Chapter 25: Laser Spiking and Mode Competition

Source: Anthony E. Siegman, Lasers (1986), Chapter 25. 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 25.1: Laser Spiking And Relaxation Oscillations

Problem 25.1.1 — Phase plane description of spiking

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 25.1.2 — Dimensionless form for the spiking 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 25.1.3 — Spiking analysis for the ruby laser

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 25.1.4 — Step response of a “spiky” laser

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 25.1.5 — Controlling spiking by an external feedback loop?

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 25.1.6 — Proof that spiking will always die out in a simple laser system

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 25.1.7 — Extended spiking analysis for a semiconductor laser

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 25.2: Laser Amplitude Modulation

Problem 25.2.1 — Linearized small-signal response to laser cavity loss modulation

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 25.2.2 — Linearized small-signal response of the laser population difference

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 25.3: Laser Frequency Modulation And Frequency Switching

Problem 25.3.1 — Verifying the step-function frequency-shift analysis

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 25.3.2 — Frequency shift analysis for a nonideal phase ramp

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 25.4: Laser Mode Competition

Problem 25.4.1 — Mode competition with partial spatial overlap

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 25.4.2 — Mode competition analysis including coupling or scattering between modes (research problem)

Normalize the variables first, evaluate the analytic limits, and then sweep the remaining dimensionless parameter so the numerical curve can be checked against both limits. 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 25.4.3 — Gain saturation with time delay (research problem)

Normalize the variables first, evaluate the analytic limits, and then sweep the remaining dimensionless parameter so the numerical curve can be checked against both limits. 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.