List of InClass Assignments Solutions
Sept. 11, 2017
List 2 properties of laser light.
Coherent (temporal and spatial), monochromatic, stable phase, ideally `TEM_{0 0}` and directional without spreading. Extracredit A laser lases at `3 mu m` with a gain profile having linewidth of 10 nm. Find bandwidth of the gain profile `nu = c_o / lambda_o = {3 times 10^8} / {3 times 10^6} = 10^{14} Hz` `{Delta nu} / nu = {Delta lambda} / lambda => Delta nu = nu {Delta lambda} / lambda =>10^14 {10 nm}/ {3000 nm} = 1/3 times 10^12 (Hz) = 1/3 (THz)` 
Sept. 13, 2017
a) List the 3 major components of a laser.
Pumping (energy) source, gain (active) medium, feedback. b) Gain profile has bandwidth of `5 THz` and the free spectral range `Delta nu_{FSR} = 20 GHz`. Estimate the number of longitudinal modes in the laser? Number of longitudinal modes = `{Delta nu_g} / {Delta nu_{FSR}}=5000/20=250`. c) A pulse laser lases at `3 mu m` with a gain profile having linewidth of 10 nm. It emits square pulse with energy of `1 mu J` per pulse, pulse width of 1 fs and repetition rate of 500 Hz. i) Find photon energy of the laser in eV. `E(ev) = 1.24 / 3 = 0.413 (eV)`. ii) Find the peak power of the laser. `P_{peak} = E_{"pulse"} / t_{"pulse"} = 10^{6} / 10^{15} = 10^9 (W) = 1 (GW)`. Extracredit iii) Find average power of the laser in c). `P_{peak} = E_{"pulse"} times` repetition rate `= 10^{6} times 500 = 5 times 10^{4} (W) = 0.5 (mW)`. iv) Estimate the number of longitudinal modes for the laser in c) if `Delta nu_{FSR} = 10 GHz`. From the extracredit of last inclass (Sept 11), `Delta nu_{g} = 1/3 (THz)`. Number of longitudinal modes = `{1/3 times 10^{12}} / 10^(10} = 100/3 = 33` modes. Notice that this number must be in integer. 
Sept. 20, 2017
`vec ccH = hat z cos (10^6 t + 1/300 y )`
a) Identify propagation direction. b) Find speed of the wave. c) Find polarization direction of the wave. Extracredit If the medium where the wave propagate has `mu = mu_o= 4 pi times 10^{7}` H/m, find the permittivity `epsilon` of the medium.
a) The propagation direction is the spatial variable(s) inside the cos or sin
functions. In this case, it is:

Sept. 27, 2017
Sept 27, 2017 solution
Oct. 4, 2017
1) Consider `n_1=1`, `n_2=1.5`. Find the power reflectivity at boundaries
1 and 2.
Boundary 1: `R =  (n_1n_2)/(n_1+n_2) ^2 = 0.5/(2.5)^2 = 1/25` Boundary 2: `R =  (n_2n_1)/(n_1+n_2) ^2 = 0.5/(2.5)^2 = 1/25`. 2) What are the two special ray conditions for finding the ray matrix. Condition 1: `r=0`; condition 2: `r prime =0`. 3) How many ray matrices are required for relating Ray in and Ray out. There are 2 boundaries and 1 space `=>` 3 ray matrices. Extracredit 4) Write down all matrices required in 3). `[(1,0),(0,n_2/n_1)] [(1,L), (0,1)] [(1,0),(0,n_1/n_2)] =[(1,0),(0,1.5)] [(1,L), (0,1)] [(1,0),(0,1/1.5)]` 
Oct. 9, 2017
Find the ray matrix for the system on right.
`[(1, d_1),(1/{f_1} , 1d_1/{f_1})] [(1, d_2),(1/{f_2} , 1d_2/{f_2})]`
`=[(1  d_1/{f_2} , d_2+ d_1 (1  d_2/{f_2})),(1/{f_1}1/{f_2} (1d_1/{f_1}) , d_2/{f_1}+(1d_1/{f_1})(1d_2/{f_2}))]` 

Ectracredit
Draw the equivalent diagram in terms of lenses for the mirrors on the right and indicates the unit cell. 

