Chapter 27: Active Laser Mode Coupling

Source: Anthony E. Siegman, Lasers (1986), Chapter 27. 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 27.1: Optical Signals: Time And Frequency Description

Problem 27.1.1 — Two-pulse laser spectrum

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 27.1.2 — Three-mode signal example

List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation. Write the field as a coherent modal sum \(E(t)=\sum_m A_m e^{j(\omega_0+m\Delta\omega)t+j\phi_m}\) and impose the modulator or saturable-absorber phase relation on adjacent modes. Check that equal modal phase produces one pulse per round trip and that the time–bandwidth product is consistent with the assumed spectrum.

Problem 27.1.3 — General three-mode signal spectrum

List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation. Write the field as a coherent modal sum \(E(t)=\sum_m A_m e^{j(\omega_0+m\Delta\omega)t+j\phi_m}\) and impose the modulator or saturable-absorber phase relation on adjacent modes. Check that equal modal phase produces one pulse per round trip and that the time–bandwidth product is consistent with the assumed spectrum.

Problem 27.1.4 — Another three-mode example

List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation. Write the field as a coherent modal sum \(E(t)=\sum_m A_m e^{j(\omega_0+m\Delta\omega)t+j\phi_m}\) and impose the modulator or saturable-absorber phase relation on adjacent modes. Check that equal modal phase produces one pulse per round trip and that the time–bandwidth product is consistent with the assumed spectrum.

Problem 27.1.5 — Phasor model with N sidebands

List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation. Write the field as a coherent modal sum \(E(t)=\sum_m A_m e^{j(\omega_0+m\Delta\omega)t+j\phi_m}\) and impose the modulator or saturable-absorber phase relation on adjacent modes. Check that equal modal phase produces one pulse per round trip and that the time–bandwidth product is consistent with the assumed spectrum.

Problem 27.1.6 — Mode-locked spectrum with random amplitudes

List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation. Write the field as a coherent modal sum \(E(t)=\sum_m A_m e^{j(\omega_0+m\Delta\omega)t+j\phi_m}\) and impose the modulator or saturable-absorber phase relation on adjacent modes. Check that equal modal phase produces one pulse per round trip and that the time–bandwidth product is consistent with the assumed spectrum.

Problem 27.1.7 — Quasi-FM signal with a square signal spectrum (research problem)

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. Write the field as a coherent modal sum \(E(t)=\sum_m A_m e^{j(\omega_0+m\Delta\omega)t+j\phi_m}\) and impose the modulator or saturable-absorber phase relation on adjacent modes. Check that equal modal phase produces one pulse per round trip and that the time–bandwidth product is consistent with the assumed spectrum.

Section 27.3: Time-Domain Analysis: Homogeneous Mode Locking

Problem 27.3.1 — Steady-state gain condition in an actively mode-locked laser

List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation. Propagate irradiance with \(dI/dz=g(I)I\), using \(g(I)=g_0/(1+I/I_s)\) when saturation matters; integrate before inserting boundary values. Verify that the small-signal limit is exponential, while extracted energy never exceeds the stored inversion energy.

Problem 27.3.2 — Physical basis ofFM mode locking

List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation. Write the field as a coherent modal sum \(E(t)=\sum_m A_m e^{j(\omega_0+m\Delta\omega)t+j\phi_m}\) and impose the modulator or saturable-absorber phase relation on adjacent modes. Check that equal modal phase produces one pulse per round trip and that the time–bandwidth product is consistent with the assumed spectrum.

Problem 27.3.3 — Evaluation ofetalon line narrowing effects

Evaluate both cases from the same symbolic expression before taking their ratio; this keeps normalization and sign conventions from obscuring the comparison. Write the field as a coherent modal sum \(E(t)=\sum_m A_m e^{j(\omega_0+m\Delta\omega)t+j\phi_m}\) and impose the modulator or saturable-absorber phase relation on adjacent modes. Check that equal modal phase produces one pulse per round trip and that the time–bandwidth product is consistent with the assumed spectrum.

Problem 27.3.4 — Changes in pulseshape produced by etalon effects

Evaluate both cases from the same symbolic expression before taking their ratio; this keeps normalization and sign conventions from obscuring the comparison. 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 27.3.5 — “Supermodes” in harmonically mode-locked lasers

List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation. Write the field as a coherent modal sum \(E(t)=\sum_m A_m e^{j(\omega_0+m\Delta\omega)t+j\phi_m}\) and impose the modulator or saturable-absorber phase relation on adjacent modes. Check that equal modal phase produces one pulse per round trip and that the time–bandwidth product is consistent with the assumed spectrum.

