Derivation of Eigenvalue Matrix Equations

Transcription

1 Drivation of Eignvalu Matrix Equations h scalar wav quations ar φ φ η + ( k + 0ξ η β ) φ 0 x y x pq ε r r whr for E mod E, 1, y pq φ φ x 1 1 ε r nr (4 36) for E mod H,, 1 x η η ξ ξ n

2 [ N ] { } i i i 1 Drivation of Eignvalu Matrix Equations Aftr th analysis rgion is dividd into many lmnts, th wav function is xpandd as M φ N φ φ (4 37) whr th numbr of th nods in th first-ordr and M scond-ordr triangular lmnt ar rspctivly 3 and 6 Substituting Eq. into Eq. η x (4 37) (4 36), [ ] { } ( )[ ] + N φ + k ξ η β N { φ } 0 y 0

3 For all th lmnts, w hav th rlation Γ η [ N ] [ N ] { φ } [ N ] { } Γ η n Wav function dγ φ [ N ] { φ } Drivativ of wav function n dγ By assuming that and η n ar continuous at th boundaris with nighboring lmnts, Γ η φ φ φ dγ n Γ Drivation of Eignvalu Matrix Equations φ φ η φ dγ n (4 38)

4 Drivation of Eignvalu Matrix Equations hn w hav φ ( ) 0 Γ n [ ] [ ] {} K β M φ + η φ dγ {} (4 39) Whn imposing Dirichlt or Numann conditions as in (4 11) w gt th ignvalu quation ([ ] K β [ M] ){} φ {} 0 on filds or thir drivativs, {} φ (4 40) whr β is an ignvalu and is an ignvctor

5 Hr and [ K ] [ N ] [ N ] [ N ] [ N ] [ N ][ N ] dxdy} ([ A ] [ B ]) k ξ [ C ] { η } 0 [ M ] η [ N ][ N ] dd xy η [ C ] {} φ { φ } ε r η constant and ξ η + dxdy x x y y + k ξ [ A ] constant Drivation of Eignvalu Matrix Equations, [ B ], [ N ] [ N ] whr dxdy, x x [ N ] [ N ] [ C ] [ ][ ] y y N N dxdy dxdy,

6 With th variabl transformation (supprssing th round-off rrors), Hr x xk, y yk 0 th ignvalu quation is rducd to ( K ){} {} ff M 0 [ N ][ N ] [ N ] [ N ] [ N ] [ N ] K η + x x y y [ N ][ N ] dxdy} 0 M η dxdy, nf f dxdy + k ξ 0 Drivation of Eignvalu Matrix Equations n φ, (4 41) β k 0

7 with Matrix Elmnts ([ ] K β [ M] ){} φ {} 0 [ A ] [ B ] [ N ] [ N ] [ N ] [ N ] [ C ] [ ][ ] [ K ] [ M ] o obtaining xplicit xprssions for and in (4 40) only th following trms hav to b calculatd x y x y N N dxdy dxdy dxdy [ N ] is xprssd by ara coordinats L, L, L 1 3 (4 4a) (4 4b) (4 4c)

8 whr Matrix Elmnts For convninc, th ara coordinats ar rwrittn as Ara in Fig 4.4 (4-3) L L L Q Q Q ( x x ) + R ( y y ) S ( x x ) + R ( y y ) S ( x x ) + R ( y y ) S (4 43a) (4 43b) (4 43c) Q y y, Q y y, Q y y ; R x x, R x x 1, R x x ; 1 [( y y )( x x ) ( x )( )] S x y y

9 h drivativs of ara coordinats ar L1 Q1 L Q L1 Q,, x S x S x S L1 R1 L R L1 R,, y S y S y S 3 3 Matrix Elmnts (4 44a) (4 44b) h intgration formula for th ara coordinats is i j LL k L dxdy i! jk!! S, ( i + j + k + )! 1 3 (4 45) i, j, k 0,1,, 3,

10 First-ordr riangular Elmnts As in (4 33) N L, N L, N L [ ] [ A ] h componnt rprsntation of th matrix is a a a [ ] [ ] N N dxdy 1 3 A a a a x x a31 a3 a 33 by using Eq. and Eq. (4 44) (4 45), Q Q a a a ,, 33 4S 4S 4S S 4S 4S (4 46) (4 47) QQ QQ QQ a a, a a, a a Q 3

