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Physical Review A

Approximate Fokker-Planck equation for a single-mode laser driven by quadratic pump noise and quantum noise with cross-correlation between real and imaginary parts of noise

Cao Li, Wu Da-Jin

Published Feb 3, 2006• 13 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.73.023802

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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 \ensuremathλq{\ensuremath{\lambda}}_{q}\ensuremathλq​ 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 \ensuremathλp{\ensuremath{\lambda}}_{p}\ensuremathλp​ 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.
We study the effects of correlations between real and imaginary parts of pump noise on the correlation function and relaxation time of laser intensity in a saturation laser model driven by quadratic pump noise and quantum noise. Using the phase–locking method and the projection–operator method, we derived the stationary probability density, the relaxation time Tc and the intensity correlation function C(s) of the laser intensity, and then performing numerical computations, we find the following: (i) Tc monotonously increases as the correlation strength between the real and imaginary parts of quadratic pump noise λp increases and (ii) C(s) decreases exponentially as decay time s increases and C(s) increases as λp increases. The correlations between the real and imaginary parts of the quadratic pump noise slow down the rate of fluctuation decay of the laser intensity.
An inhomogeneously broadened two-mode laser system with cross-correlations between the real and imaginary parts of quantum noise is considered. The Fokker–Planck equation of the system is derived by the phase-locking method. The steady-state probability distribution, the mean light intensity, the normalization autocorrelation function, and cross correlation function are calculated. The results indicate that: (i) The cross-correlation between the real and imaginary parts of quantum noise can cause the stationary probability distribution from one peak structure to two extrema structure when the laser system is operated above threshold; (ii) The cross-correlation between the real and imaginary parts of quantum noise enhance the light intensity fluctuation and decrease the laser output when the laser system is operated below or near threshold; (iii) The effect of the cross-correlation between the real and imaginary parts of quantum noise is very weak on the stationary properties when the laser system is operated far above threshold.
Based on the Fokker-Planck description of a single-mode laser model with correlations between the real and imaginary parts of the quantum noise as well as the pump noise,the dynamical properties of the laser system were studied by using of the mean relaxation time approximation method.The intensity correlation function and the associated mean relaxation time were calculated.It is found that the presence of correlations between the real and imaginary parts of the quantum noise always slows down the relaxation of the light intensity fluctuation when the laser is operated over the threshold.
We study the effects of cross-correlations between the real and imaginary parts of quantum noise on the intensity fluctuation of a saturation laser model. By virtue of the locked phase method,we derived an approximate Fokker–Planck equation and analytic expressions of the stationary probability distribution function (SPD) of the laser system. Based on the SPD, the mean, the normalized variance, and the normalized skewness of the steady-state laser intensity are calculated numerically. The results indicate that the correlation strength of the cross-correlations between the real and imaginary parts of quantum noise increases the intensity fluctuations.
A two-dimensional single-mode laser model is investigated, with cross-correlations between the real and imaginary parts of the quantum noise as well as the pump noise. The general closed form of the laser intensity Langevin equation (GILE) is obtained under a stable locked phase resulting from the cross-correlation λ q between the real and imaginary parts of the quantum noise. Because of the presence of a new term containing λ q , we can unify the two opposite intensity Langevin equations which correspond to the two special cases for |λ q |→0 and |λ q |→1 in the GILE. It is expected that the transient and stationary properties of the laser model can be changed qualitatively when λ q varies.
By means of the unified coloured noise approximation and phase lock, we study in this paper the stationary intensity distribution of the single-mode laser cubic model driven by coloured pump noise with cross-correlation between the real and imaginary parts of the quantum noise. We present a thorough discussion of how the cross-correlation λ q between the real and imaginary parts of the quantum noise and the self-correlation time τ of the pump noise determine the behaviours of the mean intensity ⟨ I ⟩ and variance λ 2 (0) for both below and above the threshold operation and many new phenomena are discovered. When the laser is operated above the threshold, increasing the cross-correlation intensity λ q makes the ⟨ I ⟩–τ curves exhibit a ``re-entrant phase transition''. Whether the laser is above or below the threshold, the whole ⟨ I ⟩–τ curve moves down as λ q increases; however, when λ q = 1 (perfect cross-correlation), the curve abruptly runs up. A nonzero value of cross-correlation time τ (τ≠0) leads to the entire λ 2 (0)– p ′ curve being suppressed (here p ′ is the pump noise intensity). This indicates the increasing precision of the laser beam due to the existence of the self-correlation time τ. The behaviour of the λ 2 (0)– p ′ curve as a function of λ q is similar to that of the ⟨ I ⟩–τ curve against λ q : that is, whether the laser is above or below the threshold, the λ 2 (0)– p ′ curve moves up as λ q increases; however, when λ q = 1, the curve suddenly moves down. Through the study in this paper, we can find a way to optimize for output laser intensity.