Nonlinear chirped interferometry for frequency-shift measurement and χ(3) spectroscopy

I. INTRODUCTION

Section:

ChooseTop of pageABSTRACTI. INTRODUCTION <<II. PRINCIPLEIII. EXPERIMENTAL METHODSIV. RESULTSV. CONCLUSIONSUPPLEMENTARY MATERIALREFERENCESPrevious sectionNext sectionThe third-order susceptibility tensor χ(3) governs both resonant and non-resonant nonlinear processes involving four-wave mixing.11. R. L. Sutherland, Handbook of Nonlinear Optics (Marcel Dekker, 2003). The real part of the diagonal terms, responsible for the well-known optical Kerr-effect (OKE), plays a fundamental role in ultrafast optics where frequency-degenerated nonlinear processes are routinely used to modify, control, or characterize the spatial, spectral, and temporal properties of intense pulses. The non-diagonal terms of the tensor χ(3) are also exploited in nonlinear processes such as cross-polarized wave generation22. A. Jullien, O. Albert, F. Burgy, G. Hamoniaux, J.-P. Rousseau, J.-P. Chambaret, F. Augé-Rochereau, G. Chériaux, J. Etchepare, N. Minkovski, and S. M. Saltiel, “10−10 temporal contrast for femtosecond ultraintense lasers by cross-polarized wave generation,” Opt. Lett. 30, 920–922 (2005). https://doi.org/10.1364/ol.30.000920 and are ubiquitous when intense waves of different polarizations are mixed in nonlinear media. This is, in particular, the case of most optical parametric amplifiers, which have become the backbone of third-generation femtosecond sources.33. H. Fattahi, H. G. Barros, M. Gorjan, T. Nubbemeyer, B. Alsaif, C. Y. Teisset, M. Schultze, S. Prinz, M. Haefner, M. Ueffing et al., “Third-generation femtosecond technology,” Optica 1, 45–63 (2014). https://doi.org/10.1364/optica.1.000045 It is thus essential to determine the coefficients of the χ(3) tensor of nonlinear media ranging from common optical glasses to exotic nonlinear birefringent crystals.Experimental methods suitable to measure the real part of the χ(3) tensor are numerous11. R. L. Sutherland, Handbook of Nonlinear Optics (Marcel Dekker, 2003). and suffice it to say that OKE spectroscopy gathers time-resolved pump–probe techniques tracking the changes (polarization, frequency, spatial phase, or temporal phase) of a weak probe pulse under the effect of an intense pump pulse in the medium of interest, either with4–84. R. DeSalvo, A. A. Said, D. J. Hagan, E. W. Van Stryland, and M. Sheik-Bahae, “Infrared to ultraviolet measurements of two-photon absorption and n2 in wide bandgap solids,” IEEE J. Quantum Electron. 32, 1324 (1996). https://doi.org/10.1109/3.5115455. C. Y. Chien, B. La Fontaine, A. Desparois, Z. Jiang, T. W. Johnston, J. C. Kieffer, H. Pépin, F. Vidal, and H. P. Mercure, “Single-shot chirped-pulse spectral interferometry used to measure the femtosecond ionization dynamics of air,” Opt. Lett. 25, 578–580 (2000). https://doi.org/10.1364/ol.25.0005786. Y.