç sC n
a, p; a2 p2 o «3 Pi a4 p4 %
-d
Fig. 2. Cross-section of the diffraction structure in O plane.
2n t \ ) = IN №.fiN)
q=1
To solve a 2D diffraction problem we calculate the total field H g (Ç, 0) which satisfies Maxwell's
equations, supplemented with Shchukin-Leontovich impedance boundary conditions. Besides, the total field must also satisfy the Sommerfeld radiation conditions and the Meixner edge condition. The H-polarized plane wave falls from infinity at an angle a :
uNnc(ï,Ç) = Hç(t,C) = cosa). (2)
The only non-zero component of magnetic field satisfies all the aforementioned conditions and also the two-dimensional Helmholtz equation off the metallic strips above flat screen reflector:
a2 a2
Hç(Ç,Ç) + ^Hç(Ç,O + kHç(ÇO) = 0, -d <0< 0. (3)
aç 6 ao 6 6
The total field u( N ) (Ç, 0) = H g (Ç, 0) is considered to be in the form of
(N) U (to)+uN (to), o> o,
{^(ÇO)+u-N(t,0), -d<o<o,
where u(N (y, z) is a known solution to the Helmholtz equation. It represents the sum of incident and reflected waves in flat reflector. The functions u;N) (Ç, 0), u-N) (Ç, 0) are considered as Fourier series:
uN (ÏO) = SCf; 0, 0> 0, (5)
s= —œ
where y(As ) = — K2 , As =~j~, s £ Z . In this case the Helmholtz equation is satisfied for uN (Ç, 0) , 0 > 0. The radiation condition will be fulfilled if ys is given by Re(^s )> 0, ImO(ys )< 0 ;
u—N(to) = I.CN— Z(As0)easÇ, — d <0< 0, (6)
s = -<x>
where
( ) = Bsh{y(As )(C + d)) + y(As )ch (y(As )(C + d)) (s Bch (/(As )d)+y(As )sh(y(As )d) '
In this case, the function u—_ (£,, £) (6) satisfies the Helmholtz equation and the following conditions for the function Z(As,£), — d <C < 0 are kept:
ZZ (As ,0) = /(As), Z Z (As ,—d)—BZ (As —d) = 0, so that the boundary condition on the flat reflector is fulfilled.
In this paper the incident angle is considered as zero. As shown in the monograph [1] and paper [2] the boundary value problem considering all mentioned conditions is reduced to two systems of coupled integral equations:
+TU1 N _ 7
A ^s + Cs-Zs
{0%ist= 0, Çg CSt( ( N)
2n
s = -T +T
+Û7s + B)CN+ -(rs + BZs(0))CN-^ = 2ôUN(Ç,0), Ç g Stg)
s=-œ
ôÇ
(7)
1rs (cN-+ c^= 0, Çg CSt2 n),
s= -T + T
rs + BC+ + (rs + BZs (0))CsN- = -2BuN (£0), Ç g Stg).
(8)
Let us introduce two new unknown functions FjN rf), FN rf) and representations of unknown coefficients CN, CN :
s + 5 s —
fNrf) = Î(cN+-CN-Zs(0psÇ, FNrf) = +irs(cN-+ CN+ (s))eisÇ,
(9)
CsN+-CsN-Zs (0) = ± J Fn rf)e-isrfdrf, CN-+ CN+=J f2N rf)e-isrfdrf s g Z. (10)
2™St(N )...... s - s + 2™rss6N)
2n
2n
Using parametric representations of hypersingular integral operators of periodic functions and integral operators with logarithmic kernels this problem is reduced [8] to hypersingular integral equation of the first kind on a set of intervals, and the Fredholm integral equation of the second kind:
- J FNrf\dn +Cl Jlnrf-Ç|F1N(rf)drf +1 JQNrfÇF(rf)drf +
n St(N )rf-Ç) n St(N) n St%)
n St(N)
+ - J QN (rfÇ)FN (rf)drf = fiN (Ç), Ç g St2n),
F2N (Ç) + C2 Jlnrf-Ç|F2Nrf)drf +1 JQ3N (rf,Ç)F2N (rf)drf +
n St(N) n St(N)
2n 2n
+ C3 FiN (Ç) + - J QN (rf,Ç)FiN (rf)drf = f2N rf), Ç g St2n),
n st2n)
(11)
where f(N) rf), £(N) rf), ^(N) rf, Ç), Q(N ) rf, Ç), N ) rf, Ç), tf(N )rf,Ç) are known functions.
