Chapter 11: Statistical Optics ============================== Source: Saleh and Teich, *Fundamentals of Photonics*, second edition, Chapter 11. In-text exercises ----------------- Exercise 11.1-1 — Coherence-time definitions ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ **Step 1 — Definitions and setup.** Symbols are local to this item and follow the chapter convention. Each physical quantity and supplied numerical value is introduced at its first use below; angles are in radians unless a degree symbol is shown, and units are retained through numerical substitution. .. _fop-exercise-11-1-1-illustration: .. figure:: /_static/knowledge_base/worked_exercises/fundamentals_of_photonics/exercise_illustrations/exercise_11_01_01.svg :alt: Illustrated calculation map for Exercise 11.1-1, Coherence-time definitions :align: center :width: 95% **Figure 69 — Exercise 11.1-1: Coherence-time definitions.** The diagram identifies the input quantities, physical operation, requested result, variable meanings, and an independent verification route. Every symbol in the variable strip is labeled on the model itself. **Step 2 — Mathematical formulas used.** The working uses :ref:`integration identities `, :ref:`exponential, logarithmic, and phasor identities `, and :ref:`algebraic rearrangement and dimensional checks `. **Step 3 — Worked derivation.** The calculation is kept in symbolic form until the governing relation has been rearranged for the requested quantity. Insert each supplied :math:`g(\tau)` in :math:`T_c=\int|g|^2d\tau`. Both integrals return the parameter :math:`T_c`. At :math:`\tau=T_c`, the exponential magnitude has fallen to :math:`e^{-1/2}` and the Gaussian to :math:`e^{-\pi/2}`. **Step 4 — State the numbered result.** The principal result obtained in the working is .. math:: :label: fop-exercise-11-1-1-result \tau=T_c **Step 5 — Check.** Equation :eq:`fop-exercise-11-1-1-result` can be checked by substituting it back into the preceding governing relation and reversing the algebraic steps. Differentiation of an antiderivative, or normalization of a definite integral, checks the integration step. Exercise 11.1-2 — Reciprocal equivalent widths ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ **Step 1 — Definitions and setup.** Symbols are local to this item and follow the chapter convention. Each physical quantity and supplied numerical value is introduced at its first use below; angles are in radians unless a degree symbol is shown, and units are retained through numerical substitution. .. _fop-exercise-11-1-2-illustration: .. figure:: /_static/knowledge_base/worked_exercises/fundamentals_of_photonics/exercise_illustrations/exercise_11_01_02.svg :alt: Illustrated calculation map for Exercise 11.1-2, Reciprocal equivalent widths :align: center :width: 95% **Figure 70 — Exercise 11.1-2: Reciprocal equivalent widths.** The diagram identifies the input quantities, physical operation, requested result, variable meanings, and an independent verification route. Every symbol in the variable strip is labeled on the model itself. **Step 2 — Mathematical formulas used.** The working uses :ref:`integration identities ` and :ref:`algebraic rearrangement and dimensional checks `. **Step 3 — Worked derivation.** The calculation is kept in symbolic form until the governing relation has been rearranged for the requested quantity. Parseval applied to the Wiener--Khinchin pair gives :math:`\int|G|^2d\tau=\int|S|^2d\nu`; dividing by :math:`G(0)^2=[\int S]^2` shows directly that the power-equivalent widths obey :math:`\boxed{T_c\Delta\nu_c=1}`. **Step 4 — State the numbered result.** The principal result obtained in the working is .. math:: :label: fop-exercise-11-1-2-result \boxed{T_c\Delta\nu_c=1} **Step 5 — Check.** Equation :eq:`fop-exercise-11-1-2-result` can be checked by substituting it back into the preceding governing relation and reversing the algebraic steps. Differentiation of an antiderivative, or normalization of a definite integral, checks the integration step. Exercise 11.1-3 — Mutual-coherence wave equations ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ **Step 1 — Definitions and setup.