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 \(G_{rt}=|G_{rt}|e^{j\Phi}\); resonance requires \(\Phi=2\pi q\), and threshold requires \(|G_{rt}|=1\). 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 \(G_{rt}=|G_{rt}|e^{j\Phi}\); resonance requires \(\Phi=2\pi q\), and threshold requires \(|G_{rt}|=1\). 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 \(G_{rt}=|G_{rt}|e^{j\Phi}\); resonance requires \(\Phi=2\pi q\), and threshold requires \(|G_{rt}|=1\). 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 \(G_{rt}=|G_{rt}|e^{j\Phi}\); resonance requires \(\Phi=2\pi q\), and threshold requires \(|G_{rt}|=1\). 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 \(G_{rt}=|G_{rt}|e^{j\Phi}\); resonance requires \(\Phi=2\pi q\), and threshold requires \(|G_{rt}|=1\). 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 \(G_{rt}=|G_{rt}|e^{j\Phi}\); resonance requires \(\Phi=2\pi q\), and threshold requires \(|G_{rt}|=1\). 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 \(G_{rt}=|G_{rt}|e^{j\Phi}\); resonance requires \(\Phi=2\pi q\), and threshold requires \(|G_{rt}|=1\). 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 \(G_{rt}=|G_{rt}|e^{j\Phi}\); resonance requires \(\Phi=2\pi q\), and threshold requires \(|G_{rt}|=1\). 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 \(G_{rt}=|G_{rt}|e^{j\Phi}\); resonance requires \(\Phi=2\pi q\), and threshold requires \(|G_{rt}|=1\). 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 \(G_{rt}=|G_{rt}|e^{j\Phi}\); resonance requires \(\Phi=2\pi q\), and threshold requires \(|G_{rt}|=1\). 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 \(G_{rt}=|G_{rt}|e^{j\Phi}\); resonance requires \(\Phi=2\pi q\), and threshold requires \(|G_{rt}|=1\). 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 \(G_{rt}=|G_{rt}|e^{j\Phi}\); resonance requires \(\Phi=2\pi q\), and threshold requires \(|G_{rt}|=1\). 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 \(G_{rt}=|G_{rt}|e^{j\Phi}\); resonance requires \(\Phi=2\pi q\), and threshold requires \(|G_{rt}|=1\). 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 \(G_{rt}=|G_{rt}|e^{j\Phi}\); resonance requires \(\Phi=2\pi q\), and threshold requires \(|G_{rt}|=1\). 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 \(G_{rt}=|G_{rt}|e^{j\Phi}\); resonance requires \(\Phi=2\pi q\), and threshold requires \(|G_{rt}|=1\). 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 \(G_{rt}=|G_{rt}|e^{j\Phi}\); resonance requires \(\Phi=2\pi q\), and threshold requires \(|G_{rt}|=1\). 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 \(G_{rt}=|G_{rt}|e^{j\Phi}\); resonance requires \(\Phi=2\pi q\), and threshold requires \(|G_{rt}|=1\). 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 \(k(\omega)\) 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 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 \(G_{rt}=|G_{rt}|e^{j\Phi}\); resonance requires \(\Phi=2\pi q\), and threshold requires \(|G_{rt}|=1\). 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 \(G_{rt}=|G_{rt}|e^{j\Phi}\); resonance requires \(\Phi=2\pi q\), and threshold requires \(|G_{rt}|=1\). 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 \(dI/dz=g(I)I\), 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 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 \(dI/dz=g(I)I\), 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 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 \(dI/dz=g(I)I\), 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 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 \(G_{rt}=|G_{rt}|e^{j\Phi}\); resonance requires \(\Phi=2\pi q\), and threshold requires \(|G_{rt}|=1\). 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 \(dI/dz=g(I)I\), 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 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 \(dI/dz=g(I)I\), 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 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 \(dI/dz=g(I)I\), 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 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 \(dI/dz=g(I)I\), 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 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 \(G_{rt}=|G_{rt}|e^{j\Phi}\); resonance requires \(\Phi=2\pi q\), and threshold requires \(|G_{rt}|=1\). 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 \(G_{rt}=|G_{rt}|e^{j\Phi}\); resonance requires \(\Phi=2\pi q\), and threshold requires \(|G_{rt}|=1\). Check the passive-cavity limit, energy conservation at every mirror, and that added loss raises rather than lowers threshold.