Homework 3

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: relative importance of stimulated emission

a. A two-level atom is placed within a cavity and is allowed to come into equilibrium with blackbody radiation of temperature T.  Show that the condition for the rate of stimulated emission from the upper level being equal to the rate of spontaneous emission is 
(Subscript[k, B])T=((h (Subscript[ν, 21]))/(ln 2)), 
where h (Subscript[ν, 21])is the energy spacing of the levels. For temperatures lower than this, the vacuum EM field has more energy density than the blackbody.

b. Find the temperature in electron volts (eV) and Kelvin when this condition is met for the transition in the following regions of the EM spectrum:
- radio frequencies at 50 Mhz
- microwaves at 1 GHz
- visible light at 500nm
- X-rays of energy 1 keV

c. Under this condition, what is the ratio of the populations per state in the upper and lower level? Explain your answer in terms of the spontaneous and stimulated transition rates between the two levels.

Problem 2.

a. What fraction of the total emission of a normalized Lorentzian lineshape of FWHM Δν occurs between (Subscript[ν, 0])-Δν and (Subscript[ν, 0])+Δν ? Numerical calculation is fine.

b. Repeat this calculation for a Gaussian lineshape.

Not sure why Mathematica returns this complicated result (an earlier version gave the cleaner version shown here)

Problem 3.

a. What is the stimulated emission rate if: (Subscript[A, 21])=(10^8)(s^(-1)), the line is homogeneously broadened with Δν=1 GHz, the input radiation is “white” with ρ(ν)=(10^(-12)) J (m^(-3))(Hz^(-1))?  Take the line center frequency to be (10^14)Hz.

b. What would be the stimulated emission rate if the input radiation were monochromatic, one FWHM from line center, and had intensity 1 W (m^(-2))? Assume the lineshape is Lorentzian.

Problem 4. 
Svelto problem 2.5.

Note that the fluorescence quantum yield is the ratio of the observed emission lifetime and the natural lifetime for the transition. The observed emission lifetime can be shorter because there are non-radiative mechanisms (e.g. collisions) that can also take population out of the upper state.

First find a relation between spontaneous emission lifetime and cross section for a simple atomic transition that is independent of the dipole moment, by taking the expressions for each, and eliminate the dipole moment.

Problem 5. 
Calculate the Doppler broadened linewidth for the following two cases:

the Doppler width of a gas absorption line is 
(SuperStar[Δν])=2(((Subscript[ν, 0])[((2 k T ln 2)/(M (c^2)))])^(1/2))

a. the 488nm transition of an argon ion laser, given that the temperature of the discharge is around 6000K.

b. the 633nm transition of a helium-neon laser, given that the temperature of the discharge is around 400K.

Problem 6: Polarization dependence of wave interference

When the polarization of the fields is included, the intensity is the dot product of the total field with itself: 
I=(1/2)(Subscript[ϵ, 0])n c E·(SuperStar[E])

a. Show that if we combine two fields with orthogonal polarization, there is no observed interference.

b. Suppose we have a waveplate in one of the arms of a Michelson interfermeter that allows us to rotate the polarization direction by an angle ϕ relative to the x-axis. Assuming the amplitude is equal in both arms, calculate an expression for the interference pattern as a function of ϕ and the relative arm length difference ΔL.

c. The fringe visibility is defined as 
V=(((Subscript[I, max])-(Subscript[I, min]))/((Subscript[I, max])+(Subscript[I, min])))
calculate the visibility as a function of the polarization angle ϕ.