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Optical Materials Express

Optical Materials Express

  • Editor: David J. Hagan
  • Vol. 3, Iss. 3 — Mar. 1, 2013
  • pp: 357–382
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The anisotropic Kerr nonlinear refractive index of the beta-barium borate (β-BaB2O4) nonlinear crystal

Morten Bache, Hairun Guo, Binbin Zhou, and Xianglong Zeng  »View Author Affiliations


Optical Materials Express, Vol. 3, Issue 3, pp. 357-382 (2013)
http://dx.doi.org/10.1364/OME.3.000357


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Abstract

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

1. Introduction

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

1. L. A. Ostrovskii, “Self-action of light in crystals,” Pisma Zh. Eksp. Teor. Fiz. 5, 331–334 (1967). [JETP Lett. 5, 272–275 (1967)].

, 2

2. J. M. R. Thomas and J. P. E. Taran, “Pulse distortions in mismatched second harmonic generation,” Opt. Commun. 4, 329–334 (1972). [CrossRef]

], 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 [3

3. R. DeSalvo, D. Hagan, M. Sheik-Bahae, G. Stegeman, E. W. Van Stryland, and H. Vanherzeele, “Self-focusing and self-defocusing by cascaded second-order effects in KTP,” Opt. Lett. 17, 28–30 (1992). [CrossRef] [PubMed]

] (for an early comprehensive review on cascading, see [4

4. G. I. Stegeman, D. J. Hagan, and L. Torner, “χ(2) cascading phenomena and their applications to all-optical signal processing, mode-locking, pulse compression and solitons,” Opt. Quantum Electron. 28, 1691–1740 (1996). [CrossRef]

]). The exciting promise of cascading nonlinearities is seen directly from the scaling properties of the Kerr-like nonlinear index it generates: n2,cascIdeff2/Δk, 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 n2,cascI<0 is accessible by having a positive phase mismatch Δk > 0.

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 [4

4. G. I. Stegeman, D. J. Hagan, and L. Torner, “χ(2) cascading phenomena and their applications to all-optical signal processing, mode-locking, pulse compression and solitons,” Opt. Quantum Electron. 28, 1691–1740 (1996). [CrossRef]

]: 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 n2,cascI given as [3

3. R. DeSalvo, D. Hagan, M. Sheik-Bahae, G. Stegeman, E. W. Van Stryland, and H. Vanherzeele, “Self-focusing and self-defocusing by cascaded second-order effects in KTP,” Opt. Lett. 17, 28–30 (1992). [CrossRef] [PubMed]

]
n2,cascI=2ω1deff2c2ε0n12n2Δk
(1)
where ω1 is the FW frequency, n1 and n2 the linear refractive indices of the FW and SH, Δk = k2 − 2k1, and kj = njωj/c are the wave vectors. The cascaded cubic nonlinearity will compete with the material cubic nonlinearity (the Kerr nonlinear index n2,KerrI), and the total nonlinear refractive index change experienced by the FW can essentially be described as
Δn=n2,totII1=(n2,cascI+n2,KerrI)I1
(2)
where I1 is the FW intensity. Due to the competing material Kerr nonlinearity a crucial requirement for a total defocusing nonlinearity n2,totI<0 is that the phase mismatch is low enough so that |n2,cascI|>n2,KerrI (the so-called Kerr limit). When this happens, cascaded SHG can generate strong spectral broadening through SPM on femtosecond pulses, which can be compensated in a dispersive element. In a pioneering experiment by Wise’s group femtosecond pulse compression was achieved this way [5

5. X. Liu, L.-J. Qian, and F. W. Wise, “High-energy pulse compression by use of negative phase shifts produced by the cascaded χ(2) : χ(2) nonlinearity,” Opt. Lett. 24, 1777–1779 (1999). [CrossRef]

]. Since the nonlinearity is self-defocusing the pulse energy is in theory unlimited as self-focusing effects are avoided [5

5. X. Liu, L.-J. Qian, and F. W. Wise, “High-energy pulse compression by use of negative phase shifts produced by the cascaded χ(2) : χ(2) nonlinearity,” Opt. Lett. 24, 1777–1779 (1999). [CrossRef]

