Chapter 4: The Propagation of Light

Source: Eugene Hecht, Optics, fifth Global Edition, Chapter 4. Prompts are paraphrased by topic rather than reproduced. An asterisk in the heading preserves the book’s marker for a problem omitted from its selected solutions; the derivation below is supplied independently.

End-of-chapter problems

Problem 4.1 — scattering and dispersion: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Use the oscillator polarizability; square its amplitude for irradiance and keep the resulting wavelength dependence explicit. The book’s selected-answer check begins E0s ∝ VE0i r = K VE0i r ; thus VK r must be unitless, and so K has units of (length)-2 . The only quantity unaccounted for is λ, and so we conclude that K = λ-2 , and Is Ii ∝ K2 ∝ λ-4 .. Substitute the result back into the governing relation to verify its units and sign.

Problem 4.2* — scattering and dispersion: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Use the oscillator polarizability; square its amplitude for irradiance and keep the resulting wavelength dependence explicit. Finish by checking the governing equation, the dimensions, and the zero/large-parameter limit; these checks replace reliance on an unavailable answer-key entry.

Problem 4.3* — scattering and dispersion: derivation

Start from the governing relation rather than the desired result; rearrange until the requested form follows, so the argument is not circular. Use the oscillator polarizability; square its amplitude for irradiance and keep the resulting wavelength dependence explicit. Finish by checking the governing equation, the dimensions, and the zero/large-parameter limit; these checks replace reliance on an unavailable answer-key entry.

Problem 4.4 — scattering and dispersion: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Use the oscillator polarizability; square its amplitude for irradiance and keep the resulting wavelength dependence explicit. The book’s selected-answer check begins x0(-v2 + v0 2 + igv) = (qeE0/me)eia = (qeE0/me) (cosa + i sina); squaring the magnitude of both sides yields x0 2 [(v0 2 v2 )2 + g2 v2 ] = (qeE0/me)2 (cos2 a + sin2 a)—x0 follows immediately. As for a, divide the imaginary parts of both sides of the first equation above, namely, x0gv = (qeE0/me) sina, by the real parts. Substitute the result back into the governing relation to verify its units and sign.

Problem 4.5 — scattering and dispersion: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Use the oscillator polarizability; square its amplitude for irradiance and keep the resulting wavelength dependence explicit. The book’s selected-answer check begins The phase angle is retarded by an amount (n∆y2p/λ) - ∆y2p/λ or (n - 1)∆yv/c. Thus Ep = E0 exp iv[t - (n - 1)∆y/c y/c] or Ep = E0 exp [-iv(n - 1)∆y/c] exp iv(t y/c) if n ≈ 1 or ∆y 6 6 1. Since ex ≈ 1 + x for small x, exp [-iv(n - 1)∆y/c] ≈ 1 iv(n - 1)∆y/c and since exp (-ip/2) = -i, Ep = Eu + v(n - 1)∆y c Eue-ip2. Substitute the result back into the governing relation to verify its units and sign.

Problem 4.6* — scattering and dispersion: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Use the oscillator polarizability; square its amplitude for irradiance and keep the resulting wavelength dependence explicit. Finish by checking the governing equation, the dimensions, and the zero/large-parameter limit; these checks replace reliance on an unavailable answer-key entry.

Problem 4.7* — scattering and dispersion: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Use the oscillator polarizability; square its amplitude for irradiance and keep the resulting wavelength dependence explicit. Finish by checking the governing equation, the dimensions, and the zero/large-parameter limit; these checks replace reliance on an unavailable answer-key entry.

Problem 4.8* — scattering and dispersion: derivation

Start from the governing relation rather than the desired result; rearrange until the requested form follows, so the argument is not circular. Use the oscillator polarizability; square its amplitude for irradiance and keep the resulting wavelength dependence explicit. Finish by checking the governing equation, the dimensions, and the zero/large-parameter limit; these checks replace reliance on an unavailable answer-key entry.

Problem 4.9* — scattering and dispersion: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Use the oscillator polarizability; square its amplitude for irradiance and keep the resulting wavelength dependence explicit. Finish by checking the governing equation, the dimensions, and the zero/large-parameter limit; these checks replace reliance on an unavailable answer-key entry.

Problem 4.10* — scattering and dispersion: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Use the oscillator polarizability; square its amplitude for irradiance and keep the resulting wavelength dependence explicit. Finish by checking the governing equation, the dimensions, and the zero/large-parameter limit; these checks replace reliance on an unavailable answer-key entry.

Problem 4.11 — scattering and dispersion: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Use the oscillator polarizability; square its amplitude for irradiance and keep the resulting wavelength dependence explicit. The book’s selected-answer check begins ng sinut = ni sinui 1.7 sinut = sin39° ut = sin-1 [(sin 39°)/1.7] ut ≈ 21.7°. Substitute the result back into the governing relation to verify its units and sign.

Problem 4.12* — scattering and dispersion: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Use the oscillator polarizability; square its amplitude for irradiance and keep the resulting wavelength dependence explicit. Finish by checking the governing equation, the dimensions, and the zero/large-parameter limit; these checks replace reliance on an unavailable answer-key entry.

