2D-RCWA 1 two dimensional rigorous coupled wave analysis [1, 2] 1 ε(x, y) = 1 ε(x, y) = ϵ mn exp [+j(mk x x + nk y y)] (1) m,n= m,n= ξ mn exp [+j(mk x

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1 2D-RCWA two dimensional rigoros copled wave analsis, 2] εx, εx, ϵ mn exp +jmk x x + nk ] m,n m,n ξ mn exp +jmk x x + nk ] 2 K x K x Λ x Λ ϵ mn ξ mn K x 2π Λ x K 2π Λ ϵ mn ξ mn Λ x Λ x Λ x Λ x Λx Λ Λx Λ εx, exp jmk x x + nk ] dxd 3 εx, exp jmk xx + nk ] dxd 4 region I incidence n I θ n gr n rd φ d g z grating region Λ x Λ x region II, sbstrate n II z : :n I n II n rd :ngr Λ x Λ x θ ϕ d g

2 2 3-4 ϵ mn εgrδ mn + ε rd εgr sinmπf x sinnπf mπ nπ ξ mn εgr δ mn + sinmπfx sinnπf ε rd εgr mπ nπ exp+jωt µ Ē H Ē jωµ H 7 H jωεx, ϵ Ē 8 z E z H z z E 7 x E z E z E z E k z jωµ H x +jωµ H x + E z jωµ H x + jωϵ jωµ H x k jωϵ k H ε x H ] x Hx ε H ] x k 2π/λ ω/c µ ϵ ω jωµ /k j µ /ϵ /jωϵ k +j/cϵ k 2 +j µ /ϵ /k 2 k x x k k z z x z f x Λ x f Λ Λ x Λ 2: 2

3 E z j µ ϵ { H x + Hx ε H ]} x 7 E x 9 E x z 8 x H z E z x E x z E x k z H z H z H k z 8 jωµ H jωµ H + E z x jωµ H + jωϵ µ j E x z j ϵ µ ϵ jωεϵ E x { H k 2 { H + x ε x H x H ] x H x ε x H ]} x Hx ε H ]} x jωεϵ E x + H z εϵ j E x + E ϵ µ k jωµ x E x ϵ j εe x + j ϵ E µ k 2 µ x E x µ H j ϵ z εe x + 2 E x 2 2 E x H x z H z x H x z H x k z jωεϵ E +jωεϵ E + H z x εϵ +j E + ϵ µ k jωµ x ϵ j εe + j ϵ µ k 2 µ x µ H x j ϵ z εe + 2 E x x 2 E x 2 E x E x E x E x E x E H x H 4 µ j j E z j E x z j µ ϵ µ ϵ H x + H + { Hx ε { Hx ε x H }] x H x }] H ϵ z εe x + 2 E x 2 2 E x 2 µ H x ϵ z εe + 2 E x x 2 E x 2 3 j µ /ϵ E/H µ /ϵ 3

4 3 x E x x, mn E x, mn ϵ H x x, j ϵ H x, j S mn x z exp jk m x x + k n ] 4 S mn z exp jk m x x + k n ] 5 µ mn µ mn U mn x z exp jk m x x + k n ] 6 U mn z exp jk m x x + k n ] 7 Sx mn z S mn z Ux mn z U mn z z z kx m kn x k n I sin θ cos ϕ, k n I sin θ sin ϕ k m x k n I sin θ cos ϕ mk x 8 k n k n I sin θ sin ϕ nk 9 n I θ ϕ mn x E z mn z mn Sx mn S mn z exp jk m x x + k n ] z z exp jkx m x + k n ] µ µ ϵ j H x j j ϵ ϵ mn µ mn U mn x z exp jk m x x + k n ] U mn x z exp jk m x x + k n ] j µ ϵ ε ε Hx H ] x U x exp jkxx p + k q ] x U exp jkxx p + k q ]] j k q Ux kp x U exp jk pxx + k q ]] ε k k j ξ st exp +jsk x x + tk ] k q Ux kp x k st j st j st j mn ξ st k q ξ st k q Ux k kp x U k U exp jk pxx + k q ]] k exp j { k px sk x x + k q tk }]] Ux kp x U k k k q ξ m p,n q Ux kp x U k k exp jkx p+s x + k q+t ]] 4 exp jk mx x + k n ]]

