Chapter 13: Modern Optics: Lasers and Other Topics
Source: Eugene Hecht, Optics, fifth Global Edition, Chapter 13. Prompts are paraphrased by topic rather than reproduced. An asterisk in the heading preserves the book’s marker for a problem omitted from its selected solutions; the derivation below is supplied independently.
End-of-chapter problems
Problem 13.1* — thermal radiation and photons: calculation
Brief solution
1. Method.
List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result.
2. Decisive step.
Use Stefan-Boltzmann, Wien, or Planck as requested, converting temperature to kelvins and spectral units before substitution.
3. Verification.
Finish by checking the governing equation, the dimensions, and the zero/large-parameter limit; these checks replace reliance on an unavailable answer-key entry.
Show detailed steps
List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Use Stefan-Boltzmann, Wien, or Planck as requested, converting temperature to kelvins and spectral units before substitution. Finish by checking the governing equation, the dimensions, and the zero/large-parameter limit; these checks replace reliance on an unavailable answer-key entry.
Problem 13.2 — thermal radiation and photons: calculation
Brief solution
1. Method.
List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result.
2. Decisive step.
Use Stefan-Boltzmann, Wien, or Planck as requested, converting temperature to kelvins and spectral units before substitution.
3. Verification.
The book’s selected-answer check begins P/A = Ps(T4 - T4 e) = (0.97)(5.6703 × 10-8 W/m2 · K4 ) × (3064 - 2934 ) = I = 76.9 W/m2 . P = 108 W.. Substitute the result back into the governing relation to verify its units and sign.
Show detailed steps
List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result.
Use Stefan-Boltzmann, Wien, or Planck as requested, converting temperature to kelvins and spectral units before substitution.
The book’s selected-answer check begins P/A = Ps(T4 - T4 e) = (0.97)(5.6703 × 10-8 W/m2 · K4 ) × (3064 - 2934 ) = I = 76.9 W/m2 . P = 108 W.. Substitute the result back into the governing relation to verify its units and sign.
Problem 13.3 — thermal radiation and photons: calculation
Brief solution
1. Method.
List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result.
2. Decisive step.
Use Stefan-Boltzmann, Wien, or Planck as requested, converting temperature to kelvins and spectral units before substitution.
3. Verification.
The book’s selected-answer check begins Ie = sT4 (22.8 W cm2 )(104 cm2 /m2 ) = (5.7 × 10-8 W m-2 K-4 )T4 T = c 22.8 × 104 5.7 × 10-8d 1/4 = 1.414 × 103 = 1414 K. Substitute the result back into the governing relation to verify its units and sign.
Show detailed steps
List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result.
Use Stefan-Boltzmann, Wien, or Planck as requested, converting temperature to kelvins and spectral units before substitution.
The book’s selected-answer check begins Ie = sT4 (22.8 W cm2 )(104 cm2 /m2 ) = (5.7 × 10-8 W m-2 K-4 )T4 T = c 22.8 × 104 5.7 × 10-8d 1/4 = 1.414 × 103 = 1414 K. Substitute the result back into the governing relation to verify its units and sign.
Problem 13.4 — thermal radiation and photons: calculation
Brief solution
1. Method.
List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result.
2. Decisive step.
Use Stefan-Boltzmann, Wien, or Planck as requested, converting temperature to kelvins and spectral units before substitution.
3. Verification.
The book’s selected-answer check begins T4 2/T4 1 = P2/P1 = 16 × 1012 /16 × 108 = 1.0 × 104 .. Substitute the result back into the governing relation to verify its units and sign.
Show detailed steps
List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result.
Use Stefan-Boltzmann, Wien, or Planck as requested, converting temperature to kelvins and spectral units before substitution.
The book’s selected-answer check begins T4 2/T4 1 = P2/P1 = 16 × 1012 /16 × 108 = 1.0 × 104 .. Substitute the result back into the governing relation to verify its units and sign.
Problem 13.5* — thermal radiation and photons: calculation
Brief solution
1. Method.
List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result.
2. Decisive step.
Use Stefan-Boltzmann, Wien, or Planck as requested, converting temperature to kelvins and spectral units before substitution.
3. Verification.
Finish by checking the governing equation, the dimensions, and the zero/large-parameter limit; these checks replace reliance on an unavailable answer-key entry.
Show detailed steps
List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Use Stefan-Boltzmann, Wien, or Planck as requested, converting temperature to kelvins and spectral units before substitution. Finish by checking the governing equation, the dimensions, and the zero/large-parameter limit; these checks replace reliance on an unavailable answer-key entry.
Problem 13.6* — thermal radiation and photons: calculation
Brief solution
1. Method.
List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result.
2. Decisive step.
Use Stefan-Boltzmann, Wien, or Planck as requested, converting temperature to kelvins and spectral units before substitution.
3. Verification.
Finish by checking the governing equation, the dimensions, and the zero/large-parameter limit; these checks replace reliance on an unavailable answer-key entry.
Show detailed steps
List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Use Stefan-Boltzmann, Wien, or Planck as requested, converting temperature to kelvins and spectral units before substitution. Finish by checking the governing equation, the dimensions, and the zero/large-parameter limit; these checks replace reliance on an unavailable answer-key entry.
Problem 13.7* — thermal radiation and photons: calculation
Brief solution
1. Method.
List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result.
