Chapter 15: Ray Optics and Ray Matrices

Source: Anthony E. Siegman, Lasers (1986), Chapter 15. 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 15.1: Paraxial Optical Rays And Ray Matrices

Problem 15.1.1 — Ray matrix for a curved dielectric interface

List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation. Multiply the element matrices in propagation order and apply \((x_2,\theta_2)^T=M(x_1,\theta_1)^T\); for a lossless first-order system verify \(AD-BC=1\). Use a chief or marginal ray as an independent sign and scaling check on the matrix result.

Problem 15.1.2 — Ray matrix elements for a curved diffraction grating

List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation. Multiply the element matrices in propagation order and apply \((x_2,\theta_2)^T=M(x_1,\theta_1)^T\); for a lossless first-order system verify \(AD-BC=1\). Use a chief or marginal ray as an independent sign and scaling check on the matrix result.

Problem 15.1.3 — Limiting case

List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation. Multiply the element matrices in propagation order and apply \((x_2,\theta_2)^T=M(x_1,\theta_1)^T\); for a lossless first-order system verify \(AD-BC=1\). Use a chief or marginal ray as an independent sign and scaling check on the matrix result.

Section 15.2: Ray Propagation Through Cascaded Elements

Problem 15.2.1 — Evaluating the focal length of a thin lens

List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation. Multiply the element matrices in propagation order and apply \((x_2,\theta_2)^T=M(x_1,\theta_1)^T\); for a lossless first-order system verify \(AD-BC=1\). Use a chief or marginal ray as an independent sign and scaling check on the matrix result.

Problem 15.2.2 — Replacing an arbitrary “black box” ray matrix with a single lens

List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation. Multiply the element matrices in propagation order and apply \((x_2,\theta_2)^T=M(x_1,\theta_1)^T\); for a lossless first-order system verify \(AD-BC=1\). Use a chief or marginal ray as an independent sign and scaling check on the matrix result.

Problem 15.2.3 — Ray matrix of cascaded elements going in the reverse direction

List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation. Multiply the element matrices in propagation order and apply \((x_2,\theta_2)^T=M(x_1,\theta_1)^T\); for a lossless first-order system verify \(AD-BC=1\). Use a chief or marginal ray as an independent sign and scaling check on the matrix result.

Problem 15.2.4 — Evaluating the total ray matrix for a reflection problem

List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation. Multiply the element matrices in propagation order and apply \((x_2,\theta_2)^T=M(x_1,\theta_1)^T\); for a lossless first-order system verify \(AD-BC=1\). Use a chief or marginal ray as an independent sign and scaling check on the matrix result.

Problem 15.2.5 — Replacing an arbitrary ray matrix system with a single mirror

List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation. Multiply the element matrices in propagation order and apply \((x_2,\theta_2)^T=M(x_1,\theta_1)^T\); for a lossless first-order system verify \(AD-BC=1\). Use a chief or marginal ray as an independent sign and scaling check on the matrix result.

Problem 15.2.6 — Focusing properties of thick-lens ABCD matrices

List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation. Multiply the element matrices in propagation order and apply \((x_2,\theta_2)^T=M(x_1,\theta_1)^T\); for a lossless first-order system verify \(AD-BC=1\). Use a chief or marginal ray as an independent sign and scaling check on the matrix result.

Problem 15.2.7 — General formulas for an arbitrary thick lens or ABCD system The focal, principal and

List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation. Multiply the element matrices in propagation order and apply \((x_2,\theta_2)^T=M(x_1,\theta_1)^T\); for a lossless first-order system verify \(AD-BC=1\). Use a chief or marginal ray as an independent sign and scaling check on the matrix result.

Section 15.3: Rays In Periodic Focusing Systems

Problem 15.3.1 — Properties of the eigenrays in a periodic system

List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation. Multiply the element matrices in propagation order and apply \((x_2,\theta_2)^T=M(x_1,\theta_1)^T\); for a lossless first-order system verify \(AD-BC=1\). Use a chief or marginal ray as an independent sign and scaling check on the matrix result.

