Optical Instrument using MATLAB Calculation

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Project Description:
these are not essay questions, they are calculation based questions:
there are 4 questions together. each question has 3 to 4 sub questions. Topic of the 4 questions are:
1. Lyot Filter 2. Laser Cavity 3. Mode{Locked Pulse 4. Radiometry.
Each question need calculations and figures and some of them need matlab. Please let the tutor read VERY carefully what the questions are, and then solve them. I don't want any thing copy from internet, please let the tutor really do them. Thanks.

Details of 1 parts : Please download attachment to see all details.
Draw a figure for each of the problems. Usually in my problems, the first step is to generate a layout of the optical system. I give points for figures. You will want to use a computer for some of the problems. You may use any language you like, but make sure that the equations and graphs are presented in such a way that I don?t need to look at your code. When I ask for a plot, I am looking for a correctly labeled one, with correct numerical values. A sketch is not sufficient. Present your work as clearly as possible. I give partial credit if I can figure out that you know what you are doing. I do not give credit for putting down everything you know and hoping I will find something correct in it. Typesetting or word–processing really helps.
1 Lyot Filter
Here we consider a Lyot Filter, or liquid–crystal tunable filter, often used in hyperspectral imaging. A single “stage” of the Lyot filter consists of a pair of crossed polarizers with a liquid crystal sandwiched between them. A voltage is applied to the liquid crystal layer to vary its birefringence. The axes of the liquid crystal layer are oriented at 45 degrees relative to those of the polarizers. If the birefringence is set so that the layer is a half–wave plate for a wavelength, λ set , then this wavelength is transmitted through the device with the only loss being from absorption and scattering in the materials. At other wavelengths, the stage will transmit less light. By adjusting the voltage, we can tune the filter. However, there are ambiguities. If the device is a half–wave plate for λ set then it is a 3/2 waveplate for λ set / 3 and so forth. Therefore, multiple stages are needed with different thicknesses. At λ set these stages have birefringence equal to odd half–multiples of the wavelength, O P D = N λ set / 2 where N = 1 , 3 , 5 . . . .
1.1 Individual Stages
Plot the transmission for unpolarized light as a function of wavelength
from 420 to 730 nm with the set wavelength at
λ set
= 650 and 480 nm, for stages,
N
= 1 ,
3 ,
5 ,
7 ,
9. For my solution, I used coherency matrices.
1.2 Performance
Plot the overall transmission of the five–stage device ( N
= 1 ,
3 ,
5 ,
7 ,
9) on the same figure. What is the linewidth (full–width at half–maximum, FWHM) for each set wavelength?
What is the maximum transmission within the plotted band at a wave- length other than the desired one?
1.3 Improvement
Using up to seven stages (not necessarily consecutive), see how narrow you can make the filter, while keeping the “leakage” of unwanted wavelengths low.
2 Laser Cavity
Let?s look at an Argon ion laser designed for operation on the 514 nm line. Now, let?s design this cavity so that the minimum beam diameter (Gaussian 1 /e 2 in the cavity is 3 mm, and the beam achieves a focus outside the cavity in front of the laser, with a waist diameter of 500 µ m. The cavity must be 30 cm long.

Skills required:
Matlab & Mathematica, Technical Writing
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