], and it can even be used to heal small-scale and whole-scale self-focusing effects [6

6. K. Beckwitt, F. W. Wise, L. Qian, L. A. Walker II, and E. Canto-Said, “Compensation for self-focusing by use of cascade quadratic nonlinearity,” Opt. Lett. 26, 1696–1698 (2001). [CrossRef]

]. 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. [7

7. S. Ashihara, J. Nishina, T. Shimura, and K. Kuroda, “Soliton compression of femtosecond pulses in quadratic media,” J. Opt. Soc. Am. B 19, 2505–2510 (2002). [CrossRef]

] 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

9. J. Moses and F. W. Wise, “Soliton compression in quadratic media: high-energy few-cycle pulses with a frequency-doubling crystal,” Opt. Lett. 31, 1881–1883 (2006). [CrossRef] [PubMed]

, 11

11. J. Moses, E. Alhammali, J. M. Eichenholz, and F. W. Wise, “Efficient high-energy femtosecond pulse compression in quadratic media with flattop beams,” Opt. Lett. 32, 2469–2471 (2007). [CrossRef] [PubMed]

, 25

25. S. Ashihara, T. Shimura, K. Kuroda, N. E. Yu, S. Kurimura, K. Kitamura, M. Cha, and T. Taira, “Optical pulse compression using cascaded quadratic nonlinearities in periodically poled lithium niobate,” Appl. Phys. Lett. 84, 1055–1057 (2004). [CrossRef]

27

27. B. B. Zhou, A. Chong, F. W. Wise, and M. Bache, “Ultrafast and octave-spanning optical nonlinearities from strongly phase-mismatched quadratic interactions,” Phys. Rev. Lett. 109, 043902 (2012). [CrossRef] [PubMed]

]. Since the defocusing limit and the soliton interaction depend critically on the n2,KerrI value, knowing an exact value of the n2,KerrI 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 ne(θ)=[no2/cos2θ+ne2/sin2θ]1/2, while the o-polarized light always has the same refractive index.

Fig. 1 (a) The definition of the crystal coordinate system XYZ relative to the beam propagation direction k. (b) Top view of the optimal crystal cut for type I ooe SHG in BBO, which has ϕ = −π/2 and θ = θc for perpendicular incidence of an o-polarized FW beam, and the e-polarized SH is generated through type I ooe SHG. The specific value of the cut angle θc depends on the wavelength and the desired application. Angle-tuning the crystal in the paper plane will change the interaction angle θ. Capital letters XYZ are traditionally used to distinguish the crystal coordinate system from the beam coordinate system xyz that has its origin in the k-vector propagation direction.

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 [21

21. P. S. Banks, M. D. Feit, and M. D. Perry, “High-intensity third-harmonic generation,” J. Opt. Soc. Am. B 19, 102–118 (2002). [CrossRef]

].

For an arbitrary input (i.e. FW) polarization various SHG processes can come into play (ooo, ooe, oee, oeo, eee, and eeo). 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 [21

21. P. S. Banks, M. D. Feit, and M. D. Perry, “High-intensity third-harmonic generation,” J. Opt. Soc. Am. B 19, 102–118 (2002). [CrossRef]

]
deffooo=d22cos3ϕ
(3)
deffooe=deffoeo=d31sin(θ+ρ)d22cos(θ+ρ)sin3ϕ
(4)
deffoee=deffeeo=d22cos2(θ+ρ)cos3ϕ
(5)
deffeee=d22cos3(θ+ρ)sin3ϕ+3d31sin(θ+ρ)cos2(θ+ρ)+d33sin3(θ+ρ)
(6)
We remark here that the correct evaluation of the effective nonlinearity requires to take into account the spatial walk-off angle ρ=arctan[tan(θ)no2/ne2]θ (for a negative uniaxial crystal) [29

29. D. N. Nikogosyan, “Beta barium borate (BBO) - A review of its properties and applications,” Appl. Phys. A 52, 359–368 (1991). [CrossRef]