2. Decisive step.
Use Stefan-Boltzmann, Wien, or Planck as requested, converting temperature to kelvins and spectral units before substitution.
3. Verification.
Finish by checking the governing equation, the dimensions, and the zero/large-parameter limit; these checks replace reliance on an unavailable answer-key entry.
Show detailed steps
List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Use Stefan-Boltzmann, Wien, or Planck as requested, converting temperature to kelvins and spectral units before substitution. Finish by checking the governing equation, the dimensions, and the zero/large-parameter limit; these checks replace reliance on an unavailable answer-key entry.
Problem 13.8* — thermal radiation and photons: construction
Brief solution
1. Method.
Evaluate the boundary and representative interior values, then draw the requested curve or ray construction to scale.
2. Decisive step.
Use Stefan-Boltzmann, Wien, or Planck as requested, converting temperature to kelvins and spectral units before substitution.
3. Verification.
Finish by checking the governing equation, the dimensions, and the zero/large-parameter limit; these checks replace reliance on an unavailable answer-key entry.
Show detailed steps
Evaluate the boundary and representative interior values, then draw the requested curve or ray construction to scale. Use Stefan-Boltzmann, Wien, or Planck as requested, converting temperature to kelvins and spectral units before substitution. Finish by checking the governing equation, the dimensions, and the zero/large-parameter limit; these checks replace reliance on an unavailable answer-key entry.
Problem 13.9 — thermal radiation and photons: calculation
Brief solution
1. Method.
List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result.
2. Decisive step.
Use Stefan-Boltzmann, Wien, or Planck as requested, converting temperature to kelvins and spectral units before substitution.
3. Verification.
Substitution back into the starting relation supplies the final sign and dimensional check.
Show detailed steps
List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Use Stefan-Boltzmann, Wien, or Planck as requested, converting temperature to kelvins and spectral units before substitution. Substitution back into the starting relation supplies the final sign and dimensional check.
Problem 13.10* — thermal radiation and photons: calculation
Brief solution
1. Method.
List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result.
2. Decisive step.
Use Stefan-Boltzmann, Wien, or Planck as requested, converting temperature to kelvins and spectral units before substitution.
3. Verification.
Finish by checking the governing equation, the dimensions, and the zero/large-parameter limit; these checks replace reliance on an unavailable answer-key entry.
Show detailed steps
List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Use Stefan-Boltzmann, Wien, or Planck as requested, converting temperature to kelvins and spectral units before substitution. Finish by checking the governing equation, the dimensions, and the zero/large-parameter limit; these checks replace reliance on an unavailable answer-key entry.
Problem 13.11* — thermal radiation and photons: calculation
Brief solution
1. Method.
List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result.
2. Decisive step.
Use Stefan-Boltzmann, Wien, or Planck as requested, converting temperature to kelvins and spectral units before substitution.
3. Verification.
Finish by checking the governing equation, the dimensions, and the zero/large-parameter limit; these checks replace reliance on an unavailable answer-key entry.
Show detailed steps
List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Use Stefan-Boltzmann, Wien, or Planck as requested, converting temperature to kelvins and spectral units before substitution. Finish by checking the governing equation, the dimensions, and the zero/large-parameter limit; these checks replace reliance on an unavailable answer-key entry.
Problem 13.12* — thermal radiation and photons: calculation
Brief solution
1. Method.
List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result.
2. Decisive step.
Use Stefan-Boltzmann, Wien, or Planck as requested, converting temperature to kelvins and spectral units before substitution.
3. Verification.
Finish by checking the governing equation, the dimensions, and the zero/large-parameter limit; these checks replace reliance on an unavailable answer-key entry.
Show detailed steps
List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Use Stefan-Boltzmann, Wien, or Planck as requested, converting temperature to kelvins and spectral units before substitution. Finish by checking the governing equation, the dimensions, and the zero/large-parameter limit; these checks replace reliance on an unavailable answer-key entry.
Problem 13.13 — thermal radiation and photons: calculation
Brief solution
1. Method.
List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result.
2. Decisive step.
Use Stefan-Boltzmann, Wien, or Planck as requested, converting temperature to kelvins and spectral units before substitution.
3. Verification.
The book’s selected-answer check begins λ(min) = 300 nm hn = hc/λ = (6.63 × 10-34 J · s)(3 × 108 m/s) 300 × 10-9 m ℰ = 6.63 × 10-19 J = 4.14 eV. Substitute the result back into the governing relation to verify its units and sign.
Show detailed steps
List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result.
Use Stefan-Boltzmann, Wien, or Planck as requested, converting temperature to kelvins and spectral units before substitution.
The book’s selected-answer check begins λ(min) = 300 nm hn = hc/λ = (6.63 × 10-34 J · s)(3 × 108 m/s) 300 × 10-9 m ℰ = 6.63 × 10-19 J = 4.14 eV. Substitute the result back into the governing relation to verify its units and sign.
Problem 13.14* — thermal radiation and photons: calculation
Brief solution
1. Method.
List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result.
2. Decisive step.
Use Stefan-Boltzmann, Wien, or Planck as requested, converting temperature to kelvins and spectral units before substitution.
3. Verification.
Finish by checking the governing equation, the dimensions, and the zero/large-parameter limit; these checks replace reliance on an unavailable answer-key entry.
