Chapter 12: Photon Optics
Source: Saleh and Teich, Fundamentals of Photonics, second edition, Chapter 12.
In-text exercises
Exercise 12.1-1 — Photons in a Gaussian beam
Brief solution
1. Method. The working uses expectation, variance, and probability identities, integration identities, and exponential, logarithmic, and phasor identities.
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 73 — Exercise 12.1-1: Photons in a Gaussian beam. 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, integration identities, and exponential, logarithmic, and phasor identities.
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. Normalize \(e^{-2\rho^2/W_0^2}\) over the plane.
Detailed step 2. The probability inside \(W_0\) is \(\boxed{1-e^{-2}=0.8647}\); for \(n\) independent photons the expected count is \(n(1-e^{-2})\).
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. For a probability result, verify the zero-to-one bounds; when a full distribution is present, also verify normalization and nonnegative variance.
Exercise 12.1-2 — Mercury recoil
Brief solution
1. Method. The working uses 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 74 — Exercise 12.1-2: Mercury recoil. 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 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. \(p_\gamma=E/c\) and \(v_r=p_\gamma/(198u)\) give \(\boxed{v_r=7.93\times10^{-3}\ \mathrm{m/s}}\).
Detailed step 2. From \(mv_{rms}^2/2=3kT/2\), \(v_{rms}=\boxed{194\ \mathrm{m/s}}\); thermal motion is about \(2.45\times10^4\) times larger.
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. 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.
Exercise 12.1-3 — One photon in a Mach–Zehnder
Brief solution
1. Method. The working uses expectation, variance, and probability identities, trigonometric and small-angle 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 75 — Exercise 12.1-3: One photon in a Mach–Zehnder. 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, trigonometric and small-angle 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. The chosen output probability is \(\boxed{P_D=\cos^2(\pi d/\lambda)}\) (the other port has sine squared; port labels may swap).
Detailed step 2. At a nonunity value the photon state is a coherent superposition of the two output possibilities,
Detailed step 3. not a classical fraction in each arm.
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 12.1-4 — Gaussian wavepacket uncertainty
Brief solution
1. Method. The working uses Fourier-transform and convolution identities, integration identities, and exponential, logarithmic, and phasor identities.
2. Reasoning and answer.
Show detailed steps
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 76 — Exercise 12.1-4: Gaussian wavepacket uncertainty. 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, integration identities, and exponential, logarithmic, and phasor identities.
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. Normalize \(|a(t)|^2=e^{-t^2/(2T^2)}\) to obtain \(\sigma_t=T\) and \(\sigma_z=cT\).
Detailed step 2. Fourier transformation gives \(\sigma_\omega=1/(2T)\),
Detailed step 3. so \(\boxed{\sigma_E\sigma_t=\hbar/2}\) and,
Detailed step 4. with \(p_z=E/c\), \(\boxed{\sigma_z\sigma_p=\hbar/2}\).
Step 4 — State the numbered result. The principal result obtained in the working is
Step 5 — Check. Equation (4) 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.
Exercise 12.2-1 — Mean thermal-mode energy
Brief solution
1. Method. The working uses exponential, logarithmic, and phasor identities and algebraic rearrangement and dimensional checks.
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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 77 — Exercise 12.2-1: Mean thermal-mode energy. 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 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. Average \(nh\nu\) over the geometric distribution to get \(\boxed{\bar E=h\nu/[e^{h\nu/kT}-1]}\).
Detailed step 2. For \(h\nu\ll kT\),
Detailed step 3. expand the exponential and recover \(\bar E\to kT\); increasing frequency or lowering temperature suppresses the mean energy exponentially.
End-of-chapter problems
Step 4 — State the numbered result. The principal result obtained in the working is
Step 5 — Check. Equation (5) 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 12.1-5 — Combining photon energies
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. (a) \(eV=hc/\lambda\) gives \(\boxed{V=1.425\ \mathrm V}\) at 0.87 micrometres. (b) Sum-frequency energy gives \(1/\lambda_3=1/1.06+1/10.6\),
Detailed step 2. hence \(\boxed{\lambda_3=0.9636\ \mathrm{\mu m}}\).
