Y x z a1 y x z 0a x n1 n2 n3 M1M2M3 k1 k2 k3 TE mode X=0 Continuous.
20
y x z a1 y x z 0 a x n1 n2 n3 M1 M2 M3 k1 k2 k3 x k e A 1 1 x k e B 1 1 x ik e A 2 2 x ik e B 2 2 TE mode x ik x ik x k x k e B e A e B e A 2 2 1 1 2 2 1 1 X=0 2 2 2 2 1 1 1 1 B ik A ik B k A k ) ( x E y Continuous ) ( x E x y Continuous 2 2 1 1 B A B A 2 1 2 1 1 1 1 1 B k A k B k A k 2 2 1 2 2 1 1 1 ) ( ) ( 2 B ik k A ik k A k 2 2 1 2 2 1 1 1 ) ( ) ( 2 B ik k A ik k B k 2 2 2 1 2 1 2 1 2 1 1 1 1 2 1 B A ik k ik k ik k ik k k B A 2 1 2 1 2 1 2 1 1 1 2 1 ik k ik k ik k ik k k M
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Transcript of Y x z a1 y x z 0a x n1 n2 n3 M1M2M3 k1 k2 k3 TE mode X=0 Continuous.
y
xz
a1
y
xz
0 a
x
n1
n2
n3
M1 M2 M3
k1
k2
k3
xkeA 11
xkeB 11
xikeA 22
xikeB 22
TE mode
xikxikxkxk eBeAeBeA 22112211
X=0
22221111 BikAikBkAk
)(xEy Continuous
)(xEx y
Continuous
2211 BABA
21211111 BkAkBkAk
22122111 )()(2 BikkAikkAk
22122111 )()(2 BikkAikkBk
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2
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2121
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Continuous @ z=0 Continuous @ z=0
Why?
y
xz
a1
0 ax
n1
n2
n3
M1 M2 M3
k1
k2
k3
xkeA 11
xkeB 11
xikeA 22
xikeB 22
TE mode
2121
2121
11 2
1
ikkikk
ikkikk
kM
xikeA 22
xikeB 22
xkeA 33
xkeB 33
xkxkxikxik eBeAeBeA 33223322
x=a
akakaikaik eBeAeBeA 33223322
akakaikaik eBkeAkeBikeAik 332233332222
akakaikaik eBikeAikeBikeAik 332232322222
akakaik eBkikeAkikeAik 33233233222 )()(2
akakaik eBkikeAkikeBik 33233233222 )()(2
3232
3232
23 2
1kikkik
kikkik
ikM
2
21
1
1
B
AM
B
A
ak
ak
aik
aik
eB
eAM
eB
eA3
3
2
2
3
33
2
2
ak
ak
aik
aik
eB
eA
kikkik
kikkik
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2
3
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2
2
1
0 ax
n1
n2
n3
M1 M2 M3
k1
k2
k3
xkeA 11
xkeB 11
xikeA 22
xikeB 22
TE mode
2121
2121
11 2
1
ikkikk
ikkikk
kM
xikeA 22
xikeB 22
xkeA 33
xkeB 33
3232
3232
23 2
1kikkik
kikkik
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2
21
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B
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0
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A
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aik
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aik
aik
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22
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2
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2
2
221
1
1
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ak
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B
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3
3
3321
1
1
0 ax
n1
n2
n3
M1 M2 M3
k1
k2
k3
xkeA 11
xkeB 11
xikeA 22
xikeB 22
TE mode
2121
2121
11 2
1
ikkikk
ikkikk
kM
xikeA 22
xikeB 22
xkeA 33
xkeB 33
3232
3232
23 2
1kikkik
kikkik
ikM
aik
aik
e
eM
2
2
0
02
ak
ak
eB
eAMMM
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A3
3
3
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1
1
ak
ak
eB
eA
MM
MM
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3
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MM
