Finite-Diffeence Methods (FDM) Finite-diffeence Appoimations Assuming a 1D f() as in Fig.4.14 witing f 1 and f as Talo seies epansions aound = 0 Appoimating deivatives at f f (1) () ( 0) = f h f + h 1 1 = 0 with diffeential epession (4 59a) h f ( h + h ) f + h f (0) = hh 1 h1+ h O h h = h 1 1 3 1 The eo is ( ) when (equidistant discetization); 1 1 h h h (4 59b) is O( ) when (nonequidistant discetization).
Wave Equations Vectoial Wave Equations The vectoial wave equation fo the electic field is ε E+ E + k0ε E = 0 ε The vectoial wave equation fo the magnetic ( ) ε H + H + k0ε H =0 ε Consideing a stuctue unifom in the z diection ε z and a D coss-sectional analsis = and = 0 z jβ (4 60) field is (4 61)
Vectoial Wave Equations 0 (4 6a) B using the in, electic field component 1 ( ) 1 0 s E E E E E k ε ε β ε ε ε + + + + = 0 (4 6b) 1 ( ) 1 0 E E E E E k ε ε β ε ε ε + + + + =
The can be ewitten as Vectoial Wave Equations 1 E ( εe) + + ( k 0ε β ) E ε E 1 ε + E = 0 ε 1 + εe + k0ε β ε ( ) ( ) E 1 ε + E = 0 ε (4 63a) (4 63b)
ε Vectoial Wave Equations B using the magnetic field components in, H 1 ε H H + + ( k0ε β ) H 1 ε H + = 0 ε (4 64a) H H H + + ( k0ε β ) H 1 ε ε 1 ε H + = 0 ε (4 64b)
Vectoial Wave Equations 0 (4 65a) The can be ewitten as 1 ( ) 1 0 k H H H H ε ε β ε ε ε + + + = 0 (4 65b) 1 ( ) 1 0 k H H H H ε ε β ε ε ε + + + =
Semivectoial Wave Equations In optical waveguides, the tems coesponding to the inteaction between E and E ae usuall small 1 ε 1 ε E and E ε ε 0 in the tems coesponding to the inteaction between H and H ae usuall small 1 ε H 1 ε H and 0 in ε ε (4 64,65) (4 6,63)
Semivectoial Wave Equations Neglecting the inteaction tems semivectoial wave equations semivectoial analses as in Fig.4.15 quasi-te mode analsis; ---- pincipal field component E o H ; quasi-tm mode analsis; ---- pincipal field component H o E.
The semivectoial wave equation fo the Semivectoial Wave Equations quasi-te mode E 1 ε E + ( E + + k 0ε β ) E = 0 ε 1 E ( εe ) + + ( k 0ε β ) E = 0 ε H 1 ε H H + + ε β H = ε ( k ) 0 is (4 66a) (4 66b) The semivectoial wave equation fo the quasi-tm mode is H 1 H + ε ( + k0ε β ) H = 0 ε 0 (4 67a) (4 67b)
Scala Wave Equations Assuming a stuctue unifom also in the, diection ε ε = = 0 and 0 The scala wave equation is φ φ + + ( k 0ε β ) φ = 0 φ whee is a wave function (scala field) (4 68)
Finite-diffeence Epessions Nonequidistant discetization as in Fig.4.16 D coss-sectional analsis of optical waveguides with (p, q) ---- (, ) of a node; the inteface of two mateials ---- midwa between two nodes; discetization widths e & w in and n & s in. SV-FDM The E epesentation semivectoial wave equation fo the quasi-te mode is E 1 ε E + E + + ( k0ε β ) E = 0 ε (4 66a)
