Photon statistics of the optical parametric oscillator including the threshold region

Starting from an effective Hamiltonian the derivation of a set of classical Langevin equations for the amplitudes of signal, idler, and pump is briefly reconsidered. From these equations all variables except those describing the signal mode are eliminated with the help of an adiabatic approximation and certain others, which are valid in the threshold region and somewhat above (i.e. photonumbers ? 10¹⁴). The signal mode amplitude then satisfies a van der Pol equation in the rotating wave approximation and is driven by a fluctuating force. With the exception of a slight difference due to the undamped phase diffusion of the pumping laser, the same Langevin equation has been derived earlier for the amplitude of a laser mode near threshold. We present the stochastically equivalent Fokker-Planck equation, whose solution is reduced to the known solution of the laser Fokker-Planck equation. Thus the complete photon statistics of the signal mode is revealed at once. In particular we obtain the stationary distribution and the amplitude and intensity correlation functions as well as the transient solution.     

Correlation function of the amplitude and of the intensity fluctuation for a laser model near threshold

In extension of a preceding paper the correlation function of the amplitude and of the intensity fluctuation are calculated in the threshold region. The laser amplitude is treated as a classical random variable obeying a van der Pol equation with a noise term. In order to get correlation functions, the method of distribution functions is employed. The distribution functions are evaluated by the Fokker-Planck equation. The lowest eigensolutions of the Fokker-Planck equation are obtained approximately by a variational method.     

Statistical Properties of the Polarized Radiation of a Dye–Solution Laser

Within the framework of the method of polarization components we obtained the Fokker–Planck equation for the intensity–distribution function of an individual component. Solutions for the Ornstein–Uhlenbeck process show that the experimentally observed special features in the behavior of the distribution function of the intensity and degree of polarization of laser radiation in the vicinity of the threshold are well described in the approximation of statistical independence of polarization components. However, since the Ornstein–Uhlenbeck process includes states not realizable physically for the given case, an exact solution of the Fokker–Planck equation is constructed by the method of expansion in eigenstates. It is shown that this solution is totally correct physically and yields virtually the same values for the distribution functions of the intensity and degree of polarization of radiation as the dependences obtained earlier for the Ornstein–Uhlenbeck process.     

Propagation characteristics of laser beams with amplitude modulations and phase fluctuations in apertured fractional Fourier transforming system

The apertured fractional Fourier transforming system is introduced and applied to treat the propagation of Gaussian beams with amplitude modulations and phase fluctuations. Based on the treatment that a rectangular function can be expanded into an approximate sum of complex Gaussian functions with finite numbers, the analytical expressions for the mutual intensity distribution of laser beams with amplitude modulations and phase fluctuations passing through the apertured fractional Fourier transforming system are obtained. Some numerical simulations are illustrated for their propagation properties.     

激光场中两维色噪声的近似计算

从色散型介质中三次激光模型出发，通过泛函导数，应用小于τ近拟计算两维色噪声，得到激光场定态强度分布，研究了定态强度分布的极值点随抽运参量和强度噪声相关时间的变化情况。应用平移1／N展开方法，得到激光场瞬态解，即本征值随抽运参量的变化情况。研究结果表明，抽运参量和噪声相关时间是激光系统出现一级相变类比的关键因素，相位噪声相关时间对强度谱分布中的线宽影响较大。     

Scattering of light in an inhomogeneous fluctuating plasma

Scattering of light in an inhomogeneous fluctuating plasma is treated whose mean density distribution is given by a solution of the Korteweg-de Vries equation. The amplitude and phase correlation functions are computed for given mean density and spectral function of the density fluctuations.The non-linear effect of the mean inhomogeneous density distribution on the scattered light leads to beatings in the spectral functions of the amplitude and phase correlations. It is shown that the covariance function for the log-amplitude (which is to first order proportional to the intensity of the scattered light) can be approximated in a large L-region by a power law whose exponent is smaller than in the homogeneous case.     

