Probability density functions of the nonlinear phase noise

Probability density functions are given for nonlinear phase noise in a photonic communication system in which the information is encoded in the optical phase, both unconditioned and conditioned to the detection of a given amount of pulse energy. It is shown that the reach of a transmission system is increased by approximately 41% by ideal postcompensation of the nonlinear phase noise.     

Effect of dispersion on nonlinear phase noise

The variance of nonlinear phase noise is analyzed by including the effect of intrachannel cross-phase modulation-induced nonlinear phase noise. Consistent with Ho and Wang [IEEE Photon. Technol. Lett.17, 1426 (2005)] for a lightwave transmission system but contrary to the conclusions of both Kumar [Opt. Lett.30, 3278 (2005)] and Green [Opt. Lett.28, 2455 (2003)] with different initial conditions, the variance of nonlinear phase noise does not decrease much with the increase of chromatic dispersion. The results are consistent with each other after a careful reexamination.     

Effect of chromatic dispersion on nonlinear phase noise

We consider the combined effects of amplified spontaneous emission noise, optical Kerr nonlinearity, and chromatic dispersion on phase noise in an optical communication system. The effect of amplified spontaneous emission noise and Kerr nonlinearity were considered previously by Gordon and Mollenauer [Opt. Lett. 15, 1351 (1990)], and the effect of nonlinearity was found to be severe. We investigate the effect of chromatic dispersion on phase noise and show that it can either enhance or suppress the nonlinear noise amplification. For large absolute values of dispersion the nonlinear effect is suppressed, and the phase noise is reduced to its linear value. For a range of negative values of dispersion, however, nonlinear phase noise is enhanced and exhibits a maximum related to the modulation instability found in amplitude fluctuations. Nonlinear phase noise is quenched by these effects even in dispersion-compensated systems; the degree of suppression is sensitively dependent on the dispersion map. We demonstrate these results analytically with a simple linearized model.     

First-order probability density function of the laser speckle phase

An expression for the first-order probability density function of the laser speckle phase is analytically derived under the assumption that the speckle field obeys a non-circular, complex Gaussian, random process with a certain correlation between the real and imaginary parts of its complex amplitude. The probability density function of the speckle phase is actually evaluated for various cases and shown three-dimensionally as a function of the standard deviation of random object phase variations. The effect of random object phase variations on the probability density function is also investigated in detail.     

Holes in the probability density of strongly colored noise driven systems

A qualitative change in the topology of the joint probability densityP(,x), which occurs for strongly colored noise in multistable systems, has recently been observed first by analog simulation (F. Moss and F. Marchesoni,Phys. Lett. A 131:322 (1988)) and confirmed by matrix continued fraction methods (Th. Leiber and H. Riskin, unpublished), and by analytic theory (P. Hänggi, P. Jung, and F. Marchesoni,J. Stat. Phys., this issue). Systems studied were of the classx=–U(x)/x+(t,), whereU(x) is a multistable potential and (t, ) is a colored, Gaussian noise of intensityD, for which =0, and (t) (s)=(D/)exp(–t–s/). When the noise correlation time is smaller than some critical value ₀, which depends onD, the two-dimensional densityP(,x) has the usual topology [P. Jung and H. Risken,Z. Phys. B 61:367 (1985); F. Moss and P. V. E. McClintock,Z. Phys. B 61:381 (1985)]: a pair of local maxima ofP(,x), which correspond to a pair of adjacent local minima ofU(x), are connected by a single saddle point which lies on thex axis. When >₀, however,the single saddle disappears and is replaced by a pair of off-axis saddles. A depression, or hole, which is bounded by the saddles and the local maxima thus appears. The most probable trajectory connecting the two potential wells therefore does not pass through the origin for >₀, but instead must detour around the local barrier. This observation implies that successful mean-first-passage-time theories of strongly colored noise driven systems must necessarily be two dimensional (Hänggiet al.). We have observed these holes for several potentialsU(x): (1)a soft, bistable potential by analog simulation (Moss and Marchesoni); (2) a periodic potential [Th. Leiber, F. Marchesoni, and H. Risken,Phys. Rev. Lett. 59:1381 (1987)] by matrix continued fractions; (3) the usual hard, bistable potential,U(x)=–ax ²/2+bx ⁴/4, by analog simulations only; and (4) a random potential for which the forcingf(x)=–U(x)/x is an approximate Gaussian with nonzero correlation length, i.e., colored spatiotemporal noise, by analog simulation. There is a critical curve ₀(D) in the versusD plane which divides the two topological behaviors. For a fixed value ofD, this curve is shifted toward larger values of ₀ for progressively weaker barriers between the wells. Therefore, strong barriers favor the observation of this topological transformation at smaller values of . Recently, an analytic expression for the critical curve, valid asymptotically in the small-D limit, has been obtained (Hänggiet al.).This paper will appear in a forthcoming issue of theJournal of Statistical Physics.     

Asymptotic stability with probability one of MDOF nonlinear oscillators with fractional derivative damping

In this paper, the asymptotic stability with probability one of multi-degree-of-freedom (MDOF) nonlinear oscillators with fractional derivative damping parametrically excited by Gaussian white noises is investigated. A stochastic averaging method and the Khasminskii’s procedure are employed to evaluate the largest Lyapunov exponent, whose sign determines the stability of the system. As an example, two coupled nonlinear oscillators with fractional derivative damping is worked out to demonstrate the proposed procedure and to examine the effect of fractional order on the stochastic stability of system. In particular, the case of factional order more than 1 is studied for the first time.     