5 mn mn k n k q ] ξ m p,n q Ux kp x U exp jkx m x + k n ] k k k S mn z z Ux mn kn k z kn k kx p ξ m p,n q U k -3 Sx mn z z km x k kx p ξ m p,n q U k U mn z z kn kx m S mn + k k k q ] ξ m p,n q Ux kp x U k k ε m p,n q S x + Ux mn z kn U mn z km x k k ξ m p,n q k q k U x 2 ξ m p,n q k q k U x 2 k n 2 Sx mn 22 k Ux mn z z k m x k 2 S mn ε m p,n q S kn k k m x k S x 23 4 RCWA coding 2-23 m s + n v p s + q s n m s S x mn Sx s 2 s 2 s 2 s 2 Sx mn Sx ξ m p,n q Ξ v S mn S ε m p,n q E v Ux mn Ux kx m /k δ v Kx v U mn U k n /k δ v K v ] m n 8-9 v m n 2-23 S ˆK ˆΞ ˆKx ˆ ˆK ˆΞ ˆK S x Ū ˆKx ˆΞ ˆKx ˆ ˆK x ˆΞ ˆK ˆK x ˆK Ê ˆK 2 ˆK x 2 Ê ˆK x ˆK Ū x 2] X ˆX TM Ê S S x Ū Ū x 5

6 3] Ê αˆξ + αê Ê 2 αê + αˆξ Ê S S x Ū Ū x ˆK ˆΞ ˆKx ˆD ˆB ˆKx ˆΞ ˆK ˆK x ˆK Ê ˆK 2 ˆK 2 x Ê2 ˆK x ˆK ˆB ˆK x ˆΞ ˆKx ˆ ˆD ˆK ˆΞ ˆK ˆ 2 S ˆK ˆΞ ˆKx z 2 ˆD Ū 2 Sx ˆB ˆK z Ūx z 2 x ˆΞ ˆK z ˆK ˆΞ ˆKx ˆD ˆKx ˆK Ê ˆK 2 ˆB ˆK S x ˆΞ ˆK ˆK x 2 Ê2 ˆK x ˆK S x ˆK ˆΞ ˆKx ˆKx ˆK ˆD ˆK x 2 + ˆDÊ2 ˆK ˆΞ ˆKx Ê ˆK 2 + ˆD ˆK x ˆK S ˆB ˆK x ˆK ˆK x ˆΞ ˆK ˆK x 2 Ê2 ˆBÊ ˆB ˆK 2 + ˆK x ˆΞ ˆK ˆKx ˆK S x ˆK x 2 + ˆDÊ2 ˆK ˆΞ ˆKx Ê ˆK x ˆK x ˆΞ ˆK Ê 2 ˆK S ˆK 2 + ˆBÊ S x S S x Ū Ū x ˆK x ˆK 2 Ū ˆKx ˆK Ê z 2 ˆK 2 S 2 Ū x ˆK 2 z 2 x Ê2 ˆK z x ˆK S x z ˆKx ˆK Ê ˆK 2 ˆK ˆΞ ˆKx ˆD Ū ˆK x 2 Ê2 ˆK x ˆK ˆB ˆK x ˆΞ ˆK Ū x ˆK 2 + Ê ˆB ˆKx Ê ˆK x ˆΞ ˆK ˆK Ê2 ˆK Ū ˆΞ ˆKx ˆK2 x + Ê2 ˆD Ū x Ŵ λ λ ˆQ S S x S S S Ŵ exp ˆQz c + + exp + ˆQz k h g c ] 27 x h g c + c +z z ˆQ ˆQ 6

