Chapter 7: Laser Amplification
Source: Anthony E. Siegman, Lasers (1986), Chapter 7. Use each section/problem identifier with the book; the original prompts are not reproduced here. Each entry gives the governing model, the decisive solution route, and a physical verification.
Section 7.2: Wave Propagation In An Atomic Medium
Problem 7.2.1 — Lineshapes for absorption and phase shift in ruby
Brief solution
1. Method.
List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation.
2. Decisive step.
Propagate irradiance with \(dI/dz=g(I)I\), using \(g(I)=g_0/(1+I/I_s)\) when saturation matters; integrate before inserting boundary values.
3. Verification.
Verify that the small-signal limit is exponential, while extracted energy never exceeds the stored inversion energy.
Show detailed steps
List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation. Propagate irradiance with \(dI/dz=g(I)I\), using \(g(I)=g_0/(1+I/I_s)\) when saturation matters; integrate before inserting boundary values. Verify that the small-signal limit is exponential, while extracted energy never exceeds the stored inversion energy.
Problem 7.2.2 — The V-E = 0 approximation in deriving the wave equation
Brief solution
1. Method.
List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation.
2. Decisive step.
Propagate irradiance with \(dI/dz=g(I)I\), using \(g(I)=g_0/(1+I/I_s)\) when saturation matters; integrate before inserting boundary values.
3. Verification.
Verify that the small-signal limit is exponential, while extracted energy never exceeds the stored inversion energy.
Show detailed steps
List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation. Propagate irradiance with \(dI/dz=g(I)I\), using \(g(I)=g_0/(1+I/I_s)\) when saturation matters; integrate before inserting boundary values. Verify that the small-signal limit is exponential, while extracted energy never exceeds the stored inversion energy.
Problem 7.2.3 — Extending the Taylor approximation to higher-order terms in high-loss materials
Brief solution
1. Method.
List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation.
2. Decisive step.
Propagate irradiance with \(dI/dz=g(I)I\), using \(g(I)=g_0/(1+I/I_s)\) when saturation matters; integrate before inserting boundary values.
3. Verification.
Verify that the small-signal limit is exponential, while extracted energy never exceeds the stored inversion energy.
Show detailed steps
List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation. Propagate irradiance with \(dI/dz=g(I)I\), using \(g(I)=g_0/(1+I/I_s)\) when saturation matters; integrate before inserting boundary values. Verify that the small-signal limit is exponential, while extracted energy never exceeds the stored inversion energy.
Section 7.3: The Paraxial Wave Equation
Problem 7.3.1 — Applying the paraxial-wave approximation to Gaussian beam propagation
Brief solution
1. Method.
List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation.
2. Decisive step.
Propagate irradiance with \(dI/dz=g(I)I\), using \(g(I)=g_0/(1+I/I_s)\) when saturation matters; integrate before inserting boundary values.
3. Verification.
Verify that the small-signal limit is exponential, while extracted energy never exceeds the stored inversion energy.
Show detailed steps
List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation. Propagate irradiance with \(dI/dz=g(I)I\), using \(g(I)=g_0/(1+I/I_s)\) when saturation matters; integrate before inserting boundary values. Verify that the small-signal limit is exponential, while extracted energy never exceeds the stored inversion energy.
Section 7.4: Single-Pass Laser Amplification
Problem 7.4.1 — Amplification bandwidth for a Gaussian atomic transition
Brief solution
1. Method.
List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation.
2. Decisive step.
Propagate irradiance with \(dI/dz=g(I)I\), using \(g(I)=g_0/(1+I/I_s)\) when saturation matters; integrate before inserting boundary values.
3. Verification.
Verify that the small-signal limit is exponential, while extracted energy never exceeds the stored inversion energy.
Show detailed steps
List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation. Propagate irradiance with \(dI/dz=g(I)I\), using \(g(I)=g_0/(1+I/I_s)\) when saturation matters; integrate before inserting boundary values. Verify that the small-signal limit is exponential, while extracted energy never exceeds the stored inversion energy.
Problem 7.4.2 — Absorption linewidth for an absorbing atomic transition
Brief solution
1. Method.
List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation.