]. 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 ooe 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 ooo 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) [29

29. D. N. Nikogosyan, “Beta barium borate (BBO) - A review of its properties and applications,” Appl. Phys. A 52, 359–368 (1991). [CrossRef]

]. 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, χeff(3)(ωj;ωj), and cross-phase modulation (XPM) nonlinearity, χeff(3)(ωj;ωk) , 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 [30

30. M. Bache and F. W. Wise, “Type-I cascaded quadratic soliton compression in lithium niobate: Compressing femtosecond pulses from high-power fiber lasers,” Phys. Rev. A 81, 053815 (2010). [CrossRef]

]. For a type I interaction (ooe), where the FW is o-polarized and the SH e-polarized, they are
χeff(3)(ω1;ω1)=c11
(7)
χeff(3)(ω2;ω2)=4c10sin(θ+ρ)cos3(θ+ρ)sin3ϕ+c11cos4(θ+ρ)+32c16sin2(2θ+2ρ)+c33sin4(θ+ρ)
(8)
χeff(3)(ω1;ω2)=13c11cos2(θ+ρ)+c16sin2(θ+ρ)+c10sin(2θ+2ρ)sin3ϕ
(9)
Instead for a type 0 eee interaction, as recently considered for cascading quadratic nonlinearities in lithium niobate [27

27. B. B. Zhou, A. Chong, F. W. Wise, and M. Bache, “Ultrafast and octave-spanning optical nonlinearities from strongly phase-mismatched quadratic interactions,” Phys. Rev. Lett. 109, 043902 (2012). [CrossRef] [PubMed]

] and periodically poled lithium niobate [31

31. C. Langrock, M. M. Fejer, I. Hartl, and M. E. Fermann, “Generation of octave-spanning spectra inside reverse-proton-exchanged periodically poled lithium niobate waveguides,” Opt. Lett. 32, 2478–2480 (2007). [CrossRef] [PubMed]

, 32

32. C. R. Phillips, C. Langrock, J. S. Pelc, M. M. Fejer, I. Hartl, and M. E. Fermann, “Supercontinuum generation in quasi-phasematched waveguides,” Opt. Express 19, 18754–18773 (2011). [CrossRef] [PubMed]

], we have
χeff(3)(ω1;ω1)=χeff(3)(ω2;ω2)=χeff(3)(ω1;ω2)
(10)
=4c10sin(θ+ρ)cos3(θ+ρ)sin3ϕ+c11cos4(θ+ρ)+32c16sin2(2θ+2ρ)+c33sin4(θ+ρ)
(11)
We have here used the contracted notation cμmχijkl(3), where the indices i, j, k, l can take the values X, Y, Z and [33

33. B. Boulanger and J. Zyss, International Tables for Crystallography (Springer, 2006), Vol. D: Physical Properties of Crystals, Chap. 1.7: Nonlinear optical properties, pp. 178–219. [CrossRef]

]
forμ:X1Y2Z3form:XXX1YYY2ZZZ3YZZ4YYZ5XZZ6XXZ7XYY8XXY9XYZ9
(12)
Note that by using Kleinman symmetry it is assumed that the chromatic dispersion of the non-linearities is negligible. However, identities like χeff(3)(ω1;ω1)=χeff(3)(ω2;ω2)=χeff(3)(ω1;ω2) as expressed by 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

34. R. C. Miller, “Optical second harmonic generation in piezoelectric crystals,” Appl. Phys. Lett. 5, 17–19 (1964). [CrossRef]

36

36. W. Ettoumi, Y. Petit, J. Kasparian, and J.-P. Wolf, “Generalized Miller formulæ,” Opt. Express 18, 6613–6620 (2010). [CrossRef] [PubMed]

] and also later in Sec. 4.4).

