Saturation of the all-optical Kerr effect

Third and fifth order nonlinear susceptibilities in thin HfO2 layers

Third harmonic generation (THG) from dielectric layers is investigated. By forming a thin gradient of HfO₂ with continuously increasing thickness, we are able to study this process in detail. This technique allows us to elucidate the influence of the substrate and to quantify the layered materials third χ⁽³⁾(3ω: ω, ω, ω) and even fifth order χ⁽⁵⁾(3ω: ω, ω, ω, ω, - ω) nonlinear susceptibility at the fundamental wavelength of 1030 nm. This is to the best of our knowledge the first measurement of the fifth order nonlinear susceptibility in thin dielectric layers.

Periodic and localized waves in parabolic-law media with third- and fourth-order dispersions

Considering the higher-order nonlinearity is essential in a broad range of real physical media as it significantly influences the wave dynamics in these systems. We study the propagation of femtosecond light pulses inside an optical fiber medium exhibiting higher-order dispersion and cubic-quintic nonlinearities. Pulse evolution in such a system is governed by a higher-order nonlinear Schrödinger equation incorporating second-, third-, and fourth-order dispersions as well as cubic and quintic nonlinearities. The periodic and solitary wave solutions are identified using the equation method. Results presented indicated the potentially rich set of periodic waves in the system under the combined influence of higher-order dispersive effects and cubic-quintic nonlinearity. The velocity of these structures is uniquely dependent on all orders of dispersion. Conditions on the optical fiber parameters for the existence of these exact stable solutions are found by analytical stability analysis.

Nonlinear focusing of supercontinuum driven by intense mid-infrared pulses in gas-filled capillaries

Strong mid-infrared light-matter interactions have attracted extensive attention as they open up new frontiers in nonlinear optics. Here we observe through simulations a novel, to the best of our knowledge, aspect of mid-infrared pulse dynamics in a high-pressure gas-filled capillary, where a pulse with a power well below the critical power for Kerr self-focusing undergoes an astonishing increase of the peak intensity following an extremely efficient spectral broadening. This intensity enhancement is attributed to the Kerr-induced focusing of the supercontinuum. Our study provides an interesting perspective for controlling the laser intensity with possible applications in nonlinear light conversion driven by mid-infrared pulses.

Non-instantaneous third-order optical response of gases in low-frequency fields

It is commonly assumed that for low-intensity short optical pulses far from resonance, the third-order optical nonlinear response is instantaneous. We solve the three-dimensional time-dependent Schrödinger equation for the hydrogen atom and show that this is not the case: the polarization is not simply proportional to the cube of the electric field even at low intensities. We analyze the fundamental-frequency and third-harmonic nonlinear susceptibilities of hydrogen, investigate their dependence on intensity, and find that the delays in the Kerr response rapidly approach the femtosecond time-scale at higher intensities, while the delays in the third harmonic generation remain much lower. We also propose an experimental scheme to detect and characterize the above effects.

Nonlinear polarization and ionization in O2: metastable electronic state model

We present a computational model for the nonlinear response of molecular oxygen exposed to strong mid-wavelength and long-wavelength infrared optical fields. Based on a non-Hermitian approach utilizing metastable electronic states, the nonlinear polarization and strong-field ionization are described as intimately connected properties. Good agreement with the measured nonlinear index and ionization rates is shown, and parameterized response functions are provided to facilitate applications in large-scale simulations of infrared optical pulses interacting with gaseous media.

True versus effective Kerr nonlinear response in optical filamentation

The optical Kerr effect, and the nonlinear polarization in general, represents an important light-matter interaction governing many regimes encountered in the nonlinear optics. We reason that in the context of optical filamentation one should distinguish the third-order Kerr effect occurring at relatively low light intensities from the effective Kerr nonlinearity relevant to higher intensity. While many properties of filaments can be captured well with a third-order nonlinear polarization model with a nonlinear index chosen somewhat higher than the true nonlinear index operative at low intensities, our comparative simulations indicate that some filamentation aspects carry significant signatures from the higher-order nonlinearity.

Bound-Electron Nonlinearity Beyond the Ionization Threshold

We present absolute space- and time-resolved measurements of the ultrafast laser-driven nonlinear polarizability in argon, krypton, xenon, nitrogen, and oxygen up to ionization fractions of a few percent. These measurements enable determination of the strongly nonperturbative bound-electron nonlinear polarizability well beyond the ionization threshold, where it is found to remain approximately quadratic in the laser field, a result normally expected at much lower intensities where perturbation theory applies.