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. 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 12.1-6 — Exponential radial beam
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. Radial normalization uses \(2\pi\int_0^\infty\rho e^{-\rho/\rho_0}d\rho\).
Detailed step 2. Inside \(\rho_0\), \(\boxed{P=1-2/e=0.26424}\); one million photons therefore give \(\boxed{2.6424\times10^5}\) on average.
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. Differentiation of an antiderivative, or normalization of a definite integral, checks the integration step.
Problem 12.1-7 — Momentum comparisons
Brief solution
1. Method. The working uses algebraic rearrangement and dimensional checks.
2. Key step.
The 10-J light pulse has \(p=E/c=\boxed{3.336\times10^{-8}}\) kg m/s; the 1-g body has \(10^{-5}\) kg m/s; a nonrelativistic electron at \(c/10\) has \(2.73\times10^{-23}\) kg m/s.
3. 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.
The 10-J light pulse has \(p=E/c=\boxed{3.336\times10^{-8}}\) kg m/s; the 1-g body has \(10^{-5}\) kg m/s; a nonrelativistic electron at \(c/10\) has \(2.73\times10^{-23}\) kg m/s.
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 12.1-8 — Gaussian photon momentum
Brief solution
1. Method. The working uses expectation, variance, and probability identities, vector-calculus identities, and integration identities.
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 expectation, variance, and probability 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. The angular-spectrum probability is Gaussian; integration inside the \(1/e^2\) divergence angle gives \(\boxed{1-e^{-2}}\).
Detailed step 2. Every plane wave component still has \(|\mathbf p|=E/c\),
Detailed step 3. but its direction is random; the mean axial momentum is slightly below \(E/c\).
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. For a probability result, verify the zero-to-one bounds; when a full distribution is present, also verify normalization and nonnegative variance.
Problem 12.1-9 — Ideal atom levitation
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. \(F_g=mg=1.63\times10^{-26}\) N.
Detailed step 2. One absorbed 1-eV photon per second gives \(5.34\times10^{-28}\) N,
Detailed step 3. so \(\boxed{30.5}\) photons/s and \(\boxed{4.88\times10^{-18}\ \mathrm W}\) balance gravity.
Detailed step 4. Perfect reflection doubles momentum transfer and halves the rate to 15.2 photons/s.
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. 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 12.1-10 — One cavity photon
Brief solution
1. Method. The working uses stationary-value condition and 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 stationary-value condition 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. \(k=10^5\pi/d\),
Detailed step 2. so the in-medium wavelength is 0.20 micrometres,
Detailed step 3. free wavelength \(\boxed{0.300\ \mathrm{\mu m}}\),
Detailed step 4. and energy \(\boxed{4.13\ \mathrm{eV}}\).
Detailed step 5. The high-order standing-wave position is nearly uniform over 1 cm, \(\sigma_x\simeq d/\sqrt{12}\); momentum has equal \(\pm\hbar k\) outcomes.
Detailed step 6. Their product greatly exceeds \(\hbar/2\); this is not a minimum-uncertainty packet.
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 12.1-11 — Single-photon beating
Brief solution
1. Method. The working uses integration identities, trigonometric and small-angle identities, and 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 integration identities, trigonometric and small-angle 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. Normalize over one beat period: \(p(t)=T^{-1}[1+\cos(2\pi t/T)]\); it vanishes at \(\boxed{t=T/2}\).
Detailed step 2. Resolving which energy to better than \(h|\nu_2-\nu_1|\) requires \(\Delta t\gtrsim1/[4\pi|\Delta\nu|]\),
Detailed step 3. of the beat-period order,
Detailed step 4. destroying the timing information needed to see the interference.
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. The zero-angle or paraxial limit supplies an independent sign and magnitude check whenever that limit is part of the model.
Problem 12.1-12 — Beamsplitter momentum
Brief solution
1. Method. The working uses expectation, variance, and probability 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 expectation, variance, and probability 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. Before incidence \(\mathbf p=\hbar\mathbf k_i\).
