Quadratic Solitons Research Articles

Dissipative quadratic solitons supported by a localized gain

We propose two models for the creation of stable dissipative solitons in optical media with the $\chi^{(2)}$ (quadratic) nonlinearity. To compensate spatially uniform loss in both the fundamental-frequency (FF) and second-harmonic (SH) components of the system, a strongly localized "hot spot", carrying the linear gain, is added, acting either on the FF component, or on the SH one. In both systems, we use numerical methods to find families of dissipative $\chi^{(2)}$ solitons pinned to the "hot spot". The shape of the existence and stability domains may be rather complex. An existence boundary for the solitons, which corresponds to the guided mode in the linearized version of the systems, is obtained in an analytical form. The solitons demonstrate noteworthy features, such as spontaneous symmetry breaking (of spatially symmetric solitons) and bistability.

Stabilization of nonlocal solitons by boundary conditions

We discovered that boundary conditions can stabilize nonlocal solitons with an oscillatory periodic response function. These solitons are the equivalent of quadratic solitons consisting of fundamental waves and oscillatory second harmonics, which are unstable unless subject to boundary confinement.

Quadratic spatial solitons generated in periodically poled lithium tantalite

The properties of quadratic spatial solitons generated in periodically poled lithium tantalite in the femtosecond regime are investigated. By using only a fundamental frequency beam as the input to excite quadratic spatial solitons, single and spatially anisotropic multiple solitons are generated. Our experiments demonstrate that the number of solitons is dependent on the input power. In the experiment, beam narrowing occurs first, leading to single-soliton generation. The threshold for multiple solitons is around 1.5 mW. The number of solitons does not increase indefinitely with input power. Multiple soliton generation only occurs in the input power range of 1.5 to 6.3 mW. The number of solitons decreases to 1 when the input power is 6.3 mW. The temporal and spectral characteristics of the spatial solitons are measured by using the GRENOUILLE technique.

Benjamin–Feir instability in nonlinear dispersive waves

In this paper, the authors extended the derivation to the nonlinear Schrödinger equation in two-dimensions, modified by the effect of non-uniformity. The authors derived several classes of soliton solutions in 2+1 dimensions. When the solution is assumed to depend on space and time only through a single argument of the function, they showed that the two-dimensional nonlinear Schrödinger equation is reduced either to the sine-Gordon for the hyperbolic case or sinh-Gordon equations for the elliptic case. Moreover, the authors extended this method to obtain analytical solutions to the nonlinear Schrödinger equation in two space dimensions plus time. This contains some interesting solutions such as the plane solitons, the N multiple solitons, the propagating breathers and quadratic solitons. The authors displayed graphically the obtained solutions by using the software Mathematica 5.

Quadratic solitons for negative effective second-harmonic diffraction as nonlocal solitons with periodic nonlocal response function

We employ the formal analogy between quadratic and nonlocal solitons to investigate analytically the properties of solitons and soliton bound states in second-harmonic generation in the regime of negative diffraction or dispersion of the second harmonic. We show that in the nonlocal description this regime corresponds to a periodic nonlocal response function. We then use the strongly nonlocal approximation to find analytical solutions of the families of single bright solitons and their bound states in terms of Mathieu functions.

Self-induced transparency quadratic solitons

We discover and theoretically explore self-induced transparency quadratic solitons (SIT-QS) supported by the media with quadratic optical nonlinearities, doped with resonant impurities. The fundamental frequency of input pulses is assumed to be close to the impurity resonance. We envision an ensemble of inhomogeneously broadened semiconductor quantum dots (QD) in the strong confinement regime grown on a substrate with a quadratic nonlinearity to be a promising candidate for the laboratory realization of SIT-QS. We also examine the influence of inhomogeneous broadening as well as wave number and group-velocity mismatches on the salient properties of the introduced solitons.

Discrete quadratic solitons with competing second-harmonic components

We describe families of discrete solitons in quadratic waveguide arrays supported by competing cascaded nonlinear interactions between one fundamental and two second-harmonic modes. We characterize the existence, stability, and excitation dynamics of these solitons and show that their features may resemble those of solitons in saturable media. Our results also demonstrate that a power threshold may appear for soliton formation, leading to a suppression of beam self-focusing which explains recent experimental observations.