Short-pulse lasers for weather control

Filamentation of ultra-short TW-class lasers recently opened new perspectives in atmospheric research. Laser filaments are self-sustained light structures of 0.1-1 mm in diameter, spanning over hundreds of meters in length, and producing a low density plasma (10¹⁵-10¹⁷ cm^-3) along their path. They stem from the dynamic balance between Kerr self-focusing and defocusing by the self-generated plasma and/or non-linear polarization saturation. While non-linearly propagating in air, these filamentary structures produce a coherent supercontinuum (from 230 nm to 4 µm, for a 800 nm laser wavelength) by self-phase modulation (SPM), which can be used for remote 3D-monitoring of atmospheric components by Lidar (Light Detection and Ranging). However, due to their high intensity (10¹³-10¹⁴ W cm^-2), they also modify the chemical composition of the air via photo-ionization and photo-dissociation of the molecules and aerosols present in the laser path. These unique properties were recently exploited for investigating the capability of modulating some key atmospheric processes, like lightning from thunderclouds, water vapor condensation, fog formation and dissipation, and light scattering (albedo) from high altitude clouds for radiative forcing management. Here we review recent spectacular advances in this context, achieved both in the laboratory and in the field, reveal their underlying mechanisms, and discuss the applicability of using these new non-linear photonic catalysts for real scale weather control.

On the manifestation of higher-order nonlinearities in a noble gas medium undergoing strong ionization

While there is a consensus that higher-order effects beyond χ⁽³⁾ are present also in high-intensity light-matter interactions, when and how they become apparent needs further study. The central question addressed in this Letter is whether it is possible to design a situation in which they show up before being completely masked by the electrons freed by a high-intensity field. The second question we attempt to answer is how much such observations, if and when feasible, can reveal about the nature of the nonlinear polarization. We answer the first question in the affirmative, but our comparative simulations indicate that distinguishing the higher-order nonlinearity from the third-order polarization can be extremely challenging. We also briefly discuss the implications for the interpretation of the measured values of the nonlinear index.

Nonlinear optical response in molecular nitrogen: from ab-initio calculations to optical pulse simulations

Using first-principle multi-electron calculations via the hybrid anti-symmetrized coupled channels method, we create a model to describe both the nonlinear polarization and ionization of the nitrogen molecule. Based on the metastable electronic state approach, it is designed for space-and-time-resolved simulations in nonlinear optics that require modeling of optical pulses that exhibit rich spectral dynamics and propagate over long distances. As a demonstration of the model's utility, we study low-order harmonic generation in mid-infrared optical filaments.

Higher-order nonlinearities revisited and their effect on harmonic generation

We report on harmonic generation experiments and calculations in air to investigate the theoretical prediction of Kolesik et al. [Opt. Lett. 35, 2550 (2010)] for testing the recently proposed higher-order Kerr effect model. Our observations show that although the fifth-order nonlinearity is non-negligible, the overall defocusing effect via the higher-order nonlinearities is sufficiently small that plasma formation should be a main defocusing mechanism in high power filamentation. We also explore cross-phase modulation via the optical Kerr effect, and find that the higher-order nonlinearities can significantly alter the phase matching of harmonic generation.

Modulational instability windows in the nonlinear Schrödinger equation involving higher-order Kerr responses

We introduce a complete analytical and numerical study of the modulational instability process in a system governed by a canonical nonlinear Schrödinger equation involving local, arbitrary nonlinear responses to the applied field. In particular, our theory accounts for the recently proposed higher-order Kerr nonlinearities, providing very simple analytical criteria for the identification of multiple regimes of stability and instability of plane-wave solutions in such systems. Moreover, we discuss a new parametric regime in the higher-order Kerr response, which allows for the observation of several, alternating stability-instability windows defining a yet unexplored instability landscape.

Measurements of the third- and fifth-order optical nonlinearities of water at 532 and 1064 nm using the D4σ method

The nonlinear response of liquid water was investigated at 1064 and 532 nm using a Nd:YAG laser delivering pulses of 17 ps and its second harmonic. The experiments were performed using the D4σ method combined with the Z-scan technique. Nonlinear refractive indices of third- and fifth-order were determined, as well as the three-photon absorption coefficient, for both wavelengths. A good agreement was found between theory and experiment.

