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
rate equations for a 3 level laser

Consider the energy level scheme for a 3-level laser shown in the figure below. 

In a conventional 3-level system such as ruby, the inversion is on the 1 to 0 transition and most of the population must be transferred out of level 0 to get a population inversion. In this case, we will be looking for an inversion on the 2 to 1 transition. 
Starting at t = 0, where the atoms are in the ground state, the atoms are raised from level 0 to level 2 at a pump rate (Subscript[R, p]). The lifetime of levels 1 and 2 are (Subscript[τ, 1]) and (Subscript[τ, 2]) respectively.

a) Write the rate equations for the population densities, (Subscript[N, 1]) and (Subscript[N, 2]), of level 1 and 2 respectively. Assume that the ground state 0 is not depleted to any significant extent and neglect stimulated emission from 2 to 0.

b) Find the steady state solution. What condition is required for a population inversion on the 2 to 1 transition?

c) Analytically calculate (Subscript[N, 1])(t) and (Subscript[N, 2])(t) for the initial conditions (Subscript[N, 1])(0)=0 and (Subscript[N, 2])(0)=0. You can use DSolve[ ] in Mathematica to check your work, but show the steps how to get the solution. Hint: the differential equation for the population of level 1 can be solved by multiplying both sides by the factor exp(t/(Subscript[τ, 1])). In this way the left-hand side of the preceding differential equation becomes a perfect differential.

d) Use Mathematica to plot the population densities ((Subscript[N, 1])(t) and (Subscript[N, 2])(t) ) and the population inversion ((Subscript[N, 2])(t)-(Subscript[N, 1])(t)) on the same plot. Make two plots using the following two input values (you may pick an arbitrary value of (Subscript[R, p])): 
ι.	(Subscript[τ, 1])=2μs, (Subscript[τ, 2])=1μs
ιι.	(Subscript[τ, 1])=1μs, (Subscript[τ, 2])=2μs
Is it possible to achieve a population inversion for each of the two cases? If so, what are the requirements on the pump pulse?

Problem 2
Generalized gain saturation

A more general derivation of the saturation in a two level system. In this version, we have pump rates (/vol/time) R1 and R2, and overall lifetimes out of the levels τ1 and τ2. We also include the effect of degeneracies on levels 1 and 2, g1 and g2. We’ll work with the beam intensity and the cross-section rather than the pump rate W=((I (Subscript[σ, 21]))/(h(Subscript[ν, L]))). With these included, the rate equations are
((d (Subscript[N, 1]))/(dt))=(Subscript[R, 1])+Δ(SuperStar[N]) (Subscript[σ, 21]) (I/(h(Subscript[ν, L])))-((Subscript[N, 1])/(Subscript[τ, 1]))+(Subscript[N, 2])(Subscript[A, 21])
((d (Subscript[N, 2]))/(dt))=(Subscript[R, 2])-(SuperStar[ΔN]) (Subscript[σ, 21]) (I/(h(Subscript[ν, L])))-((Subscript[N, 2])/(Subscript[τ, 2]))
where
Δ(SuperStar[N])=(Subscript[N, 2])-((Subscript[g, 2])/(Subscript[g, 1]))(Subscript[N, 1])=(Subscript[N, 2])-(Subscript[r, g])(Subscript[N, 1])
is the population inversion density. To make the notation more compact, I am writing (Subscript[r, g])=(Subscript[g, 2])/(Subscript[g, 1]). Remember that in this formulation, spontaneous emission out of level 2 is included in the lifetime in that level, (Subscript[τ, 2]).

a. First set the time derivatives = 0 for steady state, then calculate the steady state expressions for (Subscript[N, 1]^ss)  and (Subscript[N, 2]^ss). Then calculate the steady-state value of  Δ(SuperStar[N]), to show that
Δ(SuperStar[N])(I)=(Subscript[N, 2]^ss)-((Subscript[g, 2])/(Subscript[g, 1]))(Subscript[N, 1]^ss)=(((Subscript[R, 2])(Subscript[τ, 2])[1-(Subscript[r, g])(Subscript[A, 21])(Subscript[τ, 1])]-(Subscript[r, g])(Subscript[R, 1])(Subscript[τ, 1]))/(1+(Subscript[σ, 21])I(1/(h(Subscript[ν, L])))[(Subscript[τ, 2])+(Subscript[r, g])(Subscript[τ, 1])-(Subscript[r, g])(Subscript[A, 21])(Subscript[τ, 1])(Subscript[τ, 2])]))

b. This equation can be written in the standard form
 Δ(SuperStar[N])(I)=((Δ(SuperStar[N])(0))/(1+I/(Subscript[I, s]))) , with the saturation intensity (Subscript[I, s])=((h(Subscript[ν, L]))/((Subscript[σ, 21])(Subscript[τ, R]))). 
Write expressions for Δ(SuperStar[N])(0)  and the “recovery time” (Subscript[τ, R]) and give a description of what each of those terms mean physically.

Problem 3: wave interference

a. Write an expression for a plane wave traveling in a direction (Subscript[θ, x]) from the z-axis so that the k-vector is in the x-z plane and the wave is polarized along the y direction. Use complex (phasor) notation following the (e^(-i ω t)) convention.

b. The wave is incident on a mirror at z = L placed so that its surface normal is in the -(OverHat[z]) direction, so the component of the total E-field along the surface must be zero. Write an expression for the reflected wave and find its amplitude and phase so that the total E-field satisfies the boundary conditons.

c. Write an expression for the sum of the fields, and calculate the time averaged intensity of the combined field. Since we anticipate a standing wave in the z-direction, simplify the expression to show this explicitly (i.e. in terms of a sine or cosine function). The overall expression for the field will still be complex.

Problem 4: 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. extra credit: 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 5: 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.

c. For white-light (broadband) interferometry, the path lengths must be very close to be able to see interference. Describe how best to choose and position the lens to use this technique to get the arm lengths equal.

Problem 6: broadband interferometry

a. on the same lineout, use the 1D crossed beam expression to plot the interference pattern for several wavelengths across a 100nm bandwidth, centered at 600nm.

b. Write the expression for the interference pattern as a function of ω, and multiply it by a spectrum Exp[-2(ω-(Subscript[ω, 0]))/((Δω)^2)]. The camera integrates all the signal, so integrate across ω to get the observed spatial interference pattern. Adjust the crossing angle and/or the viewing range so that there is interference in the center but the fringe contrast decreases off to the sides. Interpret what you observe.