7 ] 27 c + c 2] ] 24 Ū ˆK ˆΞ ˆKx ˆD Ū x ˆB ˆK x ˆΞ ˆK ˆΩ S z S x z ˆΩŴ ˆQ Ū Ūx exp S z S x z ˆQz c + + exp ˆ+Qz k h g c ] ] 25 RCWA TE TM 5 Ēinc Ē inc ū exp jn I k k x ] 28 ū exp jk n I sin θ cos ϕx + sin θ sin ϕ + cos θz ] 29 k θ ϕ x x x,, z ū ū cos ψ cos θ cos ϕ sin ψ sin ϕ, cos ψ cos θ sin ϕ + sin ψ sin ϕ, cos ψ sin θ 3 ū ψ ē ē 2 k ē ē 2 ē k z ē 2 x ȳ z ē 2,, k x ȳ z sin θ cos ϕ sin θ sin ϕ cos θ sin θ sin ϕ, sin θ cos ϕ, ē 2 sin ϕ, cos ϕ, 3 ē 2 ē ē ē 2 k 32 x ȳ z sin ϕ cos ϕ sin θ cos ϕ sin θ sin ϕ cos θ cos θ cos ϕ, + cos θ sin ϕ, cos θ cos ϕ, cos θ sin ϕ, sin θ 7

8 ē ē 2 ψ ē ē 2 cos ψ sin ψ 3 ū cos ψē + sin ψē 2 33 cos ψ cos θ cos ϕ sin ψ sin ϕ cos ψ cos θ sin ϕ + sin ψ cos ϕ cos ψ sin θ ĒI Ē I Ēinc + R exp j k re x 34 ū exp jn I k k x ] + R exp j k re x 35 R k re k re k m x, k n, k I, z 36 z z k z k m x 2 + k n 2 + k I, z 2 k n I k n kz I, kz mn I 2 kx m 2 k n 2... k n I 2 > kx m 2 + k n 2 j kx m 2 + k n 2 k n I 2... k n I 2 < kx m 2 + k n 2 ] 35 R s p 35 7 jωµ H Ē H jωµ Ē ϵ µ +j µ k Ē ϵ j k Ē 6-7 k Ēinc k ū exp jn I k k x k H I j ϵ µ µ k jn I k k ū exp jn I k k x +jn I k ū exp jn I k k x R exp j k re x j k re k R exp j k re x 38 jn I k ū exp jn I k k x + j k re k R exp j k re x 8 39

9 6 z 4-7 ū x δ + R x S x 4 ū δ + R S 4 jn I k ū x δ + j k k re R x U x 42 jn I k ū δ + j k k re R U 43 m n δ δ m δ n R z x x φ φ tan k n /kx m 44 φ s TE p TM ] φ 2π 2] 44 coding φ x φ p s p cos φ, sin φ, 45 s sin φ, cos φ, 46 s r s r s s R x, R, R z R x sin φ + R cos φ ū s ū x sin φ + ū cos φ δ sin ϕcos ψ cos θ cos ϕ sin ψ sin ϕ + cos ϕcos ψ cos θ sin ϕ + sin ψ cos ϕ + sin ψ 48 δ φ ϕ s ē 2 33 s sin ψδ + r s S x sin φ + S cos φ 49 p 39 jn I k ū p 9

10 jn I k z k z cos φ + k z x k x z sin φ ] jn I { sin θ sin ϕ cos ψ sin θ cos θcos ψ cos θ sin ϕ + sin ψ cos ϕ} cos φ + {cos θcos ψ cos θ cos ϕ sin ψ sin ϕ sin θ cos ϕ cos ψ sin θ} sin φ ] jn I sin θ sin ϕ cos ψ sin θ cos ϕ cos θ cos ψ cos θ sin ϕ cos ϕ cos θ sin ψ cos ϕ cos ϕ + cos θ cos ψ cos θ cos ϕ sin ϕ cos θ sin ψ sin ϕ sin ϕ + sin θ cos ϕ cos ψ sin θ sin ϕ ] jn I cos θ sin ψ φ ϕ ] jn I k ū p jn I p k ū jn I jn I cos θ sin ϕ, cos θ sin ϕ, ū jn I cos θ sin ϕ, cos ϕ, ū jn I cos θ s ū jn I cos θ sin ψ i j k cos ϕ sin ϕ sin θ cos ϕ sin θ sin ϕ cos θ ū j k k re R p j k k re, R z k re,zr cos φ + k re,zr x k re,xr z sin φ ] j k k re,z R cos φ R x sin φ + R z k re, cos φ k re,x sin φ ] j k k re,z r s + R z k m x, k n s ] j k k re,zr s + j kz I, rs k s j k re k R p j p k k re R j i j k cos φ k sin φ R k cos φ k sin φ kre,z j k k re,z sin φ, k re,z cos φ, R j k k re,z s R j k k re,zr s

