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 :math:`\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 :math:`\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 :math:`\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 :math:`\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 :math:`\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 :math:`\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 :math:`\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 :math:`\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 :math:`\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 :math:`\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 :math:`\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 :math:`\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 :math:`\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 :math:`\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 :math:`\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.