, Litrrow. Maxwell. Helmholtz Fredholm, . 40 Maystre [4 ], Goray [5 ], Kleemann [6 ] PACC: 4210, 4110H

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1 Π2008Π57 (06) Π ACTA PHYSICA SINICA Vol. 57,No. 6,June,2008 ν 2008 Chin. Phys. Soc. 3 1) 2) 1) g 1) (, ) 2) (, ) ( ; ),Littrow,,.,., Litrrow. :,, Littrow, PACC: 4210, 4110H 11 Maxwell., Helholtz Fredhol,., Petit [1,2 ], Pavageau [3 ] TE,TM. 40 Maystre [4 ], Goray [5 ], Kleeann [6 ],,,.,,..,.,,.,,. :,.,.,,,,,.,,, [7 ], [8 10 ],.,,. [5 ],,, TM.,,.,,, 3 (: ),, (:2006BAK03A02) (: ). g. E2ail : co China Acadeic Journal Electronic Publishing House. All rights reserved.

2 6 : [6 ],,.,, Littrow Littrow,,.,,,. [5,6 ],,,,. 21 1, i p, ( x),x d, h. Helholtz Flouqet [11 ],( y > h), z : = A 0 exp [j 0 x - j 0 ( x) ] B exp (j x + j y), (1) = - Rayleigh,Helholtz, j = - 1,, 0, 0, x y,a 0, B,A 0. 1 sin = sin i + d, (2) k = 2Π, K = 2Πd, = ksin = ksin i + K, (3) = kcos = j k 2-2 ( k > ), 2 - k 2 ( k < ). (4) = B 2 cos Πcos i, (5),B. Helholtz,,, B [11 ].,Helholtz 2 u = { 2 u} + d u p + ( n u p ), (6), u,{} : p,. u d u p u d u, p p. R, R R ( R ), G R, u d u, M ( x, ( x ) ) p,g p (6) : u = G 3 P d l= d = G[ x - x, ( x) - ( x ) ]( x ) d l, (7) ( x ) d x, (8) G( x, y) = 1 1 2j d 6 exp (j x + j y ) = -,, (7) (1), d 1 B = exp [ - j 2j d ( x) - j Kx ] 0 < ( x ) d x, (9) < ( x ) = ( x ) ( x ), (1),u i, n, u i n [11 ] : China Acadeic Journal Electronic Publishing House. All rights reserved.

3 d u i [ x, ( x) ], N ( x, x ) = 1 = - j[ ( x) ] exp [ - j 0 ( x) ]. (10) = - exp [j K( x - S - ( x) x ) + j ( x) - ( x ) ], (11) N ( x, x ), S. TE TM N ( x, x ), < ( x), < ( x ), d ui : d u i d = 1 2 < ( x) - N ( x, x ) < ( x ) d x, (12) 0 Fredhol, < ( x) = ( x) ( x). (0, d) x 1, x 2, x 3,, x p,, x J, J,, < ( x p ) (9) B. 31 (11),,, M, - M : M, (11) x - x = 0,, : N ( x, x ) = - ( x) - M 1 + j 2 0 = - M 1 ( x) ( x), (13) (13),,, x - x0. (11) N ( x, x ) = 1 M = - M ( x ) S - exp [j ( x) - ( x ) + j K( x - x ) ], (14) 2 M + 1,,. M,,, M,, M,,,M (2) Littrow 2sin i = - Πd ( ), (3) (4) = (2 - ) KΠ2, = - + = - (2 - ) KΠ2, (15) = = - +, = - + x, y, Π + Π = 0, (16), c = Π2,, Π Π c, (16) ( ),,. - 1 Littrow, = - 1, c = - 1Π2 ( ), 2, - 1 0, Π Π. 2,, (3) (4) Π = tan, Π = tan (16) + = 0, 2-1 0, ,= 2 T, T, Π2 = T, T China Acadeic Journal Electronic Publishing House. All rights reserved.

4 6 : , (14),a = ( x) - ( x ), b = x - x, - M ( M - 1), (16) N ( x, x ) = 1-1 = - M N ( x, x ) = 1 M- 1 M = 0 = 0 S - ( x) exp [j a + j Kb ] S - ( x) exp [j a + j Kb ], (17) S - ( x) exp [j a + j Kb ] + S + ( x) exp [j a - j Kb ]exp ( - j Kb), (18)., 1, 2. (18) (14),, 2 M + 1 M, 1Π2.., 1) : = 10 ;2) : d = 10, b = 15 (1 ),< = 90 (1 ) ;3) : J = 202 ;4) : Matlab, Intel Pentiu 2166 GHz, 1024kbyte Cache,13316 MHz,512Mbyte,Windows XP. - 1 Litrrow TE TM, 2Π3 <Πd = 1 < 2,, 0-1, 1, 2 1 1Π ( :s) TE TM TE TM ,,, M 2 M + 1,, = - - ( M - 1) = M,,, e,. N ( x, x ) = 1 - M- 1 = - S - ( x) exp [j a + j Kb ] S - = M ( x) exp [j a + j Kb ], (19) (3) (4) China Acadeic Journal Electronic Publishing House. All rights reserved.

