Chapter 11: Laser Mirrors and Regenerative Feedback
===================================================

Source: Anthony E. Siegman, *Lasers* (1986), Chapter 11.
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 11.1: Laser Mirrors And Beam Splitters
----------------------------------------------

Problem 11.1.1 — Scattering matrix for a general dielectric slab
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation.
Form the complex round-trip factor :math:`G_{rt}=|G_{rt}|e^{j\Phi}`; resonance requires :math:`\Phi=2\pi q`, and threshold requires :math:`|G_{rt}|=1`.
Check the passive-cavity limit, energy conservation at every mirror, and that added loss raises rather than lowers threshold.

Problem 11.1.2 — Changes in the scattering matrix for different reference planes
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation.
Form the complex round-trip factor :math:`G_{rt}=|G_{rt}|e^{j\Phi}`; resonance requires :math:`\Phi=2\pi q`, and threshold requires :math:`|G_{rt}|=1`.
Check the passive-cavity limit, energy conservation at every mirror, and that added loss raises rather than lowers threshold.

Problem 11.1.3 — Derivation of the necessary matrix element relationships for a lossless reciprocal two-port
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

Begin with the stated physical law, keep the derivation symbolic, and introduce each approximation only where its limiting condition is explicit.
Form the complex round-trip factor :math:`G_{rt}=|G_{rt}|e^{j\Phi}`; resonance requires :math:`\Phi=2\pi q`, and threshold requires :math:`|G_{rt}|=1`.
Check the passive-cavity limit, energy conservation at every mirror, and that added loss raises rather than lowers threshold.

Problem 11.1.4 — Scattering matrix for a transmission line junction
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation.
Form the complex round-trip factor :math:`G_{rt}=|G_{rt}|e^{j\Phi}`; resonance requires :math:`\Phi=2\pi q`, and threshold requires :math:`|G_{rt}|=1`.
Check the passive-cavity limit, energy conservation at every mirror, and that added loss raises rather than lowers threshold.

Problem 11.1.5 — Transmission-line junction with a lumped shunt capacitance
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation.
Form the complex round-trip factor :math:`G_{rt}=|G_{rt}|e^{j\Phi}`; resonance requires :math:`\Phi=2\pi q`, and threshold requires :math:`|G_{rt}|=1`.
Check the passive-cavity limit, energy conservation at every mirror, and that added loss raises rather than lowers threshold.

Problem 11.1.6 — Three-port and five-port optical scattering systems?
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation.
Form the complex round-trip factor :math:`G_{rt}=|G_{rt}|e^{j\Phi}`; resonance requires :math:`\Phi=2\pi q`, and threshold requires :math:`|G_{rt}|=1`.
Check the passive-cavity limit, energy conservation at every mirror, and that added loss raises rather than lowers threshold.

Problem 11.1.7 — Impossibility of a completely matched three-port network
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation.
Form the complex round-trip factor :math:`G_{rt}=|G_{rt}|e^{j\Phi}`; resonance requires :math:`\Phi=2\pi q`, and threshold requires :math:`|G_{rt}|=1`.
Check the passive-cavity limit, energy conservation at every mirror, and that added loss raises rather than lowers threshold.

Problem 11.1.8 — Conditions for an N-port equal-amplitude beam splitter
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation.
Form the complex round-trip factor :math:`G_{rt}=|G_{rt}|e^{j\Phi}`; resonance requires :math:`\Phi=2\pi q`, and threshold requires :math:`|G_{rt}|=1`.
Check the passive-cavity limit, energy conservation at every mirror, and that added loss raises rather than lowers threshold.

Problem 11.1.9 — Synthesizing an arbitrary complex optical two-port (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.
Form the complex round-trip factor :math:`G_{rt}=|G_{rt}|e^{j\Phi}`; resonance requires :math:`\Phi=2\pi q`, and threshold requires :math:`|G_{rt}|=1`.
Check the passive-cavity limit, energy conservation at every mirror, and that added loss raises rather than lowers threshold.

Section 11.3: Resonance Properties Of Passive Optical Cavities
--------------------------------------------------------------

Problem 11.3.1 — Design specifications for a transmission etalon
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

Translate each performance requirement into an equality or inequality, solve the coupled constraints, and discard any root that violates a physical bound.
Form the complex round-trip factor :math:`G_{rt}=|G_{rt}|e^{j\Phi}`; resonance requires :math:`\Phi=2\pi q`, and threshold requires :math:`|G_{rt}|=1`.
Check the passive-cavity limit, energy conservation at every mirror, and that added loss raises rather than lowers threshold.

Problem 11.3.2 — Angle tuning of a transmission etalon
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation.
Form the complex round-trip factor :math:`G_{rt}=|G_{rt}|e^{j\Phi}`; resonance requires :math:`\Phi=2\pi q`, and threshold requires :math:`|G_{rt}|=1`.
Check the passive-cavity limit, energy conservation at every mirror, and that added loss raises rather than lowers threshold.

Problem 11.3.3 — Calculating cavity parameters from measured transmission-reflection curves
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation.
Form the complex round-trip factor :math:`G_{rt}=|G_{rt}|e^{j\Phi}`; resonance requires :math:`\Phi=2\pi q`, and threshold requires :math:`|G_{rt}|=1`.
Check the passive-cavity limit, energy conservation at every mirror, and that added loss raises rather than lowers threshold.

