Chapter 7: The Superposition of Waves

Source: Eugene Hecht, Optics, fifth Global Edition, Chapter 7. 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 7.1 — superposition, beats, and group velocity: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Add complex amplitudes before squaring; for a narrow packet use \(v_g=d\omega/dk\) and distinguish carrier from envelope. The book’s selected-answer check begins E2 0 = 64 + 100 + 2 · 8 · 10 cos π/3 = 244, E0 = 15.6; tan a = 10 8 , a = 51.3° 5 0.9 rad E = 15.6 sin (200pt + 0.90). Substitute the result back into the governing relation to verify its units and sign.

Problem 7.2* — superposition, beats, and group velocity: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Add complex amplitudes before squaring; for a narrow packet use \(v_g=d\omega/dk\) and distinguish carrier from envelope. 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 7.3* — superposition, beats, and group velocity: derivation

Start from the governing relation rather than the desired result; rearrange until the requested form follows, so the argument is not circular. Add complex amplitudes before squaring; for a narrow packet use \(v_g=d\omega/dk\) and distinguish carrier from envelope. 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 7.4* — superposition, beats, and group velocity: derivation

Start from the governing relation rather than the desired result; rearrange until the requested form follows, so the argument is not circular. Add complex amplitudes before squaring; for a narrow packet use \(v_g=d\omega/dk\) and distinguish carrier from envelope. 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 7.5 — superposition, beats, and group velocity: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Add complex amplitudes before squaring; for a narrow packet use \(v_g=d\omega/dk\) and distinguish carrier from envelope. The book’s selected-answer check begins (a) 0.8 m 540 nm = 0.15 × 107 waves (b) In the glass 0.1 l0/n = 0.1(1.5) 540 × 10-9 = 2.78 × 105 waves In air, 0.7/540 × 10-9 = 0.13 × 107 waves Total: 2.78 × 105 + 0.13 × 107 = 0.158 × 107 waves (c) OPD = [(1.5)(0.1) + (1)(0.7)] - (1)(0.8) OPD = (0.15 + 0.7) - 0.8 = 0.05 m (d) Λ/l0 = 0.05/540 × 10-9 = 0.9 × 105 waves. Substitute the result back into the governing relation to verify its units and sign.

Problem 7.6* — superposition, beats, and group velocity: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Add complex amplitudes before squaring; for a narrow packet use \(v_g=d\omega/dk\) and distinguish carrier from envelope. 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 7.7* — superposition, beats, and group velocity: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Add complex amplitudes before squaring; for a narrow packet use \(v_g=d\omega/dk\) and distinguish carrier from envelope. 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 7.8 — superposition, beats, and group velocity: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Add complex amplitudes before squaring; for a narrow packet use \(v_g=d\omega/dk\) and distinguish carrier from envelope. Substitution back into the starting relation supplies the final sign and dimensional check.

Problem 7.9 — superposition, beats, and group velocity: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Add complex amplitudes before squaring; for a narrow packet use \(v_g=d\omega/dk\) and distinguish carrier from envelope. Substitution back into the starting relation supplies the final sign and dimensional check.

Problem 7.10* — superposition, beats, and group velocity: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Add complex amplitudes before squaring; for a narrow packet use \(v_g=d\omega/dk\) and distinguish carrier from envelope. 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 7.11* — superposition, beats, and group velocity: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Add complex amplitudes before squaring; for a narrow packet use \(v_g=d\omega/dk\) and distinguish carrier from envelope. 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 7.12* — superposition, beats, and group velocity: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Add complex amplitudes before squaring; for a narrow packet use \(v_g=d\omega/dk\) and distinguish carrier from envelope. 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 7.13 — superposition, beats, and group velocity: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Add complex amplitudes before squaring; for a narrow packet use \(v_g=d\omega/dk\) and distinguish carrier from envelope. Substitution back into the starting relation supplies the final sign and dimensional check.

