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
Brief solution
1. Method. The working uses integration identities, exponential, logarithmic, and phasor identities, and algebraic rearrangement and dimensional checks.
2. Reasoning and answer.
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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.
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 integration identities, exponential, logarithmic, and phasor identities, and 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.
Detailed step 1. Insert each supplied \(g(\tau)\) in \(T_c=\int|g|^2d\tau\).
Detailed step 2. Both integrals return the parameter \(T_c\).
Detailed step 3. At \(\tau=T_c\),
Detailed step 4. the exponential magnitude has fallen to \(e^{-1/2}\) and the Gaussian to \(e^{-\pi/2}\).
Step 4 — State the numbered result. The principal result obtained in the working is
Step 5 — Check. Equation (1) 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
Brief solution
1. Method. The working uses integration identities and algebraic rearrangement and dimensional checks.
2. Key step.
Parseval applied to the Wiener–Khinchin pair gives \(\int|G|^2d\tau=\int|S|^2d\nu\); dividing by \(G(0)^2=[\int S]^2\) shows directly that the power-equivalent widths obey \(\boxed{T_c\Delta\nu_c=1}\).
3. Answer.
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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.
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 integration identities and 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 \(\int|G|^2d\tau=\int|S|^2d\nu\); dividing by \(G(0)^2=[\int S]^2\) shows directly that the power-equivalent widths obey \(\boxed{T_c\Delta\nu_c=1}\).
Step 4 — State the numbered result. The principal result obtained in the working is
Step 5 — Check. Equation (2) 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
Brief solution
1. Method. The working uses expectation, variance, and probability identities and algebraic rearrangement and dimensional checks.
2. Reasoning and answer.
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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.
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 expectation, variance, and probability identities and 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.
Detailed step 1. Apply the wave operator to either field inside \(G=\langle U^*(\mathbf r_1,t)U(\mathbf r_2,t+\tau)\rangle\); linearity lets it pass through the average.
Detailed step 2. This gives the two Wolf equations,
Detailed step 3. one in \(\mathbf r_1,-\tau\) and one in \(\mathbf r_2,+\tau\).
Step 4 — State the numbered result. The principal result obtained in the working is
Step 5 — Check. Equation (3) 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
Brief solution
1. Method. The working uses Fourier-transform and convolution identities and matrix multiplication and eigenvalue rules.
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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.
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 Fourier-transform and convolution identities and 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.
Detailed step 1. An unpolarized coherency matrix contributes equal diagonal terms and no cross term; the rank-one polarized matrix supplies the remaining diagonal imbalance and correlation.
Detailed step 2. Adding the stated intensities yields \(I_x,I_y\) and \(|g_{xy}|\); 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
Brief solution
1. Method. The working uses exponential, logarithmic, and phasor identities and algebraic rearrangement and dimensional checks.
2. Reasoning and answer.
Show detailed steps
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 exponential, logarithmic, and phasor identities and 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.
Detailed step 1. \(\Delta\lambda\simeq\lambda_0^2\Delta\nu/c= \boxed{1.634\ \mathrm{nm}}\).
Detailed step 2. A Lorentzian has \(T_c=1/(\pi\Delta\nu)=\boxed{0.3183\ \mathrm{ps}}\) and \(l_c=\boxed{95.4\ \mathrm{\mu m}}\).
Detailed step 3. Since \(|g|=e^{-\pi\Delta\nu|\tau|}\),
Detailed step 4. the half-coherence delay is 0.2206 ps (path \(\boxed{66.1\ \mathrm{\mu m}}\)).
Numbered result. The principal result obtained in the working is
Check. Equation (4) 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
Brief solution
1. Method. The working uses stationary-value condition, Fourier-transform and convolution identities, and integration identities.
2. Reasoning and answer.
Show detailed steps
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 stationary-value condition, Fourier-transform and convolution identities, and integration identities.
Worked derivation. The calculation is kept in symbolic form until the governing relation has been rearranged for the requested quantity.
Detailed step 1. Expand \(|V_T(\nu)|^2\) as a double time integral,
Detailed step 2. set \(\tau=t_2-t_1\),
Detailed step 3. divide by \(T\),
Detailed step 4. and take the stationary infinite window limit.
Detailed step 5. The remaining inner average is \(G(\tau)\),
Detailed step 6. giving \(S=\mathcal F\{G\}\).
Detailed step 7. Integrating over frequency produces \(\delta(\tau)\) and hence \(\int S\,d\nu=G(0)=I\).