Detailed step 2. Afterward measurement returns \(\hbar\mathbf k_t\) with probability \(T\) or \(\hbar\mathbf k_r\) with probability \(R=1-T\); before measurement the photon occupies their coherent superposition.
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. For a probability result, verify the zero-to-one bounds; when a full distribution is present, also verify normalization and nonnegative variance.
Problem 12.2-2 — One photon per cycle
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. Photon rate is \(\nu\) and photon energy \(h\nu\),
Detailed step 2. so \(\boxed{P=h\nu^2=hc^2/\lambda^2}\).
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. Check that dimensions agree term by term, then test the simplest symmetry or limiting case for the expected sign and scale.
Problem 12.2-3 — Poisson moments
Brief solution
1. Method. The working uses product, quotient, and chain rules, expectation, variance, and probability 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 product, quotient, and chain rules, expectation, variance, and probability 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. Using \(\sum\bar n^n/n!=e^{\bar n}\) proves normalization.
Detailed step 2. Differentiating that generating function once and twice gives \(\boxed{\langle n\rangle=\bar n}\) and \(\boxed{\operatorname{var}n=\bar n}\).
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. For a probability result, verify the zero-to-one bounds; when a full distribution is present, also verify normalization and nonnegative variance.
Problem 12.2-4 — Weak coherent He–Ne beam
Brief solution
1. Method. The working uses expectation, variance, and probability 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 expectation, variance, and probability 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. Total mean in 100 ns is \(PT/(hc/\lambda)=31.86\); inside \(W_0\), \(\boxed{\bar n=27.55}\).
Detailed step 2. Poisson RMS fluctuation is \(\boxed{5.25}\) and zero-count probability \(\boxed{e^{-27.55}=1.08\times10^{-12}}\).
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. For a probability result, verify the zero-to-one bounds; when a full distribution is present, also verify normalization and nonnegative variance.
Problem 12.2-5 — Bose–Einstein counts
Brief solution
1. Method. The working uses expectation, variance, and probability 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 expectation, variance, and probability 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. Geometric-series sums give normalization,
Detailed step 2. mean \(\bar n\),
Detailed step 3. and variance \(\bar n+\bar n^2\).
Detailed step 4. Mean flux one/ns over 20 ns gives \(\bar n=20\),
Detailed step 5. so \(\boxed{P(0)=1/(1+\bar n)=1/21}\).
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. For a probability result, verify the zero-to-one bounds; when a full distribution is present, also verify normalization and nonnegative variance.
Problem 12.2-6 — Negative-binomial limits
Brief solution
1. Method. The working uses expectation, variance, and probability 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 expectation, variance, and probability 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. Set \(M=1\) to recover the geometric distribution.
Detailed step 2. For fixed \(\bar n\) and \(M\to\infty\),
Detailed step 3. use \((1+\bar n/M)^{-M}\to e^{-\bar n}\) and the factorial ratio \(\to M^n/n!\); the result is Poisson.
Numbered result. The principal result obtained in the working is
Check. Equation (18) 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.
Problem 12.2-7 — Multimode thermal variance
Brief solution
1. Method. The working uses expectation, variance, and probability 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 expectation, variance, and probability 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. Independent mode means/variances add.
Detailed step 2. With total \(\bar n=M\bar n_1\), \(\boxed{\sigma_n^2=\bar n+\bar n^2/M}\); more modes average thermal bunching toward Poisson statistics.
Numbered result. The principal result obtained in the working is
Check. Equation (19) 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.
Problem 12.2-8 — Gamma-mixed Poisson process
Brief solution
1. Method. The working uses expectation, variance, and probability identities, integration identities, 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 expectation, variance, and probability identities, integration identities, 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. Insert the gamma density in Mandel’s integral \(P(n)=\int e^{-w}w^n p(w)dw/n!\).
Detailed step 2. The remaining gamma integral is \(\Gamma(n+M)\) and simplification gives exactly the negative-binomial distribution of Problem 12.2-6.
