Scalar diffraction and vector diffraction using Fourier analysis. Yasuhiro Takaki. Tokyo University of Agriculture & Technology. Faculty of Technology

Koko: px

Aloita esitys sivulta:

Download "Scalar diffraction and vector diffraction using Fourier analysis. Yasuhiro Takaki. Tokyo University of Agriculture & Technology. Faculty of Technology"

Aapo Saarinen
5 vuotta sitten
Katselukertoja:

1 Scalar diffraction and vector diffraction using Fourier analysis Yasuhiro Takaki Faculty of Technology

2 Maxwell RCWA

3 : F F I G G ; Maxwell! " # $ % & ' ( ) * +, -. / 0. 1 ' 2 3 $ , : ; < = > 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) *

4 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

5 O Ã / å Ã Þ / / / / œ Helmholtz º»h¼ ½ ¾» Â 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 6 +, -. & /! " # $ % & ' $ ( ) * # +, -. / / " $ " # / $ 3 $ " * # / ' 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

7 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~

8 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 ÁuÂwÃ 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

9 # " "! $ % $ 3 4 KxKx E $ $ & ' ( ² qm ± wm ² wi,m ) * *,+ -. / 0 ± 1 2

10 $ Y KKK KKK O 2 TMç Ú è TEç Ú è : ;=< < >@?@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): 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.

11 ` 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} ~ 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 ³ µ ÁÃÂÅÄ ÏÅÐ Æ,Ç È,É ÊÌË Í,Î ÑÅÒ uwvvvvvvvvv x

12 ú Ø ü ú Ø ü øù ý 3ý Õ Ö Ø Ù Ú Û Ü á â ã ä å æ ç è éëê@ìîí ïãð ïòñôóöõ (a) TMû øù ý 3ý á â ã ä å æ ç (b) TEû Õ Ö Ø Ù Ú Û Ü þ ÿ Ý Þ ß à

13 . 1 /0. 1 /0,,. 1 /0, 3 / TM TE <>=@?+ACBED"F+G "!$# %'&)(+* #"; (a) 9+: HJI KJL m = 2 : / TE TM WJX <>=@?+ACBED"F+G "!$# %'&)(+* #"; (c) 9+: YJZ T6U8V m = 8 : / TM TE PJQ RJS <>=@?+ACBED"F+G "!$# %'&)(+* #"; (b) 9+: M6N8O m = 4 : m

14 wx rs Š { f g z y uv t oqp [rad] ~} J ƒ TE TM h i j h k l m n F [ \ ] ^ _ ` J ˆ a b c d e Œ Ž

15 š œ ž œ Ÿ Ž ² Ž ³ µ ½ ¾ª «± ¹ º» ¼ À ¼ Á

16 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 = µm)

17 È É Img 0 amplitude Ý Þ ß p = 0.1 λ fill factor = 0à 1 Rel phase Positive grating Negative grating

18 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 , ú -. / 0 1 2! " # $ 2, ú -. / 1 % & ' ( ) *

19 9: p = λ0/4 p = λ0/ ; p = λ0/6 p = λ0/ [nm] λ0 = 550 nm d = 0.5 λ0

20 < = > <? >? A = B C D B B