HW 7 Solution

1) R_1 = R_2 = 0.9, lambda_o = 1.5 mu m, d = 1.5 mm, n = 1.46
a) Delta f_{FSR} = c / {2nd} = {3 times 10^8} / { 2 times 1.46 times 1.5 times 10^{-3}} = 6.849 times 10^{10} (Hz)
b)  ccF = {pi sqrt{R}} / { 1 - R} = {pi sqrt{0.9}} / { 1 - 0.9} = 29.81
b) ccF= {Delta f_{FSR}} / {Delta nu} => Delta nu = { 6.849 times 10^{10}}/ { 29.8} = 2.3 times 10^9 (nm)
c) {Delta lambda} / { lambda_o} = {Delta nu } / nu => Delta nu = {lambda_o^2} /c Delta nu = 2.3 times 10^9 (1.5 times 10^{-6})^2 / {3 times 10^8} = 0.01724 (nm)
d) d = {m lambda_o} / 2n => m = round ({2nd} / lambda_o) = 2920
e) Since m is an exact integer, lambda_o should be the intented wavelength, i.e. exact lambda_o = {2nd} / m = 1.5 ( mu m )

2) 16 Channels, 100 GHz spacing, n = 1.5, Delta nu = 2 GHz
i) Delta f_{FSR} approx 100 times 16 = 1600 (GHz)
Delta f_{FSR} = c / {2nd} => d = c / {2 n Delta f_{FSR}} = {3 times 10^8} / { 2 times 1.5 times 1600 times 10^9} = 0.0625 (mm)
ii) cc{F} = {Delta f_{FSR}} / {Delta nu} approx {1600 times 10^9} / {2 times 10^9} = 800
iii)  ccF = {pi sqrt{R}} / { 1 - R} = {pi x} / { 1 - x^2} where x = sqrt{R} => x^2 + pi / ccF x - 1 = 0
x = { - pi / ccF +- sqrt { (pi / ccF)^2 + 4}} / 2 approx - pi / {2ccF} +- ( 1 + 0.5 (pi / {2ccF})^2)
Choose solution of x that is less than 1, x = - pi / {2ccF} + ( 1 + 0.5 (pi / {2ccF})^2) = 1 - 1.962 times 10^{-3}
R = x^2 approx 1 -2 times 1.962 times 10 ^{-3} = 0.9961

3) a) Consider Fig. 3.21,

 At output port 2: Upper path: k_2 L + 2 times pi / 2 Lower path: k_2 ( L + Delta L) Condition for constructive interference: k_2 ( L + Delta L) - ( k_2 L + 2 times pi / 2 ) = 2 pi m k_2 Delta L - pi = 2 pi m => {2 pi} /c f_2 n Delta L = (2m + 1 ) pi At output port 1: Upper path: k_1 L + pi / 2 Lower path: k_1 ( L + Delta L) + pi / 2 Condition for constructive interference: k_1 ( L + Delta L) + pi / 2 - ( k_1 L + pi / 2 ) = 2 pi m k_1 Delta L = 2 pi m => {2 pi} /c f_1 n Delta L = 2m pi

b) Delta f = 25 GHz, n = 1.45, Delta L = ?
Delta f = f_2 - f_1 = c / {2 n Delta L} => Delta L = c / {2 n Delta f} = {3 times 10^8} / {2 times 1.45 times 2.5 times 10^{10}} = 4.138 (mm)

4) D_{"intra"} = 25 ps / {km-nm}, E_g = 0.36 + 2.012 x + 0.698 x^2, E (ev) = 1.24 / { lambda (mu m )}
a) x =0.4, E_g = 0.36 + 2.012 times 0.4 + 0.698 times 0.4^2 = 1.276 (ev)
b) At 273K, E = E_g + kT = 1.276+ 26 times 10^{-3} times 273/300 = 1.300 (ev), lambda = {1.24} / {1.300} = 0.9537 ( mu m)
At 323K, E = E_g + kT = 1.276+ 26 times 10^{-3} times 323/300 = 1.304 (ev), lambda = {1.24} / {1.304} = 0.9509 ( mu m)
c) Delta E = 3.3 kT and at 273K, Delta E = 3.3 times 26 times 10^{-3} times {273} / {300} = 0.07808 (ev)
{Delta E } / E = {Delta lambda } / lambda => Delta lambda = {Delta E } / E lambda = {Delta E } / E {1.24} /E = {0.07808 times 1.24} / {1.3^2} = 0.0574 (mu m)
At 323K, Delta E = 3.3 times 26 times 10^{-3} times {323} / {300} = 0.092378 (ev)
{Delta E } / E = {Delta lambda } / lambda => Delta lambda = {Delta E } / E lambda = {Delta E } / E {1.24} /E = {0.092378 times 1.24} / {1.304^2} = 0.0677 (mu m)
d) R_b = 1 Gb/s, Delta tau = D_{"intra"} Delta lambda L, R_b = 1 / {4 Delta tau} => L = 1 / {4 R_b D_{"intra"} Delta lambda}
At 273K, L = 1 / { 4 times 10^9 times 25 times 57.38 times 10^{-12}} = 0.174 (km)
At 323K, L = 1 / { 4 times 10^9 times 25 times 67.7 times 10^{-12 }} = 0.1477 (km)

5) a) 32 = 2^m => m =5, i.e. 5 layers or stages and number of MZ filters = 1 +2 +4 + 8 +16 = 31.
 b) c) Stage 1: Delta f = 30 GHz Delta L = c / {2 n Delta f} = {3 times 10^8} / {2 times 1.5 times 3 times 10^{10}} = 3.33 (mm) Stage 2: Delta f_2 = 60 GHz Delta L_2 = {Delta L_1} / 2 = 1.67 (mm) Stage 3: Delta f_3 = 120 GHz Delta L_3 = {Delta L_2} / 2 = 0.833 (mm) Stage 4: Delta f_4 = 240 GHz Delta L_4 = {Delta L_3} / 2 = 0.4167 (mm) Stage 5: Delta f_5 = 480 GHz Delta L_5 = {Delta L_4} / 2 = 0.2089 (mm)

6) To ensure the plot having 3-4 peaks, we choose the range for m = 2920 +- 2 and normalize the wavelength axis by the exact wavelength, i.e. lambda / lambda_o so that we get nice numbers on horizontal axis. I included scripts for "gnuplot" to show the plotting. This program can generate nice plot quickly.