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
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.
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, so \(N\simeq-N_a\). Line-centre coefficient is \(\gamma_0=N\sigma_0\); the inversion required for 0.5/cm gain is \(\boxed{N=0.5/\sigma_0}\), with \(\sigma_0\) obtained from the Lorentzian lifetime formula in Sec. 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
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.
Steady state gives \(N_2=R_2t_{sp}\) and pump depletion \(R_2=(N_a-2N_2)W\). 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
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.
Insert the lifetime inequalities in the general expression for \(T_s\); all fast nonradiative/level-1 terms drop out, 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
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.
Set the steady population-difference formulas to zero: the three-level system has a finite transparency threshold, while an ideal four-level system reaches zero difference at zero pump. 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
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
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.
Insert Lorentzian \(g(\nu)\) in \(\gamma=\gamma_0/[1+\phi/\phi_s(\nu)]\). Half maximum occurs at a detuning enlarged by \(\sqrt{1+\phi/\phi_s(\nu_0)}\), 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
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.
Solve \(d\phi/dz=\gamma_0\phi+r_{sp}\) with zero input: \(\boxed{\phi(d)=\phi_{sp}(e^{\gamma_0d}-1)}\). 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
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
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
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.
Average \(\gamma(\nu)-\alpha_s\) over the uniform two-linewidth band: \(\bar\gamma=(2\Delta\nu)^{-1} \int_{\nu_0-\Delta\nu}^{\nu_0+\Delta\nu} [0.1/(1+4\delta\nu^2/\Delta\nu^2)-0.05]d\nu\). 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
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.
The pump drives upward and stimulated downward transitions at the same rate: \(\dot N_2=W(N_1-N_2)-N_2/t_{sp}\). 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
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.
Write \(\dot N_3=R_3-N_3/T_{31}\), \(\dot N_2=R_2-N_2/T_{21}\), and \(\dot N_1=N_3/T_{31}+N_2/T_{21}-N_1/T_1\) plus stimulated terms. 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. Stimulated 2-to-1 emission raises \(N_1\), 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
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.
Stimulated decay adds \(\sigma\phi\) to \(1/T_2\); half lifetime requires \(\sigma\phi=1/T_2\). Thus \(\phi=1/(\sigma T_2)\), 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
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.
For each Table 14.3-1 row evaluate \(\boxed{\phi_s=1/(\sigma_0T_s)}\) and \(\boxed{I_s=(hc/\lambda_0)\phi_s}\). 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
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.
Integrating \(d\phi/dz=\gamma_0\phi/(1+\phi/\phi_s)\) gives \(\boxed{\ln(\phi/\phi_0)+(\phi-\phi_0)/\phi_s=\gamma_0z}\). 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
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.
For each temperature use \(N_2/N_1=e^{-2.48/kT}\) and \(N_1+N_2=10^{23}\). Then spontaneous rate is \(N_2/t_{sp}\), \(\alpha_0=(N_1-N_2)\sigma_0\), Lorentzian frequency dependence, and \(\phi_s=1/(\sigma_0T_s)\). 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
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.
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 \(\gamma_0\). A fivefold coefficient drop requires \(1+\phi/\phi_s=5\), so \(\phi=4\phi_s\). 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
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.
\(\phi_s(d)=\phi_s(0)e^x\) and \(\phi_{ASE}=\phi_{sp}(e^x-1)\), \(x=\gamma_0d\); hence \(\boxed{\phi_s(d)/\phi_{ASE}=[\phi_s(0)/\phi_{sp}]/(1-e^{-x})}\). 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
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.
Moments of the noncentral chi-square density give \(\bar w=w_s+w_{ASE}\) and \(\operatorname{var}w=w_{ASE}^2+2w_sw_{ASE}\). 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.