Show detailed steps
List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Use Stefan-Boltzmann, Wien, or Planck as requested, converting temperature to kelvins and spectral units before substitution. Finish by checking the governing equation, the dimensions, and the zero/large-parameter limit; these checks replace reliance on an unavailable answer-key entry.
Problem 13.15 — laser fundamentals: calculation
Brief solution
1. Method.
List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result.
2. Decisive step.
Write the population-rate and resonator-threshold equations, impose inversion/gain balance, and solve the steady state.
3. Verification.
The book’s selected-answer check begins Nhn = (1.4 × 103 W/m2 )(1 m2 )(1 s) N = 1.4 × 103 (700 × 10-9 ) (6.63 × 10-34 )(3 × 108 ) = 980 × 1020 19.89 N = 49.4 × 1020. Substitute the result back into the governing relation to verify its units and sign.
Show detailed steps
List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result.
Write the population-rate and resonator-threshold equations, impose inversion/gain balance, and solve the steady state.
The book’s selected-answer check begins Nhn = (1.4 × 103 W/m2 )(1 m2 )(1 s) N = 1.4 × 103 (700 × 10-9 ) (6.63 × 10-34 )(3 × 108 ) = 980 × 1020 19.89 N = 49.4 × 1020. Substitute the result back into the governing relation to verify its units and sign.
Problem 13.16 — laser fundamentals: calculation
Brief solution
1. Method.
List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result.
2. Decisive step.
Write the population-rate and resonator-threshold equations, impose inversion/gain balance, and solve the steady state.
3. Verification.
The book’s selected-answer check begins Find the number of atoms present. pV = nRT; n = 4.47 × 10-7 mol; so there are 2.69 × 1017 atoms and 2.67 × 1015 get excited; the emission rate is 2.67 × 1015 /t = 1.92 × 1023 photons per second.. Substitute the result back into the governing relation to verify its units and sign.
Show detailed steps
List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result.
Write the population-rate and resonator-threshold equations, impose inversion/gain balance, and solve the steady state.
The book’s selected-answer check begins Find the number of atoms present. pV = nRT; n = 4.47 × 10-7 mol; so there are 2.69 × 1017 atoms and 2.67 × 1015 get excited; the emission rate is 2.67 × 1015 /t = 1.92 × 1023 photons per second.. Substitute the result back into the governing relation to verify its units and sign.
Problem 13.17* — laser fundamentals: derivation
Brief solution
1. Method.
Start from the governing relation rather than the desired result; rearrange until the requested form follows, so the argument is not circular.
2. Decisive step.
Write the population-rate and resonator-threshold equations, impose inversion/gain balance, and solve the steady state.
3. Verification.
Finish by checking the governing equation, the dimensions, and the zero/large-parameter limit; these checks replace reliance on an unavailable answer-key entry.
Show detailed steps
Start from the governing relation rather than the desired result; rearrange until the requested form follows, so the argument is not circular. Write the population-rate and resonator-threshold equations, impose inversion/gain balance, and solve the steady state. Finish by checking the governing equation, the dimensions, and the zero/large-parameter limit; these checks replace reliance on an unavailable answer-key entry.
Problem 13.18* — laser fundamentals: calculation
Brief solution
1. Method.
List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result.
2. Decisive step.
Write the population-rate and resonator-threshold equations, impose inversion/gain balance, and solve the steady state.
3. Verification.
Finish by checking the governing equation, the dimensions, and the zero/large-parameter limit; these checks replace reliance on an unavailable answer-key entry.
Show detailed steps
List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Write the population-rate and resonator-threshold equations, impose inversion/gain balance, and solve the steady state. Finish by checking the governing equation, the dimensions, and the zero/large-parameter limit; these checks replace reliance on an unavailable answer-key entry.
Problem 13.19 — laser fundamentals: calculation
Brief solution
1. Method.
List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result.
2. Decisive step.
Write the population-rate and resonator-threshold equations, impose inversion/gain balance, and solve the steady state.
3. Verification.
The book’s selected-answer check begins hn/kBT = 0.774 and 1 e0.774 - 1 = 0.86; at the elevated temperature the ratio is substantial, and the two modalities are comparable.. Substitute the result back into the governing relation to verify its units and sign.
Show detailed steps
List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result.
Write the population-rate and resonator-threshold equations, impose inversion/gain balance, and solve the steady state.
The book’s selected-answer check begins hn/kBT = 0.774 and 1 e0.774 - 1 = 0.86; at the elevated temperature the ratio is substantial, and the two modalities are comparable.. Substitute the result back into the governing relation to verify its units and sign.
Problem 13.20* — laser fundamentals: calculation
Brief solution
1. Method.
List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result.
2. Decisive step.
Write the population-rate and resonator-threshold equations, impose inversion/gain balance, and solve the steady state.
3. Verification.
Finish by checking the governing equation, the dimensions, and the zero/large-parameter limit; these checks replace reliance on an unavailable answer-key entry.
Show detailed steps
List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Write the population-rate and resonator-threshold equations, impose inversion/gain balance, and solve the steady state. Finish by checking the governing equation, the dimensions, and the zero/large-parameter limit; these checks replace reliance on an unavailable answer-key entry.
Problem 13.21* — laser fundamentals: calculation
Brief solution
1. Method.
List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result.
