Chapter 6: Laser Pumping and Population Inversion

Source: Anthony E. Siegman, Lasers (1986), Chapter 6. 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 6.1: Steady-State Laser Pumping And Population Inversion

Problem 6.1.1 — Three-level system with two pumping signals applied

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 6.1.2 — Population inversion versus pumping in an uupper-leveP three-level laser system

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 6.1.3 — Cascade pumping of a four-level laser system

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 6.1.4 — Analysis of a five-level laser system

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 6.1.5 — Laser refrigeration?

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.

Section 6.2: Laser Gain Saturation

Problem 6.2.1 — Transient response in the simplified laser-pumping model

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 6.2.2 — Laser inversion and saturation including degeneracies

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 6.2.3 — Optically pumped laser 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 6.2.4 — Simultaneous pumping into both the upper and the lower laser levels

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 6.2.5 — Signal saturation behavior in the ideal three-level laser system

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.

Section 6.3: Transient Laser Pumping

Problem 6.3.1 — Transient pumping with a Gaussian time-varying pump pulse

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 6.3.2 — Ditto with an exponentially varying pump pulse

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 6.3.3 — Peak population inversion versus normalized pump pulsewidth

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.