5 = j - j k 2-2 j 8 k 4 3 +, 2 j = j K + ksin i, 1Π - jπ, Π j / j j j j j j j j / j j j j j j j j 2-1 Littrow, Π Π,,, ( - 1Π2),,, Π Π - j j., Π (19) : N ( x, x ) = [ S + j ( x ) ]exp [ ( 0 a - M Ka - j M Kb ] exp ( Ka + j Kb) [ S + j ( x ) ]exp [ ( - 0 a - ( M - 1) Ka + j ( M - 1) Kb ] exp ( Ka - j Kb) - 1, (20) = - - ( M - 1) = M. (18) e (20) 3, 1,2,3, 3 6, TE TM - 1,0 M. 4 TM TE , 1 2,; 3, 1 2, M > TE China Acadeic Journal Electronic Publishing House. All rights reserved.

6 6 : 3491., Littrow 6 TM ,, 2 1 1Π3,2 1.,, 2 1. (20),,, 3 2, 2,, 3 2,, Littrow,. = - i,,,(2) i = Π2 - sin - 1 [ Π(2 d)πcos(π2) ], (21),,,. = 8,3 Π Π., 20, ; < 20,.,, Π - Π,,.,, Π j, Littrow, (20), (14) (20) 4., 3 = 8 Littrow / j j j j j j j j / j j j j j j j j , TE, TM 4 1, 7,8., 1, 4., 7 TE TM China Acadeic Journal Electronic Publishing House. All rights reserved.

7 ) 4 1 ;2),, 4,. 41 Littrow,( 1) , e, 3 4. : 11 Littrow,, e,( 1), 3, Littrow. 31 Littrow, 4 1,. [1] Petit R 1965 C. R. Acad. Sci. Paris [2] Petit R 1966 Rev. Opt [3] Pavageau J, Eido R, KobeissH 1967 C. R. Acad. Sci. Paris [4] Maystre D 1978 J. Opt. Soc. A [5] Goray L I 2001 Proc. SPIE [6] Rathseld1 A,Schidtl G, Kleeann B H 2006 Coun. Coput. Phys [7] Zhou C H,Wang L,Wang Z H 2001 Acta Phys. Sin (in Chinese) [ ] [8 ] Bayanheshig,Zhu H C 2007 Acta Phys. Sin (in Chinese) [ ] [9 ] Zhu H C,Bayanheshig 2007 Acta Opt. Sin (in Chinese) [ ] [10] Bayanheshig, Zhu H C 2007 Opt. Precision. Eng ( in Chinese) [ ] [11] Petit R 1980 Electroagnetic Theory o Gratings ( New York : Springer2Verlag) p China Acadeic Journal Electronic Publishing House. All rights reserved.

8 6 : 3493 The kernel unctio nπs sipliicatio n and it s characteristic analysis in integral theory o diraction gratings 3 1) 2) Zhang Shan2Wen Bayanheshig 1) g 1) ( Changchun Institute o Optics, Fine Mechanics and Physics, Chinese Acadey o Sciences, Changchun , China) 2) ( Graduate School o Chinese Acadey o Sciences, Beijing , China) (Received 11 Septeber 2007 ; revised anuscript received 14 Noveber 2007) Abstract On the base o perect conductivity integral theory o diraction gratings,ater transoring the grating equation,the iage equation o diraction wave vector in Littrow ounting is given. Using the equation and in view o the power unction property, the ininite series,which represents the kernel unction,is transored into the su o a syetrical series and a geoetrical series. Copared with the priary integral ethod,ro the view o nuerical calculation,the new kernel unction can reduce the coputing tie and iprove the convergence. The extension results show that the power unction character can iprove convergence in the deviated Littrow ounting. Keywords : integral ethod o gratings, kernel unction, Littrow ounting, iage equation PACC: 4210, 4110H 3 Project supported by the National Natural Science Foundation o China ( Grant No ),the Specialized Research Foundation or the Gainer o Outstanding Doctoral Thesis and Presidential Scholarship o Chinese Acadey o Sciences,the National Key Technology R&D Progra the11th 52year Plan( Grant No. 2006BAK03A02 ) and Science and Technology Developent Project Progra o Jilin Province,China ( Grant No ). g Corresponding author. E2ail : co China Acadeic Journal Electronic Publishing House. All rights reserved.