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We applied the dielectric function method to solve analytically L-NL-L structure problems with negative Kerr nonlinearity. A damped wave in linear and a periodic standing wave in non-linear media had to be matched at boundaries. We gave a formulation of boundary conditions that did not explicitly include a film thickness. The boundary-value of a dielectric function can be expressed through the constant of non-trivial integral of motion. Using it, one generates a family of matched solutions satisfying boundary conditions. Then arbitrary film thickness can be checked against this family of solutions in search of matches. As a result, all fitted solutions are determined straightforwardly.

We experimentally investigated the third-order nonlinear optical response of absolute ethanol around 1550 nm. The real and imaginary parts of third-order susceptibilities are distinguished by Z-scan method, which demonstrated that absolute ethanol exhibited saturable absorption and self-defocusing behaviors with nonlinear refractive index n2~-10-14 m2/W. In addition, the experimental results via self-phase modulation method also verified the comparable nonlinear optical response of absolute ethanol. This work is expected to provide insights for the nonlinearities of organic solvents, and may broaden the potential of cost-effective nonlinear optoelectronic devices.

The symmetry breaking phenomenon of the parity-time (PT) symmetric solitons in self-defocusing saturable nonlinear Schrödinger equation is studied. As the soliton power increases, branches of asymmetric solitons are separated from antisymmetric solitons, and they coexist with both symmetric and antisymmetric solitons. The anti-symmetric solitons require different power thresholds when they are under different saturable nonlinear strength. The stronger the saturable nonlinearity is, the larger the power threshold is. The saturable nonlinear strength has obvious modulation effect on the symmetry breaking of antisymmetric solitons and the bifurcation of the power curve. However, when the modulation strength of PT- symmetric potential increases, the effect of this modulation effect weakens. The antisymmetric solitons are only stable in the low power region, and the stability of symmetric and asymmetric solitons is less affected by the soliton power. The increase of the saturable nonlinear strength leads to the increase of the critical power of the symmetry breaking. When a beam propagates in a PT-symmetric optical waveguide, the symmetry breaking of antisymmetric solitons can be controlled by changing the saturable nonlinear strength.

We report a large third-order nonlinear optical response of palladium diselenide (PdSe2) films – a two-dimensional (2D) noble metal dichalcogenide material. Both open-aperture (OA) and closed-aperture (CA) Z-scan measurements are performed with a femtosecond pulsed laser at 800 nm to investigate the nonlinear absorption and nonlinear refraction, respectively. In the OA experiment, we observe optical limiting behaviour originating from large two photo absorption (TPA) in the PdSe2 film of β = 3.26 ×10− 8 m/W. In the CA experiment, we measure a peak-valley response corresponding to a large and negative Kerr nonlinearity of n2 = -1.33×10− 15 m2/W – two orders of magnitude larger than bulk silicon. We also characterize the variation of n2 as a function of laser intensity, observing that n2 decreases in magnitude with incident laser intensity, becoming saturated at n2 = -9.96×10− 16 m2/W at high intensities. These results verify the large third-order nonlinear optical response of 2D PdSe2 as well as its strong potential for high performance nonlinear photonic devices.

We report detailed measurements of resonant self-focusing and self-defocusing of a continuously tunable Fe:ZnSe laser operating within the 4P-branch of the CO2 molecule. We determine the dispersion of this resonant nonlinearity near a rovibrational transition.

The long-term hydrodynamics of laser generated plasma regions are dictated by the spatial distribution of the deposited laser energy. Using an integrated chemical-optical solver to track energy deposition from single and dual nanosecond laser pulses, we show that self-defocusing results in a biased growth of the plasma region towards the focusing lens. This displacement of energy deposition scales with the Rayleigh range and can be reduced by maximizing the beam divergence.

A theory is presented to study the existence and stability properties of fundamental solitons, multi-peaked gap solitons and vortex solitons in self-defocusing media with PT-symmetric superlattice. In the first gap of the superlattice potential, fundamental, dipole, and in-phase quadruple solitons can exist stably within a wide parameter region, while in-phase two-peaked solitons, out-of-phase quadruple solitons and vortex solitons are always unstable. Compared with simple-lattice solitons in similar potentials, the gap solitons in superlattice have a wider stable range than those in simple-lattice.

