Chapter 2: Stimulated Transitions: The Classical Oscillator Model
Source: Anthony E. Siegman, Lasers (1986), Chapter 2. 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 2.1: The Classical Electron Oscillator
Problem 2.1.1 — More detailed classical electron 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 2.1.2 — Q-value for a classical electron 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 2.1.3 — Classical derivation of the radiative decay rate
Brief solution
1. Method.
Begin with the stated physical law, keep the derivation symbolic, and introduce each approximation only where its limiting condition is explicit.
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
Begin with the stated physical law, keep the derivation symbolic, and introduce each approximation only where its limiting condition is explicit. 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 2.2: Collisions And Dephasing Processes
Problem 2.2.1 — Collision broadening with a different kind of collision statistics
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 2.3: More On Atomic Dynamics And Dephasing
Problem 2.3.1 — Dephasing by random frequency modulation
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 2.4: Steady-State Response: The Atomic Susceptibility
Problem 2.4.1 — Radiative decay rate at the He-Ne laser transition frequency
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 2.4.2 — Classical-mechanics description of power transfer from field to atoms
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 2.4.3 — Lorentzian lineshape for a resonant electrical circuit
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 2.4.4 — Lorentzian lineshape locus in the complex plane
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 2.4.5 — Range of validity for the resonance approximation
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 2.4.6 — Derivative spectroscopy
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 2.4.7 — Overlapping lineshapes: maximally flat condition
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 2.5: Conversion To Real Atomic Transitions
Problem 2.5.1 — Classical oscillator model for the index of refraction in gases
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 2.5.2 — Classical oscillator model for the index of refraction in solids
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 \(\omega=\omega_0\).