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akeAM 33110
akeAMB 33211
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AM
B
1
11
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BM
B
A
0 ax
n1
n2
n3
M1 M2 M3
k1
k2
k3
xkeA 11
xkeB 11
xikeA 22
xikeB 22
TE mode
2121
2121
11 2
1
ikkikk
ikkikk
kM
xikeA 22
xikeB 22
xkeA 33
xkeB 33
3232
3232
23 2
1kikkik
kikkik
ikM
aik
aik
e
eM
2
2
0
02
ak
ak
eB
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3
3
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1
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1
11
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clear;clf;lamda = 1.5; %umn1 = 1.5;n2 = 1.6;n3 = 1.5;k0 = 2*pi/lamda;dn = n2 - n3;N = 100000;neff = n3+dn/N:dn/N:n2-dn/N;beta = k0.*neff;k1 = (beta.^2-k0^2*n1^2).^.5;k2 = (k0^2*n2^2-beta.^2).^.5;k3 = (beta.^2-k0^2*n3^2).^.5;d = 5; %umnm = 0;for m = 2:length(beta) M1(1,1) = (k1(m)+i*k2(m)); M1(1,2) = (k1(m)-i*k2(m)); M1(2,1) = (k1(m)-i*k2(m)); M1(2,2) = (k1(m)+i*k2(m)); M1 = M1./(2*k1(m)); M2(1,1) = exp(i*k2(m)*d); M2(2,1) = 0; M2(1,2) = 0; M2(2,2) = exp(-i*k2(m)*d); M3(1,1) = (k3(m)+i*k2(m)); M3(1,2) = (-k3(m)+i*k2(m)); M3(2,1) = (-k3(m)+i*k2(m)); M3(2,2) = (k3(m)+i*k2(m)); M3 = M3./(i*2*k2(m)); M = M1*M2*M3; M11(m) = M(1,1); I11(m) = abs(M11(m)); if I11(m)<1e-3 if I11(m)>I11(m-1) nm = nm +1; gneff(nm) = neff(m-1); else end else end endplot(neff, I11);zoom on;nmn = 1;ggneff(1) = gneff(1); for m = 2:nm if (gneff(m)-gneff(m-1))>1e-3 nmn = nmn+1; ggneff(nmn) = gneff(m); % effective index endend
%calculate model profilem = 1;betam = k0.*ggneff(m);k1m = (betam.^2-k0^2*n1^2).^.5;k2m = (k0^2*n2^2-betam.^2).^.5;k3m = (betam.^2-k0^2*n3^2).^.5; M1(1,1) = (k1m+i*k2m); M1(1,2) = (k1m-i*k2m); M1(2,1) = (k1m-i*k2m); M1(2,2) = (k1m+i*k2m); M1 = M1./(2*k1m); M2(1,1) = exp(i*k2m*d); M2(2,1) = 0; M2(1,2) = 0; M2(2,2) = exp(-i*k2m*d); M3(1,1) = (k3m+i*k2m); M3(1,2) = (-k3m+i*k2m); M3(2,1) = (-k3m+i*k2m); M3(2,2) = (k3m+i*k2m); M3 = M3./(i*2*k2m); M = M1*M2*M3;V1 = [0 1]';B1 = V1(2);V2 = M1^-1*V1;A2 = V2(1);B2 = V2(2);x1 = -10:0.001:0;y1 = B1.*exp(k1m.*x1);figure (2)plot(x1,y1);hold on;x2 = 0:0.001:d;y2 = A2.*exp(-i*k2m.*x2) + B2.*exp(i*k2m*x2);V3 = M^-1*V1;A3 = V3(1)*exp(k3m*d);B3 = 0; %V3(2)*exp(-k3m*d); % = 0plot(x2,y2, 'r');x3 = d:0.001:10;y3 = A3.*exp(-k3m.*x3) + B3.*exp(k3m*x3);plot(x3,y3, 'b');hold off;zoom on;
One waveguide.m
1.51 1.52 1.53 1.54 1.55 1.56 1.57 1.58 1.59
-3
-2
-1
0
1
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3
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M(1,1) = 0
-10 -8 -6 -4 -2 0 2 4 6 8 10-1.5
-1
-0.5
0
0.5
1
1.5
mode profile
01
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20
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1)()(
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n2
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M1 M2 M3
k1
k2
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xkeB 11
xikeA 22
xikeB 22
TE mode
xikeA 22
xikeB 22
xkeA 33
xkeB 33
a+b1
ba a2
a+a2+b1
M5 = M1M4M6 = M2
M7 = M3