Problem 11.3.4 — Reflection properties of uncoated dielectric etalon mirrors
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation.
Form the complex round-trip factor :math:`G_{rt}=|G_{rt}|e^{j\Phi}`; resonance requires :math:`\Phi=2\pi q`, and threshold requires :math:`|G_{rt}|=1`.
Check the passive-cavity limit, energy conservation at every mirror, and that added loss raises rather than lowers threshold.

Problem 11.3.5 — Linewidth of a power reflectivity dip
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation.
Form the complex round-trip factor :math:`G_{rt}=|G_{rt}|e^{j\Phi}`; resonance requires :math:`\Phi=2\pi q`, and threshold requires :math:`|G_{rt}|=1`.
Check the passive-cavity limit, energy conservation at every mirror, and that added loss raises rather than lowers threshold.

Problem 11.3.6 — Reflection phase angle versus frequency
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

Evaluate both cases from the same symbolic expression before taking their ratio; this keeps normalization and sign conventions from obscuring the comparison.
Form the complex round-trip factor :math:`G_{rt}=|G_{rt}|e^{j\Phi}`; resonance requires :math:`\Phi=2\pi q`, and threshold requires :math:`|G_{rt}|=1`.
Check the passive-cavity limit, energy conservation at every mirror, and that added loss raises rather than lowers threshold.

Problem 11.3.7 — Field magnification inside a resonant cavity
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation.
Form the complex round-trip factor :math:`G_{rt}=|G_{rt}|e^{j\Phi}`; resonance requires :math:`\Phi=2\pi q`, and threshold requires :math:`|G_{rt}|=1`.
Check the passive-cavity limit, energy conservation at every mirror, and that added loss raises rather than lowers threshold.

Section 11.5: Optical-Cavity Mode Frequencies
---------------------------------------------

Problem 11.5.1 — Axial-mode spectrum for an optical cavity with an internal dielectric section
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation.
Form the complex round-trip factor :math:`G_{rt}=|G_{rt}|e^{j\Phi}`; resonance requires :math:`\Phi=2\pi q`, and threshold requires :math:`|G_{rt}|=1`.
Check the passive-cavity limit, energy conservation at every mirror, and that added loss raises rather than lowers threshold.

Problem 11.5.2 — Axial-mode spectrum including dispersion
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

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 11.5.3 — Resonance properties of an equilateral triangular dielectric prism
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation.
Form the complex round-trip factor :math:`G_{rt}=|G_{rt}|e^{j\Phi}`; resonance requires :math:`\Phi=2\pi q`, and threshold requires :math:`|G_{rt}|=1`.
Check the passive-cavity limit, energy conservation at every mirror, and that added loss raises rather than lowers threshold.

Problem 11.5.4 — Mirror spacing in an optically pumped thin dye laser
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation.
Form the complex round-trip factor :math:`G_{rt}=|G_{rt}|e^{j\Phi}`; resonance requires :math:`\Phi=2\pi q`, and threshold requires :math:`|G_{rt}|=1`.
Check the passive-cavity limit, energy conservation at every mirror, and that added loss raises rather than lowers threshold.

Section 11.6: Regenerative Laser Amplification
----------------------------------------------

Problem 11.6.1 — Reflection gain of a regenerative laser amplifier
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

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 11.6.2 — Phase angle versus frequency for a regenerative laser cavity amplifier
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

Evaluate both cases from the same symbolic expression before taking their ratio; this keeps normalization and sign conventions from obscuring the comparison.
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 11.6.3 — Enhanced feedback diagram
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

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 11.7: Approaching Threshold: The Highly Regenerative Limit
------------------------------------------------------------------

Problem 11.7.1 — Power transmission through a laser cavity halfway between axial modes
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation.
Form the complex round-trip factor :math:`G_{rt}=|G_{rt}|e^{j\Phi}`; resonance requires :math:`\Phi=2\pi q`, and threshold requires :math:`|G_{rt}|=1`.
Check the passive-cavity limit, energy conservation at every mirror, and that added loss raises rather than lowers threshold.

Problem 11.7.2 — Gain sensitivity of a regenerative laser amplifier
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

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 11.7.3 — Skirt selectivity of a regenerative laser amplifier
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

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 11.7.4 — Output versus input for a regenerative laser amplifier with saturable internal gain
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

Evaluate both cases from the same symbolic expression before taking their ratio; this keeps normalization and sign conventions from obscuring the comparison.
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 11.7.5 — Regenerative gain peaks for off-line-center axial modes
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

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 11.7.6 — Transient reflection from a resonant cavity
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation.
Form the complex round-trip factor :math:`G_{rt}=|G_{rt}|e^{j\Phi}`; resonance requires :math:`\Phi=2\pi q`, and threshold requires :math:`|G_{rt}|=1`.
Check the passive-cavity limit, energy conservation at every mirror, and that added loss raises rather than lowers threshold.

Problem 11.7.7 — Approach to threshold in the Schawlow-Townes model
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation.
Form the complex round-trip factor :math:`G_{rt}=|G_{rt}|e^{j\Phi}`; resonance requires :math:`\Phi=2\pi q`, and threshold requires :math:`|G_{rt}|=1`.
Check the passive-cavity limit, energy conservation at every mirror, and that added loss raises rather than lowers threshold.