List of Homework

In class 1/25/18
a) A system has a total bandwidth of 60GHz. Each channel requires 300MHz. How many channels can be supported by the system ideally?
No. of Channels = 60G/0.3G = 200
b) A plane wave with $\omega = 10^{13}$ rad/s propagates in air along x axis. Its H field in in z directions. Find its wave number
k = $\frac \omega c = \frac {10^{13}} {3 \times 10^8} = 0.333 \times 10^5$ (rad/m)
bonus
c) Find polarization direction of the wave in b).
$\hat E = \hat H \times \hat k = \hat z \times \hat x = \hat y$

In class 1/30/18
a) A plane wave with electric field amplitude of $60 \pi$ (V/m) propagates in air along x axis. Its H field in in z directions. Find its H amplitude.
Medium is air $\eta = \eta_o$
$H_o = E_o / \eta_o = \frac {60 \pi} {120 \pi} = \frac 1 2$ (A/m)

b) A 2mW laser has wavelength of 1 micron and linewidth $\Delta \lambda$ of 1nm.
i) Find its bandwidth $\Delta \nu$ in Hz.
$\Delta \nu = \Delta \lambda \frac \nu \lambda = \Delta \lambda \frac c {\lambda^2} = 10^{-9} \frac {3 \times 10^8} {(10^{-6})^2} = 3 \times 10^{11}$ (Hz)
ii) Find the number of photon emitted per sec.
# photons/sec = $P_{op} / E$, E(ev) = 1.24 / $\lambda (\mu m)$ =1.24 (ev) and $E$ = q E(ev) = 1.24q
# photons/sec = $\frac {2 \times 10^{-3}} {1.24 q} = 10^{16} (s^{-1})$

bonus
If it takes 10 electrons to generate 1 photons, how much current is required to sustain the laser power in b)?
# electrons/sec = 10 (# photons/sec), I = q (# electrons/sec) = 10 $\frac {2 \times 10^{-3}} {1.24 } = 16.13$ (mA)

HW 1 due Feb. 1, 2018 5PM
HW 1 Solution

In class 2/1
a) Identify the polarization for the following configuration:

Since E is parallel to the screen, this is p-polarization.
b) Name THE special property of the polarization in a).
P-polarization has complete transmission at the Brewster angle.
Bonus
c) What is the fraction of the input power of an EM wave is not being transmitted at glass-air interface? Assume the wave is normal to the interface. Notice that refractive index of air = 1 and refractive index of glass = 1.5.
Not transmitted => reflected. Fraction of power reflected = R = ` (n_2 - n_1 )^2 / (n_2 + n_1 )^2`
R = `0.5^2/2.5^2` = 0.04.

In class 2/6

a) For a laser beam with the beam shape shown on the left at the laser, which dimension of the beam with be longer when it is far from the laser?

x dimension > y dimension.

b) What is the optical path difference for a Michelson interferometer with one arm having an `d_1` long air column and another arm having a `d_2` long column with refractive index `n_2`? (see diagram on the left)
`2(n_2*d_2 - d_1)`
bonus
c) Find the condition for constructive interference for the interferometer in b).
`2(n_2*d_2 - d_1)*2*pi/lambda_o = 2*m*pi => n_2*d_2 - d_1 = m*lambda_o/ 2`

HW 2 due Feb. 8, 2018 5PM
HW 2 Solution
Matlab script for problem 6
Mathcad script for problem 6

In class 2/8
Given a wave with `vec e = hat x cos ( omega t - kz ) + hat y cos ( omega t - kz + phi )`
a) What should be the value of `phi` to generate a left circularly polarized wave?
`phi = pi / 2`
bonus
b) According to the phase in a) which axis is the slow axis?
x axis since the x component of `vec E` has more negative phase than that of y component.

In class 2/13

a) What material can be used to make a quarter wave plate?

Anisotropic media with x and y basis e.g. birefringent materials

b) What material can be used to make a frequency mixer?

Nonlinear media.

