Theoretical studies on frequency doubling in glass optical fibers in constant-intensity approximation

Theoretical consideration of the second harmonic generation in the constant-intensity approximation with account of the inverse effect of the excited wave on a stimulation wave through account of the phase change of interaction waves is presented. The behavior of harmonic wave intensity in a fiber for different parameters of the problem has been studied, the self-action of the light both in glass optical fiber and inside the homogeneous medium has been analyzed. In this approximation a decrease of conversion efficiency and a change of parameters of the curves of synchronism versus the pump intensity, in particular, the change of positions of intensity minima that does not take place in the constant-field approximation, are observed. The propagation of a plane wave packet in the homogeneous medium with quadratic nonlinearity and in an optical fiber accompanied by the self-phase modulation and leading to the change of the Gaussian pulse spectrum is analyzed with account of the phase changes of all the interacting waves.     

Phase effects at second harmonic generation of powerful laser radiation in noncentrosymmetrical media

There has been developed the theory of second harmonic generation of the intensive laser fields in the existence of both quadratic and cubic polarization in medium in the constant-intensity approximation accounting for the reverse effect of the excited wave on the exciting one and simultaneously allowing us to take into account phase mismatch and the damping of all the interacting waves. It is shown that the changes of pump intensity through self-phase and cross-phase modulation processes effect optimum phase relationship between interacting waves, the change of the spatial beating period. The conditions of compensating undesirable phase shifts between interacting waves have been determined, the analytical expressions for calculation of optimum values of phase mismatch, length of noncentrosymmetrical medium and the spatial beating period are offered. It is shown that in the absence of linear phase mismatch with an increase of basic radiation intensity the spatial beating period is being reduced. The numerous analysis has been made of frequency doubling process efficiency for KDP and LiNbO₃ crystals.     

Spatial trapping of short pulses in Ti-indiffused LiNbO(3) waveguides

We show numerically and experimentally that spatial trapping can be induced in quadratic media even if the pump pulse's duration is shorter than the group-delay mismatch between fundamental wave and second-harmonic components. The influence of phase mismatch and pulse power on the trapping effect is discussed. Spatial, temporal, and spectral behaviors that accompany self-trapped propagation are highlighted.     

Efficiency of sum frequency generation by regular domain structures

We theoretically examine generation of sum-frequency radiation in a regular domain structure, taking into account phase changes for all the interacting waves in a dissipative medium. We show that in contrast to the constant-field approximation, in the constant-intensity approximation there is an optimal value for the longitudinal dimension (length) of the domains at which maximum conversion of the signal wave energy to the sum-frequency wave energy occurs. As the pump intensity increases, the optimal domain length decreases. As the losses of the interacting waves increase, both the conversion efficiency and the optimal domain dimensions decrease.     

Dynamics of the amplitude and phase of a signal in the process of quasi-phase-matched parametric interaction

The parametric amplification of a light wave in nonlinear optical crystals under a high-frequency pump and a periodic spatial modulation of the coefficient of nonlinear coupling of the waves is considered. The value of the period of modulation of the nonlinear coefficient that corresponds to the maximum gain is found. The behavior of the phase and intensity of the amplified wave is studied using the matrix approach. For the case of the optimum phase relation between the interacting waves, the expression for the intensity of the amplified wave is obtained, which, in the limit, gives the result for a homogeneous nonlinear crystal.     

Two-color nonlinear localized photonic modes

We analyze second-harmonic generation (SHG) at a thin effectively quadratic nonlinear interface between two linear optical media. We predict multistability of SHG for both plane and localized waves, and also describe two-color localized photonic modes composed of a fundamental wave and its second harmonic coupled together by parametric interaction at the interface.     

Phase Effects of the Parametric Interaction in Metamaterials

We study the phase effects of the three-wave parametric interaction in metamaterials considering the negative refraction at the frequency of a signal wave. We analyze the efficiency of energy conversions between two direct waves with respect to the energy of the backward signal wave along with the dynamics of the signal-wave amplification in metamaterials. We show that there exist optimum values of the fundamental-wave intensity and the phase mismatch at which the efficiency of conversion is maximum. We obtain an analytic expression for the optimum value of the relative length of the metamaterial and present a numerical evaluation of the expected value for efficiency of the frequency parametric conversion in dielectric waveguides. A sufficient enhancement in the signal-wave amplification, which is possible at optimum values of the pump intensity and the metamaterial total length, leads to the parametric generation of the signal wave. Changing the frequency and power of the pump wave, one can realize a regular tuning of the frequencies of parametric converters.     

