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Physical Review A

Quantum noise of an injection-locked laser oscillator

H. A. Haus, Y. Yamamoto

Published Mar 1, 1984• 46 citations
Publisher

Abstract

The quantum noise of an injection-locked laser oscillator is analyzed by the operator Langevin equation. The problem is also treated by the Fokker-Planck equation and the same results are obtained in the same regimes of applicability. The steady-state solution of the Fokker-Planck equation gives the probability distribution of amplitude and phase, the Langevin equation arrives more directly at the spectrum of amplitude and phase. The phase of the injection-locked oscillator is related to the phase of the injection signal and thus constitutes a measurement of phase. In the limit of complete inversion and zero internal loss of the laser resonator, the associated uncertainty is twice that dictated by the uncertainty principle. This result is interpreted by comparing it with the uncertainty introduced by a linear amplifier which can perform a simultaneous measurement of amplitude and phase.

DOI
10.1103/physreva.29.1261

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Communications in Theoretical Physics•Feb 10, 2025•0 citations
The quantum noise of an injection-locked laser oscillator is analyzed by the operator Langevin equation. The problem is also treated by the Fokker-Planck equation and the same results are obtained in the same regimes of applicability. The steady-state solution of the Fokker-Planck equation gives the probability distribution of amplitude and phase, the Langevin equation arrives more directly at the spectrum of amplitude and phase. The phase of the injection-locked oscillator is related to the phase of the injection signal and thus constitutes a measurement of phase. In the limit of complete inversion and zero internal loss of the laser resonator, the associated uncertainty is twice that dictated by the uncertainty principle. This result is interpreted by comparing it with the uncertainty introduced by a linear amplifier which can perform a simultaneous measurement of amplitude and phase.
We investigate the effect of injected squeezed vacuum on the phase and intensity fluctuations of a laser. By using a stochastic simulation of the Langevin equations, we find that the phase diffusion noise of a laser with injected squeezed vacuum can be transiently reduced by a factor of 2, in agreement, with earlier predictions. Further, we find that the laser linewidth is broadened. The key observation is that the phase-diffusion rate is time dependent. We find that the steady-state intensity fluctuations are increased by the injected squeezing, in qualitative (but not quantitative) agreement with an earlier prediction. The problem of locking is examined. We find that the injected squeezed vacuum does not cause the laser to lock to any particular phase. We present an approximate solution of the Fokker-Planck equation in steady state, and a simple geometrical, vector kick model from which most of our results can be obtained.
A single-mode laser noise model driven by quadratic pump noise and quantum noise with cross-correlation between the real and imaginary parts of the noises is proposed. The approximate Fokker-Planck equation (AFPE) of the model for the laser phase and the laser amplitude is derived. It is found that the laser phase is controlled intensively by the correlation between the real and imaginary parts of the pump noise and that of the quantum noise. The correlation between the real and imaginary parts of quantum noise ${\ensuremath{\lambda}}_{q}$ tends to lead the laser phase to be locked at some values and the correlation between the real and imaginary parts of the pump noise ${\ensuremath{\lambda}}_{p}$ tends to destroy or confine the laser phase lock. Quantitative results are presented and discussed in detail. As an important application of the above-mentioned results, we take a phase lock approximation to get a Langevin equation for the laser field amplitude and an AFPE of the laser intensity.
The quantum noise properties of injection-locked semiconductor laser oscillators are investigated. Included in the analysis are pump-noise suppression and squeezing of the injection signal. It is shown that the main effect of injection locking is the reduction of the phase noise of the outgoing laser light field. The influence that squeezing of the injection signal state has on the output state is shown to be strongly dependent on the value of the linewidth enhancement factor. In general, little or no noise reduction stands to be gained by squeezing the injection signal. It is also shown that under idealized conditions, the spectral uncertainty product of the outgoing field will approach, but never go below, the minimum uncertainty value given by the Heisenberg uncertainty relation.
The influence of injected atomic coherence on a laser's operation is discussed by analyzing the semiclassical equations of motion of this system. Both stationary-steady-state and time-dependent regimes of operation are investigated. We present the dependence of the stationary laser intensity and phase on the external parameters (population inversion, amplitude of atomic coherence, and detuning). For small cavity-field detuning and large atomic coherence, both frequency and phase locking occurs. The laser frequency is locked to the atomic frequency, and the phase is locked to a value determined primarily by the phase of the atomic coherence. Under certain conditions, in the inverted regime, the output intensity is an ssS-shaped function of the atomic coherence, leading to the possibility of a bistable behavior. However, only a portion of the bistable curve gives stable stationary operation, and in critical points (the location of which depends crucially on the external parameters) time-dependent instabilities branch away. When the detuning is larger than a critical value there is neither phase nor frequency locking. We find that the time-dependent behavior of the laser intensity in this case is oscillatory (quasiperiodic), and show that there is a stable limit cycle in the phase plane of the quadratures. There is a small parameter region where a stationary steady state and an oscillatory state may coexist. We also consider the nonlinear quantum theory of a laser with injected atomic coherence, and include the effect of pumping statistics. We derive the Fokker-Planck equation for the P representation, and express the noise in terms of moments. We find that in the steady state the intensity noise can be suppressed below the shot-noise limit but that the phase fluctuations are not affected by pump regularity. For nonzero detuning, we find that transient squeezing of the phase fluctuations is possible.
The intensity-noise properties of an injection-locked semiconductor laser were studied experimentally. The constant-current-driven semiconductor laser producing the amplitude-squeezed state whose intensity noise was reduced below the standard quantum limit (SQL) by 0.72 dB was injection-locked by an external master laser. The measured intensity-noise level of the injection-locked semiconductor laser was 0.91 dB below the SQL. This experimental result indicates that a phase-coherent amplitude-squeezed state or squeezed vacuum state together with a reference local oscillator wave can be generated directly by semiconductor laser systems.
Abstract The noise feature of a single-mode class-A laser amplifier is investigated by solving the Maxwell–Bloch equations of motion in the presence of the fluctuation force of cavity Langevin. The aim is to calculate the simultaneous fluctuations that are superimposed on the amplitude and phase of the cavity electric field, as well as the atomic population inversion. The correlation function of these fluctuations yields the amplitude, phase, and spontaneous emission noise fluxes, respectively. The amplitude and spontaneous emission noise fluxes exhibit the Lorentzian profiles in both the below-threshold state and the injection-locking region of the above-threshold state. While noise is typically viewed negatively in science and engineering, this research highlights its positive role as a valuable tool for measuring the optical properties of a laser amplifier. For instance, the degree of first-order temporal coherence (DFOTC) is derived by taking the Fourier transform of the amplitude noise flux. The damping rate of DFOTC is associated with the coherence time of the light emitted by the laser amplifier. Furthermore, the uncertainty relation between noise bandwidth and coherence time is confirmed. Finally, it is demonstrated that the input pumping noise flux, together with the output amplitude and spontaneous emission noise fluxes, satisfy the principle of flux conservation.