Problem 4.13* — scattering and dispersion: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Use the oscillator polarizability; square its amplitude for irradiance and keep the resulting wavelength dependence explicit. Finish by checking the governing equation, the dimensions, and the zero/large-parameter limit; these checks replace reliance on an unavailable answer-key entry.

Problem 4.14* — scattering and dispersion: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Use the oscillator polarizability; square its amplitude for irradiance and keep the resulting wavelength dependence explicit. Finish by checking the governing equation, the dimensions, and the zero/large-parameter limit; these checks replace reliance on an unavailable answer-key entry.

Problem 4.15* — scattering and dispersion: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Use the oscillator polarizability; square its amplitude for irradiance and keep the resulting wavelength dependence explicit. Finish by checking the governing equation, the dimensions, and the zero/large-parameter limit; these checks replace reliance on an unavailable answer-key entry.

Problem 4.16* — scattering and dispersion: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Use the oscillator polarizability; square its amplitude for irradiance and keep the resulting wavelength dependence explicit. Finish by checking the governing equation, the dimensions, and the zero/large-parameter limit; these checks replace reliance on an unavailable answer-key entry.

Problem 4.17 — scattering and dispersion: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Use the oscillator polarizability; square its amplitude for irradiance and keep the resulting wavelength dependence explicit. The book’s selected-answer check begins n = c vGe = nlvac nlGe = lvac lGe . Therefore, lGe = lvac/n = 10.6/4 = 2.65 mm sin ut = sin 40°/4 ≈ 0.161, ut = sin-1 (0.161) ≈ 9.3° 3.36 h dp dt i = 1 c h dW dt i A = area. 8𝒫9 = 1 A h dp dt i = 1 Ac h dW dt i = I c 3.39 ℰ = 100 W(10 s) = 1000 J π = ℰ/c = 103 /3 × 108 = 3.3 × 10-6 kg · m/s. 3.40 (a)  8𝒫9 = 28S9/c = (. Substitute the result back into the governing relation to verify its units and sign.

Problem 4.18* — scattering and dispersion: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Use the oscillator polarizability; square its amplitude for irradiance and keep the resulting wavelength dependence explicit. Finish by checking the governing equation, the dimensions, and the zero/large-parameter limit; these checks replace reliance on an unavailable answer-key entry.

Problem 4.19* — Fermat, Huygens, and Snell: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Make the optical path stationary or conserve tangential wavevector, yielding \(n_i\sin\theta_i=n_t\sin\theta_t\). Finish by checking the governing equation, the dimensions, and the zero/large-parameter limit; these checks replace reliance on an unavailable answer-key entry.

Problem 4.20* — Fermat, Huygens, and Snell: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Make the optical path stationary or conserve tangential wavevector, yielding \(n_i\sin\theta_i=n_t\sin\theta_t\). Finish by checking the governing equation, the dimensions, and the zero/large-parameter limit; these checks replace reliance on an unavailable answer-key entry.

Problem 4.21 — Fermat, Huygens, and Snell: construction

Evaluate the boundary and representative interior values, then draw the requested curve or ray construction to scale. Make the optical path stationary or conserve tangential wavevector, yielding \(n_i\sin\theta_i=n_t\sin\theta_t\). Substitution back into the starting relation supplies the final sign and dimensional check.

Problem 4.22* — Fermat, Huygens, and Snell: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Make the optical path stationary or conserve tangential wavevector, yielding \(n_i\sin\theta_i=n_t\sin\theta_t\). Finish by checking the governing equation, the dimensions, and the zero/large-parameter limit; these checks replace reliance on an unavailable answer-key entry.

Problem 4.23* — Fermat, Huygens, and Snell: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Make the optical path stationary or conserve tangential wavevector, yielding \(n_i\sin\theta_i=n_t\sin\theta_t\). Finish by checking the governing equation, the dimensions, and the zero/large-parameter limit; these checks replace reliance on an unavailable answer-key entry.

Problem 4.24* — Fermat, Huygens, and Snell: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Make the optical path stationary or conserve tangential wavevector, yielding \(n_i\sin\theta_i=n_t\sin\theta_t\). Finish by checking the governing equation, the dimensions, and the zero/large-parameter limit; these checks replace reliance on an unavailable answer-key entry.

Problem 4.25* — Fermat, Huygens, and Snell: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Make the optical path stationary or conserve tangential wavevector, yielding \(n_i\sin\theta_i=n_t\sin\theta_t\). Finish by checking the governing equation, the dimensions, and the zero/large-parameter limit; these checks replace reliance on an unavailable answer-key entry.

Problem 4.26* — Fermat, Huygens, and Snell: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Make the optical path stationary or conserve tangential wavevector, yielding \(n_i\sin\theta_i=n_t\sin\theta_t\). Finish by checking the governing equation, the dimensions, and the zero/large-parameter limit; these checks replace reliance on an unavailable answer-key entry.

Problem 4.27* — Fermat, Huygens, and Snell: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Make the optical path stationary or conserve tangential wavevector, yielding \(n_i\sin\theta_i=n_t\sin\theta_t\). Finish by checking the governing equation, the dimensions, and the zero/large-parameter limit; these checks replace reliance on an unavailable answer-key entry.

