Chapter 10: Diffraction

Source: Eugene Hecht, Optics, fifth Global Edition, Chapter 10. 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 10.1 — single-aperture diffraction: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Integrate the aperture phasors to obtain the sinc amplitude and locate extrema from its phase variable. The book’s selected-answer check begins (R + /)2 = R2 + a2 ; therefore R = (a2 - /2 )/2/ ≈ a2 /2/, /R = a2 /2, so for λ 7 7 /, lR 7 7 a2 /2 6 R = (1 × 10-3 )2 10/2l = 10 m. S a R + R. Substitute the result back into the governing relation to verify its units and sign.

Problem 10.2* — single-aperture diffraction: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Integrate the aperture phasors to obtain the sinc amplitude and locate extrema from its phase variable. 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 10.3 — single-aperture diffraction: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Integrate the aperture phasors to obtain the sinc amplitude and locate extrema from its phase variable. The book’s selected-answer check begins d sinum = ml, u = Nd/2 = π 7 sinu = (1)(0.21) d = 2p/N = kd sinu sinu = 0.03 sinu = 0.0009 u = 1.7° u = 3 min. Substitute the result back into the governing relation to verify its units and sign.

Problem 10.4 — single-aperture diffraction: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Integrate the aperture phasors to obtain the sinc amplitude and locate extrema from its phase variable. The book’s selected-answer check begins Converging spherical wave in image space is diffracted by the exit pupil. P P S Exit pupil ≈ Ll b b L. Substitute the result back into the governing relation to verify its units and sign.

Problem 10.5* — single-aperture diffraction: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Integrate the aperture phasors to obtain the sinc amplitude and locate extrema from its phase variable. 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 10.6 — single-aperture diffraction: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Integrate the aperture phasors to obtain the sinc amplitude and locate extrema from its phase variable. Substitution back into the starting relation supplies the final sign and dimensional check.

Problem 10.7* — single-aperture diffraction: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Integrate the aperture phasors to obtain the sinc amplitude and locate extrema from its phase variable. 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 10.8* — single-aperture diffraction: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Integrate the aperture phasors to obtain the sinc amplitude and locate extrema from its phase variable. 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 10.9 — single-aperture diffraction: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Integrate the aperture phasors to obtain the sinc amplitude and locate extrema from its phase variable. The book’s selected-answer check begins λ = (25 cm) sin36.87 = 15 cm.. Substitute the result back into the governing relation to verify its units and sign.

Problem 10.10* — single-aperture diffraction: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Integrate the aperture phasors to obtain the sinc amplitude and locate extrema from its phase variable. 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 10.11* — single-aperture diffraction: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Integrate the aperture phasors to obtain the sinc amplitude and locate extrema from its phase variable. 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 10.12* — single-aperture diffraction: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Integrate the aperture phasors to obtain the sinc amplitude and locate extrema from its phase variable. 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 10.13* — single-aperture diffraction: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Integrate the aperture phasors to obtain the sinc amplitude and locate extrema from its phase variable. 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 10.14 — single-aperture diffraction: derivation

Start from the governing relation rather than the desired result; rearrange until the requested form follows, so the argument is not circular. Integrate the aperture phasors to obtain the sinc amplitude and locate extrema from its phase variable. The book’s selected-answer check begins a = ka 2 sinu, b = kb 2 sinu a = mb, a = mb, a = m2p N = number of fringes = a/π = m2p/π = 2m. Substitute the result back into the governing relation to verify its units and sign.

Problem 10.15* — single-aperture diffraction: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Integrate the aperture phasors to obtain the sinc amplitude and locate extrema from its phase variable. 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 10.16* — single-aperture diffraction: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Integrate the aperture phasors to obtain the sinc amplitude and locate extrema from its phase variable. 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 10.17 — single-aperture diffraction: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Integrate the aperture phasors to obtain the sinc amplitude and locate extrema from its phase variable. The book’s selected-answer check begins a = 3p/2N = π/2[10.34] I(u) = I(0) N2 a sinb b b 2 from Eq. (10.35) and I/I(0) ≈ 1 9 . sin u λ a λ b sin u λ a λ b. Substitute the result back into the governing relation to verify its units and sign.

