Chapter 30: Hole Burning and Saturation Spectroscopy
Source: Anthony E. Siegman, Lasers (1986), Chapter 30. 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 30.2: Elementary Analysis Of Inhomogeneous Hole Burning
Problem 30.2.1 — Exact expression for inhomogeneous saturation at line center
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.
Integrate the velocity-class response with the homogeneous saturation denominator, keeping pump and probe detunings separate before evaluating the Lamb-dip or hole width.
3. Verification.
Check that the unsaturated line is recovered at zero intensity and that power broadening increases the homogeneous width.
Show detailed steps
List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation. Integrate the velocity-class response with the homogeneous saturation denominator, keeping pump and probe detunings separate before evaluating the Lamb-dip or hole width. Check that the unsaturated line is recovered at zero intensity and that power broadening increases the homogeneous width.
Problem 30.2.2 — Exact expression for hole burning in a Lorentzian inhomogeneously broadened line
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.
Integrate the velocity-class response with the homogeneous saturation denominator, keeping pump and probe detunings separate before evaluating the Lamb-dip or hole width.
3. Verification.
Check that the unsaturated line is recovered at zero intensity and that power broadening increases the homogeneous width.
Show detailed steps
List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation. Integrate the velocity-class response with the homogeneous saturation denominator, keeping pump and probe detunings separate before evaluating the Lamb-dip or hole width. Check that the unsaturated line is recovered at zero intensity and that power broadening increases the homogeneous width.
Problem 30.2.3 — Oscillation mode spectrum and power output for a strongly inhomogeneous laser 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.
Integrate the velocity-class response with the homogeneous saturation denominator, keeping pump and probe detunings separate before evaluating the Lamb-dip or hole width.
3. Verification.
Check that the unsaturated line is recovered at zero intensity and that power broadening increases the homogeneous width.
Show detailed steps
List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation. Integrate the velocity-class response with the homogeneous saturation denominator, keeping pump and probe detunings separate before evaluating the Lamb-dip or hole width. Check that the unsaturated line is recovered at zero intensity and that power broadening increases the homogeneous width.
Section 30.6: Inhomogeneous Laser Oscillation: Lamb Dips
Problem 30.6.1 — Frequency pulling in an inhomogeneously broadened laser
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.
Integrate the velocity-class response with the homogeneous saturation denominator, keeping pump and probe detunings separate before evaluating the Lamb-dip or hole width.
3. Verification.
Check that the unsaturated line is recovered at zero intensity and that power broadening increases the homogeneous width.
Show detailed steps
List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation. Integrate the velocity-class response with the homogeneous saturation denominator, keeping pump and probe detunings separate before evaluating the Lamb-dip or hole width. Check that the unsaturated line is recovered at zero intensity and that power broadening increases the homogeneous width.
Problem 30.6.2 — Oscillation spectrum and power output versus pumping in a doppler-broadened gas laser
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.
Integrate the velocity-class response with the homogeneous saturation denominator, keeping pump and probe detunings separate before evaluating the Lamb-dip or hole width.
3. Verification.
Check that the unsaturated line is recovered at zero intensity and that power broadening increases the homogeneous width.
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. Integrate the velocity-class response with the homogeneous saturation denominator, keeping pump and probe detunings separate before evaluating the Lamb-dip or hole width. Check that the unsaturated line is recovered at zero intensity and that power broadening increases the homogeneous width.
Problem 30.6.3 — Anomalous frequency pulling at line center in a doppler-broadened laser
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.
Integrate the velocity-class response with the homogeneous saturation denominator, keeping pump and probe detunings separate before evaluating the Lamb-dip or hole width.
3. Verification.
Check that the unsaturated line is recovered at zero intensity and that power broadening increases the homogeneous width.
Show detailed steps
List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation. Integrate the velocity-class response with the homogeneous saturation denominator, keeping pump and probe detunings separate before evaluating the Lamb-dip or hole width. Check that the unsaturated line is recovered at zero intensity and that power broadening increases the homogeneous width.
Problem 30.6.4 — Inverse Lamb dip analysis
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.
Integrate the velocity-class response with the homogeneous saturation denominator, keeping pump and probe detunings separate before evaluating the Lamb-dip or hole width.
3. Verification.
Check that the unsaturated line is recovered at zero intensity and that power broadening increases the homogeneous width.
Show detailed steps
List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation. Integrate the velocity-class response with the homogeneous saturation denominator, keeping pump and probe detunings separate before evaluating the Lamb-dip or hole width. Check that the unsaturated line is recovered at zero intensity and that power broadening increases the homogeneous width.