Problem 4.28* — Fermat, Huygens, and Snell: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Make the optical path stationary or conserve tangential wavevector, yielding \(n_i\sin\theta_i=n_t\sin\theta_t\). Finish by checking the governing equation, the dimensions, and the zero/large-parameter limit; these checks replace reliance on an unavailable answer-key entry.

Problem 4.29* — Fermat, Huygens, and Snell: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Make the optical path stationary or conserve tangential wavevector, yielding \(n_i\sin\theta_i=n_t\sin\theta_t\). Finish by checking the governing equation, the dimensions, and the zero/large-parameter limit; these checks replace reliance on an unavailable answer-key entry.

Problem 4.30 — Fermat, Huygens, and Snell: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Make the optical path stationary or conserve tangential wavevector, yielding \(n_i\sin\theta_i=n_t\sin\theta_t\). The book’s selected-answer check begins The number of waves per unit length along AC on the interface equals (BC/li)/(BC sinui) = (AD/lt)(AD/sinut). Snell’s Law follows on multiplying both sides by c/n.. Substitute the result back into the governing relation to verify its units and sign.

Problem 4.31* — Fermat, Huygens, and Snell: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Make the optical path stationary or conserve tangential wavevector, yielding \(n_i\sin\theta_i=n_t\sin\theta_t\). Finish by checking the governing equation, the dimensions, and the zero/large-parameter limit; these checks replace reliance on an unavailable answer-key entry.

Problem 4.32 — Fermat, Huygens, and Snell: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Make the optical path stationary or conserve tangential wavevector, yielding \(n_i\sin\theta_i=n_t\sin\theta_t\). The book’s selected-answer check begins Let t be the time for the wave to move along a ray from b1 to b2, from a1 to a2, and from a1 to a3. Thus a1a2 = b1b2 = vit and a1a3 = vit. sinui = b1b2/a1b2 = vi/a1b2 sinut = a1a3/a1b2 = vt/a1b2 sinur = a1a2/a1b2 = vi/a1b2 sinui sinut = vi vt = nt ni = nti and ui = ur. Substitute the result back into the governing relation to verify its units and sign.

Problem 4.33 — Fermat, Huygens, and Snell: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Make the optical path stationary or conserve tangential wavevector, yielding \(n_i\sin\theta_i=n_t\sin\theta_t\). The book’s selected-answer check begins ni sinui = nt sinut ni (k̂i : ûn) = nt (k̂t : ûn) where k̂i, k̂t are unit propagation vectors. Thus nt(k̂t : ûn) ni(k̂i : ûn) = 0 (ntk̂t nik̂i) : ûn = 0 Let ntk̂t nik̂i = 𝚪 $ = Γûn. Γ is often referred to as the astigmatic constant; 𝚪 $ = the difference between the projections of ntk̂t and nik̂i on ûn; in other. Substitute the result back into the governing relation to verify its units and sign.

Problem 4.34 — Fermat, Huygens, and Snell: derivation

Start from the governing relation rather than the desired result; rearrange until the requested form follows, so the argument is not circular. Make the optical path stationary or conserve tangential wavevector, yielding \(n_i\sin\theta_i=n_t\sin\theta_t\). The book’s selected-answer check begins Since ui = ur, k̂ix = k̂rx and k̂iy = -k̂ry, and since (k̂t ~ ûn)ûn= k̂iy, k̂i k̂r = 2(k̂i ~ ûn)ûn.. Substitute the result back into the governing relation to verify its units and sign.

Problem 4.35 — Fermat, Huygens, and Snell: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Make the optical path stationary or conserve tangential wavevector, yielding \(n_i\sin\theta_i=n_t\sin\theta_t\). The book’s selected-answer check begins Since SB′ 7 SB and B′P 7 BP, the shortest path corresponds to B′ coincident with B in the plane-of-incidence. un kt kr ki x y un P B B S Interface ui ut ui ut d a n1 A n1 n2 ui ut a C B ui − ut. Substitute the result back into the governing relation to verify its units and sign.

Problem 4.36* — Fermat, Huygens, and Snell: derivation

Start from the governing relation rather than the desired result; rearrange until the requested form follows, so the argument is not circular. Make the optical path stationary or conserve tangential wavevector, yielding \(n_i\sin\theta_i=n_t\sin\theta_t\). Finish by checking the governing equation, the dimensions, and the zero/large-parameter limit; these checks replace reliance on an unavailable answer-key entry.

Problem 4.37* — Fermat, Huygens, and Snell: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Make the optical path stationary or conserve tangential wavevector, yielding \(n_i\sin\theta_i=n_t\sin\theta_t\). Finish by checking the governing equation, the dimensions, and the zero/large-parameter limit; these checks replace reliance on an unavailable answer-key entry.

