Laser Spectroscopy Topics 1 through 10

Laser Spectroscopy (Classes 1 through 10)

Class 1

Resolving power of conventional spectroscopic instruments, and advantages of lasers for spectroscopy.
Some laser spectroscopic techniques.
Overview of Lorentz model of interaction of atoms with optical fields.
Solution of equation of motion in the absence of a driving field, and connection to energy loss rate of an oscillating electric dipole.
Derivation of classical expression for energy decay rate of an undriven dipole oscillator.

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

Class 2

Electric field of a dipole oscillator in the far-field (radiation zone).
Polarization property of radiation field.
Calculation of the frequency spectrum of light emitted by a decaying (undriven) dipole oscillator.
Lorentzian profile and the natural linewidth.
Distinction between the power spectrum of the field (optical frequency spectrum) and the power spectrum of intensity measured by a detector.
Autocorrelation function and the Wiener-Khinchine Theorem.
Periodic phase perturbations of a monochromatic wave train, and its autocorrelation function.
Model of transit-time broadening as the frequency spectrum of a finite section of a monochromatic wave train.

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

Comparison of some atomic transition lifetimes in alkali atoms with computed values from classical damping time.
Spectral profile for a finite duration wave train, and for transit-time broadening in a Gaussian beam: expressions for the spectral width.
Numerical example of transit time, estimate of broadening, and comparison with natural linewidth.
Natural linewidth of molecular transitions relative to atomic transitions.
Randomly distributed phase jumps in a monochromatic wave as a model for collision broadening of spectral lines (also known as Lorentz model).
Mean collision time and Poisson distribution of time intervals.
Spectrum of radiating atom undergoing random phase-perturbing collisions as a weighted average of power spectra of the form in HW1 (assigned as HW2).
Refinement of phase-perturbation model with continuous time-dependent phase during collision time: collision induced frequency shift.
Line profile for atomic radiation in the presence of collisional broadening and shifts (Fig 3.12 in Demtröder, 3rd ed.).
Pressure shifts as a source of error in atomic frequency standards.

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

Collision duration; role of interatomic potential in collision-broadening of spectra
Definition of collision cross section.
Interpretation of Table 3.1 in Demtröder, 3rd ed.; errata in ch 3. of book
Units conversions: wavenumber to Hz; cm⁻¹/cm⁻³ to MHz/Torr (at fixed temperature).
Numerical example: spectral line broadening and frequency shift of Na by Kr, at Kr density of 1×10¹⁹ cm⁻³.

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

Class 5

More about the Poisson probability function
Probability density for speed of an atom in a thermal vapor: derive from Boltzmann energy distribution, and assumption of no orientation dependence.
Define and compute characteristic measures of thermal particle speed:
- most probable speed
- mean speed
- r.m.s. speed
- mean mutual speed
- r.m.s. mutual speed
definition of homogeneous and inhomogeneous spectral broadening: classification of natural linewidth and collision broadening
Doppler effect for atom in a monochromatic beam: expression for the optical frequency in the moving atom frame (up to quadratic term in v / c).
Numerical estimate of quadratic Doppler shift at optical frequencies and room temperature.

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, ⟨ v_z² ⟩, i.e. ⟨ (v cosθ)² ⟩

Class 6

Review of HW problem 3.
Doppler profile for the absorption of a monochromatic beam by thermal atoms; Doppler width (full width at half-maximum); Example: ⁸⁷Rb at T = 300K, λ = 780 nm; Doppler width relative to natural line width at optical frequencies, and at typical temperatures.
Lorentz model dipole oscillator driven by a monochromatic wave: equation of motion
Lorentz force on electron, relative force contributions from E and B fields of the EM wave, and order of magnitude estimate of classical electron velocity in an atom at optical frequencies (10¹⁴ Hz).
Trial solution for equation of motion, expressed in terms of in-phase and in-quadrature amplitudes, and the slowly-varying approximation near resonance.

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

Atomic distance scale vs wavelength of optical driving field: justification for neglect of spatial variation of E-field in equation of motion.
Use trial solution to derive first-order coupled equations of motion for in-phase (u) and in-quadrature (v) amplitudes, in the slowly-varying and near-resonance approximations.
Solution of equations of motion on resonance (Δ = 0); steady-state solutions in the long-time limit for u and v; 90-degree phase offset between oscillator and driving field.
Units of u, v, and kE; Energy density of an EM field, and specific case of a plane EM wave in free space; time-averaged energy density for plane wave and relation to intensity (irradiance); relation between intensity, I, and E-field amplitude; units of intensity and relation to beam power.
On-resonace dipole radiation field; energy loss rate written in terms of the dipole amplitude solution on resonance.

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

On-resonance scattering of radiation from a monochromatic beam by a collection of dipole oscillators: derivation of Beer's law from dipole energy loss rate expression; definition of on-resonant absorption coefficient (α₀).
Limitation of Beer's law derived from classical electrodynamics and the linear Lorentz model: no intensity-dependent absorption (absence of saturation effect)
Intensity-dependent absorption coefficient and the saturation intensity
Solution of driven-dipole oscillator equations for arbitrary detuning; long time-limit solutions for u(t, Δ) and v(t, Δ); absorption coefficient as a function of detuning: α(Δ) (eq. 3.36b in Demtröder, 3rd ed.).
Use of Kramers-Kroning relation to obtain index of refraction from the absorption profile; expression for the index as a function of detuning: n(Δ) (eq. 3.37b in Demtröder); see external resource on K-K relations.

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

Dipole moment and polarization of a medium: sign convention in expression for dipole moment; definition of polarization as average dipole moment per unit volume; three forms for writing dipole oscillator amplitude; complex form of time-dependent polarization of a medium of identical dipole oscillators.
Classical wave propagation in a resonant dielectric medium: steady-state amplitude and phase shift as functions of detuning; show phase shift goes to zero at large detuning; wave equation in the presence of dispersion (and absorption) for a linear, isotropic, homogeneous medium; coupled-set of dynamical equations for describing wave propagation in the resonant medium.
Polarization of the medium in the presence of inhomogeneous broadening: the inhomogeneous lineshape detuning function, g(Δ); explicit expression for the polarization in the case of Doppler-broadening.
Absorption coefficient in the presence of both homogeneous and inhomogeneous (Doppler) broadening: convolution of Lorentzian and Gaussian profiles.

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

Review expressions for collision-broadening width for the cases of quenching (inelastic) collisions, and phase-perturbing (elastic) collisions, in connection with HW5, problem 3.1.
Energy-level diagram for He I and Ne I, the role of collisional transfer of energy from He to Ne, and the significance of the upper and lower level decay times in the He-Ne lasing transition at 633nm.
Absorption lineshape in a weakly-absorbing medium (αL ≪ 1); Voigt profile; opacity broadening in a strongly-absorbing medium.
End of classical description of interaction between light and atoms, and begining of quantum-mechanical description: rationale for the abstract vector representation of the state of a quantum system.

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.