Problem 7.14* — superposition, beats, and group velocity: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Add complex amplitudes before squaring; for a narrow packet use \(v_g=d\omega/dk\) and distinguish carrier from envelope. 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 7.15* — superposition, beats, and group velocity: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Add complex amplitudes before squaring; for a narrow packet use \(v_g=d\omega/dk\) and distinguish carrier from envelope. 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 7.16* — superposition, beats, and group velocity: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Add complex amplitudes before squaring; for a narrow packet use \(v_g=d\omega/dk\) and distinguish carrier from envelope. 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 7.17* — superposition, beats, and group velocity: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Add complex amplitudes before squaring; for a narrow packet use \(v_g=d\omega/dk\) and distinguish carrier from envelope. 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 7.18* — superposition, beats, and group velocity: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Add complex amplitudes before squaring; for a narrow packet use \(v_g=d\omega/dk\) and distinguish carrier from envelope. 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 7.19* — superposition, beats, and group velocity: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Add complex amplitudes before squaring; for a narrow packet use \(v_g=d\omega/dk\) and distinguish carrier from envelope. 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 7.20* — superposition, beats, and group velocity: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Add complex amplitudes before squaring; for a narrow packet use \(v_g=d\omega/dk\) and distinguish carrier from envelope. 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 7.21 — superposition, beats, and group velocity: derivation

Start from the governing relation rather than the desired result; rearrange until the requested form follows, so the argument is not circular. Add complex amplitudes before squaring; for a narrow packet use \(v_g=d\omega/dk\) and distinguish carrier from envelope. The book’s selected-answer check begins E = E0 cosvct + E0a cos vmt cosvct = E0 cosvct + E0a 2 [cos (vc vm)t + cos (vc + vm)t] Audible range nm = 20 Hz to 20 × 103 Hz. Maximum modulation frequency nm(max) = 20 × 103 Hz. nc nm(max) … n … nc + nm(max) ∆n = 2nm(max) = 40 × 103 Hz. Substitute the result back into the governing relation to verify its units and sign.

Problem 7.22 — superposition, beats, and group velocity: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Add complex amplitudes before squaring; for a narrow packet use \(v_g=d\omega/dk\) and distinguish carrier from envelope. The book’s selected-answer check begins v = v/k = 2ak2 , vg = dv/dk = (2a) (3k2 ) = 6ak2. Substitute the result back into the governing relation to verify its units and sign.

Problem 7.23* — Fourier-series decomposition: derivation

Start from the governing relation rather than the desired result; rearrange until the requested form follows, so the argument is not circular. Compute \(a_n\) and \(b_n\) over one period, use parity to eliminate terms, and reconstruct enough harmonics to verify the waveform. 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 7.24* — Fourier-series decomposition: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Compute \(a_n\) and \(b_n\) over one period, use parity to eliminate terms, and reconstruct enough harmonics to verify the waveform. 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 7.25* — Fourier-series decomposition: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Compute \(a_n\) and \(b_n\) over one period, use parity to eliminate terms, and reconstruct enough harmonics to verify the waveform. 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 7.26* — Fourier-series decomposition: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Compute \(a_n\) and \(b_n\) over one period, use parity to eliminate terms, and reconstruct enough harmonics to verify the waveform. 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 7.27* — Fourier-series decomposition: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Compute \(a_n\) and \(b_n\) over one period, use parity to eliminate terms, and reconstruct enough harmonics to verify the waveform. 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 7.28* — Fourier-series decomposition: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Compute \(a_n\) and \(b_n\) over one period, use parity to eliminate terms, and reconstruct enough harmonics to verify the waveform. 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 7.29 — Fourier-series decomposition: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Compute \(a_n\) and \(b_n\) over one period, use parity to eliminate terms, and reconstruct enough harmonics to verify the waveform. The book’s selected-answer check begins v = A gl 2p = 1g/k vg = v + k dv dk [7.38] dv dk = - 1 2kA g k = v 2k vg = v/2 Z03_HECH6933_05_GE_SOL.indd 695 08/09/16 9:14 pm 696 Solutions to Selected Problems. Substitute the result back into the governing relation to verify its units and sign.