-H. Chen, S. Varma, I. Alexeev, and H. Milchberg, “Measurement of transient nonlinear refractive index in gases using xenon supercontinuum single-shot spectral interferometry,” Opt. Express 15, 7458–7467 (2007). https://doi.org/10.1364/oe.15.0074587. Á. Börzsönyi, Z. Heiner, A. P. Kovács, M. P. Kalashnikov, and K. Osvay, “Measurement of pressure dependent nonlinear refractive index of inert gases,” Opt. Express 18, 25847–25854 (2010). https://doi.org/10.1364/oe.18.0258478. J. Burgin, C. Guillon, and P. Langot, “Femtosecond investigation of the non-instantaneous third-order nonlinear suceptibility in liquids and glasses,” Appl. Phys. Lett. 87, 211916 (2005). https://doi.org/10.1063/1.2136413 or without9,109. S. P. Le Blanc, E. W. Gaul, N. H. Matlis, A. Rundquist, and M. C. Downer, “Single-shot measurement of temporal phase shifts by frequency-domain holography,” Opt. Lett. 25, 764–766 (2000). https://doi.org/10.1364/ol.25.00076410. R. Thurston, M. M. Brister, A. Belkacem, T. Weber, N. Shivaram, and D. S. Slaughter, “Time-resolved ultrafast transient polarization spectroscopy to investigate nonlinear processes and dynamics in electronically excited molecules on the femtosecond time scale,” Rev. Sci. Instrum. 91, 053101 (2020). https://doi.org/10.1063/1.5144482 interferometric detection. Another related approach, two-beam coupling (2BC), relies on the mutual interaction between two noncollinear beams crossing in the medium.11–1511. I. Kang, T. Krauss, and F. Wise, “Sensitive measurement of nonlinear refraction and two-photon absorption by spectrally resolved two-beam coupling,” Opt. Lett. 22, 1077–1079 (1997). https://doi.org/10.1364/ol.22.00107712. S. Smolorz and F. Wise, “Femtosecond two-beam coupling energy transfer from Raman and electronic nonlinearities,” J. Opt. Soc. Am. B 17, 1636–1644 (2000). https://doi.org/10.1364/josab.17.00163613. S. Smolorz, F. Wise, and N. F. Borrelli, “Measurement of the nonlinear optical response of optical fiber materials by use of spectrally resolved two-beam coupling,” Opt. Lett. 24, 1103 (1999). https://doi.org/10.1364/ol.24.00110314. J. K. Wahlstrand, J. H. Odhner, E. T. McCole, Y.-H. Cheng, J. P. Palestro, R. J. Levis, and H. M. Milchberg, “Effect of two-beam coupling in strong-field optical pump-probe experiments,” Phys. Rev. A 87, 053801 (2013). https://doi.org/10.1103/physreva.87.05380115. G. N. Patwardhan, J. S. Ginsberg, C. Y. Chen, M. M. Jadidi, and A. L. Gaeta, “Nonlinear refractive index of solids in mid-infrared,” Opt. Lett. 46, 1824–1827 (2021). https://doi.org/10.1364/ol.421469 Chirped spectral holography has been applied to Raman spectroscopy and involves strongly chirped probe pulse, so as to create a ps-scale temporal window and measure both instantaneous and delayed phase-shifts in a single acquisition.16–1816. J. W. Wilson, P. Schlup, and R. Bartels, “Phase measurement of coherent Raman vibrational spectroscopy with chirped spectral holography,” Opt. Lett. 33, 2116–2118 (2008). https://doi.org/10.1364/ol.33.00211617. K. Hartinger and R. A. Bartels, “Single-shot measurement of ultrafast time-varying phase modulation induced by femtosecond laser pulses with arbitrary polarization,” Appl. Phys. Lett. 92, 021126 (2008). https://doi.org/10.1063/1.280151518. P. Schlup, J. W. Wilson, and R. A. Bartels, “Sensitive and selective detection of low-frequency vibrational modes through a phase-shifting fourier transform spectroscopy,” IEEE J. Quantum Electron. 45, 777 (2009). https://doi.org/10.1109/jqe.2009.2013121 All these methods have their own advantages and drawbacks, but most of them require noise reduction to isolate the contribution of weak nonlinearities (averaging and/or modulation with heterodyne detection).

