Chapter 2: Rays and Optical Beams
=================================

Source: Amnon Yariv and Pochi Yeh, *Photonics: Optical Electronics in
Modern Communications*, sixth edition (2007), Chapter
2.  Use each problem number with the book; the original prompts are
not reproduced.  Each entry supplies the governing model, a decisive solution
route, and an independent consistency check.

End-of-chapter problems
-----------------------

Problem 2.1 — ABCD ray matrices and lenslike media: derivation
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

Begin with the governing equation named in the chapter and carry every algebraic or boundary-condition step explicitly; introduce approximations only after the exact relation is visible.
Build the system matrix in propagation order.  For a lossless first-order system :math:`AD-BC=1`; its periodic eigenvalues obey :math:`\lambda^2-(A+D)\lambda+1=0`, while imaging is selected by :math:`B=0`.
Reverse the element order to recover the inverse matrix and confirm both the determinant and the thin-element limit.

Problem 2.2 — ABCD ray matrices and lenslike media: derivation
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

Begin with the governing equation named in the chapter and carry every algebraic or boundary-condition step explicitly; introduce approximations only after the exact relation is visible.
Build the system matrix in propagation order.  For a lossless first-order system :math:`AD-BC=1`; its periodic eigenvalues obey :math:`\lambda^2-(A+D)\lambda+1=0`, while imaging is selected by :math:`B=0`.
Reverse the element order to recover the inverse matrix and confirm both the determinant and the thin-element limit.

Problem 2.3 — ABCD ray matrices and lenslike media: derivation
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

Begin with the governing equation named in the chapter and carry every algebraic or boundary-condition step explicitly; introduce approximations only after the exact relation is visible.
Build the system matrix in propagation order.  For a lossless first-order system :math:`AD-BC=1`; its periodic eigenvalues obey :math:`\lambda^2-(A+D)\lambda+1=0`, while imaging is selected by :math:`B=0`.
Reverse the element order to recover the inverse matrix and confirm both the determinant and the thin-element limit.

Problem 2.4 — ABCD ray matrices and lenslike media: derivation
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

Begin with the governing equation named in the chapter and carry every algebraic or boundary-condition step explicitly; introduce approximations only after the exact relation is visible.
Build the system matrix in propagation order.  For a lossless first-order system :math:`AD-BC=1`; its periodic eigenvalues obey :math:`\lambda^2-(A+D)\lambda+1=0`, while imaging is selected by :math:`B=0`.
Reverse the element order to recover the inverse matrix and confirm both the determinant and the thin-element limit.

Problem 2.5 — ABCD ray matrices and lenslike media: derivation
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

Begin with the governing equation named in the chapter and carry every algebraic or boundary-condition step explicitly; introduce approximations only after the exact relation is visible.
Build the system matrix in propagation order.  For a lossless first-order system :math:`AD-BC=1`; its periodic eigenvalues obey :math:`\lambda^2-(A+D)\lambda+1=0`, while imaging is selected by :math:`B=0`.
Reverse the element order to recover the inverse matrix and confirm both the determinant and the thin-element limit.

Problem 2.6 — ABCD ray matrices and lenslike media: derivation
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

Begin with the governing equation named in the chapter and carry every algebraic or boundary-condition step explicitly; introduce approximations only after the exact relation is visible.
Build the system matrix in propagation order.  For a lossless first-order system :math:`AD-BC=1`; its periodic eigenvalues obey :math:`\lambda^2-(A+D)\lambda+1=0`, while imaging is selected by :math:`B=0`.
Reverse the element order to recover the inverse matrix and confirm both the determinant and the thin-element limit.

Problem 2.7 — ABCD ray matrices and lenslike media: derivation
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

Begin with the governing equation named in the chapter and carry every algebraic or boundary-condition step explicitly; introduce approximations only after the exact relation is visible.
Build the system matrix in propagation order.  For a lossless first-order system :math:`AD-BC=1`; its periodic eigenvalues obey :math:`\lambda^2-(A+D)\lambda+1=0`, while imaging is selected by :math:`B=0`.
Reverse the element order to recover the inverse matrix and confirm both the determinant and the thin-element limit.

