Chapter 18: Semiconductor Photon Detectors
Source: Saleh and Teich, Fundamentals of Photonics, second edition, Chapter 18.
In-text exercises
Exercise 18.6-1 — Shot/Johnson crossover
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 104 — Exercise 18.6-1: Shot/Johnson crossover. 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. Set \(2e(\eta e\Phi)B=4kTB/R_L\).
Detailed step 2. For the stated ideal detector, \(\boxed{\Phi=6.454\times10^{15}\ \mathrm{s^{-1}}}\) and at 1550 nm \(\boxed{P=0.827\ \mathrm{mW}}\).
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. 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 18.6-2 — APD sensitivity
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 105 — Exercise 18.6-2: APD sensitivity. 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. Signal current is \(G\eta e\Phi\); shot variance is multiplied by \(G^2F\).
Detailed step 2. Solving the resulting quadratic in photon count gives the APD analogue of Eq. (18.6-45); with circuit noise zero,
Detailed step 3. gain cancels and \(\boxed{m_0=F\,\mathrm{SNR}_0}\) detected primary electrons.
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. 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.
Exercise 18.6-3 — Efficiency and background
Brief solution
1. Method. The working uses expectation, variance, and probability identities, 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 106 — Exercise 18.6-3: Efficiency and background. 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, 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. Poisson zero-count detection gives \(\mathrm{BER}=\tfrac12e^{-2\eta n_a}\) and hence roughly ten detected electrons/bit at the stated target.
Detailed step 2. With background,
Detailed step 3. convolve independent Poisson signal/background counts and choose the integer threshold minimizing the two conditional tails; plotting those sums gives sensitivity versus \(n_B\).
End-of-chapter problems
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. Repeat the substitution with unrounded intermediate values and retain the displayed units; the final unit must have the requested dimension.
Problem 18.1-1 — Fresnel collection factor
Brief solution
1. Method. The working uses the physical definitions stated in the item; no separate calculus or algebraic identity is required for this qualitative comparison.
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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 the physical definitions stated in the item; no separate calculus or algebraic identity is required for this qualitative comparison.
Worked derivation. The calculation is kept in symbolic form until the governing relation has been rearranged for the requested quantity.
Detailed step 1. For each tabulated complex/real index compute \(R_s,R_p\) from Fresnel equations; \(1-R\) at normal incidence and \(1-(R_s+R_p)/2\) at 45 degrees are the required factors.
Detailed step 2. Using the table at the detector wavelength is essential because semiconductor index is highly dispersive near its band edge.
Check. For a qualitative conclusion, test every absolute statement against the stated assumptions and at least one limiting case or counterexample.
Problem 18.1-2 — Ideal responsivity limits
Brief solution
1. Method. The working uses stationary-value condition 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 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. Maximum useful wavelength is \(\lambda_g=hc/E_g\); with unit efficiency, \(\boxed{\mathcal R_{max}=e/E_g}\) A/W when \(E_g\) is expressed in volts/eV.
Detailed step 2. Insert the Si,
Detailed step 3. GaAs,
Detailed step 4. and InSb gaps from Table 16.2-1.
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 18.1-3 — One generated pair
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. Ramo current magnitudes are \(i_e=e v_e/w\), \(i_h=e v_h/w\).
Detailed step 2. From \(x=w/3\),
Detailed step 3. durations are distance/velocity: \(t_e=2w/(3v_e)\), \(t_h=w/(3v_h)=w/v_e\) for \(v_e=3v_h\).
Detailed step 4. Areas are \(2e/3\) and \(e/3\),
Detailed step 5. totaling \(e\); these two rectangles give the requested 10-micrometre timing sketch.
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. Check that dimensions agree term by term, then test the simplest symmetry or limiting case for the expected sign and scale.
Problem 18.1-4 — Uniform impulsive generation
Brief solution
1. Method. The working uses integration identities.
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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.
