Chapter 24: Laser Dynamics: The Laser Cavity Equations

Source: Anthony E. Siegman, Lasers (1986), Chapter 24. 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 24.1: Derivation Of The Laser Cavity Equations

Problem 24.1.1 — Cavity equations with a magnetic-dipole driving polarization

List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation. Use coupled photon–inversion equations, \(\dot n=(gN-\gamma_c)n+S\) and \(\dot N=R_p-\gamma_2N-gNn\); solve the steady state before linearizing the Jacobian. Check the threshold limit and require negative real parts for both small-signal eigenvalues when a stable operating point is claimed.

Problem 24.1.2 — Equivalent lumped circuit equations

List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation. Use coupled photon–inversion equations, \(\dot n=(gN-\gamma_c)n+S\) and \(\dot N=R_p-\gamma_2N-gNn\); solve the steady state before linearizing the Jacobian. Check the threshold limit and require negative real parts for both small-signal eigenvalues when a stable operating point is claimed.

Section 24.2: External Signal Sources

Problem 24.2.1 — Power dissipation in an externally excited cavity

List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation. Use coupled photon–inversion equations, \(\dot n=(gN-\gamma_c)n+S\) and \(\dot N=R_p-\gamma_2N-gNn\); solve the steady state before linearizing the Jacobian. Check the threshold limit and require negative real parts for both small-signal eigenvalues when a stable operating point is claimed.

Problem 24.2.2 — Calculating regenerative cavity gain from the lumped equivalent circuit model

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 24.2.3 — Alternative derivation of the external coupling term

Begin with the stated physical law, keep the derivation symbolic, and introduce each approximation only where its limiting condition is explicit. Use coupled photon–inversion equations, \(\dot n=(gN-\gamma_c)n+S\) and \(\dot N=R_p-\gamma_2N-gNn\); solve the steady state before linearizing the Jacobian. Check the threshold limit and require negative real parts for both small-signal eigenvalues when a stable operating point is claimed.

Problem 24.2.4 — Another alternative derivation of the external coupling term

Begin with the stated physical law, keep the derivation symbolic, and introduce each approximation only where its limiting condition is explicit. Use coupled photon–inversion equations, \(\dot n=(gN-\gamma_c)n+S\) and \(\dot N=R_p-\gamma_2N-gNn\); solve the steady state before linearizing the Jacobian. Check the threshold limit and require negative real parts for both small-signal eigenvalues when a stable operating point is claimed.

Section 24.4: Alternative Formulations Of The Laser Equations

Problem 24.4.1 — Obtaining previous results using the SVEA equations

List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation. Use coupled photon–inversion equations, \(\dot n=(gN-\gamma_c)n+S\) and \(\dot N=R_p-\gamma_2N-gNn\); solve the steady state before linearizing the Jacobian. Check the threshold limit and require negative real parts for both small-signal eigenvalues when a stable operating point is claimed.

Problem 24.4.2 — Obtaining previous results using the phase-amplitude equations

List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation. Use coupled photon–inversion equations, \(\dot n=(gN-\gamma_c)n+S\) and \(\dot N=R_p-\gamma_2N-gNn\); solve the steady state before linearizing the Jacobian. Check the threshold limit and require negative real parts for both small-signal eigenvalues when a stable operating point is claimed.

Problem 24.4.3 — Calculating atomic response from the phase-amplitude equations

List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation. Use coupled photon–inversion equations, \(\dot n=(gN-\gamma_c)n+S\) and \(\dot N=R_p-\gamma_2N-gNn\); solve the steady state before linearizing the Jacobian. Check the threshold limit and require negative real parts for both small-signal eigenvalues when a stable operating point is claimed.

Section 24.5: Cavity And Atomic Rate Equations

Problem 24.5.1 — Deriving the rate equations from the phase-amplitude equations

List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation. Use coupled photon–inversion equations, \(\dot n=(gN-\gamma_c)n+S\) and \(\dot N=R_p-\gamma_2N-gNn\); solve the steady state before linearizing the Jacobian. Check the threshold limit and require negative real parts for both small-signal eigenvalues when a stable operating point is claimed.