Chapter 1: Electromagnetic Fields and Waves
===========================================

Source: Amnon Yariv and Pochi Yeh, *Photonics: Optical Electronics in
Modern Communications*, sixth edition (2007), Chapter
1.  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 1.1 — Maxwell equations, momentum, and energy: 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.
Integrate :math:`\nabla\cdot\mathbf D=\rho`, :math:`\nabla\cdot\mathbf B=0`, :math:`\nabla\times\mathbf E=-\partial_t\mathbf B`, and :math:`\nabla\times\mathbf H=\mathbf J+\partial_t\mathbf D` over the stated volume or contour; then apply the divergence or Stokes theorem before taking a limiting surface.
The result must conserve charge or energy, have matching units on both sides, and reduce to the source-free Maxwell relation when charge and current vanish.

Problem 1.2 — Maxwell equations, momentum, and energy: 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.
Integrate :math:`\nabla\cdot\mathbf D=\rho`, :math:`\nabla\cdot\mathbf B=0`, :math:`\nabla\times\mathbf E=-\partial_t\mathbf B`, and :math:`\nabla\times\mathbf H=\mathbf J+\partial_t\mathbf D` over the stated volume or contour; then apply the divergence or Stokes theorem before taking a limiting surface.
The result must conserve charge or energy, have matching units on both sides, and reduce to the source-free Maxwell relation when charge and current vanish.

Problem 1.3 — Maxwell equations, momentum, and energy: 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.
Integrate :math:`\nabla\cdot\mathbf D=\rho`, :math:`\nabla\cdot\mathbf B=0`, :math:`\nabla\times\mathbf E=-\partial_t\mathbf B`, and :math:`\nabla\times\mathbf H=\mathbf J+\partial_t\mathbf D` over the stated volume or contour; then apply the divergence or Stokes theorem before taking a limiting surface.
The result must conserve charge or energy, have matching units on both sides, and reduce to the source-free Maxwell relation when charge and current vanish.

Problem 1.4 — Maxwell equations, momentum, and energy: 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.
Integrate :math:`\nabla\cdot\mathbf D=\rho`, :math:`\nabla\cdot\mathbf B=0`, :math:`\nabla\times\mathbf E=-\partial_t\mathbf B`, and :math:`\nabla\times\mathbf H=\mathbf J+\partial_t\mathbf D` over the stated volume or contour; then apply the divergence or Stokes theorem before taking a limiting surface.
The result must conserve charge or energy, have matching units on both sides, and reduce to the source-free Maxwell relation when charge and current vanish.

Problem 1.5 — Maxwell equations, momentum, and energy: 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.
Integrate :math:`\nabla\cdot\mathbf D=\rho`, :math:`\nabla\cdot\mathbf B=0`, :math:`\nabla\times\mathbf E=-\partial_t\mathbf B`, and :math:`\nabla\times\mathbf H=\mathbf J+\partial_t\mathbf D` over the stated volume or contour; then apply the divergence or Stokes theorem before taking a limiting surface.
The result must conserve charge or energy, have matching units on both sides, and reduce to the source-free Maxwell relation when charge and current vanish.

Problem 1.6 — Maxwell equations, momentum, and energy: 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.
Integrate :math:`\nabla\cdot\mathbf D=\rho`, :math:`\nabla\cdot\mathbf B=0`, :math:`\nabla\times\mathbf E=-\partial_t\mathbf B`, and :math:`\nabla\times\mathbf H=\mathbf J+\partial_t\mathbf D` over the stated volume or contour; then apply the divergence or Stokes theorem before taking a limiting surface.
The result must conserve charge or energy, have matching units on both sides, and reduce to the source-free Maxwell relation when charge and current vanish.

Problem 1.7 — material and group-velocity dispersion: 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.
Start with :math:`k=n(\omega)\omega/c`; then :math:`v_g=(dk/d\omega)^{-1}=c/[n+\omega(dn/d\omega)]`.  Differentiate once more for group-velocity dispersion and convert wavelength derivatives only after the symbolic result is fixed.
Check the nondispersive limit: constant refractive index gives :math:`v_g=c/n` and zero group-delay dispersion.