Numbered result. The principal result obtained in the working is
Check. Equation (5) 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
Brief solution
1. Method. The working uses integration identities, exponential, logarithmic, and phasor identities, and algebraic rearrangement and dimensional checks.
2. Reasoning and answer.
Show detailed steps
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 integration identities, exponential, logarithmic, and phasor identities, and 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.
Detailed step 1. Setting \(x_1=x_2=x\) gives \(I(x)=I_0e^{-2x^2/W_0^2}\).
Detailed step 2. Normalization cancels those intensity envelopes,
Detailed step 3. leaving \(\boxed{g=e^{-(x_1-x_2)^2/\rho_c^2}}\). \(I_0,W_0,\rho_c\) are peak intensity,
Detailed step 4. beam radius,
Detailed step 5. and transverse coherence distance.
Numbered result. The principal result obtained in the working is
Check. Equation (6) 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
Brief solution
1. Method. The working uses algebraic rearrangement and dimensional checks.
2. Reasoning and answer.
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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 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.
Detailed step 1. Intensity is one, \(l_c=cT_c=\boxed{300\ \mathrm m}\),
Detailed step 2. and transverse coherence distance is \(\rho_c=1\) mm.
Detailed step 3. The spectrum is centered at 5e14 Hz when \(x_1+x_2>0\) and 6e14 Hz when negative; only colour is position dependent.
Detailed step 4. Film records two uniform half-planes near 600 and 500 nm.
Numbered result. The principal result obtained in the working is
Check. Equation (7) 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
Brief solution
1. Method. The working uses algebraic rearrangement and dimensional checks.
2. Reasoning and answer.
Show detailed steps
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 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.
Detailed step 1. For a narrow band, \(\Delta\nu\simeq c\Delta\lambda/\lambda^2\),
Detailed step 2. so \(l_c=c/\Delta\nu\simeq\lambda^2/\Delta\lambda\).
Detailed step 3. A uniform spectrum from \(\lambda_{min}\) to \(2\lambda_{min}\) spans frequency \(c/(2\lambda_{min})\); inversion gives \(\boxed{l_c=2\lambda_{min}=\lambda_{max}}\).
Numbered result. The principal result obtained in the working is
Check. Equation (8) 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
Brief solution
1. Method. The working uses exponential, logarithmic, and phasor identities and algebraic rearrangement and dimensional checks.
2. Reasoning and answer.
Show detailed steps
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 exponential, logarithmic, and phasor identities and 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.
Detailed step 1. The path delay is \(\tau=[\sqrt{d^2+x^2}-d]/c\); a Lorentzian therefore gives \(\boxed{|g(x)|=\exp[-|\sqrt{d^2+x^2}-d|/(2cT_c)]}\) when \(T_c\) uses the book’s power-equivalent convention.
Detailed step 2. It is even,
Detailed step 3. unity at zero,
Detailed step 4. and falls approximately as \(e^{-x^2/(4dcT_c)}\) near the axis.
Numbered result. The principal result obtained in the working is
Check. Equation (9) 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
Brief solution
1. Method. The working uses vector-calculus identities, integration identities, and algebraic rearrangement and dimensional checks.
2. Reasoning and answer.
Show detailed steps
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 vector-calculus identities, integration identities, and 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.
Detailed step 1. Applying either spatial wave operator to \(G\) gives \((\nabla^2+k_0^2)J=0\).
Detailed step 2. The Gaussian solution has transverse coherence radius \(W(z)=W_0\sqrt{1+(z/z_0)^2}\); hence coherence area \(\boxed{A_c=\pi W^2(z)}\) up to the selected width convention and it grows with \(|z|\).
Numbered result. The principal result obtained in the working is
Check. Equation (10) 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
Brief solution
1. Method. The working uses stationary-value condition, optical path and Fermat’s principle, and exponential, logarithmic, and phasor identities.
2. Reasoning and answer.
Show detailed steps
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 stationary-value condition, optical path and Fermat’s principle, and 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.
Detailed step 1. \(V=|g|=e^{-\pi\Delta\nu|\tau|}\).
Detailed step 2. Setting \(V=1/2\) gives maximum optical path difference \(\boxed{c\ln2/(\pi\Delta\nu)=0.1323\ \mathrm{mm}}\).
Numbered result. The principal result obtained in the working is
Check. Equation (11) 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
Brief solution
1. Method. The working uses algebraic rearrangement and dimensional checks.
2. Reasoning and answer.
Show detailed steps
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 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.
Detailed step 1. For every Table 11.1-2 source,
Detailed step 2. divide coherence length by the Young path change per fringe (one wavelength): \(\boxed{N\simeq l_c/\lambda_0}\).
