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?
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

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 :math:`\dot\phi=\Delta\omega-K\sin\phi`; a steady locked phase exists only for :math:`|\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
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

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 :math:`\dot\phi=\Delta\omega-K\sin\phi`; a steady locked phase exists only for :math:`|\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
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation.
Expand :math:`k(\omega)` about the carrier, retain the requested orders, and use :math:`v_g=(dk/d\omega)^{-1}` with :math:`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?
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation.
Expand :math:`k(\omega)` about the carrier, retain the requested orders, and use :math:`v_g=(dk/d\omega)^{-1}` with :math:`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
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

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 :math:`\dot\phi=\Delta\omega-K\sin\phi`; a steady locked phase exists only for :math:`|\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
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

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 :math:`\dot\phi=\Delta\omega-K\sin\phi`; a steady locked phase exists only for :math:`|\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
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

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 :math:`\dot\phi=\Delta\omega-K\sin\phi`; a steady locked phase exists only for :math:`|\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)
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

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 :math:`\dot\phi=\Delta\omega-K\sin\phi`; a steady locked phase exists only for :math:`|\Delta\omega|\le |K|`.
Verify continuity at the locking boundary and the correct free-running beat frequency as the injected field tends to zero.