Chapter 14: Laser Amplifiers
Source: Saleh and Teich, Fundamentals of Photonics, second edition, Chapter 14.
In-text exercises
Exercise 14.1-1 — Ruby absorption and inversion
Brief solution
1. Method. The working uses exponential, logarithmic, and phasor identities and algebraic rearrangement and dimensional checks.
2. Reasoning and answer.
Show detailed steps
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 81 — Exercise 14.1-1: Ruby absorption and inversion. The diagram identifies the input quantities, physical operation, requested result, variable meanings, and an independent verification route. Every symbol in the variable strip is labeled on the model itself.
Step 2 — Mathematical formulas used. The working uses exponential, logarithmic, and phasor identities and algebraic rearrangement and dimensional checks.
Step 3 — Worked derivation. The calculation is kept in symbolic form until the governing relation has been rearranged for the requested quantity.
Detailed step 1. Use Boltzmann \(N_2/N_1=e^{-hc/(\lambda kT)}\) with \(N_1+N_2=N_a\); at 300 K the upper population is negligible,
Detailed step 2. so \(N\simeq-N_a\).
Detailed step 3. Line-centre coefficient is \(\gamma_0=N\sigma_0\); the inversion required for 0.5/cm gain is \(\boxed{N=0.5/\sigma_0}\),
Detailed step 4. with \(\sigma_0\) obtained from the Lorentzian lifetime formula in Sec.
Detailed step 5. 13.3.
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.
Exercise 14.2-1 — Optical pumping
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 82 — Exercise 14.2-1: Optical pumping. 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. Steady state gives \(N_2=R_2t_{sp}\) and pump depletion \(R_2=(N_a-2N_2)W\).
Detailed step 2. Solving yields \(\boxed{N_2=N_at_{sp}W/(1+2t_{sp}W)}\); it approaches only \(N_a/2\).
Step 4 — State the numbered result. The principal result obtained in the working is
Step 5 — Check. Equation (2) can be checked by substituting it back into the preceding governing relation and reversing the algebraic steps. Check that dimensions agree term by term, then test the simplest symmetry or limiting case for the expected sign and scale. Repeat the substitution with unrounded intermediate values and retain the displayed units; the final unit must have the requested dimension.
Exercise 14.2-2 — Saturation time
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 83 — Exercise 14.2-2: Saturation time. 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. Insert the lifetime inequalities in the general expression for \(T_s\); all fast nonradiative/level-1 terms drop out,
Detailed step 2. leaving \(\boxed{T_s\simeq t_{sp}}\).
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. Check that dimensions agree term by term, then test the simplest symmetry or limiting case for the expected sign and scale.
Exercise 14.2-3 — Three/four-level pump power
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 84 — Exercise 14.2-3: Three/four-level pump power. 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 the steady population-difference formulas to zero: the three-level system has a finite transparency threshold,
Detailed step 2. while an ideal four-level system reaches zero difference at zero pump.
Detailed step 3. Substitution of \(W=2/t_{sp}\) and \(1/(2t_{sp})\) gives \(N=N_a/3\) in both; the three-level system requires four times the transition rate and correspondingly greater pump.
Step 4 — State the numbered result. The principal result obtained in the working is
Step 5 — Check. Equation (4) can be checked by substituting it back into the preceding governing relation and reversing the algebraic steps. 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 14.4-1 — Ruby saturation
Brief solution
1. Method. The working uses algebraic rearrangement and dimensional checks.
2. Key step.
Evaluate \(\boxed{\phi_s(\nu_0)=1/[\sigma(\nu_0)T_s]}\) using \(T_s=2t_{sp}\) and Table 14.3-1; the corresponding intensity is \(\boxed{I_s=h\nu_0\phi_s}\).
3. Answer.
Show detailed steps
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 85 — Exercise 14.4-1: Ruby saturation. 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.
Evaluate \(\boxed{\phi_s(\nu_0)=1/[\sigma(\nu_0)T_s]}\) using \(T_s=2t_{sp}\) and Table 14.3-1; the corresponding intensity is \(\boxed{I_s=h\nu_0\phi_s}\).
Step 4 — State the numbered result. The principal result obtained in the working is
Step 5 — Check. Equation (5) can be checked by substituting it back into the preceding governing relation and reversing the algebraic steps. Check that dimensions agree term by term, then test the simplest symmetry or limiting case for the expected sign and scale.
Exercise 14.4-2 — Saturation broadening
Brief solution
1. Method. The working uses stationary-value condition and algebraic rearrangement and dimensional checks.
2. Reasoning and answer.
Show detailed steps
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 86 — Exercise 14.4-2: Saturation broadening. 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 stationary-value condition 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. Insert Lorentzian \(g(\nu)\) in \(\gamma=\gamma_0/[1+\phi/\phi_s(\nu)]\).
