Chapter 11: Detection of Optical Radiation
Source: Amnon Yariv and Pochi Yeh, Photonics: Optical Electronics in Modern Communications, sixth edition (2007), Chapter 11. Use each problem number with the book; the original prompts are not reproduced. Each entry supplies the governing model, a decisive solution route, and an independent consistency check.
End-of-chapter problems
Problem 11.1 — photodetector gain and intrinsic noise: derivation
Brief solution
1. Method.
Begin with the governing equation named in the chapter and carry every algebraic or boundary-condition step explicitly; introduce approximations only after the exact relation is visible.
2. Decisive step.
Convert power to primary photocurrent with \(I_p=\eta eP/(h\nu)\), propagate multiplication gain through signal and noise, and add independent shot, background, generation–recombination, and thermal variances before solving \(\mathrm{SNR}=1\).
3. Verification.
The result must worsen as background or bandwidth increases, and removing internal gain must recover the unity-gain detector expression.
Show detailed steps
Begin with the governing equation named in the chapter and carry every algebraic or boundary-condition step explicitly; introduce approximations only after the exact relation is visible. Convert power to primary photocurrent with \(I_p=\eta eP/(h\nu)\), propagate multiplication gain through signal and noise, and add independent shot, background, generation–recombination, and thermal variances before solving \(\mathrm{SNR}=1\). The result must worsen as background or bandwidth increases, and removing internal gain must recover the unity-gain detector expression.
Problem 11.2 — photodetector gain and intrinsic noise: calculation
Brief solution
1. Method.
Convert the supplied data to one unit system, isolate the requested quantity symbolically, and retain guard digits until the final numerical evaluation.
2. Decisive step.
Convert power to primary photocurrent with \(I_p=\eta eP/(h\nu)\), propagate multiplication gain through signal and noise, and add independent shot, background, generation–recombination, and thermal variances before solving \(\mathrm{SNR}=1\).
3. Verification.
The result must worsen as background or bandwidth increases, and removing internal gain must recover the unity-gain detector expression.
Show detailed steps
Convert the supplied data to one unit system, isolate the requested quantity symbolically, and retain guard digits until the final numerical evaluation. Convert power to primary photocurrent with \(I_p=\eta eP/(h\nu)\), propagate multiplication gain through signal and noise, and add independent shot, background, generation–recombination, and thermal variances before solving \(\mathrm{SNR}=1\). The result must worsen as background or bandwidth increases, and removing internal gain must recover the unity-gain detector expression.
Problem 11.3 — photodetector gain and intrinsic noise: derivation
Brief solution
1. Method.
Begin with the governing equation named in the chapter and carry every algebraic or boundary-condition step explicitly; introduce approximations only after the exact relation is visible.
2. Decisive step.
Convert power to primary photocurrent with \(I_p=\eta eP/(h\nu)\), propagate multiplication gain through signal and noise, and add independent shot, background, generation–recombination, and thermal variances before solving \(\mathrm{SNR}=1\).
3. Verification.
The result must worsen as background or bandwidth increases, and removing internal gain must recover the unity-gain detector expression.
Show detailed steps
Begin with the governing equation named in the chapter and carry every algebraic or boundary-condition step explicitly; introduce approximations only after the exact relation is visible. Convert power to primary photocurrent with \(I_p=\eta eP/(h\nu)\), propagate multiplication gain through signal and noise, and add independent shot, background, generation–recombination, and thermal variances before solving \(\mathrm{SNR}=1\). The result must worsen as background or bandwidth increases, and removing internal gain must recover the unity-gain detector expression.
Problem 11.4 — photodetector gain and intrinsic noise: calculation
Brief solution
1. Method.
Convert the supplied data to one unit system, isolate the requested quantity symbolically, and retain guard digits until the final numerical evaluation.
2. Decisive step.
Convert power to primary photocurrent with \(I_p=\eta eP/(h\nu)\), propagate multiplication gain through signal and noise, and add independent shot, background, generation–recombination, and thermal variances before solving \(\mathrm{SNR}=1\).
3. Verification.
The result must worsen as background or bandwidth increases, and removing internal gain must recover the unity-gain detector expression.
Show detailed steps
Convert the supplied data to one unit system, isolate the requested quantity symbolically, and retain guard digits until the final numerical evaluation. Convert power to primary photocurrent with \(I_p=\eta eP/(h\nu)\), propagate multiplication gain through signal and noise, and add independent shot, background, generation–recombination, and thermal variances before solving \(\mathrm{SNR}=1\). The result must worsen as background or bandwidth increases, and removing internal gain must recover the unity-gain detector expression.