Problem 27.3.6 — Research problem: Mode competition among “supermodes” in a harmonically mode-locked laser

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. Write the field as a coherent modal sum \(E(t)=\sum_m A_m e^{j(\omega_0+m\Delta\omega)t+j\phi_m}\) and impose the modulator or saturable-absorber phase relation on adjacent modes. Check that equal modal phase produces one pulse per round trip and that the time–bandwidth product is consistent with the assumed spectrum.

Section 27.4: Transient And Detuning Effects

Problem 27.4.1 — Evolution of FM mode-locked pulses

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 27.4.2 — Spectral narrowing in a mode-locked laser

List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation. Write the field as a coherent modal sum \(E(t)=\sum_m A_m e^{j(\omega_0+m\Delta\omega)t+j\phi_m}\) and impose the modulator or saturable-absorber phase relation on adjacent modes. Check that equal modal phase produces one pulse per round trip and that the time–bandwidth product is consistent with the assumed spectrum.

Section 27.6: The Modulator Polarization Term

Problem 27.6.1 — Cross-coupling between adjacent modes due to gain saturation

List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation. Propagate irradiance with \(dI/dz=g(I)I\), using \(g(I)=g_0/(1+I/I_s)\) when saturation matters; integrate before inserting boundary values. Verify that the small-signal limit is exponential, while extracted energy never exceeds the stored inversion energy.

Section 27.7: Fm Laser Operation

Problem 27.7.1 — Time-domain derivation ofFM laser operations

Begin with the stated physical law, keep the derivation symbolic, and introduce each approximation only where its limiting condition is explicit. Write the field as a coherent modal sum \(E(t)=\sum_m A_m e^{j(\omega_0+m\Delta\omega)t+j\phi_m}\) and impose the modulator or saturable-absorber phase relation on adjacent modes. Check that equal modal phase produces one pulse per round trip and that the time–bandwidth product is consistent with the assumed spectrum.

Problem 27.7.2 — Research problem: Single-sideband mode-coupled lasers?

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. Write the field as a coherent modal sum \(E(t)=\sum_m A_m e^{j(\omega_0+m\Delta\omega)t+j\phi_m}\) and impose the modulator or saturable-absorber phase relation on adjacent modes. Check that equal modal phase produces one pulse per round trip and that the time–bandwidth product is consistent with the assumed spectrum.

Problem 27.7.3 — Wigner distributions for mode-coupled lasers

List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation. Write the field as a coherent modal sum \(E(t)=\sum_m A_m e^{j(\omega_0+m\Delta\omega)t+j\phi_m}\) and impose the modulator or saturable-absorber phase relation on adjacent modes. Check that equal modal phase produces one pulse per round trip and that the time–bandwidth product is consistent with the assumed spectrum.

Problem 27.7.4 — Transient build-up of FM laser oscillation

List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation. Write the field as a coherent modal sum \(E(t)=\sum_m A_m e^{j(\omega_0+m\Delta\omega)t+j\phi_m}\) and impose the modulator or saturable-absorber phase relation on adjacent modes. Check that equal modal phase produces one pulse per round trip and that the time–bandwidth product is consistent with the assumed spectrum.

Problem 27.7.5 — Coupled-mode analysis of AM and FM mode locking

List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation. Write the field as a coherent modal sum \(E(t)=\sum_m A_m e^{j(\omega_0+m\Delta\omega)t+j\phi_m}\) and impose the modulator or saturable-absorber phase relation on adjacent modes. Check that equal modal phase produces one pulse per round trip and that the time–bandwidth product is consistent with the assumed spectrum.

Problem 27.7.6 — Research problem: Coupled mode analysis of detuning effects in mode-locked lasers

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. Write the field as a coherent modal sum \(E(t)=\sum_m A_m e^{j(\omega_0+m\Delta\omega)t+j\phi_m}\) and impose the modulator or saturable-absorber phase relation on adjacent modes. Check that equal modal phase produces one pulse per round trip and that the time–bandwidth product is consistent with the assumed spectrum.