11 + j kz I, rs k k re x k 3 R R r s s + r n n p s k re R k r s p k re,z/k p jn I cos θ sin ψ + j k k I, z r s U x cos φ + U sin φ 5 TMp s r p j k k re R s 5 ] ] 7 s jn I k ū s jn I k ū z k z ū sin φ + k z ū x k x ū z cos φ ] jn I { sin θ sin ϕ cos ψ sin θ cos θcos ψ cos θ sin ϕ + sin ψ cos ϕ} sin φ + {cos θcos ψ cos θ cos ϕ sin ψ sin ϕ sin θ cos ϕ cos ψ sin θ} cos φ ] jn I sin θ sin ϕ cos ψ sin θ sin ϕ + cos θ cos ψ cos θ sin ϕ sin ϕ + cos θ sin ψ cos ϕ sin ϕ + cos θ cos ψ cos θ cos ϕ cos ϕ cos θ sin ψ sin ϕ cos ϕ + sin θ cos ϕ cos ψ sin θ cos ϕ ] jn I sin 2 θ sin 2 ϕ cos ψ + cos 2 θ cos ψ sin 2 ϕ + cos 2 θ cos ψ cos 2 ϕ + sin 2 θ cos 2 ϕ cos ψ ] jn I cos ψ 3 5 φ ϕ R n p k re s 3:

12 ] 3 ē 2 s 32 ē ē 2 k 33 ū cos ψē + sin ψē 2 jn I k ū s jn I s k ū jn I ē 2 k cos ψē + sin ψē 2 jn I ē cos ψē + sin ψē 2 jn I cos ψ s jn I cos ψδ r p U x sin φ + U cos φ 52 p p ū p ū x cos φ + ū sin φ δ cos ϕcos ψ cos θ cos ϕ sin ψ sin ϕ + sin ϕcos ψ cos θ sin ϕ + sin ψ cos ϕ + cos ψ cos θ 53 5 Rx cos φ + R sin φ j ki, z n 2 I k rp 53 cos ψ cos θδ j ki, z n 2 I k rp Sx cos φ + U sin φ 54 r p j k k re R s j s k k re R j i j k sin φ k cos φ k cos φ k sin φ kre,z R r p j k k re,z cos φ, k re,z sin φ, k R j k k re,z cos φ R x + k re,z sin φ R k R z k j k re,z k re,z 2 cos φ R x + sin φ R k re,zk R z +jr p k k re,z k n I 2 k 2 cos φ R x + sin φ R k re,zk R z k n I 2 cos φ R x + sin φ R k k cos φ R x + k sin φ R + k re,zr z k n I 2 cos φ R x + sin φ R k k re R jr p k k re,z k n I 2 cos φ R x + sin φ R jrp kre,z k n 2 I cos φ R x + sin φ R 2