Worked derivation. The calculation is kept in symbolic form until the governing relation has been rearranged for the requested quantity.
Detailed step 1. At time \(t\),
Detailed step 2. only carriers farther than \(vt\) from their collecting contact remain.
Detailed step 3. Integrating their constant Ramo currents over that shrinking length gives the two printed triangular currents.
Detailed step 4. Time integration yields \(Ne/2\) from electrons and holes separately,
Detailed step 5. total \(Ne\).
Check. Differentiation of an antiderivative, or normalization of a definite integral, checks the integration step.
Problem 18.1-5 — Two-photon responsivity
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. \(J=\zeta\Phi^2\), \(\Phi=P/(Ah\nu)\),
Detailed step 2. and output current \(AJ\) give \(\boxed{\mathcal R=I/P=\zeta\lambda^2P/[A(hc)^2]}\).
Detailed step 3. Two-photon coincidence explains both quadratic wavelength leverage and dependence on irradiance \(P/A\).
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.
Problem 18.2-1 — Biased photoconductor
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. If conductance \(g=aP\),
Detailed step 2. divider voltage is \(\boxed{V_R=V(RaP)/(1+RaP)}\).
Detailed step 3. It is linear for \(RaP\ll1\) and saturates at the supply voltage for large optical power.
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 18.2-2 — Illuminated intrinsic silicon
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. Generation is \(\eta P_v/(hc/\lambda)=2.52\times10^{15}\) \(\mathrm{cm^{-3}s^{-1}}\); \(\Delta n=G\tau=2.52\times10^{10}\) \(\mathrm{cm^{-3}}\).
Detailed step 2. Since both carrier types rise equally,
Detailed step 3. conductivity increases by \(\boxed{\Delta n/n_i=167.8\%}\).
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. Repeat the substitution with unrounded intermediate values and retain the displayed units; the final unit must have the requested dimension.
Problem 18.3-1 — p-i-n efficiency
Brief solution
1. Method. The working uses algebraic rearrangement and dimensional checks.
2. Key step.
\(\eta=2/6=\boxed{1/3}\) and \(\mathcal R=\eta e\lambda/(hc)=\boxed{0.417\ \mathrm{A/W}}\) at 1550 nm.
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.
\(\eta=2/6=\boxed{1/3}\) and \(\mathcal R=\eta e\lambda/(hc)=\boxed{0.417\ \mathrm{A/W}}\) at 1550 nm.
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. Repeat the substitution with unrounded intermediate values and retain the displayed units; the final unit must have the requested dimension.
Problem 18.4-1 — APD efficiency/current
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. \(\eta=\mathcal R hc/(Ge\lambda)=\boxed{0.480}\).
Detailed step 2. For \(10^{10}\) photons/s, \(I=G\eta e\Phi= \boxed{15.38\ \mathrm{nA}}\).
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 18.4-2 — Equal ionization coefficients
Brief solution
1. Method. The working uses integration 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 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. With \(\alpha_e=\alpha_h\),
Detailed step 2. coupled multiplication equations integrate to a geometric feedback series; summing it gives \(\boxed{G=1/(1-\alpha_e w)}\) before breakdown.
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. Differentiation of an antiderivative, or normalization of a definite integral, checks the integration step.
Problem 18.5-1 — APD excess noise
Brief solution
1. Method. The working uses exponential, logarithmic, and phasor 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 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. McIntyre’s result \(F=kG+(1-k)(2-1/G)\) gives \(F\to2\) for \(k=0,G\gg1\); then \(G=e^{\alpha_ew}\).
Detailed step 2. Responsivity is \(\eta Ge/E_g\); for \(k=0.01,G=70\),
Detailed step 3. insert those values in the same formula to compare with two.
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 18.5-2 — Multilayer APD
Brief solution
1. Method. The working uses exponential, logarithmic, and phasor 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 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. Each stage multiplies the mean by \(1+P\); independence yields \(\boxed{G=(1+P)^l}\).