Problem 2.8 — Gaussian beams and scalar propagation: calculation
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

Convert the supplied data to one unit system, isolate the requested quantity symbolically, and retain guard digits until the final numerical evaluation.
Propagate the complex beam parameter with :math:`q_2=(Aq_1+B)/(Cq_1+D)` and use :math:`1/q=1/R-i\lambda/(\pi n w^2)`.  For a field integral, keep the quadratic phase until completing the Gaussian integral.
The imaginary part of the physical beam parameter must retain the confinement sign, and free propagation must reproduce :math:`w^2=w_0^2[1+(z/z_R)^2]`.

Problem 2.9 — Gaussian beams and scalar propagation: calculation
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

Convert the supplied data to one unit system, isolate the requested quantity symbolically, and retain guard digits until the final numerical evaluation.
Propagate the complex beam parameter with :math:`q_2=(Aq_1+B)/(Cq_1+D)` and use :math:`1/q=1/R-i\lambda/(\pi n w^2)`.  For a field integral, keep the quadratic phase until completing the Gaussian integral.
The imaginary part of the physical beam parameter must retain the confinement sign, and free propagation must reproduce :math:`w^2=w_0^2[1+(z/z_R)^2]`.

Problem 2.10 — Gaussian beams and scalar propagation: derivation
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

Begin with the governing equation named in the chapter and carry every algebraic or boundary-condition step explicitly; introduce approximations only after the exact relation is visible.
Propagate the complex beam parameter with :math:`q_2=(Aq_1+B)/(Cq_1+D)` and use :math:`1/q=1/R-i\lambda/(\pi n w^2)`.  For a field integral, keep the quadratic phase until completing the Gaussian integral.
The imaginary part of the physical beam parameter must retain the confinement sign, and free propagation must reproduce :math:`w^2=w_0^2[1+(z/z_R)^2]`.

Problem 2.11 — Gaussian beams and scalar propagation: calculation
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

Convert the supplied data to one unit system, isolate the requested quantity symbolically, and retain guard digits until the final numerical evaluation.
Propagate the complex beam parameter with :math:`q_2=(Aq_1+B)/(Cq_1+D)` and use :math:`1/q=1/R-i\lambda/(\pi n w^2)`.  For a field integral, keep the quadratic phase until completing the Gaussian integral.
The imaginary part of the physical beam parameter must retain the confinement sign, and free propagation must reproduce :math:`w^2=w_0^2[1+(z/z_R)^2]`.

Problem 2.12 — Gaussian beams and scalar propagation: calculation
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

Convert the supplied data to one unit system, isolate the requested quantity symbolically, and retain guard digits until the final numerical evaluation.
Propagate the complex beam parameter with :math:`q_2=(Aq_1+B)/(Cq_1+D)` and use :math:`1/q=1/R-i\lambda/(\pi n w^2)`.  For a field integral, keep the quadratic phase until completing the Gaussian integral.
The imaginary part of the physical beam parameter must retain the confinement sign, and free propagation must reproduce :math:`w^2=w_0^2[1+(z/z_R)^2]`.

Problem 2.13 — Gaussian beams and scalar propagation: derivation
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

Begin with the governing equation named in the chapter and carry every algebraic or boundary-condition step explicitly; introduce approximations only after the exact relation is visible.
Propagate the complex beam parameter with :math:`q_2=(Aq_1+B)/(Cq_1+D)` and use :math:`1/q=1/R-i\lambda/(\pi n w^2)`.  For a field integral, keep the quadratic phase until completing the Gaussian integral.
The imaginary part of the physical beam parameter must retain the confinement sign, and free propagation must reproduce :math:`w^2=w_0^2[1+(z/z_R)^2]`.

Problem 2.14 — Fermat principle, refraction, and periodic focusing: derivation
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

Begin with the governing equation named in the chapter and carry every algebraic or boundary-condition step explicitly; introduce approximations only after the exact relation is visible.
Vary the optical path :math:`\int n\,ds`; the Euler--Lagrange equation gives :math:`d(n\hat{\mathbf s})/ds=\nabla n`, and an interface variation gives Snell's law.  Linearize only after deriving the exact stationarity condition.
A uniform-index region must produce a straight ray, and the continuous focusing limit must agree with the corresponding periodic ABCD matrix.

Problem 2.15 — Fermat principle, refraction, and periodic focusing: derivation
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

Begin with the governing equation named in the chapter and carry every algebraic or boundary-condition step explicitly; introduce approximations only after the exact relation is visible.
Vary the optical path :math:`\int n\,ds`; the Euler--Lagrange equation gives :math:`d(n\hat{\mathbf s})/ds=\nabla n`, and an interface variation gives Snell's law.  Linearize only after deriving the exact stationarity condition.
A uniform-index region must produce a straight ray, and the continuous focusing limit must agree with the corresponding periodic ABCD matrix.

