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In this paper we numerically study supercontinuum generation by pumping a silicon nitride waveguide, with two zero-dispersion wavelengths, with femtosecond pulses. The waveguide dispersion is designed so that the pump pulse is in the normal-dispersion regime. We show that because of self-phase modulation, the initial pulse broadens into the anomalous-dispersion regime, which is sandwiched between the two normal-dispersion regimes, and here a soliton is formed. The interaction of the soliton and the broadened pulse in the normal-dispersion regime causes additional spectral broadening through formation of dispersive waves by non-degenerate four-wave mixing and cross-phase modulation. This broadening occurs mainly towards the second normal-dispersion regime. We show that pumping in either normal-dispersion regime allows broadening towards the other normal-dispersion regime. This ability to steer the continuum extension towards the direction of the other normal-dispersion regime beyond the sandwiched anomalous-dispersion regime underlies the directional supercontinuum notation. We numerically confirm the approach in a standard silica microstructured fiber geometry with two zero-dispersion wavelengths.

A pure silica photonic crystal fiber with a group velocity dispersion ($\beta_2$) of 4 ps$^2$/km at 1.55 $\mu$m and less than 7 ps$^2$/km from 1.32 $\mu$m to the zero dispersion wavelength (ZDW) 1.80 $\mu$m was designed and fabricated. The dispersion of the fiber was measured experimentally and found to agree with the fiber design, which also provides low loss below 1.83 $\mu$m due to eight outer rings with increased hole diameter. The fiber was pumped with a 1.55 $\mu$m, 125 fs laser and, at the maximum in-coupled peak power (P$_0$) of 9 kW, a 1.34$-$1.82 $\mu$m low-noise spectrum with a relative intensity noise below 2.2\% was measured. The numerical modeling agreed very well with the experiments and showed that P$_0$ could be increased to 26 kW before noise from solitons above the ZDW started to influence the spectrum by pushing high-noise dispersive waves through the spectrum.

We investigate various methods for extending the simple analytical capillary model to describe the dispersion and loss of anti-resonant hollow-core fibers without the need of detailed finite-element simulations across the desired wavelength range. This poor-man's model can with a single fitting parameter quite accurately mimic dispersion and loss resonances and anti-resonances from full finite-element simulations. Due to the analytical basis of the model it is easy to explore variations in core size and cladding wall thickness, and should therefore provide a valuable tool for numerical simulations of the ultrafast nonlinear dynamics of gas-filled hollow-core fibers., Comment: In preparation

In this work, we numerically investigate an experimentally feasible design of a tapered Ne-filled hollow-core anti-resonant fiber and we report the generation of multiple dispersive waves (DWs) in the range 90-120 nm, well into the extreme ultraviolet (UV) region. The simulations assume an 800 nm pump pulse with 30 fs 10 $\mu$J pulse energy, launched into a 9 bar Ne-filled fiber with $34~\mu$m initial core diameter that is then tapered to a $10~\mu$m core diameter. The simulations were performed using a new model that provides a realistic description of both loss and dispersion of the resonant and anti-resonant spectral bands of the fiber, and also importantly includes the material loss of silica in the UV. We show that by first generating solitons that emit DWs in the far-UV region in the pre-taper section, optimization of the following taper structure can allow re-collision with the solitons and further up-conversion of the far-UV DWs to the extreme-UV with energies up to 190 nJ in the 90-120 nm range. This process provides a new way to generate light in the extreme-UV spectral range using relatively low gas pressure.

Using long-range surface plasmon polaritons light can propagate in metal nano-scale waveguides for ultracompact opto-electronic devices. Gold is an important material for plasmonic waveguides, but although its linear optical properties are fairly well understood, the nonlinear response is still under investigation. We consider propagation of pulses in ultrathin gold strip waveguides, modeled by the nonlinear Schr\"odinger equation. The nonlinear response of gold is accounted for by the two-temperature model, revealing it as a delayed nonlinearity intrinsic in gold. The consequence is that the measured nonlinearities are strongly dependent on pulse duration. This issue has so far only been addressed phenomenologically, but we provide an accurate estimate of the quantitative connection as well as a phenomenological theory to understand the enhanced nonlinear response as the gold thickness is reduced. In comparison with the previous works, the analytical model for the power-loss equation has been improved, and can be applied now to cases with a high laser peak power. We show new fits to experimental data from literature and provide updated values for the real and imaginary part of the nonlinear susceptibility of gold for various pulse durations and gold layer thicknesses. Our simulations show that the nonlinear loss is inhibiting efficient nonlinear interaction with low-power laser pulses. We therefore propose to design waveguides suitable for the mid-IR, where the ponderomotive instantaneous nonlinearity can dominate over the delayed hot-electron nonlinearity and provide a suitable plasmonics platform for efficient ultrafast nonlinear optics., Comment: J. Opt., in press

