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
Brief solution
1. Method.
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.
2. Decisive step.
Integrate \(\nabla\cdot\mathbf D=\rho\), \(\nabla\cdot\mathbf B=0\), \(\nabla\times\mathbf E=-\partial_t\mathbf B\), and \(\nabla\times\mathbf H=\mathbf J+\partial_t\mathbf D\) over the stated volume or contour; then apply the divergence or Stokes theorem before taking a limiting surface.
3. Verification.
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.
Show detailed steps
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 \(\nabla\cdot\mathbf D=\rho\), \(\nabla\cdot\mathbf B=0\), \(\nabla\times\mathbf E=-\partial_t\mathbf B\), and \(\nabla\times\mathbf H=\mathbf J+\partial_t\mathbf D\) 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
Brief solution
1. Method.
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.
2. Decisive step.
Integrate \(\nabla\cdot\mathbf D=\rho\), \(\nabla\cdot\mathbf B=0\), \(\nabla\times\mathbf E=-\partial_t\mathbf B\), and \(\nabla\times\mathbf H=\mathbf J+\partial_t\mathbf D\) over the stated volume or contour; then apply the divergence or Stokes theorem before taking a limiting surface.
3. Verification.
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.
Show detailed steps
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 \(\nabla\cdot\mathbf D=\rho\), \(\nabla\cdot\mathbf B=0\), \(\nabla\times\mathbf E=-\partial_t\mathbf B\), and \(\nabla\times\mathbf H=\mathbf J+\partial_t\mathbf D\) 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
Brief solution
1. Method.
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.
2. Decisive step.
Integrate \(\nabla\cdot\mathbf D=\rho\), \(\nabla\cdot\mathbf B=0\), \(\nabla\times\mathbf E=-\partial_t\mathbf B\), and \(\nabla\times\mathbf H=\mathbf J+\partial_t\mathbf D\) over the stated volume or contour; then apply the divergence or Stokes theorem before taking a limiting surface.
3. Verification.
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.
Show detailed steps
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 \(\nabla\cdot\mathbf D=\rho\), \(\nabla\cdot\mathbf B=0\), \(\nabla\times\mathbf E=-\partial_t\mathbf B\), and \(\nabla\times\mathbf H=\mathbf J+\partial_t\mathbf D\) 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
Brief solution
1. Method.
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.
2. Decisive step.
Integrate \(\nabla\cdot\mathbf D=\rho\), \(\nabla\cdot\mathbf B=0\), \(\nabla\times\mathbf E=-\partial_t\mathbf B\), and \(\nabla\times\mathbf H=\mathbf J+\partial_t\mathbf D\) over the stated volume or contour; then apply the divergence or Stokes theorem before taking a limiting surface.
3. Verification.
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.
Show detailed steps
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 \(\nabla\cdot\mathbf D=\rho\), \(\nabla\cdot\mathbf B=0\), \(\nabla\times\mathbf E=-\partial_t\mathbf B\), and \(\nabla\times\mathbf H=\mathbf J+\partial_t\mathbf D\) 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
Brief solution
1. Method.
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.
2. Decisive step.
Integrate \(\nabla\cdot\mathbf D=\rho\), \(\nabla\cdot\mathbf B=0\), \(\nabla\times\mathbf E=-\partial_t\mathbf B\), and \(\nabla\times\mathbf H=\mathbf J+\partial_t\mathbf D\) over the stated volume or contour; then apply the divergence or Stokes theorem before taking a limiting surface.
3. Verification.
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.
Show detailed steps
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 \(\nabla\cdot\mathbf D=\rho\), \(\nabla\cdot\mathbf B=0\), \(\nabla\times\mathbf E=-\partial_t\mathbf B\), and \(\nabla\times\mathbf H=\mathbf J+\partial_t\mathbf D\) 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
Brief solution
1. Method.
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.
2. Decisive step.
Integrate \(\nabla\cdot\mathbf D=\rho\), \(\nabla\cdot\mathbf B=0\), \(\nabla\times\mathbf E=-\partial_t\mathbf B\), and \(\nabla\times\mathbf H=\mathbf J+\partial_t\mathbf D\) over the stated volume or contour; then apply the divergence or Stokes theorem before taking a limiting surface.
