Chapter 11: Fourier Optics

Source: Eugene Hecht, Optics, fifth Global Edition, Chapter 11. 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 11.1 — Fourier transforms: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Insert the function in the chapter’s transform convention, exploit parity/shift/scaling, and check the inverse transform. The book’s selected-answer check begins E0 sin kpx = E0(eikpx e-ikpx )/2i F(k) = E0 2i c3 +L -L ei(k + kp)x dx - 3 +L -L ei(k kp)x dxd F(k) = iE0 sin (k + kp)L (k + kp) + iE0 sin (k + kp)L (k kp) F(k) = iE0L[sinc (k kp)L sinc (k + kp)L] v 2vp 2vp −2vp −2vp v 0 0 T ∞ f(x) 0 x d(x − x0) 0 x f(x − x0) 0 x f(x) d(x − x0) x0 x0. Substitute the result back into the governing relation to verify its units and sign.

Problem 11.2* — Fourier transforms: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Insert the function in the chapter’s transform convention, exploit parity/shift/scaling, and check the inverse transform. 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 11.3 — Fourier transforms: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Insert the function in the chapter’s transform convention, exploit parity/shift/scaling, and check the inverse transform. The book’s selected-answer check begins cos2 vpt = 1 2 + 1 2 cos 2vpt = 1 2 + e2ivpt + e-2ivpt 4 F(v) = 1 23 +T -T eivt dt + 1 43ei(v+2vp)t dt + 1 43ei(v-2vp)t dt F(v) = 1 v sin vT + 1 2(v + 2vp) sin (v + 2vp)T + 1 2(v - 2vp) sin(v - 2vp)T F(v) = T sinc vT + T 2 sinc(v + 2vp)T + T 2 sinc(v - 2vp)T 1.25 0 5 10.88 Z03_HECH6933_05_GE_SOL.indd 704 08/09/16 9:15. Substitute the result back into the governing relation to verify its units and sign.

Problem 11.4* — Fourier transforms: derivation

Start from the governing relation rather than the desired result; rearrange until the requested form follows, so the argument is not circular. Insert the function in the chapter’s transform convention, exploit parity/shift/scaling, and check the inverse transform. 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 11.5* — Fourier transforms: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Insert the function in the chapter’s transform convention, exploit parity/shift/scaling, and check the inverse transform. 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 11.6* — Fourier transforms: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Insert the function in the chapter’s transform convention, exploit parity/shift/scaling, and check the inverse transform. 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 11.7* — Fourier transforms: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Insert the function in the chapter’s transform convention, exploit parity/shift/scaling, and check the inverse transform. 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 11.8* — Fourier transforms: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Insert the function in the chapter’s transform convention, exploit parity/shift/scaling, and check the inverse transform. 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 11.9 — Fourier transforms: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Insert the function in the chapter’s transform convention, exploit parity/shift/scaling, and check the inverse transform. Substitution back into the starting relation supplies the final sign and dimensional check.

Problem 11.10* — Fourier transforms: derivation

Start from the governing relation rather than the desired result; rearrange until the requested form follows, so the argument is not circular. Insert the function in the chapter’s transform convention, exploit parity/shift/scaling, and check the inverse transform. 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 11.11 — Fourier transforms: derivation

Start from the governing relation rather than the desired result; rearrange until the requested form follows, so the argument is not circular. Insert the function in the chapter’s transform convention, exploit parity/shift/scaling, and check the inverse transform. Substitution back into the starting relation supplies the final sign and dimensional check.

Problem 11.12* — Fourier transforms: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Insert the function in the chapter’s transform convention, exploit parity/shift/scaling, and check the inverse transform. 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 11.13* — Fourier transforms: derivation

Start from the governing relation rather than the desired result; rearrange until the requested form follows, so the argument is not circular. Insert the function in the chapter’s transform convention, exploit parity/shift/scaling, and check the inverse transform. 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 11.14* — Fourier transforms: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Insert the function in the chapter’s transform convention, exploit parity/shift/scaling, and check the inverse transform. 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 11.15* — Fourier transforms: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Insert the function in the chapter’s transform convention, exploit parity/shift/scaling, and check the inverse transform. 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 11.16* — Fourier transforms: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Insert the function in the chapter’s transform convention, exploit parity/shift/scaling, and check the inverse transform. 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 11.17* — Fourier transforms: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Insert the function in the chapter’s transform convention, exploit parity/shift/scaling, and check the inverse transform. 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 11.18 — Fourier transforms: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Insert the function in the chapter’s transform convention, exploit parity/shift/scaling, and check the inverse transform. Substitution back into the starting relation supplies the final sign and dimensional check.