Problem 7.30* — Fourier-series decomposition: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Compute \(a_n\) and \(b_n\) over one period, use parity to eliminate terms, and reconstruct enough harmonics to verify the waveform. 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 7.31 — Fourier-series decomposition: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Compute \(a_n\) and \(b_n\) over one period, use parity to eliminate terms, and reconstruct enough harmonics to verify the waveform. The book’s selected-answer check begins vg = v + k dv dk and dv dk = dv dv dv dk = vg dv dv . Since v = c/n, dv dv = dv dn dn dv = c n2 dn dv vg = v vgck n2 dn dv = v 1 + (ck/n2 )(dn/dv) = c n + v(dn/dv). Substitute the result back into the governing relation to verify its units and sign.

Problem 7.32* — Fourier-series decomposition: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Compute \(a_n\) and \(b_n\) over one period, use parity to eliminate terms, and reconstruct enough harmonics to verify the waveform. 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 7.33* — Fourier-series decomposition: derivation

Start from the governing relation rather than the desired result; rearrange until the requested form follows, so the argument is not circular. Compute \(a_n\) and \(b_n\) over one period, use parity to eliminate terms, and reconstruct enough harmonics to verify the waveform. 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 7.34* — Fourier-series decomposition: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Compute \(a_n\) and \(b_n\) over one period, use parity to eliminate terms, and reconstruct enough harmonics to verify the waveform. 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 7.35* — Fourier-series decomposition: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Compute \(a_n\) and \(b_n\) over one period, use parity to eliminate terms, and reconstruct enough harmonics to verify the waveform. 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 7.36* — Fourier-series decomposition: derivation

Start from the governing relation rather than the desired result; rearrange until the requested form follows, so the argument is not circular. Compute \(a_n\) and \(b_n\) over one period, use parity to eliminate terms, and reconstruct enough harmonics to verify the waveform. 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 7.37* — Fourier-series decomposition: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Compute \(a_n\) and \(b_n\) over one period, use parity to eliminate terms, and reconstruct enough harmonics to verify the waveform. 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 7.38* — Fourier-series decomposition: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Compute \(a_n\) and \(b_n\) over one period, use parity to eliminate terms, and reconstruct enough harmonics to verify the waveform. 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 7.39* — Fourier-series decomposition: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Compute \(a_n\) and \(b_n\) over one period, use parity to eliminate terms, and reconstruct enough harmonics to verify the waveform. 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 7.40 — Fourier-series decomposition: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Compute \(a_n\) and \(b_n\) over one period, use parity to eliminate terms, and reconstruct enough harmonics to verify the waveform. The book’s selected-answer check begins v 7 7 vi, n2 = 1 - Nqe 2 v2 P0me ^ƒi = 1 - Nqe 2 v2 P0me . Using the binomial expansion, we have (1 x)1/2 ≈ 1 - 1 2 x for x 6 6 1 n = 1 - Nqe 2 /v2 P0me2, dn/dv = Nqe 2 /P0mev3 vg = c n + v(dn/dv) = c 1 - Nqe 2 /v2 P0me2 + Nqe 2 /P0mev2 = c 1 + Nqe 2 /P0mev2 2 and vg 6 c, v = c/n = c 1 - Nqe 2 /P0mev2 2 Binomial expans. Substitute the result back into the governing relation to verify its units and sign.

Problem 7.41* — Fourier-series decomposition: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Compute \(a_n\) and \(b_n\) over one period, use parity to eliminate terms, and reconstruct enough harmonics to verify the waveform. 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 7.42* — Fourier-series decomposition: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Compute \(a_n\) and \(b_n\) over one period, use parity to eliminate terms, and reconstruct enough harmonics to verify the waveform. 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 7.43 — Fourier-series decomposition: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Compute \(a_n\) and \(b_n\) over one period, use parity to eliminate terms, and reconstruct enough harmonics to verify the waveform. The book’s selected-answer check begins 3 λ 0 sinakx sinbkx dx = 1 2k c3 λ 0 cos [(a b)kx]k dx - 3 λ 0 cos [(a + b)kx]k dxd = 1 2k sin(a b)kx a b ' λ 0 - 1 2k sin (a + b)kx a + b ' λ 0 = 0 if a Z b Whereas if a = b 3 λ 0 sin2 akx dx = 1 2k3 λ 0 (1 + cos 2akx)k dx = λ 2 The other integrals are similar.. Substitute the result back into the governing relation to verify its units and sign.

