We study the anisotropic nature of the Kerr nonlinear response in a beta-barium borate (β-BaB2O4, BBO) nonlinear crystal. The focus is on determining the relevant χ(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 also perform an additional experimental measurement of the Kerr nonlinear tensor component responsible for self-phase modulation in cascading, and we show that the average value of 14 different measurements is considerably larger than what has been used to date. Our own measurements are consistent with this average value. We also treat data measurements for mixtures of tensor components, and by disentangling them we present for the first time a complete list that we propose as reference of the four major cubic tensor components in BBO. We finally discuss the impact of using the cubic anisotropic response in ultrafast cascading experiments in BBO.
© 2013 OSA
Cascaded second-harmonic generation (SHG) describes the case where the frequency-converted second harmonic (SH) is strongly phase mismatched, and this may create a Kerr-like nonlinear phase shift on the fundamental wave (FW). Since the early discovery of this cascaded self-action nonlinearity [1, 2], cascaded SHG was for a long time overlooked until DeSalvo et al. measured a negative nonlinear phase shift from phase-mismatched SHG using the Z-scan method  (for an early comprehensive review on cascading, see ). The exciting promise of cascading nonlinearities is seen directly from the scaling properties of the Kerr-like nonlinear index it generates: , where deff is the effective nonlinearity of the quadratic interaction. By simply changing the phase mismatch parameter Δk we therefore have a Kerr-like nonlinearity that can be tuned in sign and strength. A particularly interesting and attractive property is that a self-defocusing Kerr-like nonlinearity is accessible by having a positive phase mismatch Δk > 0.
A popular crystal for cascading experiments is beta-barium-borate (β-BaB2O4, BBO), see e.g. [5–15]. It has a decent quadratic nonlinear coefficient, and because the crystal is anisotropic it can be birefringence phase-matched for type I (oo → e) SHG. For femtosecond experiments it has the important properties of a low dispersion, a high damage threshold, and a quite low Kerr self-focusing nonlinearity; the latter is important for cascading as we discuss later, because the material Kerr nonlinearity will compete with the induced cascading nonlinearity. The main goal of this paper is to analyze experiments in the literature where the Kerr nonlinear refractive index was measured in BBO [13–21], and for the fist time extract all four tensor components that are relevant to describe the anisotropic nature of the cubic nonlinearity.
While the anisotropy of the quadratic nonlinearity is extensively used for optimizing the phase-matching properties and maximizing the quadratic nonlinear coefficients in nonlinear crystals, the case is different when it comes to treating the cubic nonlinearities: the anisotropic nature has often been neglected when studying self-phase modulation (SPM) effects from the Kerr nonlinear refractive index, in contrast to the case of third-harmonic generation, as evidenced even in early studies [22, 23]. Here we show which anisotropic nonlinear susceptibility components the experimental literature data represent. The experimental data will be analyzed and we correct the reported values for cascading contributions if necessary. We eventually obtain complete information of all four relevant tensor components for the BBO cubic nonlinear susceptibility. Our results show that the tensor component affecting the SPM of the ordinary wave, , is very well documented, and the corrected literature data agree extremely well in the near-IR with the popular two-band model . We also perform our own experimental measurement of c11 with a very accurate technique based on balancing out the self-focusing SPM from the Kerr effect with the self-defocusing effect from cascading (originally used in ). Both this measurement and the corrected data from the literature show that c11 is substantially larger than what has been used so far in simulations in the literature, and an important consequence for cascaded SHG is that this reduces the range of phase-mismatch values that gives a total defocusing nonlinearity. In contrast experiments measuring the tensor components that affect the extraordinary wave interaction are more scarce, so new measurements are necessary in order to get more accurate and reliable values. We also assess the impact of using the anisotropic Kerr response in simulations of cascaded SHG, and it turns out that for BBO the optimal interaction angles are such that the anisotropy plays a minor role in the wave dynamics.
As to analyze the experiments properly and rule out any misunderstandings in the definitions, the foundation of the paper is a careful formulation of the propagation equations of the FW and SH waves under slowly-varying envelope approximation, see Appendix A. These include the anisotropic nature of the frequency conversion crystal in formulating the effective cubic nonlinear coefficients. This theoretical background is important in order to analyze and understand the experiments from the literature that report Kerr nonlinearities for BBO. The analysis presented here should help understanding what exactly has been measured, and put the results into the context of cascaded quadratic soliton compression.
2. Background for cascading nonlinearities
In order to understand the importance of knowing accurately the Kerr nonlinear index, let us explain first how cascading works. The cascaded Kerr-like nonlinearity can intuitively be understood from investigating the cascade of frequency-conversion steps that occur in a strongly phase-mismatched medium : After two coherence lengths 2π/|Δk| back-conversion to the FW is complete, and as Δk ≠ 0 the SH has a different phase velocity than the FW. Thus, the back-converted FW photons has a different phase than the unconverted FW photons, simply because in the brief passage when they traveled as converted SH photons the nonzero phase mismatch implies that their phase velocities are different. For a strong phase mismatch ΔkL ≫ 2π, this process repeats many times during the nonlinear interaction length, and in this limit the FW therefore effectively experiences a nonlinear phase shift parameterized by a higher-order (cubic) nonlinear index given as 5]. Since the nonlinearity is self-defocusing the pulse energy is in theory unlimited as self-focusing effects are avoided , and it can even be used to heal small-scale and whole-scale self-focusing effects . Another very attractive feature of the negative self-defocusing nonlinearity is that temporal solitons can be excited in presence of normal (positive) group-velocity dispersion, which means anywhere in the visible and near-IR. With this approach Ashihara et al.  achieved soliton compression of longer femtosecond pulses. Numerous experiments over the past decade have since been motivated by soliton compression of energetic pulses to few-cycle duration [9, 11, 25–27]. Since the defocusing limit and the soliton interaction depend critically on the value, knowing an exact value of the coefficient is crucial.