We numerically and experimentally investigated the deflection of a laser beam in a photo-induced one dimensional (1D) photonic lattice (PL) in a pure LiNbO3 crystal accompanied with self-focusing and self-defocusing nonlinearities, respectively. The results show that the probe beam will be self-focused or self-defocused when the sample temperature is increased or decreased. Moreover, the light beam deflects to the direction of heat flux if there exists an additional temperature difference between the two lateral sides of the sample. The probe beam forms a deflected discrete circle soliton in the PL when the sample temperature is increased from 25 °C to 45°C and the additional temperature difference is ±5 °C. Our simulation results are in agreement with the experimental observation.

Femtosecond modelocked lasers with multi-gigahertz pulse repetition rates are attractive sources for all applications that require individually resolvable frequency comb lines or high sampling rates. However, the modelocked laser architectures demonstrated so far have several issues, including the need for single-mode pump lasers, limited output power, Q-switching instabilities and challenging cavity geometries. Here, we introduce a technique that solves these issues. In a two-dimensionally patterned quasi-phase-matching (QPM) device, we create a large, low-loss self-defocusing nonlinearity, which simultaneously provides SESAM-assisted soliton modelocking in the normal dispersion regime and suppresses Q-switching induced damage. We demonstrate femtosecond passive modelocking at 10-GHz pulse repetition rates from a simple straight laser cavity, directly pumped by a low-cost highly spatially multimode pump diode. The 10.6-GHz Yb:CaGdAlO4 (Yb:CALGO) laser delivers 166-fs pulses at 1.2 W of average output power. This enables a new class of femtosecond modelocked diode-pumped solid-state lasers with repetition rates at 10 GHz and beyond., Ultrafast lasers with multi-gigahertz repetition rates are desirable for applications requiring high sampling rates or resolvable frequency comb lines. Here, Mayer et al. use cascading of quadratic nonlinearities to passively modelock a femtosecond solid-state laser at a repetition rate of 10 GHz.

We put forward the existence and stability of defect surface gap solitons at the interface between uniform media and an superlattice with self-defocusing nonlinearity. We reveal that the defect plays the significant role in controlling the region of solitons existing. Various solitons are found to be existed in different gaps for different defects. For positive defects, fundamental solitons can exist stably in the semi-infinite gap, and dipole solitons can exist stably in the first gap but they are unstable in the second gap. For zero or negative defects, fundamental and dipole solitons can exist stably in the first gap and the second gap, respectively.

All the characteristics of the self-defocusing and “position dispersion” in KNbO3:Fe crystal are experimentally investigated. The results show that the scattering from KNbO3:Fe crystal is mainly due to the photo-induced lensing.

The third order nonlinear optical properties of methyl blue dye was studied using He–Ne laser light operated at 632.8 nm wavelength, employing the standard Z-scan technique. The dye sample exhibited negative nonlinear refraction (self defocusing) with large nonlinear refractive index value and also the dye possesses saturable nonlinear absorption under the experimental conditions. The effective values of the nonlinear refractive index n2, the nonlinear absorption coefficient β, and the real and imaginary parts of the third order optical nonlinearity χ(3) were evaluated for two different concentrations which were found to be several times greater than the reported chalcone derivatives. The results imply that methyl blue dye can be used as a potential medium for various optoelectronic applications.

The existence and stability of defect solitons in parity-time (PT) symmetric optical lattices with self-defocusing nonlinearity are reported. It is found that various solitons can exist in different gaps for different defects. All of the solitons have real and imaginary parts of opposite parities. For positive defects, fundamental and multipole solitons can exist stably in the semi-infinite gap and the first gap, respectively. For zero or negative defects, fundamental and multipole solitons can exist stably in the first gap and the second gap, respectively. For solitons with positive and negative defects, there exists a cutoff point of the propagation constant above which the defect solitons vanish. The relation with the cutoff point and the depth of PT potentials is studied. The influence of the imaginary part of the PT-symmetric potentials on soliton stability is given.