Numbered result. The principal result obtained in the working is
Check. Equation (20) 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.
Problem 12.2-9 — Doubly stochastic moments
Brief solution
1. Method. The working uses expectation, variance, and probability 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 expectation, variance, and probability 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. Conditional Poisson moments are \(E[n|W]=W\) and \(\operatorname{var}(n|W)=W\).
Detailed step 2. Total expectation/variance therefore give \(\boxed{E[n]=E[W]}\) and \(\boxed{\operatorname{var}n=E[W]+\operatorname{var}W}\).
Numbered result. The principal result obtained in the working is
Check. Equation (21) 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. Repeat the substitution with unrounded intermediate values and retain the displayed units; the final unit must have the requested dimension.
Problem 12.2-10 — Partitioned coherent light
Brief solution
1. Method. The working uses expectation, variance, and probability 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 expectation, variance, and probability 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. Binomial thinning of a Poisson generating function \(G(z)=e^{\bar n(z-1)}\) gives \(G_R(z)=G[T+Rz]=e^{R\bar n(z-1)}\).
Detailed step 2. Thus reflected counts remain Poisson with mean/variance \(\boxed{R\bar n}\).
Numbered result. The principal result obtained in the working is
Check. Equation (22) 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.
Problem 12.2-11 — Partitioned thermal light
Brief solution
1. Method. The working uses expectation, variance, and probability 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 expectation, variance, and probability 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. Apply the same substitution to geometric \(G(z)=[1+\bar n(1-z)]^{-1}\).
Detailed step 2. It remains geometric with mean \(R\bar n\) and variance \(R\bar n+(R\bar n)^2\),
Detailed step 3. proving all three parts.
Numbered result. The principal result obtained in the working is
Check. Equation (23) 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.
Problem 12.2-12 — Absorber thinning
Brief solution
1. Method. The working uses expectation, variance, and probability 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 expectation, variance, and probability 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. \(d\bar n/dx=-\alpha\bar n\) gives \(\bar n(x)=\bar n_0e^{-\alpha x}\).
Detailed step 2. Coherent input remains Poisson with that mean under independent absorption.
Detailed step 3. A single photon survives thickness \(d\) with probability \(\boxed{e^{-\alpha d}}\).
Numbered result. The principal result obtained in the working is
Check. Equation (24) 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.
Problem 12.3-1 — Binomial light
Brief solution
1. Method. The working uses expectation, variance, and probability 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 expectation, variance, and probability 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. Send an \(M\)-photon number state through a beamsplitter of transmission \(p\).
Detailed step 2. The binomial theorem proves normalization and gives \(\bar n=Mp\), \(\sigma^2=Mp(1-p)\),
Detailed step 3. and \(\boxed{\mathrm{SNR}=\bar n/(1-p)}\) for power SNR \(\bar n^2/\sigma^2\). \(p\to0\) approaches Poisson thinning; \(p\to1\) approaches a noiseless number state.
Numbered result. The principal result obtained in the working is
Check. Equation (25) 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. Repeat the substitution with unrounded intermediate values and retain the displayed units; the final unit must have the requested dimension.
Problem 12.3-2 — Discrete-uniform source
Brief solution
1. Method. The working uses expectation, variance, and probability 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 expectation, variance, and probability 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. For uniform integers \(0,\ldots,2\bar n\),
Detailed step 2. finite sums give \(\sigma^2=\bar n(\bar n+1)/3\) and \(\boxed{\mathrm{SNR}=3\bar n/(\bar n+1)}\).
Detailed step 3. Relative to Poisson SNR \(\bar n\),
Detailed step 4. it is quieter for \(\bar n<2\),
Detailed step 5. equal at 2,
Detailed step 6. and noisier above 2; its SNR is exactly three times the single-mode thermal value \(\bar n/(1+\bar n)\).
Numbered result. The principal result obtained in the working is
Check. Equation (26) 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. Repeat the substitution with unrounded intermediate values and retain the displayed units; the final unit must have the requested dimension.