2. Decisive step.
Propagate irradiance with \(dI/dz=g(I)I\), using \(g(I)=g_0/(1+I/I_s)\) when saturation matters; integrate before inserting boundary values.
3. Verification.
Verify that the small-signal limit is exponential, while extracted energy never exceeds the stored inversion energy.
Show detailed steps
List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation. Propagate irradiance with \(dI/dz=g(I)I\), using \(g(I)=g_0/(1+I/I_s)\) when saturation matters; integrate before inserting boundary values. Verify that the small-signal limit is exponential, while extracted energy never exceeds the stored inversion energy.
Problem 7.4.3 — An alternative bandwidth definition for low-gain amplifiers
Brief solution
1. Method.
List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation.
2. Decisive step.
Propagate irradiance with \(dI/dz=g(I)I\), using \(g(I)=g_0/(1+I/I_s)\) when saturation matters; integrate before inserting boundary values.
3. Verification.
Verify that the small-signal limit is exponential, while extracted energy never exceeds the stored inversion energy.
Show detailed steps
List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation. Propagate irradiance with \(dI/dz=g(I)I\), using \(g(I)=g_0/(1+I/I_s)\) when saturation matters; integrate before inserting boundary values. Verify that the small-signal limit is exponential, while extracted energy never exceeds the stored inversion energy.
Problem 7.4.4 — Testing for a Gaussian atomic lineshape
Brief solution
1. Method.
List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation.
2. Decisive step.
Propagate irradiance with \(dI/dz=g(I)I\), using \(g(I)=g_0/(1+I/I_s)\) when saturation matters; integrate before inserting boundary values.
3. Verification.
Verify that the small-signal limit is exponential, while extracted energy never exceeds the stored inversion energy.
Show detailed steps
List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation. Propagate irradiance with \(dI/dz=g(I)I\), using \(g(I)=g_0/(1+I/I_s)\) when saturation matters; integrate before inserting boundary values. Verify that the small-signal limit is exponential, while extracted energy never exceeds the stored inversion energy.
Problem 7.4.5 — Gain versus frequency for a cascaded amplifier plus absorber
Brief solution
1. Method.
Evaluate both cases from the same symbolic expression before taking their ratio; this keeps normalization and sign conventions from obscuring the comparison.
2. Decisive step.
Propagate irradiance with \(dI/dz=g(I)I\), using \(g(I)=g_0/(1+I/I_s)\) when saturation matters; integrate before inserting boundary values.
3. Verification.
Verify that the small-signal limit is exponential, while extracted energy never exceeds the stored inversion energy.
Show detailed steps
Evaluate both cases from the same symbolic expression before taking their ratio; this keeps normalization and sign conventions from obscuring the comparison. Propagate irradiance with \(dI/dz=g(I)I\), using \(g(I)=g_0/(1+I/I_s)\) when saturation matters; integrate before inserting boundary values. Verify that the small-signal limit is exponential, while extracted energy never exceeds the stored inversion energy.
Problem 7.4.6 — Continuation of the previous problem: general evaluation ofpassband broadening in a laser amplifier
Brief solution
1. Method.
List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation.
2. Decisive step.
Propagate irradiance with \(dI/dz=g(I)I\), using \(g(I)=g_0/(1+I/I_s)\) when saturation matters; integrate before inserting boundary values.
3. Verification.
Verify that the small-signal limit is exponential, while extracted energy never exceeds the stored inversion energy.
Show detailed steps
List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation. Propagate irradiance with \(dI/dz=g(I)I\), using \(g(I)=g_0/(1+I/I_s)\) when saturation matters; integrate before inserting boundary values. Verify that the small-signal limit is exponential, while extracted energy never exceeds the stored inversion energy.
Problem 7.4.7 — Continuation of the previous problem: relationship between midband gain and phase shift derivatives?
Brief solution
1. Method.
List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation.
2. Decisive step.
Propagate irradiance with \(dI/dz=g(I)I\), using \(g(I)=g_0/(1+I/I_s)\) when saturation matters; integrate before inserting boundary values.