Problem 1.8 — material and group-velocity dispersion: 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.
Start with :math:`k=n(\omega)\omega/c`; then :math:`v_g=(dk/d\omega)^{-1}=c/[n+\omega(dn/d\omega)]`.  Differentiate once more for group-velocity dispersion and convert wavelength derivatives only after the symbolic result is fixed.
Check the nondispersive limit: constant refractive index gives :math:`v_g=c/n` and zero group-delay dispersion.

Problem 1.9 — Jones vectors, polarization ellipses, and wave plates: 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.
Represent the field by :math:`\mathbf e=(\cos\psi,\,e^{i\delta}\sin\psi)^T`, remove its common phase, and use the Jones matrix of each plate or polarizer.  Obtain ellipse axes from the real-field quadratic form or equivalently from the Stokes parameters.
Verify normalization and :math:`S_0^2\geq S_1^2+S_2^2+S_3^2`; lossless retarders must preserve :math:`S_0`, and orthogonal Jones states must have zero inner product.

Problem 1.10 — Jones vectors, polarization ellipses, and wave plates: 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.
Represent the field by :math:`\mathbf e=(\cos\psi,\,e^{i\delta}\sin\psi)^T`, remove its common phase, and use the Jones matrix of each plate or polarizer.  Obtain ellipse axes from the real-field quadratic form or equivalently from the Stokes parameters.
Verify normalization and :math:`S_0^2\geq S_1^2+S_2^2+S_3^2`; lossless retarders must preserve :math:`S_0`, and orthogonal Jones states must have zero inner product.

Problem 1.11 — Jones vectors, polarization ellipses, and wave plates: 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.
Represent the field by :math:`\mathbf e=(\cos\psi,\,e^{i\delta}\sin\psi)^T`, remove its common phase, and use the Jones matrix of each plate or polarizer.  Obtain ellipse axes from the real-field quadratic form or equivalently from the Stokes parameters.
Verify normalization and :math:`S_0^2\geq S_1^2+S_2^2+S_3^2`; lossless retarders must preserve :math:`S_0`, and orthogonal Jones states must have zero inner product.

Problem 1.12 — Jones vectors, polarization ellipses, and wave plates: calculation
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

Convert the supplied data to one unit system, isolate the requested quantity symbolically, and retain guard digits until the final numerical evaluation.
Represent the field by :math:`\mathbf e=(\cos\psi,\,e^{i\delta}\sin\psi)^T`, remove its common phase, and use the Jones matrix of each plate or polarizer.  Obtain ellipse axes from the real-field quadratic form or equivalently from the Stokes parameters.
Verify normalization and :math:`S_0^2\geq S_1^2+S_2^2+S_3^2`; lossless retarders must preserve :math:`S_0`, and orthogonal Jones states must have zero inner product.

Problem 1.13 — Jones vectors, polarization ellipses, and wave plates: 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.
Represent the field by :math:`\mathbf e=(\cos\psi,\,e^{i\delta}\sin\psi)^T`, remove its common phase, and use the Jones matrix of each plate or polarizer.  Obtain ellipse axes from the real-field quadratic form or equivalently from the Stokes parameters.
Verify normalization and :math:`S_0^2\geq S_1^2+S_2^2+S_3^2`; lossless retarders must preserve :math:`S_0`, and orthogonal Jones states must have zero inner product.

Problem 1.14 — Jones vectors, polarization ellipses, and wave plates: 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.
Represent the field by :math:`\mathbf e=(\cos\psi,\,e^{i\delta}\sin\psi)^T`, remove its common phase, and use the Jones matrix of each plate or polarizer.  Obtain ellipse axes from the real-field quadratic form or equivalently from the Stokes parameters.
Verify normalization and :math:`S_0^2\geq S_1^2+S_2^2+S_3^2`; lossless retarders must preserve :math:`S_0`, and orthogonal Jones states must have zero inner product.

Problem 1.15 — Jones vectors, polarization ellipses, and wave plates: 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.
Represent the field by :math:`\mathbf e=(\cos\psi,\,e^{i\delta}\sin\psi)^T`, remove its common phase, and use the Jones matrix of each plate or polarizer.  Obtain ellipse axes from the real-field quadratic form or equivalently from the Stokes parameters.
Verify normalization and :math:`S_0^2\geq S_1^2+S_2^2+S_3^2`; lossless retarders must preserve :math:`S_0`, and orthogonal Jones states must have zero inner product.