Problem 10.18* — single-aperture diffraction: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Integrate the aperture phasors to obtain the sinc amplitude and locate extrema from its phase variable. 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 10.19* — single-aperture diffraction: design

Translate every stated requirement into an equation, solve the simultaneous constraints, and reject roots that violate the geometry. Integrate the aperture phasors to obtain the sinc amplitude and locate extrema from its phase variable. 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 10.20* — single-aperture diffraction: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Integrate the aperture phasors to obtain the sinc amplitude and locate extrema from its phase variable. 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 10.21* — two-dimensional apertures: 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 aperture function; separable rectangles factor, while circular symmetry gives the Airy/Bessel form. 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 10.22* — two-dimensional apertures: 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 aperture function; separable rectangles factor, while circular symmetry gives the Airy/Bessel form. 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 10.23* — two-dimensional apertures: 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 aperture function; separable rectangles factor, while circular symmetry gives the Airy/Bessel form. 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 10.24* — two-dimensional apertures: 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 aperture function; separable rectangles factor, while circular symmetry gives the Airy/Bessel form. 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 10.25* — two-dimensional apertures: 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 aperture function; separable rectangles factor, while circular symmetry gives the Airy/Bessel form. 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 10.26 — two-dimensional apertures: 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 aperture function; separable rectangles factor, while circular symmetry gives the Airy/Bessel form. The book’s selected-answer check begins If the aperture is symmetrical about a line, the pattern will be symmetrical about a line parallel to it. Moreover, the pattern will be symmetrical about yet another line perpendicular to the aperture’s symmetry axis. This follows from the fact that Fraunhofer patterns have a center of symmetry. Z03_HECH6933_05_GE_SOL.. Substitute the result back into the governing relation to verify its units and sign.

Problem 10.27 — two-dimensional apertures: 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 aperture function; separable rectangles factor, while circular symmetry gives the Airy/Bessel form. Substitution back into the starting relation supplies the final sign and dimensional check.

Problem 10.28 — two-dimensional apertures: 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 aperture function; separable rectangles factor, while circular symmetry gives the Airy/Bessel form. The book’s selected-answer check begins Three parallel short slits.. Substitute the result back into the governing relation to verify its units and sign.

Problem 10.29 — two-dimensional apertures: 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 aperture function; separable rectangles factor, while circular symmetry gives the Airy/Bessel form. The book’s selected-answer check begins Two parallel short slits.. Substitute the result back into the governing relation to verify its units and sign.

Problem 10.30 — two-dimensional apertures: 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 aperture function; separable rectangles factor, while circular symmetry gives the Airy/Bessel form. The book’s selected-answer check begins An equilateral triangular hole.. Substitute the result back into the governing relation to verify its units and sign.

Problem 10.31 — two-dimensional apertures: 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 aperture function; separable rectangles factor, while circular symmetry gives the Airy/Bessel form. The book’s selected-answer check begins A cross-shaped hole.. Substitute the result back into the governing relation to verify its units and sign.

Problem 10.32 — two-dimensional apertures: physical interpretation

State the controlling conservation or symmetry principle first, then use it to determine signs, directions, and limiting behavior. Fourier transform the aperture function; separable rectangles factor, while circular symmetry gives the Airy/Bessel form. The book’s selected-answer check begins The E-field of a rectangular hole.. Substitute the result back into the governing relation to verify its units and sign.

Problem 10.33* — two-dimensional apertures: 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 aperture function; separable rectangles factor, while circular symmetry gives the Airy/Bessel form. 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 10.34* — two-dimensional apertures: 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 aperture function; separable rectangles factor, while circular symmetry gives the Airy/Bessel form. 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 10.35* — two-dimensional apertures: 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 aperture function; separable rectangles factor, while circular symmetry gives the Airy/Bessel form. 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 10.36* — two-dimensional apertures: 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 aperture function; separable rectangles factor, while circular symmetry gives the Airy/Bessel form. 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 10.37* — two-dimensional apertures: 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 aperture function; separable rectangles factor, while circular symmetry gives the Airy/Bessel form. 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 10.38 — two-dimensional apertures: 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 aperture function; separable rectangles factor, while circular symmetry gives the Airy/Bessel form. The book’s selected-answer check begins From Eq. (10.58), q1 ≈ 1.22(ƒ/D)λ ≈ λ.. Substitute the result back into the governing relation to verify its units and sign.

