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
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

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 :math:`q^{-1}=R^{-1}-j\lambda/(\pi w^2)` and propagate it with :math:`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) —
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

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 :math:`q^{-1}=R^{-1}-j\lambda/(\pi w^2)` and propagate it with :math:`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
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

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 :math:`q^{-1}=R^{-1}-j\lambda/(\pi w^2)` and propagate it with :math:`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
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

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 :math:`q^{-1}=R^{-1}-j\lambda/(\pi w^2)` and propagate it with :math:`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
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

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 :math:`q^{-1}=R^{-1}-j\lambda/(\pi w^2)` and propagate it with :math:`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
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

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 :math:`q^{-1}=R^{-1}-j\lambda/(\pi w^2)` and propagate it with :math:`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
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

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 :math:`q^{-1}=R^{-1}-j\lambda/(\pi w^2)` and propagate it with :math:`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)
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

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 :math:`q^{-1}=R^{-1}-j\lambda/(\pi w^2)` and propagate it with :math:`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)
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

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 :math:`q^{-1}=R^{-1}-j\lambda/(\pi w^2)` and propagate it with :math:`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
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

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 :math:`q^{-1}=R^{-1}-j\lambda/(\pi w^2)` and propagate it with :math:`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
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

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 :math:`dI/dz=g(I)I`, using :math:`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
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

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 :math:`q^{-1}=R^{-1}-j\lambda/(\pi w^2)` and propagate it with :math:`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
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

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 :math:`q^{-1}=R^{-1}-j\lambda/(\pi w^2)` and propagate it with :math:`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
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

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 :math:`q^{-1}=R^{-1}-j\lambda/(\pi w^2)` and propagate it with :math:`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.