Detailed step 2. With \(P=\alpha w/l\) and \(l\to\infty\),
Detailed step 3. this tends to \(e^{\alpha w}\),
Detailed step 4. the continuous single-carrier APD.
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. Check that dimensions agree term by term, then test the simplest symmetry or limiting case for the expected sign and scale.
Problem 18.5-3 — One-stage PMT noise
Brief solution
1. Method. The working uses expectation, variance, and probability identities and algebraic rearrangement and dimensional checks.
2. Key step.
For Poisson secondary yield \(\bar G=\delta\) and \(\langle G^2\rangle=\delta^2+\delta\); hence \(\boxed{F=1+1/\delta}\).
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 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 Poisson secondary yield \(\bar G=\delta\) and \(\langle G^2\rangle=\delta^2+\delta\); hence \(\boxed{F=1+1/\delta}\).
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. For a probability result, verify the zero-to-one bounds; when a full distribution is present, also verify normalization and nonnegative variance.
Problem 18.5-4 — Photoconductor gain noise
Brief solution
1. Method. The working uses expectation, variance, and probability identities, exponential, logarithmic, and phasor identities, and algebraic rearrangement and dimensional checks.
2. Key step.
With \(G=\tau/t_e\) and exponential lifetime, \(\langle\tau^2\rangle=2\bar\tau^2\); therefore \(\boxed{F=\langle G^2\rangle/\bar G^2=2}\).
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 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.
With \(G=\tau/t_e\) and exponential lifetime, \(\langle\tau^2\rangle=2\bar\tau^2\); therefore \(\boxed{F=\langle G^2\rangle/\bar G^2=2}\).
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 18.5-5 — RC noise bandwidth
Brief solution
1. Method. The working uses Fourier-transform and convolution identities, integration identities, and exponential, logarithmic, and phasor 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 Fourier-transform and convolution 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. Fourier transforming \(h=e^{-t/\tau}/\tau\) and integrating \(|H|^2\) gives \(B=1/(4\tau)=1/(4RC)\).
Detailed step 2. For 1 kohm and 5 pF, \(\boxed{B=50\ \mathrm{MHz}}\) and Johnson RMS current \(\boxed{28.8\ \mathrm{nA}}\) at 300 K.
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. Repeat the substitution with unrounded intermediate values and retain the displayed units; the final unit must have the requested dimension.
Problem 18.5-6 — Changing APD ionization ratio
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. With negligible circuit noise,
Detailed step 2. SNR is inversely proportional to \(F=kG+(1-k)(2-1/G)\).
Detailed step 3. Evaluate \(F(0.1,100)\) and \(F(0.2,100)\); their inverse ratio is the SNR factor.
Detailed step 4. For \(G\gg2(1-k)/k\), \(F\simeq kG\),
Detailed step 5. proving \(\mathrm{SNR}\propto1/G\).
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. 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 18.5-7 — APD receiver noise budget
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. Compute signal primary current \(I_p=\eta eP/(h\nu)\),
Detailed step 2. multiplied output \(GI_p\),
Detailed step 3. APD shot RMS \([2eBFG^2(I_p+I_d)]^{1/2}\),
Detailed step 4. and Johnson RMS \((4kTB/R_L)^{1/2}\).
Detailed step 5. Add variances,
Detailed step 6. not RMS values;
Detailed step 7. SNR is \((GI_p)^2\) divided by their sum.
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. Check that dimensions agree term by term, then test the simplest symmetry or limiting case for the expected sign and scale.
Problem 18.5-8 — Optimum APD gain
Brief solution
1. Method. The working uses product, quotient, and chain rules, integration 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 product, quotient, and chain rules, 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. Write normalized noise denominator \(F(G)+100/G^2\) with \(k=0.2\); differentiate and solve its positive root.