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n2
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xkeB 11
xikeA 22
xikeB 22
TE mode
xikeA 22
xikeB 22
xkeA 33
xkeB 33
a+b1
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xikeA 22
xikeB 22
TE mode
xikeA 22
xikeB 22
xkeA 33
xkeB 33
a+b1
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x
n1
n2
n3
M1 M2 M3
k1
k2
k3
xkeA 11
xkeB 11
xikeA 22
xikeB 22
TE mode
xikeA 22
xikeB 22
xkeA 33
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ba a2M5 = M1M4 M7 = M3
)(4
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clear;clf;lamda = 1.5; %umn1 = 1.5;n2 = 1.6;n3 = 1.5;k0 = 2*pi/lamda;dn = n2 - n3;N = 100000;neff = n3+dn/N:dn/N:n2-dn/N;beta = k0.*neff;k1 = (beta.^2-k0^2*n1^2).^.5;k2 = (k0^2*n2^2-beta.^2).^.5;k3 = (beta.^2-k0^2*n3^2).^.5;d = 5; %umb = 2; %umnm = 0;for m = 2:length(beta) M1(1,1) = (k1(m)+i*k2(m)); M1(1,2) = (k1(m)-i*k2(m)); M1(2,1) = (k1(m)-i*k2(m)); M1(2,2) = (k1(m)+i*k2(m)); M1 = M1./(2*k1(m)); M2(1,1) = exp(i*k2(m)*d); M2(2,1) = 0; M2(1,2) = 0; M2(2,2) = exp(-i*k2(m)*d); M3(1,1) = (k3(m)+i*k2(m)); M3(1,2) = (-k3(m)+i*k2(m)); M3(2,1) = (-k3(m)+i*k2(m)); M3(2,2) = (k3(m)+i*k2(m)); M3 = M3./(i*2*k2(m)); M4(1,1) = exp(k3(m)*b); M4(2,1) = 0; M4(1,2) = 0; M4(2,2) = exp(-k3(m)*b); M5 = M1; M6 = M2; M7 = M3; M = M1*M2*M3*M4*M5*M6*M7; M11(m) = M(1,1); I11(m) = abs(M11(m)); if I11(m)<1e-3 if I11(m)>I11(m-1) nm = nm +1; gneff(nm) = neff(m-1); else end else end endplot(neff, I11);zoom on;nmn = 1;ggneff(1) = gneff(1); for m = 2:nm if (gneff(m)-gneff(m-1))>1e-5 nmn = nmn+1; ggneff(nmn) = gneff(m); % effective index endend
%calculate model profilem = 2;betam = k0.*ggneff(m);k1m = (betam.^2-k0^2*n1^2).^.5;k2m = (k0^2*n2^2-betam.^2).^.5;k3m = (betam.^2-k0^2*n3^2).^.5; M1(1,1) = (k1m+i*k2m); M1(1,2) = (k1m-i*k2m); M1(2,1) = (k1m-i*k2m); M1(2,2) = (k1m+i*k2m); M1 = M1./(2*k1m); M2(1,1) = exp(i*k2m*d); M2(2,1) = 0; M2(1,2) = 0; M2(2,2) = exp(-i*k2m*d); M3(1,1) = (k3m+i*k2m); M3(1,2) = (-k3m+i*k2m); M3(2,1) = (-k3m+i*k2m); M3(2,2) = (k3m+i*k2m); M3 = M3./(i*2*k2m); M4(1,1) = exp(k3m*b); M4(2,1) = 0; M4(1,2) = 0; M4(2,2) = exp(-k3m*b); M5 = M1; M6 = M2; M7 = M3; M = M1*M2*M3*M4*M5*M6*M7;V1 = [0 1]';B1 = V1(2);V2 = M1^-1*V1;A2 = V2(1);B2 = V2(2);x1 = -10:0.001:0;y1 = B1.*exp(k1m.*x1);figure (2)plot(x1,y1);hold on;x2 = 0:0.001:d;y2 = A2.*exp(-i*k2m.*x2) + B2.*exp(i*k2m*x2);V3 = (M1*M2*M3)^-1*V1;A3 = V3(1)*exp(k3m*d);B3 = V3(2)*exp(-k3m*d); % = 0plot(x2,y2, 'r');x3 = d:0.001:d+b;y3 = A3.*exp(-k3m.*x3) + B3.*exp(k3m*x3);plot(x3,y3, 'b');V4 = (M1*M2*M3*M4*M5)^-1*V1;A4 = V4(1);B4 = V4(2); % = 0x4 = d+b:0.001:d+b+d;y4 = A4.*exp(-i*k2m.*(x4-d-b)) + B4.*exp(i*k2m.*(x4-d-b));plot(x4,y4,'r');V5 = (M1*M2*M3*M4*M5*M6*M7)^-1*V1;A5 = V5(1);B5 = 0; %V5(2); % = 0x5 = d+b+d:0.001:d+b+d+d;y5 = A5.*exp(-k3m.*(x5-d-b-d)) + B5.*exp(k3m.*(x5-d-b-d));plot(x5,y5,'b');hold off;zoom on;
Twowaveguide.m
M(1,1) = 0
1.5 1.51 1.52 1.53 1.54 1.55 1.56 1.57 1.58 1.59
-50
0
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mode profile
-10 -5 0 5 10 15 20-1.5
-1
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