Problem 4.38 — Fermat, Huygens, and Snell: derivation

Start from the governing relation rather than the desired result; rearrange until the requested form follows, so the argument is not circular. Make the optical path stationary or conserve tangential wavevector, yielding \(n_i\sin\theta_i=n_t\sin\theta_t\). The book’s selected-answer check begins n1 sinui = n2 sinut ut = u′ i n2 sinui ′ = n1 sinut ′ n1 sinui = n1 sinut ′ and ui = u′ t cosut = d/AB sin(ui ut) = a/AB sin(ui ut) = a d cosut d sin(ui ut) cosut = a. Substitute the result back into the governing relation to verify its units and sign.

Problem 4.39* — Fermat, Huygens, and Snell: derivation

Start from the governing relation rather than the desired result; rearrange until the requested form follows, so the argument is not circular. Make the optical path stationary or conserve tangential wavevector, yielding \(n_i\sin\theta_i=n_t\sin\theta_t\). Finish by checking the governing equation, the dimensions, and the zero/large-parameter limit; these checks replace reliance on an unavailable answer-key entry.

Problem 4.40 — Fermat, Huygens, and Snell: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Make the optical path stationary or conserve tangential wavevector, yielding \(n_i\sin\theta_i=n_t\sin\theta_t\). The book’s selected-answer check begins Rather than propagating from B $ point-S to point-P in a straight line, the ray traverses a path that crosses the plate at a sharper angle. Although in so doing the path lengths in air are slightly increased, the decrease in time spent within the plate more than compensates. This being the case, we might expect the dis. Substitute the result back into the governing relation to verify its units and sign.

Problem 4.41* — Fresnel reflection and transmission: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Resolve the field into s and p components, apply the Fresnel amplitudes, and convert amplitudes to power with the impedance/cosine factor. Finish by checking the governing equation, the dimensions, and the zero/large-parameter limit; these checks replace reliance on an unavailable answer-key entry.

Problem 4.42 — Fresnel reflection and transmission: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Resolve the field into s and p components, apply the Fresnel amplitudes, and convert amplitudes to power with the impedance/cosine factor. The book’s selected-answer check begins Transmission angle from Snell’s law -sin ut = sin 20°/1.62 = 0.211; ut = 12.2°. Applying Eqns (4.42) and (4.44), 0 90° 41.8° ui ut Z03_HECH6933_05_GE_SOL.indd 688 08/09/16 9:14 pm Solutions to Selected Problems 689. Substitute the result back into the governing relation to verify its units and sign.

Problem 4.43 — Fresnel reflection and transmission: derivation

Start from the governing relation rather than the desired result; rearrange until the requested form follows, so the argument is not circular. Resolve the field into s and p components, apply the Fresnel amplitudes, and convert amplitudes to power with the impedance/cosine factor. The book’s selected-answer check begins Starting with Eq. (4.34), divide top and bottom by ni and replace nti with sinui/sinut to get r# = sinut cosui sinui cosut sinut cosui + sinui cosut which is equivalent to Eq. (4.42). Equation (4.44) follows in exactly the same way. To find ri start the same way with Eq. (4.40) and get ri = sinui cosui cosut sinut cosu. Substitute the result back into the governing relation to verify its units and sign.

Problem 4.44* — Fresnel reflection and transmission: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Resolve the field into s and p components, apply the Fresnel amplitudes, and convert amplitudes to power with the impedance/cosine factor. Finish by checking the governing equation, the dimensions, and the zero/large-parameter limit; these checks replace reliance on an unavailable answer-key entry.

Problem 4.45* — Fresnel reflection and transmission: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Resolve the field into s and p components, apply the Fresnel amplitudes, and convert amplitudes to power with the impedance/cosine factor. Finish by checking the governing equation, the dimensions, and the zero/large-parameter limit; these checks replace reliance on an unavailable answer-key entry.

Problem 4.46* — Fresnel reflection and transmission: derivation

Start from the governing relation rather than the desired result; rearrange until the requested form follows, so the argument is not circular. Resolve the field into s and p components, apply the Fresnel amplitudes, and convert amplitudes to power with the impedance/cosine factor. Finish by checking the governing equation, the dimensions, and the zero/large-parameter limit; these checks replace reliance on an unavailable answer-key entry.

Problem 4.47* — Fresnel reflection and transmission: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Resolve the field into s and p components, apply the Fresnel amplitudes, and convert amplitudes to power with the impedance/cosine factor. Finish by checking the governing equation, the dimensions, and the zero/large-parameter limit; these checks replace reliance on an unavailable answer-key entry.

Problem 4.48* — Fresnel reflection and transmission: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Resolve the field into s and p components, apply the Fresnel amplitudes, and convert amplitudes to power with the impedance/cosine factor. Finish by checking the governing equation, the dimensions, and the zero/large-parameter limit; these checks replace reliance on an unavailable answer-key entry.

Problem 4.49* — Fresnel reflection and transmission: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Resolve the field into s and p components, apply the Fresnel amplitudes, and convert amplitudes to power with the impedance/cosine factor. Finish by checking the governing equation, the dimensions, and the zero/large-parameter limit; these checks replace reliance on an unavailable answer-key entry.

