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
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.
Normalize \(e^{-2\rho^2/W_0^2}\) over the plane. 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
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.
\(p_\gamma=E/c\) and \(v_r=p_\gamma/(198u)\) give \(\boxed{v_r=7.93\times10^{-3}\ \mathrm{m/s}}\). 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
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.
The chosen output probability is \(\boxed{P_D=\cos^2(\pi d/\lambda)}\) (the other port has sine squared; port labels may swap). At a nonunity value the photon state is a coherent superposition of the two output possibilities, 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
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.
Normalize \(|a(t)|^2=e^{-t^2/(2T^2)}\) to obtain \(\sigma_t=T\) and \(\sigma_z=cT\). Fourier transformation gives \(\sigma_\omega=1/(2T)\), so \(\boxed{\sigma_E\sigma_t=\hbar/2}\) and, 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
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.
Average \(nh\nu\) over the geometric distribution to get \(\boxed{\bar E=h\nu/[e^{h\nu/kT}-1]}\). For \(h\nu\ll kT\), 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
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.
(a) \(eV=hc/\lambda\) gives \(\boxed{V=1.425\ \mathrm V}\) at 0.87 micrometres. (b) Sum-frequency energy gives \(1/\lambda_3=1/1.06+1/10.6\), 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
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.
Radial normalization uses \(2\pi\int_0^\infty\rho e^{-\rho/\rho_0}d\rho\). 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
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
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.
The angular-spectrum probability is Gaussian; integration inside the \(1/e^2\) divergence angle gives \(\boxed{1-e^{-2}}\). Every plane wave component still has \(|\mathbf p|=E/c\), 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
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.
\(F_g=mg=1.63\times10^{-26}\) N. One absorbed 1-eV photon per second gives \(5.34\times10^{-28}\) N, so \(\boxed{30.5}\) photons/s and \(\boxed{4.88\times10^{-18}\ \mathrm W}\) balance gravity. 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
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.
\(k=10^5\pi/d\), so the in-medium wavelength is 0.20 micrometres, free wavelength \(\boxed{0.300\ \mathrm{\mu m}}\), and energy \(\boxed{4.13\ \mathrm{eV}}\). 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. 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
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.
Normalize over one beat period: \(p(t)=T^{-1}[1+\cos(2\pi t/T)]\); it vanishes at \(\boxed{t=T/2}\). Resolving which energy to better than \(h|\nu_2-\nu_1|\) requires \(\Delta t\gtrsim1/[4\pi|\Delta\nu|]\), of the beat-period order, 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
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.
Before incidence \(\mathbf p=\hbar\mathbf k_i\). 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
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.
Photon rate is \(\nu\) and photon energy \(h\nu\), 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
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.
Using \(\sum\bar n^n/n!=e^{\bar n}\) proves normalization. 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
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.
Total mean in 100 ns is \(PT/(hc/\lambda)=31.86\); inside \(W_0\), \(\boxed{\bar n=27.55}\). 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
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.
Geometric-series sums give normalization, mean \(\bar n\), and variance \(\bar n+\bar n^2\). Mean flux one/ns over 20 ns gives \(\bar n=20\), 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
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.
Set \(M=1\) to recover the geometric distribution. For fixed \(\bar n\) and \(M\to\infty\), 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
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.
Independent mode means/variances add. 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
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.
Insert the gamma density in Mandel’s integral \(P(n)=\int e^{-w}w^n p(w)dw/n!\). 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
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.
Conditional Poisson moments are \(E[n|W]=W\) and \(\operatorname{var}(n|W)=W\). 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
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.
Binomial thinning of a Poisson generating function \(G(z)=e^{\bar n(z-1)}\) gives \(G_R(z)=G[T+Rz]=e^{R\bar n(z-1)}\). 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
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.
Apply the same substitution to geometric \(G(z)=[1+\bar n(1-z)]^{-1}\). It remains geometric with mean \(R\bar n\) and variance \(R\bar n+(R\bar n)^2\), 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
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.
\(d\bar n/dx=-\alpha\bar n\) gives \(\bar n(x)=\bar n_0e^{-\alpha x}\). Coherent input remains Poisson with that mean under independent absorption. 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
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.
Send an \(M\)-photon number state through a beamsplitter of transmission \(p\). The binomial theorem proves normalization and gives \(\bar n=Mp\), \(\sigma^2=Mp(1-p)\), 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
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.
For uniform integers \(0,\ldots,2\bar n\), finite sums give \(\sigma^2=\bar n(\bar n+1)/3\) and \(\boxed{\mathrm{SNR}=3\bar n/(\bar n+1)}\). Relative to Poisson SNR \(\bar n\), it is quieter for \(\bar n<2\), equal at 2, 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.