3. Verification.
Verify that the small-signal limit is exponential, while extracted energy never exceeds the stored inversion energy.
Show detailed steps
List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation. Propagate irradiance with \(dI/dz=g(I)I\), using \(g(I)=g_0/(1+I/I_s)\) when saturation matters; integrate before inserting boundary values. Verify that the small-signal limit is exponential, while extracted energy never exceeds the stored inversion energy.
Problem 7.4.8 — Linewidth modulation spectroscopyv: a new experimental technique
Brief solution
1. Method.
List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation.
2. Decisive step.
Propagate irradiance with \(dI/dz=g(I)I\), using \(g(I)=g_0/(1+I/I_s)\) when saturation matters; integrate before inserting boundary values.
3. Verification.
Verify that the small-signal limit is exponential, while extracted energy never exceeds the stored inversion energy.
Show detailed steps
List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation. Propagate irradiance with \(dI/dz=g(I)I\), using \(g(I)=g_0/(1+I/I_s)\) when saturation matters; integrate before inserting boundary values. Verify that the small-signal limit is exponential, while extracted energy never exceeds the stored inversion energy.
Section 7.5: Stimulated-Transition Cross Sections
Problem 7.5.1 — Practical expression for atomic oscillator strength
Brief solution
1. Method.
List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation.
2. Decisive step.
Propagate irradiance with \(dI/dz=g(I)I\), using \(g(I)=g_0/(1+I/I_s)\) when saturation matters; integrate before inserting boundary values.
3. Verification.
Verify that the small-signal limit is exponential, while extracted energy never exceeds the stored inversion energy.
Show detailed steps
List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation. Propagate irradiance with \(dI/dz=g(I)I\), using \(g(I)=g_0/(1+I/I_s)\) when saturation matters; integrate before inserting boundary values. Verify that the small-signal limit is exponential, while extracted energy never exceeds the stored inversion energy.
Problem 7.5.2 — Design considerations for a high-energy-storage laser medium
Brief solution
1. Method.
Translate each performance requirement into an equality or inequality, solve the coupled constraints, and discard any root that violates a physical bound.
2. Decisive step.
Propagate irradiance with \(dI/dz=g(I)I\), using \(g(I)=g_0/(1+I/I_s)\) when saturation matters; integrate before inserting boundary values.
3. Verification.
Verify that the small-signal limit is exponential, while extracted energy never exceeds the stored inversion energy.
Show detailed steps
Translate each performance requirement into an equality or inequality, solve the coupled constraints, and discard any root that violates a physical bound. Propagate irradiance with \(dI/dz=g(I)I\), using \(g(I)=g_0/(1+I/I_s)\) when saturation matters; integrate before inserting boundary values. Verify that the small-signal limit is exponential, while extracted energy never exceeds the stored inversion energy.
Problem 7.5.3 — Measuring an inverted laser transition cross section
Brief solution
1. Method.
List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation.
2. Decisive step.
Propagate irradiance with \(dI/dz=g(I)I\), using \(g(I)=g_0/(1+I/I_s)\) when saturation matters; integrate before inserting boundary values.
3. Verification.
Verify that the small-signal limit is exponential, while extracted energy never exceeds the stored inversion energy.
Show detailed steps
List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation. Propagate irradiance with \(dI/dz=g(I)I\), using \(g(I)=g_0/(1+I/I_s)\) when saturation matters; integrate before inserting boundary values. Verify that the small-signal limit is exponential, while extracted energy never exceeds the stored inversion energy.
Problem 7.5.4 — Energy storage in a Nd.YAG rod
Brief solution
1. Method.
List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation.
2. Decisive step.
Propagate irradiance with \(dI/dz=g(I)I\), using \(g(I)=g_0/(1+I/I_s)\) when saturation matters; integrate before inserting boundary values.
3. Verification.
Verify that the small-signal limit is exponential, while extracted energy never exceeds the stored inversion energy.
Show detailed steps
List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation. Propagate irradiance with \(dI/dz=g(I)I\), using \(g(I)=g_0/(1+I/I_s)\) when saturation matters; integrate before inserting boundary values. Verify that the small-signal limit is exponential, while extracted energy never exceeds the stored inversion energy.