** Symbols are local to this item and follow the chapter convention. Each physical quantity and supplied numerical value is introduced at its first use below; angles are in radians unless a degree symbol is shown, and units are retained through numerical substitution. .. _fop-exercise-11-1-3-illustration: .. figure:: /_static/knowledge_base/worked_exercises/fundamentals_of_photonics/exercise_illustrations/exercise_11_01_03.svg :alt: Illustrated calculation map for Exercise 11.1-3, Mutual-coherence wave equations :align: center :width: 95% **Figure 71 — Exercise 11.1-3: Mutual-coherence wave equations.** The diagram identifies the input quantities, physical operation, requested result, variable meanings, and an independent verification route. Every symbol in the variable strip is labeled on the model itself. **Step 2 — Mathematical formulas used.** The working uses :ref:`expectation, variance, and probability identities ` and :ref:`algebraic rearrangement and dimensional checks `. **Step 3 — Worked derivation.** The calculation is kept in symbolic form until the governing relation has been rearranged for the requested quantity. Apply the wave operator to either field inside :math:`G=\langle U^*(\mathbf r_1,t)U(\mathbf r_2,t+\tau)\rangle`; linearity lets it pass through the average. This gives the two Wolf equations, one in :math:`\mathbf r_1,-\tau` and one in :math:`\mathbf r_2,+\tau`. **Step 4 — State the numbered result.** The principal result obtained in the working is .. math:: :label: fop-exercise-11-1-3-result G=\langle U^*(\mathbf r_1,t)U(\mathbf r_2,t+\tau)\rangle **Step 5 — Check.** Equation :eq:`fop-exercise-11-1-3-result` can be checked by substituting it back into the preceding governing relation and reversing the algebraic steps. For a probability result, verify the zero-to-one bounds; when a full distribution is present, also verify normalization and nonnegative variance. Exercise 11.4-1 — Polarized plus unpolarized decomposition ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ **Step 1 — Definitions and setup.** Symbols are local to this item and follow the chapter convention. Each physical quantity and supplied numerical value is introduced at its first use below; angles are in radians unless a degree symbol is shown, and units are retained through numerical substitution. .. _fop-exercise-11-4-1-illustration: .. figure:: /_static/knowledge_base/worked_exercises/fundamentals_of_photonics/exercise_illustrations/exercise_11_04_01.svg :alt: Illustrated calculation map for Exercise 11.4-1, Polarized plus unpolarized decomposition :align: center :width: 95% **Figure 72 — Exercise 11.4-1: Polarized plus unpolarized decomposition.** The diagram identifies the input quantities, physical operation, requested result, variable meanings, and an independent verification route. Every symbol in the variable strip is labeled on the model itself. **Step 2 — Mathematical formulas used.** The working uses :ref:`Fourier-transform and convolution identities ` and :ref:`matrix multiplication and eigenvalue rules `. **Step 3 — Worked derivation.** The calculation is kept in symbolic form until the governing relation has been rearranged for the requested quantity. An unpolarized coherency matrix contributes equal diagonal terms and no cross term; the rank-one polarized matrix supplies the remaining diagonal imbalance and correlation. Adding the stated intensities yields :math:`I_x,I_y` and :math:`|g_{xy}|`; its rank-one weight is exactly the degree of polarization. End-of-chapter problems ----------------------- **Step 4 — Interpret the result.** The final relation or conclusion in Step 3 is the requested result. Read its sign, scale, or physical classification using the conventions fixed in Step 1. **Step 5 — Check.** Multiply the matrices independently in the stated input-to-output order and verify that every product has compatible dimensions. Problem 11.1-4 — Lorentzian LED spectrum ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ **Definitions and setup.** Symbols are local to this item and follow the chapter convention. Each physical quantity and supplied numerical value is introduced at its first use below; angles are in radians unless a degree symbol is shown, and units are retained through numerical substitution. **Mathematical formulas used.