Laser Spectroscopy (Classes 1 through 10)

Copyright © 2009 Krishna Myneni

Classes:  1  2  3  4  5  6  7  8  9  10

Homework:  1  2  3  4  5  6  7

Class 1

Exercise: For an optical frequency of 4×1014 Hz, compute the classical damping time (lifetime) of the oscillator.

Class 2

Exercise: Compute the power spectrum of the decaying field of an undriven dipole oscillator (eqn. 3.9 in Demtröder, 3rd ed.).

HW1: Calculate the power spectrum of a finite duration monochromatic wave (eqn. 3.61 in Demtröder, 3rd ed.).

Class 3

HW2: Derive the power spectrum for a monochromatic wave undergoing random phase jumps due to collisions, by using a weighted average of the power spectra of finite duration monochromatic wave trains (HW 1), with a Poisson distribution of time intervals.

Class 4

HW3: Use data in Table 3.1 to compute the collision broadening cross section of Na by Kr at T = 300 K.

Class 5

Exercise: Find the normalization constant in the probability density function for the speed of a thermal atom.

Exercise: Show ⟨ v2 ⟩ = 3kT / m

HW4: Evaluate the average of the square of the z-component of velocity, ⟨ vz2 ⟩, i.e. ⟨ (v cosθ)2 ⟩

Class 6

Exercise: Use Maxwell's equations to derive the relationship between the relative magnitudes of the E and B fields for a plane monochromatic wave in free space.

HW5: Demtröder, ch. 3, problems 3.1 and 3.3 (a)--(c).

Class 7

Exercise: Write the dipole oscillation amplitude, A(t), and phase, φ(t), in terms of the in-phase and in-quadrature amplitudes: u(t) and v(t).

Exercise: Use a Taylor-series expansion to show that the quantity, (ω2 − ω02) may be approximated as 2ω0(ω − ω0), or 2ω0Δ, near resonance.

Exercise: Solve the dv/dt equation for the case of on-resonant driving to find the solution for v(t), given in class.

Class 8

Exercise: Following our method of solving for u(t, Δ), write the second-order differential equation for v in the non-resonant case, and show that its solution is the same as given in class.

HW6: Problem 1.1 from Steck (auxiliary text): Estimate the absorption of a room-temperature (T = 300K) vapor cell of length 10 cm, containing a vapor of 87Rb, and illuminated by a weak monochromatic beam at the resonant frequency of the D2 transition (wavelength of 780nm). Assume the vapor pressure of 87Rb at T = 300K to be 3×10-7 Torr.

Hint: Due to Doppler-broadening, not all atoms will be resonant with the beam. Find the effective density of atoms which have a Doppler shift within one natural linewidth of the resonant frequency.

Class 9

Exercise: Derive the steady-state amplitude and phase-shift vs detuning, using the steady-state solutions for u and v.

Exercise: Problem 1.6a) from Steck, Derive the electro-magnetic wave equation in the presence of a polarized medium, starting from Maxwell's equations in a dielectric medium, and the constitutive relation.

Class 10

HW7: a) Using the NIST Atomic Spectral Database, find the 632.8 nm line of Ne I, the corresponding upper and lower levels of the transition, and the transition rate.

b) Compare the (inverse of the) transition rate to the level lifetimes given in HW5, 3.1, and explain the discrepancy.

c) Next, find the closest level in He I to the upper level of the transition. What is the frequency difference between these two levels? Convert this frequency difference to an energy difference, and compare with the average kinetic energy of an atom in thermal equilibrium at T = 500K.