In this paper, we report a novel time-resolved ultrafast transient spectroscopy method to characterize third-order nonlinearity on the femtosecond time-scale. We coin this method as “nonlinear chirped interferometry.”

The approach consists of measuring the variations of the first derivative of the nonlinear temporal phase, i.e., the optical group delay, rather than the phase changes, induced on a probe pulse by an intense pulse in a non-linear medium. We demonstrate that, under certain small chirp conditions, two distinct physical effects, spectral and temporal shifts, add up and that monitoring the optical group delay, via spectral interferometry, gives access to the nonlinear phase value and, therefore, to the nonlinear tensor terms of the medium of interest. We show, both theoretically and experimentally, that the method is(i)

interferometric, but by design, less sensitive to environmental phase fluctuations and drifts: no active stabilization or shielding of the interferometer is required, even for meter-scale interferometer,

(ii)

sensitive: 10 mrad nonlinear phase-shifts can be detected without heterodyne detection,

(iii)

selective: self-focusing although visible on the experimental data does not affect the measurement,

(iv)

polarization sensitive: non-diagonal terms of the χ(3) tensor can be independently measured, and

(v)

temporally-resolved: both instantaneous and delayed nonlinear processes can be investigated.

Features (iv) and (v) are common to non-collinear pump–probe spectroscopy setups. The paper is organized as follows. The theoretical line-out of nonlinear chirped interferometry is described in Sec. . The experimental setup and nonlinear materials are then described in Sec. . Section  presents measurement results: validation of the technique with known isotropic and χ(3)-anisotropic materials, followed by soft vibration mode measurement in an anisotropic crystal (KTA).

II. PRINCIPLE

Section:

ChooseTop of pageABSTRACTI. INTRODUCTIONII. PRINCIPLE <<III. EXPERIMENTAL METHODSIV. RESULTSV. CONCLUSIONSUPPLEMENTARY MATERIALREFERENCESPrevious sectionNext section

The setup is essentially a frequency-degenerated pump–probe experiment with an interferometric detection. As justified below, the probe optical group delay as a function of pump–probe delay is monitored, rather than the phase.

