57 6 2008 6 100023290Π2008Π57 (06) Π3486208 ACTA PHYSICA SINICA Vol. 57,No. 6,June,2008 ν 2008 Chin. Phys. Soc. 3 1) 2) 1) g 1) (, 130033) 2) (, 100049) (2007 9 11 ;2007 11 14 ),Littrow,,.,., Litrrow. :,, Littrow, PACC: 4210, 4110H 11 Maxwell., Helholtz Fredhol,., 20 60 Petit [1,2 ], Pavageau [3 ] TE,TM. 40 Maystre [4 ], Goray [5 ], Kleeann [6 ],,,.,,..,.,,.,,. :,.,.,,,,,.,,, [7 ], [8 10 ],.,,. [5 ],,, TM.,,.,,, 3 (:60478034),, (:2006BAK03A02) (:20070523). g. E2ail : bayin888 @sina. co 1994-2008 China Acadeic Journal Electronic Publishing House. All rights reserved. http://www.cnki.net
6 : 3487. [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) ] + + 6 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) 0 1 + 2 ( 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 ) 1 + 2 ( x ), (1),u i, n, u i n [11 ] : 1994-2008 China Acadeic Journal Electronic Publishing House. All rights reserved. http://www.cnki.net
3488 57 d u i [ x, ( x) ], N ( x, x ) = 1 = - j[ 0 + 0 ( 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) 1 + 2 ( 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) 41 + 2 ( 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. 3111 (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, - 2 + 1 Π Π. 2,, (3) (4) Π = tan, Π = tan (16) + = 0, 2-1 0, - 2 + 1.,= 2 T, T, Π2 = T, T. 1994-2008 China Acadeic Journal Electronic Publishing House. All rights reserved. http://www.cnki.net
6 : 3489 3121, (14),a = ( x) - ( x ), b = x - x, - M - 1 0 ( M - 1), (16) N ( x, x ) = 1-1 = - M N ( x, x ) = 1 M- 1 M- 1 + 6 = 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 1Π3. 1 1 2 ( :s) - 1 0 TE TM TE TM 1 2881594 2911313 2891146 2921543 2 1011456 1021157 1011785 1031589,,,. 3131 M 2 M + 1,, = - - ( M - 1) = M,,, e,. N ( x, x ) = 1 - M- 1 = - S - ( x) exp [j a + j Kb ] + + 6 S - = M ( x) exp [j a + j Kb ], (19) (3) (4) 1994-2008 China Acadeic Journal Electronic Publishing House. All rights reserved. http://www.cnki.net
3490 57 = j - j k 2-2 j 8 k 4 3 +, 2 j = j K + ksin i, 1Π - jπ, Π j. 0 1 5 10 20 40 60 80 100 / 015774-113416j - 110170j - 110046j - 110012j - 110003j - 110001j - 110001j - 110000j - 1-2 - 6-11 - 21-41 - 61-81 - 101 / - 015774 113416j 110170j 110046j 110012j 110003j 110001j 110001j 110000j 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) - 1 + [ 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 - 1 3 TE - 1 3 6, 1 2,; 3, 1 2, M > 80. 5 TE 0 1994-2008 China Acadeic Journal Electronic Publishing House. All rights reserved. http://www.cnki.net
6 : 3491.,. 3151 Littrow 6 TM 0 3141. 2 1,, 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 0 1 5 10 20 40 60 80 100 / 016766-113027j - 110166j - 110045j - 110012j - 110003j - 110001j - 110001j - 110000j - 1-2 - 6-11 - 21-41 - 61-81 - 101 / - 014895 113901j 110173j 110046j 110012j 110003j 110001j 110001j 110000j. 10 30, TE, TM 4 1, 7,8., 1, 4., 7 TE - 1 8 TM - 1 1994-2008 China Acadeic Journal Electronic Publishing House. All rights reserved. http://www.cnki.net
3492 57 1) 4 1 ;2),, 4,. 41 Littrow,( 1) 2. 2 1, e, 3 4. : 11 Littrow,, e,( 1), 3,. 21 3 Littrow. 31 Littrow, 4 1,. [1] Petit R 1965 C. R. Acad. Sci. Paris 260 4454 [2] Petit R 1966 Rev. Opt. 45 249 [3] Pavageau J, Eido R, KobeissH 1967 C. R. Acad. Sci. Paris 264 424 [4] Maystre D 1978 J. Opt. Soc. A. 68 490 [5] Goray L I 2001 Proc. SPIE 4291 1 [6] Rathseld1 A,Schidtl G, Kleeann B H 2006 Coun. Coput. Phys. 1 98 [7] Zhou C H,Wang L,Wang Z H 2001 Acta Phys. Sin. 50 1046 (in Chinese) [ 2001 50 1046] [8 ] Bayanheshig,Zhu H C 2007 Acta Phys. Sin. 56 3893 (in Chinese) [ 2007 56 3893 ] [9 ] Zhu H C,Bayanheshig 2007 Acta Opt. Sin. 27 1151 (in Chinese) [ 2007 27 1151 ] [10] Bayanheshig, Zhu H C 2007 Opt. Precision. Eng. 15 1 ( in Chinese) [ 2007 15 1 ] [11] Petit R 1980 Electroagnetic Theory o Gratings ( New York : Springer2Verlag) p35 1994-2008 China Acadeic Journal Electronic Publishing House. All rights reserved. http://www.cnki.net
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 130033, China) 2) ( Graduate School o Chinese Acadey o Sciences, Beijing 100049, 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. 60478034),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. 20070523). g Corresponding author. E2ail : bayin888 @sina. co 1994-2008 China Acadeic Journal Electronic Publishing House. All rights reserved. http://www.cnki.net