** The working uses :ref:`exponential, logarithmic, and phasor identities ` and :ref:`algebraic rearrangement and dimensional checks `. **Worked derivation.** The calculation is kept in symbolic form until the governing relation has been rearranged for the requested quantity. :math:`\Delta\lambda\simeq\lambda_0^2\Delta\nu/c= \boxed{1.634\ \mathrm{nm}}`. A Lorentzian has :math:`T_c=1/(\pi\Delta\nu)=\boxed{0.3183\ \mathrm{ps}}` and :math:`l_c=\boxed{95.4\ \mathrm{\mu m}}`. Since :math:`|g|=e^{-\pi\Delta\nu|\tau|}`, the half-coherence delay is 0.2206 ps (path :math:`\boxed{66.1\ \mathrm{\mu m}}`). **Numbered result.** The principal result obtained in the working is .. math:: :label: fop-problem-11-1-4-result \boxed{66.1\ \mathrm{\mu m}} **Check.** Equation :eq:`fop-problem-11-1-4-result` can be checked by substituting it back into the preceding governing relation and reversing the algebraic steps. Check that dimensions agree term by term, then test the simplest symmetry or limiting case for the expected sign and scale. Repeat the substitution with unrounded intermediate values and retain the displayed units; the final unit must have the requested dimension. Problem 11.1-5 — Wiener--Khinchin theorem ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ **Definitions and setup.** Symbols are local to this item and follow the chapter convention. Each physical quantity and supplied numerical value is introduced at its first use below; angles are in radians unless a degree symbol is shown, and units are retained through numerical substitution. **Mathematical formulas used.** The working uses :ref:`stationary-value condition `, :ref:`Fourier-transform and convolution identities `, and :ref:`integration identities `. **Worked derivation.** The calculation is kept in symbolic form until the governing relation has been rearranged for the requested quantity. Expand :math:`|V_T(\nu)|^2` as a double time integral, set :math:`\tau=t_2-t_1`, divide by :math:`T`, and take the stationary infinite window limit. The remaining inner average is :math:`G(\tau)`, giving :math:`S=\mathcal F\{G\}`. Integrating over frequency produces :math:`\delta(\tau)` and hence :math:`\int S\,d\nu=G(0)=I`. **Numbered result.** The principal result obtained in the working is .. math:: :label: fop-problem-11-1-5-result \int S\,d\nu=G(0)=I **Check.** Equation :eq:`fop-problem-11-1-5-result` can be checked by substituting it back into the preceding governing relation and reversing the algebraic steps. Transform dimensions and the expected even/odd or conjugate symmetry provide an independent check on signs and scale factors. Problem 11.1-6 — Gaussian mutual intensity ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ **Definitions and setup.** Symbols are local to this item and follow the chapter convention. Each physical quantity and supplied numerical value is introduced at its first use below; angles are in radians unless a degree symbol is shown, and units are retained through numerical substitution. **Mathematical formulas used.** The working uses :ref:`integration identities `, :ref:`exponential, logarithmic, and phasor identities `, and :ref:`algebraic rearrangement and dimensional checks `. **Worked derivation.** The calculation is kept in symbolic form until the governing relation has been rearranged for the requested quantity. Setting :math:`x_1=x_2=x` gives :math:`I(x)=I_0e^{-2x^2/W_0^2}`. Normalization cancels those intensity envelopes, leaving :math:`\boxed{g=e^{-(x_1-x_2)^2/\rho_c^2}}`. :math:`I_0,W_0,\rho_c` are peak intensity, beam radius, and transverse coherence distance. **Numbered result.** The principal result obtained in the working is .. math:: :label: fop-problem-11-1-6-result \boxed{g=e^{-(x_1-x_2)^2/\rho_c^2}} **Check.** Equation :eq:`fop-problem-11-1-6-result` can be checked by substituting it back into the preceding governing relation and reversing the algebraic steps. Differentiation of an antiderivative, or normalization of a definite integral, checks the integration step. Problem 11.1-7 — Position-dependent colour ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ **Definitions and setup.