3. Verification.
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.
Show detailed steps
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 \(\nabla\cdot\mathbf D=\rho\), \(\nabla\cdot\mathbf B=0\), \(\nabla\times\mathbf E=-\partial_t\mathbf B\), and \(\nabla\times\mathbf H=\mathbf J+\partial_t\mathbf D\) 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
Brief solution
1. Method.
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.
2. Decisive step.
Start with \(k=n(\omega)\omega/c\); then \(v_g=(dk/d\omega)^{-1}=c/[n+\omega(dn/d\omega)]\). Differentiate once more for group-velocity dispersion and convert wavelength derivatives only after the symbolic result is fixed.
3. Verification.
Check the nondispersive limit: constant refractive index gives \(v_g=c/n\) and zero group-delay dispersion.
Show detailed steps
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 \(k=n(\omega)\omega/c\); then \(v_g=(dk/d\omega)^{-1}=c/[n+\omega(dn/d\omega)]\). 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 \(v_g=c/n\) and zero group-delay dispersion.
Problem 1.8 — material and group-velocity dispersion: plot
Brief solution
1. Method.
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.
2. Decisive step.
Start with \(k=n(\omega)\omega/c\); then \(v_g=(dk/d\omega)^{-1}=c/[n+\omega(dn/d\omega)]\). Differentiate once more for group-velocity dispersion and convert wavelength derivatives only after the symbolic result is fixed.
3. Verification.
Check the nondispersive limit: constant refractive index gives \(v_g=c/n\) and zero group-delay dispersion.
Show detailed steps
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 \(k=n(\omega)\omega/c\); then \(v_g=(dk/d\omega)^{-1}=c/[n+\omega(dn/d\omega)]\). 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 \(v_g=c/n\) and zero group-delay dispersion.
Problem 1.9 — Jones vectors, polarization ellipses, and wave plates: derivation
Brief solution
1. Method.
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.
2. Decisive step.
Represent the field by \(\mathbf e=(\cos\psi,\,e^{i\delta}\sin\psi)^T\), 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.
3. Verification.
Verify normalization and \(S_0^2\geq S_1^2+S_2^2+S_3^2\); lossless retarders must preserve \(S_0\), and orthogonal Jones states must have zero inner product.
Show detailed steps
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 \(\mathbf e=(\cos\psi,\,e^{i\delta}\sin\psi)^T\), 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 \(S_0^2\geq S_1^2+S_2^2+S_3^2\); lossless retarders must preserve \(S_0\), and orthogonal Jones states must have zero inner product.
Problem 1.10 — Jones vectors, polarization ellipses, and wave plates: derivation
Brief solution
1. Method.
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.
2. Decisive step.
Represent the field by \(\mathbf e=(\cos\psi,\,e^{i\delta}\sin\psi)^T\), 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.
3. Verification.
Verify normalization and \(S_0^2\geq S_1^2+S_2^2+S_3^2\); lossless retarders must preserve \(S_0\), and orthogonal Jones states must have zero inner product.
Show detailed steps
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 \(\mathbf e=(\cos\psi,\,e^{i\delta}\sin\psi)^T\), 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 \(S_0^2\geq S_1^2+S_2^2+S_3^2\); lossless retarders must preserve \(S_0\), and orthogonal Jones states must have zero inner product.
Problem 1.11 — Jones vectors, polarization ellipses, and wave plates: derivation
Brief solution
1. Method.
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.
2. Decisive step.
Represent the field by \(\mathbf e=(\cos\psi,\,e^{i\delta}\sin\psi)^T\), 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.
3. Verification.
Verify normalization and \(S_0^2\geq S_1^2+S_2^2+S_3^2\); lossless retarders must preserve \(S_0\), and orthogonal Jones states must have zero inner product.