Problem 7.44 — wave packets and temporal coherence: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Fourier transform the spectrum or pulse, use \(\Delta t\,\Delta\nu\) consistently, and check the monochromatic limit. The book’s selected-answer check begins Even function, therefore Bm = 0. A0 = 2 λ 3 λ/a -λ/a dx = 2 λ a λ a + λ a b = 4 a Am = 2 λ 3 λ/a -λ/a (1) cos mkx dx Am = 2 mkl sin mkxd λ/a -λ/a Am = 2 mp sin m2p a π 2p π 2 −π 2 u Si(u). Substitute the result back into the governing relation to verify its units and sign.

Problem 7.45* — wave packets and temporal coherence: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Fourier transform the spectrum or pulse, use \(\Delta t\,\Delta\nu\) consistently, and check the monochromatic limit. 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 7.46* — wave packets and temporal coherence: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Fourier transform the spectrum or pulse, use \(\Delta t\,\Delta\nu\) consistently, and check the monochromatic limit. 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 7.47* — wave packets and temporal coherence: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Fourier transform the spectrum or pulse, use \(\Delta t\,\Delta\nu\) consistently, and check the monochromatic limit. 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 7.48* — wave packets and temporal coherence: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Fourier transform the spectrum or pulse, use \(\Delta t\,\Delta\nu\) consistently, and check the monochromatic limit. 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 7.49* — wave packets and temporal coherence: derivation

Start from the governing relation rather than the desired result; rearrange until the requested form follows, so the argument is not circular. Fourier transform the spectrum or pulse, use \(\Delta t\,\Delta\nu\) consistently, and check the monochromatic limit. 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 7.50 — wave packets and temporal coherence: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Fourier transform the spectrum or pulse, use \(\Delta t\,\Delta\nu\) consistently, and check the monochromatic limit. The book’s selected-answer check begins ƒ′(x) = 1 p3 a 0 E0L sinkL/2 kL/2 cos kx dk = E0L p2 3 b 0 sin(kL/2 + kx) kL/2 dk + E0L p2 3 b 0 sin(kL/2 kx) kL/2 dk Let kL/2 = w, (L/2) dk = dw, kx = wx′. ƒ′(x) = E0 π 3 b 0 sin(w + wx′) w dw + E0 π 3 b 0 sin(w wx′) w dw where b = aL/2. Let w + wx′ = t, dw/w = dt/t. 0 … w … b and 0 … t … (x′ + 1)b. Let w wx′ = -t in. Substitute the result back into the governing relation to verify its units and sign.

Problem 7.51* — wave packets and temporal coherence: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Fourier transform the spectrum or pulse, use \(\Delta t\,\Delta\nu\) consistently, and check the monochromatic limit. 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 7.52* — wave packets and temporal coherence: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Fourier transform the spectrum or pulse, use \(\Delta t\,\Delta\nu\) consistently, and check the monochromatic limit. 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 7.53* — wave packets and temporal coherence: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Fourier transform the spectrum or pulse, use \(\Delta t\,\Delta\nu\) consistently, and check the monochromatic limit. 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 7.54 — wave packets and temporal coherence: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Fourier transform the spectrum or pulse, use \(\Delta t\,\Delta\nu\) consistently, and check the monochromatic limit. The book’s selected-answer check begins By analogy with Eq. (7.61), A(v) = ∆t 2 E0 sinc(vp v) ∆t 2 From Table 1 (π. 681) sinc(π/2) = 63.7%. Not quite 50% actually, sinc a π 1.65 b = 49.8% ' (vp v) ∆t 2 ' 6 π 2 or π ∆t 6 (vp v) 6 π ∆t Thus appreciable values of A(v) lie in a range ∆v ∼ 2p/∆t and ∆n ∆t ≈ 1. The power spectrum is proportional to A2 (v), and [si. Substitute the result back into the governing relation to verify its units and sign.

