Chapter 21: Generalized Paraxial Resonator Theory
Source: Anthony E. Siegman, Lasers (1986), Chapter 21. 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 21.1: Complex Paraxial Resonator Analysis
Problem 21.1.1 — Resonator analysis using symmetrized matrices
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.
Section 21.3: Real And Geometrically Unstable Resonators
Problem 21.3.1 — Gaussian-beam evolution in a geometrically unstable system
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.
Represent the beam by \(q^{-1}=R^{-1}-j\lambda/(\pi w^2)\) and propagate it with \(q_2=(Aq_1+B)/(Cq_1+D)\); separate real and imaginary parts only at the end.
3. Verification.
Check that free propagation reproduces the Rayleigh-range formulas and that every computed spot size is real and positive.
Show detailed steps
List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation. Represent the beam by \(q^{-1}=R^{-1}-j\lambda/(\pi w^2)\) and propagate it with \(q_2=(Aq_1+B)/(Cq_1+D)\); separate real and imaginary parts only at the end. Check that free propagation reproduces the Rayleigh-range formulas and that every computed spot size is real and positive.
Section 21.4: Complex Stable And Unstable Resonators
Problem 21.4.1 — Inserting a weak Gaussian aperture into an arbitrary ring resonator
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.
Represent the beam by \(q^{-1}=R^{-1}-j\lambda/(\pi w^2)\) and propagate it with \(q_2=(Aq_1+B)/(Cq_1+D)\); separate real and imaginary parts only at the end.
3. Verification.
Check that free propagation reproduces the Rayleigh-range formulas and that every computed spot size is real and positive.
Show detailed steps
List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation. Represent the beam by \(q^{-1}=R^{-1}-j\lambda/(\pi w^2)\) and propagate it with \(q_2=(Aq_1+B)/(Cq_1+D)\); separate real and imaginary parts only at the end. Check that free propagation reproduces the Rayleigh-range formulas and that every computed spot size is real and positive.
Problem 21.4.2 — Practical design of a large-mode-volume resonator
Brief solution
1. Method.
Translate each performance requirement into an equality or inequality, solve the coupled constraints, and discard any root that violates a physical bound.
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
Translate each performance requirement into an equality or inequality, solve the coupled constraints, and discard any root that violates a physical bound. 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 21.4.3 — Convergence to the final stable mode in a Gaussian-aperture-stabilized resonator
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.
Represent the beam by \(q^{-1}=R^{-1}-j\lambda/(\pi w^2)\) and propagate it with \(q_2=(Aq_1+B)/(Cq_1+D)\); separate real and imaginary parts only at the end.
3. Verification.
Check that free propagation reproduces the Rayleigh-range formulas and that every computed spot size is real and positive.
Show detailed steps
List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation. Represent the beam by \(q^{-1}=R^{-1}-j\lambda/(\pi w^2)\) and propagate it with \(q_2=(Aq_1+B)/(Cq_1+D)\); separate real and imaginary parts only at the end. Check that free propagation reproduces the Rayleigh-range formulas and that every computed spot size is real and positive.
Problem 21.4.4 — Same problem, but for a negative branch resonator
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 21.4.5 — Mode parameters in a Gaussian-aperture-stabilized two-mirror standing-wave resonator
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.
Represent the beam by \(q^{-1}=R^{-1}-j\lambda/(\pi w^2)\) and propagate it with \(q_2=(Aq_1+B)/(Cq_1+D)\); separate real and imaginary parts only at the end.
3. Verification.
Check that free propagation reproduces the Rayleigh-range formulas and that every computed spot size is real and positive.
Show detailed steps
List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation. Represent the beam by \(q^{-1}=R^{-1}-j\lambda/(\pi w^2)\) and propagate it with \(q_2=(Aq_1+B)/(Cq_1+D)\); separate real and imaginary parts only at the end. Check that free propagation reproduces the Rayleigh-range formulas and that every computed spot size is real and positive.
Problem 21.4.6 — Gaussian aperture stabilization in a plane-mirror resonator
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.
Represent the beam by \(q^{-1}=R^{-1}-j\lambda/(\pi w^2)\) and propagate it with \(q_2=(Aq_1+B)/(Cq_1+D)\); separate real and imaginary parts only at the end.
3. Verification.
Check that free propagation reproduces the Rayleigh-range formulas and that every computed spot size is real and positive.
Show detailed steps
List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation. Represent the beam by \(q^{-1}=R^{-1}-j\lambda/(\pi w^2)\) and propagate it with \(q_2=(Aq_1+B)/(Cq_1+D)\); separate real and imaginary parts only at the end. Check that free propagation reproduces the Rayleigh-range formulas and that every computed spot size is real and positive.
Section 21.5: Other General Properties Of Paraxial Resonators
Problem 21.5.1 — Magnifying and demagnifying eigenwaves in an unsymmetric ring unstable resonator
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.
Section 21.6: Multielement Stable Resonator Designs
Problem 21.6.1 — Longitudinal magnification of a telescope
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 21.6.2 — Contours of constant m in the stability diagram
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 21.6.3 — Perturbation-insensitive cavity design
Brief solution
1. Method.
Translate each performance requirement into an equality or inequality, solve the coupled constraints, and discard any root that violates a physical bound.
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
Translate each performance requirement into an equality or inequality, solve the coupled constraints, and discard any root that violates a physical bound. 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 21.6.4 — Planar resonator with a single intracavity lens
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 21.6.5 — Mode profile in a stable multielement resonator
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 21.6.6 — Spot size stability against focusing at a different plane
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.
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.
3. Verification.
Verify that the small-signal limit is exponential, while extracted energy never exceeds the stored inversion energy.
Show detailed steps
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 21.6.7 — Characteristics of various types of ray systems
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.
Section 21.7: Orthogonality Properties Of Optical Resonator Modes
Problem 21.7.1 — Demonstrating the transpose relationship for propagation in opposite directions through a multiaperture resonator
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.
Insert the aperture field in the Huygens–Fresnel integral; retain the quadratic phase in the near field and use the aperture Fourier transform only after the Fraunhofer condition holds.
3. Verification.
Verify the on-axis limit and conservation of total transmitted power; aperture enlargement must narrow the far-field pattern.
Show detailed steps
List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation. Insert the aperture field in the Huygens–Fresnel integral; retain the quadratic phase in the near field and use the aperture Fourier transform only after the Fraunhofer condition holds. Verify the on-axis limit and conservation of total transmitted power; aperture enlargement must narrow the far-field pattern.
Problem 21.7.2 — Symmetrizing the optical resonator eigenproblem
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.