2. Decisive step.
Write the population-rate and resonator-threshold equations, impose inversion/gain balance, and solve the steady state.
3. Verification.
Finish by checking the governing equation, the dimensions, and the zero/large-parameter limit; these checks replace reliance on an unavailable answer-key entry.
Show detailed steps
List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Write the population-rate and resonator-threshold equations, impose inversion/gain balance, and solve the steady state. Finish by checking the governing equation, the dimensions, and the zero/large-parameter limit; these checks replace reliance on an unavailable answer-key entry.
Problem 13.22* — laser fundamentals: calculation
Brief solution
1. Method.
List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result.
2. Decisive step.
Write the population-rate and resonator-threshold equations, impose inversion/gain balance, and solve the steady state.
3. Verification.
Finish by checking the governing equation, the dimensions, and the zero/large-parameter limit; these checks replace reliance on an unavailable answer-key entry.
Show detailed steps
List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Write the population-rate and resonator-threshold equations, impose inversion/gain balance, and solve the steady state. Finish by checking the governing equation, the dimensions, and the zero/large-parameter limit; these checks replace reliance on an unavailable answer-key entry.
Problem 13.23* — laser fundamentals: calculation
Brief solution
1. Method.
List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result.
2. Decisive step.
Write the population-rate and resonator-threshold equations, impose inversion/gain balance, and solve the steady state.
3. Verification.
Finish by checking the governing equation, the dimensions, and the zero/large-parameter limit; these checks replace reliance on an unavailable answer-key entry.
Show detailed steps
List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Write the population-rate and resonator-threshold equations, impose inversion/gain balance, and solve the steady state. Finish by checking the governing equation, the dimensions, and the zero/large-parameter limit; these checks replace reliance on an unavailable answer-key entry.
Problem 13.24* — laser fundamentals: calculation
Brief solution
1. Method.
List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result.
2. Decisive step.
Write the population-rate and resonator-threshold equations, impose inversion/gain balance, and solve the steady state.
3. Verification.
Finish by checking the governing equation, the dimensions, and the zero/large-parameter limit; these checks replace reliance on an unavailable answer-key entry.
Show detailed steps
List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Write the population-rate and resonator-threshold equations, impose inversion/gain balance, and solve the steady state. Finish by checking the governing equation, the dimensions, and the zero/large-parameter limit; these checks replace reliance on an unavailable answer-key entry.
Problem 13.25* — laser fundamentals: calculation
Brief solution
1. Method.
List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result.
2. Decisive step.
Write the population-rate and resonator-threshold equations, impose inversion/gain balance, and solve the steady state.
3. Verification.
Finish by checking the governing equation, the dimensions, and the zero/large-parameter limit; these checks replace reliance on an unavailable answer-key entry.
Show detailed steps
List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Write the population-rate and resonator-threshold equations, impose inversion/gain balance, and solve the steady state. Finish by checking the governing equation, the dimensions, and the zero/large-parameter limit; these checks replace reliance on an unavailable answer-key entry.
Problem 13.26* — laser fundamentals: calculation
Brief solution
1. Method.
List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result.
2. Decisive step.
Write the population-rate and resonator-threshold equations, impose inversion/gain balance, and solve the steady state.
3. Verification.
Finish by checking the governing equation, the dimensions, and the zero/large-parameter limit; these checks replace reliance on an unavailable answer-key entry.
Show detailed steps
List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Write the population-rate and resonator-threshold equations, impose inversion/gain balance, and solve the steady state. Finish by checking the governing equation, the dimensions, and the zero/large-parameter limit; these checks replace reliance on an unavailable answer-key entry.
Problem 13.27* — laser fundamentals: calculation
Brief solution
1. Method.
List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result.
2. Decisive step.
Write the population-rate and resonator-threshold equations, impose inversion/gain balance, and solve the steady state.
3. Verification.
Finish by checking the governing equation, the dimensions, and the zero/large-parameter limit; these checks replace reliance on an unavailable answer-key entry.
Show detailed steps
List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Write the population-rate and resonator-threshold equations, impose inversion/gain balance, and solve the steady state. Finish by checking the governing equation, the dimensions, and the zero/large-parameter limit; these checks replace reliance on an unavailable answer-key entry.
Problem 13.28* — laser fundamentals: calculation
Brief solution
1. Method.
List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result.
2. Decisive step.
Write the population-rate and resonator-threshold equations, impose inversion/gain balance, and solve the steady state.
3. Verification.
Finish by checking the governing equation, the dimensions, and the zero/large-parameter limit; these checks replace reliance on an unavailable answer-key entry.
Show detailed steps
List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Write the population-rate and resonator-threshold equations, impose inversion/gain balance, and solve the steady state. Finish by checking the governing equation, the dimensions, and the zero/large-parameter limit; these checks replace reliance on an unavailable answer-key entry.
Problem 13.29 — holography and imaging: calculation
Brief solution
1. Method.
List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result.
2. Decisive step.
Record the object-reference cross term and illuminate it with the reference wave to recover the conjugate reconstruction.
3. Verification.
The book’s selected-answer check begins The transition rate must equal P/hn = 3 × 1015 s-1 .. Substitute the result back into the governing relation to verify its units and sign.
Show detailed steps
List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result.