Nonlinear and thermo optical properties of the gold and silver nano-colloids were investigated using CW Z-scan experiments. Sign of nonlinear refraction coefficient of colloids are negative at this condition. Negative sign implying thermal self defocusing is the most important mechanism of nonlinearity in our experiments. Diffraction pattern shows that nonlinear refraction of gold nanocolloid must be at least three times greater. The linear and nonlinear absorption coefficients of gold nano-colloid and silver nano-colloid is determined to be 0.9 cm � 1 , 2.3 � 10 � 3 cm/W and 0.22 cm � 1 , 1.9 � 10 � 4 cm/W, respectively. These values produce temperature changes of about 90 K in gold nanocolloid and 10 K in silver nano-colloid at focus. Thermo optical constants of samples are determined at low and high temperature changes at focus.

We report on the experimental observation of the modulation instability in self-defocusing photorefractive LiNbO 3 : Fe crystal. Experimental results show that the modulation instability is related to the illumination time; and it is also related to the angle that between the direction of c-axis and the direction of the threadlike beam that illuminate the crystal. And the modulation instability is also related to the shape of the incident light.

We present two-dimensional numerical simulations of a nonlinear optical system made of a bright probe and a dark pump beam that counter propagate in a thin slice of self-defocusing Kerr media, simultaneously. The numerical results show that although the medium is self-defocusing to both pump and probe beam, a profile compression occurs under some conditions, and the weak bright probe beam is focused due to cross-phase modulation. The influence of the characteristics of beams and medium length on the probe beam compression is also investigated.

We report on the experimental observation of the modulation instability of uniform beam and frequency-doubling of interference fringes spatial frequency in self-defocusing photorefractive crystal LiNbO3:Fe by the same experimental setup. Frequency-doubling appears if the nonlinearity of the c-axis is bigger than the modulation of the cylindrical lenses, otherwise the modulation instability appears.

We investigate the stability of incoherently coupled two-color vector dark and grey solitons in a nonlinear self-defocusing media. We employ the variational technique to derive approximate solutions of the two-color vector dark solitons. The relations between the soliton parameters and the representation of the transverse velocities of the vector solitons are analytically obtained. We also demonstrate numerically the propagation dynamics of the two-color vector solitons, which shows that the two-color vector dark and grey solitons with different wavelengths are always unstable.

We investigate Airy-soliton interactions in self-defocusing media with PT potentials in one transverse dimension. We discuss different potentials in which the interacting beams with different phases are launched into the media at different separation distances. During interactions, there exist a primary collision region and a relaxation region accompanied by continuous interaction with the dispersed Airy tail. In the relaxation region, the beams exist soliton-like and breathers-like propagation. The beam width and mean power are influenced by initial separation, phase shift and modulation depth of PT potentials. Especially, the collision distance decreases with the spatial beam separation and the mean power possesses sinusoidal dependence on the phase shift., 16 pages, 5 figures

We find theoretically that the optical modulation instability can spontaneously happen in the noninstantaneous defocusing medium when there is a coherent feedback. It requires both above-threshold gain and phase matching for the instability to happen, very similar to the oscillation condition of a laser. The theoretical result is in good agreement with our experimental observation.

Using a Fourier transform method we fabricate experimentally the square and hexagonal two dimensional light-induce nonlinear photonic lattices of over 20 × 20 segments with different period in self-defocusing LiNbO 3 :Fe crystal. The experimental setup of this method is very simple, the photonic lattices are quite regular and robust, and the period of the lattice can be dominated easily.

We reveal the relevance between the nonlocality and the focusing/defocusing states in nonlocally nonlinear media, and predict a novel phenomenon that the self-focusing/self-defocusing property of the optical beam in the nonlocally nonlinear medium with a sine-oscillation response function depends on its degree of nonlocality. The transition from the focusing nonlinearity to the defocusing nonlinearity of the nonlinear refractive index will happen when the degree of nonlocality of the system goes cross a critical value, and vise verse. Bright and dark soliton solutions are obtained, respectively, in the focusing state and in the defocusing state, and their stabilities are also discussed. It is mentioned that such a phenomenon might be experimentally realized in the nematic liquid crystal with negative dielectric anisotropy or in the quadratic nonlinear medium., 6 pages, 4 figures