Detailed step 2. The ratio of the original p-i-n denominator 101 to the minimized APD denominator is the SNR improvement.
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. Differentiation of an antiderivative, or normalization of a definite integral, checks the integration step.
Problem 18.5-9 — Shot-noise sensitivity
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. With no circuit noise, \(\mathrm{SNR}=\eta\Phi/(2B)\) for the chapter’s one-sided bandwidth convention.
Detailed step 2. Thus \(\boxed{P=(2B\,\mathrm{SNR}/\eta)(hc/\lambda)}\); insert \(B=100\) MHz,
Detailed step 3. SNR \(10^3\),
Detailed step 4. eta 0.8.
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. 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 18.5-10 — Three-detector comparison
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.
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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, 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. For each device compute mean output \(G\eta e\Phi\); detector variance \(2e^2B\eta\Phi\langle G^2\rangle\) and common Johnson variance \(4kTB/R\).
Detailed step 2. Use \(\langle G^2\rangle=G^2F\); a device is detectable when resulting power SNR exceeds one.
Detailed step 3. This common table prevents confusing large gain with improved photon-limited SNR.
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.
Problem 18.5-11 — Wavelength scaling
Brief solution
1. Method. The working uses power and decibel conversions 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 power and decibel conversions 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 same photons/bit require power proportional to photon energy \(1/\lambda\).
Detailed step 2. Therefore sensitivity improves by \(10\log_{10}(1300/870)=1.744\) dB: \(\boxed{-77.74\ \mathrm{dBm}}\).
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. Converting the final decibel value back to a linear power ratio checks the logarithm, sign, and accumulated loss budget. Repeat the substitution with unrounded intermediate values and retain the displayed units; the final unit must have the requested dimension.
Problem 18.5-12 — Changed zero-count BER
Brief solution
1. Method. The working uses expectation, variance, and probability identities and exponential, logarithmic, and phasor identities.
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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 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. Use \(P_e\propto e^{-\eta PT/(hc/\lambda)}\).
Detailed step 2. Relative exponent factors are 1300/870, 2, 0.5, 1,
Detailed step 3. and (with a nonzero decision threshold) the APD excess-noise-tail calculation.
Detailed step 4. Ideal noiseless gain alone does not change zero-primary-count probability; doubling power squares the original \(10^{-10}\) exponential factor.
Check. For a probability result, verify the zero-to-one bounds; when a full distribution is present, also verify normalization and nonnegative variance.
Problem 18.5-13 — Detecting AM on background
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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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. Signal mean-square current is \((\mathcal R P_s)^2/2\); background shot variance is \(2e\mathcal RP_0B\).
Detailed step 2. Solving gives \(\boxed{P_{s,min}=2\sqrt{eBP_0\mathrm{SNR}_0/\mathcal R}}\); sensitivity degrades as \(\sqrt{P_0}\).
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. Repeat the substitution with unrounded intermediate values and retain the displayed units; the final unit must have the requested dimension.
Problem 18.5-14 — Counting sensitivity
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. Poisson power SNR equals detected mean,
Detailed step 2. so SNR \(10^3\) requires 1000 photoelectrons or \(\boxed{2000}\) photons at eta 0.5.
Detailed step 3. At 870 nm over 1 microsecond, \(\boxed{P=4.57\times10^{-10}\ \mathrm W}\); zero-count probability is \(e^{-1000}\).
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. Repeat the substitution with unrounded intermediate values and retain the displayed units; the final unit must have the requested dimension.
Problem 18.5-15 — One-dynode PMT
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. Condition on input \(n=0,1\); the zero branch remains zero output and the one-photon branch has the stated dynode secondary distribution.
Detailed step 2. Therefore the output law is a 50:50 mixture of a delta at zero and that secondary law; use total expectation/variance to obtain its mean,
Detailed step 3. excess-noise factor,
Detailed step 4. and SNR without treating the input as Poisson.
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.