Show detailed steps
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 \(\mathbf e=(\cos\psi,\,e^{i\delta}\sin\psi)^T\), 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 \(S_0^2\geq S_1^2+S_2^2+S_3^2\); lossless retarders must preserve \(S_0\), and orthogonal Jones states must have zero inner product.
Problem 1.12 — Jones vectors, polarization ellipses, and wave plates: calculation
Brief solution
1. Method.
Convert the supplied data to one unit system, isolate the requested quantity symbolically, and retain guard digits until the final numerical evaluation.
2. Decisive step.
Represent the field by \(\mathbf e=(\cos\psi,\,e^{i\delta}\sin\psi)^T\), 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.
3. Verification.
Verify normalization and \(S_0^2\geq S_1^2+S_2^2+S_3^2\); lossless retarders must preserve \(S_0\), and orthogonal Jones states must have zero inner product.
Show detailed steps
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 \(\mathbf e=(\cos\psi,\,e^{i\delta}\sin\psi)^T\), 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 \(S_0^2\geq S_1^2+S_2^2+S_3^2\); lossless retarders must preserve \(S_0\), and orthogonal Jones states must have zero inner product.
Problem 1.13 — Jones vectors, polarization ellipses, and wave plates: derivation
Brief solution
1. Method.
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.
2. Decisive step.
Represent the field by \(\mathbf e=(\cos\psi,\,e^{i\delta}\sin\psi)^T\), 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.
3. Verification.
Verify normalization and \(S_0^2\geq S_1^2+S_2^2+S_3^2\); lossless retarders must preserve \(S_0\), and orthogonal Jones states must have zero inner product.
Show detailed steps
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 \(\mathbf e=(\cos\psi,\,e^{i\delta}\sin\psi)^T\), 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 \(S_0^2\geq S_1^2+S_2^2+S_3^2\); lossless retarders must preserve \(S_0\), and orthogonal Jones states must have zero inner product.
Problem 1.14 — Jones vectors, polarization ellipses, and wave plates: derivation
Brief solution
1. Method.
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.
2. Decisive step.
Represent the field by \(\mathbf e=(\cos\psi,\,e^{i\delta}\sin\psi)^T\), 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.
3. Verification.
Verify normalization and \(S_0^2\geq S_1^2+S_2^2+S_3^2\); lossless retarders must preserve \(S_0\), and orthogonal Jones states must have zero inner product.
Show detailed steps
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 \(\mathbf e=(\cos\psi,\,e^{i\delta}\sin\psi)^T\), 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 \(S_0^2\geq S_1^2+S_2^2+S_3^2\); lossless retarders must preserve \(S_0\), and orthogonal Jones states must have zero inner product.
Problem 1.15 — Jones vectors, polarization ellipses, and wave plates: derivation
Brief solution
1. Method.
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.
2. Decisive step.
Represent the field by \(\mathbf e=(\cos\psi,\,e^{i\delta}\sin\psi)^T\), 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.
3. Verification.
Verify normalization and \(S_0^2\geq S_1^2+S_2^2+S_3^2\); lossless retarders must preserve \(S_0\), and orthogonal Jones states must have zero inner product.
Show detailed steps
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 \(\mathbf e=(\cos\psi,\,e^{i\delta}\sin\psi)^T\), 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 \(S_0^2\geq S_1^2+S_2^2+S_3^2\); lossless retarders must preserve \(S_0\), and orthogonal Jones states must have zero inner product.
Problem 1.16 — Jones vectors, polarization ellipses, and wave plates: derivation
Brief solution
1. Method.
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.
2. Decisive step.
Represent the field by \(\mathbf e=(\cos\psi,\,e^{i\delta}\sin\psi)^T\), 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.
3. Verification.
Verify normalization and \(S_0^2\geq S_1^2+S_2^2+S_3^2\); lossless retarders must preserve \(S_0\), and orthogonal Jones states must have zero inner product.
Show detailed steps
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 \(\mathbf e=(\cos\psi,\,e^{i\delta}\sin\psi)^T\), 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 \(S_0^2\geq S_1^2+S_2^2+S_3^2\); lossless retarders must preserve \(S_0\), and orthogonal Jones states must have zero inner product.
