Homework 4

constants and unit definitions

This is a block of code I use to define my constants. The internal units for calculations are SI, but the numbers can be entered in other units.

s=1;ms=(10^(-3))s;us=(10^(-6))s;ns=(10^(-9))s;ps=(10^(-12))s; fs=(10^(-15))s;
Hz=1;GHz=(10^9)Hz;THz=(10^12)Hz;
m=1;cm=(10^(-2))m;mm=(10^(-3))m;um=(10^(-6))m;nm=(10^(-9))m;angs=(10^(-10))m;
kg=1;g=(10^(-3));
J=1;mJ=(10^(-3))J;eV=e J;keV=(10^3)eV;MeV=(10^6)eV;
W=1;kW=(10^3)W;GW=(10^9)W;mW=(10^(-3))W;
Torr=1/760;
K=1;

c=299792458 m/s;      
eps0=8.854187817 (10^(-12));   (*permittivity of free space*)
re=2.81794092 (10^(-15))m;
e=1.60217733 (10^(-19));		(*Coul electron charge*)
Natm=760 3.3 (10^16)/(cm^3);  (*no.particles in 1 atm*)
me=9.1093897 (10^(-31))kg;
mp=me/0.000544617013;
hbar=1.05457266 (10^(-34))J s;
h=2π hbar;
kB=1.3806503 (10^(-23)) J/K;

Problem 1: Michelson interferometer output

a. As in the lab, we consider a Michelson interferometer that has one mirror detuned so that the output beams are at a slight angle. First calculate the interference intensity pattern in just the x-direction. Do this two ways: 
- first choose the z-axis along the beam direction that has not been adjusted, so that the other beam is at an angle (Subscript[γ, x]) to the other beam. 
- next calculate the interference expression using a reference axis that lies along the bisector between the the beams, so that the half angle is (Subscript[θ, x])=(Subscript[γ, x])/2. Show that these give the same result. Hint: use the exponential notation for the fields to make the analysis simpler.

b. Derive an expression for a 2D interference pattern, with half angles (Subscript[θ, x]) and (Subscript[θ, y]). Use DensityPlot[ ] to make an image of an interference pattern with (Subscript[θ, x])=2(Subscript[θ, y]). Make the plot range dimensions similar to the size of the CCD camera, 5mm x 5mm, use the HeNe wavelength. Try using Manipulate[ ] with sliders for the two angles to simulate adjustment of the mirror mount.

c. Put a Gaussian intensity envelope on each beam, and make the center of the beams move as the angles are changed. Note how the fringe contrast depends on the local power balance between the beams.

Problem 2: diverging beam in a Michelson

a. When a lens is placed in front of the interferometer, the input wave is paraxial spherical, with a radius of curvature that is equal to the distance to the focal point. If the arm lengths are unequal, the camera sees an interference between two waves with different wavefront curvature. Write an expression for the sum of the two fields, using x,y coordinates, then use Manipulate[ ] and DensityPlot[ ] with a slider for the path length difference. Use realistic numbers similar to your lab set up.

b. Add as a variable an angle between the beams in the x-direction, and show how the fringe pattern varies as the arm length difference changes from positive to negative.

Problem 3: pump laser optics

A Ti:sapphire laser (wavelength 800nm) is pumped with a collimated green laser beam (532nm, diameter d=2mm). To get good pump overlap with the laser mode, the pump beam is focused (with lens (Subscript[L, P])) through the curved dielectric mirror ((Subscript[F, 1])) that is near the laser crystal. This curved optic is coated to reflect 100% at 800nm, but is transparent at 532nm. Therefore it acts like a negative lens for the pump beam. The substrate of the curved mirror has a refractive index of 1.5.

a. If the curved mirror radius is 100mm, and the pump lens focal length is 75mm, determine the distance the lens must be from the curved mirror to place the pump beam focus at the focal point of the curved mirror.  Treat both optics as “thin”, so you can use paraxial approximations.  You can solve this either with ABCD or the imaging equation.

b. What is the f-number of the final beam focus? (f-number = dist to focus/beam diameter). If we now treat the beam as a Gaussian laser beam (d = 1/(e^2) diameter), what would the diffraction-limited focal spot diameter be? You can use geometric optics to get the beam size at the curved mirror.

Problem 4:

a. A CW laser beam has a Gaussian intensity profile I(r)=(Subscript[I, pk])exp(-2(r^2)/((Subscript[w, 0])^2)). Calculate the power in the beam, in terms of (Subscript[I, pk]) and (Subscript[w, 0]).

b. Suppose we set up a 1W laser beam (λ=514nm) here in Golden to signal to a team at DIA airport 35 miles away. To get the minimum spread in the beam, we would have a slight focus to put the beam waist at half of the distance.  What is the smallest initial beam size that we can use under this condition?

c. What is the intensity of the beam at the target?