Detailed step 2. Half maximum occurs at a detuning enlarged by \(\sqrt{1+\phi/\phi_s(\nu_0)}\),
Detailed step 3. so \(\boxed{\Delta\nu_{sat}=\Delta\nu \sqrt{1+\phi/\phi_s(\nu_0)}}\).
Step 4 — State the numbered result. The principal result obtained in the working is
Step 5 — 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.
Exercise 14.5-1 — Amplified spontaneous emission
Brief solution
1. Method. The working uses exponential, logarithmic, and phasor identities and algebraic rearrangement and dimensional checks.
Show detailed steps
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 87 — Exercise 14.5-1: Amplified spontaneous emission. The diagram identifies the input quantities, physical operation, requested result, variable meanings, and an independent verification route. Every symbol in the variable strip is labeled on the model itself.
Step 2 — Mathematical formulas used. The working uses exponential, logarithmic, and phasor identities and algebraic rearrangement and dimensional checks.
Step 3 — Worked derivation. The calculation is kept in symbolic form until the governing relation has been rearranged for the requested quantity.
Detailed step 1. Solve \(d\phi/dz=\gamma_0\phi+r_{sp}\) with zero input: \(\boxed{\phi(d)=\phi_{sp}(e^{\gamma_0d}-1)}\).
Detailed step 2. For large gain a Lorentzian exponent narrows near line centre by approximately \(1/\sqrt{\gamma_0d}\).
End-of-chapter problems
Step 4 — State the numbered result. The principal result obtained in the working is
Step 5 — 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.
Problem 14.1-2 — Longer ruby rod
Brief solution
1. Method. The working uses exponential, logarithmic, and phasor identities and algebraic rearrangement and dimensional checks.
2. Key step.
\(G=e^{\gamma d}\); therefore \(\boxed{G_{20}=12^{20/15}=27.47}\).
3. 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 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.
\(G=e^{\gamma d}\); therefore \(\boxed{G_{20}=12^{20/15}=27.47}\).
Numbered result. The principal result obtained in the working is
Check. Equation (8) can be checked by substituting it back into the preceding governing relation and reversing the algebraic steps. Check that dimensions agree term by term, then test the simplest symmetry or limiting case for the expected sign and scale.
Problem 14.1-3 — Nd:glass inversion
Brief solution
1. Method. The working uses exponential, logarithmic, and phasor identities and algebraic rearrangement and dimensional checks.
2. Key step.
\(\gamma_0=\ln10/(15\ \mathrm{cm})=0.1535\ \mathrm{cm^{-1}}\); use the Table 14.3-1 peak cross section to obtain \(\boxed{N=0.1535/\sigma_0\ \mathrm{cm^{-3}}}\).
3. 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 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.
\(\gamma_0=\ln10/(15\ \mathrm{cm})=0.1535\ \mathrm{cm^{-1}}\); use the Table 14.3-1 peak cross section to obtain \(\boxed{N=0.1535/\sigma_0\ \mathrm{cm^{-3}}}\).
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 14.1-4 — Broadband signal
Brief solution
1. Method. The working uses integration 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 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. Average \(\gamma(\nu)-\alpha_s\) over the uniform two-linewidth band: \(\bar\gamma=(2\Delta\nu)^{-1} \int_{\nu_0-\Delta\nu}^{\nu_0+\Delta\nu} [0.1/(1+4\delta\nu^2/\Delta\nu^2)-0.05]d\nu\).
Detailed step 2. This yields the net logarithmic one-centimetre gain; exponentiation gives the power ratio.
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. Differentiation of an antiderivative, or normalization of a definite integral, checks the integration step.
Problem 14.2-4 — Why two-level pumping fails
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. The pump drives upward and stimulated downward transitions at the same rate: \(\dot N_2=W(N_1-N_2)-N_2/t_{sp}\).
Detailed step 2. Steady state gives \(N_2/N_1=Wt_{sp}/(1+Wt_{sp})<1\); inversion is impossible.
Numbered result. The principal result obtained in the working is
Check. Equation (11) can be checked by substituting it back into the preceding governing relation and reversing the algebraic steps. Check that dimensions agree term by term, then test the simplest symmetry or limiting case for the expected sign and scale. Repeat the substitution with unrounded intermediate values and retain the displayed units; the final unit must have the requested dimension.
Problem 14.2-5 — Two simultaneous laser lines
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. Write \(\dot N_3=R_3-N_3/T_{31}\), \(\dot N_2=R_2-N_2/T_{21}\),
Detailed step 2. and \(\dot N_1=N_3/T_{31}+N_2/T_{21}-N_1/T_1\) plus stimulated terms.