Problem 1.17 — Jones vectors, polarization ellipses, and wave plates: derivation
Brief solution
1. Method.
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.
2. Decisive step.
Represent the field by \(\mathbf e=(\cos\psi,\,e^{i\delta}\sin\psi)^T\), 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.
3. Verification.
Verify normalization and \(S_0^2\geq S_1^2+S_2^2+S_3^2\); lossless retarders must preserve \(S_0\), and orthogonal Jones states must have zero inner product.
Show detailed steps
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 \(\mathbf e=(\cos\psi,\,e^{i\delta}\sin\psi)^T\), 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 \(S_0^2\geq S_1^2+S_2^2+S_3^2\); lossless retarders must preserve \(S_0\), and orthogonal Jones states must have zero inner product.
Problem 1.18 — Jones vectors, polarization ellipses, and wave plates: derivation
Brief solution
1. Method.
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.
2. Decisive step.
Represent the field by \(\mathbf e=(\cos\psi,\,e^{i\delta}\sin\psi)^T\), 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.
3. Verification.
Verify normalization and \(S_0^2\geq S_1^2+S_2^2+S_3^2\); lossless retarders must preserve \(S_0\), and orthogonal Jones states must have zero inner product.
Show detailed steps
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 \(\mathbf e=(\cos\psi,\,e^{i\delta}\sin\psi)^T\), 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 \(S_0^2\geq S_1^2+S_2^2+S_3^2\); lossless retarders must preserve \(S_0\), and orthogonal Jones states must have zero inner product.
Problem 1.19 — Jones vectors, polarization ellipses, and wave plates: derivation
Brief solution
1. Method.
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.
2. Decisive step.
Represent the field by \(\mathbf e=(\cos\psi,\,e^{i\delta}\sin\psi)^T\), 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.
3. Verification.
Verify normalization and \(S_0^2\geq S_1^2+S_2^2+S_3^2\); lossless retarders must preserve \(S_0\), and orthogonal Jones states must have zero inner product.
Show detailed steps
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 \(\mathbf e=(\cos\psi,\,e^{i\delta}\sin\psi)^T\), 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 \(S_0^2\geq S_1^2+S_2^2+S_3^2\); lossless retarders must preserve \(S_0\), and orthogonal Jones states must have zero inner product.
Problem 1.20 — Jones vectors, polarization ellipses, and wave plates: derivation
Brief solution
1. Method.
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.
2. Decisive step.
Represent the field by \(\mathbf e=(\cos\psi,\,e^{i\delta}\sin\psi)^T\), 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.
3. Verification.
Verify normalization and \(S_0^2\geq S_1^2+S_2^2+S_3^2\); lossless retarders must preserve \(S_0\), and orthogonal Jones states must have zero inner product.
Show detailed steps
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 \(\mathbf e=(\cos\psi,\,e^{i\delta}\sin\psi)^T\), 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 \(S_0^2\geq S_1^2+S_2^2+S_3^2\); lossless retarders must preserve \(S_0\), and orthogonal Jones states must have zero inner product.
Problem 1.21 — Jones vectors, polarization ellipses, and wave plates: derivation
Brief solution
1. Method.
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.
2. Decisive step.
Represent the field by \(\mathbf e=(\cos\psi,\,e^{i\delta}\sin\psi)^T\), 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.
3. Verification.
Verify normalization and \(S_0^2\geq S_1^2+S_2^2+S_3^2\); lossless retarders must preserve \(S_0\), and orthogonal Jones states must have zero inner product.
Show detailed steps
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 \(\mathbf e=(\cos\psi,\,e^{i\delta}\sin\psi)^T\), 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 \(S_0^2\geq S_1^2+S_2^2+S_3^2\); lossless retarders must preserve \(S_0\), and orthogonal Jones states must have zero inner product.
Problem 1.22 — Jones vectors, polarization ellipses, and wave plates: derivation
Brief solution
1. Method.
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.