Problem 1.16 — Jones vectors, polarization ellipses, and wave plates: 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.
Represent the field by :math:`\mathbf e=(\cos\psi,\,e^{i\delta}\sin\psi)^T`, remove its common phase, and use the Jones matrix of each plate or polarizer.  Obtain ellipse axes from the real-field quadratic form or equivalently from the Stokes parameters.
Verify normalization and :math:`S_0^2\geq S_1^2+S_2^2+S_3^2`; lossless retarders must preserve :math:`S_0`, and orthogonal Jones states must have zero inner product.

Problem 1.17 — Jones vectors, polarization ellipses, and wave plates: 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.
Represent the field by :math:`\mathbf e=(\cos\psi,\,e^{i\delta}\sin\psi)^T`, remove its common phase, and use the Jones matrix of each plate or polarizer.  Obtain ellipse axes from the real-field quadratic form or equivalently from the Stokes parameters.
Verify normalization and :math:`S_0^2\geq S_1^2+S_2^2+S_3^2`; lossless retarders must preserve :math:`S_0`, and orthogonal Jones states must have zero inner product.

Problem 1.18 — Jones vectors, polarization ellipses, and wave plates: 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.
Represent the field by :math:`\mathbf e=(\cos\psi,\,e^{i\delta}\sin\psi)^T`, remove its common phase, and use the Jones matrix of each plate or polarizer.  Obtain ellipse axes from the real-field quadratic form or equivalently from the Stokes parameters.
Verify normalization and :math:`S_0^2\geq S_1^2+S_2^2+S_3^2`; lossless retarders must preserve :math:`S_0`, and orthogonal Jones states must have zero inner product.

Problem 1.19 — Jones vectors, polarization ellipses, and wave plates: 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.
Represent the field by :math:`\mathbf e=(\cos\psi,\,e^{i\delta}\sin\psi)^T`, remove its common phase, and use the Jones matrix of each plate or polarizer.  Obtain ellipse axes from the real-field quadratic form or equivalently from the Stokes parameters.
Verify normalization and :math:`S_0^2\geq S_1^2+S_2^2+S_3^2`; lossless retarders must preserve :math:`S_0`, and orthogonal Jones states must have zero inner product.

Problem 1.20 — Jones vectors, polarization ellipses, and wave plates: 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.
Represent the field by :math:`\mathbf e=(\cos\psi,\,e^{i\delta}\sin\psi)^T`, remove its common phase, and use the Jones matrix of each plate or polarizer.  Obtain ellipse axes from the real-field quadratic form or equivalently from the Stokes parameters.
Verify normalization and :math:`S_0^2\geq S_1^2+S_2^2+S_3^2`; lossless retarders must preserve :math:`S_0`, and orthogonal Jones states must have zero inner product.

Problem 1.21 — Jones vectors, polarization ellipses, and wave plates: 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.
Represent the field by :math:`\mathbf e=(\cos\psi,\,e^{i\delta}\sin\psi)^T`, remove its common phase, and use the Jones matrix of each plate or polarizer.  Obtain ellipse axes from the real-field quadratic form or equivalently from the Stokes parameters.
Verify normalization and :math:`S_0^2\geq S_1^2+S_2^2+S_3^2`; lossless retarders must preserve :math:`S_0`, and orthogonal Jones states must have zero inner product.

Problem 1.22 — Jones vectors, polarization ellipses, and wave plates: 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.
Represent the field by :math:`\mathbf e=(\cos\psi,\,e^{i\delta}\sin\psi)^T`, remove its common phase, and use the Jones matrix of each plate or polarizer.  Obtain ellipse axes from the real-field quadratic form or equivalently from the Stokes parameters.
Verify normalization and :math:`S_0^2\geq S_1^2+S_2^2+S_3^2`; lossless retarders must preserve :math:`S_0`, and orthogonal Jones states must have zero inner product.