Detailed step 3. 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
Check. Equation (12) 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?
Brief solution
1. Method. The working uses stationary-value condition, Fourier-transform and convolution identities, and integration identities.
2. Reasoning and answer.
Show detailed steps
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 stationary-value condition, Fourier-transform and convolution identities, and integration identities.
Worked derivation. The calculation is kept in symbolic form until the governing relation has been rearranged for the requested quantity.
Detailed step 1. \(S=S_1+S_2+2\Re S_{12}\) and the spectral Cauchy–Schwarz bound is \(|S_{12}(\nu)|\leq\sqrt{S_1S_2}=S_1\).
Detailed step 2. Correlation can reshape or cancel parts of the common Gaussian support,
Detailed step 3. but a stationary linear superposition cannot translate every frequency to create an exact same-width Gaussian at a new carrier.
Detailed step 4. A genuine shift requires time variation/nonlinearity (e.g.
Detailed step 5. Doppler modulation).
Numbered result. The principal result obtained in the working is
Check. Equation (13) 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
Brief solution
1. Method. The working uses Fourier-transform and convolution identities, vector-calculus identities, and integration identities.
2. Reasoning and answer.
Show detailed steps
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 Fourier-transform and convolution identities, vector-calculus identities, and integration identities.
Worked derivation. The calculation is kept in symbolic form until the governing relation has been rearranged for the requested quantity.
Detailed step 1. Double Fourier propagation of the Gaussian mutual intensity remains Gaussian.
Detailed step 2. Its far-field width is \(\boxed{W^2(z)=W_0^2+(\lambda z/\pi)^2(1/W_0^2+2/\rho_c^2)}\) under the stated \(1/e^2\) convention.
Detailed step 3. Smaller coherence distance increases angular divergence.
Numbered result. The principal result obtained in the working is
Check. Equation (14) 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
Brief solution
1. Method. The working uses Fourier-transform and convolution identities.
Show detailed steps
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 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.
Detailed step 1. Each source point forms its own shifted Fourier intensity and mutually incoherent contributions add.
Detailed step 2. For spatially uniform incoherent illumination,
Detailed step 3. the result is the convolution of \(|F|^2\) with the source angular intensity; in the ideal infinite-uniform limit it is flat.
Detailed step 4. Unlike coherent illumination,
Detailed step 5. 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
Brief solution
1. Method. The working uses Fourier-transform and convolution identities, integration identities, and trigonometric and small-angle identities.
2. Reasoning and answer.
Show detailed steps
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 Fourier-transform and convolution identities, integration identities, and 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.
Detailed step 1. Van Cittert–Zernike gives the normalized transform of two delta sources: \(\boxed{g(x_1,x_2)=\cos[2\pi a(x_1-x_2)/(\lambda d)]}\) up to a common phase.
Detailed step 2. Coherence alternates between magnitude one and zero across separation.
Numbered result. The principal result obtained in the working is
Check. Equation (15) 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
Brief solution
1. Method. The working uses Fourier-transform and convolution identities, integration identities, and algebraic rearrangement and dimensional checks.
2. Key step.
The normalized transform of a uniform slit of width \(2a\) is \(\boxed{g(x_1,x_2)=\operatorname{sinc}[2a(x_1-x_2)/(\lambda f)]}\) up to quadratic phase; coherence width is inversely proportional to slit width.
3. Answer.
Show detailed steps
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 Fourier-transform and convolution identities, integration identities, and 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 \(2a\) is \(\boxed{g(x_1,x_2)=\operatorname{sinc}[2a(x_1-x_2)/(\lambda f)]}\) up to quadratic phase; coherence width is inversely proportional to slit width.
Numbered result. The principal result obtained in the working is
Check. Equation (16) 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
Brief solution
1. Method. The working uses matrix multiplication and eigenvalue rules and algebraic rearrangement and dimensional checks.
2. Reasoning and answer.
Show detailed steps
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 matrix multiplication and eigenvalue rules and 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.
Detailed step 1. Here \(P=|g_{xy}|\).
Detailed step 2. The coherency matrix is \(\boxed{J=\tfrac12\begin{bmatrix}1&-jP\\jP&1\end{bmatrix}}\) for \(P=0,0.5,1\): respectively unpolarized,
Detailed step 3. mixed,
Detailed step 4. and circularly polarized light.
Detailed step 5. An x polarizer transmits \(\boxed{I_x=1/2}\) in every case.
Numbered result. The principal result obtained in the working is
Check. Equation (17) 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.