pq, with the eo is O( ). Finite-diffeence Epessions B using the finite-diffeence appoimation, E = E + E E ee ( + w) we ( + w) ew e w p+ 1, q p 1, q p, q (4 69a) E = E + E E + + pq, + 1 pq, 1 pq, sn ( s) nn ( s) sn pq, s n with the eo is O( ). (4 69b)
1 ε ε E pq, with the eo is O( ). Finite-diffeence Epessions 1 ε 1 ε E e w ε E = + e w p+ 1/, q p 1/, q ε p+ 1/, q p 1/, q 1 ε 1ε( p+ 1, q) ε( p, q) hee E = ( Ep+ 1, q + Ep, q) + ε e ε( p+ 1, q) + ε ( p, q) 1 ε 1 ε ( pq, ) ε( p 1, q) E = ( Epq, + Ep 1, q) + ε w ε ( pq, ) + ε( p 1, q) e O( ) w O( )
Thus 1 ε ε E pq, Finite-diffeence Epessions 1ε( p+ 1, q) ε( p, q) = ( Ep+ 1, q + Ep, q) e + w e ε( p+ 1, q) + ε( p, q) 1 ε( pq, ) ε( p 1, q) ( Ep q + Ep q) w ε( pq, ) + ε( p 1, q), 1, (4 69c)
p 1, q e p+ 1, q n p, q 1 s p, q+ 1 pq, { p } Finite-diffeence Epessions B substituting Eqs.(4-69) into Eq.(4-66a), the finite diffeence epession α E + α E + α E + α E w + ( α + α ) E + k ε (, q) β E = 0 0 pq, (4 70) whee α α α e n w ε ( p 1, q) =, we ( + w) ε ( pq, ) + ε ( p 1, q) ε ( p+ 1, q) =, ee ( + w) ε ( p, q) + ε ( p+ 1, q) =, αs =, nn ( + s) sn ( + s) 4 α = + αe + αw, α = = αn αs. ew ns
Finite-diffeence Epessions The H epesentation wave equation fo quasi-te mode is 1 H H ε + + ( k 0ε β ) H = 0 ε B using the finite-diffeence appoimation, with the eo is O( ). (4 71) 1 H 1 H 1 H = ε e w ε ε + pq, p+ 1/, q p 1/, q e w 1 H 1 hee = ( H p+ 1, q H p, q) + ε e ε ( p+ 1, q) + ε ( p, q) p+ 1/, q p 1/, q O( ) 1 H 1 = ( H pq, H p 1, q) + w ε w ε ( pq, ) + ε ( p 1, q) e O( )
Thus 1 ε H pq, Finite-diffeence Epessions 1 = ( Hp+ 1, q H p, q) e + w eε( p+ 1, q) + ε( p, q) 1 ( H pq H p q) w ε( pq, ) + ε( p 1, q) H = H + H H sn ( + s) nn ( + s) sn, 1, (4 7a) pq, + 1 pq, 1 pq, pq, s n with the eo is O( ). (4 7b)
w p 1, q e p+ 1, q n pq, 1 s pq, + 1 { } 0 p q Finite-diffeence Epessions B substituting Eqs.(4-6) into Eq.(4-61), the finite diffeence epession α H + α H + α H + α H + ( α + α ) H + k ε (, ) β H = 0 pq, p, q (4 73) whee α α α e n w ε ( pq, ) =, we ( + w) ε ( pq, ) + ε ( p 1, q) ε ( pq, ) =, ee ( + w) ε ( p, q) + ε ( p+ 1, q) =, αs =, nn ( + s) sn ( + s) α = αe αw, α = = αn αs. ns
Finite-diffeence Epessions Fo E epesentation wave equation fo the quasi-tm mode is E 1 ε + E + + ( k ε β ) E = 0, 0 ε the finite diffeence epession is just like Eq. E (4 70) but whee α α α n s w =, αe =, we ( + w) ee ( + w) ε ( pq, 1) =, nn ( + s) ε ( pq, ) + ε ( pq, 1) = ε ( pq, + 1) sn ( + s) ε ( pq, ) + ε ( pq,, + 1) 4 α = = αe αw, α = + αn + αs. ew ns
