Chapter 12: Basics of Coherence Theory

Source: Eugene Hecht, Optics, fifth Global Edition, Chapter 12. 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 12.1* — visibility and coherence: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Use \(\mathcal V=(I_{max}-I_{min})/(I_{max}+I_{min})\) and identify the normalized mutual-coherence factor. Finish by checking the governing equation, the dimensions, and the zero/large-parameter limit; these checks replace reliance on an unavailable answer-key entry.

Problem 12.2* — visibility and coherence: derivation

Start from the governing relation rather than the desired result; rearrange until the requested form follows, so the argument is not circular. Use \(\mathcal V=(I_{max}-I_{min})/(I_{max}+I_{min})\) and identify the normalized mutual-coherence factor. Finish by checking the governing equation, the dimensions, and the zero/large-parameter limit; these checks replace reliance on an unavailable answer-key entry.

Problem 12.3* — visibility and coherence: derivation

Start from the governing relation rather than the desired result; rearrange until the requested form follows, so the argument is not circular. Use \(\mathcal V=(I_{max}-I_{min})/(I_{max}+I_{min})\) and identify the normalized mutual-coherence factor. Finish by checking the governing equation, the dimensions, and the zero/large-parameter limit; these checks replace reliance on an unavailable answer-key entry.

Problem 12.4* — visibility and coherence: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Use \(\mathcal V=(I_{max}-I_{min})/(I_{max}+I_{min})\) and identify the normalized mutual-coherence factor. Finish by checking the governing equation, the dimensions, and the zero/large-parameter limit; these checks replace reliance on an unavailable answer-key entry.

Problem 12.5* — visibility and coherence: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Use \(\mathcal V=(I_{max}-I_{min})/(I_{max}+I_{min})\) and identify the normalized mutual-coherence factor. Finish by checking the governing equation, the dimensions, and the zero/large-parameter limit; these checks replace reliance on an unavailable answer-key entry.

Problem 12.6* — visibility and coherence: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Use \(\mathcal V=(I_{max}-I_{min})/(I_{max}+I_{min})\) and identify the normalized mutual-coherence factor. Finish by checking the governing equation, the dimensions, and the zero/large-parameter limit; these checks replace reliance on an unavailable answer-key entry.

Problem 12.7* — visibility and coherence: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Use \(\mathcal V=(I_{max}-I_{min})/(I_{max}+I_{min})\) and identify the normalized mutual-coherence factor. Finish by checking the governing equation, the dimensions, and the zero/large-parameter limit; these checks replace reliance on an unavailable answer-key entry.

Problem 12.8 — visibility and coherence: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Use \(\mathcal V=(I_{max}-I_{min})/(I_{max}+I_{min})\) and identify the normalized mutual-coherence factor. The book’s selected-answer check begins At low pressures, the intensity emitted from the lamp is low, the bandwidth is narrow, and the coherence length is large. The fringes will initially display a high contrast, although they’ll be fairly faint. As the pressure builds, the coherence length will decrease, the contrast will drop off, and the fringes might ev. Substitute the result back into the governing relation to verify its units and sign.

Problem 12.9* — visibility and coherence: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Use \(\mathcal V=(I_{max}-I_{min})/(I_{max}+I_{min})\) and identify the normalized mutual-coherence factor. Finish by checking the governing equation, the dimensions, and the zero/large-parameter limit; these checks replace reliance on an unavailable answer-key entry.

Problem 12.10* — visibility and coherence: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Use \(\mathcal V=(I_{max}-I_{min})/(I_{max}+I_{min})\) and identify the normalized mutual-coherence factor. Finish by checking the governing equation, the dimensions, and the zero/large-parameter limit; these checks replace reliance on an unavailable answer-key entry.

Problem 12.11 — visibility and coherence: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Use \(\mathcal V=(I_{max}-I_{min})/(I_{max}+I_{min})\) and identify the normalized mutual-coherence factor. The book’s selected-answer check begins Each sine function in the signal produces a cosinusoidal autocorrelation function with its own wavelength and amplitude. All of these are in phase at the zero delay point corresponding to t = 0. Beyond that origin, the cosines soon fall out of phase, producing a jumble where destructive interference is more likely. (Th. Substitute the result back into the governing relation to verify its units and sign.