Problem 10.39 — two-dimensional apertures: 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 aperture function; separable rectangles factor, while circular symmetry gives the Airy/Bessel form. The book’s selected-answer check begins Z03_HECH6933_05_GE_SOL.indd 702 08/09/16 9:14 pm Solutions to Selected Problems 703. Substitute the result back into the governing relation to verify its units and sign.

Problem 10.40* — two-dimensional apertures: 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 aperture function; separable rectangles factor, while circular symmetry gives the Airy/Bessel form. 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 10.41* — resolution and gratings: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Combine the Rayleigh criterion with \(d\sin\theta=m\lambda\) and use the number of illuminated grooves for resolving power. 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 10.42* — resolution and gratings: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Combine the Rayleigh criterion with \(d\sin\theta=m\lambda\) and use the number of illuminated grooves for resolving power. 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 10.43* — resolution and gratings: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Combine the Rayleigh criterion with \(d\sin\theta=m\lambda\) and use the number of illuminated grooves for resolving power. 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 10.44* — resolution and gratings: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Combine the Rayleigh criterion with \(d\sin\theta=m\lambda\) and use the number of illuminated grooves for resolving power. 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 10.45 — resolution and gratings: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Combine the Rayleigh criterion with \(d\sin\theta=m\lambda\) and use the number of illuminated grooves for resolving power. The book’s selected-answer check begins 1 part in 1000. 3 yd ≈ 100 inches. (See figure below.) inch 1 10 inch 1 10 inch 1 10. Substitute the result back into the governing relation to verify its units and sign.

Problem 10.46* — resolution and gratings: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Combine the Rayleigh criterion with \(d\sin\theta=m\lambda\) and use the number of illuminated grooves for resolving power. 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 10.47* — resolution and gratings: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Combine the Rayleigh criterion with \(d\sin\theta=m\lambda\) and use the number of illuminated grooves for resolving power. 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 10.48* — resolution and gratings: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Combine the Rayleigh criterion with \(d\sin\theta=m\lambda\) and use the number of illuminated grooves for resolving power. 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 10.49* — resolution and gratings: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Combine the Rayleigh criterion with \(d\sin\theta=m\lambda\) and use the number of illuminated grooves for resolving power. 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 10.50* — resolution and gratings: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Combine the Rayleigh criterion with \(d\sin\theta=m\lambda\) and use the number of illuminated grooves for resolving power. 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 10.51* — resolution and gratings: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Combine the Rayleigh criterion with \(d\sin\theta=m\lambda\) and use the number of illuminated grooves for resolving power. 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 10.52* — resolution and gratings: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Combine the Rayleigh criterion with \(d\sin\theta=m\lambda\) and use the number of illuminated grooves for resolving power. 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 10.53* — resolution and gratings: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Combine the Rayleigh criterion with \(d\sin\theta=m\lambda\) and use the number of illuminated grooves for resolving power. 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 10.54* — resolution and gratings: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Combine the Rayleigh criterion with \(d\sin\theta=m\lambda\) and use the number of illuminated grooves for resolving power. 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 10.55 — resolution and gratings: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Combine the Rayleigh criterion with \(d\sin\theta=m\lambda\) and use the number of illuminated grooves for resolving power. The book’s selected-answer check begins FromEq.(10.32),wherea = (1/1000 lines per cm) = 0.001cm per line (center-to-center), sinum = 1(620 × 10-9 m)/(0.01 × 10-2 m) = 6.2 × 10-2 m and u1 = 3.56°.. Substitute the result back into the governing relation to verify its units and sign.

Problem 10.56* — resolution and gratings: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Combine the Rayleigh criterion with \(d\sin\theta=m\lambda\) and use the number of illuminated grooves for resolving power. 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 10.57* — resolution and gratings: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Combine the Rayleigh criterion with \(d\sin\theta=m\lambda\) and use the number of illuminated grooves for resolving power. 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 10.58* — resolution and gratings: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Combine the Rayleigh criterion with \(d\sin\theta=m\lambda\) and use the number of illuminated grooves for resolving power. 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 10.59* — resolution and gratings: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Combine the Rayleigh criterion with \(d\sin\theta=m\lambda\) and use the number of illuminated grooves for resolving power. 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 10.60* — resolution and gratings: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Combine the Rayleigh criterion with \(d\sin\theta=m\lambda\) and use the number of illuminated grooves for resolving power. 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 10.61 — spectrometers and diffraction systems: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Differentiate the grating equation for angular dispersion and propagate it through the stated focal length or detector geometry. The book’s selected-answer check begins The largest value of m in Eq. (10.32) occurs when the sine function is equal to 1, making the left side of the equation as large as possible; then m = a/λ = (1/9 × 105 )/(3.0 × 108 m/s , 4.0 × 1014 Hz) = 1.4, and only the first-order spectrum is visible.. Substitute the result back into the governing relation to verify its units and sign.