Problem 11.19* — Fourier transforms: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Insert the function in the chapter’s transform convention, exploit parity/shift/scaling, and check the inverse transform. 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 11.20* — Fourier transforms: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Insert the function in the chapter’s transform convention, exploit parity/shift/scaling, and check the inverse transform. 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 11.21* — convolution and imaging: 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 convolution theorem and represent apertures with rect, sinc, or delta functions before simplifying. 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 11.22 — convolution and imaging: 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 convolution theorem and represent apertures with rect, sinc, or delta functions before simplifying. Substitution back into the starting relation supplies the final sign and dimensional check.

Problem 11.23* — convolution and imaging: 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 convolution theorem and represent apertures with rect, sinc, or delta functions before simplifying. 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 11.24 — convolution and imaging: 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 convolution theorem and represent apertures with rect, sinc, or delta functions before simplifying. Substitution back into the starting relation supplies the final sign and dimensional check.

Problem 11.25* — convolution and imaging: 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 convolution theorem and represent apertures with rect, sinc, or delta functions before simplifying. 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 11.26* — convolution and imaging: construction

Evaluate the boundary and representative interior values, then draw the requested curve or ray construction to scale. Use the convolution theorem and represent apertures with rect, sinc, or delta functions before simplifying. 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 11.27* — convolution and imaging: 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 convolution theorem and represent apertures with rect, sinc, or delta functions before simplifying. 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 11.28 — convolution and imaging: 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 convolution theorem and represent apertures with rect, sinc, or delta functions before simplifying. Substitution back into the starting relation supplies the final sign and dimensional check.

Problem 11.29* — convolution and imaging: 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 convolution theorem and represent apertures with rect, sinc, or delta functions before simplifying. 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 11.30* — convolution and imaging: 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 convolution theorem and represent apertures with rect, sinc, or delta functions before simplifying. 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 11.31* — convolution and imaging: 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 convolution theorem and represent apertures with rect, sinc, or delta functions before simplifying. 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 11.32 — convolution and imaging: 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 convolution theorem and represent apertures with rect, sinc, or delta functions before simplifying. Substitution back into the starting relation supplies the final sign and dimensional check.

Problem 11.33 — convolution and imaging: 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 convolution theorem and represent apertures with rect, sinc, or delta functions before simplifying. The book’s selected-answer check begins ƒ(x)àh(x) = [d(x + 4) + d(x - 3) + d(x - 6)]àh(x) = h(x + 4) + h(x - 3) + h(x - 6).. Substitute the result back into the governing relation to verify its units and sign.

Problem 11.34* — convolution and imaging: 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 convolution theorem and represent apertures with rect, sinc, or delta functions before simplifying. 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 11.35* — convolution and imaging: 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 convolution theorem and represent apertures with rect, sinc, or delta functions before simplifying. 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 11.36 — transfer functions and spatial filtering: derivation

Start from the governing relation rather than the desired result; rearrange until the requested form follows, so the argument is not circular. Multiply the object spectrum by the coherent or incoherent transfer function, then inverse transform and inspect passed spatial frequencies. Substitution back into the starting relation supplies the final sign and dimensional check.

Problem 11.37 — transfer functions and spatial filtering: derivation

Start from the governing relation rather than the desired result; rearrange until the requested form follows, so the argument is not circular. Multiply the object spectrum by the coherent or incoherent transfer function, then inverse transform and inspect passed spatial frequencies. The book’s selected-answer check begins 𝒜(y, z) = 𝒜(-y, -z) E(Y, Z, t) ∝ 6𝒜(y, z)ei(kYy+kZz) dy dz Change Y to -Y, Z to -Z, y to -y, z to -z; then kY goes to -kY and kZ to -kZ. E(-Y, -Z) ∝ 6𝒜(-y, -z)ei(kYy+kZz) dy dz 6E(-Y, -Z) = E(Y, Z). Substitute the result back into the governing relation to verify its units and sign.