Problem 7.5.5 — Gain through a thin atomic layer near a mirror
Brief solution
1. Method.
List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation.
2. Decisive step.
Propagate irradiance with \(dI/dz=g(I)I\), using \(g(I)=g_0/(1+I/I_s)\) when saturation matters; integrate before inserting boundary values.
3. Verification.
Verify that the small-signal limit is exponential, while extracted energy never exceeds the stored inversion energy.
Show detailed steps
List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation. Propagate irradiance with \(dI/dz=g(I)I\), using \(g(I)=g_0/(1+I/I_s)\) when saturation matters; integrate before inserting boundary values. Verify that the small-signal limit is exponential, while extracted energy never exceeds the stored inversion energy.
Section 7.6: Saturation Intensities In Laser Materials
Problem 7.6.1 — Saturation intensity for a three-level atomic absorber
Brief solution
1. Method.
List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation.
2. Decisive step.
Write one balance equation per level, \(\dot N_i=\sum_j(W_{ji}N_j-W_{ij}N_i)-N_i/\tau_i\), add population conservation, and solve the resulting linear steady-state system.
3. Verification.
Check that every population is nonnegative, their sum is conserved, and the unpumped and strongly pumped limits are sensible.
Show detailed steps
List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation. Write one balance equation per level, \(\dot N_i=\sum_j(W_{ji}N_j-W_{ij}N_i)-N_i/\tau_i\), add population conservation, and solve the resulting linear steady-state system. Check that every population is nonnegative, their sum is conserved, and the unpumped and strongly pumped limits are sensible.
Problem 7.6.2 — Saturation lineshape for the reactive part of a homogeneous two-level atomic transition
Brief solution
1. Method.
List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation.
2. Decisive step.
Write one balance equation per level, \(\dot N_i=\sum_j(W_{ji}N_j-W_{ij}N_i)-N_i/\tau_i\), add population conservation, and solve the resulting linear steady-state system.
3. Verification.
Check that every population is nonnegative, their sum is conserved, and the unpumped and strongly pumped limits are sensible.
Show detailed steps
List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation. Write one balance equation per level, \(\dot N_i=\sum_j(W_{ji}N_j-W_{ij}N_i)-N_i/\tau_i\), add population conservation, and solve the resulting linear steady-state system. Check that every population is nonnegative, their sum is conserved, and the unpumped and strongly pumped limits are sensible.
Problem 7.6.3 — Saturation behavior in a two-level absorber including excited-state absorption
Brief solution
1. Method.
List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation.
2. Decisive step.
Write one balance equation per level, \(\dot N_i=\sum_j(W_{ji}N_j-W_{ij}N_i)-N_i/\tau_i\), add population conservation, and solve the resulting linear steady-state system.
3. Verification.
Check that every population is nonnegative, their sum is conserved, and the unpumped and strongly pumped limits are sensible.
Show detailed steps
List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation. Write one balance equation per level, \(\dot N_i=\sum_j(W_{ji}N_j-W_{ij}N_i)-N_i/\tau_i\), add population conservation, and solve the resulting linear steady-state system. Check that every population is nonnegative, their sum is conserved, and the unpumped and strongly pumped limits are sensible.
Problem 7.6.4 — Power balance versus intensity in a two-level saturable absorber
Brief solution
1. Method.
Evaluate both cases from the same symbolic expression before taking their ratio; this keeps normalization and sign conventions from obscuring the comparison.
2. Decisive step.
Write one balance equation per level, \(\dot N_i=\sum_j(W_{ji}N_j-W_{ij}N_i)-N_i/\tau_i\), add population conservation, and solve the resulting linear steady-state system.
3. Verification.
Check that every population is nonnegative, their sum is conserved, and the unpumped and strongly pumped limits are sensible.