Detailed step 3. Their zero derivatives give \(N_3=R_3T_{31}\), \(N_2=R_2T_{21}\) and \(N_1=T_1(R_2+R_3)\) before lasing.
Detailed step 4. Stimulated 2-to-1 emission raises \(N_1\),
Detailed step 5. thereby reducing \(N_3-N_1\) and competing with the other line.
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 14.4-3 — Meaning of saturation flux
Brief solution
1. Method. The working uses 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 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. Stimulated decay adds \(\sigma\phi\) to \(1/T_2\); half lifetime requires \(\sigma\phi=1/T_2\).
Detailed step 2. Thus \(\phi=1/(\sigma T_2)\),
Detailed step 3. equal to the saturation flux when the lower-level relaxation terms make \(T_s=T_2\) (otherwise scaled by \(T_s/T_2\)).
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 14.4-4 — Ruby and Nd:YAG saturation
Brief solution
1. Method. The working uses 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 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 Table 14.3-1 row evaluate \(\boxed{\phi_s=1/(\sigma_0T_s)}\) and \(\boxed{I_s=(hc/\lambda_0)\phi_s}\).
Detailed step 2. Keep square-centimetre units for the tabulated cross sections before converting intensity.
Numbered result. The principal result obtained in the working is
Check. Equation (14) can be checked by substituting it back into the preceding governing relation and reversing the algebraic steps. Check that dimensions agree term by term, then test the simplest symmetry or limiting case for the expected sign and scale.
Problem 14.4-5 — Saturated growth plot
Brief solution
1. Method. The working uses integration 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 integration identities, exponential, logarithmic, and phasor identities, and algebraic rearrangement and dimensional checks.
Worked derivation. The calculation is kept in symbolic form until the governing relation has been rearranged for the requested quantity.
Detailed step 1. Integrating \(d\phi/dz=\gamma_0\phi/(1+\phi/\phi_s)\) gives \(\boxed{\ln(\phi/\phi_0)+(\phi-\phi_0)/\phi_s=\gamma_0z}\).
Detailed step 2. Plot this implicit relation for \(\phi_0/\phi_s=0.05\); saturation begins near \(\phi/\phi_s=1\).
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. Differentiation of an antiderivative, or normalization of a definite integral, checks the integration step.
Problem 14.4-6 — Hot two-level absorber
Brief solution
1. Method. The working uses 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 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 temperature use \(N_2/N_1=e^{-2.48/kT}\) and \(N_1+N_2=10^{23}\).
Detailed step 2. Then spontaneous rate is \(N_2/t_{sp}\), \(\alpha_0=(N_1-N_2)\sigma_0\),
Detailed step 3. Lorentzian frequency dependence,
Detailed step 4. and \(\phi_s=1/(\sigma_0T_s)\).
Detailed step 5. Transmitted flux follows the implicit saturable-absorber equation obtained from Problem 14.4-5 with negative gain; at one linewidth detuning insert the Lorentzian quarter-peak cross section.
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. Check that dimensions agree term by term, then test the simplest symmetry or limiting case for the expected sign and scale.
Problem 14.4-7 — Measured saturated amplifier
Brief solution
1. Method. The working uses 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 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. Insert the stated input/output in \(\ln(\phi_d/\phi_0)+(\phi_d-\phi_0)/\phi_s=\gamma_0d\) to obtain \(G_0=e^{\gamma_0d}\) and \(\gamma_0\).
Detailed step 2. A fivefold coefficient drop requires \(1+\phi/\phi_s=5\),
Detailed step 3. so \(\phi=4\phi_s\).
Detailed step 4. At the final high input, \(\gamma=\gamma_0/(1+10)\) and total gain is below small-signal gain.
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.
Problem 14.5-2 — Signal-to-ASE ratio
Brief solution
1. Method. The working uses 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 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. \(\phi_s(d)=\phi_s(0)e^x\) and \(\phi_{ASE}=\phi_{sp}(e^x-1)\), \(x=\gamma_0d\); hence \(\boxed{\phi_s(d)/\phi_{ASE}=[\phi_s(0)/\phi_{sp}]/(1-e^{-x})}\).
Detailed step 2. It falls from infinity and asymptotes to the input/spontaneous ratio.
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 14.5-3 — Amplified coherent-light statistics
Brief solution
1. Method. The working uses 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 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. Moments of the noncentral chi-square density give \(\bar w=w_s+w_{ASE}\) and \(\operatorname{var}w=w_{ASE}^2+2w_sw_{ASE}\).
Detailed step 2. Mandel’s identities then give \(\bar n=\bar w/(h\nu)\) and \(\operatorname{var}n=\bar n+operatorname{var}w/(h\nu)^2\).
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. Check that dimensions agree term by term, then test the simplest symmetry or limiting case for the expected sign and scale.