We construct families of discrete solitons (DSs) in an array of self-defocusing waveguides with an embedded parity-time- ($\mathcal{PT}$-) symmetric dimer, which is represented by a pair of waveguides carrying mutually balanced gain and loss. Four types of states attached to the embedded defect are found, namely, staggered and unstaggered bright localized modes and gray or antigray DSs. Their existence and stability regions expand with the increase of the strength of the coupling between the dimer-forming sites. The existence of the gray and staggered bright DSs is qualitatively explained by dint of the continuum limit. All the gray and antigray DSs are stable (some of them are unstable if the dimer carries the nonlinear $\mathcal{PT}$ symmetry, represented by balanced nonlinear gain and loss; in that case, the instability does not lead to a blowup, but rather creates oscillatory dynamical states). The boundary between the gray and antigray DSs is predicted in an approximate analytical form.

The properties of a light beam passed through an NLC layer with positive orientational nonlinearity and then reflected from the NLC–substrate interface have been experimentally studied. The divergence and the intensity profile of this beam were found to be quite different from those of the beam transmitted through the NLC cell. It is shown that the aberrational pattern in the cross-section of the reflected beam and the dynamics of its formation can be used to study the orientational nonlinearity even in the presence of thermal effects.

Two novel cluster polymers {[NMe4]2[MoOS3Cu3(μ2-I)3]} 1 and {[NMe4]2[WOS3Cu3(μ2-I)3]}n 2 have been prepared. Bithiometalates MO2S22− (M=Mo,W), CuI, Me4NBr and 4,4′-trimethylene-dipyridine(tdp) reacted in solid state under low-heating temperature resulting in these two clusters. X-ray diffraction analysis reveals that the two polymers have the same two-dimensional network structures, which are formed from nest-shaped units along crystallographic b and c directions. I atoms take on a μ2-fashion in bridging all the units. An alternative view along crystallographic a direction shows that they are layered features. Their third-order non-linear optical (NLO) properties were determined by Z-scan techniques. Both compounds possess very strong NLO refractive effect with n2 value of −5.02 × 10−17 m2W−1 for 1 and 1.20 × 10−16 m2W−1 for 2, respectively. Different from routine work, there is an alternation from cluster polymer 1–2 in their optical properties: 1 shows self-defocusing behavior, while 2 gives self-focusing effect. Effective third-order NLO susceptibilities χ(3) were calculated to be 4.90 × 10−11 esu 1 and 1.48 × 10−9 esu 2. The corresponding hyperpolarizabilities γ are 3.39 × 10−28 and 1.02 × 10−26 esu, respectively.

A three-wave resonant interaction for nonlinear excitations created from a continuous-wave background is shown to be possible in an isotropic optical medium with a self-defocusing cubic nonlinearity. Under suitable phase-matching conditions the nonlinear envelope equations for the resonant interaction are derived by using a method of multiple-scales. Some explicit three-wave solitary wave and lump solutions are discussed.

This letter addresses a numerical study on evolution of a dark soliton-like beam in self-defocusing media with measurable fifth-order focusing nonlinearity. Modified by the competing nonlinearities the incident dark soliton-like beam of appropriate parameters evolves into a dark solitonic channel twinned with its bright companion born of the additional radiation. Stable propagation takes shape of the divergent dark and bright solitonic channels when energy redistribution among the solitonic channels and the background keeps up to a dynamical equilibrium.

A spatial optical switch phenomenon caused by the induced focusing of a weak probe beam occurring in self-defocusing nonlinear media is discussed theoretically. A weak beam is induced to focus when it copropagates with an intense pump beam under the conditions that the probe and pump beams peak at different positions and propagate in different directions. Due to the effect of cross-phase modulation, the weak beam can not only be focused but also be deflected. The phenomenon is discussed by numerically solving the coupled amplitude equations.

Based on the opposite response of self-defocusing and self-focusing media to the light intensity changes, a novel design is investigated in detail. Numerical analyses show the device can be used at relatively low threshold. At the same time, the analyses also show some factors influence the threshold, such as the nonlinear index coefficient and the linear index difference.