HW 3 due Feb. 15, 2018 5PM
HW 3 Solution

In class 2/15
A fiber with `n_1`=1.5, `n_2`=1.49 `lambda_o = 1 mu m`, a = 2.5 micron.
a) Find V and NA.
`NA = sqrt { 1.5^2 - 1.49^2} = 0.1729`
`V = 2 pi / lambda_o a NA = 2.707`

b) Is this a single mode fiber? Show steps for your answer.
V > 2.405 `=>` not single mode

c) A laser beam with `w_o = 1 mu m` at `lambda_o = 1 mu m`. What is its divergence angle?
`theta = 2 lambda_o / ( pi w_o ) = 2 / pi ` (rad)

Bonus
d) Can the beam in c) be optimally coupled into the fiber? Explain.

`alpha_m = sin^{-1} ( 0.1729 ) = 0.1738` (rad)
`theta / 2 > alpha_m` (bad), `w_o < a` (good)
Overall, the situation is not optimal.

HW 4 due Feb. 22, 2018 5PM
HW 4 Solution

In class 2/22/2018
A glass fiber with typical refractive index, `Delta` = 0.01 and its step index radius of 3 micron.
a) What dispersion (`D_{"intra"}` or `D_{"modal"}`) be considered if it operates at `lambda` = 1.5 micron. Justify your answer.
`V = {2 pi } / lambda a n sqrt{2 Delta} = {2 pi } / 1.5 3 times 1.45 sqrt{2 times 0.01} = 2.5768 > 2.405 => ` multi mode, Consider `D_{"modal"}`

b) What dispersion (`D_{"intra"}` or `D_{"modal"}`) be considered if it operates at `lambda` = 0.8 micron. Justify your answer.
`V = {2 pi } / lambda a n sqrt{2 Delta} = {2 pi } / 0.8 3 times 1.45 sqrt{2 times 0.01} = 4.83 > 2.405 => ` multi mode, Consider `D_{"modal"}`

bonus
Write down the formula for dispersion assuming the fiber is a DSF. for
i) `lambda` = 1.5 micron
ii) `lambda` = 0.8 micron.

i) Multimode `D_{"modal"} = n Delta / c`
ii) Multimode `D_{"modal"} = n Delta / c`

2/27/18 (1st assignment)
A fiber optics communication system is 5 km long. It has $D_{modal}$ = 1 ns/km and $D_{intra}$= 2 ps/km-nm It uses a laser with `Delta lambda` = 3nm.
a) If we use an appropriate `lambda`, we can choose the system to have V=2.3. Which dispersion parameter should be used? ($D_{modal}$ or $D_{intra}$)
V < 2.405 -> Single mode and `D_{"modal"}` = 0. So we consider $D_{intra}$

b) If we make a wrong choice of `lambda`, we have V=10. What is the time delay for the system that is introduced by the fiber?
`Delta tau = D_{"modal"} L = 1 ({ns}/{km}) times 5 (km) = 5 (ns)`

Bonus
c) What is the maximum data rate that can be achieved based on conservative estimate for situation in a)?

`Delta tau = D_{"intra"} Delta lambda L = 2 ({ps} / {km-nm}) 3 (mn) 5 (km) = 30 (ps)`
`R = 1 / { 4 Delta tau} = 1 / { 120 (ps)} = 0.00833 ({Tb}/s) = 8.33 ({Gb}/s)`

In class 2/27/18 (2nd assignment)
A laser has power of 3mW. The fiber link has 1dB loss for attenuation and 2dB loss of connectors.
a) Find laser power in dBm.
b) Total losses in dB.
c) Received power at the other end of the fiber link in dBm.
bonus
d) Express answer for c) in mW.
a) 10 log (3) = 4.77(dBm)
b) -3dB
c) `P_{rec} =` 4.77 -3 = 1.77 (dBm)
d) `P_{rec} = 10^0.177` = 1.503 (mW)

HW 5 due Mar. 1, 2018 5PM
HW 5 Solution

In class 3/1/18


a) The threshold power for SRS is 0.5W. You proposed a multi-channel system with 1mW per channel. How many channels can be supported by the system without incurring substantial SRS?
# of Channels = 500(mW)/1(mW) = 500.

