Chapter 5: Electromagnetic Optics
Source: Saleh and Teich, Fundamentals of Photonics, second edition, Chapter 5.
In-text exercise
Exercise 5.5-1 — Dilute absorbing impurities
Brief solution
1. Method. The working uses common differential-equation solutions and algebraic rearrangement and dimensional checks.
Show detailed steps
Step 1 — Definitions and setup. Symbols are local to this item and follow the chapter convention. Each physical quantity and supplied numerical value is introduced at its first use below; angles are in radians unless a degree symbol is shown, and units are retained through numerical substitution.
Figure 47 — Exercise 5.5-1: Dilute absorbing impurities. The diagram identifies the input quantities, physical operation, requested result, variable meanings, and an independent verification route. Every symbol in the variable strip is labeled on the model itself.
Step 2 — Mathematical formulas used. The working uses common differential-equation solutions and algebraic rearrangement and dimensional checks.
Step 3 — Worked derivation. The calculation is kept in symbolic form until the governing relation has been rearranged for the requested quantity.
Detailed step 1. The host has \(\epsilon_r=n_0^2\); adding dilute susceptibility gives \(\epsilon_r=n_0^2+\chi'+j\chi''\).
Detailed step 2. Expanding \(\tilde n=\sqrt{\epsilon_r}\) to first order gives \(n\simeq n_0+\chi'/(2n_0)\) and extinction part \(\kappa\simeq\chi''/(2n_0)\).
Detailed step 3. Therefore \(\boxed{\alpha=2k_0\kappa=k_0\chi''/n_0}\).
End-of-chapter problems
Step 4 — State the numbered result. The principal result obtained in the working is
Step 5 — Check. Equation (1) can be checked by substituting it back into the preceding governing relation and reversing the algebraic steps. Differentiating the proposed solution and substituting it into the original differential equation verifies the functional form.
Problem 5.1-1 — Gaussian electromagnetic pulse
Brief solution
1. Method. The working uses vector-calculus identities, integration identities, and algebraic rearrangement and dimensional checks.
2. Reasoning and answer.
Show detailed steps
Definitions and setup. Symbols are local to this item and follow the chapter convention. Each physical quantity and supplied numerical value is introduced at its first use below; angles are in radians unless a degree symbol is shown, and units are retained through numerical substitution.
Mathematical formulas used. The working uses vector-calculus identities, integration identities, and algebraic rearrangement and dimensional checks.
Worked derivation. The calculation is kept in symbolic form until the governing relation has been rearranged for the requested quantity.
Detailed step 1. This is an \(x\)-polarized Gaussian-envelope carrier traveling in \(+z\) at \(c_0\).
Detailed step 2. Maxwell’s plane-wave relation gives \(\boxed{\mathbf H=\hat{\mathbf y},f(t-z/c_0)/\eta_0}\); the Poynting vector points along \(+z\).
Numbered result. The principal result obtained in the working is
Check. Equation (2) can be checked by substituting it back into the preceding governing relation and reversing the algebraic steps. Differentiation of an antiderivative, or normalization of a definite integral, checks the integration step.
Problem 5.2-1 — Constitutive-law classification
Brief solution
1. Method. The working uses product, quotient, and chain rules and algebraic rearrangement and dimensional checks.
2. Reasoning and answer.
Show detailed steps
Definitions and setup. Symbols are local to this item and follow the chapter convention. Each physical quantity and supplied numerical value is introduced at its first use below; angles are in radians unless a degree symbol is shown, and units are retained through numerical substitution.
Mathematical formulas used. The working uses product, quotient, and chain rules and algebraic rearrangement and dimensional checks.
Worked derivation. The calculation is kept in symbolic form until the governing relation has been rearranged for the requested quantity.
Detailed step 1. (a) The spatial derivative makes the medium linear,
Detailed step 2. homogeneous,
Detailed step 3. temporally nondispersive but spatially dispersive. (b) \(P+aP^2=\epsilon_0\chi E\) is nonlinear,
Detailed step 4. instantaneous,
Detailed step 5. homogeneous,
Detailed step 6. and local. (c) time derivatives make it linear,
Detailed step 7. homogeneous,
Detailed step 8. local,
Detailed step 9. and temporally dispersive. (d) the position-dependent coefficient is linear,
Detailed step 10. instantaneous,
Detailed step 11. local,
Detailed step 12. and inhomogeneous.
Numbered result. The principal result obtained in the working is
Check. Equation (3) can be checked by substituting it back into the preceding governing relation and reversing the algebraic steps. Check that dimensions agree term by term, then test the simplest symmetry or limiting case for the expected sign and scale.