Problem 15.3.2 — Ray properties of an elementary periodic lensguide

List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation. Multiply the element matrices in propagation order and apply \((x_2,\theta_2)^T=M(x_1,\theta_1)^T\); for a lossless first-order system verify \(AD-BC=1\). Use a chief or marginal ray as an independent sign and scaling check on the matrix result.

Problem 15.3.3 — Computer plotting of periodic ray positions

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. Multiply the element matrices in propagation order and apply \((x_2,\theta_2)^T=M(x_1,\theta_1)^T\); for a lossless first-order system verify \(AD-BC=1\). Use a chief or marginal ray as an independent sign and scaling check on the matrix result.

Problem 15.3.4 — Periodic systems with integer numbers of spots

List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation. Multiply the element matrices in propagation order and apply \((x_2,\theta_2)^T=M(x_1,\theta_1)^T\); for a lossless first-order system verify \(AD-BC=1\). Use a chief or marginal ray as an independent sign and scaling check on the matrix result.

Problem 15.3.5 — Alignment procedure for the periodic delay line demonstration

List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation. Multiply the element matrices in propagation order and apply \((x_2,\theta_2)^T=M(x_1,\theta_1)^T\); for a lossless first-order system verify \(AD-BC=1\). Use a chief or marginal ray as an independent sign and scaling check on the matrix result.

Problem 15.3.6 — Eigenray solutions for a near-spherical optical resonator

List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation. Multiply the element matrices in propagation order and apply \((x_2,\theta_2)^T=M(x_1,\theta_1)^T\); for a lossless first-order system verify \(AD-BC=1\). Use a chief or marginal ray as an independent sign and scaling check on the matrix result.

Problem 15.3.7 — Perturbation stability of periodic focusing eigenrays

List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation. Multiply the element matrices in propagation order and apply \((x_2,\theta_2)^T=M(x_1,\theta_1)^T\); for a lossless first-order system verify \(AD-BC=1\). Use a chief or marginal ray as an independent sign and scaling check on the matrix result.

Problem 15.3.8 — Ray intersections inside an optical resonator

List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation. Multiply the element matrices in propagation order and apply \((x_2,\theta_2)^T=M(x_1,\theta_1)^T\); for a lossless first-order system verify \(AD-BC=1\). Use a chief or marginal ray as an independent sign and scaling check on the matrix result.

Section 15.4: Ray Optics With Misaligned Elements

Problem 15.4.1 — Error vector for a tilted Hat mirror

List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation. Multiply the element matrices in propagation order and apply \((x_2,\theta_2)^T=M(x_1,\theta_1)^T\); for a lossless first-order system verify \(AD-BC=1\). Use a chief or marginal ray as an independent sign and scaling check on the matrix result.

Problem 15.4.2 — Misaligned optical resonator

List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation. Multiply the element matrices in propagation order and apply \((x_2,\theta_2)^T=M(x_1,\theta_1)^T\); for a lossless first-order system verify \(AD-BC=1\). Use a chief or marginal ray as an independent sign and scaling check on the matrix result.

Problem 15.4.3 — More misaligned resonators

List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation. Multiply the element matrices in propagation order and apply \((x_2,\theta_2)^T=M(x_1,\theta_1)^T\); for a lossless first-order system verify \(AD-BC=1\). Use a chief or marginal ray as an independent sign and scaling check on the matrix result.

Problem 15.4.4 — Finding the axis ray in another optical resonator with misaligned elements

List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation. Multiply the element matrices in propagation order and apply \((x_2,\theta_2)^T=M(x_1,\theta_1)^T\); for a lossless first-order system verify \(AD-BC=1\). Use a chief or marginal ray as an independent sign and scaling check on the matrix result.

Section 15.6: Nonorthogonal Ray Matrices

Problem 15.6.1 — Image rotation in a Dove prism

List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation. Multiply the element matrices in propagation order and apply \((x_2,\theta_2)^T=M(x_1,\theta_1)^T\); for a lossless first-order system verify \(AD-BC=1\). Use a chief or marginal ray as an independent sign and scaling check on the matrix result.