Show detailed steps
Evaluate both cases from the same symbolic expression before taking their ratio; this keeps normalization and sign conventions from obscuring the comparison. Write one balance equation per level, \(\dot N_i=\sum_j(W_{ji}N_j-W_{ij}N_i)-N_i/\tau_i\), add population conservation, and solve the resulting linear steady-state system. Check that every population is nonnegative, their sum is conserved, and the unpumped and strongly pumped limits are sensible.
Section 7.7: Homogeneous Saturation In Laser Amplifiers
Problem 7.7.1 — Input-output intensity curves for a saturable amplifier offline center
Brief solution
1. Method.
List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation.
2. Decisive step.
Write one balance equation per level, \(\dot N_i=\sum_j(W_{ji}N_j-W_{ij}N_i)-N_i/\tau_i\), add population conservation, and solve the resulting linear steady-state system.
3. Verification.
Check that every population is nonnegative, their sum is conserved, and the unpumped and strongly pumped limits are sensible.
Show detailed steps
List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation. Write one balance equation per level, \(\dot N_i=\sum_j(W_{ji}N_j-W_{ij}N_i)-N_i/\tau_i\), add population conservation, and solve the resulting linear steady-state system. Check that every population is nonnegative, their sum is conserved, and the unpumped and strongly pumped limits are sensible.
Problem 7.7.2 — Input versus output intensities for a saturable atomic absorber
Brief solution
1. Method.
Evaluate both cases from the same symbolic expression before taking their ratio; this keeps normalization and sign conventions from obscuring the comparison.
2. Decisive step.
Write one balance equation per level, \(\dot N_i=\sum_j(W_{ji}N_j-W_{ij}N_i)-N_i/\tau_i\), add population conservation, and solve the resulting linear steady-state system.
3. Verification.
Check that every population is nonnegative, their sum is conserved, and the unpumped and strongly pumped limits are sensible.
Show detailed steps
Evaluate both cases from the same symbolic expression before taking their ratio; this keeps normalization and sign conventions from obscuring the comparison. Write one balance equation per level, \(\dot N_i=\sum_j(W_{ji}N_j-W_{ij}N_i)-N_i/\tau_i\), add population conservation, and solve the resulting linear steady-state system. Check that every population is nonnegative, their sum is conserved, and the unpumped and strongly pumped limits are sensible.
Problem 7.7.3 — Signal penetration and saturation depth versus signal intensity in a homogeneous saturable absorber
Brief solution
1. Method.
Evaluate both cases from the same symbolic expression before taking their ratio; this keeps normalization and sign conventions from obscuring the comparison.
2. Decisive step.
Write one balance equation per level, \(\dot N_i=\sum_j(W_{ji}N_j-W_{ij}N_i)-N_i/\tau_i\), add population conservation, and solve the resulting linear steady-state system.
3. Verification.
Check that every population is nonnegative, their sum is conserved, and the unpumped and strongly pumped limits are sensible.
Show detailed steps
Evaluate both cases from the same symbolic expression before taking their ratio; this keeps normalization and sign conventions from obscuring the comparison. Write one balance equation per level, \(\dot N_i=\sum_j(W_{ji}N_j-W_{ij}N_i)-N_i/\tau_i\), add population conservation, and solve the resulting linear steady-state system. Check that every population is nonnegative, their sum is conserved, and the unpumped and strongly pumped limits are sensible.
Problem 7.7.4 — Obtaining an amplifier output intensity just equal to the available intensity of the laser medium
Brief solution
1. Method.
List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation.
2. Decisive step.
Write one balance equation per level, \(\dot N_i=\sum_j(W_{ji}N_j-W_{ij}N_i)-N_i/\tau_i\), add population conservation, and solve the resulting linear steady-state system.
3. Verification.
Check that every population is nonnegative, their sum is conserved, and the unpumped and strongly pumped limits are sensible.
Show detailed steps
List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation. Write one balance equation per level, \(\dot N_i=\sum_j(W_{ji}N_j-W_{ij}N_i)-N_i/\tau_i\), add population conservation, and solve the resulting linear steady-state system. Check that every population is nonnegative, their sum is conserved, and the unpumped and strongly pumped limits are sensible.
Problem 7.7.5 — Power output stabilization factor in a partially saturated laser amplifier
Brief solution
1. Method.
List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation.