Dynamic recording of a digital hologram with single exposure by a wave-splitting phase-shifting method

The wave-splitting phase-shifting digital holography using a pixelated microretarder array is proposed. By using four intensity images recorded by the pixelated microretarder array, Stokes parameters of the incident wave can be calculated. The fully complex amplitude distribution of the object wave can be obtained by using Stokes parameters, if the polarization states of the object and reference waves are linear and orthogonal. Two sets of experimental results are provided to verify the feasibility of the proposed method. One is the result for a movie recording using a CW laser and the other is the result for instant recording using a single pulsed laser.     

Convergence of general beams into Gaussian intensity profiles after propagation in turbulent atmosphere

It is shown that a general shaped laser beam will eventually approach a Gaussian average intensity profile after propagation in turbulent atmosphere. In our formulation, source field at the exit plane of the laser is taken as the product of arbitrary functions of source transverse coordinates with Gaussian exponential modulations. Following the expansion of the arbitrary functions in terms of Hermite polynomials, the average receiver intensity expression is derived using the extended Huygens-Fresnel principle and the conditions for the intensity profile to assume a Gaussian shape are stated. The results are illustrated by simulating various source field distributions.     

Fast Analysis for the Focusing of a Laser Beam with Amplitude Modulation and Phase Fluctuation Through an Aperture Lens

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Fast Analysis for the Focusing of a Laser Beam with Amplitude Modulation and Phase Fluctuation Through an Aperture Lens

Based on the treatment that a rectangular function can be expanded into an approximate sum of complex Gaussian functions with finite numbers, the analytical expression for the focusing intensity distribution of a laser beam with amplitude modulation (AM) and phase fluctuation (PF) through an aperture lens is derived. The typical numerical examples are given and compared with those obtained from numerically integral calculation. The results show that our method can significantly improve the numerical calculation efficiency.     

Spectral phase distribution retrieval through coherent control of harmonic generation

The temporal intensity distribution of the third harmonic of a Ti:sapphire laser generated in Xe gas is fully reconstructed from its spectral phase and amplitude distributions. The spectral phases are retrieved by cross correlating the fundamental laser frequency field with that of the third harmonic, in a three laser versus one harmonic photon coupling scheme. The third harmonic spectral amplitude distribution is extracted from its field autocorrelation. The measured pulse duration is found to be in agreement with that expected from lowest order perturbation theory both for unstretched and chirped pulses.     

三维高斯拟合激光光强快速衰减算法

为缩短衰减倍率调整的时间,提高激光参数测试的效率,提出激光光强快速衰减算法。衰减倍率精确调整量由当前衰减倍率和采集到的光斑光强真实的最大灰度值共同决定。当因光电接收器件(CCD)饱和造成采集光斑图像失真时,即衰减倍率过小时,由于激光光斑的光强通常满足高斯分布,通过对光斑图像进行处理,去除饱和部分光强信息,对剩余部分光强信息利用最小二乘法进行三维高斯拟合,还原出激光光斑光强的真实分布并获得最大灰度值;当衰减倍率过大时,根据采集光斑图像可以直接获得当前最大灰度值,最后通过计算获得最佳的衰减倍率调整值,实现了激光光强快速准确的调整。算法的有效性通过步进电机带动的双轮可变衰减器及CCD配合得到验证。     

Geometrical form factor calculation using Monte Carlo integration for lidar

We proposed a geometrical form factor (GFF) calculation using Monte Carlo integration (GFF–MC) for lidar that is practical and can be applied to any laser intensity distribution. Theoretical results have been calculated with our method based on the functions of measured, uniform and Gaussian laser intensity distribution. Two experimental GFF traces on clear days are obtained to verify the validity of the theoretical results. The results indicated that the measured distribution function outperformed the Gaussian and uniform functions. That means that the deviation of the measured laser intensity distribution from an ideal one can be too large to neglect. In addition, the theoretical GFF of the uniform distribution had a larger error than that of the Gaussian distribution. Furthermore, the effects of the inclination angle of the laser beam and the central obstruction of the support structure of the second mirror of the telescope are discussed in this study.     