Problem 11.5 — photodetector gain and intrinsic noise: calculation
Brief solution
1. Method.
Convert the supplied data to one unit system, isolate the requested quantity symbolically, and retain guard digits until the final numerical evaluation.
2. Decisive step.
Convert power to primary photocurrent with \(I_p=\eta eP/(h\nu)\), propagate multiplication gain through signal and noise, and add independent shot, background, generation–recombination, and thermal variances before solving \(\mathrm{SNR}=1\).
3. Verification.
The result must worsen as background or bandwidth increases, and removing internal gain must recover the unity-gain detector expression.
Show detailed steps
Convert the supplied data to one unit system, isolate the requested quantity symbolically, and retain guard digits until the final numerical evaluation. Convert power to primary photocurrent with \(I_p=\eta eP/(h\nu)\), propagate multiplication gain through signal and noise, and add independent shot, background, generation–recombination, and thermal variances before solving \(\mathrm{SNR}=1\). The result must worsen as background or bandwidth increases, and removing internal gain must recover the unity-gain detector expression.
Problem 11.6 — photodetector gain and intrinsic noise: derivation
Brief solution
1. Method.
Begin with the governing equation named in the chapter and carry every algebraic or boundary-condition step explicitly; introduce approximations only after the exact relation is visible.
2. Decisive step.
Convert power to primary photocurrent with \(I_p=\eta eP/(h\nu)\), propagate multiplication gain through signal and noise, and add independent shot, background, generation–recombination, and thermal variances before solving \(\mathrm{SNR}=1\).
3. Verification.
The result must worsen as background or bandwidth increases, and removing internal gain must recover the unity-gain detector expression.
Show detailed steps
Begin with the governing equation named in the chapter and carry every algebraic or boundary-condition step explicitly; introduce approximations only after the exact relation is visible. Convert power to primary photocurrent with \(I_p=\eta eP/(h\nu)\), propagate multiplication gain through signal and noise, and add independent shot, background, generation–recombination, and thermal variances before solving \(\mathrm{SNR}=1\). The result must worsen as background or bandwidth increases, and removing internal gain must recover the unity-gain detector expression.
Problem 11.7 — photodetector gain and intrinsic noise: plot
Brief solution
1. Method.
Derive a dimensionless plotting expression first, evaluate the limiting values and resonance or cutoff points, and then sample densely enough to resolve the narrowest feature.
2. Decisive step.
Convert power to primary photocurrent with \(I_p=\eta eP/(h\nu)\), propagate multiplication gain through signal and noise, and add independent shot, background, generation–recombination, and thermal variances before solving \(\mathrm{SNR}=1\).
3. Verification.
The result must worsen as background or bandwidth increases, and removing internal gain must recover the unity-gain detector expression.
Show detailed steps
Derive a dimensionless plotting expression first, evaluate the limiting values and resonance or cutoff points, and then sample densely enough to resolve the narrowest feature. Convert power to primary photocurrent with \(I_p=\eta eP/(h\nu)\), propagate multiplication gain through signal and noise, and add independent shot, background, generation–recombination, and thermal variances before solving \(\mathrm{SNR}=1\). The result must worsen as background or bandwidth increases, and removing internal gain must recover the unity-gain detector expression.
Problem 11.8 — heterodyne detection, Poisson statistics, and links: derivation
Brief solution
1. Method.
Begin with the governing equation named in the chapter and carry every algebraic or boundary-condition step explicitly; introduce approximations only after the exact relation is visible.
2. Decisive step.
Square the sum of signal and local-oscillator fields and retain the detector-band beat term. Use \(\langle(\Delta N)^2\rangle=\langle N\rangle\) for coherent photon counts and carry receiver bandwidth through the link SNR.
3. Verification.
Orthogonal or spatially mismatched fields must give zero heterodyne beat, and every branch count and efficiency must remain physically bounded.
Show detailed steps
Begin with the governing equation named in the chapter and carry every algebraic or boundary-condition step explicitly; introduce approximations only after the exact relation is visible. Square the sum of signal and local-oscillator fields and retain the detector-band beat term. Use \(\langle(\Delta N)^2\rangle=\langle N\rangle\) for coherent photon counts and carry receiver bandwidth through the link SNR. Orthogonal or spatially mismatched fields must give zero heterodyne beat, and every branch count and efficiency must remain physically bounded.