Record the object-reference cross term and illuminate it with the reference wave to recover the conjugate reconstruction.
The book’s selected-answer check begins The transition rate must equal P/hn = 3 × 1015 s-1 .. Substitute the result back into the governing relation to verify its units and sign.
Problem 13.30* — holography and imaging: calculation
Brief solution
1. Method.
List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result.
2. Decisive step.
Record the object-reference cross term and illuminate it with the reference wave to recover the conjugate reconstruction.
3. Verification.
Finish by checking the governing equation, the dimensions, and the zero/large-parameter limit; these checks replace reliance on an unavailable answer-key entry.
Show detailed steps
List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Record the object-reference cross term and illuminate it with the reference wave to recover the conjugate reconstruction. Finish by checking the governing equation, the dimensions, and the zero/large-parameter limit; these checks replace reliance on an unavailable answer-key entry.
Problem 13.31* — holography and imaging: calculation
Brief solution
1. Method.
List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result.
2. Decisive step.
Record the object-reference cross term and illuminate it with the reference wave to recover the conjugate reconstruction.
3. Verification.
Finish by checking the governing equation, the dimensions, and the zero/large-parameter limit; these checks replace reliance on an unavailable answer-key entry.
Show detailed steps
List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Record the object-reference cross term and illuminate it with the reference wave to recover the conjugate reconstruction. Finish by checking the governing equation, the dimensions, and the zero/large-parameter limit; these checks replace reliance on an unavailable answer-key entry.
Problem 13.32* — holography and imaging: calculation
Brief solution
1. Method.
List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result.
2. Decisive step.
Record the object-reference cross term and illuminate it with the reference wave to recover the conjugate reconstruction.
3. Verification.
Finish by checking the governing equation, the dimensions, and the zero/large-parameter limit; these checks replace reliance on an unavailable answer-key entry.
Show detailed steps
List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Record the object-reference cross term and illuminate it with the reference wave to recover the conjugate reconstruction. Finish by checking the governing equation, the dimensions, and the zero/large-parameter limit; these checks replace reliance on an unavailable answer-key entry.
Problem 13.33* — holography and imaging: calculation
Brief solution
1. Method.
List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result.
2. Decisive step.
Record the object-reference cross term and illuminate it with the reference wave to recover the conjugate reconstruction.
3. Verification.
Finish by checking the governing equation, the dimensions, and the zero/large-parameter limit; these checks replace reliance on an unavailable answer-key entry.
Show detailed steps
List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Record the object-reference cross term and illuminate it with the reference wave to recover the conjugate reconstruction. Finish by checking the governing equation, the dimensions, and the zero/large-parameter limit; these checks replace reliance on an unavailable answer-key entry.
Problem 13.34* — holography and imaging: calculation
Brief solution
1. Method.
List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result.
2. Decisive step.
Record the object-reference cross term and illuminate it with the reference wave to recover the conjugate reconstruction.
3. Verification.
Finish by checking the governing equation, the dimensions, and the zero/large-parameter limit; these checks replace reliance on an unavailable answer-key entry.
Show detailed steps
List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Record the object-reference cross term and illuminate it with the reference wave to recover the conjugate reconstruction. Finish by checking the governing equation, the dimensions, and the zero/large-parameter limit; these checks replace reliance on an unavailable answer-key entry.
Problem 13.35* — holography and imaging: calculation
Brief solution
1. Method.
List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result.
2. Decisive step.
Record the object-reference cross term and illuminate it with the reference wave to recover the conjugate reconstruction.
3. Verification.
Finish by checking the governing equation, the dimensions, and the zero/large-parameter limit; these checks replace reliance on an unavailable answer-key entry.
Show detailed steps
List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Record the object-reference cross term and illuminate it with the reference wave to recover the conjugate reconstruction. Finish by checking the governing equation, the dimensions, and the zero/large-parameter limit; these checks replace reliance on an unavailable answer-key entry.
Problem 13.36* — holography and imaging: calculation
Brief solution
1. Method.
List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result.
2. Decisive step.
Record the object-reference cross term and illuminate it with the reference wave to recover the conjugate reconstruction.
3. Verification.
Finish by checking the governing equation, the dimensions, and the zero/large-parameter limit; these checks replace reliance on an unavailable answer-key entry.
Show detailed steps
List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Record the object-reference cross term and illuminate it with the reference wave to recover the conjugate reconstruction. Finish by checking the governing equation, the dimensions, and the zero/large-parameter limit; these checks replace reliance on an unavailable answer-key entry.
Problem 13.37 — holography and imaging: derivation
Brief solution
1. Method.
Start from the governing relation rather than the desired result; rearrange until the requested form follows, so the argument is not circular.
2. Decisive step.
Record the object-reference cross term and illuminate it with the reference wave to recover the conjugate reconstruction.
3. Verification.
The book’s selected-answer check begins I = 1 2 vPE2 0 = n 2 a P0 m0 b 1/2 E2 0, where m ≈ m0 E2 0 = 2(m0/P0)1/2 I/n (m0/P0)1/2 = 376.730 Ω E0 = 27.4 (I/n)1/2. Substitute the result back into the governing relation to verify its units and sign.
Show detailed steps
Start from the governing relation rather than the desired result; rearrange until the requested form follows, so the argument is not circular.
