Chapter 16: Wave Optics and Gaussian Beams
Source: Anthony E. Siegman, Lasers (1986), Chapter 16. 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 16.2: Huygens’ Integral
Problem 16.2.1 — Solid angular spread from a uniformly illuminated aperture
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 16.2.2 — Cascade properties of the Huygens-Fresnel integral Suppose we use the operator notation ii(xi,zi) —
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 16.3: Gaussian Spherical Waves
Problem 16.3.1 — Changes in wavefront curvature on reflection from a curved mirror
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 16.3.2 — Using a reversed coordinate 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.
Problem 16.3.3 — Spherical waves and circular interference rings
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 16.3.4 — Complex transverse source point coordinates
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 16.4: Higher-Order Gaussian Modes
Problem 16.4.1 — Intensity contours for a higher-order Hermite-Gaussian mode
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 16.4.2 — Finding the mode content of an arbitrary optical beam (research problem)
Brief solution
1. Method.
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.
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
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. 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 16.4.3 — Recursion relation for Hermite-Gaussian modes A standard recursion relation for the Hermite polynomials Hn(x)
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 16.6: Gaussian Beam Propagation In Ducts
Problem 16.6.1 — Practical criteria for trapping a Gaussian beam in a duct
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 16.6.2 — Lensing or ducting effects in a saturated laser amplifier
Brief solution
1. Method.
Evaluate both cases from the same symbolic expression before taking their ratio; this keeps normalization and sign conventions from obscuring the comparison.
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
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 16.6.3 — Higher-order eigenmodes in ducts
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 16.7: Numerical Beam Propagation Methods
Problem 16.7.1 — Center of gravity of a paraxial optical beam
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 16.7.2 — Second moment of a paraxial optical beam
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.