Chapter 3: Electric Dipole Transitions in Real Atoms
Source: Anthony E. Siegman, Lasers (1986), Chapter 3. Use each section/problem identifier with the book; the original prompts are not reproduced here. Each entry gives the governing model, the decisive solution route, and a physical verification.
Section 3.1: Decay Rates And Transition Strengths In Real Atoms
Problem 3.1.1 — Quantum calculation: Hydrogen-atom oscillator strengths
Brief solution
1. Method.
List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation.
2. Decisive step.
Use the driven-oscillator response \(\chi(\omega)\propto[\omega_0^2-\omega^2-j\gamma\omega]^{-1}\); its real part gives dispersion and its imaginary part gives absorption.
3. Verification.
Normalize the line profile to unit area and check its value and symmetry at \(\omega=\omega_0\).
Show detailed steps
List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation. Use the driven-oscillator response \(\chi(\omega)\propto[\omega_0^2-\omega^2-j\gamma\omega]^{-1}\); its real part gives dispersion and its imaginary part gives absorption. Normalize the line profile to unit area and check its value and symmetry at \(\omega=\omega_0\).
Section 3.2: Line-Broadening Mechanisms In Real Atoms
Problem 3.2.1 — Derivative spectroscopy on a variable-pressure gas sample
Brief solution
1. Method.
List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation.
2. Decisive step.
Use the driven-oscillator response \(\chi(\omega)\propto[\omega_0^2-\omega^2-j\gamma\omega]^{-1}\); its real part gives dispersion and its imaginary part gives absorption.
3. Verification.
Normalize the line profile to unit area and check its value and symmetry at \(\omega=\omega_0\).
Show detailed steps
List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation. Use the driven-oscillator response \(\chi(\omega)\propto[\omega_0^2-\omega^2-j\gamma\omega]^{-1}\); its real part gives dispersion and its imaginary part gives absorption. Normalize the line profile to unit area and check its value and symmetry at \(\omega=\omega_0\).
Section 3.3: Polarization Properties Of Atomic Transitions
Problem 3.3.1 — Two-dimensional Zeeman-split classical oscillator model
Brief solution
1. Method.
List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation.
2. Decisive step.
Use the driven-oscillator response \(\chi(\omega)\propto[\omega_0^2-\omega^2-j\gamma\omega]^{-1}\); its real part gives dispersion and its imaginary part gives absorption.
3. Verification.
Normalize the line profile to unit area and check its value and symmetry at \(\omega=\omega_0\).
Show detailed steps
List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation. Use the driven-oscillator response \(\chi(\omega)\propto[\omega_0^2-\omega^2-j\gamma\omega]^{-1}\); its real part gives dispersion and its imaginary part gives absorption. Normalize the line profile to unit area and check its value and symmetry at \(\omega=\omega_0\).
Problem 3.3.2 — Computer plots of oscillating atomic charge distributions (research problem)
Brief solution
1. Method.
Normalize the variables first, evaluate the analytic limits, and then sweep the remaining dimensionless parameter so the numerical curve can be checked against both limits.
2. Decisive step.
Relate microscopic strength to decay with \(A_{21}=\omega_{21}^3|\boldsymbol\mu_{21}|^2/(3\pi\epsilon_0\hbar c^3)\) and include degeneracy and polarization projections before summing sublevels.
3. Verification.
Check selection rules, normalization over polarization/orientation, and the cubic frequency scaling of spontaneous decay.
Show detailed steps
Normalize the variables first, evaluate the analytic limits, and then sweep the remaining dimensionless parameter so the numerical curve can be checked against both limits. Relate microscopic strength to decay with \(A_{21}=\omega_{21}^3|\boldsymbol\mu_{21}|^2/(3\pi\epsilon_0\hbar c^3)\) and include degeneracy and polarization projections before summing sublevels. Check selection rules, normalization over polarization/orientation, and the cubic frequency scaling of spontaneous decay.
Section 3.4: Tensor Susceptibilities
Problem 3.4.1 — Negative circular polarization response of a gyrotropic tensor
Brief solution
1. Method.
List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation.
2. Decisive step.
Use the driven-oscillator response \(\chi(\omega)\propto[\omega_0^2-\omega^2-j\gamma\omega]^{-1}\); its real part gives dispersion and its imaginary part gives absorption.
3. Verification.
Normalize the line profile to unit area and check its value and symmetry at \(\omega=\omega_0\).
Show detailed steps
List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation. Use the driven-oscillator response \(\chi(\omega)\propto[\omega_0^2-\omega^2-j\gamma\omega]^{-1}\); its real part gives dispersion and its imaginary part gives absorption. Normalize the line profile to unit area and check its value and symmetry at \(\omega=\omega_0\).