** Symbols are local to this item and follow the chapter convention. Each physical quantity and supplied numerical value is introduced at its first use below; angles are in radians unless a degree symbol is shown, and units are retained through numerical substitution. **Mathematical formulas used.** The working uses :ref:`algebraic rearrangement and dimensional checks `. **Worked derivation.** The calculation is kept in symbolic form until the governing relation has been rearranged for the requested quantity. Intensity is one, :math:`l_c=cT_c=\boxed{300\ \mathrm m}`, and transverse coherence distance is :math:`\rho_c=1` mm. The spectrum is centered at 5e14 Hz when :math:`x_1+x_2>0` and 6e14 Hz when negative; only colour is position dependent. Film records two uniform half-planes near 600 and 500 nm. **Numbered result.** The principal result obtained in the working is .. math:: :label: fop-problem-11-1-7-result l_c=cT_c=\boxed{300\ \mathrm m} **Check.** Equation :eq:`fop-problem-11-1-7-result` can be checked by substituting it back into the preceding governing relation and reversing the algebraic steps. Check that dimensions agree term by term, then test the simplest symmetry or limiting case for the expected sign and scale. Repeat the substitution with unrounded intermediate values and retain the displayed units; the final unit must have the requested dimension. Problem 11.1-8 — Coherence length estimates ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ **Definitions and setup.** Symbols are local to this item and follow the chapter convention. Each physical quantity and supplied numerical value is introduced at its first use below; angles are in radians unless a degree symbol is shown, and units are retained through numerical substitution. **Mathematical formulas used.** The working uses :ref:`algebraic rearrangement and dimensional checks `. **Worked derivation.** The calculation is kept in symbolic form until the governing relation has been rearranged for the requested quantity. For a narrow band, :math:`\Delta\nu\simeq c\Delta\lambda/\lambda^2`, so :math:`l_c=c/\Delta\nu\simeq\lambda^2/\Delta\lambda`. A uniform spectrum from :math:`\lambda_{min}` to :math:`2\lambda_{min}` spans frequency :math:`c/(2\lambda_{min})`; inversion gives :math:`\boxed{l_c=2\lambda_{min}=\lambda_{max}}`. **Numbered result.** The principal result obtained in the working is .. math:: :label: fop-problem-11-1-8-result \boxed{l_c=2\lambda_{min}=\lambda_{max}} **Check.** Equation :eq:`fop-problem-11-1-8-result` can be checked by substituting it back into the preceding governing relation and reversing the algebraic steps. Check that dimensions agree term by term, then test the simplest symmetry or limiting case for the expected sign and scale. Problem 11.1-9 — Spatial coherence from a point source ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ **Definitions and setup.** Symbols are local to this item and follow the chapter convention. Each physical quantity and supplied numerical value is introduced at its first use below; angles are in radians unless a degree symbol is shown, and units are retained through numerical substitution. **Mathematical formulas used.** The working uses :ref:`exponential, logarithmic, and phasor identities ` and :ref:`algebraic rearrangement and dimensional checks `. **Worked derivation.** The calculation is kept in symbolic form until the governing relation has been rearranged for the requested quantity. The path delay is :math:`\tau=[\sqrt{d^2+x^2}-d]/c`; a Lorentzian therefore gives :math:`\boxed{|g(x)|=\exp[-|\sqrt{d^2+x^2}-d|/(2cT_c)]}` when :math:`T_c` uses the book's power-equivalent convention. It is even, unity at zero, and falls approximately as :math:`e^{-x^2/(4dcT_c)}` near the axis. **Numbered result.** The principal result obtained in the working is .. math:: :label: fop-problem-11-1-9-result \boxed{|g(x)|=\exp[-|\sqrt{d^2+x^2}-d|/(2cT_c)]} **Check.** Equation :eq:`fop-problem-11-1-9-result` can be checked by substituting it back into the preceding governing relation and reversing the algebraic steps. Check that dimensions agree term by term, then test the simplest symmetry or limiting case for the expected sign and scale. Problem 11.1-10 — Gaussian coherence area ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ **Definitions and setup.