Analytical formulae for the optimization of the process of low-power phase modulation in a quadratic nonlinear medium

We show that using the approximation of fixed intensity analytical formulae, describing the process of induced phase modulation for the beams involved in second-order nonlinear optical processes can be derived. Expressions that allow the optimization of the phase shifts experienced by the fundamental and generated waves are presented for nonlinear quadratic processes, second-harmonic generation and sum-frequency mixing. In the case of seeding at the generated wavelength, the phase shift of the fundamental wave is due to two interactions: (i) a cubic one, based on coupled second-order processes (cascade cubic nonlinearity) and (ii) single quadratic interaction with participation of the seeding wave. By comparison with the exact numerical solution, we defined the input parameters of the beams for which this analytical approach is valid. It is shown that phase shifts exceeding /2 can be correctly predicted using the expressions obtained.     

Suppression of radiation in a momentum-nonconserving nonlinear interaction

We have considered the quadratic nonlinear radiation from a thin dipole sheet in front of a mirror. Radiation at the second-harmonic (SH) frequency on incidence of a fundamental field can be inhibited or enhanced independently of the SH field intensity stored between the nonlinear layer and the mirror. We have shown that this apparent contradiction can be fully understood only if the quadratic nonlinear interaction includes terms with a momentum mismatch equal to the magnitude of the SH field wave vector, such as the term that accounts for transfer of energy from the reflected SH field back to the incident fundamental.     

Dynamic holograms and phase conjugation in multilevel resonant media

The generation of dynamic holograms and four-wave phase conjugation in resonant media has been investigated under conditions of interaction between radiation and excited singlet and triplet states. Two mechanisms of optical control of resonant media non-linearities using independent pump-up to increase the diffraction efficiency of dynamic holograms have been considered. The peculiarities of non-linear recording of holograms, and a variant of quadratic recording, have been investigated. The dependence of the efficiency of diffraction by dynamic holograms on the intensity of interacting waves and the spectroscopic characteristics of the medium has been analysed.     

Highly localized discrete quadratic solitons

We observe highly localized solitons in periodically poled lithium niobate waveguide arrays close to phase matching for second-harmonic generation. With fundamental and second-harmonic input in one channel the response indicates two distinguishable propagation schemes. Depending on the relative phase between the two input waves, a self-trapped beam emerges, resembling closely either the in- or the out-of-phase quadratic eigenmode of a single waveguide. A stable soliton propagates when the input waves are in phase.     

Optimum conditions for the generation of the second harmonic of intense laser radiation

A generalized approximation of strong interactions of waves is suggested for the theoretical analysis of second-harmonic generation under conditions of self-action. On the basis of the method suggested, approximate solutions for the efficiency of second-harmonic generation are obtained with regard to the influence of higher nonlinearities, depletion of pumping radiation, and linear phase mismatch. The effects of the phase mismatch and the spatial distribution of the amplitude of the fundamental harmonic on the efficiency of second harmonic generation by intense laser radiation is analyzed. The results obtained with the approximate method developed are shown to be in good agreement with known experimental data and numerical calculations. Optimum conditions for second harmonic generation are determined in a wide range of laser radiation intensity and at different spatial distributions of the fundamental harmonic amplitude.     

Second-harmonic generation by intense beams containing phase dislocations:self-breaking into sets of solitons

Second-harmonic generation of light in bulk quadratic non-linear media with intense input beams containing phase dislocations is studied numerically under conditions for TypeI phase-matching. We investigate how, above a threshold light intensity, the input beams self-split along the azimuthal direction into a pattern of separate beams which then form a set of spatial solitons. The mechanism of such a behaviour is the azimuthal modulational instability of the ring-shaped, mutually trapped fundamental and second-harmonic beams containing the phase dislocations. This revised version was published online in November 2006 with corrections to the Cover Date.     

On reconstructing photon statistics by on/off detectors: Toward the multi-partite case

The propagation of pulses of the pump and its second harmonic in a quadratically nonlinear medium whose linear properties are characterized by a negative refractive index at the pump frequency and by a positive refractive index at the harmonic frequency is considered theoretically. In the case of a low intensity of the interacting waves, the pump and second-harmonic pulses propagate in the opposite directions, but sufficiently powerful pulses can form a simulton—a solitary two-frequency wave propagating in a certain direction as a single whole. Solutions to a set of equations are found which describe the steady-state propagation of a solitary wave and of a nonlinear periodic (cnoidal) wave.     

