3-dB Couplers/splitters 50% 50%

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

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

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.

Mode–coupling theory index perturbations Generalized propagation equation

perturbation theory, TE mode Mode–coupling equation

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 .

Couplers/splitters, working principle – optical mode coupling 1 2 3 Un-connected port A1(z) A2(z) z L Without coupling Mode-coupling equations, phase matched case

Couplers/splitters, working principle – optical mode coupling 1 2 3 Un-connected port A1(z) A2(z) z Energy conservation

WDM Couplers/splitters, working principle – optical mode coupling 1 3 A1(z) A2(z) 2 Un-connected port Fused fiber z L A, B, C, D are determined by the initial conditions

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

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

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

No coupling between two different wavelengths 1 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

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

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)

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

Waveguide gratings A1(z) A2(z)

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

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

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

Ring resonators

Ring resonators P

Ring resonators T

Ring resonators

Ring resonators

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;

Group velocity Recall velocity of a wave

Group velocity Group velocity dispersion