3. Anisotropic quadratic and cubic nonlinearities in uniaxial crystals
In a uniaxial crystal, the isotropic base-plane is spanned by the crystal XY axes, and light polarized in this plane is ordinary (o-polarized) and has the linear refractive index no. The optical axis (crystal Z-axis, also called the c-axis) lies perpendicular to this plane, and light polarized in this plane is extraordinary (e-polarized) and has the linear refractive index ne. In a negative uniaxial crystal like BBO, no > ne. The propagation vector k in this crystal coordinate system has the angle θ from the Z-axis and the angle ϕ relative to the X-axis, cf. Fig. 1(a), and the e-polarized component will therefore experience the refractive index , while the o-polarized light always has the same refractive index.
Degenerate SHG can either be noncritical (type 0) interaction, where the FW and SH fields are polarized along the same direction, or critical (type I) interaction, where the FW and SH are cross-polarized along arbitrary directions in the (θ, ϕ) parameter space. In type 0 the FW and SH are usually polarized along the crystal axes (θ = 0 or π/2) since this turns out to maximize the nonlinearity, and it is called noncritical because the interaction does not depend critically on the propagation angles (both nonlinearity and phase matching parameters vary little with angle). A great advantage of this interaction is that there is little or no spatial walk off (a consequence of the choice θ = 0 or π/2). In type I θ and ϕ values can often be found where phase matching is achieved. This interaction is very angle-sensitive, which is why it is called critical interaction.
The slowly-varying envelope equations (SVEA) for degenerate SHG are derived in Appendix A. There the quadratic and cubic “effective” anisotropic nonlinear coefficients were symbolically introduced. Below we show the expressions for these coefficients relevant for SHG in BBO, which belongs to the crystal class 3m, and in all cases we consider the reduced numbers of tensor components that come from adopting Kleinman symmetry (which assumes dispersionless nonlinear susceptibilities [28, Ch. 1.5]); thus BBO has 3 independent χ(2) tensor components and 4 independent χ(3) tensor components. An excellent overview of the quadratic and cubic nonlinear coefficients in other anisotropic nonlinear crystal classes is found in .
For an arbitrary input (i.e. FW) polarization various SHG processes can come into play (oo → o, oo → e, oe → e, oe → o, ee → e, and ee → o). This includes both type 0, type I and type II (nondegenerate SHG, where the FW photons are cross-polarized). Their respective deff-values for a crystal in the 3m point group (which also includes lithium niobate) are 29]. In BBO ρ is around 3 – 4° for the typical angles used, which gives changes in deff of a few percent, and as this is well within the standard errors on the experimental dij values it is typically ignored for qualitative simulations. However, when we later evaluate the cascading contributions and compare them to the competing material Kerr nonlinearity, an exclusion of ρ would lead to a systematic error so we choose to include it in the analysis in Sec. 4.
BBO is usually pumped with o-polarized light as the quadratic nonlinearity is largest for this configuration (d22 is the largest tensor component, see Appendix C), and because the oo → e interaction can be phase matched by a suitable angle θ. For such an interaction, the crystal is cut so ϕ = −π/2 as to optimize the nonlinearity (it is here relevant to mention that d22 and d31 has opposite signs in BBO). This has the direct consequence that oo → o interaction is zero. For this reason, when studying BBO pumped with o-polarized light in a crystal with ϕ = −π/2, Eq. (4) shows that deff = d31 sin(θ + ρ) − d22 cos(θ + ρ).
Note that BBO has historically been misplaced in the point group 3, and in addition there has been some confusion about the assignment of the crystal axes (where the mirror plane of the crystal was taken parallel instead of perpendicular to the crystal X-axis) . Unfortunately this means that even today crystal company web sites operate with d11 as being the largest tensor component (in the 3m point group d11 = 0) and supply crystals apparently cut with ϕ = 0 (because using the point group 3 combined with a nonstandard crystal axes definition means that d11 cos3ϕ must be maximized, while with the correct point group, 3m, and correct crystal axes assignments, d22 is the largest tensor component and sin3ϕ = 1 maximizes the effective nonlinearity). In order to sort out any confusion, the optimal crystal cut is shown in Fig. 1(b), and we hereby urge crystal companies to follow the standard notation.
In Appendix A the SVEA propagation equations included also an “effective” third-order SPM nonlinearity, , and cross-phase modulation (XPM) nonlinearity, , which due to the anisotropy had to be calculated specifically for a given crystal class and input polarization. The SPM and XPM anisotropic cubic nonlinearities for a uniaxial crystal in the point group 3m were found in Eqs. (B13), (B14) and (B15) in . For a type I interaction (oo → e), where the FW is o-polarized and the SH e-polarized, they are27] and periodically poled lithium niobate [31, 32], we have 33] Eq. (10) have empirically been found not to hold. Specifically, the tensor components they are calculated from turn out to obey slight frequency variations that can be predicted by frequency scaling rules, like Miller’s rule (see [34–36] and also later in Sec. 4.4).
If the Kerr nonlinearity is considered isotropic, we have [type I oo → e interaction, taking θ = 0 in Eqs. (7)–(9)], or [type 0 ee → e interaction, cf. Eq. (10)]. These properties underlie the parameter B that we used in our previous isotropic model .
A mistake often seen in the early literature on type I interaction is to take , and this mistake gives a 3 times too large XPM term. As shown above the identity only holds in the type 0 configuration, where the FW and SH have identical polarizations, but importantly it is not an identity that is restricted to isotropic nonlinearities as it holds for an anisotropic medium as well. Thus, the error made in the past for type I could either come from using directly the propagation equations for an isotropic medium and where two pulses with the same polarization interact, but it could also come from generalizing type 0 SHG propagation equations to type I, and forgetting the XPM properties for cross-polarized interaction.