Problem 2.16 — Fermat principle, refraction, and periodic focusing: derivation
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

Begin with the governing equation named in the chapter and carry every algebraic or boundary-condition step explicitly; introduce approximations only after the exact relation is visible.
Vary the optical path :math:`\int n\,ds`; the Euler--Lagrange equation gives :math:`d(n\hat{\mathbf s})/ds=\nabla n`, and an interface variation gives Snell's law.  Linearize only after deriving the exact stationarity condition.
A uniform-index region must produce a straight ray, and the continuous focusing limit must agree with the corresponding periodic ABCD matrix.

Problem 2.17 — Fermat principle, refraction, and periodic focusing: calculation
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

Convert the supplied data to one unit system, isolate the requested quantity symbolically, and retain guard digits until the final numerical evaluation.
Vary the optical path :math:`\int n\,ds`; the Euler--Lagrange equation gives :math:`d(n\hat{\mathbf s})/ds=\nabla n`, and an interface variation gives Snell's law.  Linearize only after deriving the exact stationarity condition.
A uniform-index region must produce a straight ray, and the continuous focusing limit must agree with the corresponding periodic ABCD matrix.

Problem 2.18 — Fermat principle, refraction, and periodic focusing: derivation
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

Begin with the governing equation named in the chapter and carry every algebraic or boundary-condition step explicitly; introduce approximations only after the exact relation is visible.
Vary the optical path :math:`\int n\,ds`; the Euler--Lagrange equation gives :math:`d(n\hat{\mathbf s})/ds=\nabla n`, and an interface variation gives Snell's law.  Linearize only after deriving the exact stationarity condition.
A uniform-index region must produce a straight ray, and the continuous focusing limit must agree with the corresponding periodic ABCD matrix.

Problem 2.19 — Fermat principle, refraction, and periodic focusing: derivation
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

Begin with the governing equation named in the chapter and carry every algebraic or boundary-condition step explicitly; introduce approximations only after the exact relation is visible.
Vary the optical path :math:`\int n\,ds`; the Euler--Lagrange equation gives :math:`d(n\hat{\mathbf s})/ds=\nabla n`, and an interface variation gives Snell's law.  Linearize only after deriving the exact stationarity condition.
A uniform-index region must produce a straight ray, and the continuous focusing limit must agree with the corresponding periodic ABCD matrix.

Problem 2.20 — Fermat principle, refraction, and periodic focusing: derivation
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

Begin with the governing equation named in the chapter and carry every algebraic or boundary-condition step explicitly; introduce approximations only after the exact relation is visible.
Vary the optical path :math:`\int n\,ds`; the Euler--Lagrange equation gives :math:`d(n\hat{\mathbf s})/ds=\nabla n`, and an interface variation gives Snell's law.  Linearize only after deriving the exact stationarity condition.
A uniform-index region must produce a straight ray, and the continuous focusing limit must agree with the corresponding periodic ABCD matrix.

Problem 2.21 — Gaussian modes and the slowly varying envelope: plot
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

Derive a dimensionless plotting expression first, evaluate the limiting values and resonance or cutoff points, and then sample densely enough to resolve the narrowest feature.
Insert the Laguerre-- or Hermite--Gaussian envelope in the paraxial equation :math:`2ik\,\partial_z u+\nabla_\perp^2u=0`; use orthogonality for superpositions and scale transverse coordinates by the spot radius.
Mode power must be finite and conserved, and neglected longitudinal derivatives must be smaller than :math:`k\partial_z u` in the paraxial regime.

Problem 2.22 — Gaussian modes and the slowly varying envelope: derivation
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

Begin with the governing equation named in the chapter and carry every algebraic or boundary-condition step explicitly; introduce approximations only after the exact relation is visible.
Insert the Laguerre-- or Hermite--Gaussian envelope in the paraxial equation :math:`2ik\,\partial_z u+\nabla_\perp^2u=0`; use orthogonality for superpositions and scale transverse coordinates by the spot radius.
Mode power must be finite and conserved, and neglected longitudinal derivatives must be smaller than :math:`k\partial_z u` in the paraxial regime.