Laser Physics: HW2

Problem 1. Practice with QM wavefunctions
Consider a mixture of two states of a particle in a one-dimensional infinite square well, so that ψ(x,t)=C (3 (Subscript[ψ, 2])(x,t)+4 (Subscript[ψ, 3])(x,t)).

a. Using Mathematica or calculating analytically, calculate the normalization constant C. Assume the eigenfunctions are normalized. Hint: using Dirac notation, so that (Subscript[u, n])(x)→|n⟩, makes this calculation much easier.

b. Calculate an expression for the charge density and make a plot of it at 3 representative times to illustrate the motion of the charge.

c. Calculate and expression for the time-dependent dipole moment.

Problem 2. 
Find a relation between spontaneous emission lifetime and cross section for a simple atomic transition that is independent of the dipole moment (take the expressions for each, and eliminate the dipole moment).

Problem 3.

a. What fraction of the total emission of a normalized Lorentzian lineshape of FWHM Δν occurs between ν-Δν and ν+Δν ? Numerical calculation is fine.

b. Repeat this calculation for a Gaussian lineshape.

Problem 4.

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)=1 J (m^(-3))(Hz^(-1))?

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))? Take the line center frequency to be (10^14)Hz.

Problem 5. 
Svelto problem 2.5.

Problem 6. 
Calculate the Doppler broadened linewidth for the following two cases:

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 7. 
Svelto problem 2.12. The saturation intensity is for saturation of absorption.