The dynamics of ring dark solitary waves propagating in saturable self-defocusing media is studied numerically. The results show that the saturation leads to a reduction of the ring's transverse velocity when compared to this in Kerr media. At high saturation levels, however, in a formal contradiction but in actual agreement to the one-dimensional case, a strong snake-type instability and creation of optical vortices of alternate topological charges are to be expected.

Propagation of dark and grey spatial solitary waves in slab waveguides using self-defocusing non-linear media is studied and an exact analytical solution is obtained.

The thermally induced optical nonlinearity in a chlorophyll ethanol solution is examined. A theory is presented which shows good agreement with experiment. The theory models an optically thick but physically thin medium whose only heat transport mechanism is conduction. An average n2 was found for the medium which corresponds well with previous results but the thermal medium is shown to be fundamentally different from a Kerr medium.

We show analytically that the bright background of a class of two-frequency dark solitons is modulationally unstable, leading to the instability of the dark wave itself. The analytical result is confirmed numerically.

The existence and stability of spatial solitons in one-dimensional binary photonic lattices with alternating spacing and a saturable defocusing type of nonlinearity are investigated. Five types of nonlinear localized structures are found to exist: two in the mini-gap in the energy spectrum and others in the regular gap. It is proved that some of them are stable in certain ranges of the system parameters. Interactions between two identical localized structures propagating parallel to each other are investigated, too. It is shown that this interaction leads to formation of different localized patterns, such as solitons, breather-like modes, and breather complexes. The interaction output depends on the power and type of interacting identical solitons, the separation between them, the width of the mini-gap, and the phase relation between the tails of interacting solitons.

Three-dimensional spatiotemporal solitons for a bidispersive three-dimensional nonlinear Schrodinger equation are constructed by using Hirota bilinear method. The limiting behaviour of the solutions is shown by analytic expressions . The results show that there exist singular and nonsingular three-dimensional spatiotemporal solitons in media with the cubic defocusing nonlinearity and normal group velocity dispersion.

We demonstrate that self-defocusing photonic lattices support various shapes of soliton clusters (such as Y-shape and H-shape) which remain robust during propagation. We propose soliton-based image transmission through photonic structures induced in bulk materials.

We demonstrate that self-defocusing photonic lattices support various shapes of soliton states (such as Y-shape and H-shape) which remain robust during propagation. Based on this finding, we propose to transmit images through these self-defocusing photonic lattices.

TE 0 modes in symmetric nonlinear planar waveguides with a self-defocusing core and self-focusing cladding and substrate are studied. A finite element method is used to calculate the modes and their power-dispersion curves. New stability results are presented.

We have investigated the properties of p-polarized nonlinear surface polaritons (NLSP) near the interface of two media, one of which is optically linear and the other having an optical nonlinearity with saturation and a dielectric function of the form ϵ=ϵ xx =ϵ zz =ϵ 0 +a| E | 2 /( 1 +b| E | 2 ) , where E is the amplitude of the electric field, for the case ϵ00, a

Three-dimensional dark spatial solitory waves of the TM type are found to exist in a uniform saturable self defocusing nonlinear medium only within a certain range of the beam effective index n eff which may be wider or narrower compared with the corresponding one in the Kerr law medium, depending on nonlinear saturation index. This contrasts with the TE type, the existence of which in a saturable nonlinear medium is limited in the range of n eff equal to or narrower than that in the Kerr law nonlinear medium. The physical mechanism behind the distinct contrast is also discussed.

Self-focusing and self-defocusing are studied in a near resonance pulse propagation in a saturable amplifier. Spatial ring structure is observed in the self-focusing case preventing the catastrophic axial intensity increase.

We report the experimental observation of bright temporal solitonlike pulses propagating in a material with a negative nonlinearity and positive (normal) dispersion. We believe this to be the first observation of bright solitonic pulses in a material with a negative Kerr nonlinearity. We have observed a gradual narrowing of the output pulse width as a function of increasing input power, and our results agree well with simulations.

We demonstrate that steerable, two-dimensional, single-mode optical waveguides can be formed in defocusing media by exciting dark spatial solitons.