11 [ ] First-ordr riangular Elmnts [ B ] h componnt rprsntation of th matrix is b b b [ ] [ ] N N dxdy 1 3 B b b b y y b31 b3 b 33 by using Eq. and Eq. (4 44) (4 45), R R b b b ,, 33 4S 4S 4S S 4S 4S (4 48) (4 49) RR RR RR b b, b b, b b R 3

12 N N dxdy 1 c 3 [ ] [ ][ ] [ C ] h componnt rprsntation of th matrix is c c c C c c c c c by using Eq. and Eq. (4 44) (4 45), First-ordr riangular Elmnts S S S c11, c, c S S S c c, c c, c c (4 50) (4 51)

13 Scond-ordr riangular Elmnts N must b quadratic functions of, i L i N N N N N N L L L (L 1) L L L (L 1) L L L (L 1) LL 1 4LL 3 3 4LL 1 (4 35a) (4 35b) (4 35c) (4 35d) (4 35) (4 35f )

14 [ A ] a a a a a a a a a a a a a a a a a a a41 a4 a43 a44 a45 a46 a a a a a a a a a a a a Scond-ordr riangular Elmnts (4 5) Q Q Q a, a, a, a Q QQ Q, S 4S 4S 3S a Q + Q Q + Q a Q + QQ + Q ( ) ( ) (, ) S 3S (4 53)

15 QQ QQ a a, a a S 1S QQ QQ a a, a 0 a, a a S 3S QQ QQ QQ a a, a a, a a, a 0 a S 3S 3S QQ QQ a 0 a, a a, a a S 3S 1 1 a QQ + Q + QQ + QQ a a QQ + QQ + Q + QQ a ( ), ( ) S 3S 1 a QQ + Q + QQ + Q Q a ( ) S Scond-ordr riangular Elmnts

16 [ B ] b b b b b b b b b b b b b b b b b b b41 b4 b43 b44 b45 b46 b b b b b b b b b b b b Scond-ordr riangular Elmnts (4 54) R R R b, b, b, b R RR R, S 4S 4S 3S b R + R R + R b R + RR + R ( ) ( ) (, ) S 3S (4 55)

17 RR RR b b, b b S 1S RR RR b b, b 0 b, b b S 3S RR RR RR b b, b b, b b, b 0 b S 3S 3S RR RR b 0 b, b b, b b S 3S 1 1 b RR + R + RR + RR b b RR + RR + R + RR b ( ), ( ) S 3S 1 b RR + R + RR + R R b ( ) S Scond-ordr riangular Elmnts

18 [ C ] c c c c c c c c c c c c c c c c c c c41 c4 c43 c44 c45 c46 c c c c c c c c c c c c Scond-ordr riangular Elmnts (4 56) S 8S c c c, c a c (4 57)

19 S S c1 c1, c13 c S c 0 c, c c, c 0 c S S c c, a 0 c, c 0 c, c c S 34 43, , c c c c c c 45 Scond-ordr riangular Elmnts 4S 4S 4S c c, c c, c c

20 Programming By using th ignvalu matrix quation ([ ] K β [ M] ){} φ {} 0 { ( ) [ ]} A B k0ξ C [ K] η [ ] [ ] [ ] M η [ C ] {} { } (4 40) whr + +, φ φ,

21 First-ordr riangular Elmnts An optical wavguid as shown in Fig burid structur 18 first-ordr triangular lmnts 1,, 18 cor lmnts 9 and 10 Elmnt 9 as shown in Fig local coordinats of th nod numbrs 6, 7 and 10 ar 1, and 3

22 First-ordr riangular Elmnts h actual programming flow Gnral mshs as in Fig.4.9 Glob matrixs as in Fig.4.10

23 Scond-ordr riangular Elmnts Gnral mshs as in Fig.4.11 Glob matrixs as in Fig.4.1

24 Boundary conditions h ignvalu matrix quation is φ ( ) 0 Γ n ( ) 0 (4 40) [ ] [ ] {} K β M φ + η φ dγ {} [ ] K β [ M ] {} φ {} with boundary conditions, Dirichlt φi 0 ( Kii β Mi i ) φ i 0 φi or Numann 0 n {} (4 39) (4 58)

25 Boundary conditions h dfinit mods of optical wavguids th solutions for only th vn/odd mods by analyzing th half-plan structur as shown in Fig boundary condition at th mirror-symmtrical plan at th cntr Exrcis Four P.113 Problms.

26 S you Latr!