Scalar diffraction and vector diffraction using Fourier analysis Yasuhiro Takaki Faculty of Technology
Maxwell RCWA
: F F I G G ; Maxwell! " # $ % & ' ( ) * +, -. / 0. 1 ' 2 3 $ 4 5 6 7 8 9, : ; < = > 2? @ 4 A B C %! D E < A B C -. / 0. * 2 J, K C < L C M N % P # B Q, x,y,zh Helmholtz O H I R ) " ( ) g(x,y,z) *
S T R U _ b ` _ a s r / u t Helmholtz z * R U g(x, y; z) " V W X D Y G(νx, νy; z) " Z C 4 # 9 [ \ νx νy & ' ] R ) Helmholtz _ ` 2 P ^ _ a ` a ced c fhg k l $ % % G(νx, νy; 0) z=0 * vxwzy {}w~ g(x, y; 0) V W X D Y npo q m ƒx }ˆŠ Œ Ž x } Š c iej
O à / å Ã Þ / / / / œ Helmholtz º»h¼ ½ ¾»  ¼ ½ ¾ À Á» À Á x á â Þ ß 2 & ' ] R ) " Ä Å Æ ) Ç U R È É Ê Ë Q y νx νy š š & ' ] R ) " Ä Å Ç U R 2 3 $ 4 l Í Î C 4 Ï Ð * % νx Ñ νy xì yì θx θy Ò / Ó Ô Õ, G(νx, νy; 0) Ç U R Ö S Ø Ù Ú Ù Û ÜzÝŠÙ á ã Þ à á Ö ä Ê æhç æ è é æ êhë Ö ä Ø ì í î ï ð Ï ñ ò ó ô õ Ø ö î ï ð ø ò ù ú û ü ý þ ÿ pž Ÿ ª «x } Š Œ ± x z } ò ó ô õ ² ³x µ} Š ¹
6 +, -. & /! " # $ % & ' $ ( ) * # +, -. / / " $ 0 1 2 " # / $ 3 $ " * 4 1 5 # / ' 5 / 8. 9 : g(x, y; 0) 6 7 G(νx, νy; 0) $ ; ) * / 8. 9 : E & ' 3 $ ) * 8 F < = > =? @ D >? @ A B 8 - G H. 8 I / J = ACB
L S K M L M š š S T r q p o l k n m T ON P Q R ï U õ V ó W X ð Y ä ó Z ä P x, y, z[ \ ] ^ _ ` ò ó e f a U õ dn _ ` g h. j TE(Transverse Electric) i X œ Y ž Ÿ θ y[ \ P sutwv TM(Transverse Magnetic) i X œ Z ž Ÿ θ. j y[ \ P TM V ó a ï b ô c xzy TE TE,TM ª «± ² ˆ Š Œ Ž sut~ xzy {w} ƒt~
ô õ å Ø å ú Ý æ å Þ é À ü ø ¾ ½ ê º ¹ ¼» Ö Õ Rigorous Coupled-Wave Analysis (RCWA) ³ µ TE(Transverse Electric) ç Ú è é ê ú û ü ö ø ë ì ù ßáàãâ ý þ ÿ ² Ù Ú Û Ü Λ: ë ì í εh: î ï ð ñ ò ó Si: i ó ÁuÂwà TM TE ä ä ä ü ó coupling ü î ï ð ñ ò ó εi ó Si ÿ εi-p ÄzÅ Ñ Ò ÁuÂÈ É Ë ÌˆÍŠÎŒÏŽÐ ÄzÅ ÆwÇ Ó Ô ÊƒÂÈ É
# " "! $ % $ 3 4 KxKx E $ $ & ' ( ² qm ± wm ² wi,m ) * *,+ -. / 0 ± 1 2
$ Y 5 6 6 5 2 KKK KKK O 2 TMç Ú è TEç Ú è 7 8 9 : ;=< < >@?@A@BDCFE G HIIIIIIIIIJ $ S 0 T U V W Ø S U X 7 Z [ \ [ ² Û Ú 2 ] LNM PQQQQQQQQQ R U 9 LAPACK (CLAPACK): http://www.netlib.org, http://www.phase.etl.go.jp/netlib The Netlib repository contains freely available software, documents, and databases of interest to the numerical, scientific computing, and other communities. The repository is maintained by AT&T Bell Laboratories, the University of Tennessee and Oak Ridge National Laboratory, and by colleagues world-wide. The collection is replicated at several sites around the world, automatically synchronized, to provide reliable and network efficient service to the global community.
