Evolution of effective characteristics of laser beam of femtosecond duration upon self-action in a gas medium

Based on numerical solution of the nonlinear Schrödinger equation for complex envelope of the light-wave electric field strength, we study regularities in variation of effective parameters of a femtosecond laser beam (energy transfer ratio, root-mean-square radius, duration, and width of temporal spectrum) propagating in the air with formation of a single filament. We derive approximate formulas for the effective radius of the femtosecond laser beam in the regime of solitary filamentation for three characteristic zones corresponding to different stages of transient self-focusing of the beam. We also analyze the problem of effective radius of the laser beam with a nonideal Gaussian profile in the regime of multiple filamentation. At the later stages of evolution of the effective radius, for identical initial energy and dimensional characteristics of the ideal and nonideal light beams, we establish similarity of their limiting angular divergences.     

X and Y waves in the spatiotemporal Kerr dynamics of a self-guided light beam

The nonlinear stage of development of the spatiotemporal instability of the monochromatic Townes beam in a medium with self-focusing nonlinearity and normal dispersion is studied by analytical and numerical means. Small perturbations to the self-guided light beam are found to grow into two giant, splitting Y pulses featuring shock fronts on opposite sides. Each shocking pulse amplifies a co-propagating X wave, or dispersion- and diffraction-free linear wave mode of the medium, with super-broad spectrum.     

An improved nonlinear optical pulse propagation equation

A new equation for self-focusing of extremely focused short-duration intense pulses is derived using a method that treats diffraction and dispersion to all orders with nonlinearity present, and self-consistently determines the nonlinear derivative terms present in the propagation equation. It generalizes both the previous formulation of linear optical pulse propagation to the nonlinear regime, and the cw nonlinear regime propagation to the pulsed regime by including temporal characteristics of the pulse. We apply the new equation and propagate a tightly focused picosecond pulse in silica and explicitly show the effects of spatial-derivative nonlinear coupling terms.     

Lie-optics, geometrical phase and nonlinear dynamics of self-focusing and soliton evolution in a plasma

It is useful to state propagation laws for a self-focusing laser beam or a soliton in group-theoretical form to be called Lie-optical form for being able to predict self-focusing dynamics conveniently and amongst other things, the geometrical phase. It is shown that the propagation of the gaussian laser beam is governed by a rotation group in a non-absorbing medium and by the Lorentz group in an absorbing medium if the additional symmetry of paraxial propagation is imposed on the laser beam. This latter symmetry, however, needs care in its implementation because the electromagnetic wave of the laser sees a different refractive index profile than the laboratory observer in this approximation. It is explained how to estimate this non-Taylor paraxial power series approximation. The group theoretical laws so-stated are used to predict the geometrical or Berry phase of the laser beam by a technique developed by one of us elsewhere. The group-theoretical Lie-optic (or ABCD) laws are also useful in predicting the laser behavior in a more complex optical arrangement like in a laser cavity etc. The nonlinear dynamical consequences of these laws for long distance (or time) predictions are also dealt with. Ergodic dynamics of an ensemble of laser beams on the torus during absorptionless self-focusing is discussed in this context. From the point of view of new physics concepts, we introduce a stroboscopic invariant torus and a stroboscopic generating function in classical mechanics that is useful for long-distance predictions of absorptionless self-focusing.     

强激光超高斯光束形成的自聚焦环的分裂与抑制

通过数值求解非线性近轴波方程,详细描述了在钕玻璃介质中传输的强激光圆对称超高斯光束形成的自聚焦环,并展示了初始光束受到扰动引起环分裂的现象。对于峰值光强一定的光束,合理地选择其阶数和宽度,可以降低自聚焦成环效应,从而降低因自聚焦环分裂在光束边缘出现的细光束对介质造成的成丝破坏。     

Wideband modulation instability at combination frequencies in wave beams

We study theoretically wideband modulation instability at combination frequencies in media having cubic nonlinearity of self-focusing type along with the higher-order defocusing nonlinearity. It is assumed that in a medium with a purely cubic nonlinearity, the medium dispersion does not permit modulation instability. In this case, a collapse of the wave field exists if the beam power is higher than the critical power of self-focusing. The higher-order nonlinearity limits the field at the nonlinear focus, and the instability at combination frequencies becomes possible. It turns out that the field at the nonlinear focus increases with increasing excess of the beam power over the critical power of self-focusing. The obtained values of the nonlinear dielectric permittivity are used for determination of the growth rates of instability at combination frequencies. These growth rates ensure an increase in the combination fields from noise levels up to values comparable with the field of the high-power beam. Such an increase takes place if the beam power is severalfold higher than the critical one. The developed theory can be used for explanation of spectrum superbroadening during self-focusing of sufficiently short laser pulses and high-harmonic generation. __________ Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 50, No. 6, pp. 522–532, June 2007.     