4. Measurement of cubic nonlinearities of BBO
4.1. Experimental conditions for the literature measurements
Most experiments aiming to measure the cubic nonlinearities have used the Z-scan method , which uses a Gaussian laser beam in a tight-focus limiting geometry to measure the Kerr nonlinear refractive index. The Z-scan method measures the transmittance of a nonlinear medium passing through a finite aperture placed in the far field as a function of the sample position (z) measured with respect to the focal plane. As the sample is Z-scanned (i.e. translated) through the focus of the beam, the lens effect from the Kerr nonlinear index change will change the amount of light recorded by the detector; this gives information about the intensity-induced nonlinear index change Δn and thus the coefficient, defined phenomenologically from the general expansion
There are some issues with the Z-scan method that may affect the measured nonlinear refractive index. If the repetition rate is too high, there are contributions to the measured from thermal effects as well as two-photon excited free carriers  (for more on these issues, see e.g. ). Similarly, a long pulse duration can also lead to more contributions to the measured than just the electronic response that we aim to model, and in particular the static (DC) Raman contribution is measured as well. However, for BBO the fraction of the delayed Raman effects is believed to be quite small, so we will assume that it can be neglected making the measured nonlinearities correspond to the electronic response.
In the following we will analyze measurements in the literature in order to extract the four BBO cubic tensor components in Eqs. (7)–(9). When converting from the nonlinear susceptibilities cμm to the intensity nonlinear refractive index we can use Eq. (53). There is a small caveat here, because the nonlinear index is defined for SPM and XPM terms, where only two waves interact through their intensities, while the cμm coefficients are generally defined for four-wave mixing. Thus, it does not always make sense to define a nonlinear refractive index (consider e.g. the c10 component) until the total effective susceptibility is calculated, e.g. through Eqs. (7)–(11). Thus, it is safer to keep the susceptibility notation as long as possible when calculating the effective nonlinearities. The exception is when measuring the nonlinear refractive index of the oooo SPM interaction because the relevant nonlinear tensor component that is measured is c11, see Eq. (7), and thus the connection between c11 (i.e. ) and is unambiguous.
In the experiments in the literature the cascaded quadratic contributions were often forgotten or assumed negligible. The exact relations for the evaluating cascading nonlinearities are derived in Appendix B. We will address these issues below by in each case calculating where possible the cascading nonlinearity.
4.2. The c11 tensor component
In this section we present measurements using o-polarized input light, which means that the c11 tensor component is accessed by measuring the SPM component of the o-polarized light . The connection to the Kerr nonlinear index is .
To our knowledge, the first measurement of any of the cubic susceptibilities was performed by Tan et al. . They investigated the cascaded nonlinearity in a 10 mm BBO crystal cut at θ = 22.8° for phase matching at 1064 nm (they claim also that the crystal is cut with ϕ = 0, which is probably due to the confusion related to placing BBO in the point group 3 instead of 3m, cf. the discussion on p. 6). Pumping with 30 ps o-polarized pulses from a Nd:YAG mode-locked laser at two nearly degenerate wavelengths (a strong wave at λa = 1.064 μm and a weaker wave at λb = 1.090 μm), they rotated the BBO crystal around Δk = 0 in a type I oo → e SHG setup, where both the SH 2ωa of the strong beam was generated and four-wave mixing of the two pumps ωc = 2ωa −ωb was generated. By recording the impinging and generated pulse energies vs. angle they found by a fitting analysis the coefficient g = 220, where was the only free parameter in the fit. For θ = 22.8° we have ρ = 3.2° and deff = 1.99 pm/V, so this gives the measured nonlinear susceptibility . They also performed a Z-scan measurement, and the setup was calibrated towards a BK7 glass sample. They found . By using the value they propose , one therefore finds . We report these results with a bar because their definition of the cubic susceptibility is different than what we use: they define the SPM polarization as , which replaces our definition Eq. (47), so the connection between them is . To check this, from the literature we find for BK7 the value measured at 804 nm by Nibbering et al. , which corresponds to χ(3) = 3.03 × 10−22 m2/V2. This is precisely a factor 3/2 smaller than the value Tan et al. mentions, and this confirms the relationship between the cubic susceptibilities. We now apply Miller’s rule to the Nibbering et al. result to get for BK7 χ(3) = 2.90 × 10−22 m2/V2 at 1064 nm, so we get a corrected value . In our notation their cascading measurements gave
Hache et al.  used a 1 mm BBO cut at θc = 29.2° with 800 nm 100 fs pulses from a MHz-repetition rate Ti:sapphire oscillator. The pump was o-polarized allowing for oo → e cascading interaction. They carried out Z-scan measurements both close to the phase-matching point θc as well as “far from SHG phase matching”. In the latter case they found
DeSalvo et al.  used the Z-scan method with a single-shot 30 ps pulse at 1064 nm, which had k parallel to the optical axis c, so θ = 0 and consequently ρ = 0 as well. Thus, (a) the pump is always o-polarized no matter how the input beam is polarized, and (b) the input polarization determines the angle ϕ. Since the specific orientation of the crystal with respect to the input polarization was not reported, we take it as unknown. The relevant Kerr contribution at the pump wavelength is oooo interaction, which is given by Eq. (7) as simply one tensor component c11. The experiment reported at 1064 nm, where the nonlinear index change is defined as , and using Eq. (C1) in  this corresponds to χ(3) = 2.70±0.49·10−22 m2/V2. The cascading contributions are from oo → e and oo → o processes. Since θ = 0 they have the same phase mismatch values as e-polarized light in this case has the same refractive index as o-polarized light. At 1064 nm the value is Δkooe = Δkooo = 234 mm−1. Taking into account the two cascading channels Eqs. (4) and (3), and using Eq. (55), we get the total contribution , i.e. independent on the propagation angle ϕ. For these calculations we used d22 = −2.2 pm/V. The deff values are typically reported with a 5% uncertainty , giving a 7% uncertainty on the cascading estimate. This means that when correcting for the cascading contributions we get42]), so 43] puts the poles at 135 nm and 129 nm for o-and e-polarized light, respectively). Finally, at 266 nm they measured . Here cascading is estimated to be insignificant (certainly we cannot estimate accurately it using the Sellmeier equations as the SH lies right at the UV pole), so converting to SI we have