Effect of nonlinear gain on the phase noise of Y-branch lasers

Effect of nonlinear gain on the phase noise of modulated grating Y-branch (MGY) lasers is investigated by experiments and simulations. The phase noise is first characterized by measuring the frequency modulation noise spectrum of the MGY laser, and some interesting phenomena are observed. In order to understand the underlying physic mechanism of those phenomena, simulations are performed with taking Langevin noise sources and nonlinear gain terms into account. Simulated results show that, in the presence of the nonlinear gain, fluctuations of side-modes can bring excess phase noise to the lasing mode, especially when the side-modes are on the long-wavelength side of the main mode. Furthermore, it has been found that the phase noise hopping phenomenon can be induced by a bi-stable state of the laser, which is also closely related to the nonlinear gain. The investigation is helpful for explaining the complicated phase noise characteristics of the MGY laser.     

非线性相位噪声对DQPSK调制系统性能影响研究

为了研究非线性相位噪声对差分正交相移键控（DQPSK）调制系统性能的影响,在理论推导非线性相位噪声数学模型的基础上,通过固定接收端信噪比不变,仿真分析了40Gb/s速率时DQPSK调制系统误码率随输入信噪比的变化情况。结果表明:与二进制差分相移键控（DPSK）调制相比,DQPSK调制对非线性相位噪声的影响更为敏感,当非线性效应较大时,非线性相位噪声将使系统误码率显著增大,严重影响系统通信质量。因此,当采用DQPSK调制时,必须考虑非线性相位噪声对系统性能的影响。     

Effect of dispersion on nonlinear phase noise in optical transmission systems

An analytic expression for the variance of nonlinear phase noise that uses a first-order perturbation technique is obtained. The results show that for highly dispersive transmission systems, amplified spontaneous emission-induced phase noise due to self-phase modulation becomes much smaller than that for the systems with no dispersion.     

Compensation of nonlinear phase noise in an in-plane-magnetized anisotropic spin-torque oscillator

A theory of generation linewidth of a spin-torque oscillator (STO) based on an in-plane-magnetized nano-pillar with an anisotropic “free” magnetic layer has been developed. It is predicted that by choosing the direction of the in-plane bias magnetic field H₀ along the “hard” anisotropy axis of the STO “free” layer and the magnitude of this field to be four times larger than the anisotropy field H_A (H₀=4H_A) it would be possible to compensate the nonlinear phase noise and to achieve the minimum value of the generation linewidth, characteristic for an auto-oscillator without a nonlinear frequency shift.     

也谈概率密度和概率流密度算符

针对<大学物理>2008年第8期上关于粒子概率密度算符和概率流密度算符的讨论一文进行了补充,对一般空间(如动量空间)中的概率和概率流进行了较系统的阐述,并纠正了该文中的有关错误论断.     

The quantum-mechanical probability density

The evolution of the probability density for a quantum-mechanical particle is derived from a variational principle given earlier. It is shown that this derivation leads to the identification of the probability density with Ψ¹Ψ.     

Analysis of nonlinear phase noise in single-channel return-to-zero differential phase-shift keying transmission systems

Self-phase modulation and intrachannel cross-phase-modulation- (IXPM) induced nonlinear phase noise is investigated by the variational method. IXPM can cause a considerable increase of phase noise. We show, however, that IXPM leads to a partial correlation between the phase noises of adjacent pulses, which tends to reduce the influence of nonlinear phase noise in return-to-zero differential phase-shift keying transmission. In highly dispersive transmission systems, intrachannel four-wave mixing-induced differential phase fluctuations are typically larger than differential nonlinear phase noise. The analysis is validated by Monte Carlo simulation.     

Asymptotic expansion for eigenvalue density

The density of eigenvalues of a potential well is calculated in an asymptotic expansion for large geometrical size. Explicit, readily calculable expressions are obtained for volume and surface contributions. The resulting expressions are numerically applied to the case of a spherical Woods-Saxon well.     

Integro-differential equation for joint probability density in phase space associated with continuous-time random walk

We derive an integro-differential equation for the joint probability density function in phase space associated with the continuous-time random walk, with generic waiting time probability density function and external force. This equation permits us to investigate whole diffusion processes covering initial-, intermediate-, and long-time ranges, which can distinguish the evolution details for systems having the same behavior in the long-time limit with different initial- and intermediate-time behaviors. Moreover, we obtained analytic solutions for probability density functions both in velocity and phase spaces, and interesting dynamic behaviors are discovered.     

Phase-shifting interferometry in the presence of nonlinear phase steps, harmonics, and noise

A phase-shifting piezo device commonly employed in phase-shifting interferometry exhibits a nonlinear response to applied voltage. Hence, a method for estimation of phase distribution in the presence of nonlinear phase steps is presented. The proposed method compensates for the harmonics present in the intensity fringe, allows the use of arbitrary phase-step values between 0 and tau rad, and does not impose constraints on the selection of particular phase-step values for minimizing nonlinearity and compensating for the harmonics. The comparison of the proposed method with other well-known benchmarking algorithms shows that our method is highly efficient and also works well in the presence of noise.     

Asymptotic expansions in nonlinear regression theory

Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 9, pp. 20–25, September, 1995.     

Theory of nonlinear Gaussian noise

We consider stochastic differential equations of the Langevin type in which the noise enters nonlinearly. In particular we study quadratic gaussian noise and we derive equations for the probability density under different approximations. In the limit of small intensity and small correlation time of the noise we obtain a Fokker-Planck equation which accounts for the main effects of the nonlinear noise. We present some examples and we discuss the consequences of our results in the analysis of an electrohydrodynamic instability in liquid crystals in the presence of external noise.