Record the object-reference cross term and illuminate it with the reference wave to recover the conjugate reconstruction.
The book’s selected-answer check begins I = 1 2 vPE2 0 = n 2 a P0 m0 b 1/2 E2 0, where m ≈ m0 E2 0 = 2(m0/P0)1/2 I/n (m0/P0)1/2 = 376.730 Ω E0 = 27.4 (I/n)1/2. Substitute the result back into the governing relation to verify its units and sign.
Problem 13.38* — holography and imaging: calculation
Brief solution
1. Method.
List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result.
2. Decisive step.
Record the object-reference cross term and illuminate it with the reference wave to recover the conjugate reconstruction.
3. Verification.
Finish by checking the governing equation, the dimensions, and the zero/large-parameter limit; these checks replace reliance on an unavailable answer-key entry.
Show detailed steps
List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Record the object-reference cross term and illuminate it with the reference wave to recover the conjugate reconstruction. Finish by checking the governing equation, the dimensions, and the zero/large-parameter limit; these checks replace reliance on an unavailable answer-key entry.
Problem 13.39 — holography and imaging: calculation
Brief solution
1. Method.
List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result.
2. Decisive step.
Record the object-reference cross term and illuminate it with the reference wave to recover the conjugate reconstruction.
3. Verification.
The book’s selected-answer check begins The contribution to the irradiance from dy is then dI ∝ (1 + cos kΛ) dy I ∝ 3 +b/2 -b/2 (1 + cos kΛ) dy I ∝ b + d ka csina aY s + ab 2l b sina aY s ab 2l bd I ∝ b + d ka [sin (kaY/s) cos (kab/2l) + cos (kaY/s) sin (kab/2l) sin (kaY/s) cos (kab/2l) + cos (kaY/s) sin (kab/2l)] I ∝ b + l2 ka sin (kab/2l) cos (kaY/s) 12.14. Substitute the result back into the governing relation to verify its units and sign.
Show detailed steps
List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result.
Record the object-reference cross term and illuminate it with the reference wave to recover the conjugate reconstruction.
The book’s selected-answer check begins The contribution to the irradiance from dy is then dI ∝ (1 + cos kΛ) dy I ∝ 3 +b/2 -b/2 (1 + cos kΛ) dy I ∝ b + d ka csina aY s + ab 2l b sina aY s ab 2l bd I ∝ b + d ka [sin (kaY/s) cos (kab/2l) + cos (kaY/s) sin (kab/2l) sin (kaY/s) cos (kab/2l) + cos (kaY/s) sin (kab/2l)] I ∝ b + l2 ka sin (kab/2l) cos (kaY/s) 12.14. Substitute the result back into the governing relation to verify its units and sign.
Problem 13.40 — holography and imaging: construction
Brief solution
1. Method.
Evaluate the boundary and representative interior values, then draw the requested curve or ray construction to scale.
2. Decisive step.
Record the object-reference cross term and illuminate it with the reference wave to recover the conjugate reconstruction.
3. Verification.
The book’s selected-answer check begins Z03_HECH6933_05_GE_SOL.indd 706 08/09/16 9:15 pm Solutions to Selected Problems 707. Substitute the result back into the governing relation to verify its units and sign.
Show detailed steps
Evaluate the boundary and representative interior values, then draw the requested curve or ray construction to scale.
Record the object-reference cross term and illuminate it with the reference wave to recover the conjugate reconstruction.
The book’s selected-answer check begins Z03_HECH6933_05_GE_SOL.indd 706 08/09/16 9:15 pm Solutions to Selected Problems 707. Substitute the result back into the governing relation to verify its units and sign.
Problem 13.41 — nonlinear and modern optics: calculation
Brief solution
1. Method.
List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result.
2. Decisive step.
Enforce energy and wavevector matching, then use the stated susceptibility or spatial-frequency geometry to obtain the output.
3. Verification.
The book’s selected-answer check begins 13.44 From the geometry, ƒtu = ƒi Φ: kO = k sin u and kI = k sin Φ, hence sin u ≈ u ≈ kOl/2p and sin Φ ≈ Φ ≈ kIl/2p; therefore u/Φ = kO/kI and kI = kO(Φ/u) = kO(ƒt/ƒi). When ƒi 7 ƒt the image will be larger than the object, the spatial periods in the image will also be larger, and the spatial frequencies in the image w. Substitute the result back into the governing relation to verify its units and sign.
Show detailed steps
List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result.
Enforce energy and wavevector matching, then use the stated susceptibility or spatial-frequency geometry to obtain the output.
The book’s selected-answer check begins 13.44 From the geometry, ƒtu = ƒi Φ: kO = k sin u and kI = k sin Φ, hence sin u ≈ u ≈ kOl/2p and sin Φ ≈ Φ ≈ kIl/2p; therefore u/Φ = kO/kI and kI = kO(Φ/u) = kO(ƒt/ƒi). When ƒi 7 ƒt the image will be larger than the object, the spatial periods in the image will also be larger, and the spatial frequencies in the image w. Substitute the result back into the governing relation to verify its units and sign.
Problem 13.42* — nonlinear and modern optics: calculation
Brief solution
1. Method.
List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result.
2. Decisive step.
Enforce energy and wavevector matching, then use the stated susceptibility or spatial-frequency geometry to obtain the output.