Li et al. [18, 19] performed two different Z-scan experiments with similar conditions: a thin Z-cut BBO crystal was pumped with 25 ps 10 Hz pulses at 532 nm and 1064 nm  and later with 150 fs 780 nm pulses from a 76 MHz Ti:sapphire oscillator . The propagation was along the optical axis, implying θ = 0 and consequently ρ = 0 as well, and also that for both cascading and Kerr nonlinearities the angle ϕ should not matter as the pump always will be o-polarized. Nonetheless, they measured using the Z-scan method different results with the input polarization along the [1 0 0] direction (X-axis, i.e. ϕ = 0), and with input polarization along the [0 1 0] direction (Y-axis, i.e. ϕ = π/2). Cascading was in both papers ignored as it was considered too small, so let us assess whether this is true. The cascading contributions are from oo → e and oo → o, both having the same Δkooe = Δkooo. However, only one come into play in each case: for ϕ = 0 the only nonzero nonlinearity is while for ϕ = π/2 the only nonzero nonlinearity is . Thus, they turn out to have the same cascading contribution, which from Eq. (1) becomes (532 nm), (780 nm), and (1064 nm). When correcting the measured nonlinearities with this value we get for the picosecond experiments at 532 nm 19]
Ganeev et al.  used a 2 Hz 55 ps 1064 nm pump pulse and a BBO crystal with θ = 51° and consequently ρ = 4.1°. The procedure they used was to calculate the “critical” Δkc where defocusing cascading exactly balances the intrinsic focusing Kerr, i.e. where . By angle tuning well away from this point (so Δk ≫ Δkc) they tried to make cascading insignificant. We estimate this to be true: specifically, the pump was o-polarized and the crystal was cut with ϕ = −π/2 to optimize SHG . Then the only cascading channel is oo → e, with Δkooe = −655 mm−1, and using Eq. (1) we get that is therefore self-focusing (as they also note themselves). Using the Z-scan method they measured with the uncertainty estimated to be 30%, so correcting for cascading
Moses et al.  pumped a BBO crystal at 800 nm with 110 fs o-polarized light from a 1 kHz Ti:Sapphire regenerative amplifier. By angle-tuning the crystal they achieved zero nonlinear refraction: this happens when the cascading from the oo → e interaction exactly balances the Kerr nonlinear refraction. Investigating a range of intensities they determined this critical phase mismatch value to Δkc = 31 ± 5 π/mm. Using this value and assuming that it corresponds to then a reverse calculation though Eq. (1) gives . Using the value deff = 1.8 pm/V they inferred the value (the main sources of the uncertainty are to determine the precise phase-mismatch value as well as the uncertainty on deff). Since o-polarized pump light was used, this contains only the c11 tensor contribution from an oooo interaction. However, the value deff = 1.8 pm/V they used to infer this nonlinearity was probably taken too low . Let us therefore estimate it again using a more accurate value: Δk = 31 π/mm is achieved at 800 nm with θ = 26.0°, giving ρ = 3.6° and deff = 2.05 pm/V. With this corrected value we get
In the same spirit, we performed an experiment using the technique employed by Moses et al. The crystal was a 25 mm BBO from Castech with a 10×7 mm2 aperture, cut so θ = 21° and ϕ = −π/2, and antireflection coated. The crystal was pumped with pulses from a 1 kHz Ti:Sapphire regenerative amplifier followed by a commercial OPA, giving λ = 1.032 μm pulses with around 40 μJ pulse energy. The pulses were Gaussian with 39.3 nm FWHM bandwidth and an intensity autocorrelation trace revealed a Gaussian pulse with an intensity FWHM of 46 fs; the pulses were therefore nearly transform-limited Gaussian pulses (the time-bandwidth product was 16% above transform limit). The pump was collimated with a telescope before the crystal and the spot size was measured to 0.5 mm FWHM, and the intensity was controlled by a neutral-density filter. We chose to pump at this wavelength because the GVM-induced self-steepening is smaller than at 800 nm, and besides the transition from negative to positive effective nonlinearity occurs in the so-called stationary regime, where the cascading nonlinearity is non-resonant (see also Sec. 4.5 later). We first located the phase-matching angle and then angle-tuned the crystal until any visible nonlinear components of the spectrum had vanished (i.e. it returns close to its input state) as in this range. By fine-tuning (with 1/6° precision) θ in this range we measured the trends shown in Fig. 2(a): this plots the bandwidth 10 dB below the peak value of the recorded spectra for two different intensities, and the curves are seen to cross each other at Δk ≃ 88 mm−1. We investigated numerically whether this is a good measure of the transition from negative to positive total nonlinearity; for the parameters used in the numerics the critical point was calculated analytically to be Δkc = 88.5 mm−1 (neglecting XPM effects, see later). Figure 2(b) shows the equivalent numerical results, and clearly the two different intensities cross each other very close to the theoretical transition. Two comments are in order: firstly, the model includes XPM effects that give a self-focusing contribution  (see also discussion later in Appendix B, where the cascading quintic term is calculated) that for the chosen intensities is quite minimal. We can judge the XPM impact by turning off XPM in the code: the transition now occurs practically at the theoretical value, which is a bit higher than the result with XPM. Thus, we can expect that XPM gives a small correction on the order of −1 mm−1. Secondly, the reason why the spectrum does not return to the input value is that when cascading cancels Kerr SPM effects there are still self-steepening effects that remain (see the detailed experiments by Moses et al. , and also the recent discussion by Guo et al. ). Plots (c) and (d) show the spectra right at the transition, and the red edge is quite similar for the two intensities while the blue edge is always higher for