Finite-diffeence Epessions Fo H epesentation wave equation fo the quasi-tm mode is H 1 H + ε ( k ) 0 + ε β H = 0 ε the finite diffeence epession is just like Eq. but whee α α α n s w =, αe =, we ( + w) ee ( + w) ε ( pq, ) =, nn ( + s) ε ( pq, ) + ε ( pq, 1) = ε ( pq, ) sn ( + s) ε ( pq, ) + ε ( pq, + 1) α = = αe αw, α = αn αs. ew, (4 73),
Finite-diffeence Epessions Fo φ scala wave equation fo the scala mode is whee φ φ + + ( k ε β ) φ = 0 (4 68) 0 the finite diffeence epession is α φ + α φ + α φ + α φ w p 1, q e p+ 1, q n p, q 1 s p, q+ 1 + ( α + α) φpq, + k0ε ( pq, ) β φp, q = 0 αw =, αe =, we ( + w) ee ( + w) α n { } =, αs =, nn ( + s) sn ( + s) (4 74) SC-FDM α = = αe αw, α = = αn αs. ew ns
{ α α k } 0 pq pq, Pogamming w p 1, q e p+ 1, q n pq, 1 s pq, + 1 pq, pq, whee ( pq, ) (, ); o H fo the quasi-te mode, o H fo the quasi-tm mode. SV-FDM as in (70,73) The finite diffeence epession fo SV wave equation is α φ + α φ + α φ + α φ + ( + ) + ε (, ) φ β φ = 0 φ E E A mesh model in a finite-diffeence scheme as in Fig.4.17 ---- the whole analsis aea a numbe of meshes; The nodes top-bottom & left-ight. (4 75)
[ ]{} A φ = β {} φ {} 1 Pogammin g B calcuating Eq. (4 75) fo each node in an mati, we obtain the eigenvalue mati equation β hee is an eigenvalue; φ =( φ φ φ ) M T (4 76) M M is an eigenvecto with is. M M M With the vaiable tansfomation (suppessing the ound-off eos), = k, = the eigenvalue is eff 0 n k 0 n, whee ( β / k ) is the effective inde. eff 0
no summation needed as in FEM The global mati [A] as in Fig.4.18 ---- the SV-FDM a nonsmmetic spase mati with bandwidth M +1; the SC-FDM a smmetic spase mati with bandwidth M +1. Pogammin g In actual pogamming, node numbe is used instead of ( p, q) = ( p 1) M + q p M whee 1 and 1. q M (4 77) The th ow in the eigenvalue mati equation can be as a φ + a φ + a φ + a φ, M M, 1 1,, + 1 + 1, + M + a φ β φ = 0 + M fo pogamming (4 78)
Pogamming the eigenvalue mati equation [ ]{} A φ = β {} φ φ =( φφ φ ) T {} hee. Fo simplicit, we conside the finite diffeence epession α φ + α φ + α φ + α φ w p 1, q e p+ 1, q n p, q 1 s p, q+ 1 { α } α k0ε p q + + + φ β φ = ( ) (, ) pq, pq 0 The th ow in the eigenvalue mati equation can be as a φ + a φ + a φ + a φ, M M, 1 1,, + 1 + 1, + M 1 + a φ β φ = 0 + M M (4 76), (4 75) (4 78)
Bounda conditions Diichlet conditions: A wave function outside the analsis window φ = 0 (4 79a) Neumann conditions: Nomal deivative of a wave function at the edge of the analsis window φ = 0 n (4 79b)
Analtical bounda conditions: Bounda conditions The analtical wave function outside the analsis window to be connected with a wave function at the edge is assumed to deca eponentiall ( - k n ε ) 0 eff p, q) Δ ep ( (4 79c) k n Δ ε ( pq, ) whee,, and ae wave numbe in a vacuum, 0 eff effective inde, discetization width and elative pemittivit.