The temporal development of the intensity distribution in a focussed laser beam

This paper extends the steady-state calculation of the intensity distribution near the focus of a laser beam, to the case when the laser intensity is changing rapidly, over a time of order 1 picosecond. An efficient procedure for computation is outlined, allowing for the primary spherical aberration of the focussing lens and arbitrary dependence of the incident amplitude on position and time, and results are given for two trial calculations.     

基于自适应光学技术控制光束近场场强

在双变形镜自适应光学系统中,需要主激光出射时与信标光的振幅分布一致、相位共轭,当主激光到达目标时光波的分布与目标上发射的信标光光波分布相同,主激光的振幅和相位都得到校正。根据双变形镜自适应光学技术的概念,提出一种基于自适应光学技术控制光束近场场强的方法,从而实验验证双变形镜技术的可行性。该方法利用哈特曼传感器探测到的波前信息,对变形镜进行控制,实现了对光束近场场强的控制。仿真结果表明该系统对光束近场强度能进行较好地校正,使校正前后光束振幅的残余均方差值从0.310 0降为0.052 2,同时实验也验证了这种方法的可行性。     

Algorithm for near-field reconstruction based on radial-shearing interferometry

A new iterative algorithm to be used to precisely reconstruct near-field distribution from an interferogram of a laser output generated by a cyclic radial-shearing interferometer is proposed. First, by use of a window function around the zero-frequency part of the Fourier transform of the interferogram and calculation of the inverse Fourier transform of the zero-frequency part, we obtain the background intensity distribution of the interferogram. Then, according to the iterative algorithm, the near-field distribution of the laser output is precisely reconstructed from the background intensity distribution obtained in the first step. A numerical simulation and an actual experiment of the near-field reconstruction of the laser output with arbitrary amplitude distribution are implemented successfully.     

Weak versus strong harmonic-expansion analyses of self-pulsing lasers: I—the Laser Lorenz model

We apply a simple harmonic expansion method to the single-mode laser equations to analyze their dynamic properties. First, we extend the well-known small signal analysis to predict the transient pulsations of the relaxation oscillations. Such transients are characteristic of the laser signal relaxing towards its long-term solution, at any level of excitation, both beyond and below the instability threshold. Secondly, we extend the method to a strong harmonic expansion to analyze the properties of the long-term solutions. These properties are derived for typical examples, extending well beyond the boundary region of the instability domain, for which the laser field amplitude undergoes regular pulsations around zero-mean values.     

Radiation Polarization Distribution Function of a Dye Laser

The Fokker–Planck equation for the distribution function of the intensity of an individual component has been solved in the approximation of the Ornstein–Uhlenbeck process within the framework of the formalism of the method of polarization components. Based on this solution, we have constructed the distribution functions of the degree of lasing radiation polarization, analyzed experimental data for a certain geometry of laser pumping, and determined the values of the distribution parameters, including the loss coefficients for the polarization component.     

Quantum mechanical masterequation and Fokker-Planck equation for the damped harmonic oscillator

For the statistical operator of the damped harmonic oscillator a Masterequation is given in operator form describing both inelastic and elastic, purely phase destroying processes. By expressing the statistical operator in the diagonal representation with respect toGlauber's coherent states the Masterequation is transformed into a Fokker-Planck equation forGlauber's quasiprobability distribution function. The general solution of this Fokker-Planck equation is calculated. It is shown how the solution of a Masterequation can be used for calculating correlation functions and expressions are given for the amplitude and intensity correlation functions which are in complete formal agreement with the corresponding classical formulae.     

Statistical properties in the chaotic regime of the Maxwell-Bloch equations

The statistical properties of the electric field solution of the Maxwell-Bloch equations describing a single mode, homogeneously broadened laser in the mean field limit are investigated in the strange attractor regime. The electric field distribution sis calculated and it is found that the low order intensity moments are somewhat higher than those for thermal-chaotic light whilst the higher order moments are substantially lower. The field and intensity correlation functions are also calculated and found to exhibit radically different behaviour. The results are interpreted in terms of iterative map which is dederived using multiple time-scale perturbation theory. It is shown that a simple random phasor model is compatible with the numerical data.