Problem 1.23 — Jones vectors, polarization ellipses, and wave plates: calculation
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

Convert the supplied data to one unit system, isolate the requested quantity symbolically, and retain guard digits until the final numerical evaluation.
Represent the field by :math:`\mathbf e=(\cos\psi,\,e^{i\delta}\sin\psi)^T`, remove its common phase, and use the Jones matrix of each plate or polarizer.  Obtain ellipse axes from the real-field quadratic form or equivalently from the Stokes parameters.
Verify normalization and :math:`S_0^2\geq S_1^2+S_2^2+S_3^2`; lossless retarders must preserve :math:`S_0`, and orthogonal Jones states must have zero inner product.

Problem 1.24 — Jones vectors, polarization ellipses, and wave plates: 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.
Represent the field by :math:`\mathbf e=(\cos\psi,\,e^{i\delta}\sin\psi)^T`, remove its common phase, and use the Jones matrix of each plate or polarizer.  Obtain ellipse axes from the real-field quadratic form or equivalently from the Stokes parameters.
Verify normalization and :math:`S_0^2\geq S_1^2+S_2^2+S_3^2`; lossless retarders must preserve :math:`S_0`, and orthogonal Jones states must have zero inner product.

Problem 1.25 — Jones vectors, polarization ellipses, and wave plates: 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.
Represent the field by :math:`\mathbf e=(\cos\psi,\,e^{i\delta}\sin\psi)^T`, remove its common phase, and use the Jones matrix of each plate or polarizer.  Obtain ellipse axes from the real-field quadratic form or equivalently from the Stokes parameters.
Verify normalization and :math:`S_0^2\geq S_1^2+S_2^2+S_3^2`; lossless retarders must preserve :math:`S_0`, and orthogonal Jones states must have zero inner product.

Problem 1.26 — Jones vectors, polarization ellipses, and wave plates: 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.
Represent the field by :math:`\mathbf e=(\cos\psi,\,e^{i\delta}\sin\psi)^T`, remove its common phase, and use the Jones matrix of each plate or polarizer.  Obtain ellipse axes from the real-field quadratic form or equivalently from the Stokes parameters.
Verify normalization and :math:`S_0^2\geq S_1^2+S_2^2+S_3^2`; lossless retarders must preserve :math:`S_0`, and orthogonal Jones states must have zero inner product.

Problem 1.27 — Jones vectors, polarization ellipses, and wave plates: 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.
Represent the field by :math:`\mathbf e=(\cos\psi,\,e^{i\delta}\sin\psi)^T`, remove its common phase, and use the Jones matrix of each plate or polarizer.  Obtain ellipse axes from the real-field quadratic form or equivalently from the Stokes parameters.
Verify normalization and :math:`S_0^2\geq S_1^2+S_2^2+S_3^2`; lossless retarders must preserve :math:`S_0`, and orthogonal Jones states must have zero inner product.

Problem 1.28 — waves and energy flow in anisotropic 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.
Insert a plane wave into Maxwell's equations to obtain :math:`\mathbf k\times(\mathbf k\times\mathbf E)+(\omega^2/c^2)\boldsymbol\epsilon_r\mathbf E=0`.  Set the determinant to zero for the Fresnel surface, then find energy or group direction from :math:`\nabla_{\mathbf k}\omega`.
Test the isotropic limit, confirm that the displacement is transverse to the wave normal, and verify positive time-averaged energy flow for a passive crystal.

Problem 1.29 — waves and energy flow in anisotropic 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.
Insert a plane wave into Maxwell's equations to obtain :math:`\mathbf k\times(\mathbf k\times\mathbf E)+(\omega^2/c^2)\boldsymbol\epsilon_r\mathbf E=0`.  Set the determinant to zero for the Fresnel surface, then find energy or group direction from :math:`\nabla_{\mathbf k}\omega`.
Test the isotropic limit, confirm that the displacement is transverse to the wave normal, and verify positive time-averaged energy flow for a passive crystal.

Problem 1.30 — waves and energy flow in anisotropic 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.
Insert a plane wave into Maxwell's equations to obtain :math:`\mathbf k\times(\mathbf k\times\mathbf E)+(\omega^2/c^2)\boldsymbol\epsilon_r\mathbf E=0`.  Set the determinant to zero for the Fresnel surface, then find energy or group direction from :math:`\nabla_{\mathbf k}\omega`.
Test the isotropic limit, confirm that the displacement is transverse to the wave normal, and verify positive time-averaged energy flow for a passive crystal.

