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 \(AD-BC=1\); its periodic eigenvalues obey \(\lambda^2-(A+D)\lambda+1=0\), while imaging is selected by \(B=0\). 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 \(AD-BC=1\); its periodic eigenvalues obey \(\lambda^2-(A+D)\lambda+1=0\), while imaging is selected by \(B=0\). 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 \(AD-BC=1\); its periodic eigenvalues obey \(\lambda^2-(A+D)\lambda+1=0\), while imaging is selected by \(B=0\). 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 \(AD-BC=1\); its periodic eigenvalues obey \(\lambda^2-(A+D)\lambda+1=0\), while imaging is selected by \(B=0\). 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 \(AD-BC=1\); its periodic eigenvalues obey \(\lambda^2-(A+D)\lambda+1=0\), while imaging is selected by \(B=0\). 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 \(AD-BC=1\); its periodic eigenvalues obey \(\lambda^2-(A+D)\lambda+1=0\), while imaging is selected by \(B=0\). 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 \(AD-BC=1\); its periodic eigenvalues obey \(\lambda^2-(A+D)\lambda+1=0\), while imaging is selected by \(B=0\). 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 \(q_2=(Aq_1+B)/(Cq_1+D)\) and use \(1/q=1/R-i\lambda/(\pi n w^2)\). 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 \(w^2=w_0^2[1+(z/z_R)^2]\).

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 \(q_2=(Aq_1+B)/(Cq_1+D)\) and use \(1/q=1/R-i\lambda/(\pi n w^2)\). 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 \(w^2=w_0^2[1+(z/z_R)^2]\).

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 \(q_2=(Aq_1+B)/(Cq_1+D)\) and use \(1/q=1/R-i\lambda/(\pi n w^2)\). 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 \(w^2=w_0^2[1+(z/z_R)^2]\).

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 \(q_2=(Aq_1+B)/(Cq_1+D)\) and use \(1/q=1/R-i\lambda/(\pi n w^2)\). 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 \(w^2=w_0^2[1+(z/z_R)^2]\).

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 \(q_2=(Aq_1+B)/(Cq_1+D)\) and use \(1/q=1/R-i\lambda/(\pi n w^2)\). 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 \(w^2=w_0^2[1+(z/z_R)^2]\).

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 \(q_2=(Aq_1+B)/(Cq_1+D)\) and use \(1/q=1/R-i\lambda/(\pi n w^2)\). 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 \(w^2=w_0^2[1+(z/z_R)^2]\).

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 \(\int n\,ds\); the Euler–Lagrange equation gives \(d(n\hat{\mathbf s})/ds=\nabla n\), 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 \(\int n\,ds\); the Euler–Lagrange equation gives \(d(n\hat{\mathbf s})/ds=\nabla n\), 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 \(\int n\,ds\); the Euler–Lagrange equation gives \(d(n\hat{\mathbf s})/ds=\nabla n\), 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 \(\int n\,ds\); the Euler–Lagrange equation gives \(d(n\hat{\mathbf s})/ds=\nabla n\), 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 \(\int n\,ds\); the Euler–Lagrange equation gives \(d(n\hat{\mathbf s})/ds=\nabla n\), 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 \(\int n\,ds\); the Euler–Lagrange equation gives \(d(n\hat{\mathbf s})/ds=\nabla n\), 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 \(\int n\,ds\); the Euler–Lagrange equation gives \(d(n\hat{\mathbf s})/ds=\nabla n\), 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 \(2ik\,\partial_z u+\nabla_\perp^2u=0\); 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 \(k\partial_z u\) 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 \(2ik\,\partial_z u+\nabla_\perp^2u=0\); 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 \(k\partial_z u\) in the paraxial regime.