List of In-Class 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.
Extra-credit
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)`.
Extra-credit
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 extra-credit of last in-class (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.
Extra-credit
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:
`hat k = - hat y`.
In general, the argument inside the cos or sin function is `omega t - vec k cdot vec r = omega t - k hat {k} cdot vec r`. For this problem `hat k cdot vec r = - hat y cdot (x hat x + y hat y + z hat z)=-y`. However, in general `hat k cdot vec r = {vec k}/ | vec k| cdot ( x hat x + y hat y + z hat z )`
`= 1/k (k_x hat x + k_y hat y + k_z hat z) cdot ( x hat {x} + y hat {y} + z hat {z} )` `= {k_x x + k_y y + k_z z}/k` where `k = | vec k | = sqrt {k_x^2+k_y^2+k_z^2}`. For example, `cos ( t + y + x)` states that `omega = 1`, `k_y = 1` and `k_x =1 `. Hence, `vec k = - hat x - hat y` and `hat k = -1/sqrt{2} hat x - 1/sqrt{2} hat y`, i.e. propagating in a direction making angle of `225^o ` from the positive x axis.
b) `c = omega / k = {10^6} / (1/300} = 3 times 10^8` m/s `= c_o`
c) Polarization direction is `hat E = hat {H} times hat k = hat z times - hat y = hat x`
If the propagation direction is along one of the coordinate axes, polarization direction should be the axis not along the H direction and the propagation direction, i.e. the axis not mentioned in the H expression. In general, we should use `hat {H} times hat k` to find `hat {E}`, particularly for propagation direction between axes or off-axis.
Extra-credit
Since `c = c_o` and `mu = mu_o`, `epsilon = epsilon_o`.
`c = 1 / sqrt{ mu epsilon} => epsilon = 1 / {mu c^2} = 1/ {36 pi times 10^{9}} = 8.84 times 10^{-12}` F/m

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_1-n_2)/(n_1+n_2) |^2 = |0.5/(2.5)|^2 = 1/25`
Boundary 2: `R = | (n_2-n_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.
Extra-credit
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} , 1-d_1/{f_1})] [(1, d_2),(-1/{f_2} , 1-d_2/{f_2})]`
`=[(1 - d_1/{f_2} , d_2+ d_1 (1 - d_2/{f_2})),(-1/{f_1}-1/{f_2} (1-d_1/{f_1}) , -d_2/{f_1}+(1-d_1/{f_1})(1-d_2/{f_2}))]`
Ectra-credit
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
Extra-credit
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`.
Extra-credit
For 1), Finesse = ?
`frF = {pi (frR frR)^(1/4))/(1-sqrt(frR frR)} = {pi (frR )^(1/2))/(1-frR }=pi( 2/3)/ (5/9)={6pi}/5`.
Oct. 18, 2017
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`
Extra-credit
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`
Oct. 23, 2017
1) a) Find the fractional power transmitted in one round trip for the block of material inside a FP-cavity 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`
Extra-credit
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`

Nov. 20, 2017
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?
rate = `N / tau = 10^9/(0.5 times 10^{-9}} = 2 times 10^(18)`

c) Consider a 0.5 m F-P 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?
Note that `gamma = 0.1 cm^{-1} = 100 times 0.1 m^{-1}`
`I = frR I_o e^(10 times 2 times 0.5} = 0.8 e^(10} = 17.6 ((kW) /(cm^2))`

Extra-credit
For c), what is the intensity leaving the cavity after 1 round trip?
Note that the beam only travel one pass before exiting
`I_(out) = (1- frR) I_o e^(10 times 0.5} = 0.2 e^(5) = 29.68 ((W) /(cm^2))`


Nov. 27, 2017
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).
`Delta E = 0.001 (ev) => Delta nu = 0.01 times 241.9 = 2.419` (THz).

Extra-credit
Find the value of `g_o` with the definition of `g( nu )`, `int_0^oo g (nu) d nu=1`. Also specify the unit of `g_o`.

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


Dec. 4, 2017
Write a reate equation for level 3.

`(d N_3)/(dt) = R_3 - (N_3)/(tau_(30)) - (N_3)/(tau_(31))`

Extra-credit
Write a rate equation for level 1.

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




Last Modified: Oct. 12, 2017
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