13 jrp kz I, k n 2 I cos φ R x + sin φ R k re R 3 rp 5 s s rn x 53 kre,z z sin ψδ + r s S cos φ S x sin φ 55 cos ψ cos θδ j ki, z n 2 I k rp S sin φ + Sx cos φ 56 jn I cos ψδ + rp U cos φ Ux ] sin φ 57 jn I cos θ sin ψ j kz I, rs U sin φ + U ] x cos φ 58 k 27 S S x Ŵ c + + exp ˆQk h g c ] ˆX exp ˆQk h g exp h g k λ ] cos φ s 2 cos ˆΦ sin φ sin ˆΦ 2s 2 cos ˆF ˆΦ sin ˆΦ sin ˆΦ cos ˆΦ 59 s 2 ˆ ˆ ˆ ˆ ˆR S ˆ jkz I, /k n 2 I ˆ ˆ ˆR U jkz I, /k ˆ S inc Ū inc r sin ψδ cos ψ cos θδ jn I cos ψδ jn I sin ψ cos θδ r s r p S inc ˆR S r ˆF S ˆF Ŵ c+ + ˆX c S inc ˆR S r + ˆF Ŵ c+ + ˆX c 6 3

14 57-58 Ū inc ˆR U r ˆF Ū ˆF ˆΩŴ ˆQ c + + ˆX c Ū inc ˆR U r + ˆF ˆΩŴ ˆQ c + ˆX c 6 7 z d g z d g z Ē II T exp j k tr x d 62 x d x,, z d g z d g k tr z k II, z k II, z kz mn k tr k m x, k n, k II, z 63 n II + k n II 2 kx m 2 k n 2... k n II 2 > kx m 2 + k n 2 j kx m 2 + k n 2 k n II 2... k n II 2 < kx m 2 + k n 2 H II j ϵ µ ϵ j µ ϵ j j µ k k ĒII { } T exp j k tr x d k { } k tr T exp j k re x d z d g T x S x d g 64 T S d g 65 j k tr k R x Ux d g 66 j k tr k R U d g 67 s s t s T s sin φ T x + cos φ T 68 p µ j HII p j k tr ϵ k T p cos φ S d g sin φ S x d g 69 j k p k tr T 4

15 j k i j k cos φ sin φ k cos φ k sin φ k tr,z T j k k tr,z sin φ, k tr,z cos φ, T j k k tr,z s T j k k tr,zt s j kz II, t s k j kii, z t s cos φ Ux d g + sin φ U d g 7 k p s ] t p j k k tr T s 7 t p sin φ U x d g + cos φ U d g t p t p j k k tr T s j s k k tr T j i j k sin φ k cos φ k cos φ k sin φ ktr,z T j k k tr,z cos φ, k tr,z sin φ, k T j k k tr,z cos φ T x + k tr,z sin φ T k T z k j k tr,zt p k tr,z 2 cos φ T x + sin φ T k tr,zk T z +jt pk k tr,z { k n II 2 k 2} cos φ T x + sin φ T k tr,zk T z k n II 2 cos φ T x + sin φ T k k cos φ T x + k sin φ T + k tr,zt z k n II 2 cos φ Tx + sin φ T k k tr T k n II 2 cos φ T x + sin φ T k tr,z +jt p k n 2 II cos φ T x + sin φ T j kii, z k n 2 t p II cos φ Tx + sin φ T p T j kii, z k n 2 t p cos φ S x d g + sin φ S d g II 5

16 z d g t s cos φ S d g sin φ S x d g 72 j kii, z k n 2 t p II sin φ S d g + cos φ S x d g 73 t p cos φ U d g sin φ U x d g ] 74 j kii, z t s sin φ U d g + cos φ Ux d g ] 75 k ˆ ˆ ˆT S ˆ jkz II, /k n 2 II ˆ ˆ ˆT U jkz II, /k ˆ t s t t p ˆT S t ˆF Sd g ˆT S t + ˆF Ŵ ˆX c + + c ˆT U t ˆF Ūd g ˆF ˆΩŴ ˆQ ˆX c + + c ˆT U t ˆF ˆΩŴ ˆQ ˆX c + c ˆR S ˆ ˆF Ŵ ˆF Ŵ ˆX r S inc ˆR U ˆ ˆF ˆΩŴ ˆQ ˆF ˆΩŴ ˆQ ˆX ˆ ˆTS ˆF Ŵ ˆX t ˆF Ŵ c + Ū inc ˆ ˆTU ˆF ˆΩŴ ˆQ ˆX ˆF ˆΩŴ ˆQ c 78 Â ˆF Ŵ Ĉ ˆF ˆΩŴ ˆQ ˆR S ˆ Â Â ˆX ˆR U ˆ Ĉ Ĉ ˆX ˆ ˆTS Â ˆX Â ˆ ˆTU Ĉ ˆX Ĉ r t c + c S inc Ū inc 79 r t c + c 6