the larger intensity. We should mention that we found similar transition “indicators” at very similar Δk values by observing the −20 dB bandwidth, FWHM bandwidth, in tracking the wavelength position of the spectral edges, in tracking the center wavelength etc. Especially the latter is the most precise indicator numerically as cascading gives a very pronounced blue-shift of the spectrum – see graph (e) – but experimentally the peak was too erratic in the fine-tuning around the transition, which is probably related to the fact that the pump is generated as the SH of the idler of our OPA. We conclude that after correcting for a small XPM contribution the critical transition is found the be Δkc = 89 ± 5 mm−1, where the uncertainty lies mostly in finding a suitable transition value to track between the different intensities and phase mismatch values. This critical value corresponds to θ = 18.5°, ρ = 2.7° and deff = 2.08 pm/V, so calculating the cascading nonlinearity and employing the hypothesis, we get
This completes the c11 measurements. A summary of the data is shown in Fig. 3, together with the predicted electronic nonlinearity from the two-band model (2BM) . The experimental near-IR data are amazingly well predicted on an absolute scale by the 2BM; we must here clarify that for the 2BM we chose the material constant K = 3100 eV3/2cm/GW, which was found appropriate as a single material parameter for dielectrics , and by using the BBO band-gap value Eg = 6.2 eV . Thus, in practice there are no free parameters in the model, which underlines the incredible agreement obtained on an absolute scale (and not just the translation from one wavelength to another). We mention here that in DeSalvo et al.  they proposed to “rescale” the bandgap do obtain a better fit with the experimental data (mainly because the two-photon absorption values β were not accurately predicted with 6.2 eV), and eventually suggested using 6.8 eV instead of 6.2 eV for BBO. This gave a much better β agreement at 266 nm, while the 352 nm value was still off. Here we see that the corrected DeSalvo near-IR values actually agree extremely well with the 2BM value when using Eg = 6.2 eV, while the agreement is less accurate with Eg = 6.8 eV. A common issue whether one uses 6.2 or 6.8 eV is that the predicted strong enhancement just below 400 nm due to two-photon absorption of the FW is not reflected in the experimental data. This is of little importance for our purpose, because in cascaded SHG the FW wavelength is usually never below 600 nm. In order to extract a suitable value for the c11 coefficient, we calculated first the Miller’s delta from all the data. It is defined as [35, 36]Fig. 3 that this average is a quite good representative of the experimental data; all except two are within two standard deviations. We also note that the value we measured in this work fits extremely well with this proposed average value.
4.3. Other tensor components
Banks et al.  measured the c10 tensor component of BBO using third-harmonic generation (THG) measurements at 1053 nm using 350 fs pulses from a 10 Hz Ti:Sapphire regenerative amplifier. The propagation angle was θ = 37.7°, and from the data recorded while ϕ was varied around zero they extracted C10 = −6 · 10−24 m2/V2 using the quadratic nonlinear coefficients of Shoji et al. . (They extracted other values as well, but we choose to use this one because it was calculated from the experimental data using the same effective quadratic nonlinearities of Shoji et al. as we use.) In fact, Banks et al. reported that this value came with a quite large uncertainty due to the uncertainty in the d31 – and d15, in absence of Kleinman symmetry – coefficients in the literature. Since  define Cμm = χ(3)/4, we have Cμm = cμm/4, so
Banks et al.  also measured the mixture at θm = 47.7°. The TH walk-off angle is ρ = 4.5°, and replacing in the above expression θ → θ + ρ will not change significantly the following numbers; therefore we choose to ignore the walk-off contribution because the fit done by Banks et al. also ignored it. If we now use Eq. (31), and the apply Miller’s scaling to get the c11 component appropriate for ω + ω + ω → 3ω interaction, see Eq. (30), we get c11(−3ω; ω, ω, ω) = c11(THG@λ = 1.053 μm) = 5.24 · 10−22 m2/V2, and therefore47]. Therefore using these values to model the SPM and XPM effects in BBO is an approximation.
Sheik-Bahae and Ebrahimzadeh  used the Kerr-lens autocorrelation method  to measure (χ(3) = 3.43 ± 0.56 · 10−22 m2/V2) using a 76 MHz Ti:Sapphire oscillator giving 120 fs pulses at λ = 0.850 μm. The BBO crystal was cut for phase matching at 800 nm, i.e. with θ = 29.2°, and consequently ρ = 3.9°. The crystal cut was optimized for SHG, so ϕ = −π/2. The pump pulse was e-polarized as to ensure lack of phase matching and thereby no cascading. We will now assess whether this assumption is fulfilled. The cascading contributions that might occur are ee → o and ee → e, however the former has zero nonlinearity when ϕ = −π/2, cf. Eq. (5). The phase mismatch for the latter interaction is Δkeee = 396 mm−1 for θ = 29.2°. The cascading contribution at ϕ = −π/2 then becomes . Thus, the choice of using e-polarized light does make the cascading contribution small, but not insignificant (it is roughly equal to the reported uncertainty). If we now correct the measured value with the cascading contributions we get . Let us now understand what tensor components this value represents. By using e-polarized light several tensor components come into play for the cubic nonlinearity: the effective Kerr nonlinearity experienced by the pump is an eeee interaction, which is given by Eq. (11). For θ = 29.2° the TH walk-off angle is ρ = 4.3°, and with ϕ = −π/2 the effective nonlinearity isEq. (31), (32) and (33), all suitably converted to 850 nm with Miller’s rule, we can from the corrected value of and Eq. (34) calculate Eq. (34). Apart from that, it also relies on three other separate measurements, making it very sensible to the c11, c10 and c16 values used. Here we mention that the large uncertainty of c10 reported in  plays a role; with the various possibilities reported there for the c10 value we always get a negative c33, and its value can vary by a factor of two from that reported in Eq. (35).
4.4. Summary of experiments
A summary of the data reported in the literature along with our corrected or updated values is shown in Table 1. This table also gives an overview of the crystal angles, pulse duration, repetition rates, wavelengths and the tensor components accessed in the measurements.