< N )
( N )/
(N)
Discrete mathematical model of the hypersingular integral equations (HSIE)
To reduce the equation (11) on St(N) to equations on a set of intervals StqN = (aq ,ftq )•
q = 1,2N, we introduce restrictive conditions to functions:
<
<
s = -T
s = -T
s = -T
fN h) = FN h)^N), FNp (t) = FN (t)
T7N,
teSt<qN )s
fNp(t)=fN(t), „(N),i=1,2,p,q=1,2N.
N
(12)
teStqN )
The Meixner condition will be satisfied if unknown functions (8) are in the form of:
FNq h)=vNq n/(fiN—nh—aN ), q=12N,t=1,2, (13)
FNp (t) = vNNp (t)J (fiPN —t\t—aN ), P = = 12 (14)
where functions vNq (tf), vNp (£),i = 1,2, p, q = 1,2N, are Holder continuous.
By choosing a normalized interval (-1,1) the variables transform accordingly:
,q -V ' V", .r, - ■ - o, W 2 2 .
nN _ N nN t
g™ ■ (-1,1) ■-> ' » sf\t) + ^—«
h=g?\t\ t=gqN\t0), it < 1 k| < 1, h £ stqN), t £ StpN\
Then, for < 1, |t0| < 1, p, q = 1,2N :
nN _ N nN _ N
FNq (gqN\t)) = vNq (<)^^ fi-?2, FNp (&%)) = vNp(t)fi-tl (16)
Considering (12)-(16) and excluding logarithmic singularity at p=q from equations (11), we obtain a system of boundary hypersingular integral equations of the 1st kind and Fredholm integral equation of the 2nd kind on a
normalized interval for p, q = 1,2N :
1} vjAgï^l^dt + Cl) ln| t —10 \viP ( g f )(.t ))& n —1 (t —10) x—1
+1} PN (t,t 0)vNp ( g pN\t))41—^dt +
n .
1}pN(tt0)vNp(gpN)(t))^dt=fN(gpN\t)), \t0\< 1, n—1 (16)
fiN—aN 1 _
^^vNp ( g (pN ) (t0 ))+C21 in 11—10 \vNp ( g (pN ) (t ))41—ïdt+
2 n —1
J ONN N
+if p3N (tt 0)vNq ( g'N\t ))J1—Sdt C ^^a^vNp ( g pN )(t0)+
n —1 2
1}pN(tt0)vNq(gqN\t))41—Ndt=fN(gpn\t0)\ \t0
+- J PN (t,t o)vNq ()(t))^1—t2 dt = f2N (g p )(to)), 10 < 1
n —1
Discrete mathematical model has been developed with the help of an efficient numerical method DSM [1], [5], [6]. The unknown functions and smooth functions are interpolated by a Lagrange polynomial of (n-2) -th degree in the nodes which are the zeros of Chebyshev polynomials of the 2st kind. From (16) we have obtained a system of approximate solutions. In the next step, using quadrature formulas [7] for integrals with hypersingular and logarithmic singularity and integrals of smooth functions, we derive a system of linear algebraic equations (SLAE) for the values of unknown functions in the node points. Therefore we have obtained
a SLAE for p = 1,2N :
f
2 N z
q=1
n-1
v vN '(n-2>(cr(N )(t n \\ (1 ( 1 )(1 (t°j) ) 1 vN,(n-2Ucr(N Ufn Wn z v1,p (gp (toj))-———---V1,p (gp (tok))-
N ,(n - 2), ( N)(. n
j=1 V j * k
(tok -'oj)2
n
n-2WN)ftn )) n
J
r 2N n-1
-^ z z vNf-2)(gpN)(tonj))(1 -(tonj)2)
n q=1 j = 1
1 2N n-1
n-1 Tk (t" ) (-1)j
ln2 + Tk (tn0k ) + Tk (tn0k )
k=1 k n
+1 z z < p (tonj, tonk )vNNq(n - 2)(gqN)(tonj ))(1 - (tonj )2)+
+
n q =1 j =1
1 2N n-1
+1 z z Piiq,p(tn0j,tonk-2)(g(qN)(tonj))(1 -(tonj)2) = f1N,(n-2)(gpN)(tonk)),
n q =1 j =1
pN -*n
-vNjn - 2)( g (N )(tonk)) -
a 2N n-1
- ^ z z vN(n - 2)( g pN )(tonj ))(1 - (tonj )2) n q=1 j=1 'p p j
n-1 Tk (tSj) (-1)j
~ k (j Tk (tn0k) + ^ Tk (tn0k)
ln2 + 2 z
k =1 k
+
?N _i rN _ N
+1 z "z PN,p ('ij. & )vNqr -2,(gf)('Si ))(1 - ('Sj )2)+a ^--^-v*«*-2'(gpV)(tSk))+
n q =1 j =1 2
1 2N n-1
1 z z P
n q =1 j =1
+ 1 z z PNq, p (t0j , tik )vWq(0 - 2)( gqN )(t0j ))(1 - (t0j )2) = fN,(0 - 2)( g p )(tSk )).