A mistake often seen in the early literature on type I interaction is to take χSPM(3)=χXPM(3), and this mistake gives a 3 times too large XPM term. As shown above the identity χSPM(3)=χXPM(3) 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 [38

38. M. Sheik-Bahae, A. Said, T.-H. Wei, D. Hagan, and E. Van Stryland, “Sensitive measurement of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. 26, 760–769 (1990). [CrossRef]

], 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 n2I coefficient, defined phenomenologically from the general expansion
Δn=n2II+n4II2+
(13)
The intensity is rarely high enough to allow for any contributions but the n2I term. The superscript I underlines that the n2I parameter is the intensity-dependent nonlinear index, as one can also define an electric-field dependent index, typically as Δn=12n2E|E|2. In the other sections of this paper we use n2,KerrI to denote the material Kerr nonlinear refractive index and n2,cascI the Kerr-like cascading nonlinear index from cascaded SHG; this notation should suffice to avoid confusion with the linear refractive index of the SH, n2. As the result of a closed-aperture measurement can be influenced by contributions from multi-photon absorption, the aperture can be removed (open aperture scan). The n2I contributions then vanish, and only multi-photon absorption effects remain. Thus it is possible to separate the two contributions.

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 n2I from thermal effects as well as two-photon excited free carriers [39

39. T. D. Krauss and F. W. Wise, “Femtosecond measurement of nonlinear absorption and refraction in CdS, ZnSe, and ZnS,” Appl. Phys. Lett. 65, 1739–1741 (1994). [CrossRef]

] (for more on these issues, see e.g. [40

40. A. Gnoli, L. Razzari, and M. Righini, “Z-scan measurements using high repetition rate lasers: how to manage thermal effects,” Opt. Express 13, 7976–7981 (2005). [CrossRef] [PubMed]

]). Similarly, a long pulse duration can also lead to more contributions to the measured n2,KerrI 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. χoooo(3)) and n2,KerrI 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 χeff(3)(ω1;ω1)=χoooo(3)(ω1;ω1,ω1,ω1). The connection to the Kerr nonlinear index is n2,KerrI(ω1;ω1)=3χeff(3)(ω1;ω1)/4n12ε0c=3c11/4n12ε0c.

Hache et al. [14

14. F. Hache, A. Zéboulon, G. Gallot, and G. M. Gale, “Cascaded second-order effects in the femtosecond regime in β-barium borate: self-compression in a visible femtosecond optical parametric oscillator,” Opt. Lett. 20, 1556–1558 (1995). [CrossRef] [PubMed]

] 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 ooe 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
n2,KerrI(ω1;ω1)=4.5±1.01020m2/W,λ=0.8μm
(16)
which corresponds to c11 = 4.36 ± 0.97·10−22 m2/V2. They claim that cascading contributions did not contribute for this measurement, but whether this is true cannot be judged with the information at hand, as they did not specify the angle they used for this measurement.

DeSalvo et al. [16

16. R. DeSalvo, A. A. Said, D. 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, 1324–1333 (1996). [CrossRef]

] 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 n2E=11±21014esu at 1064 nm, where the nonlinear index change is defined as Δn=12n2E|E|2, and using Eq. (C1) in [30

30. M. Bache and F. W. Wise, “Type-I cascaded quadratic soliton compression in lithium niobate: Compressing femtosecond pulses from high-power fiber lasers,” Phys. Rev. A 81, 053815 (2010). [CrossRef]

] this corresponds to χ(3) = 2.70±0.49·10−22 m2/V2. The cascading contributions are from ooe and ooo 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 χcasc(3)=[sin2(3ϕ)+cos2(3ϕ)]1.951022m2/V2=1.951022m2/V2, 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 [42

42. I. Shoji, H. Nakamura, K. Ohdaira, T. Kondo, R. Ito, T. Okamoto, K. Tatsuki, and S. Kubota, “Absolute measurement of second-order nonlinear-optical coefficients of β-BaB2O4 for visible to ultraviolet second-harmonic wavelengths,” J. Opt. Soc. Am. B 16, 620–624 (1999). [CrossRef]

], giving a 7% uncertainty on the cascading estimate. This means that when correcting for the cascading contributions we get
c11=4.65±0.511022m2/V2,λ=1.064μm
(17)
corresponding to n2,KerrI(ω1;ω1)=4.80±0.531020m2/W. At 532 nm n2E=21±41014esu was measured, corresponding to χ(3) = 5.22±0.99·10−20 m2/V2. The cascading is smaller here, using Δkooe = Δkooo = 1, 933 mm−1 we get χcasc(3)=0.611022m2/V2 (where we used d22 = 2.6 pm/V as measured at 532 nm [42