Problem 3.4.2 — Tensor response of an anisotropic two-dimensional classical oscillator
Brief solution
1. Method.
List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation.
2. Decisive step.
Use the driven-oscillator response \(\chi(\omega)\propto[\omega_0^2-\omega^2-j\gamma\omega]^{-1}\); its real part gives dispersion and its imaginary part gives absorption.
3. Verification.
Normalize the line profile to unit area and check its value and symmetry at \(\omega=\omega_0\).
Show detailed steps
List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation. Use the driven-oscillator response \(\chi(\omega)\propto[\omega_0^2-\omega^2-j\gamma\omega]^{-1}\); its real part gives dispersion and its imaginary part gives absorption. Normalize the line profile to unit area and check its value and symmetry at \(\omega=\omega_0\).
Problem 3.4.3 — Tensor response of a three-dimensional Zeeman-split classical oscillator
Brief solution
1. Method.
List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation.
2. Decisive step.
Use the driven-oscillator response \(\chi(\omega)\propto[\omega_0^2-\omega^2-j\gamma\omega]^{-1}\); its real part gives dispersion and its imaginary part gives absorption.
3. Verification.
Normalize the line profile to unit area and check its value and symmetry at \(\omega=\omega_0\).
Show detailed steps
List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation. Use the driven-oscillator response \(\chi(\omega)\propto[\omega_0^2-\omega^2-j\gamma\omega]^{-1}\); its real part gives dispersion and its imaginary part gives absorption. Normalize the line profile to unit area and check its value and symmetry at \(\omega=\omega_0\).
Problem 3.4.4 — Field patterns in a “twisted-mode” laser cavity
Brief solution
1. Method.
List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation.
2. Decisive step.
Use the driven-oscillator response \(\chi(\omega)\propto[\omega_0^2-\omega^2-j\gamma\omega]^{-1}\); its real part gives dispersion and its imaginary part gives absorption.
3. Verification.
Normalize the line profile to unit area and check its value and symmetry at \(\omega=\omega_0\).
Show detailed steps
List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation. Use the driven-oscillator response \(\chi(\omega)\propto[\omega_0^2-\omega^2-j\gamma\omega]^{-1}\); its real part gives dispersion and its imaginary part gives absorption. Normalize the line profile to unit area and check its value and symmetry at \(\omega=\omega_0\).
Problem 3.4.5 — More on the twisted-mode cavity
Brief solution
1. Method.
List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation.
2. Decisive step.
Use the driven-oscillator response \(\chi(\omega)\propto[\omega_0^2-\omega^2-j\gamma\omega]^{-1}\); its real part gives dispersion and its imaginary part gives absorption.
3. Verification.
Normalize the line profile to unit area and check its value and symmetry at \(\omega=\omega_0\).
Show detailed steps
List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation. Use the driven-oscillator response \(\chi(\omega)\propto[\omega_0^2-\omega^2-j\gamma\omega]^{-1}\); its real part gives dispersion and its imaginary part gives absorption. Normalize the line profile to unit area and check its value and symmetry at \(\omega=\omega_0\).
Section 3.5: The “Factor Of Three”
Problem 3.5.1 — Averaging cos2 0 over 47r steradians
Brief solution
1. Method.
List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation.
2. Decisive step.
Relate microscopic strength to decay with \(A_{21}=\omega_{21}^3|\boldsymbol\mu_{21}|^2/(3\pi\epsilon_0\hbar c^3)\) and include degeneracy and polarization projections before summing sublevels.
3. Verification.
Check selection rules, normalization over polarization/orientation, and the cubic frequency scaling of spontaneous decay.
Show detailed steps
List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation. Relate microscopic strength to decay with \(A_{21}=\omega_{21}^3|\boldsymbol\mu_{21}|^2/(3\pi\epsilon_0\hbar c^3)\) and include degeneracy and polarization projections before summing sublevels. Check selection rules, normalization over polarization/orientation, and the cubic frequency scaling of spontaneous decay.
Section 3.7: Inhomogeneous Line Broadening
Problem 3.7.1 — Inhomogeneous broadening with a Lorentzian (rather than Gaussian) inhomogeneous distribution
Brief solution
1. Method.
List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation.
2. Decisive step.
Use the driven-oscillator response \(\chi(\omega)\propto[\omega_0^2-\omega^2-j\gamma\omega]^{-1}\); its real part gives dispersion and its imaginary part gives absorption.
3. Verification.
Normalize the line profile to unit area and check its value and symmetry at \(\omega=\omega_0\).