2. Decisive step.
Represent the field by \(\mathbf e=(\cos\psi,\,e^{i\delta}\sin\psi)^T\), 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.
3. Verification.
Verify normalization and \(S_0^2\geq S_1^2+S_2^2+S_3^2\); lossless retarders must preserve \(S_0\), and orthogonal Jones states must have zero inner product.
Show detailed steps
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 \(\mathbf e=(\cos\psi,\,e^{i\delta}\sin\psi)^T\), 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 \(S_0^2\geq S_1^2+S_2^2+S_3^2\); lossless retarders must preserve \(S_0\), and orthogonal Jones states must have zero inner product.
Problem 1.23 — Jones vectors, polarization ellipses, and wave plates: calculation
Brief solution
1. Method.
Convert the supplied data to one unit system, isolate the requested quantity symbolically, and retain guard digits until the final numerical evaluation.
2. Decisive step.
Represent the field by \(\mathbf e=(\cos\psi,\,e^{i\delta}\sin\psi)^T\), 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.
3. Verification.
Verify normalization and \(S_0^2\geq S_1^2+S_2^2+S_3^2\); lossless retarders must preserve \(S_0\), and orthogonal Jones states must have zero inner product.
Show detailed steps
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 \(\mathbf e=(\cos\psi,\,e^{i\delta}\sin\psi)^T\), 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 \(S_0^2\geq S_1^2+S_2^2+S_3^2\); lossless retarders must preserve \(S_0\), and orthogonal Jones states must have zero inner product.
Problem 1.24 — Jones vectors, polarization ellipses, and wave plates: derivation
Brief solution
1. Method.
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.
2. Decisive step.
Represent the field by \(\mathbf e=(\cos\psi,\,e^{i\delta}\sin\psi)^T\), 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.
3. Verification.
Verify normalization and \(S_0^2\geq S_1^2+S_2^2+S_3^2\); lossless retarders must preserve \(S_0\), and orthogonal Jones states must have zero inner product.
Show detailed steps
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 \(\mathbf e=(\cos\psi,\,e^{i\delta}\sin\psi)^T\), 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 \(S_0^2\geq S_1^2+S_2^2+S_3^2\); lossless retarders must preserve \(S_0\), and orthogonal Jones states must have zero inner product.
Problem 1.25 — Jones vectors, polarization ellipses, and wave plates: derivation
Brief solution
1. Method.
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.
2. Decisive step.
Represent the field by \(\mathbf e=(\cos\psi,\,e^{i\delta}\sin\psi)^T\), 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.
3. Verification.
Verify normalization and \(S_0^2\geq S_1^2+S_2^2+S_3^2\); lossless retarders must preserve \(S_0\), and orthogonal Jones states must have zero inner product.
Show detailed steps
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 \(\mathbf e=(\cos\psi,\,e^{i\delta}\sin\psi)^T\), 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 \(S_0^2\geq S_1^2+S_2^2+S_3^2\); lossless retarders must preserve \(S_0\), and orthogonal Jones states must have zero inner product.
Problem 1.26 — Jones vectors, polarization ellipses, and wave plates: derivation
Brief solution
1. Method.
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.
2. Decisive step.
Represent the field by \(\mathbf e=(\cos\psi,\,e^{i\delta}\sin\psi)^T\), 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.
3. Verification.
Verify normalization and \(S_0^2\geq S_1^2+S_2^2+S_3^2\); lossless retarders must preserve \(S_0\), and orthogonal Jones states must have zero inner product.
Show detailed steps
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 \(\mathbf e=(\cos\psi,\,e^{i\delta}\sin\psi)^T\), 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 \(S_0^2\geq S_1^2+S_2^2+S_3^2\); lossless retarders must preserve \(S_0\), and orthogonal Jones states must have zero inner product.
Problem 1.27 — Jones vectors, polarization ellipses, and wave plates: derivation
Brief solution
1. Method.
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.
2. Decisive step.
Represent the field by \(\mathbf e=(\cos\psi,\,e^{i\delta}\sin\psi)^T\), 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.
