Chapter 10: Noise in Optical Detection and Generation
Source: Amnon Yariv and Pochi Yeh, Photonics: Optical Electronics in Modern Communications, sixth edition (2007), Chapter 10. 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 10.1 — shot noise, carrier response, and thermal 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.
Construct the single-carrier current impulse, Fourier-transform it, and apply the Poisson spectral-density rule for shot noise. For thermal noise, enforce equipartition in the equivalent RLC bandwidth to recover \(\langle i_n^2\rangle=4kT\Delta f/R\).
3. Verification.
Noise variances must add, remain nonnegative, and reduce to the low-frequency white-noise formulas when transit or scattering times vanish.
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. Construct the single-carrier current impulse, Fourier-transform it, and apply the Poisson spectral-density rule for shot noise. For thermal noise, enforce equipartition in the equivalent RLC bandwidth to recover \(\langle i_n^2\rangle=4kT\Delta f/R\). Noise variances must add, remain nonnegative, and reduce to the low-frequency white-noise formulas when transit or scattering times vanish.
Problem 10.2 — shot noise, carrier response, and thermal 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.
Construct the single-carrier current impulse, Fourier-transform it, and apply the Poisson spectral-density rule for shot noise. For thermal noise, enforce equipartition in the equivalent RLC bandwidth to recover \(\langle i_n^2\rangle=4kT\Delta f/R\).
3. Verification.
Noise variances must add, remain nonnegative, and reduce to the low-frequency white-noise formulas when transit or scattering times vanish.
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. Construct the single-carrier current impulse, Fourier-transform it, and apply the Poisson spectral-density rule for shot noise. For thermal noise, enforce equipartition in the equivalent RLC bandwidth to recover \(\langle i_n^2\rangle=4kT\Delta f/R\). Noise variances must add, remain nonnegative, and reduce to the low-frequency white-noise formulas when transit or scattering times vanish.
Problem 10.3 — shot noise, carrier response, and thermal 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.
Construct the single-carrier current impulse, Fourier-transform it, and apply the Poisson spectral-density rule for shot noise. For thermal noise, enforce equipartition in the equivalent RLC bandwidth to recover \(\langle i_n^2\rangle=4kT\Delta f/R\).
3. Verification.
Noise variances must add, remain nonnegative, and reduce to the low-frequency white-noise formulas when transit or scattering times vanish.
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. Construct the single-carrier current impulse, Fourier-transform it, and apply the Poisson spectral-density rule for shot noise. For thermal noise, enforce equipartition in the equivalent RLC bandwidth to recover \(\langle i_n^2\rangle=4kT\Delta f/R\). Noise variances must add, remain nonnegative, and reduce to the low-frequency white-noise formulas when transit or scattering times vanish.
Problem 10.4 — shot noise, carrier response, and thermal 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.
Construct the single-carrier current impulse, Fourier-transform it, and apply the Poisson spectral-density rule for shot noise. For thermal noise, enforce equipartition in the equivalent RLC bandwidth to recover \(\langle i_n^2\rangle=4kT\Delta f/R\).
3. Verification.
Noise variances must add, remain nonnegative, and reduce to the low-frequency white-noise formulas when transit or scattering times vanish.
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. Construct the single-carrier current impulse, Fourier-transform it, and apply the Poisson spectral-density rule for shot noise. For thermal noise, enforce equipartition in the equivalent RLC bandwidth to recover \(\langle i_n^2\rangle=4kT\Delta f/R\). Noise variances must add, remain nonnegative, and reduce to the low-frequency white-noise formulas when transit or scattering times vanish.
Problem 10.5 — shot noise, carrier response, and thermal 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.
Construct the single-carrier current impulse, Fourier-transform it, and apply the Poisson spectral-density rule for shot noise. For thermal noise, enforce equipartition in the equivalent RLC bandwidth to recover \(\langle i_n^2\rangle=4kT\Delta f/R\).
3. Verification.
Noise variances must add, remain nonnegative, and reduce to the low-frequency white-noise formulas when transit or scattering times vanish.
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. Construct the single-carrier current impulse, Fourier-transform it, and apply the Poisson spectral-density rule for shot noise. For thermal noise, enforce equipartition in the equivalent RLC bandwidth to recover \(\langle i_n^2\rangle=4kT\Delta f/R\). Noise variances must add, remain nonnegative, and reduce to the low-frequency white-noise formulas when transit or scattering times vanish.
Problem 10.6 — shot noise, carrier response, and thermal 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.
Construct the single-carrier current impulse, Fourier-transform it, and apply the Poisson spectral-density rule for shot noise. For thermal noise, enforce equipartition in the equivalent RLC bandwidth to recover \(\langle i_n^2\rangle=4kT\Delta f/R\).
3. Verification.
Noise variances must add, remain nonnegative, and reduce to the low-frequency white-noise formulas when transit or scattering times vanish.