Our analysis of the experiments [14–17, 19–21] points towards using the Kerr nonlinearities summarized in Table 2. Only the susceptibilities are reported in order to minimize the errors when using the values for composite nonlinear indices. We have there also indicated the Miller’s delta calculated from Eq. (30). Since the experiments are performed at different wavelengths, using the Δijkl values makes it easier to evaluate a linear combination of the nonlinear coefficient at some particular wavelength. [Another popular model for scaling the cubic nonlinearity is the Boling-Glass-Owyoung (BGO) model , but we did not see a big difference in using that model compared to Miller’s rule. Besides, extending the BGO model to anisotropic nonlinearities is not straightforward, so we prefer to use Miller’s rule for frequency scaling.] The values are reported with more significant digits than supported by the uncertainties, but this is done on purpose so it is easier to cross-check the coefficients as well as frequency scaling them.
4.5. Implications for cascaded pulse compression in BBO
In context of cascaded quadratic nonlinearities by far most important component is the FW SPM coefficient , and in a type I oo → e configuration it is given by the c11 component. In some of our previous work related to BBO [37, 51, 52], we assumed an isotropic Kerr nonlinearity, and have used the value taken from , and in a later publication, we have used the value given by Eq. (28). An implication of this larger value is that the “compression window”  becomes smaller. With this we imply a range of phase-mismatch values, where compression is optimal. We first need Δk to be small enough so , i.e. . This is the upper end of the window, also denoted the Kerr limit. Note that as mentioned in Sec. 3 the Kerr limit is quite sensitive to whether one includes the spatial walk-off angle ρ in calculating the deff-value in the cascading expression. Once below the Kerr limit, decreasing Δk strengthens the total defocusing nonlinearity. However, at a certain point the group-velocity mismatch (GVM) becomes too strong and the compression quality is strongly reduced. Several things happen: (a) the GVM-induced self-steepening term [9, 46] is increased as it scales as d12/Δk, where d12 is the GVM parameter, so the compressed pulse experiences a strong pulse-front shock. (b) When Δk becomes too low (specified more accurately below) the SH will experience a resonant phase matching condition of a sideband frequency to the center frequency of the FW spectrum . This is damaging to pulse compression because in the cascading process effectively this is the cascading bandwidth felt by the FW, and in the transition from the nonresonant to the resonant regime, the bandwidth essentially goes from being octave spanning to becoming resonant and thereby narrow . The criterion for being in the nonresonant regime, also denoted the stationary regime, can in the simple case where only second-order dispersion is considered  be expressed as , where is the group-velocity dispersion (GVD) coefficient of the SH. Thus, in the case where the SH GVD is normal (positive) we have the stationary (nonresonant) regime with broadband cascading when . Below this threshold, in the so-called nonstationary regime, the cascaded nonlinearity is as mentioned resonant: the poor bandwidth implies that there is no possibility to achieve few-cycle duration . At the same time the compressed pulse quality is low as there is a strong pulse shock front: this stems from the d12/Δk ratio being large, and this term controls the cascading-induced self-steepening, as mentioned above. As a consequence one can only achieve a decent pulse quality by keeping very low soliton orders (roughly below 2) . The implications of the larger nonlinear SPM Kerr nonlinearity for the FW is visualized in Fig. 4(a). With the lower value, , a large compression window is predicted, almost even encompassing Ti:Sapphire laser wavelengths. With the new larger value, Δ1111 = 52.3 m2/V2 corresponding to , the compression window starts opening around 900 nm. To judge the impact of using this larger Kerr value, Fig. 4(b) shows a simulation at 1030 nm, where the phase mismatch is chosen to lie inside the compression window using the new larger Kerr value. After 25 mm of propagation the short (50 fs) and intense (Iin = 500 GW/cm2) input pulse is compressed somewhat due to the formation of a self-defocusing soliton, and the spectrum shows SPM-like broadening and even formation of a soliton-induced Cherenkov wave (dispersive wave) around 2900 nm .
When instead using the ’old’ lower Kerr value (dashed lines) the result is changed quite a lot: this is because |n2,tot| is now much larger so the intensity gives a larger effective soliton order (the specific numbers are Neff = 3.4 in case (1) and Neff = 1.7 in case (2). The soliton is therefore compressed more and earlier in case (1), so after 25 mm soliton splitting has already occurred and in the spectrum the broadening is more pronounced and the dispersive wave is much stronger. These results clearly show how sensible cascading is to the value of the FW Kerr SPM parameter.
The experiments [5, 7] carried out at 800 nm were performed well into the nonstationary region, which was necessary to achieve a defocusing nonlinearity. No few-cycle compressed solitons were observed in , as the intensity had to be kept low in order to avoid severe GVM-induced self-steepening. (Note that the simulations in  used χ(3) = 5 · 10−22 m2/V2@800 nm, i.e. identical to the value we suggest.) Instead the experiment carried out at 1250 nm  was done in the stationary regime, and indeed a few-cycle soliton was observed.
A question is whether pumping with e-polarized light could give any advantages. As the ee → o interaction is heavily phase mismatched in a negative uniaxial crystal, the main cascading channel would be the ee → e interaction. As expected this interaction can never be phase matched. The maximum deff is for θ = 0, and this turns out to give the maximum cascading strength as well. However, our calculations show that it is too small to overcome the nonlinearity, and the total effective cubic nonlinearity would be focusing. If θ = π/2 we get deff = d33, which for BBO is very small (this is typical for borates, while the niobates instead have very large d33 components). Nevertheless, for θ = π/2 the c33 component determines the SPM coefficient, and as we found it could be negative. The total effective cubic nonlinearity would therefore be defocusing. Investigating a BBO crystal pumped with θ = π/2 using e-polarized light would therefore be an interesting next step to resolve this issue.