3. Verification.
Verify normalization and \(S_0^2\geq S_1^2+S_2^2+S_3^2\); lossless retarders must preserve \(S_0\), and orthogonal Jones states must have zero inner product.
Show detailed steps
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 \(\mathbf e=(\cos\psi,\,e^{i\delta}\sin\psi)^T\), 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 \(S_0^2\geq S_1^2+S_2^2+S_3^2\); lossless retarders must preserve \(S_0\), and orthogonal Jones states must have zero inner product.
Problem 1.28 — waves and energy flow in anisotropic media: derivation
Brief solution
1. Method.
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.
2. Decisive step.
Insert a plane wave into Maxwell’s equations to obtain \(\mathbf k\times(\mathbf k\times\mathbf E)+(\omega^2/c^2)\boldsymbol\epsilon_r\mathbf E=0\). Set the determinant to zero for the Fresnel surface, then find energy or group direction from \(\nabla_{\mathbf k}\omega\).
3. Verification.
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.
Show detailed steps
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 \(\mathbf k\times(\mathbf k\times\mathbf E)+(\omega^2/c^2)\boldsymbol\epsilon_r\mathbf E=0\). Set the determinant to zero for the Fresnel surface, then find energy or group direction from \(\nabla_{\mathbf k}\omega\). 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
Brief solution
1. Method.
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.
2. Decisive step.
Insert a plane wave into Maxwell’s equations to obtain \(\mathbf k\times(\mathbf k\times\mathbf E)+(\omega^2/c^2)\boldsymbol\epsilon_r\mathbf E=0\). Set the determinant to zero for the Fresnel surface, then find energy or group direction from \(\nabla_{\mathbf k}\omega\).
3. Verification.
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.
Show detailed steps
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 \(\mathbf k\times(\mathbf k\times\mathbf E)+(\omega^2/c^2)\boldsymbol\epsilon_r\mathbf E=0\). Set the determinant to zero for the Fresnel surface, then find energy or group direction from \(\nabla_{\mathbf k}\omega\). 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
Brief solution
1. Method.
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.
2. Decisive step.
Insert a plane wave into Maxwell’s equations to obtain \(\mathbf k\times(\mathbf k\times\mathbf E)+(\omega^2/c^2)\boldsymbol\epsilon_r\mathbf E=0\). Set the determinant to zero for the Fresnel surface, then find energy or group direction from \(\nabla_{\mathbf k}\omega\).
3. Verification.
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.
Show detailed steps
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 \(\mathbf k\times(\mathbf k\times\mathbf E)+(\omega^2/c^2)\boldsymbol\epsilon_r\mathbf E=0\). Set the determinant to zero for the Fresnel surface, then find energy or group direction from \(\nabla_{\mathbf k}\omega\). 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
Brief solution
1. Method.
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.
2. Decisive step.
Insert a plane wave into Maxwell’s equations to obtain \(\mathbf k\times(\mathbf k\times\mathbf E)+(\omega^2/c^2)\boldsymbol\epsilon_r\mathbf E=0\). Set the determinant to zero for the Fresnel surface, then find energy or group direction from \(\nabla_{\mathbf k}\omega\).
3. Verification.
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.
Show detailed steps
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 \(\mathbf k\times(\mathbf k\times\mathbf E)+(\omega^2/c^2)\boldsymbol\epsilon_r\mathbf E=0\). Set the determinant to zero for the Fresnel surface, then find energy or group direction from \(\nabla_{\mathbf k}\omega\). 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
Brief solution
1. Method.
Convert the supplied data to one unit system, isolate the requested quantity symbolically, and retain guard digits until the final numerical evaluation.
2. Decisive step.
Insert a plane wave into Maxwell’s equations to obtain \(\mathbf k\times(\mathbf k\times\mathbf E)+(\omega^2/c^2)\boldsymbol\epsilon_r\mathbf E=0\). Set the determinant to zero for the Fresnel surface, then find energy or group direction from \(\nabla_{\mathbf k}\omega\).
3. Verification.
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.