Problem 12.12* — visibility and coherence: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Use \(\mathcal V=(I_{max}-I_{min})/(I_{max}+I_{min})\) and identify the normalized mutual-coherence factor. Finish by checking the governing equation, the dimensions, and the zero/large-parameter limit; these checks replace reliance on an unavailable answer-key entry.

Problem 12.13 — visibility and coherence: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Use \(\mathcal V=(I_{max}-I_{min})/(I_{max}+I_{min})\) and identify the normalized mutual-coherence factor. The book’s selected-answer check begins The irradiance at g0 arising from a point source is 4I0 cos2 (d/2) = 2I0(1 + cos d) For a differential source element of width dy at point S′, y from axis, the OPD to P at Y via the two slits is Λ = (S′S1 + S1P) - (S′S2 + S2P) = (S′S1 - S′S2) + (S1P + S2P) = ay/λ + aY/s from Section 9.3. 11.32 We see that ƒ(x) is the c. Substitute the result back into the governing relation to verify its units and sign.

Problem 12.14 — visibility and coherence: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Use \(\mathcal V=(I_{max}-I_{min})/(I_{max}+I_{min})\) and identify the normalized mutual-coherence factor. Substitution back into the starting relation supplies the final sign and dimensional check.

Problem 12.15 — visibility and coherence: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Use \(\mathcal V=(I_{max}-I_{min})/(I_{max}+I_{min})\) and identify the normalized mutual-coherence factor. Substitution back into the starting relation supplies the final sign and dimensional check.

Problem 12.16* — mutual coherence and source extent: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Apply the mutual-coherence integral or Van Cittert-Zernike transform and compare detector spacing with the coherence width. 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 12.17* — mutual coherence and source extent: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Apply the mutual-coherence integral or Van Cittert-Zernike transform and compare detector spacing with the coherence width. 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 12.18 — mutual coherence and source extent: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Apply the mutual-coherence integral or Van Cittert-Zernike transform and compare detector spacing with the coherence width. Substitution back into the starting relation supplies the final sign and dimensional check.

Problem 12.19* — mutual coherence and source extent: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Apply the mutual-coherence integral or Van Cittert-Zernike transform and compare detector spacing with the coherence width. 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 12.20 — mutual coherence and source extent: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Apply the mutual-coherence integral or Van Cittert-Zernike transform and compare detector spacing with the coherence width. Substitution back into the starting relation supplies the final sign and dimensional check.

Problem 12.21* — mutual coherence and source extent: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Apply the mutual-coherence integral or Van Cittert-Zernike transform and compare detector spacing with the coherence width. 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 12.22* — mutual coherence and source extent: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Apply the mutual-coherence integral or Van Cittert-Zernike transform and compare detector spacing with the coherence width. 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 12.23* — mutual coherence and source extent: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Apply the mutual-coherence integral or Van Cittert-Zernike transform and compare detector spacing with the coherence width. 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 12.24 — mutual coherence and source extent: design

Translate every stated requirement into an equation, solve the simultaneous constraints, and reject roots that violate the geometry. Apply the mutual-coherence integral or Van Cittert-Zernike transform and compare detector spacing with the coherence width. Substitution back into the starting relation supplies the final sign and dimensional check.

Problem 12.25* — mutual coherence and source extent: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Apply the mutual-coherence integral or Van Cittert-Zernike transform and compare detector spacing with the coherence width. 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 12.26* — mutual coherence and source extent: derivation

Start from the governing relation rather than the desired result; rearrange until the requested form follows, so the argument is not circular. Apply the mutual-coherence integral or Van Cittert-Zernike transform and compare detector spacing with the coherence width. 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 12.27 — mutual coherence and source extent: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Apply the mutual-coherence integral or Van Cittert-Zernike transform and compare detector spacing with the coherence width. Substitution back into the starting relation supplies the final sign and dimensional check.

Problem 12.28* — mutual coherence and source extent: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Apply the mutual-coherence integral or Van Cittert-Zernike transform and compare detector spacing with the coherence width. 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 12.29* — mutual coherence and source extent: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Apply the mutual-coherence integral or Van Cittert-Zernike transform and compare detector spacing with the coherence width. 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 12.30* — mutual coherence and source extent: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Apply the mutual-coherence integral or Van Cittert-Zernike transform and compare detector spacing with the coherence width. Finish by checking the governing equation, the dimensions, and the zero/large-parameter limit; these checks replace reliance on an unavailable answer-key entry.