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 :math:`k(\omega)` about the carrier, retain the requested orders, and use :math:`v_g=(dk/d\omega)^{-1}` with :math:`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 :math:`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 :math:`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 :math:`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 :math:`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 :math:`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 :math:`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 :math:`dI/dz=g(I)I`, using :math:`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 :math:`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 :math:`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 :math:`k(\omega)` about the carrier, retain the requested orders, and use :math:`v_g=(dk/d\omega)^{-1}` with :math:`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 :math:`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 :math:`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 :math:`k(\omega)` about the carrier, retain the requested orders, and use :math:`v_g=(dk/d\omega)^{-1}` with :math:`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 :math:`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 :math:`dI/dz=g(I)I`, using :math:`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 :math:`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 :math:`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 :math:`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 :math:`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 :math:`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 :math:`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.