b) Consider FWM of 2 channels with frequencies `f_a` and `f_b`. List 3 possible frequencies.
Single frequency: `f_1 = f_2 = f_3 = f_a => f_a + f_a - f_a = f_a`
`f_1 = f_2 = f_3 = f_b => f_b + f_b - f_b = f_b`
Two frequencies: `f_1=f_2=f_a and f_3=f_b => f_a + f_a - f_b = 2 f_a - f_b`
`f_1=f_3=f_a and f_2=f_b => f_a + f_b - f_a = f_b`

Bonus
c) Give the 4th frequency from FWM in b).
Two frequencies: `f_1=f_2=f_b and f_3=f_a => f_b + f_b - f_a = 2 f_b - f_a`
`f_1=f_3=f_b and f_2=f_a => f_b + f_a - f_b = f_a`

In class 3/6/18
During the lecture we consider the case `n_2 > 0` (positive nonlinearity). Now try to understand what will happen if `n_2 < 0` (negative nonlinearity).
Consider a Gaussian pulse
a) Draw `{dI} / {dt}` versus time.

b) Draw frequency chirp (`delta omega`) versus time.
Since `n_2 < 0`, `{dI}/{dt}` has the same polarity of `delta omega`. Therefore, the plot in a) can be applied to b)
c) Based on b), what is your interpretation for the freq shift (red shift or blue shift) at leading and trailing edges of the pulse?
See the colored labels in a). That means long wavelength (red) travels slower than short wavelength (blue).
Bonus
What should be the polarity of `D_{"intra"}` to compensate for the frequency chirp in b)?
To counter the SPM effect, `D_{"intra"}` needs to enable long wavelength (red) traveling faster (less delay) than short wavelength (blue), i.e. as `lambda` increases, `Delta tau` decreases (opposite tendency), `=> D_{"intra"} < 0`

HW 6 due Mar. 8, 2018 5PM
HW 6 Solution

In class 3/13
A piece of etalon is made of glass (n=1.5). It is 1mm thick. Find
a) free spectral range of the etalon,
`Delta f_{FSR} = c / {2nd} = {3 times 10^8}/{2 times 1.5 times 10^{-3}} = 10^{11}`(Hz)

b) power reflectivity between air and glass

`R = ({n_2 - n_1}/ {n_2 + n_1))^2 = ({1.5 - 1}/ {1.5 + 1))^2= 1 / {25}`

Bonus
c) Finesse of the etalon.

`F = {pi sqrt{R}}/ {1 - R} = {pi sqrt{0.04}}/ {1 - 0.04}= {5pi}/{24}= 0.6545`

In class 3/27/18
a) We consider to use an etalon as a filter for a multichannels optical communication system. If the etalon has a free spectral range of 100 GHz and Finesse of 50. Estimate the maximum number of channels that can be used with this filter and the bandwidth of each channel.
Max number of channels = 50
Channel bandwidth = 100/50 = 2(GHz)

b) How many 2x2 MZ filters are required for construction of a 1x4 filter?
There are two layers (`2^2 = 4`) and 3 MZ filters.

In class 3/29/18
a) What are the 3 radiative processes that happen in a laser?
Spontaneous emission, stimulated emission and absorption

b) Which one is responsible for lasing?
Stimulated emission

Bonus
c) GaAs has bandgap energy of 1.4eV. What will be the output wavelength of a GaAs LED at `0^o C`?
`E(eV) = 1.4 + 26 times 10^{-3} times 273 / 300 = 1.4237(eV)`
`lambda = 1.24 / 1.4237 = 0.871 ( mu m )`

HW 7 due Apr. 3, 2018 5PM
HW7 solution ( pdf version)
Gnuplot script for extra-credit a) and b)
Gnuplot script for extra-credit c), d) and e)

In class 4/3/18
a) How should we bias a laser diode? Same question for biasing a photodiode.
Forward bias for LD.
Reverse bias for photodiode.

b) The active layer of a LD is 0.2mm long with refractive index of 3 and attenuation coefficient of `1000 (m^{-1})`. Find mode spacing.
mode spacing = `Delta nu_{FSR} = c/{2nd} = {3 times 10^8} /{2*3*0.2 times 10^{-3}}` = 250(GHz)
Bonus
Find threshold gain coefficient for the LD in b).
`g_{th} = alpha + 1 / (2d} ln(1/{R^2})`
`R= ((3-1}/{3+1})^2= 0.25`
`g_(th} = 10 ^ 3 + {ln (16)}/ {2*0.2 times 10^{-3}}= 7932(1/m)`