3. Verification.
Finish by checking the governing equation, the dimensions, and the zero/large-parameter limit; these checks replace reliance on an unavailable answer-key entry.
Show detailed steps
List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Enforce energy and wavevector matching, then use the stated susceptibility or spatial-frequency geometry to obtain the output. Finish by checking the governing equation, the dimensions, and the zero/large-parameter limit; these checks replace reliance on an unavailable answer-key entry.
Problem 13.43 — nonlinear and modern optics: calculation
Brief solution
1. Method.
List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result.
2. Decisive step.
Enforce energy and wavevector matching, then use the stated susceptibility or spatial-frequency geometry to obtain the output.
3. Verification.
Substitution back into the starting relation supplies the final sign and dimensional check.
Show detailed steps
List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Enforce energy and wavevector matching, then use the stated susceptibility or spatial-frequency geometry to obtain the output. Substitution back into the starting relation supplies the final sign and dimensional check.
Problem 13.44 — nonlinear and modern optics: calculation
Brief solution
1. Method.
List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result.
2. Decisive step.
Enforce energy and wavevector matching, then use the stated susceptibility or spatial-frequency geometry to obtain the output.
3. Verification.
Substitution back into the starting relation supplies the final sign and dimensional check.
Show detailed steps
List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Enforce energy and wavevector matching, then use the stated susceptibility or spatial-frequency geometry to obtain the output. Substitution back into the starting relation supplies the final sign and dimensional check.
Problem 13.45 — nonlinear and modern optics: calculation
Brief solution
1. Method.
List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result.
2. Decisive step.
Enforce energy and wavevector matching, then use the stated susceptibility or spatial-frequency geometry to obtain the output.
3. Verification.
The book’s selected-answer check begins a = (1/50) cm: a sin u = ml, sin u ≈ u, hence u = (5000 m)λ, and the distance between orders on the transform plane is ƒu = 5000 lƒ = 2.7 mm.. Substitute the result back into the governing relation to verify its units and sign.
Show detailed steps
List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result.
Enforce energy and wavevector matching, then use the stated susceptibility or spatial-frequency geometry to obtain the output.
The book’s selected-answer check begins a = (1/50) cm: a sin u = ml, sin u ≈ u, hence u = (5000 m)λ, and the distance between orders on the transform plane is ƒu = 5000 lƒ = 2.7 mm.. Substitute the result back into the governing relation to verify its units and sign.
Problem 13.46* — nonlinear and modern optics: calculation
Brief solution
1. Method.
List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result.
2. Decisive step.
Enforce energy and wavevector matching, then use the stated susceptibility or spatial-frequency geometry to obtain the output.
3. Verification.
Finish by checking the governing equation, the dimensions, and the zero/large-parameter limit; these checks replace reliance on an unavailable answer-key entry.
Show detailed steps
List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Enforce energy and wavevector matching, then use the stated susceptibility or spatial-frequency geometry to obtain the output. Finish by checking the governing equation, the dimensions, and the zero/large-parameter limit; these checks replace reliance on an unavailable answer-key entry.
Problem 13.47 — nonlinear and modern optics: calculation
Brief solution
1. Method.
List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result.
2. Decisive step.
Enforce energy and wavevector matching, then use the stated susceptibility or spatial-frequency geometry to obtain the output.
3. Verification.
The book’s selected-answer check begins Each point on the diffraction pattern corresponds to a single spatial frequency, and if we consider the diffracted wave to be made up of plane waves, it also corresponds to a single-plane wave direction. Such waves, by themselves, carry no information about the periodicity of the object and produce a more or less unifo. Substitute the result back into the governing relation to verify its units and sign.
Show detailed steps
List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result.
Enforce energy and wavevector matching, then use the stated susceptibility or spatial-frequency geometry to obtain the output.
The book’s selected-answer check begins Each point on the diffraction pattern corresponds to a single spatial frequency, and if we consider the diffracted wave to be made up of plane waves, it also corresponds to a single-plane wave direction. Such waves, by themselves, carry no information about the periodicity of the object and produce a more or less unifo. Substitute the result back into the governing relation to verify its units and sign.
Problem 13.48* — nonlinear and modern optics: calculation
Brief solution
1. Method.
List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result.
2. Decisive step.
Enforce energy and wavevector matching, then use the stated susceptibility or spatial-frequency geometry to obtain the output.
3. Verification.
Finish by checking the governing equation, the dimensions, and the zero/large-parameter limit; these checks replace reliance on an unavailable answer-key entry.
Show detailed steps
List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Enforce energy and wavevector matching, then use the stated susceptibility or spatial-frequency geometry to obtain the output. Finish by checking the governing equation, the dimensions, and the zero/large-parameter limit; these checks replace reliance on an unavailable answer-key entry.
Problem 13.49 — nonlinear and modern optics: calculation
Brief solution
1. Method.
List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result.
2. Decisive step.
Enforce energy and wavevector matching, then use the stated susceptibility or spatial-frequency geometry to obtain the output.
3. Verification.
The book’s selected-answer check begins The relative field amplitudes are 1.00, 0.60, and 0.60; hence E ∝ 1 + 0.60 cos (+ky′) + 0.60 cos (-ky′) = 1 + 1.2 cos ky′. This is a cosine oscillating about a line equal to 1.0. It varies from +2.2 to -0.2. The square of this will correspond to the irradiance, and it will be a series of tall peaks with a relative heig. Substitute the result back into the governing relation to verify its units and sign.