Problem 7.55 — wave packets and temporal coherence: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Fourier transform the spectrum or pulse, use \(\Delta t\,\Delta\nu\) consistently, and check the monochromatic limit. The book’s selected-answer check begins ∆lc = c ∆tc, ∆lc ≈ c/∆n. But ∆v/∆k0 = v/k0 = c; thus ∆n/∆l0  = n/l0, ∆lc ≈ cl0 ∆l0n ∆lc ≈ l0 2 /∆l0 Or try using the uncertainty principle: ∆λ ≈ h ∆π where π = h/λ and ∆l0 6 6 l0 Z03_HECH6933_05_GE_SOL.indd 696 08/09/16 9:14 pm Solutions to Selected Problems 697 Let E0 + E′ 0 = E″ 0x and E0 - E′ 0 = E″ 0y; then E $ =. Substitute the result back into the governing relation to verify its units and sign.

Problem 7.56* — wave packets and temporal coherence: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Fourier transform the spectrum or pulse, use \(\Delta t\,\Delta\nu\) consistently, and check the monochromatic limit. 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 7.57 — wave packets and temporal coherence: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Fourier transform the spectrum or pulse, use \(\Delta t\,\Delta\nu\) consistently, and check the monochromatic limit. The book’s selected-answer check begins ∆lc = c ∆tc = 3 × 108 m/s × 10-8 s = 3 m ∆l0 ≈ l0 2 /∆lc = (500 × 10-9 m)2 /3 m ∆l0 ≈ 8.3 × 10-14 m = 8.3 × 10-5 nm ∆l0/l0 = ∆n/n = 8.3 × 10-5 /500 = 1.6 × 10-7 ≈ 1 part in 107. Substitute the result back into the governing relation to verify its units and sign.

Problem 7.58 — wave packets and temporal coherence: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Fourier transform the spectrum or pulse, use \(\Delta t\,\Delta\nu\) consistently, and check the monochromatic limit. The book’s selected-answer check begins ∆n = 54 × 103 Hz ∆n/n = (54 × 103 )(10600 × 10-9 m) (3 × 108 m/s) = 1.91 × 10-9 ∆lc = c ∆tc ≈ c/∆n ∆lc ≈ (3 × 108 m/s) (54 × 103 Hz) = 5.55 × 103 m. Substitute the result back into the governing relation to verify its units and sign.

Problem 7.59* — wave packets and temporal coherence: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Fourier transform the spectrum or pulse, use \(\Delta t\,\Delta\nu\) consistently, and check the monochromatic limit. 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 7.60 — wave packets and temporal coherence: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Fourier transform the spectrum or pulse, use \(\Delta t\,\Delta\nu\) consistently, and check the monochromatic limit. The book’s selected-answer check begins ∆lc = c ∆tc = 3 × 108 × 10-10 = 3 × 10-2 m ∆n ≈ 1/∆tc = 1010 Hz ∆l0 ≈ l0 2 /∆lc (see Problem 7.55) = (632.8 nm)2 /3 × 10-2 m = 0.013 nm ∆n = 1015 Hz, ∆lc = c × 10-15 = 300 nm ∆l0 ≈ l0 2 /∆lc = 1334.78 nm. Substitute the result back into the governing relation to verify its units and sign.

Problem 7.61* — wave packets and temporal coherence: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Fourier transform the spectrum or pulse, use \(\Delta t\,\Delta\nu\) consistently, and check the monochromatic limit. 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 7.62* — wave packets and temporal coherence: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Fourier transform the spectrum or pulse, use \(\Delta t\,\Delta\nu\) consistently, and check the monochromatic limit. 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 7.63* — wave packets and temporal coherence: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Fourier transform the spectrum or pulse, use \(\Delta t\,\Delta\nu\) consistently, and check the monochromatic limit. Finish by checking the governing equation, the dimensions, and the zero/large-parameter limit; these checks replace reliance on an unavailable answer-key entry.