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