We should finally mention that the error made in assuming an isotropic response for the cascaded soliton compression is not large: most importantly, the crucial parameter is the FW SPM coefficient, which as we saw is the same in the isotropic and in the anisotropic cases for this type I interaction. The XPM and SH SPM coefficients change when anisotropy is taken into account, but they play a minor role and rather tend to perturb the result more than shape it. In order to make a more quantitative statement, we calculated the θ-ranges where self-defocusing solitons can be excited in BBO. A necessary (but not sufficient ) criterion for this to happen is Δk > 0 as well as Δk low enough for . The range is shown vs. FW wavelength in Fig. 5(a). Inside this θ range we can now for each wavelength calculate the anisotropic Kerr nonlinearities. As we know the FW SPM coefficient does not change with θ (but it does change across the wavelength range shown, which we here estimate using Miller’s rule), so we can normalize the results to this value. We then get the results shown in Fig. 5(b). The XPM term is seen to lie close to the isotropic value 1/3 (found by taking , dashed grey line). Instead the SH SPM term lies well below the isotropic value [found taking , dashed blue line]. The main reason for this is the large negative c33 component. In summary, assuming an isotropic response in BBO is a good approximation for the FW XPM term, mainly because the working θ range lies close to zero (the isotropic limit). This result cannot be generalized to other nonlinear crystals, where the working range might lie closer to π/2; an example is lithium niobate that operates quite close to θ = π/2 in the type I configuration, see e.g. . For the SH SPM term its value is quite far from the isotropic case, but off all the parameters this is the least important one because the SH SPM is negligible in the cascading limit compared to FW XPM and in particular FW SPM. To confirm this we checked that the simulations of case (2) in Fig. 4(b) were almost identical results when using isotropic Kerr nonlinearities. A similar conclusion was drawn in our recent work, where the anisotropy was also taken into account .
We have analyzed the Kerr nonlinear index in BBO from a number of experiments [13–21], and we argued that in many of them contributions from cascaded SHG nonlinearities need to be taken into account. We also performed a very accurate measurement of the main tensor component c11 by balancing the Kerr nonlinearity with a negative cascading nonlinearity. Our analysis points towards using the Kerr nonlinearities summarized in Table 2. They encompass all four relevant tensor components for describing the anisotropic nature of the BBO Kerr nonlinearity under Kleinman symmetry. The most reliable value is the c11 component, relevant for describing the oooo SPM coefficient, while the other three are more uncertain: the c16 and c10 values are measured for THG, so using them for describing SPM and XPM effects is an approximation. Finally, the c33 coefficient is very uncertain as it was measured in a non-ideal way (since the experiment used a crystal angle where its importance is very small) and on top of that all the three other coefficients in the table were used to deduce its value. In all cases the measured coefficients had contributions from both instantaneous electronic and delayed Raman effects, but we assumed that the Raman contributions to the measured nonlinearities were vanishing.
The proposed c11 value was calculated as a weighted average over 14 different experiments in the 532–1064 nm range, and is therefore expressed as a wavelength-independent Miller’s delta Δ1111 = 52.3 ± 7.7 × 10−24 m2/V2, cf. Eq. (31); at 800 nm this value corresponds to c11 = 5.0 × 10−22 m2/V2 and , which is larger than typical values used in the literature. We stress that the c11 value we measured in this work agrees extremely well with this proposed value, and we believe that this method – where the focusing Kerr nonlinearity is nulled by an equivalent but defocusing cascading nonlinearity – is quite precise and can be used in many other contexts as an alternative to Z-scan measurements. When plotting the (corrected) Kerr nonlinear refractive indices vs. wavelength, see Fig. 3, we came to a surprisingly good absolute agreement with the 2-band model , except in the short end of the visible range.
Finally we showed the predicted consequences for cascaded femtosecond pulse compression exploiting type I interaction in BBO when using this larger value: the operation range where few-cycle pulse compression can be achieved will shift to longer wavelengths, roughly above 1.0 μm. We also showed that the anisotropic XPM coefficient of the FW is quite similar to the isotropic one, but this relies on the particular range of phase-mismatch rotation angles used in BBO, so this result cannot be generalized to other crystals exploiting type I interaction. Instead the anisotropic SH SPM coefficient is quite different from the one found assuming an isotropic Kerr nonlinearity, but since in cascading the SH is weak this difference will be insignificant in a simulation of cascaded SHG.
A. The propagation equations under the slowly-varying envelope approximation
We have previously touched upon the issue of anisotropic cubic nonlinearities in quadratic nonlinear crystals , where we focused on type I interaction and the measurements in the literature for the lithium niobate crystal. Here we formulate the complete propagation equations that hold for both type 0 and type I, and discuss the anisotropic cubic tensor components relevant for the BBO crystal class.
Let us show briefly how to derive the basic slowly-varying envelope equations (SVEA) for degenerate SHG (i.e. where the FW photons are degenerate), thus describing both type 0 and type I SHG. We do this in order to avoid confusion about how to define the relation between the electric field and the susceptibility tensors. Working in mks units, from Maxwell’s equations and applying the paraxial approximation we immediately get the time-domain wave equation
For SHG we define the electric field envelopes ℰj for the FW (frequency ω1) and the SH (frequency ω2 = 2ω1), as well as envelopes for the nonlinear polarization response 𝒫NL,j
Besides a trivial transformation to a suitably chosen co-moving frame, what is left now is to evaluate the tensorial contributions to the nonlinear coefficients. Due to the vast variety of tensor combination possibilities for anisotropic nonlinearities it is here convenient to specify the problem very specifically: we therefore constrict ourselves to SHG in a uniaxial crystal, i.e. where light can be o-polarized with unit vector eo or e-polarized with unit vector ee [30, Eq. (B6)], and we also restrict ourselves to degenerate SHG, where the two FW photons are indistinguishable. Let us define the quadratic nonlinear polarization response as , where we have assumed that the quadratic nonlinearity is instantaneous in time. The procedure is then to insert the field envelope in this nonlinear polarization response and then calculate the polarization response for a specific field by applying the dot product with the fields unit vector: . The only relevant terms are those that oscillate with the fast carrier frequency oscillations of the field e−iωjt, as this defines the polarization envelope 𝒫. Doing this we get for the FW and SH fields33]. Depending on the crystal symmetry class (point group) many of these are zero. It is therefore fruitful to consider a specific point group, and perform the calculations. We show in Sec. 3 the results for the point group 3m. We note that the nonlinear polarizations would have been a factor of 2 larger had we not used a factor in front of our envelope definition.