Problem 1.31 — waves and energy flow in anisotropic 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.
Insert a plane wave into Maxwell's equations to obtain :math:`\mathbf k\times(\mathbf k\times\mathbf E)+(\omega^2/c^2)\boldsymbol\epsilon_r\mathbf E=0`.  Set the determinant to zero for the Fresnel surface, then find energy or group direction from :math:`\nabla_{\mathbf k}\omega`.
Test the isotropic limit, confirm that the displacement is transverse to the wave normal, and verify positive time-averaged energy flow for a passive crystal.

Problem 1.32 — waves and energy flow in anisotropic media: calculation
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

Convert the supplied data to one unit system, isolate the requested quantity symbolically, and retain guard digits until the final numerical evaluation.
Insert a plane wave into Maxwell's equations to obtain :math:`\mathbf k\times(\mathbf k\times\mathbf E)+(\omega^2/c^2)\boldsymbol\epsilon_r\mathbf E=0`.  Set the determinant to zero for the Fresnel surface, then find energy or group direction from :math:`\nabla_{\mathbf k}\omega`.
Test the isotropic limit, confirm that the displacement is transverse to the wave normal, and verify positive time-averaged energy flow for a passive crystal.

Problem 1.33 — waves and energy flow in anisotropic media: calculation
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

Convert the supplied data to one unit system, isolate the requested quantity symbolically, and retain guard digits until the final numerical evaluation.
Insert a plane wave into Maxwell's equations to obtain :math:`\mathbf k\times(\mathbf k\times\mathbf E)+(\omega^2/c^2)\boldsymbol\epsilon_r\mathbf E=0`.  Set the determinant to zero for the Fresnel surface, then find energy or group direction from :math:`\nabla_{\mathbf k}\omega`.
Test the isotropic limit, confirm that the displacement is transverse to the wave normal, and verify positive time-averaged energy flow for a passive crystal.

Problem 1.34 — waves and energy flow in anisotropic 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.
Insert a plane wave into Maxwell's equations to obtain :math:`\mathbf k\times(\mathbf k\times\mathbf E)+(\omega^2/c^2)\boldsymbol\epsilon_r\mathbf E=0`.  Set the determinant to zero for the Fresnel surface, then find energy or group direction from :math:`\nabla_{\mathbf k}\omega`.
Test the isotropic limit, confirm that the displacement is transverse to the wave normal, and verify positive time-averaged energy flow for a passive crystal.

Problem 1.35 — waves and energy flow in anisotropic 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.
Insert a plane wave into Maxwell's equations to obtain :math:`\mathbf k\times(\mathbf k\times\mathbf E)+(\omega^2/c^2)\boldsymbol\epsilon_r\mathbf E=0`.  Set the determinant to zero for the Fresnel surface, then find energy or group direction from :math:`\nabla_{\mathbf k}\omega`.
Test the isotropic limit, confirm that the displacement is transverse to the wave normal, and verify positive time-averaged energy flow for a passive crystal.

Problem 1.36 — waves and energy flow in anisotropic 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.
Insert a plane wave into Maxwell's equations to obtain :math:`\mathbf k\times(\mathbf k\times\mathbf E)+(\omega^2/c^2)\boldsymbol\epsilon_r\mathbf E=0`.  Set the determinant to zero for the Fresnel surface, then find energy or group direction from :math:`\nabla_{\mathbf k}\omega`.
Test the isotropic limit, confirm that the displacement is transverse to the wave normal, and verify positive time-averaged energy flow for a passive crystal.

Problem 1.37 — waves and energy flow in anisotropic 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.
Insert a plane wave into Maxwell's equations to obtain :math:`\mathbf k\times(\mathbf k\times\mathbf E)+(\omega^2/c^2)\boldsymbol\epsilon_r\mathbf E=0`.  Set the determinant to zero for the Fresnel surface, then find energy or group direction from :math:`\nabla_{\mathbf k}\omega`.
Test the isotropic limit, confirm that the displacement is transverse to the wave normal, and verify positive time-averaged energy flow for a passive crystal.