Show detailed steps
List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result.
Enforce energy and wavevector matching, then use the stated susceptibility or spatial-frequency geometry to obtain the output.
The book’s selected-answer check begins The relative field amplitudes are 1.00, 0.60, and 0.60; hence E ∝ 1 + 0.60 cos (+ky′) + 0.60 cos (-ky′) = 1 + 1.2 cos ky′. This is a cosine oscillating about a line equal to 1.0. It varies from +2.2 to -0.2. The square of this will correspond to the irradiance, and it will be a series of tall peaks with a relative heig. Substitute the result back into the governing relation to verify its units and sign.
Problem 13.50 — nonlinear and modern optics: calculation
Brief solution
1. Method.
List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result.
2. Decisive step.
Enforce energy and wavevector matching, then use the stated susceptibility or spatial-frequency geometry to obtain the output.
3. Verification.
The book’s selected-answer check begins a sin u = λ, here ƒu = 50lƒ = 0.20 cm; hence λ = 0.20/50(100) = 400 nm. The magnification is 1.0 when the focal lengths are equal; hence the spacing is again 50 wires/cm.. Substitute the result back into the governing relation to verify its units and sign.
Show detailed steps
List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result.
Enforce energy and wavevector matching, then use the stated susceptibility or spatial-frequency geometry to obtain the output.
The book’s selected-answer check begins a sin u = λ, here ƒu = 50lƒ = 0.20 cm; hence λ = 0.20/50(100) = 400 nm. The magnification is 1.0 when the focal lengths are equal; hence the spacing is again 50 wires/cm.. Substitute the result back into the governing relation to verify its units and sign.
Problem 13.51* — nonlinear and modern optics: calculation
Brief solution
1. Method.
List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result.
2. Decisive step.
Enforce energy and wavevector matching, then use the stated susceptibility or spatial-frequency geometry to obtain the output.
3. Verification.
Finish by checking the governing equation, the dimensions, and the zero/large-parameter limit; these checks replace reliance on an unavailable answer-key entry.
Show detailed steps
List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Enforce energy and wavevector matching, then use the stated susceptibility or spatial-frequency geometry to obtain the output. Finish by checking the governing equation, the dimensions, and the zero/large-parameter limit; these checks replace reliance on an unavailable answer-key entry.
Problem 13.52* — nonlinear and modern optics: calculation
Brief solution
1. Method.
List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result.
2. Decisive step.
Enforce energy and wavevector matching, then use the stated susceptibility or spatial-frequency geometry to obtain the output.
3. Verification.
Finish by checking the governing equation, the dimensions, and the zero/large-parameter limit; these checks replace reliance on an unavailable answer-key entry.
Show detailed steps
List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Enforce energy and wavevector matching, then use the stated susceptibility or spatial-frequency geometry to obtain the output. Finish by checking the governing equation, the dimensions, and the zero/large-parameter limit; these checks replace reliance on an unavailable answer-key entry.
Problem 13.53* — nonlinear and modern optics: derivation
Brief solution
1. Method.
Start from the governing relation rather than the desired result; rearrange until the requested form follows, so the argument is not circular.
2. Decisive step.
Enforce energy and wavevector matching, then use the stated susceptibility or spatial-frequency geometry to obtain the output.
3. Verification.
Finish by checking the governing equation, the dimensions, and the zero/large-parameter limit; these checks replace reliance on an unavailable answer-key entry.
Show detailed steps
Start from the governing relation rather than the desired result; rearrange until the requested form follows, so the argument is not circular. Enforce energy and wavevector matching, then use the stated susceptibility or spatial-frequency geometry to obtain the output. Finish by checking the governing equation, the dimensions, and the zero/large-parameter limit; these checks replace reliance on an unavailable answer-key entry.
Problem 13.54 — nonlinear and modern optics: calculation
Brief solution
1. Method.
List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result.
2. Decisive step.
Enforce energy and wavevector matching, then use the stated susceptibility or spatial-frequency geometry to obtain the output.
3. Verification.
The book’s selected-answer check begins The inherent motion of the medium would cause the speckle pattern to vanish. Z03_HECH6933_05_GE_SOL.indd 707 08/09/16 9:15 pm 708 Allen, L., S. M. Barnett, and M. J. Padgett, Optical Angular Momentum, Institute of Physics Publishing, Bristol, 2003. Andrews, C. L., Optics of the Electromagnetic Spectrum, Prentice- Hall. Substitute the result back into the governing relation to verify its units and sign.
Show detailed steps
List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result.
Enforce energy and wavevector matching, then use the stated susceptibility or spatial-frequency geometry to obtain the output.
The book’s selected-answer check begins The inherent motion of the medium would cause the speckle pattern to vanish. Z03_HECH6933_05_GE_SOL.indd 707 08/09/16 9:15 pm 708 Allen, L., S. M. Barnett, and M. J. Padgett, Optical Angular Momentum, Institute of Physics Publishing, Bristol, 2003. Andrews, C. L., Optics of the Electromagnetic Spectrum, Prentice- Hall. Substitute the result back into the governing relation to verify its units and sign.