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
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.
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 \(AD-BC=1\).
3. Verification.
Use a chief or marginal ray as an independent sign and scaling check on the matrix result.
Show detailed steps
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
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.
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 \(AD-BC=1\).
3. Verification.
Use a chief or marginal ray as an independent sign and scaling check on the matrix result.
Show detailed steps
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
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.
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 \(AD-BC=1\).
3. Verification.
Use a chief or marginal ray as an independent sign and scaling check on the matrix result.
Show detailed steps
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
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.
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 \(AD-BC=1\).
3. Verification.
Use a chief or marginal ray as an independent sign and scaling check on the matrix result.
Show detailed steps
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
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.
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 \(AD-BC=1\).
3. Verification.
Use a chief or marginal ray as an independent sign and scaling check on the matrix result.
Show detailed steps
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
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.
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 \(AD-BC=1\).
3. Verification.
Use a chief or marginal ray as an independent sign and scaling check on the matrix result.
Show detailed steps
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
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.
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 \(AD-BC=1\).
3. Verification.
Use a chief or marginal ray as an independent sign and scaling check on the matrix result.
Show detailed steps
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
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.
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 \(AD-BC=1\).
3. Verification.
Use a chief or marginal ray as an independent sign and scaling check on the matrix result.
Show detailed steps
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
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.
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 \(AD-BC=1\).
3. Verification.
Use a chief or marginal ray as an independent sign and scaling check on the matrix result.
Show detailed steps
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
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.
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 \(AD-BC=1\).
3. Verification.
Use a chief or marginal ray as an independent sign and scaling check on the matrix result.
Show detailed steps
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
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.
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 \(AD-BC=1\).
3. Verification.
Use a chief or marginal ray as an independent sign and scaling check on the matrix result.
Show detailed steps
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
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.
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 \(AD-BC=1\).
3. Verification.
Use a chief or marginal ray as an independent sign and scaling check on the matrix result.
Show detailed steps
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
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.
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 \(AD-BC=1\).
3. Verification.
Use a chief or marginal ray as an independent sign and scaling check on the matrix result.
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. 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
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.
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 \(AD-BC=1\).
3. Verification.
Use a chief or marginal ray as an independent sign and scaling check on the matrix result.
Show detailed steps
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
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.
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 \(AD-BC=1\).
3. Verification.
Use a chief or marginal ray as an independent sign and scaling check on the matrix result.
Show detailed steps
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
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.
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 \(AD-BC=1\).
3. Verification.
Use a chief or marginal ray as an independent sign and scaling check on the matrix result.
Show detailed steps
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
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.
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 \(AD-BC=1\).
3. Verification.
Use a chief or marginal ray as an independent sign and scaling check on the matrix result.
Show detailed steps
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
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.
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 \(AD-BC=1\).
3. Verification.
Use a chief or marginal ray as an independent sign and scaling check on the matrix result.
Show detailed steps
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
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.
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 \(AD-BC=1\).
3. Verification.
Use a chief or marginal ray as an independent sign and scaling check on the matrix result.
Show detailed steps
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
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.
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 \(AD-BC=1\).
3. Verification.
Use a chief or marginal ray as an independent sign and scaling check on the matrix result.
Show detailed steps
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
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.
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 \(AD-BC=1\).
3. Verification.
Use a chief or marginal ray as an independent sign and scaling check on the matrix result.
Show detailed steps
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
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.
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 \(AD-BC=1\).
3. Verification.
Use a chief or marginal ray as an independent sign and scaling check on the matrix result.
Show detailed steps
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
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.
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 \(AD-BC=1\).
3. Verification.
Use a chief or marginal ray as an independent sign and scaling check on the matrix result.
Show detailed steps
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.