**Summary of Lectures**

Jan. 29. 2019

HW 2 due 2/7/2019

1) Show that the field

`vec E = 20 sin ( k_x x ) cos (k_y y) hat z`

where `k_x^2 + k_y^2 = omega^2 mu_o epsilon_o` , can be represented as the super
position of four propagating
plane waves. Find the corresponding `vec H`. (10 points)

2) A dielectric material is characterized by the `vec D` to `vec E` relationship

`[(D_x),(D_y),(D_z)]=[(epsilon_{x\x},epsilon_{xy},0),(epsilon_{yx},epsilon_{y\y}
,0),(0,0,9)] [(E_x),(E_y),(E_z)]`

a) Name the property for this material. ( 1 points)

b) For `vec E = E_x hat x + E_y hat y`, find the value(s) of `E_y/E_x` for whic
h `vec D` is parallel to `vec E`. ( 7 points)

c) Find the effective permittivity for each case. ( 2 points)

Feb. 5. 2019

HW 3 due 2/14/2019

1. An opaque screen is used to intercept a Gaussian beam. At the screen,
the beam radius is `w`.
A ring-shaped aperture with an inner radius `a = w` and an outer radius
`b = 1.5 w` is cut onto the screen.
Calculate the percentage of power transmitted through the aperture.

2. A Gaussian beam (`lambda_o approx 1.5708 mu m`) with a beam waist
radius of 0.25 mm is to be transformed to a beam with a beam waist radius
of `0.10` mm. The separation of the two beam waists is 12 cm.
Determine the focal length f and the position x of the thin lens
relative to the input beam waist.

3. Find the peak intensity and beam radius of a Gaussian beam by solving
paraxial wave equation (Eq. (3.1-2) in the textbook) numerically.
More precisely, you will solve the normalized paraxial wave equation

`-j {partial A} / {partial Z}+ grad _{_|_}^2 A =0` where
`Z=z/ (4 z_o )`, `(X,Y)=(x / w _ o , y / w_ o )`.

Recall the paraxial wave equation can be expressed in Fourier domain as:

`{partial A_F} /{partial Z} =j ( K_x^2+ K_y^2 ) A_F`

where `A_F = FFT (A)` Now the form of solution of `A _F` is just that of a 1st order ODE.
Consider a Gaussian beam with `lambda _o =1 mu m` propagating in air has `I _o= 1 W/ m ^2` and `w_o=10 mu m` at z=0.
a) Plot intensity distribution at z=0, b) plot intensity distribution at
`z =2 z_o`, c) find peak intensity and beam radius at `z =2 z_o`.

Your results should be obtained from Matlab and compared to analytical solutions.

Matlab function to be used: fft2, ifft2, fftshift, image, colormap(gray(256)).
(Hint: to plot 8bit gray scale image, data must be normalized to 0 -255, i.e.
input data array of image should be 255*A/max(A).)

4. Start from Maxwell equations in phasor form with permittivity `mu = mu_o`,
`vec D = epsilon_o ( 1+chi_h+chi) vec E = epsilon_h vec E + epsilon_o chi vec E`and conductivity `sigma`.
Show that for a plane wave (phasor), the Helmholtz equation is
`grad^2 vec E + k_c^2 vec E =0` where `k_c = omega sqrt{mu_o epsilon_h}
(1+chi-j sigma/(omega epsilon))^(1/2)`.

Extra-Credit

Spatial filters consisting of a lens and a pinhole with a small aperture are
often used to reduce higher-order modes.
Suppose the input is a Gaussian beam with wavelength `lambda_o` that
diverges with half angle `theta_o`.

a) The distance `z_a` between the lens and the beam waist of the incoming beam is larger than the focal length `f` of the lens.
Where should the pinhole be placed if the power passing through the pinhole is to be maximized?

b) Assuming that the pinhole of radius `a` is positioned precisely at the point
where it is supposed to be, derive an expression for the fractional power
power passing through the pinhole as a function of `z_a`, `lambda_o`, `f`, `theta_o`
and `a`.

c) Assume `lambda_o = 0.633 mu m`, `theta_o = 1.5 mrad`, and `z_a = 20 cm`.
As the focal length changes from 2mm to 40mm, plot the fractional power passing
through a pinhole with radius of `2.5 mu m`.
Repeat for pinholes with radii of 5 and 10 `mu m`.

Feb. 12. 2019

HW 4 due 2/21/2019

1. Determine an expression for the group velocity `v_g` of a resonant
medium with refractive index given by the near resonance
equations in Eq. (5.5-21), (5.5-19) and (5.5-20) of the 1st Ed. [equivalent to
Eq. (5.5-27), (5.5-23) and (5.5-24) of the 2nd Ed.].
Sketch `v_g` as a function of the frequency `nu`.
(identify regions of
normal and anomalous dispersion)
[Problem 5.6-2 (page 192) in 1st Ed.] (10 points)

2. A Gaussian pulse of width `tau_0=100` ps travels a distance of 1km through an optical fiber made of silica with the characteristics shown in Fig. 5.6-5 (on page 190 in 1st Ed and page 189 in 2nd Ed]. Estimate the time delays `tau_d` and the width of the received pulse if the wavelength is (a) 0.8 `mu m`, (b) `1.312 mu m`, (c) `1.55 mu m`. [Problem 5.6-3 (page (192) ] (10 points)

Extra-Credit

Write a program (in Matlab or other language) to solve
`{partial A} / {partial z}= j {D_nu} / {4 pi} {partial^2 A} / {partial t'^2}`
[Eq. (5.6-18) in 1st Ed of text book] with
the method outlined by
`A ( z, t ) = F^-1 [ H( f , z ) F[ A( 0 , t ) ] ]` where
`H(f)= exp ( - alpha ( f+ nu_o ) z / 2 - j [ beta ( f+ nu_o ) - beta_o ] z )`,
[Eq. (5.6-2) and (5.6-4) in 1st Ed of textbook] and apply FFT.
Test your program with the following cases with the amplitude of pulse plotted
versus time at z=0, z=0.2km, z=0.4km, z=0.6km, z=0.8km and z=1km :
a) Input Gaussian pulse with width = 10ps and wavelength `0.8 mu m`.

b) Input Gaussian pulse with width = 10ps and wavelength `1.55 mu m`.

c) Input Gaussian pulse with width = 1ps and wavelength `0.8 mu m`.

d) Input Gaussian pulse with width = 1ps and wavelength `1.55 mu m`.

Use the dispersion parameter in Fig. 5.6-5 and calculations similar to
problem 5.6-3 to guide your calculations.

As a rule of thumb, at least 50 samples falls within the pulse.
The entire window should be at least 5 times of the pulse width.
This prevents aliasing caused by expansion of the pulse and gives you an
idea on total number of samples within the window.
Please submit your program with the homework. (15 points)

Last Modified: January 24, 2019

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