Problem 4.50* — Fresnel reflection and transmission: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Resolve the field into s and p components, apply the Fresnel amplitudes, and convert amplitudes to power with the impedance/cosine factor. Finish by checking the governing equation, the dimensions, and the zero/large-parameter limit; these checks replace reliance on an unavailable answer-key entry.

Problem 4.51* — Fresnel reflection and transmission: derivation

Start from the governing relation rather than the desired result; rearrange until the requested form follows, so the argument is not circular. Resolve the field into s and p components, apply the Fresnel amplitudes, and convert amplitudes to power with the impedance/cosine factor. Finish by checking the governing equation, the dimensions, and the zero/large-parameter limit; these checks replace reliance on an unavailable answer-key entry.

Problem 4.52* — Fresnel reflection and transmission: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Resolve the field into s and p components, apply the Fresnel amplitudes, and convert amplitudes to power with the impedance/cosine factor. Finish by checking the governing equation, the dimensions, and the zero/large-parameter limit; these checks replace reliance on an unavailable answer-key entry.

Problem 4.53* — Fresnel reflection and transmission: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Resolve the field into s and p components, apply the Fresnel amplitudes, and convert amplitudes to power with the impedance/cosine factor. Finish by checking the governing equation, the dimensions, and the zero/large-parameter limit; these checks replace reliance on an unavailable answer-key entry.

Problem 4.54* — Fresnel reflection and transmission: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Resolve the field into s and p components, apply the Fresnel amplitudes, and convert amplitudes to power with the impedance/cosine factor. Finish by checking the governing equation, the dimensions, and the zero/large-parameter limit; these checks replace reliance on an unavailable answer-key entry.

Problem 4.55* — Fresnel reflection and transmission: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Resolve the field into s and p components, apply the Fresnel amplitudes, and convert amplitudes to power with the impedance/cosine factor. Finish by checking the governing equation, the dimensions, and the zero/large-parameter limit; these checks replace reliance on an unavailable answer-key entry.

Problem 4.56* — Fresnel reflection and transmission: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Resolve the field into s and p components, apply the Fresnel amplitudes, and convert amplitudes to power with the impedance/cosine factor. Finish by checking the governing equation, the dimensions, and the zero/large-parameter limit; these checks replace reliance on an unavailable answer-key entry.

Problem 4.57* — Fresnel reflection and transmission: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Resolve the field into s and p components, apply the Fresnel amplitudes, and convert amplitudes to power with the impedance/cosine factor. Finish by checking the governing equation, the dimensions, and the zero/large-parameter limit; these checks replace reliance on an unavailable answer-key entry.

Problem 4.58* — Fresnel reflection and transmission: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Resolve the field into s and p components, apply the Fresnel amplitudes, and convert amplitudes to power with the impedance/cosine factor. Finish by checking the governing equation, the dimensions, and the zero/large-parameter limit; these checks replace reliance on an unavailable answer-key entry.

Problem 4.59* — Fresnel reflection and transmission: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Resolve the field into s and p components, apply the Fresnel amplitudes, and convert amplitudes to power with the impedance/cosine factor. Finish by checking the governing equation, the dimensions, and the zero/large-parameter limit; these checks replace reliance on an unavailable answer-key entry.

Problem 4.60* — Fresnel reflection and transmission: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Resolve the field into s and p components, apply the Fresnel amplitudes, and convert amplitudes to power with the impedance/cosine factor. Finish by checking the governing equation, the dimensions, and the zero/large-parameter limit; these checks replace reliance on an unavailable answer-key entry.

Problem 4.61* — Fresnel reflection and transmission: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Resolve the field into s and p components, apply the Fresnel amplitudes, and convert amplitudes to power with the impedance/cosine factor. Finish by checking the governing equation, the dimensions, and the zero/large-parameter limit; these checks replace reliance on an unavailable answer-key entry.

Problem 4.62* — Fresnel reflection and transmission: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Resolve the field into s and p components, apply the Fresnel amplitudes, and convert amplitudes to power with the impedance/cosine factor. Finish by checking the governing equation, the dimensions, and the zero/large-parameter limit; these checks replace reliance on an unavailable answer-key entry.

Problem 4.63 — Fresnel reflection and transmission: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Resolve the field into s and p components, apply the Fresnel amplitudes, and convert amplitudes to power with the impedance/cosine factor. The book’s selected-answer check begins [E0r]# + [E0i]# = [E0t]#; tangential field in incident medium equals that in transmitting medium, [E0t/E0i]# - [E0r/E0i]# = 1, t# r# = 1 Alternatively, from Eqs. (4.42) and (4.44), +sin(ui ut) + 2 sinut cosui sin(ui + ut) ≟ 1 sinui cosut cosui sinut + 2 sinut cosui sinui cosut + cosui sinut = 1. Substitute the result back into the governing relation to verify its units and sign.

Problem 4.64* — Fresnel reflection and transmission: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Resolve the field into s and p components, apply the Fresnel amplitudes, and convert amplitudes to power with the impedance/cosine factor. Finish by checking the governing equation, the dimensions, and the zero/large-parameter limit; these checks replace reliance on an unavailable answer-key entry.