Show detailed steps
List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation. Use the driven-oscillator response \(\chi(\omega)\propto[\omega_0^2-\omega^2-j\gamma\omega]^{-1}\); its real part gives dispersion and its imaginary part gives absorption. Normalize the line profile to unit area and check its value and symmetry at \(\omega=\omega_0\).
Problem 3.7.2 — Inhomogeneous broadening with a uniform inhomogeneous distribution
Brief solution
1. Method.
List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation.
2. Decisive step.
Use the driven-oscillator response \(\chi(\omega)\propto[\omega_0^2-\omega^2-j\gamma\omega]^{-1}\); its real part gives dispersion and its imaginary part gives absorption.
3. Verification.
Normalize the line profile to unit area and check its value and symmetry at \(\omega=\omega_0\).
Show detailed steps
List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation. Use the driven-oscillator response \(\chi(\omega)\propto[\omega_0^2-\omega^2-j\gamma\omega]^{-1}\); its real part gives dispersion and its imaginary part gives absorption. Normalize the line profile to unit area and check its value and symmetry at \(\omega=\omega_0\).
Problem 3.7.3 — Ditto with a triangular distribution
Brief solution
1. Method.
List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation.
2. Decisive step.
Use the driven-oscillator response \(\chi(\omega)\propto[\omega_0^2-\omega^2-j\gamma\omega]^{-1}\); its real part gives dispersion and its imaginary part gives absorption.
3. Verification.
Normalize the line profile to unit area and check its value and symmetry at \(\omega=\omega_0\).
Show detailed steps
List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation. Use the driven-oscillator response \(\chi(\omega)\propto[\omega_0^2-\omega^2-j\gamma\omega]^{-1}\); its real part gives dispersion and its imaginary part gives absorption. Normalize the line profile to unit area and check its value and symmetry at \(\omega=\omega_0\).
Problem 3.7.4 — Midband absorption versus pressure in a gas
Brief solution
1. Method.
Evaluate both cases from the same symbolic expression before taking their ratio; this keeps normalization and sign conventions from obscuring the comparison.
2. Decisive step.
Use the driven-oscillator response \(\chi(\omega)\propto[\omega_0^2-\omega^2-j\gamma\omega]^{-1}\); its real part gives dispersion and its imaginary part gives absorption.
3. Verification.
Normalize the line profile to unit area and check its value and symmetry at \(\omega=\omega_0\).
Show detailed steps
Evaluate both cases from the same symbolic expression before taking their ratio; this keeps normalization and sign conventions from obscuring the comparison. Use the driven-oscillator response \(\chi(\omega)\propto[\omega_0^2-\omega^2-j\gamma\omega]^{-1}\); its real part gives dispersion and its imaginary part gives absorption. Normalize the line profile to unit area and check its value and symmetry at \(\omega=\omega_0\).
Problem 3.7.5 — Chemical lasers, and absorption versus pressure in a deuterium fluoride gas cell
Brief solution
1. Method.
Evaluate both cases from the same symbolic expression before taking their ratio; this keeps normalization and sign conventions from obscuring the comparison.
2. Decisive step.
Use the driven-oscillator response \(\chi(\omega)\propto[\omega_0^2-\omega^2-j\gamma\omega]^{-1}\); its real part gives dispersion and its imaginary part gives absorption.
3. Verification.
Normalize the line profile to unit area and check its value and symmetry at \(\omega=\omega_0\).
Show detailed steps
Evaluate both cases from the same symbolic expression before taking their ratio; this keeps normalization and sign conventions from obscuring the comparison. Use the driven-oscillator response \(\chi(\omega)\propto[\omega_0^2-\omega^2-j\gamma\omega]^{-1}\); its real part gives dispersion and its imaginary part gives absorption. Normalize the line profile to unit area and check its value and symmetry at \(\omega=\omega_0\).
Problem 3.7.6 — Inhomogeneous Voight profiles far out in the wings
Brief solution
1. Method.
List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation.
2. Decisive step.
Use the driven-oscillator response \(\chi(\omega)\propto[\omega_0^2-\omega^2-j\gamma\omega]^{-1}\); its real part gives dispersion and its imaginary part gives absorption.
3. Verification.
Normalize the line profile to unit area and check its value and symmetry at \(\omega=\omega_0\).
Show detailed steps
List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation. Use the driven-oscillator response \(\chi(\omega)\propto[\omega_0^2-\omega^2-j\gamma\omega]^{-1}\); its real part gives dispersion and its imaginary part gives absorption. Normalize the line profile to unit area and check its value and symmetry at \(\omega=\omega_0\).