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

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

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

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

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

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

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

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

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

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

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?

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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.