Discrete mathematical model of the diffraction problem of H-polarized EM waves on the gratings, consisting of periodic impedance pre-Cantor strips with an impedance flat reflector underneath, has been developed with the help of an efficient DSM. As a next step we plan to perform numerical simulations and analyze the performance of this approach for both E- and H-polarization cases.
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REFERENCES:
1. Gandel Yu. V., Dushkin V. D., 2012, Mathematical models of two-dimensional diffraction problems: singular integral equations and numerical discrete singularities method (in Russian). Monograph. Kharkov: Academy of Internal Troops of the Ministry of Internal Affairs of Ukraine.
2. Gandel Yu.V., 2010, Boundary - Value Problems for the Helmholtz Equation and their Discrete Mathematical Models. Journal of Mathematical Sciences, Springer Science + Business Media Inc., Vol. 171, No. 1, pp. 74 - 88.
3. Gandel Yu. V., 2004, Parametric Representations of Integral and Psevdodifferential Operators in diffraction Problems. Conference Proceeding 10th International Conference on Mathematical Methods in Electromagnetic Theory. Dnipropetrovsk, Ukraine, pp. 57-62.
4. Gandel Yu. V., 2003, Double and hypersingular integral equation of the diffraction problem of electromagnetic waves on the plane gratings and screens (in Russian). Proceedings of the XI International Symposium on "Methods of discrete singularities in mathematical physics", pp. 53-58
5. Lifanov I.K., Poltavskii L.N., Vainikko G.M. Hypersingular Integral Equations and Their Application. - London: CRC Press, 2004.
6. Gandel Yu.V., Eremenko S.V., Polyanskaya T.S., 1992, Mathematical problems in the method of discrete currents. Justification of the numerical method of discrete singularities of solutions of two-dimensional problems of diffraction of electromagnetic waves (in Russian). Educational aid. Part II. Kharkov: Kharkov State University.
7. Gandel Yu. V., 2002, Introduction to methods of evaluation of singular and hypersingular integrals (in Russian). Textbook. - Kharkov.
8. Benoit B. Mandelbrot, 1983, The Fractal Geometry of Nature, W.H. Freeman and Company, New York.
9. Nesvit K. V., 2012, Hypersingular integral equations of diffraction problems on impedance periodic gratings above the impedance plane (in Russian). Bulletin of Kherson National Technical University 2 (45), pp.261-266.
10. Nesvit K. V., 2012, Hypersingular integral equation of wave diffraction problem on pre-cantor grating and its discrete mathematical models. Proceedings of the International Conference "Days on Diffraction" 2012, St. Petersburg, Russia. IEEE Catalog No.: CFP12489-ART (Electronic media), CFP12489-PRT (Print). ISBN: 978-1-4673-4419-7 (Electronic), 978-1-4673-4417-3 (Print). pp. 183188.
NESVIT K. V. - PhD student of Department of Mathematical Physics and Computational Mathematics, Karazin Kharkiv National University.
Research Interests - mathematical and computer modeling, computational methods, singular and hypersingular integral equations