Spectral pulse transformations and phase transitions in quadratic nonlinear waveguide arrays

We study experimentally and numerically the dynamics of a recently found topological phase transition for discrete quadratic solitons with linearly coupled SH waves. We find that, although no stationary states are excited in the experimental situation, the generic feature of the phase transition of the SH is preserved. By utilizing simulations of the coupled mode equations we identify the complex processes leading to the phase transition involving spatial focusing and the generation of new frequency components. These distinct signatures of the dynamic phase transition are also demonstrated experimentally.

Spatial quadratic solitons guided by narrow layers of a nonlinear material

We report analytical solutions for spatial solitons supported by layers of a quadratically nonlinear material embedded into a linear planar waveguide. A full set of symmetric, asymmetric, and antisymmetric modes pinned to a symmetric pair of the nonlinear layers is obtained. The solutions describe a bifurcation of the subcritical type, which accounts for the transition from the symmetric to asymmetric modes. The antisymmetric states (which do not undergo the bifurcation) are completely stable (the stability of the solitons pinned to the embedded layers is tested by means of numerical simulations). Exact solutions are also found for nonlinear layers embedded into a nonlinear waveguide, including the case when the uniform and localized nonlinearities have opposite signs (competing nonlinearities). For the layers embedded into the nonlinear medium, stability properties are explained by comparison to the respective cascading limit.

Publisher’s Note: Symmetry-breaking instability of quadratic soliton bound states [Phys. Rev. A83, 013807 (2011)

Received 13 January 2011DOI:https://doi.org/10.1103/PhysRevA.83.019904© 2011 American Physical Society

Symmetry-breaking instability of quadratic soliton bound states

We study both numerically and experimentally two-dimensional soliton bound states in quadratic media and demonstrate their symmetry-breaking instability. The experiment is performed in a potassium titanyl phosphate crystal in a type-II configuration. The bound state is generated by the copropagation of the antisymmetric fundamental beam locked in phase with the symmetrical second harmonic one. Experimental results are in good agreement with numerical simulations of the nonlinear wave equations.

Designing microstructured polymer optical fibers for cascaded quadratic soliton compression of femtosecond pulses: erratum

Designing microstructured polymer optical fibers for cascaded quadratic soliton compression of femtosecond pulses: erratum

Group Velocity Control of Ultrafast Pulses Based on Electro-Optic Effect and Quadratic Cascading Nonlinearity

In this paper we propose and analyze a method to control the group velocity of an ultrafast pulse based on the electro-optic effect and quadratic cascading nonlinearity. The evolution of the input pulse in a quasi-phase matching (QPM) grating behaves like quadratic slow light solitons, whose group delay can be tuned by changing the applied external electric field along the z-axis of the crystal. Our simulation also shows that an asymmetric QPM domain structure is necessary for the group delay to be tunable by an external electric field. A potential application in dispersion management when constructing a delay line in high-speed system is suggested at the end of discussion.

Type-I cascaded quadratic soliton compression in lithium niobate: Compressing femtosecond pulses from high-power fiber lasers

The output pulses of a commercial high-power femtosecond fiber laser or amplifier are typically around 300-500 fs with a wavelength around 1030 nm and 10s of $\mu$J pulse energy. Here we present a numerical study of cascaded quadratic soliton compression of such pulses in LiNbO$_3$ using a type I phase matching configuration. We find that because of competing cubic material nonlinearities compression can only occur in the nonstationary regime, where group-velocity mismatch induced Raman-like nonlocal effects prevent compression to below 100 fs. However, the strong group velocity dispersion implies that the pulses can achieve moderate compression to sub-130 fs duration in available crystal lengths. Most of the pulse energy is conserved because the compression is moderate. The effects of diffraction and spatial walk-off is addressed, and in particular the latter could become an issue when compressing in such long crystals (around 10 cm long). We finally show that the second harmonic contains a short pulse locked to the pump and a long multi-ps red-shifted detrimental component. The latter is caused by the nonlocal effects in the nonstationary regime, but because it is strongly red-shifted to a position that can be predicted, we show that it can be removed using a bandpass filter, leaving a sub-100 fs visible component at $\lambda=515$ nm with excellent pulse quality.

Periodic and solitary waves in systems of coherently coupled nonlinear envelope equations

Exact solutions for two classes of coherently coupled nonlinear envelope equations are derived in terms of products of Jacobi elliptic functions. Physical applications are illustrated in the context of nonlinear optics, namely, polarization of light beams and quadratic (or parametric) solitons. Stabilities of these double-humped solitary pulses are studied by direct numerical simulations. The use of computer is crucial, both in terms of symbolic manipulation in the derivation process and in the implementation of numerical schemes in stability consideration.