Problem 11.38 — transfer functions and spatial filtering: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Multiply the object spectrum by the coherent or incoherent transfer function, then inverse transform and inspect passed spatial frequencies. The book’s selected-answer check begins As given E(X, Y) = 6𝒜(x, y)eik(Xx+Yy)/R dx dy E′(X, Y) = 6𝒜(ax, by)eik(Xx+Yy)/R dx dy Now, let x′ = xa, y′ = by E′(X, Y) = 1 ab6𝒜(x′, y′)eik[(X/a)x′+(Y/b)y′]/R dx′dy′ or E′(X, Y) = (1/ab) E(X/a, Y/b).. Substitute the result back into the governing relation to verify its units and sign.

Problem 11.39 — transfer functions and spatial filtering: derivation

Start from the governing relation rather than the desired result; rearrange until the requested form follows, so the argument is not circular. Multiply the object spectrum by the coherent or incoherent transfer function, then inverse transform and inspect passed spatial frequencies. The book’s selected-answer check begins CBB = lim TS ∞ 1 2T3 T -T A sin (vt + e) A sin (vt vt + e) dt = lim TS ∞ A2 2T3 T -T c 1 2 cos (vt) - 1 2 cos (2vt vt + 2e)ddt Since cos a cos b = -2 sin (1/2)(a + b) sin (1/2)(a b). Thus, CBB = (A2 /2) cos (vt). {h(x)} {f(x)} 0 = = 1 2 {E(x)}. Substitute the result back into the governing relation to verify its units and sign.

Problem 11.40 — transfer functions and spatial filtering: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Multiply the object spectrum by the coherent or incoherent transfer function, then inverse transform and inspect passed spatial frequencies. The book’s selected-answer check begins E(kZ) = 3 t/2 -t/2 𝒜0 cos (pz/t)eikZz dz = 𝒜0 3cos(pz/t) cos(kZz) dz + i𝒜03cos(pz/t) sin(kZz) dz E(kZ) = 𝒜 cosa tkZ 2 b c 1 π/t kZ + 1 π/t + kZ d ·. Substitute the result back into the governing relation to verify its units and sign.

Problem 11.41* — transfer functions and spatial filtering: construction

Evaluate the boundary and representative interior values, then draw the requested curve or ray construction to scale. Multiply the object spectrum by the coherent or incoherent transfer function, then inverse transform and inspect passed spatial frequencies. 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 11.42* — transfer functions and spatial filtering: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Multiply the object spectrum by the coherent or incoherent transfer function, then inverse transform and inspect passed spatial frequencies. 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 11.43* — transfer functions and spatial filtering: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Multiply the object spectrum by the coherent or incoherent transfer function, then inverse transform and inspect passed spatial frequencies. 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 11.44* — transfer functions and spatial filtering: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Multiply the object spectrum by the coherent or incoherent transfer function, then inverse transform and inspect passed spatial frequencies. 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 11.45* — transfer functions and spatial filtering: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Multiply the object spectrum by the coherent or incoherent transfer function, then inverse transform and inspect passed spatial frequencies. 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 11.46* — transfer functions and spatial filtering: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Multiply the object spectrum by the coherent or incoherent transfer function, then inverse transform and inspect passed spatial frequencies. 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 11.47* — transfer functions and spatial filtering: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Multiply the object spectrum by the coherent or incoherent transfer function, then inverse transform and inspect passed spatial frequencies. 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 11.48* — transfer functions and spatial filtering: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Multiply the object spectrum by the coherent or incoherent transfer function, then inverse transform and inspect passed spatial frequencies. 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 11.49* — transfer functions and spatial filtering: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Multiply the object spectrum by the coherent or incoherent transfer function, then inverse transform and inspect passed spatial frequencies. 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 11.50* — transfer functions and spatial filtering: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Multiply the object spectrum by the coherent or incoherent transfer function, then inverse transform and inspect passed spatial frequencies. Finish by checking the governing equation, the dimensions, and the zero/large-parameter limit; these checks replace reliance on an unavailable answer-key entry.