Problem 5.3-1 — Traveling standing wave
Brief solution
1. Method. The working uses vector-calculus identities, trigonometric and small-angle identities, and algebraic rearrangement and dimensional checks.
2. Reasoning and answer.
Show detailed steps
Definitions and setup. Symbols are local to this item and follow the chapter convention. Each physical quantity and supplied numerical value is introduced at its first use below; angles are in radians unless a degree symbol is shown, and units are retained through numerical substitution.
Mathematical formulas used. The working uses vector-calculus identities, trigonometric and small-angle identities, and algebraic rearrangement and dimensional checks.
Worked derivation. The calculation is kept in symbolic form until the governing relation has been rearranged for the requested quantity.
Detailed step 1. The Helmholtz equation requires \(2\beta^2=k_0^2\),
Detailed step 2. so \(\boxed{\beta=k_0/\sqrt2}\).
Detailed step 3. From \(\mathbf H=(j\omega\mu_0)^{-1}\nabla\times\mathbf E\),
Detailed step 4. obtain the \(y\) and \(z\) magnetic components.
Detailed step 5. Expanding \(\sin\beta y\) into exponentials shows two equal TEM waves with \(\mathbf k_\pm=(0,\pm\beta,\beta)\),
Detailed step 6. i.e. at \(\pm45^\circ\) in the \(y-z\) plane.
Detailed step 7. Their transverse power cancels and mean power flows in \(+z\).
Numbered result. The principal result obtained in the working is
Check. Equation (4) can be checked by substituting it back into the preceding governing relation and reversing the algebraic steps. The zero-angle or paraxial limit supplies an independent sign and magnitude check whenever that limit is part of the model. Repeat the substitution with unrounded intermediate values and retain the displayed units; the final unit must have the requested dimension.
Problem 5.4-1 — Focused electric-field strength
Brief solution
1. Method. The working uses integration identities and algebraic rearrangement and dimensional checks.
2. Reasoning and answer.
Show detailed steps
Definitions and setup. Symbols are local to this item and follow the chapter convention. Each physical quantity and supplied numerical value is introduced at its first use below; angles are in radians unless a degree symbol is shown, and units are retained through numerical substitution.
Mathematical formulas used. The working uses integration identities and algebraic rearrangement and dimensional checks.
Worked derivation. The calculation is kept in symbolic form until the governing relation has been rearranged for the requested quantity.
Detailed step 1. For uniform area \(10^{-8}\ \mathrm{m^2}\), \(I=10^8\) \(\mathrm{W,m^{-2}}\) and \(E_0=\sqrt{2\eta_0I}=\boxed{2.75\times10^5\ \mathrm{V/m}}\).
Detailed step 2. For the Gaussian, \(I(0)=2P/(\pi W_0^2)=6.37\times10^7\) \(\mathrm{W,m^{-2}}\),
Detailed step 3. hence \(\boxed{E_0=2.19\times10^5\ \mathrm{V/m}}\).
Numbered result. The principal result obtained in the working is
Check. Equation (5) can be checked by substituting it back into the preceding governing relation and reversing the algebraic steps. Differentiation of an antiderivative, or normalization of a definite integral, checks the integration step. Repeat the substitution with unrounded intermediate values and retain the displayed units; the final unit must have the requested dimension.
Problem 5.5-2 — Modulation in dispersion
Brief solution
1. Method. The working uses exponential, logarithmic, and phasor identities and algebraic rearrangement and dimensional checks.
2. Key step.
Resolve the input into carrier and two sidebands. After distance \(z\),
3. Answer.
The bracket is purely real apart from a common phase when \((\beta_2+\beta_1-2\beta_0)z=2\pi q\); at those distances the wave is again pure AM (with a dispersion-dependent RF phase delay).
Show detailed steps
Definitions and setup. Symbols are local to this item and follow the chapter convention. Each physical quantity and supplied numerical value is introduced at its first use below; angles are in radians unless a degree symbol is shown, and units are retained through numerical substitution.
Mathematical formulas used. The working uses exponential, logarithmic, and phasor identities and algebraic rearrangement and dimensional checks.
Worked derivation. The calculation is kept in symbolic form until the governing relation has been rearranged for the requested quantity.
Resolve the input into carrier and two sidebands. After distance \(z\),
The bracket is purely real apart from a common phase when \((\beta_2+\beta_1-2\beta_0)z=2\pi q\); at those distances the wave is again pure AM (with a dispersion-dependent RF phase delay).