Bounda conditions Fo simplicit, we conside the finite diffeence epession α φ + α φ + α φ + α φ w p 1, q e p+ 1, q n p, q 1 s p, q+ 1 { α } α k0ε p q + + + φ β φ = ( ) (, ) pq, pq 0, (4 75) Left-Hand Bounda (p = 1 and ecept at cones) as 1 in Fig.4.19 ---- the discetization width is Δ.
with = 1, 0, e p+ 1, q n pq, 1 s pq, + 1 { } L k pq Bounda conditions When assuming that ( pq, ) is a node on the bounda; the hpothetical node outside the analsis widow is ( p 1, q), φ = γ φ γ L B substituting Eq. (4 80) into Eq. (4 75), α φ + α φ + α φ p 1, q L p, q (4 80) ( ) -k n ε pq Δ ep (, 0 eff Diichlet Neumann ), Analtical αγ + ( α + α ) + ε (, ) φ β φ = 0 w 0 pq, pq, (4 81)
Bounda conditions Right-Hand Bounda (p = M and ecept at cones) as in Fig.4.19 ---- the discetization width is again Δ. When assuming that ( pq, ) is a node on the bounda; the hpothetical node outside the analsis widow is ( p + 1, q), γ with = R p+ 1, q R p, q (4 83) 0, Diichlet 1, φ = γ φ ( ) -k n ε p q Δ 0 eff Neumann ep (, ), Analtical
Bounda conditions B substituting Eq. (4 8) into Eq. (4 75), α φ + α φ + α φ e p 1, q n pq, 1 s pq, + 1 { } R k pq αγ + ( α + α ) + ε (, ) φ β φ = 0 w 0 pq, pq, (4 83) Top Bounda (q = 1 and ecept at cones) as 3 in Fig.4.19 ---- the discetization width is Δ.
with = 1, 0, pq, U pq, w p 1, q e p+ 1, q s p, q+ 1 { } U k ε pq Bounda conditions When assuming that ( pq, ) is a node on the bounda; the hpothetical node outside the analsis widow is ( pq, 1), φ = γ φ γ U B substituting Eq. (4 84) into Eq. (4 75), α φ + α φ + α φ 1 (4 84) ( ) -k n ε pq Δ ep (, 0 eff Diichlet Neumann ), Analtical αγ + ( α + α ) + (, ) φ β φ = 0 n 0 pq, pq, (4 85)
Bottom Bounda (q = M and ecept at cones) as 4 in Fig.4.19 ---- the discetization width is again Δ. with = 1, 0, pq, + D pq, Bounda conditions When assuming that ( pq, ) is a node on the bounda; the hpothetical node outside the analsis widow is ( pq, + 1), φ = γ φ γ D 1 (4 86) ( ) -k n ε pq Δ ep (, 0 eff Diichlet Neumann ), Analtical
Bounda conditions B substituting Eq. (4 86) into Eq. (4 75), α φ + α φ + α φ w p 1, q e p+ 1, q n p, q 1 { } D k ε pq αγ + ( α + α ) + (, ) φ β φ = 0 s 0 pq, pq, (4 87) Left-Top Cone (p = q =1) as 5 in Fig.4.19 ---- the discetization widths ae Δ and Δ.