We investigate numerically soliton-plasma interaction in a noble-gas-filled silica hollow-core anti-resonant fiber pumped in the mid-IR at 3.0 {\mu}m. We observe multiple soliton self-compression stages due to distinct stages where either the self-focusing or the self-defocusing nonlinearity dominates. Specifically, the parameters may be tuned so the competing plasma self-defocusing nonlinearity only dominates over the Kerr self-focusing nonlinearity around the soliton self-compression stage, where the increasing peak intensity on the leading pulse edge initiates a competing self-defocusing plasma nonlinearity acting nonlocally on the trailing edge, effectively preventing soliton-formation there. As the plasma switches off after the self-compression stage, self-focusing dominates again, initiating another soliton self-compression stage in the trailing edge. This process is accompanied by supercontinuum generation spanning 1-4 {\mu}m. The technique could be exploited to generate an ultrafast sequence of several few-cycle pulses., Comment: submitted

We experimentally show octave-spanning supercontinuum generation in a non-stoichiometric silicon-rich nitride waveguide when pumped by femtosecond pulses from an erbium fiber laser. The pulse energy and bandwidth are comparable to results achieved in stoichiometric silicon nitride waveguides, but our material platform is simpler to manufacture. We also observe wave-breaking supercontinuum generation by using orthogonal pumping in the same waveguide. Additional analysis reveals that the waveguide height is a powerful tuning parameter for generating mid-infrared dispersive waves while keeping the pump in the telecom band., Comment: Accepted for OL, https://www.osapublishing.org/ol/upcoming_pdf.cfm?id=263052

Bright and broadband coherent mid-IR radiation is important for exciting and probing molecular vibrations. Using cascaded nonlinearities in conventional quadratic nonlinear crystal like lithium niobate, self-defocusing near-IR solitons have been demonstrated that led to very broadband supercontinuum generation in the visible, near-IR and short-wavelength mid-IR. Here we conduct an experiment where a mid-IR crystal pumped in the mid-IR gives multiple-octave spanning supercontinua. The crystal is cut for noncritical interaction, so the three-wave mixing of a single mid-IR femtosecond pump source leads to highly phase-mismatched second-harmonic generation. This self-acting cascaded process leads to the formation of a self-defocusing soliton at the mid-IR pump wavelength and after the self-compression point multiple octave-spanning supercontinua are observed (covering 1.6-$7.0~\mu$m). The results were recorded in a commercially available crystal LiInS$_2$ pumped in the 3-$4~\mu$m range, but other mid-IR crystals can readily be used as well., Comment: submitted to APL Photonics

We here derive the so-called Nonlinear Analytical Envelope Equation (NAEE) inspired by the work of Conforti et al. [M. Conforti, A. Marini, T. X. Tran, D. Faccio, and F. Biancalana, "Interaction between optical fields and their conjugates in nonlinear media," Opt. Express 21, 31239-31252 (2013)], whose notation we follow. We present a complete model that includes $\chi^{(2)}$ terms [M. Conforti, F. Baronio, and C. De Angelis, "Nonlinear envelope equation for broadband optical pulses in quadratic media," Phys. Rev. A 81, 053841 (2010)], $\chi^{(3)}$ terms, and then extend the model to delayed Raman effects in the $\chi^{(3)}$ term. We therefore get a complete model for ultrafast pulse propagation in quadratic nonlinear crystals similar to the Nonlinear Wave Equation in Frequency domain [H. Guo, X. Zeng, B. Zhou, and M. Bache, "Nonlinear wave equation in frequency domain: accurate modeling of ultrafast interaction in anisotropic nonlinear media," J. Opt. Soc. Am. B 30, 494-504 (2013)], but where the envelope is modelled rather than the electrical field while still keeping a sub-carrier level resolution. The advantage of the envelope formation is that the physical origin of the additional terms that are included to model the physics at the carrier level becomes more clear, in contrast to the electric field equations that are more "black box" expansions of the electrical field. We also point out that by comparing our results to a very similar model and widely used model [G. Genty, P. Kinsler, B. Kibler, and J. M. Dudley, "Nonlinear envelope equation modeling of sub-cycle dynamics and harmonic generation in nonlinear waveguides," Opt. Express 15, 5382-5387 (2007)], the Raman terms presented there will most likely lead to an artificially lower Raman effect.