17 ˆRS ˆR U Â Â r + ˆX Ĉ Ĉ ˆX c + c Sinc Ū inc 8 r ˆR S S inc Â c+ Â ˆX c 8 ˆR S ˆR U r + Ĉ c+ Ĉ ˆX c Ūinc ˆR U ˆR S S inc Â c+ Â ˆX c + Ĉ c+ Ĉ ˆX c Ūinc Ĉ ˆR ˆR U S Â c+ + Ĉ ˆX ˆR ˆR U S Â ˆX c ˆR ˆR U S ˆR ˆR U S Â Ĉ c+ + ˆR ˆR U S Â ˆX + Ĉ ˆX c ˆR ˆR U S 8 ˆTS Â ˆX Â c t + + ˆT U Ĉ ˆX Ĉ c t ˆRU ˆR ˆT U ˆT S ˆT U ˆT S t ˆT S Â ˆX c + + Â c ˆT U t Ĉ ˆX c + + Ĉ c ˆT U ˆT S Â ˆX c + + Â c Ĉ ˆX c + + Ĉ c ˆT U ˆT S S Â Ĉ ˆRU ˆR S Â ˆX + Ĉ ˆX Â ˆX Ĉ ˆX c + + ˆT U ˆT S Â + Ĉ c Â ˆX + Ĉ ˆX c + + ˆT U ˆT S Â Ĉ c ˆTU ˆT S Â ˆX + Ĉ ˆX Â Ĉ c + c S inc + Ūinc S inc Ūinc ˆRU ˆR S S inc Ūinc c + c r t s z p z R s Rk I, z n I k cos θ r s 2 Rp RkI, z /n 2 I rp 2 k cos θ/n I T s Rk II, z n I k cos θ t s 2 Tp RkII, z /n 2 II t k cos θ/n p 2 I ] ] 2] n I n I 334 7

18 ] Formlation of stable and efficient implementation of the rigoros copled-wave analsis of binar gratings, M. G. Moharam, E. B. Grann, and D. A. Pommet, J. Opt. Soc. Am. A 2, ] Rigros Copled-wave analsis for two-dimensional gratings, Petre Cǎtǎlin Logofǎt, Proc. SPIE 5972, ] Highl improved convergence of the copled-wave method for TM polarization, P. Lalanne and G. M. Morris, J. Opt. Soc. Am. A, 3,

19 L fx fx a 2 + 2πn 2πn a n cos L x + b n sin L x n n a n 2 L 2πn fx cos L L x dx b n 2 L 2πn fx sin L L x dx fx c L c n L + n L L c n exp j 2πn L x dx fxdx fx exp j 2πn L x a n b n 2 { ε rd < x < Lf/2 εx ε gr Lf/2 < x < L/2 k x 2π/L n ϵ n n ϵ n L ε gr L ε gr L L/2 L/2 εx exp jnk x x dx Lf/2< x <L/2 L/2 L/2 exp jnk x x dx + ε rd L exp jnk x x dx + ε rd ε gr L Lf/2 + 2 ε rd ε gr cos nk x x dx L 2 ε rd ε gr sin nk x x] Lf/2 L nk x 2 ε rd ε gr L L n2π sin n 2π Lf L 2 sin nπf ε rd ε gr nπ ε L εxdx fε rd + fε gr < x <Lf/2 Lf/2 Lf/2 exp jnk x x dx exp jnk x x dx 82 9