Moving to the cubic nonlinear tensor component, it is done in much the same way, and we refer to [30, App. B] for details. We again consider an instantaneous response, and the cubic nonlinear polarization response that oscillates with e−iωjt turns out to be related to SPM and XPM; we could have had contributions from third-harmonic generation here if we had included such a harmonic field ℰ3(z,t)eik3 (ω3)z−iω3t in the envelope definition. We then get [30, Eq. (B8)]Eqs. (40)–(41); if we do not have this factor , as in e.g. Boyd , it is important to note that the values (and consequently also the -values) in the two cases remain the same. The χ̱(3) is a rank 4 tensor with 81 different elements, but due to permutation symmetry in the the crystal axes indices there are only 30 nonzero elements . The calculations of the specific to a uniaxial crystal in the 3m point group was done in [30, App. B], and in Sec. 3 we summarize the results.
We can now express the plane-wave SVEA equations for the electric field envelopes as30]. Both and χ(3) are in mks (SI) units, while conversion to and from the esu system is reported in .
The SVEA equations can be extended to the slowly-evolving wave equations (SEWA) [9, 37] by including self-steepening effects, and also the non-instantaneous (delayed) Raman effect can straightforwardly be included [37, 46]. For simplicity such effects are neglected here.
B. Reduced nonlinear Schrödinger equation in the strong cascading limit
The cascading nonlinear contribution in the strong cascading limit (ΔkL ≫ 2π) can quickly be found by taking assuming an undepleted FW. Thus E1 is taken independent on z in Eq. (49), and the SH can therefore be solved directly when the Kerr SPM and XPM terms are neglected (which follows directly from the undepleted FW assumption). In absence of dispersion (the effect of dispersion is discussed in Sec. 4.5) we get that the SH is slaved to the FW: . When inserting this in the FW equation (48) we arrive at the following nonlinear Schrödinger equation for the FWEq. (2): in the strong cascading limit the FW experiences a cubic Kerr self-action nonlinearity ∝|ℰ1|2, where both cascading and material Kerr nonlinearities contribute. The cascading Kerr-like nonlinearity is described by the parameters Eq. (1) and it comes from using Eq. (53), or simply performing the same exercise with the intensity version of the SVEA equations. These will be used in Sec. 4 to estimate the cascading contribution in the various experiments. A practical note: this lowest order cascading nonlinearity seemingly diverges at Δk = 0, but this is because the assumption of an undepleted FW breaks down in this limit. To the next order a correction term 1 − sinc(ΔkL) must be applied , which resolves the divergence at Δk = 0 and from this the maximum cascading strength is obtained at ΔkL ≃ π .
There is also a quintic nonlinearity from cascading mixed with the XPM term of the FWEq. (13). This term gives a nonlinear index change that is positive and ∝ I2, and becomes important for high pulse fluences [37, 53]. However, we evaluate it to be negligible in the experiments treated in Sec. 4.
Note that these simple cascading expressions only hold in the strong cascading limit, which essentially means that the characteristic length for the phase mismatch (i.e. the coherence length) is much shorter than any other characteristic length scale in the system . This can be achieved by fulfillment of two criteria: (a) a strong phase mismatch ΔkL ≫ 2π (i.e. the many up- and down-conversion steps must occur inside the interaction length). (b) A phase mismatch that dominates over the quadratic nonlinearity. To express this we introduce the traditional  quadratic nonlinear strength parameter , where |ℰin| is the peak input electric field. Criterion (b) can then be expressed as Δk ≫ Γ (which essentially ensures an un-depleted FW, see also the discussion in ). In all the cases in Sec. 4 where we correct for cascading contributions, we evaluate that the strong cascading limit is fulfilled.
C. BBO crystal parameters
The crystal beta-barium-borate (β-BaB2O4, BBO) is a negative uniaxial crystal of the point group 3m, which has a transmission range from 189–3500 nm [29, 57]. The Sellmeier equations were taken from Zhang et al. . We used the quadratic nonlinear tensor components measured by Shoji et al. : at λ = 1064 nm |d22| = 2.2 pm/V, |d31| = |d22|0.018 = 0.04 pm/V, d15 = 0.03 pm/V, and d33 = 0.04 pm/V; at λ = 852 nm |d22| = 2.3 pm/V; at λ = 532 nm |d22| = 2.6 pm/V; at λ = 1313 nm |d22| = 1.9 pm/V. Note that Kleinman symmetry has d31 = d15, which is only slightly violated at 1064 nm by these measurements; we therefore throughout this paper assume Kleinman symmetry. The sign of the product d31d22 has been shown to be negative , which means that flipping the crystal 180° gives a changed effective nonlinearity (see also discussion for Fig. 1). An overview of other measurements is given in ; in particular note that the 1064 nm d22 value is similar in most measurements, but the d15 value was historically measured to be higher. This stems from the initial measurements of BBO  that gave d15 = 0.07d22, which then with the d22 = 2.2 pm/V value gives d15 ≃ 0.15 pm/V. A users note of the Shoji et al. values: the 852 and 1064 nm measurements of the |d22| value agree well with Miller’s scaling (they have the same Miller’s delta ), while the 1313 nm measurement seems to lie too low. Therefore when calculating the effective nonlinearity at an arbitrary wavelength the safest option generally is to use the 1064 nm values, where all the nonlinear components were measured, and employ Miller’s scaling to go to other wavelengths.
The Danish Council for Independent Research (grants no. 21-04-0506, 274-08-0479, and 11-106702) is acknowledged for support. Xianglong Zeng acknowledges the support of Marie Curie International Incoming Fellowship from EU (grant No. PIIF-GA-2009–253289) and the financial support from National Natural Science Foundation of China ( 60978004) and Shanghai Shuguang Program ( 10SG38). Frank W. Wise, Jeffrey Moses and Satoshi Ashihara are acknowledged for useful discussions. We would also like to thank some of the authors behind the measurements of the Kerr nonlinearity in the literature for comments and additional details: Mansoor Sheik-Bahae, François Hache, Ji Wei, Tan Huiming, and Rashid Ganeev.
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