An illustration is available in Fig. 1(a). A strong pump beam (P) and a weak probe beam (Pr), of identical carrier frequency ω0, are weakly focused and cross in a thin optical sample. A small angle (≃1) is introduced between the two beams to allow a spatial separation before/after the sample and a delay line controls the relative group delay between the two pulses (τPPr). Through cross-phase modulation (XPM), the probe pulse undergoes a transient nonlinear phase shift φNL(t). The temporal dependence of φNL causes a shift of the probe’s instantaneous carrier frequency, ±Ω [Fig. 1(b)]: a down-shift (shift toward the “red”) on the rising edge of the pump pulse and an up-shift (shift toward the “blue”) on the trailing edge of the pump pulse (for a medium characterized by a positive nonlinear index n2 > 0).1919. R. W. Boyd, Nonlinear Optics (Academic Press, 1992). Ω is proportional to the nonlinear phase and, thus, to the involved χ(3) term.1919. R. W. Boyd, Nonlinear Optics (Academic Press, 1992). In nonlinear spectroscopy, φNL can be extracted from the transient variation of Ω, although this frequency shift is usually too small to be measured with precision2020. J.-F. Ripoche, G. Grillon, B. Prade, M. Franco, E. Nibbering, R. Lange, and A. Mysyrowicz, “Determination of the time dependence of n2 in air,” Opt. Commun. 135, 310 (1997). https://doi.org/10.1016/s0030-4018(96)00675-x and/or might be entangled with spectral distortions induced by other effects such as self-phase modulation. We assume in the following that this frequency shift, noted Ω = Ω(τPPr), is small compared to the optical bandwidth Δω such that |Ω| ≪ Δω.Let EP(ω), ER(ω), and EPr(ω) be the complex spectral amplitudes of, respectively, the pump, reference, and probe pulses. Their respective spectral phases are φP(ω), φR(ω), and φPr(ω) with the convention Ek(ω)=|Ek(ω)|exp−iφk(ω), where k is the wave label. Spectral phases are hereafter assumed as purely quadratic, with chirp coefficients, respectively, labeled φk(2). The spectral phases thus areφk(ω)=φk(ω0)+τk(ω0)(ω−ω0)+φk(2)(ω−ω0)2/2,(1)where τk = τk(ω0) stands for the group delay.The coupling between the pump and probe beams is treated elsewhere and can be described by the following propagation equation for the probe field1414. J. K. Wahlstrand, J. H. Odhner, E. T. McCole, Y.-H. Cheng, J. P. Palestro, R. J. Levis, and H. M. Milchberg, “Effect of two-beam coupling in strong-field optical pump-probe experiments,” Phys. Rev. A 87, 053801 (2013). https://doi.org/10.1103/physreva.87.053801 in the limit of a purely electronic nonlinearity,c∂EPr∂z+ng,0+4γIP∂EPr∂t=2iω0γIPEPr−4γ∂IP∂tEPr,(2)where IP is the time-dependent pump intensity, ng,0 is the group index at ω0, and γ is the nonlinear coupling coefficient. This expression holds as long as the input polarization states of the pump and probe beams are either parallel or perpendicular with respect to each other. The general expression of γ is rather complex and depends on the polarization states of the pump and probe pulses as well as on the crystallographic orientation and symmetry of the sample. For an isotropic medium, far away from any resonance, the expressions of γ in SI units, for, respectively, parallel and perpendicular polarizations, areγ‖=34ε0n02cχxxxx(3)=n2,(3)γ⊥=14ε0n02cχyyyy(3).