Show detailed steps
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 \(\mathbf k\times(\mathbf k\times\mathbf E)+(\omega^2/c^2)\boldsymbol\epsilon_r\mathbf E=0\). Set the determinant to zero for the Fresnel surface, then find energy or group direction from \(\nabla_{\mathbf k}\omega\). 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
Brief solution
1. Method.
Convert the supplied data to one unit system, isolate the requested quantity symbolically, and retain guard digits until the final numerical evaluation.
2. Decisive step.
Insert a plane wave into Maxwell’s equations to obtain \(\mathbf k\times(\mathbf k\times\mathbf E)+(\omega^2/c^2)\boldsymbol\epsilon_r\mathbf E=0\). Set the determinant to zero for the Fresnel surface, then find energy or group direction from \(\nabla_{\mathbf k}\omega\).
3. Verification.
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.
Show detailed steps
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 \(\mathbf k\times(\mathbf k\times\mathbf E)+(\omega^2/c^2)\boldsymbol\epsilon_r\mathbf E=0\). Set the determinant to zero for the Fresnel surface, then find energy or group direction from \(\nabla_{\mathbf k}\omega\). 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
Brief solution
1. Method.
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.
2. Decisive step.
Insert a plane wave into Maxwell’s equations to obtain \(\mathbf k\times(\mathbf k\times\mathbf E)+(\omega^2/c^2)\boldsymbol\epsilon_r\mathbf E=0\). Set the determinant to zero for the Fresnel surface, then find energy or group direction from \(\nabla_{\mathbf k}\omega\).
3. Verification.
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.
Show detailed steps
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 \(\mathbf k\times(\mathbf k\times\mathbf E)+(\omega^2/c^2)\boldsymbol\epsilon_r\mathbf E=0\). Set the determinant to zero for the Fresnel surface, then find energy or group direction from \(\nabla_{\mathbf k}\omega\). 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
Brief solution
1. Method.
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.
2. Decisive step.
Insert a plane wave into Maxwell’s equations to obtain \(\mathbf k\times(\mathbf k\times\mathbf E)+(\omega^2/c^2)\boldsymbol\epsilon_r\mathbf E=0\). Set the determinant to zero for the Fresnel surface, then find energy or group direction from \(\nabla_{\mathbf k}\omega\).
3. Verification.
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.
Show detailed steps
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 \(\mathbf k\times(\mathbf k\times\mathbf E)+(\omega^2/c^2)\boldsymbol\epsilon_r\mathbf E=0\). Set the determinant to zero for the Fresnel surface, then find energy or group direction from \(\nabla_{\mathbf k}\omega\). 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
Brief solution
1. Method.
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.
2. Decisive step.
Insert a plane wave into Maxwell’s equations to obtain \(\mathbf k\times(\mathbf k\times\mathbf E)+(\omega^2/c^2)\boldsymbol\epsilon_r\mathbf E=0\). Set the determinant to zero for the Fresnel surface, then find energy or group direction from \(\nabla_{\mathbf k}\omega\).
3. Verification.
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.
Show detailed steps
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 \(\mathbf k\times(\mathbf k\times\mathbf E)+(\omega^2/c^2)\boldsymbol\epsilon_r\mathbf E=0\). Set the determinant to zero for the Fresnel surface, then find energy or group direction from \(\nabla_{\mathbf k}\omega\). 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
Brief solution
1. Method.
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.
2. Decisive step.
Insert a plane wave into Maxwell’s equations to obtain \(\mathbf k\times(\mathbf k\times\mathbf E)+(\omega^2/c^2)\boldsymbol\epsilon_r\mathbf E=0\). Set the determinant to zero for the Fresnel surface, then find energy or group direction from \(\nabla_{\mathbf k}\omega\).
3. Verification.
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.
Show detailed steps
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 \(\mathbf k\times(\mathbf k\times\mathbf E)+(\omega^2/c^2)\boldsymbol\epsilon_r\mathbf E=0\). Set the determinant to zero for the Fresnel surface, then find energy or group direction from \(\nabla_{\mathbf k}\omega\). 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.