Gas-induced solitons

The self-guiding of femtosecond laser pulses in air is investigated. For powers close to the threshold for self-focusing, we show that the balance between the nonlinear focusing of the beam and its defocusing by multiphoton sources produces a new kind of solitonlike structure. Over considerably long distances, the radial profile of the pulse relaxes to a steady-state shape, whereas its temporal profile shrinks along propagation. The influence of the normal group-velocity dispersion is finally discussed.     

Formation of elliptically polarized ring-shaped electric field structures on the self-focusing of light in an isotropic medium featuring a spatially disperse nonlinearity

The possibility is demonstrated that the self-focusing of an elliptically polarized beam in a medium featuring a spatially disperse cubic nonlinearity can give rise to several radially symmetric ring-shaped regions of the same sense of rotation (right or left) of the electric-field vector in the cross-section of the beam. The formation dynamics and propagation features of such electric field structures are studied for various parameters of the incident radiation and nonlinear medium.     

Nonsteady-state self-interaction in a medium with striction nonlinearity

An analytic and numerical investigation is made of the self-focusing of a wave beam allowing for the inertia of the nonlinear response of the medium described by an acoustic type of equation. Some characteristics of the dynamics of self-interaction of the wave fields are analyzed in the paraxial optics approximation and the self-similar structures and space-time instability of a plane wave are considered. The stages of instability buildup, structure formation, and the establishment of a steady state are studied numerically.     

强非局域克尔介质中光束传输的变分问题

 在非局域克尔介质中,光束的演化规律服从非局域非线性薛定谔方程。用变分法对此问题进行了重新表述。在强非局域的情况下,通过对介质响应函数进行泰勒展开,可以解析地表示变分问题。束宽的演化规律也可以定性地从光束束宽变分势得出。运用瑞利-里兹方法求解其变分方程,分别求出光束在自散焦和自聚焦介质中的变分解。对于自聚焦介质,当输入功率为某一特定值时,可以得到空间孤子,其束宽在传输过程中保持不变。通过与其他方法得到的解比较表明,变分法是解析讨论光束在非局域非线性介质中演化规律的方法之一。     

Formation of Hot Images in Laser Beams through a Self-defocusing Kerr Medium Slab

A nonlinear hot image is usually thought as of a special case of self-focusing, and thus occurs when a laser beam propagates through a slab of self-focusing medium. Here we show theoretically that a hot image may also be formed by a thin slab of self-defocusing medium. The physical origin for this hot image formation is akin to the in-line volume-phase holographic imaging due to the intensity-dependent refractive-index modulation of the self- defocusing medium. NumericM simulations confirm the theoretical prediction and further identify the dependence of the hot image on the beam power, the modulation depth of obscuration and the thickness of self-defocusing medium. The analysis presented here brings new insight into the physics of hot image formation in the high power laser system.     

Nonlinear unbalanced Bessel beams in the collapse of Gaussian beams arrested by nonlinear losses

Collapse of a Gaussian beam in self-focusing Kerr media arrested by nonlinear losses may lead to the spontaneous formation of a quasi-stationary nonlinear unbalanced Bessel beam with finite energy, which can propagate without significant distortion over tens of diffraction lengths, and without peak intensity attenuation while the beam power is drastically diminishing.     

Kerr非线性介质中聚焦像散高斯光束的传输特性

当高功率激光通过Kerr非线性介质传输时,Kerr效应会严重影响激光的传输特性.实际应用中常遇到像散光束.迄今为止,像散光束传输特性的研究大都局限于在线性介质中的传输,而在非线性介质中传输的研究较少,且还未涉及像散激光束通过含光学系统的Kerr非线性介质传输变换的研究.本文主要研究Kerr效应对聚焦光束像散特性和焦移特性的影响,以及聚焦像散高斯光束的自聚焦焦距和光束焦点调控.在光束扩展情况下,推导出了聚焦像散高斯光束在Kerr非线性介质中传输的束宽、束腰位置和焦移的解析公式,研究表明:在自聚焦介质中,随着自聚焦作用增强(如光束功率增强),光束像散越强,但焦移越小;在自散焦介质中,随着自散焦作用增强(如光束功率增强),光束像散越弱,但焦移越大.另一方面,在光束自聚焦情况下,推导出了自聚焦焦距的解析公式,研究表明利用光束像散可以调控光束焦点个数.     

Pattern formation in a passive incoherent ring resonator system based on nonlinear material with the self-focusing non-instantaneous Kerr response

We have analyzed the pattern formation in a nonlinear medium with self-focusing non-instantaneous Kerr response by employing the passive incoherent ring resonator system. In such a system, coherence time of the light is much shorter than the time of one round trip in the resonator. This delayed response of the nonlinearity can amplify the noise of certain spatial frequencies of the perturbed wave field and thus patterns can form when nonlinear gain (i.e., amplification of the noises) overcomes the loss (i.e., a well defined cavity threshold set by the coherence properties) in a single pass. The expression for the spatial spectral density of the perturbed wave field, which is the characteristic parameters of the pattern formation, have been derived in the case of lowest order approximation. It is found that for a specific value of the spatial frequency of the perturbed wave field, the intensity feedback of the cavity is much effective factor rather than the crystal thickness of the nonlinear media and amplitude of the incoming beam in the cavity for the enhancement of the spatial spectral density of the intensity pattern, which greatly improved the performance and applications of the pattern formation such as information processing, symmetry-breaking, and dynamics in non-equilibrium systems.     