Spectral investigation of higher-order Kerr effects in a tight-focusing geometry

The role of the Higher-Order Kerr Effects (HOKE) in intensity clamping is experimentally investigated. We fail to observe any evidence of HOKE-based intensity clamping in a tight geometrical focusing experiment. We introduce a polarization-based technique that can distinguish between spectral components from the leading and trailing edges of the pulse. The results of this time-resolved measurement support the ionization theory of intensity clamping.

Combined action of the bound-electron nonlinearity and the tunnel-ionization current in low-order harmonic generation in noble gases

We study numerically low-order harmonic generation in noble gases pumped by intense femtosecond laser pulses in the tunneling ionization regime. We analyze the influence of the phase-mismatching on this process, caused by the generated plasma, and study in dependence on the pump intensity the origin of harmonic generation arising either from the bound-electron nonlinearity or the tunnel-ionization current. It is shown that in argon the optimum pump intensity of about 100 TW/cm² leads to the maximum efficiency, where the main contribution to low-order harmonics originates from the bound-electron third and fifth order susceptibilities, while for intensities higher than 300 TW/cm² the tunnel-ionization current plays the dominant role. Besides, we predict that VUV pulses at 133 nm can be generated with relatively high efficiency of about 1.5 × 10⁻³ by 400 nm pump pulses.

The role of stimulated Raman scattering in supercontinuum generation in bulk diamond

We report on experimental results of supercontinuum generation in bulk diamond. The spectrum of supercontinuum generated with 800 nm pump extends up to 600 nm towards short wavelengths. We present the numerical model explaining the phenomenon, in which the role of different nonlinear effects including stimulated Raman scattering is discussed. Unlike in other materials, in diamond the feature of supercontinuum due to stimulated Raman response is apparently visible.

Seventh-harmonic generation from tightly focused 2 μm ultrashort pulses in air

We report generation of third, fifth and seventh harmonics from air by using tightly focused, ultrashort pulses of short-wave infrared (2 μm) radiation. We have measured the third- and fifth-harmonic efficiencies to be 5×10(-5) and ~1.4×10(-5), respectively, with the ratio of fifth-to-third-harmonic efficiency being close to 0.28. Our experimental results provide confirmation of expectations of the higher-order Kerr effect model.

Filamentation of arbitrary polarized femtosecond laser pulses in case of high-order Kerr effect

We developed a model of femtosecond filamentation which includes high-order Kerr effect and an arbitrary polarization of a laser pulse. We show that a circularly polarized pulse has maximum filament intensity. Also, we show that, independently of the initial pulse polarization, the value of a maximum filament intensity tends to the maximum intensity of either linearly or circularly polarized pulse.

High-field quantum calculation reveals time-dependent negative Kerr contribution

The exact quantum time-dependent optical response of hydrogen under strong-field near-infrared excitation is investigated and compared to the perturbative model widely used for describing the effective atomic polarization induced by intense laser fields. By solving the full 3D time-dependent Schrödinger equation, we exhibit a supplementary, quasi-instantaneous defocusing contribution missing in the weak-field model of polarization. We show that this effect is far from being negligible, in particular when closures of ionization channels occur and stems from the interaction of electrons with their parent ions. It provides an interpretation of the higher-order Kerr effect recently observed in various gases.

Temporal switching induced by cascaded third order nonlinearity

We investigate the impact of cascaded third harmonic generation and the intrinsic n4 material nonlinearity on the propagation of ultrashort pulses in noble-gas filled Kagome fibers. We show that the pressure tunability of the cascade allows for the implementation of temporal switching. We also investigate the relative strengths of both effects and show their ratio to be pressure tunable.

Postionization medium evolution in a laser filament: A uniquely nonplasma response

Theoretical consideration of the optical response of nascent free electrons in the process of laser filamentation reveals that the initial microscopically inhomogeneous charge distribution causes a transient electromagnetic response of the medium that differs drastically from that of a homogeneous plasma with the same degree of ionization. An analytical model, describing the forced oscillations of virtually isolated and expanding electron clouds, predicts considerable enhancement of these oscillations caused by transient resonance with the laser field. The transient resonance processes should play a role in the currently accepted picture of filament formation dynamics.

Higher-order Kerr effect and harmonic cascading in gases

The higher-order Kerr effect (HOKE) has recently been advocated to explain measurements of the saturation of the nonlinear refractive index in gases. Here we show that cascaded third-harmonic generation results in an effective fifth-order nonlinearity that is negative and significant. Higher-order harmonic cascading will also occur from the HOKE, and the cascading contributions may significantly modify the observed nonlinear index change. At lower wavelengths, cascading increases the HOKE saturation intensity, while for longer wavelengths cascading will decrease the HOKE saturation intensity.