2. Decisive step.
Write one balance equation per level, \(\dot N_i=\sum_j(W_{ji}N_j-W_{ij}N_i)-N_i/\tau_i\), add population conservation, and solve the resulting linear steady-state system.
3. Verification.
Check that every population is nonnegative, their sum is conserved, and the unpumped and strongly pumped limits are sensible.
Show detailed steps
List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation. Write one balance equation per level, \(\dot N_i=\sum_j(W_{ji}N_j-W_{ij}N_i)-N_i/\tau_i\), add population conservation, and solve the resulting linear steady-state system. Check that every population is nonnegative, their sum is conserved, and the unpumped and strongly pumped limits are sensible.
Problem 7.7.6 — Cross-saturation of a transversely double-pass laser amplifier
Brief solution
1. Method.
List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation.
2. Decisive step.
Write one balance equation per level, \(\dot N_i=\sum_j(W_{ji}N_j-W_{ij}N_i)-N_i/\tau_i\), add population conservation, and solve the resulting linear steady-state system.
3. Verification.
Check that every population is nonnegative, their sum is conserved, and the unpumped and strongly pumped limits are sensible.
Show detailed steps
List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation. Write one balance equation per level, \(\dot N_i=\sum_j(W_{ji}N_j-W_{ij}N_i)-N_i/\tau_i\), add population conservation, and solve the resulting linear steady-state system. Check that every population is nonnegative, their sum is conserved, and the unpumped and strongly pumped limits are sensible.
Problem 7.7.7 — Amplifier input-output curves for other forms of laser saturation
Brief solution
1. Method.
List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation.
2. Decisive step.
Write one balance equation per level, \(\dot N_i=\sum_j(W_{ji}N_j-W_{ij}N_i)-N_i/\tau_i\), add population conservation, and solve the resulting linear steady-state system.
3. Verification.
Check that every population is nonnegative, their sum is conserved, and the unpumped and strongly pumped limits are sensible.
Show detailed steps
List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation. Write one balance equation per level, \(\dot N_i=\sum_j(W_{ji}N_j-W_{ij}N_i)-N_i/\tau_i\), add population conservation, and solve the resulting linear steady-state system. Check that every population is nonnegative, their sum is conserved, and the unpumped and strongly pumped limits are sensible.
Problem 7.7.8 — Signal transmission through two intermingled saturable absorber transitions
Brief solution
1. Method.
List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation.
2. Decisive step.
Write one balance equation per level, \(\dot N_i=\sum_j(W_{ji}N_j-W_{ij}N_i)-N_i/\tau_i\), add population conservation, and solve the resulting linear steady-state system.
3. Verification.
Check that every population is nonnegative, their sum is conserved, and the unpumped and strongly pumped limits are sensible.
Show detailed steps
List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation. Write one balance equation per level, \(\dot N_i=\sum_j(W_{ji}N_j-W_{ij}N_i)-N_i/\tau_i\), add population conservation, and solve the resulting linear steady-state system. Check that every population is nonnegative, their sum is conserved, and the unpumped and strongly pumped limits are sensible.
Problem 7.7.9 — Saturation effects on transverse beam profiles
Brief solution
1. Method.
Evaluate both cases from the same symbolic expression before taking their ratio; this keeps normalization and sign conventions from obscuring the comparison.
2. Decisive step.
Write one balance equation per level, \(\dot N_i=\sum_j(W_{ji}N_j-W_{ij}N_i)-N_i/\tau_i\), add population conservation, and solve the resulting linear steady-state system.
3. Verification.
Check that every population is nonnegative, their sum is conserved, and the unpumped and strongly pumped limits are sensible.
Show detailed steps
Evaluate both cases from the same symbolic expression before taking their ratio; this keeps normalization and sign conventions from obscuring the comparison. Write one balance equation per level, \(\dot N_i=\sum_j(W_{ji}N_j-W_{ij}N_i)-N_i/\tau_i\), add population conservation, and solve the resulting linear steady-state system. Check that every population is nonnegative, their sum is conserved, and the unpumped and strongly pumped limits are sensible.