20 x Λ x Λ { ε rd < x < Λ x f x /2 < < Λ f /2 εx, ε gr ϵ mn dxdεx, exp jmk x x + nk Λ x Λ ε rd ε gr Λ x Λ Λx f /2 Λ xf x/2 ε rd ε gr sin mπf x nπ m n ϵ Λ f /2 exp jmk x x dx dεx, exp jnk Λ f /2 sin nπf nπ Λ x Λ εx, dxd f x f ε rd + f x f ε gr { ε rd x εx, < r ε gr ϵ mn dxdεx, exp jmk x x + nk Λ x Λ { dxdε gr exp jmk x x + nk Λ x Λ } + dxdε rd ε gr exp jmk x x + nk r 2π + ε rd ε gr Λ x Λ dr dθr exp jkr cos θ K mk x, nk r x, K K K r θ θ J n x J x 2π 2π d dx xn J n x] x n J n x d dx xj x] xj x ϵ mn ε rd ε gr Λ x Λ expjx sin θdθ 2π r 2πε rd ε gr Λ x Λ dr 2π r 2π dθr exp jkr cos θ drrj Kr 2πε rd ε gr Λ x Λ K 2 KrJ Kr] r 2πε rd ε gr r 2 J Kr Λ x Λ Kr ε rd ε gr 2πr2 J Kr Λ x Λ Kr 2 expjx cos θdθ 83 84

21 J Λ x Λ 2D-RCWA s 2 : Sx, S, Ūx, Ū, r s, r p, t s, t p 2s 2 : c +, c, S, S inc, Ūinc, r, t s 2 s 2 : Ê, ˆΞ, ˆK x, ˆK, ˆB, ˆD, cos ˆΦ, sin ˆΦ 2s 2 2s 2 : Ŵ, ˆΩ, ˆQ, ˆX, ˆF, ˆR S, ˆR U, ˆT S, ˆT U, Â, Ĉ mn Ŝx mn S S 2 S 3 S 2 S 22 S 23 S 3 S 32 S 33 S S 2 S 3 S 2 S 22 S 23 S 3 S 32 S 33 S S 2 S 3 S 4 S 5 S 6 S 7 S 8 S 9 S ˆK v x ˆK v x v ˆK k x k x k x... k 2 x k 2 x k 2 x... k 3 x k 3 x k 3 x 2

22 ˆK v k k 2 k 3... k k 2 k 3... k k 2 k 3 ˆF ˆR S ˆT S ˆR U ˆT U ˆR ˆR U S : ˆR U ˆR S ˆ ˆ ˆ ˆ jkz I, /k ˆ ˆ jkz I, /k n 2 I ˆ ˆ ˆ ˆ jkz I, /k ˆ ˆ jk n 2 I /ki, z ˆ +jk n 2 I /ki, z jkz I, /k ˆ ˆT U ˆT S ˆ ˆ jk II, z /k ˆ ˆ ˆ 22 ˆ jk II, z /k n 2 II

23 ˆ ˆ jk II, z /k ˆ ˆ ˆ ˆ ˆ +jk n 2 II /kii, z jk II, z /k ˆ +jk n 2 II /kii, z 23

II No.01 [n/2] [1]H n (x) H n (x) = ( 1) r n! r!(n 2r)! (2x)n 2r. r=0 [2]H n (x) n,, H n ( x) = ( 1) n H n (x). [3] H n (x) = ( 1) n dn x2 e dx n e x2

II No.01 [n/2] [1]H n (x) H n (x) = ( 1) r n! r!(n 2r)! (2x)n 2r. r=0 [2]H n (x) n,, H n ( x) = ( 1) n H n (x). [3] H n (x) = ( 1) n dn x2 e dx n e x2 II No.1 [n/] [1]H n x) H n x) = 1) r n! r!n r)! x)n r r= []H n x) n,, H n x) = 1) n H n x) [3] H n x) = 1) n dn x e dx n e x [4] H n+1 x) = xh n x) nh n 1 x) ) d dx x H n x) = H n+1 x) d dx H nx) = nh