We experimentally observe long-wavelength dispersive waves generation in a BBO crystal. A soliton was formed in normal GVD regime of the crystal by a self-defocusing and negative nonlinearity through phase-mismatched quatradic interaction. Strong temporal pulse compression confirmed the formation of soliton during the pulse propagation inside the crystal. Significant dispersive wave radiation was measured in the anomalous GVD regime of the BBO crystal. With the pump wavelengths from 1.24 to 1.4 $\mu$m, tunable dispersive waves are generated around 1.9 to 2.2 $\mu$m. The observed dispersive wave generation is well understood by simulations., Comment: in preparation

We show numerically that ultrashort self-defocusing temporal solitons colliding with a weak pulsed probe in the near-IR can convert the probe to the mid-IR. A near-perfect conversion efficiency is possible for a high effective soliton order. The near-IR self-defocusing soliton can form in a quadratic nonlinear crystal (beta-barium borate) in the normal dispersion regime due to cascaded (phase-mismatched) second-harmonic generation, and the mid-IR converted wave is formed in the anomalous dispersion regime between $\lambda=2.2-2.4~\mu\rm m$ as a resonant dispersive wave. This process relies on non-degenerate four-wave mixing mediated by an effective negative cross-phase modulation term caused by cascaded soliton-probe sum-frequency generation., Comment: Longer version with appendix

Efficient supercontinuum generation (SCG) requires excitation of solitons at the pump laser wavelength. Quadratic nonlinear waveguides may support an effective self-defocusing nonlinearity so solitons can directly be generated at common ultrafast laser wavelengths without any waveguide dispersion engineering. We here experimentally demonstrate efficient SCG in a standard lithium niobate (LN) waveguide without using quasi-phase matching (QPM). By using femtosecond pumps with wavelengths in the $1.25-1.5 \mu\rm m$ range, where LN has normal dispersion and thus supports self-defocusing solitons, octave-spanning SCG is observed. An optimized mid-IR waveguide design is expected to support even broader spectra. The QPM-free design reduces production complexity, allows longer waveguides, limits undesired spectral resonances and effectively allows using nonlinear crystals where QPM is inefficient or impossible. This result is important for mid-IR SCG, where QPM-free self-defocusing waveguides in common mid-IR nonlinear crystals can support solitons directly at mid-IR ultrafast laser wavelengths, where these waveguides have normal dispersion., Comment: To appear in Optics Letters

Experimental data of femtosecond thick-crystal second-harmonic generation shows that when tuning away from phase matching, a dominating narrow spectral peak appears in the second harmonic that can be tuned over 100's of nm by changing the phase-mismatch parameter. Traditional theory explains this as phase matching between a sideband in the broadband pump to its second-harmonic. However, our experiment is conducted under high input intensities and instead shows excellent quantitative agreement with a nonlocal theory describing cascaded quadratic nonlinearities. This theory explains the detuned peak as a nonlocal resonance that arises due to phase-matching between the pump and a detuned second-harmonic frequency, but where in contrast to the traditional theory the pump is assumed dispersion-free. As a soliton is inherently dispersion-free, the agreement between our experiment and the nonlocal theory indirectly proves that we have observed a soliton-induced nonlocal resonance. The soliton exists in the self-defocusing regime of the cascaded nonlinear interaction and in the normal dispersion regime of the crystal, and needs high input intensities to become excited., Comment: submitted, revised version

Concept of spectral soliton coupling is proposed to explain the group-velocity-matching dispersive wave generation and the soliton spectral tunneling effect. Soliton eigen state is found corresponding to a spectral phase profile under which the soliton phase shifts induced by the dispersion and the nonlinearity are instantaneously counterbalanced. A local but non-eigen soliton pulse will shed off energy and form radiations at other wavelengths, during the coupling to the soliton eigen state. Group-velocity-mathing between the local and generated wave is necessary not only as it produces a coupler-like phase profile which supports soliton coupling but also because it physically supports a soliton phase matching condition, which means the generated wave has a soliton state phase matched to the local exciting pulse. Examples in realistic photonic crystal fiber structures are also presented.

