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
[ 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
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)
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
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 0 + + [ A ] constant Drivation of Eignvalu Matrix Equations, [ B ], [ N ] [ N ] whr dxdy, x x [ N ] [ N ] [ C ] [ ][ ] y y N N dxdy dxdy,
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
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)
whr 1 3 1 1 3 3 3 1 3 1 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 1 3 3 1 3 1 1 3 3 1 (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 3 1 1 3 1 3 1
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,
First-ordr riangular Elmnts As in (4 33) N L, N L, N L [ ] 1 1 3 3 [ A ] h componnt rprsntation of th matrix is a a a [ ] [ ] 11 1 13 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 1 3 11,, 33 4S 4S 4S 1 1 3 3 1 1 13 31 3 4S 4S 4S (4 46) (4 47) QQ QQ QQ a a, a a, a a Q 3
[ ] First-ordr riangular Elmnts [ B ] h componnt rprsntation of th matrix is b b b [ ] [ ] 11 1 13 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 1 3 11,, 33 4S 4S 4S 1 1 3 3 1 1 13 31 3 4S 4S 4S (4 48) (4 49) RR RR RR b b, b b, b b R 3
11 1 13 N N dxdy 1 c 3 [ ] [ ][ ] 31 3 33 [ 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, c33 6 6 6 S S S c c, c c, c c 1 1 1 1 1 13 31 3 3 (4 50) (4 51)
Scond-ordr riangular Elmnts N must b quadratic functions of, i L i N N N N N N 1 3 4 5 6 L L L (L 1) 1 1 1 1 L L L (L 1) L L L (L 1) 3 3 3 3 4LL 1 4LL 3 3 4LL 1 (4 35a) (4 35b) (4 35c) (4 35d) (4 35) (4 35f )
[ A ] a a a a a a a a a a a a a a a a a a 11 1 13 14 15 16 1 3 4 5 6 31 3 33 34 35 36 a41 a4 a43 a44 a45 a46 a a a a a a a a a a a a 51 5 53 54 55 56 61 6 63 64 65 66 Scond-ordr riangular Elmnts (4 5) Q Q Q a, a, a, a Q QQ Q, 1 3 11 33 44 1 + 1 + 4S 4S 4S 3S a Q + Q Q + Q a Q + QQ + Q ( ) ( ) (, ) 3 55 3 3 3 66 1 1 3 3S 3S (4 53)
QQ QQ a a, a a 1 1 3 1 1 13 31 1S 1S QQ QQ a a, a 0 a, a a 1 1 3 14 41 15 51 16 61 3S 3S QQ QQ QQ a a, a a, a a, a 0 a 3 1 3 3 3 4 4 5 5 6 6 1S 3S 3S QQ QQ a 0 a, a a, a a 3 3 1 34 43 35 53 36 63 3S 3S 1 1 a QQ + Q + QQ + QQ a a QQ + QQ + Q + QQ a ( ), ( ) 45 3 1 3 1 54 46 1 3S 3S 1 a QQ + Q + QQ + Q Q a ( ) 56 1 3 3 1 3 65 3S Scond-ordr riangular Elmnts 3 1 1 3 64
[ B ] b b b b b b b b b b b b b b b b b b 11 1 13 14 15 16 1 3 4 5 6 31 3 33 34 35 36 b41 b4 b43 b44 b45 b46 b b b b b b b b b b b b 51 5 53 54 55 56 61 6 63 64 65 66 Scond-ordr riangular Elmnts (4 54) R R R b, b, b, b R RR R, 1 3 11 33 44 1 + 1 + 4S 4S 4S 3S b R + R R + R b R + RR + R ( ) ( ) (, ) 3 55 3 3 3 66 1 1 3 3S 3S (4 55)
RR RR b b, b b 1 1 3 1 1 13 31 1S 1S RR RR b b, b 0 b, b b 1 1 3 14 41 15 51 16 61 3S 3S RR RR RR b b, b b, b b, b 0 b 3 1 3 3 3 4 4 5 5 6 6 1S 3S 3S RR RR b 0 b, b b, b b 3 3 1 34 43 35 53 36 63 3S 3S 1 1 b RR + R + RR + RR b b RR + RR + R + RR b ( ), ( ) 45 3 1 3 1 54 46 1 3S 3S 1 b RR + R + RR + R R b ( ) 56 1 3 3 1 3 65 3S Scond-ordr riangular Elmnts 3 1 1 3 64
[ C ] c c c c c c c c c c c c c c c c c c 11 1 13 14 15 16 1 3 4 5 6 31 3 33 34 35 36 c41 c4 c43 c44 c45 c46 c c c c c c c c c c c c 51 5 53 54 55 56 61 6 63 64 65 66 Scond-ordr riangular Elmnts (4 56) S 8S c c c, c a c 30 45 11 33 44 55 66 (4 57)
S S c1 c1, c13 c31 180 180 S c 0 c, c c, c 0 c 45 14 41 15 51 16 61 S S c c, a 0 c, c 0 c, c c 180 45 3 3 4 4 5 5 6 6 S 34 43, 35 0 53, 36 0 63 c c c c c c 45 Scond-ordr riangular Elmnts 4S 4S 4S c c, c c, c c 45 45 45 45 54 46 64 56 65
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,
First-ordr riangular Elmnts An optical wavguid as shown in Fig.4.7 ---- burid structur 18 first-ordr triangular lmnts 1,, 18 cor lmnts 9 and 10 Elmnt 9 as shown in Fig.4.8 ---- local coordinats of th nod numbrs 6, 7 and 10 ar 1, and 3
First-ordr riangular Elmnts h actual programming flow Gnral mshs as in Fig.4.9 Glob matrixs as in Fig.4.10
Scond-ordr riangular Elmnts Gnral mshs as in Fig.4.11 Glob matrixs as in Fig.4.1
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)
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.4.13 ----boundary condition at th mirror-symmtrical plan at th cntr Exrcis Four P.113 Problms.
S you Latr!