High field optical nonlinearity and the Kramers-Kronig relations

The nonlinear optical response to high fields is absolutely measured for the noble gas atoms He, Ne, Ar, Kr, and Xe. We find that the response is quadratic in the laser field magnitude up to the ionization threshold of each gas. Its size and quadratic dependence are well predicted by a Kramers-Kronig analysis employing known ionization probabilities, and the results are consistent with calculations using the time-dependent Schrödinger equation.

On the relative roles of higher-order nonlinearity and ionization in ultrafast light-matter interactions

Far off-resonant ultrafast and nonlinear light-matter interactions are studied using a one-dimensional atomic model. Results from a pump-probe diagnostic reveal that any higher-order nonlinear refraction is masked by ionization-induced defocusing before it becomes significant. On the other hand, we show that signatures of a higher-order nonlinearity may still be manifest via low-order harmonics of the pump center frequency. Implications for filamentation of femtosecond pulses are pointed out.

Saturation of the all-optical Kerr effect in solids

We discuss the influence of the higher-order Kerr effect (HOKE) in wide bandgap solids at extreme intensities below the onset of optically induced damage. Using different theoretical models, we employ multiphoton absorption rates to compute the nonlinear refractive index by a Kramers-Kronig transform. Within this theoretical framework we provide an estimate for the appearance of significant deviations from the standard optical Kerr effect predicting a linear index change with intensity. We discuss the role of the observed saturation behavior in practically relevant situations, including Kerr lens mode-locking and supercontinuum generation in photonic crystal fibers. Furthermore, we present experimental data from a multiwave mixing experiment in BaF2, which can be explained by the appearance of the HOKE.

Loss of phase of collapsing beams

We experimentally investigate the phase of an optical field after it has undergone wave collapse. We confirm the theoretical prediction that it acquires a large cumulative nonlinear phase shift that is highly sensitive to small fluctuations of the laser input power. This results in an effective postcollapse "loss of phase," whereby the phase of the transmitted beam shows a significant increase in sensitivity to the input fluctuations of the pulse energy. We also investigate interactions between two beams that each undergoes collapse and observe large fluctuations in the output mode profiles, which are due to the postcollapse loss of their relative phase difference. Such effects should occur in all systems that exhibit wave collapse.

Off-resonance and non-resonant dispersion of Kerr nonlinearity for symmetric molecules [Invited]

The exact formula is derived from the "sum over states" (SOS) quantum mechanical model for the frequency dispersion of the nonlinear refractive index coefficient n₂ for centrosymmetric molecules in the off-resonance and non-resonant regimes. This expression is characterized by interference between terms from two-photon transitions from the ground state to the even-symmetry excited states and one-photon transitions between the ground state and odd-symmetry excited states. When contributions from the two-photon terms exceed those from the one-photon terms, the non-resonant intensity-dependent refractive index n₂>0, and vice versa. Examples of the frequency dispersion for the three-level SOS model are given. Comparison is made with other existing theories.

Optical nonlinearity in Ar and N2 near the ionization threshold

We directly measure the nonlinear optical response in argon and nitrogen in a thin gas target to laser intensities near the ionization threshold. No instantaneous negative nonlinear refractive index is observed, nor is saturation, in contrast with a previous measurement [Opt. Express 17, 13429 (2009)] and calculations [Phys. Rev. Lett. 106, 183902 (2011)]. In addition, we are able to cleanly separate the instantaneous and rotational components of the nonlinear response in nitrogen. In both Ar and N2, the peak instantaneous index response scales linearly with the laser intensity until the point of ionization, whereupon the response turns abruptly negative and ∼constant, consistent with plasma generation.

Transition from plasma-driven to Kerr-driven laser filamentation

While filaments are generally interpreted as a dynamic balance between Kerr focusing and plasma defocusing, the role of the higher-order Kerr effect (HOKE) is actively debated as a potentially dominant defocusing contribution to filament stabilization. In a pump-probe experiment supported by numerical simulations, we demonstrate the transition between two distinct filamentation regimes at 800 nm. For long pulses (1.2 ps), the plasma substantially contributes to filamentation, while this contribution vanishes for short pulses (70 fs). These results confirm the occurrence, in adequate conditions, of filamentation driven by the HOKE rather than by plasma.