Chapter 17: Physical Properties of Gaussian Beams

Source: Anthony E. Siegman, Lasers (1986), Chapter 17. 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 17.1: Gaussian Beam Propagation

Problem 17.1.1 — Gaussian beam transmission through a square aperture

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 17.1.2 — Criteria for centering accuracy of a circular aperture

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 17.1.3 — Setting tolerances on beam collimation and far-field beam angle

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 17.1.4 — Simulating an annular beam with positive and negative Gaussians

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 17.1.5 — Locating the center of curvature of a Gaussian beam

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 17.1.6 — Beam spot size at long distances

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 17.2: Gaussian Beam Focusing

Problem 17.2.1 — Focusing for absolute minimum spot size

Translate each performance requirement into an equality or inequality, solve the coupled constraints, and discard any root that violates a physical bound. 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 17.2.2 — Focusing a Gaussian beam with an astigmatic lens

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 17.2.3 — Focusing into a dielectric medium

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 17.3: Lens Laws And Gaussian Mode Matching

Problem 17.3.1 — Thin-lens imaging formulas for Gaussian beams

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 17.5: Higher-Order Gaussian Modes

Problem 17.5.1 — Instantaneous profile of the “donut” mode

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 17.5.2 — Near-field beam with a central hole

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 17.5.3 — Recursion relation for expanding a half-rectangle in Hermite-Gaussian functions

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 17.6: Multimode Optical Beams

Problem 17.6.1 — Beam expansion criteria using Laguerre-Gaussian functions

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.