We interpret the forward Maxwell equation with up to third order induced polarizations and get so called nonlinear wave equation in frequency domain (NWEF), which is based on Maxwell wave equation and using slowly varying spectral amplitude approximation. The NWEF is generalized in concept as it directly describes the electric field dynamics rather than the envelope dynamics and because it concludes most current-interested nonlinear processes such as three-wave mixing, four-wave-mixing and material Raman effects. We give two sets of NWEF, one is a 1+1D equation describing the (approximated) planar wave propagation in nonlinear bulk material and the other corresponds to the propagation in a waveguide structure., Comment: Equation Derivations

We review measurements in the literature regarding Kerr nonlinearity of lithium niobate. Of particular interest is to separate the value of the electronic Kerr nonlinearity and the strength of the delayed Raman nonlinearity. The spectral shape of the Raman response in LN is discussed and we restrict ourselves to the case where the pump is polarized along the optical axis. This configuration is important for so-called type 0 noncritical interaction, which is used in QPM devices and in recent supercontinuum generation and few-cycle soliton compression experiments., Comment: in progress, comments welcome

We propose an efficient approach to improve few-cycle soliton compression with cascaded quadratic nonlinearities by using an engineered multi-section structure of the nonlinear crystal. By exploiting engineering of the cascaded quadratic nonlinearities, in each section soliton compression with a low effective order is realized, and high-quality few-cycle pulses with large compression factors are feasible. Each subsequent section is designed so that the compressed pulse exiting the previous section experiences an overall effective self-defocusing cubic nonlinearity corresponding to a modest soliton order, which is kept larger than unity to ensure further compression. This is done by increasing the cascaded quadratic nonlinearity in the new section with an engineered reduced residual phase mismatch. The low soliton orders in each section ensure excellent pulse quality and high efficiency. Numerical results show that compressed pulses with less than three-cycle duration can be achieved even when the compression factor is very large, and in contrast to standard soliton compression, these compressed pulses have minimal pedestal and high quality factor.

We interpret the purely spectral forward Maxwell equation with up to 3${^{\rm rd}}$ order induced polarizations for pulse propagation and interactions in quadratic nonlinear crystals. The interpreted equation, also named nonlinear wave equation in frequency domain, includes both quadratic and cubic nonlinearities, delayed Raman effects and anisotropic nonlinearities. The full potential of this wave equation is demonstrated by investigating simulations of solitons generated in the process of ultrafast cascaded second-harmonic generation. We show that a balance in the soliton delay can be achieved due to competition between self-steepening, Raman effects and self-steepening-like effects from cascading originating in the group-velocity mismatch between the pump and second harmonic. We analyze the first-order contributions, and show that this balance can be broken to create fast or slow pulses. Through further simulations we demonstrate few-cycle compressed solitons in extremely short crystals, where spectral phenomena such as blue/red shifting, non-stationary radiation in accordance with the non-local phase matching condition and dispersive-wave generation are observed and marked, which help improving the experimental knowledge of cascading nonlinear soliton pulse compression.

We study the anisotropic nature of the Kerr nonlinear response in a beta-barium borate (\beta-BaB2O4, BBO) nonlinear crystal. The focus is on determining the relevant $\chi^{(3)}$ cubic tensor components that affect interaction of type I cascaded second-harmonic generation. Various experiments in the literature are analyzed and we correct the data from some of the experiments for contributions from cascading as well as for updated material parameters. We find that the Kerr nonlinear tensor component responsible for self-phase modulation in cascading is considerably larger than what has been used to date. We evaluate the impact of using such a cubic anisotropic response in ultrafast cascading experiments., Comment: Updated version, comments on experiments from the literature welcome

The higher-order Kerr effect (HOKE) has been recently 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., Comment: Optics Letters, in press

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., Comment: 14 pages, 10 figures, 1 table, submitted to PRA

We consider an index-guiding silica photonic crystal fiber with a triangular air-hole structure and a poled quadratic nonlinearity. By tuning the pitch and the relative hole size, second-harmonic generation with zero group-velocity mismatch is found for any fundamental wavelength above 780 nm. The phase-velocity mismatch has a lower limit with coherence lengths in the micron range. The dimensionless nonlinear parameter is inversely proportional to the pitch and proportional to the relative hole size. Selected cases show bandwidths suitable for 20 fs pulse-conversion with conversion efficiencies as high as 25%/mW., Comment: 3 pages, 5 figures, submitted to Optics Letters

High-resolution ghost image and ghost diffraction experiments are performed by using a single source of thermal-like speckle light divided by a beam splitter. Passing from the image to the diffraction result solely relies on changing the optical setup in the reference arm, while leaving untouched the object arm. The product of spatial resolutions of the ghost image and ghost diffraction experiments is shown to overcome a limit which was formerly thought to be achievable only with entangled photons., Comment: 5 pages, 4 figures

We analytically show that it is possible to perform coherent imaging by using the classical correlation of two beams obtained by splitting incoherent thermal radiation. The case of such two classically correlated beams is treated in parallel with the configuration based on two entangled beams produced by parametric down-conversion, and a basic analogy is pointed out. The results are compared in a specific numerical example., Comment: 6 pages, 3 figures