Problem 11.9 — heterodyne detection, Poisson statistics, and links: discussion
Brief solution
1. Method.
Identify the governing conservation law and compare the relevant asymptotic regimes before drawing the qualitative conclusion.
2. Decisive step.
Square the sum of signal and local-oscillator fields and retain the detector-band beat term. Use \(\langle(\Delta N)^2\rangle=\langle N\rangle\) for coherent photon counts and carry receiver bandwidth through the link SNR.
3. Verification.
Orthogonal or spatially mismatched fields must give zero heterodyne beat, and every branch count and efficiency must remain physically bounded.
Show detailed steps
Identify the governing conservation law and compare the relevant asymptotic regimes before drawing the qualitative conclusion. Square the sum of signal and local-oscillator fields and retain the detector-band beat term. Use \(\langle(\Delta N)^2\rangle=\langle N\rangle\) for coherent photon counts and carry receiver bandwidth through the link SNR. Orthogonal or spatially mismatched fields must give zero heterodyne beat, and every branch count and efficiency must remain physically bounded.
Problem 11.10 — heterodyne detection, Poisson statistics, and links: derivation
Brief solution
1. Method.
Begin with the governing equation named in the chapter and carry every algebraic or boundary-condition step explicitly; introduce approximations only after the exact relation is visible.
2. Decisive step.
Square the sum of signal and local-oscillator fields and retain the detector-band beat term. Use \(\langle(\Delta N)^2\rangle=\langle N\rangle\) for coherent photon counts and carry receiver bandwidth through the link SNR.
3. Verification.
Orthogonal or spatially mismatched fields must give zero heterodyne beat, and every branch count and efficiency must remain physically bounded.
Show detailed steps
Begin with the governing equation named in the chapter and carry every algebraic or boundary-condition step explicitly; introduce approximations only after the exact relation is visible. Square the sum of signal and local-oscillator fields and retain the detector-band beat term. Use \(\langle(\Delta N)^2\rangle=\langle N\rangle\) for coherent photon counts and carry receiver bandwidth through the link SNR. Orthogonal or spatially mismatched fields must give zero heterodyne beat, and every branch count and efficiency must remain physically bounded.
Problem 11.11 — heterodyne detection, Poisson statistics, and links: derivation
Brief solution
1. Method.
Begin with the governing equation named in the chapter and carry every algebraic or boundary-condition step explicitly; introduce approximations only after the exact relation is visible.
2. Decisive step.
Square the sum of signal and local-oscillator fields and retain the detector-band beat term. Use \(\langle(\Delta N)^2\rangle=\langle N\rangle\) for coherent photon counts and carry receiver bandwidth through the link SNR.
3. Verification.
Orthogonal or spatially mismatched fields must give zero heterodyne beat, and every branch count and efficiency must remain physically bounded.
Show detailed steps
Begin with the governing equation named in the chapter and carry every algebraic or boundary-condition step explicitly; introduce approximations only after the exact relation is visible. Square the sum of signal and local-oscillator fields and retain the detector-band beat term. Use \(\langle(\Delta N)^2\rangle=\langle N\rangle\) for coherent photon counts and carry receiver bandwidth through the link SNR. Orthogonal or spatially mismatched fields must give zero heterodyne beat, and every branch count and efficiency must remain physically bounded.
Problem 11.12 — heterodyne detection, Poisson statistics, and links: design
Brief solution
1. Method.
Translate each performance requirement into an equality or inequality, solve the coupled constraints, and reject any component value that violates power, bandwidth, or material limits.
2. Decisive step.
Square the sum of signal and local-oscillator fields and retain the detector-band beat term. Use \(\langle(\Delta N)^2\rangle=\langle N\rangle\) for coherent photon counts and carry receiver bandwidth through the link SNR.
3. Verification.
Orthogonal or spatially mismatched fields must give zero heterodyne beat, and every branch count and efficiency must remain physically bounded.
Show detailed steps
Translate each performance requirement into an equality or inequality, solve the coupled constraints, and reject any component value that violates power, bandwidth, or material limits. Square the sum of signal and local-oscillator fields and retain the detector-band beat term. Use \(\langle(\Delta N)^2\rangle=\langle N\rangle\) for coherent photon counts and carry receiver bandwidth through the link SNR. Orthogonal or spatially mismatched fields must give zero heterodyne beat, and every branch count and efficiency must remain physically bounded.