` a i g e» ¾ ¼½ f ¹ t Ó Ô t TMç o è TEç o è ë ì g h h ^ _ i j k l m l n b c d À ¼ º» 0.50 y{z} ~ 0.45 0.40 0.35 0.30 0.25 0.20 0.15 TM TE TM TE p qrrrrrrrrrs ƒ z} ~, ƒˆš {Œ} Ž Š ƒ Œ} Ž, Š ƒ Š { } š Šœƒ ž }Ÿ š œƒ Š { } Š ƒ ª }«ƒ Š 2 4 6 8 10 12 14 16 18 20,± ² ³ µ ÁÃÂÅÄ ÏÅÐ Æ,Ç È,É ÊÌË Í,Î ÑÅÒ +3 +1 0-1 -3 uwvvvvvvvvv x
ú Ø ü ú Ø ü øù 0.45 0.40 0.35 0.30 0.25 0.20 0.15 0.10 0.05 1ý 3ý Õ Ö Ø Ù Ú Û Ü 0 0 1 2 3 4 5 6 á â ã ä å æ ç è éëê@ìîí ïãð ïòñôóöõ (a) TMû øù 0.45 0.40 0.35 0.30 0.25 0.20 0.15 0.10 0.05 1ý 3ý á â ã ä å æ ç (b) TEû Õ Ö Ø Ù Ú Û Ü 0 0 1 2 3 4 5 6 þ ÿ Ý Þ ß à
. 1 /0. 1 /0,,. 1 /0, 3 /2 -. 1.0 0.8 0.6 0.4 0.2 0 TM TE <>=@?+ACBED"F+G "!$# %'&)(+* #"; 46587 2 4 6 8 10 12 14 16 18 20 (a) 9+: HJI KJL m = 2 : 0.389 3 /2 -. 1.0 0.8 0.6 0.4 0.2 0 TE TM WJX <>=@?+ACBED"F+G "!$# %'&)(+* #"; 2 4 6 8 10 12 14 16 18 20 (c) 9+: YJZ T6U8V m = 8 : 0.912 3 /2 -. 1.0 0.8 0.6 0.4 0.2 0 TM TE PJQ RJS <>=@?+ACBED"F+G "!$# %'&)(+* #"; 2 4 6 8 10 12 14 16 18 20 (b) 9+: M6N8O m = 4 : 0.778 m
wx rs Š { f g z y uv t oqp [rad] 7 6 5 4 3 2 1 ~} J ƒ TE TM 0 0 0.2 0.4 0.6 0.8 1.0 h i j h k l m n F [ \ ] ^ _ ` J ˆ a b c d e Œ Ž
š œ ž œ Ÿ Ž ² Ž ³ µ ½ ¾ª «± ¹ º» ¼ À ¼ Á
È Ì ÊË É Â Ã Ç Í Î Ï Ð Ñ Ò Ó Ô Õ Ö Ï Ø Ù Ú Û Ü Normal incidence Ä Å Æ Normal incidence Positive grating Negative grating (Si, λ = 0.6199 µm)
È É 1 1 0.5 0.8 Img 0 amplitude 0.6 0.4-0.5 0.2-1 -1-0.5 0 0.5 1 Ý Þ ß p = 0.1 λ fill factor = 0à 1 Rel 0-3 -2-1 0 1 2 3 phase Positive grating Negative grating
á â ã ä å æ ç è é ê ë ì í á î ï ð î ñ ò ó ô õ ö ë ø ù ú û ü ð ý þ ö ÿ á â ã ä ú ã ä λ λ 1 1 2 +, ú -. / 0 1 2! " # $ 2, ú -. / 1 % & ' ( ) *
9: p = λ0/4 p = λ0/8 1.00 ; 0.98 0.96 p = λ0/6 p = λ0/2 0.94 0.92 3 4 5 6 7 8 0.90 400 450 500 5 6 550 600 650 700 [nm] λ0 = 550 nm d = 0.5 λ0
@ @ < = > <? <? @ >? A = B C D B B