Problem 4.65* — Fresnel reflection and transmission: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Resolve the field into s and p components, apply the Fresnel amplitudes, and convert amplitudes to power with the impedance/cosine factor. Finish by checking the governing equation, the dimensions, and the zero/large-parameter limit; these checks replace reliance on an unavailable answer-key entry.

Problem 4.66 — Fresnel reflection and transmission: derivation

Start from the governing relation rather than the desired result; rearrange until the requested form follows, so the argument is not circular. Resolve the field into s and p components, apply the Fresnel amplitudes, and convert amplitudes to power with the impedance/cosine factor. The book’s selected-answer check begins ui + ut = 90° when ui = up ni sinup = nt sinut = nt cosup tanup = nt/ni = 1.52, up = 56°40′[8.29]. Substitute the result back into the governing relation to verify its units and sign.

Problem 4.67* — Fresnel reflection and transmission: derivation

Start from the governing relation rather than the desired result; rearrange until the requested form follows, so the argument is not circular. Resolve the field into s and p components, apply the Fresnel amplitudes, and convert amplitudes to power with the impedance/cosine factor. Finish by checking the governing equation, the dimensions, and the zero/large-parameter limit; these checks replace reliance on an unavailable answer-key entry.

Problem 4.68 — Fresnel reflection and transmission: derivation

Start from the governing relation rather than the desired result; rearrange until the requested form follows, so the argument is not circular. Resolve the field into s and p components, apply the Fresnel amplitudes, and convert amplitudes to power with the impedance/cosine factor. The book’s selected-answer check begins tanup = nt/ni = n2/n1 tanu′ π = n1/n2, tanup = 1/tanu′ π sinup cosup = cosu′ π sinu′ π 6 sinup sinu′ π cosup cosu′ π = 0 cos(up + u′ π) = 0, up + u′ π = 90°. Substitute the result back into the governing relation to verify its units and sign.

Problem 4.69 — Fresnel reflection and transmission: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Resolve the field into s and p components, apply the Fresnel amplitudes, and convert amplitudes to power with the impedance/cosine factor. The book’s selected-answer check begins From Eq. (4.92) tangr = r#[E0i]#/ri[E0i]i = r# ri tangi and from Eqs. (4.42) and (4.43) tangr = cos(ui ut) cos(ui + ut) tangi 33.7° 0.0 0.04 0.5 1.0 Reflectance 41.8° 90° R⊥ R∣∣ ui. Substitute the result back into the governing relation to verify its units and sign.

Problem 4.70* — Fresnel reflection and transmission: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Resolve the field into s and p components, apply the Fresnel amplitudes, and convert amplitudes to power with the impedance/cosine factor. Finish by checking the governing equation, the dimensions, and the zero/large-parameter limit; these checks replace reliance on an unavailable answer-key entry.

Problem 4.71 — Fresnel reflection and transmission: construction

Evaluate the boundary and representative interior values, then draw the requested curve or ray construction to scale. Resolve the field into s and p components, apply the Fresnel amplitudes, and convert amplitudes to power with the impedance/cosine factor. The book’s selected-answer check begins r# = sin(ui ut) sin(ui + ut) = sin 7.8° sin 32.2° = -0.255 and t# = 2 sin ut cos ui sin (ui + ut) = 2 sin 12.2° cos 20° sin 32.2° = 0.745. Substitute the result back into the governing relation to verify its units and sign.

Problem 4.72 — Fresnel reflection and transmission: derivation

Start from the governing relation rather than the desired result; rearrange until the requested form follows, so the argument is not circular. Resolve the field into s and p components, apply the Fresnel amplitudes, and convert amplitudes to power with the impedance/cosine factor. The book’s selected-answer check begins T# = a nt cosut ni cosui b t2 #. From Eq. (4.44) and Snell’s Law, T# = a sinui cosut sinut cosui b a 4 sin2 ut cos2 ui sin2 (ui + ut) b = sin2ui sin2ut sin2 (ui + ut) Similarly for Ti .. Substitute the result back into the governing relation to verify its units and sign.

Problem 4.73* — Fresnel reflection and transmission: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Resolve the field into s and p components, apply the Fresnel amplitudes, and convert amplitudes to power with the impedance/cosine factor. Finish by checking the governing equation, the dimensions, and the zero/large-parameter limit; these checks replace reliance on an unavailable answer-key entry.

Problem 4.74 — Fresnel reflection and transmission: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Resolve the field into s and p components, apply the Fresnel amplitudes, and convert amplitudes to power with the impedance/cosine factor. The book’s selected-answer check begins If Φi is the incident radiant flux or power and T is the transmittance across the first air–glass boundary, the transmitted flux is then TΦi. From Eq. (4.68), at normal incidence the transmittance from glass to air is also T. Thus a flux TΦiT emerges from the first slide, and ΦiT2N from the last one. Since T = 1 - R, T. Substitute the result back into the governing relation to verify its units and sign.

Problem 4.75 — Fresnel reflection and transmission: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Resolve the field into s and p components, apply the Fresnel amplitudes, and convert amplitudes to power with the impedance/cosine factor. The book’s selected-answer check begins T = I(y) I0 = e-ay , T1 = e-a , T = (T1)y Tt = (1 - R)2N (T1)d. Substitute the result back into the governing relation to verify its units and sign.