Oct. 11, 2017
a) Draw the equivalent lens diagram for the cavity on the right.
b) Identify unit cell at the sample plane 

Extracredit
Is the resonator stable? `[(A,B),(C,D)]=[(1,1),(0,1)][(1,1),(1/2,1  1/2)]=[(1/2,3/2),(1/2,1/2)]` `{A+D}/2 = 1/2 (1/2 + 1/2) = 1/2 < 1 =>` The resonator is stable. 
Oct. 16, 2017
1) Find `Delta nu_{FSR}` and power reflectivity at the end of the etalon (FP
cavity) on the right.
`Delta nu_{FSR}` or `nu_F = c_o / {2nd} = {3 times 10^{8}}/{2 times 5 times0.01} = 3 times 10^9` (Hz). Power reflectivity `frR_1 = frR_2 = frR = n_1  n_2^2/n_1 + n_2^2 = 4^2/6^2 = 4/9`. 

Extracredit
For 1), Finesse = ? `frF = {pi (frR frR)^(1/4))/(1sqrt(frR frR)} = {pi (frR )^(1/2))/(1frR }=pi( 2/3)/ (5/9)={6pi}/5`. 
1) Find the fractional power transmitted from one end to the other end.
`T = (1 frR)^2 = (1 4/9)^2 = 25/(81)=0.309` 

Extracredit
If the block of material has attenuation coefficient `alpha = 50 (m^(1))`, now what is the fractional power transmitted? `T = (1 frR)^2 e^(50 times 0.01) = 25/(81) e^{0.5}=0.1872` 
1) a) Find the fractional power transmitted in one round trip for the block of
material inside a FPcavity with 2 plane mirrors that have power reflectivity of
0.9.
`T_{Total} = T^4 frR_m^2 = (5/9)^4 times 0.9^2 =0.07716` b) Find the fractional power loss per round trip. `F_l = 1  T_{Total} = 0.9228` 

Extracredit
If the block of material has attenuation coefficient `alpha = 50 (m^(1))`, now what is the fractional power loss per round trip? `F_l = 1  T_{Total} times e^{50 times 2 times 0.01} = 0.9716` 
Consider the life time for nonradiative emission is 1 ns and the life time for
radiative emission emission is 1 ns for transition from state 2 to state 1.
a) What is the life time of excited atoms in state 2? `1 /(tau_("total")) = 1/(tau_r) + 1/(tau_(rn)) => 1/(tau_("total")) = 2/(10^(9))` Hence, `tau_("total")= 0.5` (ns).
b) Assume number atoms in state 2 is `10^9`. What is the rate of change in
number of atoms in state 2?
c) Consider a 0.5 m FP cavity, one mirror has 100% reflectivity, another mirror
has 80% reflectivity. If the cavity is filled with a
gain medium with gain coefficient of `0.1 cm^{1}`, now an input optical beam
with
intensity of `1 W/{cm^2}` at the 100% mirror, what is the intensity of the
beam after 1 round trip?
Extracredit

Consider an isosceles triangle shaped line shape function in energy (ev) scale.
Redraw the diagram in `nu` (Hz).
`E_o=1 (ev) => nu_o=c_o/(1.24 times 10 ^(6)) = 241.9` (THz).


Extracredit
`(g_o Delta nu )/2 = 1 => g_o = 2/(Delta nu) = 8.268 times 10^(13)` (s), i.e.
`g_o=0.8268` (ps).

Write a reate equation for level 3.
`(d N_3)/(dt) = R_3  (N_3)/(tau_(30))  (N_3)/(tau_(31))` 

Extracredit
`(d N_1)/(dt) =  (N_1)/(tau_(1)) + (N_3)/(tau_(31)) + W_i N_2  W_i N_1`