** Symbols are local to this item and follow the chapter convention. Each physical quantity and supplied numerical value is introduced at its first use below; angles are in radians unless a degree symbol is shown, and units are retained through numerical substitution. **Mathematical formulas used.** The working uses :ref:`vector-calculus identities `, :ref:`integration identities `, and :ref:`algebraic rearrangement and dimensional checks `. **Worked derivation.** The calculation is kept in symbolic form until the governing relation has been rearranged for the requested quantity. Applying either spatial wave operator to :math:`G` gives :math:`(\nabla^2+k_0^2)J=0`. The Gaussian solution has transverse coherence radius :math:`W(z)=W_0\sqrt{1+(z/z_0)^2}`; hence coherence area :math:`\boxed{A_c=\pi W^2(z)}` up to the selected width convention and it grows with :math:`|z|`. **Numbered result.** The principal result obtained in the working is .. math:: :label: fop-problem-11-1-10-result \boxed{A_c=\pi W^2(z)} **Check.** Equation :eq:`fop-problem-11-1-10-result` can be checked by substituting it back into the preceding governing relation and reversing the algebraic steps. Differentiation of an antiderivative, or normalization of a definite integral, checks the integration step. Repeat the substitution with unrounded intermediate values and retain the displayed units; the final unit must have the requested dimension. Problem 11.2-1 — Sodium interferogram visibility ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ **Definitions and setup.** Symbols are local to this item and follow the chapter convention. Each physical quantity and supplied numerical value is introduced at its first use below; angles are in radians unless a degree symbol is shown, and units are retained through numerical substitution. **Mathematical formulas used.** The working uses :ref:`stationary-value condition `, :ref:`optical path and Fermat's principle `, and :ref:`exponential, logarithmic, and phasor identities `. **Worked derivation.** The calculation is kept in symbolic form until the governing relation has been rearranged for the requested quantity. :math:`V=|g|=e^{-\pi\Delta\nu|\tau|}`. Setting :math:`V=1/2` gives maximum optical path difference :math:`\boxed{c\ln2/(\pi\Delta\nu)=0.1323\ \mathrm{mm}}`. **Numbered result.** The principal result obtained in the working is .. math:: :label: fop-problem-11-2-1-result \boxed{c\ln2/(\pi\Delta\nu)=0.1323\ \mathrm{mm}} **Check.** Equation :eq:`fop-problem-11-2-1-result` can be checked by substituting it back into the preceding governing relation and reversing the algebraic steps. Check that dimensions agree term by term, then test the simplest symmetry or limiting case for the expected sign and scale. Repeat the substitution with unrounded intermediate values and retain the displayed units; the final unit must have the requested dimension. Problem 11.2-2 — Observable Young fringes ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ **Definitions and setup.** Symbols are local to this item and follow the chapter convention. Each physical quantity and supplied numerical value is introduced at its first use below; angles are in radians unless a degree symbol is shown, and units are retained through numerical substitution. **Mathematical formulas used.** The working uses :ref:`algebraic rearrangement and dimensional checks `. **Worked derivation.** The calculation is kept in symbolic form until the governing relation has been rearranged for the requested quantity. For every Table 11.1-2 source, divide coherence length by the Young path change per fringe (one wavelength): :math:`\boxed{N\simeq l_c/\lambda_0}`. This gives the requested values without mixing temporal coherence with the assumed-perfect spatial coherence. **Numbered result.** The principal result obtained in the working is .. math:: :label: fop-problem-11-2-2-result \boxed{N\simeq l_c/\lambda_0} **Check.** Equation :eq:`fop-problem-11-2-2-result` can be checked by substituting it back into the preceding governing relation and reversing the algebraic steps. Check that dimensions agree term by term, then test the simplest symmetry or limiting case for the expected sign and scale. Problem 11.2-3 — Can correlation shift a spectrum? ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ **Definitions and setup.** Symbols are local to this item and follow the chapter convention. Each physical quantity and supplied numerical value is introduced at its first use below; angles are in radians unless a degree symbol is shown, and units are retained through numerical substitution. **Mathematical formulas used.** The working uses :ref:`stationary-value condition `, :ref:`Fourier-transform and convolution identities `, and :ref:`integration identities `. **Worked derivation.** The calculation is kept in symbolic form until the governing relation has been rearranged for the requested quantity. :math:`S=S_1+S_2+2\Re S_{12}` and the spectral Cauchy--Schwarz bound is :math:`|S_{12}(\nu)|\leq\sqrt{S_1S_2}=S_1`. Correlation can reshape or cancel parts of the common Gaussian support, but a stationary linear superposition cannot translate every frequency to create an exact same-width Gaussian at a new carrier. A genuine shift requires time variation/nonlinearity (e.g. Doppler modulation). **Numbered result.** The principal result obtained in the working is .. math:: :label: fop-problem-11-2-3-result |S_{12}(\nu)|\leq\sqrt{S_1S_2}=S_1 **Check.** Equation :eq:`fop-problem-11-2-3-result` can be checked by substituting it back into the preceding governing relation and reversing the algebraic steps. Transform dimensions and the expected even/odd or conjugate symmetry provide an independent check on signs and scale factors. Problem 11.3-1 — Partially coherent Gaussian beam ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ **Definitions and setup.** Symbols are local to this item and follow the chapter convention. Each physical quantity and supplied numerical value is introduced at its first use below; angles are in radians unless a degree symbol is shown, and units are retained through numerical substitution. **Mathematical formulas used.** The working uses :ref:`Fourier-transform and convolution identities `, :ref:`vector-calculus identities `, and :ref:`integration identities `. **Worked derivation.** The calculation is kept in symbolic form until the governing relation has been rearranged for the requested quantity. Double Fourier propagation of the Gaussian mutual intensity remains Gaussian. Its far-field width is :math:`\boxed{W^2(z)=W_0^2+(\lambda z/\pi)^2(1/W_0^2+2/\rho_c^2)}` under the stated :math:`1/e^2` convention. Smaller coherence distance increases angular divergence. **Numbered result.** The principal result obtained in the working is .. math:: :label: fop-problem-11-3-1-result \boxed{W^2(z)=W_0^2+(\lambda z/\pi)^2(1/W_0^2+2/\rho_c^2)} **Check.** Equation :eq:`fop-problem-11-3-1-result` can be checked by substituting it back into the preceding governing relation and reversing the algebraic steps. Transform dimensions and the expected even/odd or conjugate symmetry provide an independent check on signs and scale factors. Repeat the substitution with unrounded intermediate values and retain the displayed units; the final unit must have the requested dimension. Problem 11.3-2 — Incoherent Fourier illumination ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ **Definitions and setup.** Symbols are local to this item and follow the chapter convention. Each physical quantity and supplied numerical value is introduced at its first use below; angles are in radians unless a degree symbol is shown, and units are retained through numerical substitution. **Mathematical formulas used.** The working uses :ref:`Fourier-transform and convolution identities `. **Worked derivation.** The calculation is kept in symbolic form until the governing relation has been rearranged for the requested quantity. Each source point forms its own shifted Fourier intensity and mutually incoherent contributions add. For spatially uniform incoherent illumination, the result is the convolution of :math:`|F|^2` with the source angular intensity; in the ideal infinite-uniform limit it is flat. Unlike coherent illumination, complex Fourier amplitudes do not add. **Check.