Problem 10.62* — spectrometers and diffraction systems: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Differentiate the grating equation for angular dispersion and propagate it through the stated focal length or detector geometry. 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 10.63 — spectrometers and diffraction systems: derivation

Start from the governing relation rather than the desired result; rearrange until the requested form follows, so the argument is not circular. Differentiate the grating equation for angular dispersion and propagate it through the stated focal length or detector geometry. The book’s selected-answer check begins sinui = n sinun Optical path length difference = ml a sinum na sinun = ml a(sinum sinui) = ml. Substitute the result back into the governing relation to verify its units and sign.

Problem 10.64* — spectrometers and diffraction systems: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Differentiate the grating equation for angular dispersion and propagate it through the stated focal length or detector geometry. 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 10.65 — spectrometers and diffraction systems: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Differentiate the grating equation for angular dispersion and propagate it through the stated focal length or detector geometry. The book’s selected-answer check begins ℛ = mN = 106 , N = 78 × 103 , 6 m = 106 /78 × 103 ∆lfsr = λ/m = 550 nm/(106 /78 × 103 ) = 43 nm ℛ = ℱm = ℱ 2nƒd λ = 106 ∆lfsr = l2 /wnƒd = 0.01512 nm. Substitute the result back into the governing relation to verify its units and sign.

Problem 10.66 — spectrometers and diffraction systems: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Differentiate the grating equation for angular dispersion and propagate it through the stated focal length or detector geometry. The book’s selected-answer check begins ℛ = λ/∆λ = 5893/6 = 982 N = ℛ/m = 982/3 = 327.. Substitute the result back into the governing relation to verify its units and sign.

Problem 10.67* — spectrometers and diffraction systems: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Differentiate the grating equation for angular dispersion and propagate it through the stated focal length or detector geometry. 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 10.68 — spectrometers and diffraction systems: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Differentiate the grating equation for angular dispersion and propagate it through the stated focal length or detector geometry. The book’s selected-answer check begins y = Ll/d d = (20 m) × (5.5 × 10-7 m)/(12 × 10-2 m) = 9.16 × 10-5 m. Substitute the result back into the governing relation to verify its units and sign.

Problem 10.69* — spectrometers and diffraction systems: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Differentiate the grating equation for angular dispersion and propagate it through the stated focal length or detector geometry. 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 10.70 — spectrometers and diffraction systems: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Differentiate the grating equation for angular dispersion and propagate it through the stated focal length or detector geometry. The book’s selected-answer check begins A = 2pr2 3 w 0 sinwdw = 2pr2 (1 cosw) cosw = [r2 + (r + r0)2 r2 λ ]/2r(r + r0) rl = r0 + ll/2 Area of first λ zones A = 2pr2 pr(2r2 + 2rr0 llr0 l2 l2 /4)/(r + r0) Al = A - Al-1 = lpr r + r0 cr0 + (2l - 1)λ 4 d. Substitute the result back into the governing relation to verify its units and sign.