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. Construct the single-carrier current impulse, Fourier-transform it, and apply the Poisson spectral-density rule for shot noise. For thermal noise, enforce equipartition in the equivalent RLC bandwidth to recover \(\langle i_n^2\rangle=4kT\Delta f/R\). Noise variances must add, remain nonnegative, and reduce to the low-frequency white-noise formulas when transit or scattering times vanish.
Problem 10.7 — laser frequency noise, error probability, and sensing: 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.
Differentiate the resonance condition \(2nL\nu/c=m\) to obtain fractional frequency sensitivity. Convert Gaussian decision thresholds to complementary-error functions, and express sensor output as a calibrated phase or beat-frequency change.
3. Verification.
Common fractional changes in optical length must give the opposite fractional change in frequency, and predicted probabilities must remain between zero and one.
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. Differentiate the resonance condition \(2nL\nu/c=m\) to obtain fractional frequency sensitivity. Convert Gaussian decision thresholds to complementary-error functions, and express sensor output as a calibrated phase or beat-frequency change. Common fractional changes in optical length must give the opposite fractional change in frequency, and predicted probabilities must remain between zero and one.
Problem 10.8 — laser frequency noise, error probability, and sensing: 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.
Differentiate the resonance condition \(2nL\nu/c=m\) to obtain fractional frequency sensitivity. Convert Gaussian decision thresholds to complementary-error functions, and express sensor output as a calibrated phase or beat-frequency change.
3. Verification.
Common fractional changes in optical length must give the opposite fractional change in frequency, and predicted probabilities must remain between zero and one.
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. Differentiate the resonance condition \(2nL\nu/c=m\) to obtain fractional frequency sensitivity. Convert Gaussian decision thresholds to complementary-error functions, and express sensor output as a calibrated phase or beat-frequency change. Common fractional changes in optical length must give the opposite fractional change in frequency, and predicted probabilities must remain between zero and one.
Problem 10.9 — laser frequency noise, error probability, and sensing: 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.
Differentiate the resonance condition \(2nL\nu/c=m\) to obtain fractional frequency sensitivity. Convert Gaussian decision thresholds to complementary-error functions, and express sensor output as a calibrated phase or beat-frequency change.
3. Verification.
Common fractional changes in optical length must give the opposite fractional change in frequency, and predicted probabilities must remain between zero and one.
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. Differentiate the resonance condition \(2nL\nu/c=m\) to obtain fractional frequency sensitivity. Convert Gaussian decision thresholds to complementary-error functions, and express sensor output as a calibrated phase or beat-frequency change. Common fractional changes in optical length must give the opposite fractional change in frequency, and predicted probabilities must remain between zero and one.
Problem 10.10 — laser frequency noise, error probability, and sensing: 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.
Differentiate the resonance condition \(2nL\nu/c=m\) to obtain fractional frequency sensitivity. Convert Gaussian decision thresholds to complementary-error functions, and express sensor output as a calibrated phase or beat-frequency change.
3. Verification.
Common fractional changes in optical length must give the opposite fractional change in frequency, and predicted probabilities must remain between zero and one.
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. Differentiate the resonance condition \(2nL\nu/c=m\) to obtain fractional frequency sensitivity. Convert Gaussian decision thresholds to complementary-error functions, and express sensor output as a calibrated phase or beat-frequency change. Common fractional changes in optical length must give the opposite fractional change in frequency, and predicted probabilities must remain between zero and one.
Problem 10.11 — laser frequency noise, error probability, and sensing: 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.
Differentiate the resonance condition \(2nL\nu/c=m\) to obtain fractional frequency sensitivity. Convert Gaussian decision thresholds to complementary-error functions, and express sensor output as a calibrated phase or beat-frequency change.
3. Verification.
Common fractional changes in optical length must give the opposite fractional change in frequency, and predicted probabilities must remain between zero and one.
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. Differentiate the resonance condition \(2nL\nu/c=m\) to obtain fractional frequency sensitivity. Convert Gaussian decision thresholds to complementary-error functions, and express sensor output as a calibrated phase or beat-frequency change. Common fractional changes in optical length must give the opposite fractional change in frequency, and predicted probabilities must remain between zero and one.
Problem 10.12 — laser frequency noise, error probability, and sensing: discussion
Brief solution
1. Method.
Identify the governing conservation law and compare the relevant asymptotic regimes before drawing the qualitative conclusion.
2. Decisive step.
Differentiate the resonance condition \(2nL\nu/c=m\) to obtain fractional frequency sensitivity. Convert Gaussian decision thresholds to complementary-error functions, and express sensor output as a calibrated phase or beat-frequency change.
3. Verification.
Common fractional changes in optical length must give the opposite fractional change in frequency, and predicted probabilities must remain between zero and one.
Show detailed steps
Identify the governing conservation law and compare the relevant asymptotic regimes before drawing the qualitative conclusion. Differentiate the resonance condition \(2nL\nu/c=m\) to obtain fractional frequency sensitivity. Convert Gaussian decision thresholds to complementary-error functions, and express sensor output as a calibrated phase or beat-frequency change. Common fractional changes in optical length must give the opposite fractional change in frequency, and predicted probabilities must remain between zero and one.