Chapter 29: Laser Injection Locking
Source: Anthony E. Siegman, Lasers (1986), Chapter 29. 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 29.3: The Locked-Oscillator Regime
Problem 29.3.1 — Can the change in instantaneous frequency during the frequency lock-up transient be measured?
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.
Reduce the phase dynamics to Adler’s equation \(\dot\phi=\Delta\omega-K\sin\phi\); a steady locked phase exists only for \(|\Delta\omega|\le |K|\).
3. Verification.
Verify continuity at the locking boundary and the correct free-running beat frequency as the injected field tends to zero.
Show detailed steps
List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation. Reduce the phase dynamics to Adler’s equation \(\dot\phi=\Delta\omega-K\sin\phi\); a steady locked phase exists only for \(|\Delta\omega|\le |K|\). Verify continuity at the locking boundary and the correct free-running beat frequency as the injected field tends to zero.
Section 29.4: Solutions Outside The Locking Range
Problem 29.4.1 — Fourier signal components near the edge of the locking range
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.
Reduce the phase dynamics to Adler’s equation \(\dot\phi=\Delta\omega-K\sin\phi\); a steady locked phase exists only for \(|\Delta\omega|\le |K|\).
3. Verification.
Verify continuity at the locking boundary and the correct free-running beat frequency as the injected field tends to zero.
Show detailed steps
List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation. Reduce the phase dynamics to Adler’s equation \(\dot\phi=\Delta\omega-K\sin\phi\); a steady locked phase exists only for \(|\Delta\omega|\le |K|\). Verify continuity at the locking boundary and the correct free-running beat frequency as the injected field tends to zero.
Section 29.5: Pulsed Injection Locking: A Phasor Description
Problem 29.5.1 — Using the phasor model to obtain additional steady-state locking results
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.
Expand \(k(\omega)\) about the carrier, retain the requested orders, and use \(v_g=(dk/d\omega)^{-1}\) with \(k''\) controlling quadratic dispersive broadening.
3. Verification.
Check the transform-limited and zero-dispersion limits, and conserve pulse energy when only phase is changed.
Show detailed steps
List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation. Expand \(k(\omega)\) about the carrier, retain the requested orders, and use \(v_g=(dk/d\omega)^{-1}\) with \(k''\) controlling quadratic dispersive broadening. Check the transform-limited and zero-dispersion limits, and conserve pulse energy when only phase is changed.
Problem 29.5.2 — Ambiguity in the phase relationship between injected and oscillating signals?
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.
Expand \(k(\omega)\) about the carrier, retain the requested orders, and use \(v_g=(dk/d\omega)^{-1}\) with \(k''\) controlling quadratic dispersive broadening.
3. Verification.
Check the transform-limited and zero-dispersion limits, and conserve pulse energy when only phase is changed.
Show detailed steps
List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation. Expand \(k(\omega)\) about the carrier, retain the requested orders, and use \(v_g=(dk/d\omega)^{-1}\) with \(k''\) controlling quadratic dispersive broadening. Check the transform-limited and zero-dispersion limits, and conserve pulse energy when only phase is changed.
Section 29.6: Applications: The Ring-Laser Gyroscope
Problem 29.6.1 — Locking together of two weakly coupled oscillators
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.
Reduce the phase dynamics to Adler’s equation \(\dot\phi=\Delta\omega-K\sin\phi\); a steady locked phase exists only for \(|\Delta\omega|\le |K|\).
3. Verification.
Verify continuity at the locking boundary and the correct free-running beat frequency as the injected field tends to zero.
Show detailed steps
List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation. Reduce the phase dynamics to Adler’s equation \(\dot\phi=\Delta\omega-K\sin\phi\); a steady locked phase exists only for \(|\Delta\omega|\le |K|\). Verify continuity at the locking boundary and the correct free-running beat frequency as the injected field tends to zero.
Problem 29.6.2 — Frequency pulling of two coupled but unlocked oscillators
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.
Reduce the phase dynamics to Adler’s equation \(\dot\phi=\Delta\omega-K\sin\phi\); a steady locked phase exists only for \(|\Delta\omega|\le |K|\).
3. Verification.
Verify continuity at the locking boundary and the correct free-running beat frequency as the injected field tends to zero.
Show detailed steps
List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation. Reduce the phase dynamics to Adler’s equation \(\dot\phi=\Delta\omega-K\sin\phi\); a steady locked phase exists only for \(|\Delta\omega|\le |K|\). Verify continuity at the locking boundary and the correct free-running beat frequency as the injected field tends to zero.
Problem 29.6.3 — Mode coupling due to backseat tering
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.
Reduce the phase dynamics to Adler’s equation \(\dot\phi=\Delta\omega-K\sin\phi\); a steady locked phase exists only for \(|\Delta\omega|\le |K|\).
3. Verification.
Verify continuity at the locking boundary and the correct free-running beat frequency as the injected field tends to zero.
Show detailed steps
List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation. Reduce the phase dynamics to Adler’s equation \(\dot\phi=\Delta\omega-K\sin\phi\); a steady locked phase exists only for \(|\Delta\omega|\le |K|\). Verify continuity at the locking boundary and the correct free-running beat frequency as the injected field tends to zero.
Problem 29.6.4 — Simultaneous injection of signals at two different frequencies into a free-running oscillator (research problem)
Brief solution
1. Method.
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.
2. Decisive step.
Reduce the phase dynamics to Adler’s equation \(\dot\phi=\Delta\omega-K\sin\phi\); a steady locked phase exists only for \(|\Delta\omega|\le |K|\).
3. Verification.
Verify continuity at the locking boundary and the correct free-running beat frequency as the injected field tends to zero.
Show detailed steps
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. Reduce the phase dynamics to Adler’s equation \(\dot\phi=\Delta\omega-K\sin\phi\); a steady locked phase exists only for \(|\Delta\omega|\le |K|\). Verify continuity at the locking boundary and the correct free-running beat frequency as the injected field tends to zero.