(4)The propagation Eq. (2) assumes slowly varying envelopes and neglects temporal dispersion, which is compatible with the thin medium assumption and/or narrowband pulses, as considered here. The effect of the pump field is three-fold: the group index is increased by 4γIP [the left member of Eq. (2)], and two nonlinear source terms contribute to the propagation [the right member of Eq. (2)]. The first source term corresponds to XPM and is responsible for the frequency shift Ω, while the second term induces gain and loss via an energy transfer between the two beams (2BC). The interplay between these three effects is rather complex but, to simplify, the change in wave velocity can be neglected while the XPM and 2BC may significantly reshape the transmitted probe pulse in both the spectral and time domains. As a general result, the optical group delay of the transmitted probe pulse τPr is altered when the pump and probe pulses overlap. For the sake of clarity, the probe group delays with and without the pump beam are, respectively, noted as τPr and τPr0.As τPr−τPr0 cannot be measured directly, the transmitted probe pulse is recombined with a reference pulse (in this case, a replica selected upstream) and the relative group delay τRPr = τR − τPr between the reference and probe is measured instead. We note τRPr0=τR−τPr0 so that τPr−τPr0=τRPr−τRPr0. The relative delay τRPr0, which is kept constant during the experiment, is chosen adequately so as to be able to resolve the phase difference between the probe and reference pulses by spectral interferometry: Δω≫1/|τRPr0|>δωsp, where δωsp is the spectral resolution of the spectrometer. The spectral interference pattern between the transmitted probe and the reference isS(ω)=|ER(ω)|2+|EPr(ω)|2+2ReER(ω)EPr*(ω)eiωτRPr.(5)S(ω) contains one non-oscillating term (DC term) and two conjugate oscillating terms (AC terms), the Fourier-transform of which isŜAC(t−τ0)=∫ER(ω)EPr*(ω)eiωtdω.(6)If EPr(ω) = ER(ω) (i.e., in the case without the pump wave), then ŜAC(t−τRPr0) is equal to the Fourier-transform of |ER(ω)|2, and the AC terms are centered at t=±τRPr0 and well separated from the DC term at t = 0. To anticipate on the following paragraphs, four-wave mixing introduces additional contributions to the optical group delay of the probe pulse and tends to shift the location of the AC terms from ±τRPr0 to ±τRPr (our observable). As shown below, this definition actually aggregates two distinct physical effects.A numerical resolution of Eq. (2), described in the supplementary material, is proposed for a pump pulse duration of 180 fs FWHM, IP = 300 GW/cm2, and a crystal of 1 mm length characterized by a nonlinear index γ = n2 = 2.8 10−16 cm2/W. The corresponding nonlinear XPM phase is ≃300 mrad. Pump and probe pulses are equally chirped, with a chirp coefficient of either φP(2) = φPr(2) = 0 fs2 or φP(2) = φPr(2) = +5000 fs 2. The latter corresponds to a minor change in the pump pulse duration (195 fs). The reference pulse has a chirp coefficient of φR(2) = φPr(2) + ΔφRPr(2), with either ΔφRPr(2) = 0 fs2 or ΔφRPr(2) = +2000 fs2. The initial group delay between the probe and the reference pulses is τPr0 = 3 ps. As our model is unidirectional, spatial effects such as Kerr lens and self-diffraction are not simulated.We first consider the case φP(2)=φR(2)=φPr(2) = 0 fs 2, i.e., all involved pulses are limited by Fourier transform. The spectral interferogram as a function of pump–probe delay, τPPr=τP−τPr0, is plotted in Fig. 1(c). The transient frequency shift Ω appears for ΔΩτPPr ≲ 1, when the pump and probe pulses temporally overlap. For each pump–probe delay, the discrete Fourier transform of the spectral interferogram is computed. The relative group delay between the reference and probe pulses (τRPr) is then the center of mass of the AC peak |ŜAC(t−τRPr0)|, retrieved by fitting the peak with a Gaussian function. As shown in Fig. 1(c), τRPr−τRPr0 varies with the pump–probe delay τPPr. When none of the three pulses are chirped, weak variations are observed, indicating that, to the first order, the optical group delay of the transmitted probe pulse is constant despite the spectral/temporal reshaping effects.We then consider φP(2)=φR(2)=φPr(2) = +5000 fs 2, i.e., the three pulses are equally chirped. Because of this chirp, the instantaneous frequencies of the pump and probe pulses are detuned with respect to each other when τPPr ≠ 0. 