In class 4/5
A single mode communication link has a 10km fiber with `D_{"intra"} =20 {ps}/{km-nm}`, a detector with a modulation bandwidth of 1GHz and a laser diode with a rise time of 0.3ns and `Delta lambda` (linewidth) of 1nm.
a) Find fiber delay time,
`tau_f = D_{"intra"} times L times Delta lambda = 20 times 10 times 1 = 200` (ps) = 0.2 (ns)

b) Find rise time of detector
`tau_r = 0.35 / {10^9} = 0.35` (ns)

c) Find the data rate of the system.
`tau_{sys} = sqrt{ tau_f^2 + tau_r^2 + tau_t^2} = sqrt {0.2^2 + 0.35^2 + 0.3^2} = 0.5025` (ns)
` R_b = 1 / {4 tau_{sys}} = 1 / { 4 times 0.5025}` (Gb/s) `= 497.5` (Mb/s)

bonus
d) Name two methods to increase the effective data rate without changing the fiber.
Reduce the rise time of laser, reduce rise time of the detector and reduce linewidth of the laser.

HW 8 due Apr. 11, 2018 5PM
HW 8 solution

In class 4/12
An APD with responsivity=0.3A/W, M=30, `I_d = 2nA`, `P_{"in"}= {10^{-5}} /3` (mW)
a) Find a) `I_{apd}`.
`I_{apd} = M times I_{ph} = M times R P_{"in"} = 30 times 0.3 times 0.333 times 10^{-5}=30`(nA)

b) Find mean square value of shot noise for the APD if `Delta f = 20`MHz and `F_{apd} =1`,
`\bar {i_S^2} ` = `2q (R P_"in" + I_d) Delta f times M^2 times F_{apd} = 1.728 times 10^{-17}(A^2)`

c) Find mean square value of thermal noise if `Delta f`=20MHz and F=1, T=300K and `R_L`= 500 Ohm.
`\bar {i_T^2} ` = `{4 kT Delta f times F}/{R_L} = 6.656 times 10^{-16} (A^2)`

bonus
d) Find electrical power detected by APD.
`P_{detected} = (M ( I_{ph} + I_d ) )^2 R_L = 900 times 9 times 10^{-18} times 500 = 4.05 times 10^{-12} (W)`

In class 4/17
For a thermal-noise limited digital system, we have SNR=17dB.
a) What is its BER?
SNR = `10^{-17/10} = 50.12`
`x = 0.354 times sqrt{SNR} = 2.506`
BER = `0.5 times (1-erf(x)) = 0.5 times (1-0.99959) = 2.05 times 10^{-4}`
b) What should be the minimum receiver power if the mean square current of thermal noise is `10^{-15} (A^2)` for a detector with responsivity=0.5A/W.
For NEP calculation, SNR=1 but this problem gives SNR=17dB = 50.12. We need to use this SNR
`I_{ph}^2 = SNR times 10^{-15} = 5.012 times 10^{-14} (A^2)`
`P_{"in"} = I_{ph}/R = 2 I_{ph} = 4.48 times 10^{-7}` (W)

HW 9 due Apr. 19, 2018 5PM
HW 9 solution

In class 4/19
a) What is the difference between a 1R regenerator and a 2R regenerator?
2R regenerator has pulse reshaping.
b) For EDFA, which pumping wavelength (`lambda_p`) has the highest efficiency? (5 possible `lambda_p`s).
1480 nm (the longest pumping wavelength)

In class 4/24

a) If we limit the optical bandwidth which noise term will be dominant in EDFA?

b) What is the best pumping configuration for amplification based on SRS?
sig-ASE noise.

b) What is the best pumping configuration for amplification based on SRS?
Counter pump or backward pump.

HW 10 due Apr. 26, 2018 5PM
HW 10 solution

HW 11 due May 3, 2018 5PM
HW 11 solution


Last Modified: April 11, 2018
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