Problem 10.71* — spectrometers and diffraction systems: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Differentiate the grating equation for angular dispersion and propagate it through the stated focal length or detector geometry. 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 10.72* — spectrometers and diffraction systems: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Differentiate the grating equation for angular dispersion and propagate it through the stated focal length or detector geometry. 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 10.73* — spectrometers and diffraction systems: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Differentiate the grating equation for angular dispersion and propagate it through the stated focal length or detector geometry. 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 10.74* — spectrometers and diffraction systems: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Differentiate the grating equation for angular dispersion and propagate it through the stated focal length or detector geometry. 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 10.75* — spectrometers and diffraction systems: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Differentiate the grating equation for angular dispersion and propagate it through the stated focal length or detector geometry. 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 10.76* — spectrometers and diffraction systems: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Differentiate the grating equation for angular dispersion and propagate it through the stated focal length or detector geometry. 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 10.77* — Fresnel diffraction: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Evaluate the Fresnel number or zone construction, retaining the quadratic phase until the far-field approximation is justified. 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 10.78* — Fresnel diffraction: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Evaluate the Fresnel number or zone construction, retaining the quadratic phase until the far-field approximation is justified. 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 10.79* — Fresnel diffraction: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Evaluate the Fresnel number or zone construction, retaining the quadratic phase until the far-field approximation is justified. 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 10.80* — Fresnel diffraction: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Evaluate the Fresnel number or zone construction, retaining the quadratic phase until the far-field approximation is justified. 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 10.81* — Fresnel diffraction: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Evaluate the Fresnel number or zone construction, retaining the quadratic phase until the far-field approximation is justified. 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 10.82* — Fresnel diffraction: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Evaluate the Fresnel number or zone construction, retaining the quadratic phase until the far-field approximation is justified. 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 10.83* — Fresnel diffraction: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Evaluate the Fresnel number or zone construction, retaining the quadratic phase until the far-field approximation is justified. 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 10.84 — Fresnel diffraction: construction

Evaluate the boundary and representative interior values, then draw the requested curve or ray construction to scale. Evaluate the Fresnel number or zone construction, retaining the quadratic phase until the far-field approximation is justified. The book’s selected-answer check begins a n ui um un un ∆w = 5.5 1 1 3 0 2 3. Substitute the result back into the governing relation to verify its units and sign.

Problem 10.85 — Fresnel diffraction: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Evaluate the Fresnel number or zone construction, retaining the quadratic phase until the far-field approximation is justified. The book’s selected-answer check begins I = I0 2 5[1 2 - 𝒞(v1)]2 + [1 2 - 𝒮(v1)]2 6 I = I0 2 a 1 pv1 b 2 csin2 a pv2 1 2 b + cos2 a pv2 1 2 bd I = I0 2 a 1 pv1 b 2 Z03_HECH6933_05_GE_SOL.indd 703 08/09/16 9:15 pm 704 Solutions to Selected Problems 11.9 ℱ[pf(y) + qh(y)] = pF(k) + qH(k). 11.11 F(k) = L sinc2 kL/2 at k = 0, F(0) = L, and F(±2p/L) = 0. 11.18 3 ∞. Substitute the result back into the governing relation to verify its units and sign.

Problem 10.86 — Fresnel diffraction: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Evaluate the Fresnel number or zone construction, retaining the quadratic phase until the far-field approximation is justified. The book’s selected-answer check begins Fringes in both the clear and shadow region [(see M. P. Givens and W. L. Goffe, Am. J. Phys. 34, 248 (1966)].. Substitute the result back into the governing relation to verify its units and sign.

Problem 10.87 — Fresnel diffraction: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Evaluate the Fresnel number or zone construction, retaining the quadratic phase until the far-field approximation is justified. The book’s selected-answer check begins u = y[2/lr0]1/2 ; ∆u = ∆y × 103 = 2.5. +kp −kp F(k) 0 k. Substitute the result back into the governing relation to verify its units and sign.

Problem 10.88 — Fresnel diffraction: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Evaluate the Fresnel number or zone construction, retaining the quadratic phase until the far-field approximation is justified. Substitution back into the starting relation supplies the final sign and dimensional check.

Problem 10.89* — Fresnel diffraction: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Evaluate the Fresnel number or zone construction, retaining the quadratic phase until the far-field approximation is justified. 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 10.90* — Fresnel diffraction: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Evaluate the Fresnel number or zone construction, retaining the quadratic phase until the far-field approximation is justified. 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 10.91* — Fresnel diffraction: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Evaluate the Fresnel number or zone construction, retaining the quadratic phase until the far-field approximation is justified. 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 10.92* — Fresnel diffraction: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Evaluate the Fresnel number or zone construction, retaining the quadratic phase until the far-field approximation is justified. 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 10.93* — Fresnel diffraction: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Evaluate the Fresnel number or zone construction, retaining the quadratic phase until the far-field approximation is justified. Finish by checking the governing equation, the dimensions, and the zero/large-parameter limit; these checks replace reliance on an unavailable answer-key entry.