2BC induces energy flows from one wave to the other during the nonlinear interaction. If the chirp coefficient is positive, the probe will gain energy for negative pump–probe delays and vice versa. As this energy transfer also scales with the pump intensity, the general effect is a reshaping of the temporal profile, which is indistinguishable from an additional optical group delay [Fig. 1(d)]. For negative pump–probe delays (resp. positive), the rear edge (resp. the leading edge) of the probe is strengthened, resulting in an overall increase (resp. decrease) of the group delay. As a result, τRPr exhibits a Z-shape, similar to the transient probe transmission reported for two-beam coupling.11–13,1511. I. Kang, T. Krauss, and F. Wise, “Sensitive measurement of nonlinear refraction and two-photon absorption by spectrally resolved two-beam coupling,” Opt. Lett. 22, 1077–1079 (1997). https://doi.org/10.1364/ol.22.00107712. S. Smolorz and F. Wise, “Femtosecond two-beam coupling energy transfer from Raman and electronic nonlinearities,” J. Opt. Soc. Am. B 17, 1636–1644 (2000). https://doi.org/10.1364/josab.17.00163613. S. Smolorz, F. Wise, and N. F. Borrelli, “Measurement of the nonlinear optical response of optical fiber materials by use of spectrally resolved two-beam coupling,” Opt. Lett. 24, 1103 (1999). https://doi.org/10.1364/ol.24.00110315. G. N. Patwardhan, J. S. Ginsberg, C. Y. Chen, M. M. Jadidi, and A. L. Gaeta, “Nonlinear refractive index of solids in mid-infrared,” Opt. Lett. 46, 1824–1827 (2021). https://doi.org/10.1364/ol.421469We now consider the case of unchirped pump and probe pulses φP(2)=φPr(2) = 0 fs 2 with a positively chirped reference pulse φR(2) = +2000 fs 2 [Fig. 1(e)]. The reference pulse being chirped, the frequency-shift Ω is temporally encoded in the interferogram and (also) appears as a delay τRPr. As plotted in Fig. 1(e), this effect produces a similar Z-shape behaviour, although less pronounced than in the former case—but scaling linearly with ΔφRPr(2)=φR(2)−φPr(2). To illustrate the principle of spectral encoding, we present in Fig. 1 the Wigner–Ville distributions of the transmitted probe and of the delayed and chirped reference pulse. The relative chirp ΔφRPr(2) makes the time–frequency slope different for each pulse, which is sufficient to encode a spectral shift as a temporal shift. As provided in the supplementary material, the linear relationship between the nonlinear frequency shift (Ω) and the relative chirp between the probe and reference (ΔφRPr(2)) can be retrieved analytically from Eq. (6). We emphasize the fact that if ΔφRPr(2)=0, this encoding cannot occur according to our model, in the small chirp configuration considered here.We have evidenced here two phenomena: temporal reshaping (due to 2BC and triggered by φP(2)=φPr(2)≠0) and temporal encoding of the nonlinear spectral shift (triggered by ΔφRPr(2)≠0). Although the differences between the spectrograms in Fig. 1 are not visible to the naked eye, the Fourier analysis shows that both mechanisms result in similar transient shifts of τRPr, over the same temporal scale (the correlation width of the pump pulse), and with similar amplitudes (a few fs). With the right chirp parameters, these two contributions may add up and increase significantly the global signal-to-noise ratio of the measurement, as shown in Fig. 1(f). The delay swing δτ = max(τRPr) − min(τRPr) is then our metric that evolves linearly with the nonlinear phase amount, as plotted in Fig. 1(g). As will be demonstrated experimentally in Sec. , measuring τRPr instead of phase changes not only makes the detection less sensitive to phase fluctuations but also gives additional means to enhance the sensitivity and specificity [Fig. 1(g)] of the detection, without resorting to heterodyne detection.