with = 0, p 1, q L p, q p, q 1 p, q Bounda conditions When assuming that ( p, q) is a node at the cone; the hpothetical nodes outside the analsis widow ae ( p 1, q) to the left and ( p, q 1) to the top, φ = γ φ (4 88a) γ L 1, φ = γ U φ ( ) -k n ε p q Δ 0 eff (4 88b) Diichlet Neumann ep (, ), Analtical
Bounda conditions γ U = 1, 0, Diichlet ( ) -k n ε pq Δ ep (, ), 0 eff Neumann Analtical B substituting Eq. (4 88) into Eq. (4 75), α φ e + α φ p+ 1, q s p, q+ 1 { } w L n k0 pq αγ + αγ + ( α + α) + ε(, ) φ βφ = 0 U pq, p, q (4 89)
with = 0, 1, p 1 q L p q pq, + 1 D pq, Bounda conditions Left-Bottom Cone (p = 1, q = M ) as 6 in Fig.4.19 ---- the discetization widths ae again Δ and Δ. When assuming that ( p, q) is a node at the cone; the hpothetical nodes outside the analsis widow ae ( p 1, q) to the left and ( p, q+ 1) to the bottom, φ = γ φ γ L φ,, (4 90a) = γ φ ( ) -k n ε p q Δ ep (, ), 0 eff (4 90b) Diichlet Neumann Analtical
Bounda conditions γ D = 1, 0, Diichlet ( ) -k n ε pq Δ ep (, ), 0 eff Neumann Analtical B substituting Eq. (4 90) into Eq. (4 75), α φ e + α φ p+ 1, q n p, q 1 { } w L s k0 pq αγ + αγ + ( α + α) + ε(, ) φ βφ = 0 D pq, p, q (4 91)
with = 0, p+ 1, q R p, q pq, 1 U pq, Bounda conditions Right-Top Cone (p = M, q =1) as 7 in Fig.4.19 ---- the discetization widths ae again Δ and Δ. When assuming that ( p, q) is a node at the cone; the hpothetical nodes outside the analsis widow ae ( p+ 1, q) to the ight and ( p, q 1) to the top, φ = γ φ (4 9a) γ R 1, φ = γ φ ( ) -k n ε p q Δ 0 eff (4 9b) Diichlet Neumann ep (, ), Analtical
Bounda conditions γ U = 1, 0, Diichlet ( ) -k n ε pq Δ ep (, ), 0 eff Neumann Analtical B substituting Eq. (4 9) into Eq. (4 75), α w φ + α φ p 1, q s p, q+ 1 { } e R n k0 pq αγ + αγ + ( α + α ) + ε (, ) φ βφ = 0 U pq, p, q (4 93)
with = 0, Bounda conditions Right-Bottom Cone (p = M, q = M ) as 8 in Fig.4.19 ---- the discetization widths ae again Δ and Δ. When assuming that ( p, q) is a node at the cone; the hpothetical nodes outside the analsis widow ae ( p+ 1, q) to the ight and ( p, q+ 1 ) to the bottom, φ = γ φ γ R 1, φ p+ 1, q R p, q (4 94a) p, q+ 1 = γ D φ p, q ( ) -k n ε p q Δ ep (, ), 0 eff (4 94b) Diichlet Neumann Analtical
γ D = 1, 0, ( ) -k n ε pq Δ ep (, ), 0 eff Bounda conditions Diichlet Neumann Analtical B substituting Eq. (4 94) into Eq. (4 75), α w φ + α φ p 1, q n p, q 1 { } e R s k0 pq αγ + αγ + ( α + α ) + ε (, ) φ βφ = 0 D pq, p, q (4 95) Simila to that fo the FEM, we can calculate the even o odd mode b assuming Neumann o Diichlet conditions at the smmet plane.
Numeical Eample A calculation model (0.4 μm coe) as in Fig.4.0 and esults calculated b SV-FDM softwae n coe =3.5, n cladding =3.1693 at 1.55 μm; equidistant discetization scheme ---- the numbe of nodes M = M = 96, the discetization widths W min = 0.05 μm, W ma = 0.05 μm The calculated n eff = 3.17 fo quasi-te and quasi-tm models; A thee-dimensional plot of E calculated fo quasi-te mode as in Figue.4.8. at Page 161. Eecise Fou P.161 Poblems 3,4*.
Discussion Questions 4 1. 针对 FEM 中的一阶三角元, 如何利用面坐标 L i 来理解形状函数 N i?. FEM 中如下两个本征矩阵方程有何联系? ( [ A] λ [ B] ){ φ} = 0 (4 1) ([ K] [ M] ){ } = { 0} 3. 简述 FEM 中本征矩阵方程的全局矩阵 K 和 M 是如何构成的 4. 为什么在光波导中 E 和 E 之间的相互作用可忽略? 即 1 ε 1 ε E and E 0 in ε ε β φ (4 6,63) 5. 为什么将本征矩阵方程的变量进行如下转换后就可以降低舍入误差? = k, = 0 k 0. 6. 简述 FDM 中本征矩阵方程中的全局矩阵 A 是如何构成的, 为什么公式 (4-75) 可表示为公式 (4-78)? (4 40)
Be Be!