Designing microstructured polymer optical fibers for cascaded quadratic soliton compression of femtosecond pulses

The dispersion of index-guiding microstructured polymer optical fibers is calculated for second-harmonic generation. The quadratic nonlinearity is assumed to come from poling of the polymer, which in this study is chosen to be the cyclic olefin copolymer Topas. We found a very large phase mismatch between the pump and the second-harmonic waves. Therefore the potential for cascaded quadratic second-harmonic generation is investigated in particular for soliton compression of fs pulses. We found that excitation of temporal solitons from cascaded quadratic nonlinearities requires an effective quadratic nonlinearity of 5 pm/V or more. This might be reduced if a polymer with a low Kerr nonlinear refractive index is used. We also found that the group-velocity mismatch could be minimized if the design parameters of the microstructured fiber are chosen so the relative hole size is large and the hole pitch is on the order of the pump wavelength. Almost all design-parameter combinations resulted in cascaded effects in the stationary regime, where efficient and clean soliton compression can be found. We therefore did not see any benefit from choosing a fiber design where the group-velocity mismatch was minimized. Instead numerical simulations showed excellent compression of $\lambda=800$ nm 120 fs pulses with nJ pulse energy to few-cycle duration using a standard endlessly single-mode design with a relative hole size of 0.4.

The evolution of two-frequency solitons in an optical fiber with a longitudinally nonuniform nonlinearity

The evolution of two-frequency solitons in an optical fiber, as well as the practically important special case of absence of the second-harmonic wave, in the presence of a longitudinal nonuniformity of the coefficients characterizing the propagation nonlinearity are considered. The solitons found for media with constant values of the nonlinearity coefficients are used as initial distributions for media with a periodic dependence of the nonlinearity coefficients on the longitudinal coordinate. Modulation of the coefficient of cubic or quadratic nonlinearity is shown to result in oscillations of the peak intensity of the solitons (in both their components if two-color solitons are considered). In the case of a weak modulation of the nonlinearity coefficients, oscillations of the peak intensity occur at the frequency coinciding with the frequency of modulation of the nonlinearity coefficients. Under the weak influence of a periodically modulated cubic nonlinearity, parameters of quadratic solitons also oscillate upon the propagation. Regions of stability of solitons in the space of the modulation parameters are established.

Two-color pulse compression in aperiodically-poled lithium niobate

We experimentally demonstrate engineerable compression of two-colored pulses in a linearly-chirped quasi-phase-matching grating. Quadratic solitons generated from fundamental input are reshaped through cascaded parametric processes of second-harmonic generation (SHG) and the back-conversion. We use type-I (e: o+o) SHG geometry in a 50-mm-long aperiodically-poled MgO:LiNbO3 device to satisfy the group-velocity matching condition. Simultaneously compressed fundamental and SH pulses of about 55-fs duration with small pedestal are generated from the fundamental input pulses of 95-fs duration.

Limits to compression with cascaded quadratic soliton compressors

We study cascaded quadratic soliton compressors and address the physical mechanisms that limit the compression. A nonlocal model is derived, and the nonlocal response is shown to have an additional oscillatory component in the nonstationary regime when the group-velocity mismatch (GVM) is strong. This inhibits efficient compression. Raman-like perturbations from the cascaded nonlinearity, competing cubic nonlinearities, higher-order dispersion, and soliton energy may also limit compression, and through realistic numerical simulations we point out when each factor becomes important. We find that it is theoretically possible to reach the single-cycle regime by compressing high-energy fs pulses for wavelengths lambda = 1.0-1.3 microm in a beta -barium-borate crystal, and it requires that the system is in the stationary regime, where the phase mismatch is large enough to overcome the detrimental GVM effects. however, the simulations show that reaching single-cycle duration is ultimately inhibited by competing cubic nonlinearities as well as dispersive waves, that only show up when taking higher-order dispersion into account.

Mobility of Discrete Solitons in Quadratically Nonlinear Media

We study the mobility of solitons in lattices with quadratic (chi(2), alias second-harmonic-generating) nonlinearity. Using the notion of the Peierls-Nabarro potential and systematic numerical simulations, we demonstrate that, in contrast with their cubic (chi(3)) counterparts, the discrete quadratic solitons are mobile not only in the one-dimensional (1D) setting, but also in two dimensions (2D), in any direction. We identify parametric regions where an initial kick applied to a soliton leads to three possible outcomes: staying put, persistent motion, or destruction. On the 2D lattice, the solitons survive the largest kick and attain the largest speed along the diagonal direction.