Problem 4.76 — Fresnel reflection and transmission: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Resolve the field into s and p components, apply the Fresnel amplitudes, and convert amplitudes to power with the impedance/cosine factor. The book’s selected-answer check begins At ui = 0, R = Ri = R# = a nt ni nt + ni b 2 .[4.67] As nti S 1, nt S ni and clearly R S 0. At ui = 0, T = Ti = T# 4ntni (nt + ni)2 and since nt S ni , lim nti S1 T = 4ni 2 /(2ni)2 = 1. From Problem 4.91, and the fact that as nt S ni Snell’s Law says that ut S ui, we have lim nti S1 Ti = sin2 2ui sin2 2ui = 1, lim nti. Substitute the result back into the governing relation to verify its units and sign.

Problem 4.77* — Fresnel reflection and transmission: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Resolve the field into s and p components, apply the Fresnel amplitudes, and convert amplitudes to power with the impedance/cosine factor. Finish by checking the governing equation, the dimensions, and the zero/large-parameter limit; these checks replace reliance on an unavailable answer-key entry.

Problem 4.78 — Fresnel reflection and transmission: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Resolve the field into s and p components, apply the Fresnel amplitudes, and convert amplitudes to power with the impedance/cosine factor. The book’s selected-answer check begins For ui 7 uc, Eq. (4.70) can be written r# = cosui i(sin2 ui nti 2 )12 cosui + i(sin2 ui nti 2 )12 r#r# × = cos2 ui + sin2 ui nti 2 cos2 ui + sin2 ui nti 2 = 1 Similarly r‘ r# × = 1. Z03_HECH6933_05_GE_SOL.indd 689 08/09/16 9:14 pm 690 Solutions to Selected Problems Similarly, t#t′ # = T# r2 i = c tan(u1 u2) tan(u1 +. Substitute the result back into the governing relation to verify its units and sign.

Problem 4.79* — total internal reflection and evanescence: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Compare incidence with \(\theta_c=\sin^{-1}(n_t/n_i)\) and, above critical angle, use the imaginary normal wavevector to obtain decay and phase. Finish by checking the governing equation, the dimensions, and the zero/large-parameter limit; these checks replace reliance on an unavailable answer-key entry.

Problem 4.80* — total internal reflection and evanescence: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Compare incidence with \(\theta_c=\sin^{-1}(n_t/n_i)\) and, above critical angle, use the imaginary normal wavevector to obtain decay and phase. Finish by checking the governing equation, the dimensions, and the zero/large-parameter limit; these checks replace reliance on an unavailable answer-key entry.

Problem 4.81* — total internal reflection and evanescence: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Compare incidence with \(\theta_c=\sin^{-1}(n_t/n_i)\) and, above critical angle, use the imaginary normal wavevector to obtain decay and phase. Finish by checking the governing equation, the dimensions, and the zero/large-parameter limit; these checks replace reliance on an unavailable answer-key entry.

Problem 4.82* — total internal reflection and evanescence: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Compare incidence with \(\theta_c=\sin^{-1}(n_t/n_i)\) and, above critical angle, use the imaginary normal wavevector to obtain decay and phase. Finish by checking the governing equation, the dimensions, and the zero/large-parameter limit; these checks replace reliance on an unavailable answer-key entry.

Problem 4.83* — total internal reflection and evanescence: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Compare incidence with \(\theta_c=\sin^{-1}(n_t/n_i)\) and, above critical angle, use the imaginary normal wavevector to obtain decay and phase. Finish by checking the governing equation, the dimensions, and the zero/large-parameter limit; these checks replace reliance on an unavailable answer-key entry.

Problem 4.84* — total internal reflection and evanescence: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Compare incidence with \(\theta_c=\sin^{-1}(n_t/n_i)\) and, above critical angle, use the imaginary normal wavevector to obtain decay and phase. Finish by checking the governing equation, the dimensions, and the zero/large-parameter limit; these checks replace reliance on an unavailable answer-key entry.

Problem 4.85* — total internal reflection and evanescence: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Compare incidence with \(\theta_c=\sin^{-1}(n_t/n_i)\) and, above critical angle, use the imaginary normal wavevector to obtain decay and phase. Finish by checking the governing equation, the dimensions, and the zero/large-parameter limit; these checks replace reliance on an unavailable answer-key entry.

Problem 4.86 — total internal reflection and evanescence: derivation

Start from the governing relation rather than the desired result; rearrange until the requested form follows, so the argument is not circular. Compare incidence with \(\theta_c=\sin^{-1}(n_t/n_i)\) and, above critical angle, use the imaginary normal wavevector to obtain decay and phase. Substitution back into the starting relation supplies the final sign and dimensional check.

Problem 4.87 — total internal reflection and evanescence: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Compare incidence with \(\theta_c=\sin^{-1}(n_t/n_i)\) and, above critical angle, use the imaginary normal wavevector to obtain decay and phase. Substitution back into the starting relation supplies the final sign and dimensional check.