Check. Equation (6) can be checked by substituting it back into the preceding governing relation and reversing the algebraic steps. Check that dimensions agree term by term, then test the simplest symmetry or limiting case for the expected sign and scale.
Problem 5.6-1 — Sellmeier dispersion
Brief solution
1. Method. The working uses product, quotient, and chain rules and algebraic rearrangement and dimensional checks.
2. Reasoning and answer.
Show detailed steps
Definitions and setup. Symbols are local to this item and follow the chapter convention. Each physical quantity and supplied numerical value is introduced at its first use below; angles are in radians unless a degree symbol is shown, and units are retained through numerical substitution.
Mathematical formulas used. The working uses product, quotient, and chain rules and algebraic rearrangement and dimensional checks.
Worked derivation. The calculation is kept in symbolic form until the governing relation has been rearranged for the requested quantity.
Detailed step 1. For \(n^2=1+\sum_i A_i\lambda^2/(\lambda^2-\lambda_i^2)\),
Detailed step 2. differentiate analytically and use \(\boxed{N=n-\lambda,dn/d\lambda}\) and \(\boxed{D_\lambda=-(\lambda/c_0)d^2n/d\lambda^2}\).
Detailed step 3. Evaluating these expressions with the table coefficients reproduces the silica curves and,
Detailed step 4. with the three listed GaAs terms,
Detailed step 5. the GaAs curves.
Detailed step 6. GaAs has much larger index and stronger,
Detailed step 7. resonance-proximate dispersion; silica has a broad low-dispersion telecommunications region.
Numbered result. The principal result obtained in the working is
Check. Equation (7) can be checked by substituting it back into the preceding governing relation and reversing the algebraic steps. Check that dimensions agree term by term, then test the simplest symmetry or limiting case for the expected sign and scale.
Problem 5.6-2 — Air dispersion from three measurements
Brief solution
1. Method. The working uses algebraic rearrangement and dimensional checks.
2. Reasoning and answer.
Show detailed steps
Definitions and setup. Symbols are local to this item and follow the chapter convention. Each physical quantity and supplied numerical value is introduced at its first use below; angles are in radians unless a degree symbol is shown, and units are retained through numerical substitution.
Mathematical formulas used. The working uses algebraic rearrangement and dimensional checks.
Worked derivation. The calculation is kept in symbolic form until the governing relation has been rearranged for the requested quantity.
Detailed step 1. With \(\lambda\) in micrometres,
Detailed step 2. the exact quadratic through the data is \(n-1=-2.0000\times10^{-5}\lambda^2+2.5400\times10^{-5}\lambda +2.59448\times10^{-4}\).
Detailed step 3. Then \(v_g=c_0/[n-\lambda n']\); at 0.76, 0.81,
Detailed step 4. and 0.86 micrometres it is \(2.9971124\), \(2.9971077\),
Detailed step 5. and \(2.9971027\) \(\times10^8\ \mathrm{m/s}\).
Detailed step 6. The constant curvature gives \(D_\lambda=0.1014\), 0.1081,
Detailed step 7. and 0.1147 \(\mathrm{ps/(km,nm)}\),
Detailed step 8. hundreds of times smaller than ordinary silica fibre dispersion.
Numbered result. The principal result obtained in the working is
Check. Equation (8) can be checked by substituting it back into the preceding governing relation and reversing the algebraic steps. Check that dimensions agree term by term, then test the simplest symmetry or limiting case for the expected sign and scale. Repeat the substitution with unrounded intermediate values and retain the displayed units; the final unit must have the requested dimension.
Problem 5.6-3 — Drude phase/group velocities
Brief solution
1. Method. The working uses algebraic rearrangement and dimensional checks.
2. Reasoning and answer.
Show detailed steps
Definitions and setup. Symbols are local to this item and follow the chapter convention. Each physical quantity and supplied numerical value is introduced at its first use below; angles are in radians unless a degree symbol is shown, and units are retained through numerical substitution.
Mathematical formulas used. The working uses algebraic rearrangement and dimensional checks.
Worked derivation. The calculation is kept in symbolic form until the governing relation has been rearranged for the requested quantity.
Detailed step 1. For the lossless Drude law \(k^2=(\omega^2-\omega_p^2)/c_0^2\), \(v_p=\omega/k\) and \(v_g=d\omega/dk=c_0^2k/\omega\).
Detailed step 2. Therefore \(\boxed{v_pv_g=c_0^2}\).
Numbered result. The principal result obtained in the working is
Check. Equation (9) can be checked by substituting it back into the preceding governing relation and reversing the algebraic steps. Check that dimensions agree term by term, then test the simplest symmetry or limiting case for the expected sign and scale.