1. 3-dB Couplers/splitters
50%
50%

2. WDM Couplers/splitters
1, 980nm
2, 1550nm 1 3
1 + 2 2
1 + 2
2, 1550nm
1, 980nm
1, 980nm
2, 1550nm
Insertion loss
Port 1 to 3
0.2dB
Port 2 to 3
20dB
Insertion loss
Port 1 to 3
20dB
Port 2 to 3
0.3dB

3. WDM Couplers/splitters
1, 980nm 1 3 2
1 + 2
2, 1550nm
1, 980nm
2, 1550nm
Insertion loss
Port 3 to 1
0.2dB
Port 3 to 2
20dB
Insertion loss
Port 3 to 1
20dB
Port 3 to 2
0.3dB

4. Mode–coupling theory
Recall mode equations
TE Mode:
Notice:
Mode profile E(x, y) doesn’t change with z if no index perturbation.
No coupling between different modes if no index perturbation.

5. Mode–coupling theory
index perturbations
Generalized propagation equation

6. perturbation theory, TE mode
Mode–coupling equation

7. discussion of the mode-coupling equation
if no index perturbation, A(z) is an constant, goes back to the normal equation.
Cn measures how much the index perturbation changes the propagation constant n.
If index perturbation doesn’t vary with z, coupling between the mode depends on the

8. Couplers/splitters, working principle – optical mode coupling1 2 3
Un-connected port
A1(z)
A2(z) z L
Without coupling
Mode-coupling equations, phase matched case

9. Couplers/splitters, working principle – optical mode coupling1 2 3
Un-connected port
A1(z)
A2(z) z
Energy conservation

10. WDM Couplers/splitters, working principle – optical mode coupling1 3
A1(z)
A2(z) 2
Un-connected port
Fused fiber z L
A, B, C, D are determined by the initial conditions

11. Splitter 1 3
For A1(0) = A1, A2(0) =0
A1(z)
A2(z)
Un-connected port 4

12. 2 to 1 combiner
Initially, A1(0), A2(0)
A1(0) = A2(0) 1 3
A1(z)
A2(z) 2
Un-connected port
z = 0
Similarly

13. 2 to 1 combiner
Initially, A1(0), A2(0)
A1(0) not eq. A2(0) 1 3
A1(z)
A2(z) 2
Un-connected port
z = 0
Similarly

14. No coupling between two different wavelengths1 3
A1(z)
WDM coupler
1, 980nm
2, 1550nm
A2(z) 2
Un-connected port
Fused fiber
No coupling between two different wavelengths
For 1, 980nm
Initially, A1(0, 1), A2(0, 1) =0
z = 0
z = 0

15. 1 3
A1(z)
1, 980nm
2, 1550nm
A2(z) 2
Un-connected port
For 2, 1550nm
Initially, A1(0, 2)=0, A2(0, 2)
z = 0

16. 2, 1550nm 1 3
A1(z)
1, 980nm
A2(z) 2
Un-connected port
For 2, 1550nm
Initially, A1(0, 2), A2(0, 2)=0
For 1, 980nm
Initially, A1(0, 1)=0, A2(0, 1)

17. 2, 1550nm
For 2, 1550nm 1 3
A1(z)
1, 980nm
Initially, A1(0, 2), A2(0, 2)=0
Un-connected port
A2(z) 4
For 1, 980nm
Initially, A1(0, 1), A2(0, 1) = 0

18. Waveguide gratings
A1(z)
A2(z)

19. Waveguide gratings A1(z) A2(z)
A, B, C, D are determined by the initial conditions

20. Waveguide gratings
DBR grating
A1(z)|z=0 = A1(0)
A1(z)
Waveguide
A2(z) )|z=L = 0
A2(z)

21. Waveguide gratings
A1(z)|z=0 = A1(0)
A1(z)
A2(z) )|z=L = 0
A2(z)

22. Ring resonators

23. Ring resonators P

24. Ring resonators T

25. Ring resonators

26. Ring resonators

27. Ring resonators n1 = 1; lamda = 1.538:0.0000001:1.539; %in um
beta = 2*pi./lamda;
R = 10/(2*pi*n1);
kapa = 0.005;
F = 1/kapa^2;
pha = beta*pi*R*n1;
t = sqrt(1-kapa^2);
P(1,1) = t;
P(1,2) = 1;
P(2,1) = 1;
P(2,2) = t;
P = P/kapa;
for m = 1:length(lamda)
T(1,1) = exp(-j*pha(m));
T(1,2) = 0;
T(2,1) = 0;
T(2,2) = exp(j*pha(m));
M = P*T*P;
Tr(m) = (abs(M(1,2)/M(2,2)))^2;
Trr(m) = Tr(m)*(abs(1/M(1,2)))^2;
end
plot(lamda,Tr);
hold on;
plot(lamda,Trr,'r');
zoom on;
hold off;

28. Group velocity
Recall velocity of a wave

29. Group velocity
Group velocity dispersion