** Transform dimensions and the expected even/odd or conjugate symmetry provide an independent check on signs and scale factors. Problem 11.3-3 — Two incoherent points ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ **Definitions and setup.** Symbols are local to this item and follow the chapter convention. Each physical quantity and supplied numerical value is introduced at its first use below; angles are in radians unless a degree symbol is shown, and units are retained through numerical substitution. **Mathematical formulas used.** The working uses :ref:`Fourier-transform and convolution identities `, :ref:`integration identities `, and :ref:`trigonometric and small-angle identities `. **Worked derivation.** The calculation is kept in symbolic form until the governing relation has been rearranged for the requested quantity. Van Cittert--Zernike gives the normalized transform of two delta sources: :math:`\boxed{g(x_1,x_2)=\cos[2\pi a(x_1-x_2)/(\lambda d)]}` up to a common phase. Coherence alternates between magnitude one and zero across separation. **Numbered result.** The principal result obtained in the working is .. math:: :label: fop-problem-11-3-3-result \boxed{g(x_1,x_2)=\cos[2\pi a(x_1-x_2)/(\lambda d)]} **Check.** Equation :eq:`fop-problem-11-3-3-result` can be checked by substituting it back into the preceding governing relation and reversing the algebraic steps. Transform dimensions and the expected even/odd or conjugate symmetry provide an independent check on signs and scale factors. Problem 11.3-4 — Slit-generated coherence ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ **Definitions and setup.** Symbols are local to this item and follow the chapter convention. Each physical quantity and supplied numerical value is introduced at its first use below; angles are in radians unless a degree symbol is shown, and units are retained through numerical substitution. **Mathematical formulas used.** The working uses :ref:`Fourier-transform and convolution identities `, :ref:`integration identities `, and :ref:`algebraic rearrangement and dimensional checks `. **Worked derivation.** The calculation is kept in symbolic form until the governing relation has been rearranged for the requested quantity. The normalized transform of a uniform slit of width :math:`2a` is :math:`\boxed{g(x_1,x_2)=\operatorname{sinc}[2a(x_1-x_2)/(\lambda f)]}` up to quadratic phase; coherence width is inversely proportional to slit width. **Numbered result.** The principal result obtained in the working is .. math:: :label: fop-problem-11-3-4-result \boxed{g(x_1,x_2)=\operatorname{sinc}[2a(x_1-x_2)/(\lambda f)]} **Check.** Equation :eq:`fop-problem-11-3-4-result` can be checked by substituting it back into the preceding governing relation and reversing the algebraic steps. Transform dimensions and the expected even/odd or conjugate symmetry provide an independent check on signs and scale factors. Problem 11.4-2 — Equal-component partial polarization ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ **Definitions and setup.** Symbols are local to this item and follow the chapter convention. Each physical quantity and supplied numerical value is introduced at its first use below; angles are in radians unless a degree symbol is shown, and units are retained through numerical substitution. **Mathematical formulas used.** The working uses :ref:`matrix multiplication and eigenvalue rules ` and :ref:`algebraic rearrangement and dimensional checks `. **Worked derivation.** The calculation is kept in symbolic form until the governing relation has been rearranged for the requested quantity. Here :math:`P=|g_{xy}|`. The coherency matrix is :math:`\boxed{J=\tfrac12\begin{bmatrix}1&-jP\\jP&1\end{bmatrix}}` for :math:`P=0,0.5,1`: respectively unpolarized, mixed, and circularly polarized light. An x polarizer transmits :math:`\boxed{I_x=1/2}` in every case. **Numbered result.** The principal result obtained in the working is .. math:: :label: fop-problem-11-4-2-result \boxed{I_x=1/2} **Check.** Equation :eq:`fop-problem-11-4-2-result` can be checked by substituting it back into the preceding governing relation and reversing the algebraic steps. Multiply the matrices independently in the stated input-to-output order and verify that every product has compatible dimensions.