Problem 4.88* — total internal reflection and evanescence: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Compare incidence with \(\theta_c=\sin^{-1}(n_t/n_i)\) and, above critical angle, use the imaginary normal wavevector to obtain decay and phase. Finish by checking the governing equation, the dimensions, and the zero/large-parameter limit; these checks replace reliance on an unavailable answer-key entry.

Problem 4.89* — total internal reflection and evanescence: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Compare incidence with \(\theta_c=\sin^{-1}(n_t/n_i)\) and, above critical angle, use the imaginary normal wavevector to obtain decay and phase. Finish by checking the governing equation, the dimensions, and the zero/large-parameter limit; these checks replace reliance on an unavailable answer-key entry.

Problem 4.90* — total internal reflection and evanescence: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Compare incidence with \(\theta_c=\sin^{-1}(n_t/n_i)\) and, above critical angle, use the imaginary normal wavevector to obtain decay and phase. Finish by checking the governing equation, the dimensions, and the zero/large-parameter limit; these checks replace reliance on an unavailable answer-key entry.

Problem 4.91 — total internal reflection and evanescence: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Compare incidence with \(\theta_c=\sin^{-1}(n_t/n_i)\) and, above critical angle, use the imaginary normal wavevector to obtain decay and phase. Substitution back into the starting relation supplies the final sign and dimensional check.

Problem 4.92 — total internal reflection and evanescence: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Compare incidence with \(\theta_c=\sin^{-1}(n_t/n_i)\) and, above critical angle, use the imaginary normal wavevector to obtain decay and phase. Substitution back into the starting relation supplies the final sign and dimensional check.

Problem 4.93 — total internal reflection and evanescence: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Compare incidence with \(\theta_c=\sin^{-1}(n_t/n_i)\) and, above critical angle, use the imaginary normal wavevector to obtain decay and phase. Substitution back into the starting relation supplies the final sign and dimensional check.

Problem 4.94 — total internal reflection and evanescence: derivation

Start from the governing relation rather than the desired result; rearrange until the requested form follows, so the argument is not circular. Compare incidence with \(\theta_c=\sin^{-1}(n_t/n_i)\) and, above critical angle, use the imaginary normal wavevector to obtain decay and phase. Substitution back into the starting relation supplies the final sign and dimensional check.

Problem 4.95 — total internal reflection and evanescence: construction

Evaluate the boundary and representative interior values, then draw the requested curve or ray construction to scale. Compare incidence with \(\theta_c=\sin^{-1}(n_t/n_i)\) and, above critical angle, use the imaginary normal wavevector to obtain decay and phase. Substitution back into the starting relation supplies the final sign and dimensional check.

Problem 4.96* — total internal reflection and evanescence: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Compare incidence with \(\theta_c=\sin^{-1}(n_t/n_i)\) and, above critical angle, use the imaginary normal wavevector to obtain decay and phase. Finish by checking the governing equation, the dimensions, and the zero/large-parameter limit; these checks replace reliance on an unavailable answer-key entry.

Problem 4.97* — total internal reflection and evanescence: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Compare incidence with \(\theta_c=\sin^{-1}(n_t/n_i)\) and, above critical angle, use the imaginary normal wavevector to obtain decay and phase. Finish by checking the governing equation, the dimensions, and the zero/large-parameter limit; these checks replace reliance on an unavailable answer-key entry.

Problem 4.98* — total internal reflection and evanescence: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Compare incidence with \(\theta_c=\sin^{-1}(n_t/n_i)\) and, above critical angle, use the imaginary normal wavevector to obtain decay and phase. Finish by checking the governing equation, the dimensions, and the zero/large-parameter limit; these checks replace reliance on an unavailable answer-key entry.

Problem 4.99 — total internal reflection and evanescence: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Compare incidence with \(\theta_c=\sin^{-1}(n_t/n_i)\) and, above critical angle, use the imaginary normal wavevector to obtain decay and phase. Substitution back into the starting relation supplies the final sign and dimensional check.

Problem 4.100* — total internal reflection and evanescence: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Compare incidence with \(\theta_c=\sin^{-1}(n_t/n_i)\) and, above critical angle, use the imaginary normal wavevector to obtain decay and phase. Finish by checking the governing equation, the dimensions, and the zero/large-parameter limit; these checks replace reliance on an unavailable answer-key entry.

Problem 4.101 — total internal reflection and evanescence: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Compare incidence with \(\theta_c=\sin^{-1}(n_t/n_i)\) and, above critical angle, use the imaginary normal wavevector to obtain decay and phase. The book’s selected-answer check begins From Eq. (4.45) t′ i(u′ π)ti(up) = c 2 sinup cosu′ π sin(up + u′ π) cos(u′ π up) d × c 2 sinu′ π cosup sin(up + u′ π) cos(up u′ π) d = sin2u′ π sin2up cos2 (up u′ π) , since up + u′ π = 90° = sin2 2up cos2 (up u′ π) , since sin2u′ π = sin2up = sin2 2up cos2 (2up - 90°) = 1. Substitute the result back into the governing relation to verify its units and sign.