Impact of light absorption and temperature on self-focusing of zeroth-order Bessel–Gauss beams in a plasma with relativistic–ponderomotive regime

The influence of light absorption and temperature on self-focusing of zeroth-order Bessel–Gauss beams through plasma, with relativistic–ponderomotive regime, is investigated in this paper. The nonlinear differential equation for beam-width is established by using parabolic equation approach under Wentzel–Kramers–Brillouin (WKB) paraxial approximation and solved numerically. The numerical results show the effects of beam parameter, relative density plasma, intensity parameter, absorption coefficient and plasma electron temperature on self-focusing of zeroth-order Bessel–Gauss beams in plasma. The self-focusing of Gaussian beams in the considered plasma is also deduced as a particular case in the present work.     

Investigation of self-focusing a Gaussian laser pulse in a weakly relativistic and ponderomotive regime inside a collisionless warm quantum plasma

This article investigates nonlinear self-focusing of an intense right hand circularly polarized Gaussian profile laser pulse in a weakly relativistic and ponderomotive regime inside a collisionless and unmagnetized warm quantum plasma. The nonlinear propagation equation for laser pulse in plasma has been derived. Then, the evolution differential equation for laser spot-size was obtained with considering the parabolic equation approach under the Wentzel-Kramers-Brillouin and paraxial ray approximations. This differential equation was solved numerically by fourth-order Runge-Kutta method. It is shown that our solution confirms the results of the self-focusing of the laser pulse in a weakly relativistic ponderomotive regime in cold quantum plasma in extreme conditions. Numerical results indicate that self-focusing of the laser pulse in the presence of relativistic and ponderomotive nonlinearity inside warm quantum plasma is improved in comparison with relativistic and ponderomotive cold quantum plasma.     

The spatiotemporal evolution of sinusoidal phase modulated laser and its competitions with noise perturbed laser during small-scale self-focusing

Smoothing by spectral dispersion (SSD) is employed to smooth the laser irradiation in inertial confinement fusion and its effectiveness can be affected by nonlinear effects because of some special modulation of itself. In the paper, the spatial evolution dependent on the time period of the sinusoidal phase modulated laser from SSD is investigated theoretically and numerically when the modulated laser is affected by the small-scale self-focusing(SSSF). Although the effects of SSSF for SSD's modulation and noise can be eliminated by temporal average, the time-dependent modulation intensity of laser with SSD or noise will increased quickly. The results show that the low-frequency spatial modulation from SSD sometimes can affect beam quality bigger than the small-scale noise because of its controlled small initial amplitude. The simulation results agree with the theoretical analysis.     

Regularities of nonstationary self-focusing of profiled laser beams: Averaged description

Nonstationary self-focusing of powerful laser radiation whose transverse intensity profile is Gaussian, super-Gaussian, or annular is theoretically investigated in air in the single filamentation regime using the formalism of averaged parameters of the radiation. Analytical relations are obtained that make it possible to calculate the basic characteristics of nonstationary self-focusing of beams with non-Gaussian profiles. It is found that, at distances of up to the global nonlinear focus, the initial stages of the dependences of the effective radius of these beams on the generalized evolutional variable are qualitatively and quantitatively similar. After the focal waist, this similarity is not observed, and the spatial behavior of the effective beam radius depends on the initial radiation profile.     

Relativistic channel-coupling focusing enhanced nonlinear guiding of an intense laser beam in a plasma channel

We employ the variational method to study the optical guiding of an intense laser beam in a preformed plasma channel without using the weakly relativistic approximation. Apart from the dependence on the laser power and the nonlinear channel strength parameter, the beam focusing properties is shown also to be governed by the laser intensity. Relativistic channel-coupling focusing, arising from the coupling between relativistic self-focusing and linear channel focusing, can enhance relativistic self-focusing but its strength is weaker than that of linear channel focusing.     

Light propagation in media with a highly nonlinear response: An analytical study

The problem of light propagation in highly nonlinear media is studied with the help of a recently introduced systematic approach to the analytical solution of equations of nonlinear optics [L.L. Tatarinova, M.E. Garcia, Exact solutions of the eikonal equations describing self-focusing in highly nonlinear geometrical optics, Phys. Rev. A 78 (2008) 021806(R)(1—4)]. Numerous particular cases of media exhibiting high-order nonlinear refractive indices are considered. We obtain analytical expressions for determining the self-focusing position and a new exact expression for calculating the filament intensity. The constructed solutions allowed us to revise a so-called self-focusing scaling law, i.e., the functional dependence of the self-focusing position on the initial light peak intensity. It was demonstrated that this dependence is governed by the form of the nonlinear refractive index and not by the laser beam shape at the boundary.