Simultaneous measurement of amplitude and phase in surface second-harmonic generation

The intensity and the phase in surface second-harmonic generation were simultaneously measured by use of an ac optical balanced homodyne detection system. In this system the surface second-harmonic wave was superimposed upon a local oscillator wave generated by a barium borate nonlinear optical crystal and then detected as an interference signal. Extremely high sensitivity of 3 aW (6 photons/s) and high precision were achieved by use of a lock-in amplifier, in which an interference signal of fundamental waves was used as a reference signal. Simultaneous measurement of the intensity and the phase in surface second-harmonic waves generated from native-oxidized Si(111) surfaces is demonstrated.     

Effect of the group velocity mismatch and dispersion of nonlinear susceptibility on the modulation instability of electromagnetic waves in a medium with quadratic nonlinearity

Modulation instability of continuous plane waves of the pump beams and the second harmonic that are described by the set of equations for the process of second-harmonic generation with allowance made for the dispersion of nonlinear susceptibility is considered. The case of type I phase matching is analyzed. In the regions of anomalous and normal group velocity dispersion, the instability increments are determined as functions of the frequency and the wave numbers of perturbations. It was found that the difference in the group velocities of the interacting waves exerts the most substantial effect on the evolution of the modulation instability.     

Phase effects at second harmonic generation in the layer media

An analysis is made of quasi-phase-matched interaction at second harmonic generation in a regular domain structure with accounts for losses and change of phases of all the interacting waves. The constant-intensity approximation of basic radiation is applied for this purpose, not to regular domain structure as a whole, but to each separately taken domain. It allows to carry out more strict analysis of quasi-synchronous interaction during the frequency conversion to the polydomain consisting of n layers, forming “grating” periods of modulation of the nonlinear susceptibility. With this, the values of complex amplitudes of basic radiation and second harmonic wave at the outlet of each domain are entrance values of the corresponding complex amplitudes to the following domain. The analytical expression is given for the case of n domains and the factors limiting the efficiency of the process of frequency conversion are analyzed. In the constant-intensity approximation, in contrast to the constant field approximation, the coherent length of domain depends on pump intensity. With increasing pump intensity the optimum length decreases. In a regular domain structure at frequency conversion from a layer to a layer, intensity of the basic radiation changes. Also the optimum length of domains at which conversion efficiency is maximal therefore changes. Thus, it is possible to obtain the high values of conversion frequency at the outlet of a regular domain structure by choice of optimum parameters of a task (length of domains, phase mismatch, pump intensity), as well as using the layers-domains of high quality.     

一种评价固体板材表面性质的非线性超声兰姆波方法

在基频与二倍频兰姆波相速度匹配但群速度失配的条件下, 通过选择适当的兰姆波二次谐波时域信号的测量起止时间, 可完全扣除换能器对二次谐波积分振幅测量所带来的影响。本文提出采用兰姆波二次谐波的积分振幅作为评价参量, 以实现对板材表面性质的准确评价。当板材表面性质发生改变时, 原本在理想表面条件下成立的基频与二倍频兰姆波相速度匹配的条件不再严格满足, 这将显著地影响到兰姆波的二次谐波发生效率, 相应的二次谐波积分振幅随表面性质的改变也将发生非常敏感的单调变化。实验结果表明,利用扣除换能器影响之后所测得的兰姆波二次谐波的积分振幅,可对板材表面性质的变化情况进行准确评价。     

基于群速度失配的超声兰姆波二次谐波的时域测量方法

考虑到发射和接收换能器对超声兰姆波时域二次谐波信号所带来的不可避免的影响,提出一种基于基频与二倍频兰姆波群速度失配的超声兰姆波二次谐波的时域测量方法。当基频与二倍频超声兰姆波的相速度匹配而群速度失配时,在超声兰姆波传播过程中所发生的二次谐波信号,在时域上可与源于斜劈换能器的二次谐波信号相分离。采用仅源自于基频兰姆波的时域二次谐波的积分振幅,定量描述兰姆波二次谐波的发生效率。以铝板中传播的兰姆波为例,给出了时域二次谐波的具体测量过程。本文提出的测量方法放宽了超声兰姆波二次谐波的测量条件,且扣除了换能器对二次谐波信号所带来的影响,所测得的二次谐波信号完全来自于基频兰姆波时域信号的二次谐波发生效应。      

Spontaneously generated walking X-shaped light bullets

In quadratic nonlinear media with normal dispersion and nonvanishing group velocity mismatch between fundamental wave and second-harmonic pulses, the wave packets of two harmonics can be locked together in propagation in the form of walking X-shaped light bullets. The output wave packets are developed into X-shaped light bullets with significant group delay time due to mutual dragging and significant shifting in spatiotemporal spectrum due to delayed nonlinear phase shift. We also show that spontaneously generated phase-front tilting can balance the dragging force induced by group velocity mismatch and lead to zero-velocity walking X-shaped light bullets.