III. EXPERIMENTAL METHODS

Section:

ChooseTop of pageABSTRACTI. INTRODUCTIONII. PRINCIPLEIII. EXPERIMENTAL METHODS <<IV. RESULTSV. CONCLUSIONSUPPLEMENTARY MATERIALREFERENCESPrevious sectionNext sectionIn the present experiments, a Pharos laser system (PH1-SP-1mJ, Light Conversion) delivers 180 fs FTL pulses, with a central wavelength of 1034 nm, a repetition rate of 10 kHz, and pulse energy up to 500 μJ. The laser chirp can be tuned by adjusting the compressor. Each pulse is split into three separated pulses: an excitation pump pulse, a probe pulse (≃20% of the pump energy), and a reference pulse selected before the nonlinear stage (Fig. 2). The pump and probe pulses are focused (f = 1.5 m) and overlap in the focal plane under a small angle (τPPr) is controlled with a delay stage equipped by a motorized actuator with K-Cube controller (models Z825B and KDC101, Thorlabs). After the interaction, the probe is selected and recombined with the reference pulse. The nominal group delay between the reference and the probe pulses (τRPr0) is set to 3.5 ± 0.15 ps for all measurements. The interferometer extends over about 1–2 m. As shown in Sec. , the chirp of all involved pulses has to be controlled independently. The laser compressor tunes the pump, probe, and reference chirp. Furthermore, the reference pulse can experience an additional chirp through the addition of various bulk plates: SF11 (20 mm, 2500 fs2), CaCO3(10 mm, 430 fs2),2121. G. Ghosh, “Dispersion-equation coefficients for the refractive index and birefringence of calcite and quartz crystals,” Opt. Commun. 163, 95–102 (1999). https://doi.org/10.1016/s0030-4018(99)00091-7 and Al2O3 (5 mm, 156 fs2).2222. I. H. Malitson and M. J. Dodge, “Refractive index and birefringence of synthetic sapphire,” J. Opt. Soc. Am. 62, 1405 (1972). https://doi.org/10.1364/JOSA.52.001377 The resulting interference pattern is collected by a spectrometer (Avantes, spectral resolution 0.07 nm) for each step of the optical delay stage in the pump arm. The polarization and energy of each pulse are controlled by half-wave plates and thin-film polarizers (TFP). The different components of the χ(3) tensor can then be measured by changing the polarization state of the three pulses.The detailed procedure for data acquisition and analysis can be found in the supplementary material. To summarize, for each acquisition scan, τPPr is scanned (single scan) from −2.6 to 2.6 ps with temporal steps of 13 fs (400 spectra per scan). The integration time of the spectrometer is 1 ms (total acquisition time >1 mn, limited by the delay stage). The probe–reference delay and relative chirp are measured before each acquisition, out of pump–probe temporal overlap. τRPr is computed by fitting the modulus of the AC peak with a Gaussian function after discrete Fourier transform. The relative phase between the probe and reference pulses was also retrieved by Fourier filtering, so as to compare our analysis with usual nonlinear phase measurements.The method was validated against a set of isotropic materials for which the nonlinear refractive indices are well known: fused silica [1 mm, (2.2 ± 0.3) × 10−16 cm2/W at 1030 nm],2323. P. Kabaciński, T. M. Kardaś, Y. Stepanenko, and C. Radzewicz, “Nonlinear refractive index measurement by SPM-induced phase regression,” Opt. Express 27, 11018–11028 (2019). https://doi.org/10.1364/OE.27.011018 barium fluoride [0.5 mm, (2.2 ± 0.2) × 10−16 cm2/W at 1064 nm],2424. R. Adair, L. L. Chase, and S. A. Payne, “Nonlinear refractive index of optical crystals,” Phys. Rev. B 39, 3337–3350 (1989). https://doi.org/10.1103/physrevb.39.3337 sapphire [1 mm, (2.8 ± 0.7) × 10−16 cm2/W at 1030 nm],2525. A. Major, F. Yoshino, I. Nikolakakos, J. S. Aitchison, and P. W. E. Smith, “Dispersion of the nonlinear refractive index in sapphire,” Opt. Lett. 29, 602–604 (2004). https://doi.org/10.1364/ol.29.000602 and calcium fluoride [1 mm, (1.2 ± 0.3) × 10−16 cm2/W at 1030 nm].2323. P. Kabaciński, T. M. Kardaś, Y. Stepanenko, and C. Radzewicz, “Nonlinear refractive index measurement by SPM-induced phase regression,” Opt. Express 27, 11018–11028 (2019). https://doi.org/10.1364/OE.27.011018 All plates are uncoated. In order to check the validity of various polarization configurations, we also characterized a barium fluoride crystal, with a holographic [011] crystallographic orientation, which is known to present an anisotropy of its third-order nonlinearity, σ, defined as follows:26,2726. N. Minkovski, G. I. Petrov, S. M. Saltiel, O. Albert, and J. Etchepare, “Nonlinear polarization rotation and orthogonal polarization generation experienced in a single-beam configuration,” J. Opt. Soc. Am. B 21, 1659–1664 (2004). https://doi.org/10.1364/josab.21.00165927. L. Canova, S. Kourtev, N. Minkovski, A. Jullien, R. Lopez-Martens, O. Albert, and S. M. Saltiel, “Efficient generation of cross-polarized femtosecond pulses in cubic crystals with holographic cut orientation,” Appl. Phys. Lett. 92, 231102 (2008). https://doi.org/10.1063/1.2939584σ=χxxxx3−2χxyyx3−χxxyy3χxxxx3.(7)

V. CONCLUSION

Section:

ChooseTop of pageABSTRACTI. INTRODUCTIONII. PRINCIPLEIII. EXPERIMENTAL METHODSIV. RESULTSV. CONCLUSION <<SUPPLEMENTARY MATERIALREFERENCESPrevious sectionNext section

To conclude, we have introduced a novel time-resolved spectroscopic method to characterize the third-order nonlinearity on the femtose

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