Chapter 14: Optical Beams and Resonators: An Introduction
Source: Anthony E. Siegman, Lasers (1986), Chapter 14. 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 14.3: Build-Up And Oscillation Of Optical Resonator Modes
Problem 14.3.1 — Higher-order mode suppression during laser turn-on
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.
Build the round-trip ABCD matrix and solve \(q=(Aq+B)/(Cq+D)\); for real systems apply \(|A+D|<2\) before selecting the physical root with \(\operatorname{Im}(1/q)<0\).
3. Verification.
Back-propagate the eigenmode through one round trip to verify self-consistency and compare its aperture loss with the assumed stability regime.
Show detailed steps
List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation. Build the round-trip ABCD matrix and solve \(q=(Aq+B)/(Cq+D)\); for real systems apply \(|A+D|<2\) before selecting the physical root with \(\operatorname{Im}(1/q)<0\). Back-propagate the eigenmode through one round trip to verify self-consistency and compare its aperture loss with the assumed stability regime.
Problem 14.3.2 — Finding the next higher-order transverse mode
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.
Build the round-trip ABCD matrix and solve \(q=(Aq+B)/(Cq+D)\); for real systems apply \(|A+D|<2\) before selecting the physical root with \(\operatorname{Im}(1/q)<0\).
3. Verification.
Back-propagate the eigenmode through one round trip to verify self-consistency and compare its aperture loss with the assumed stability regime.
Show detailed steps
List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation. Build the round-trip ABCD matrix and solve \(q=(Aq+B)/(Cq+D)\); for real systems apply \(|A+D|<2\) before selecting the physical root with \(\operatorname{Im}(1/q)<0\). Back-propagate the eigenmode through one round trip to verify self-consistency and compare its aperture loss with the assumed stability regime.
Problem 14.3.3 — Higher-order transverse mode beats
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.
Build the round-trip ABCD matrix and solve \(q=(Aq+B)/(Cq+D)\); for real systems apply \(|A+D|<2\) before selecting the physical root with \(\operatorname{Im}(1/q)<0\).
3. Verification.
Back-propagate the eigenmode through one round trip to verify self-consistency and compare its aperture loss with the assumed stability regime.
Show detailed steps
List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation. Build the round-trip ABCD matrix and solve \(q=(Aq+B)/(Cq+D)\); for real systems apply \(|A+D|<2\) before selecting the physical root with \(\operatorname{Im}(1/q)<0\). Back-propagate the eigenmode through one round trip to verify self-consistency and compare its aperture loss with the assumed stability regime.
Problem 14.3.4 — Spectral content of a circulating pulse
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.
Expand \(k(\omega)\) about the carrier, retain the requested orders, and use \(v_g=(dk/d\omega)^{-1}\) with \(k''\) controlling quadratic dispersive broadening.
3. Verification.
Check the transform-limited and zero-dispersion limits, and conserve pulse energy when only phase is changed.
Show detailed steps
List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation. Expand \(k(\omega)\) about the carrier, retain the requested orders, and use \(v_g=(dk/d\omega)^{-1}\) with \(k''\) controlling quadratic dispersive broadening. Check the transform-limited and zero-dispersion limits, and conserve pulse energy when only phase is changed.
Problem 14.3.5 — Output beam characteristics of a multi-transverse-mode 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.
Build the round-trip ABCD matrix and solve \(q=(Aq+B)/(Cq+D)\); for real systems apply \(|A+D|<2\) before selecting the physical root with \(\operatorname{Im}(1/q)<0\).
3. Verification.
Back-propagate the eigenmode through one round trip to verify self-consistency and compare its aperture loss with the assumed stability regime.
Show detailed steps
List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation. Build the round-trip ABCD matrix and solve \(q=(Aq+B)/(Cq+D)\); for real systems apply \(|A+D|<2\) before selecting the physical root with \(\operatorname{Im}(1/q)<0\). Back-propagate the eigenmode through one round trip to verify self-consistency and compare its aperture loss with the assumed stability regime.
Problem 14.3.6 — Transverse modes that self-reproduce after several round trips?
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.
Build the round-trip ABCD matrix and solve \(q=(Aq+B)/(Cq+D)\); for real systems apply \(|A+D|<2\) before selecting the physical root with \(\operatorname{Im}(1/q)<0\).
3. Verification.
Back-propagate the eigenmode through one round trip to verify self-consistency and compare its aperture loss with the assumed stability regime.
Show detailed steps
List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation. Build the round-trip ABCD matrix and solve \(q=(Aq+B)/(Cq+D)\); for real systems apply \(|A+D|<2\) before selecting the physical root with \(\operatorname{Im}(1/q)<0\). Back-propagate the eigenmode through one round trip to verify self-consistency and compare its aperture loss with the assumed stability regime.