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.