42. I. Shoji, H. Nakamura, K. Ohdaira, T. Kondo, R. Ito, T. Okamoto, K. Tatsuki, and S. Kubota, “Absolute measurement of second-order nonlinear-optical coefficients of β-BaB2O4 for visible to ultraviolet second-harmonic wavelengths,” J. Opt. Soc. Am. B 16, 620–624 (1999). [CrossRef]

]), so
c11=5.82±0.991022m2/V2,λ=0.532μm
(18)
corresponding to n2,KerrI(ω1;ω1)=5.87±1.001020m2/W. At 355 nm n2E=14±31014esu was measured, corresponding to χ(3) = 3.54±0.67·10−22 m2/V2. The cascading is small, using Δkooe = Δkooo = 11, 736 mm−1 we get χcasc(3)=0.381022m2/V2 (where Miller’s scaling was used to get the value d22 = −4.4 pm/V), so
c11=3.92±0.681022m2/V2,λ=0.355μm
(19)
corresponding to n2,KerrI(ω1;ω1)=3.81±0.741020m2/W. We must mention that the cascading value is quite uncertain: the SH wavelength is here 177.5 nm, which is very close to the UV poles in the Sellmeier equations (the fit in [43

43. D. Zhang, Y. Kong, and J. Zhang, “Optical parametric properties of 532-nm-pumped beta-barium-borate near the infrared absorption edge,” Opt. Commun. 184, 485–491 (2000). [CrossRef]

] puts the poles at 135 nm and 129 nm for o-and e-polarized light, respectively). Finally, at 266 nm they measured n2E=1±0.31014esu. 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
c11=0.26±0.0781022m2/V2,λ=0.266μm
(20)
corresponding to n2,KerrI(ω1;ω1)=0.24±0.071020m2/W.

Li et al. [18

18. H. Li, F. Zhou, X. Zhang, and W. Ji, “Bound electronic kerr effect and self-focusing induced damage in second-harmonic-generation crystals,” Opt. Commun. 144, 75–81 (1997). [CrossRef]

, 19

19. H. P. Li, C. H. Kam, Y. L. Lam, and W. Ji, “Femtosecond Z-scan measurements of nonlinear refraction in nonlinear optical crystals,” Opt. Mater. 15, 237–242 (2001). [CrossRef]

] 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 [18

18. H. Li, F. Zhou, X. Zhang, and W. Ji, “Bound electronic kerr effect and self-focusing induced damage in second-harmonic-generation crystals,” Opt. Commun. 144, 75–81 (1997). [CrossRef]

] and later with 150 fs 780 nm pulses from a 76 MHz Ti:sapphire oscillator [19

19. H. P. Li, C. H. Kam, Y. L. Lam, and W. Ji, “Femtosecond Z-scan measurements of nonlinear refraction in nonlinear optical crystals,” Opt. Mater. 15, 237–242 (2001). [CrossRef]

]. 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 ooe and ooo, both having the same Δkooe = Δkooo. However, only one come into play in each case: for ϕ = 0 the only nonzero nonlinearity is deffooo=d22 while for ϕ = π/2 the only nonzero nonlinearity is deffooe=d22. Thus, they turn out to have the same cascading contribution, which from Eq. (1) becomes n2,cascI=0.611020m2/W (532 nm), n2,cascI=1.301020m2/W (780 nm), and n2,cascI=2.011020m2/W (1064 nm). When correcting the measured nonlinearities with this value we get for the picosecond experiments at 532 nm [18

18. H. Li, F. Zhou, X. Zhang, and W. Ji, “Bound electronic kerr effect and self-focusing induced damage in second-harmonic-generation crystals,” Opt. Commun. 144, 75–81 (1997). [CrossRef]

]
n2,KerrI(ω1;ω1)=5.41±0.901020m2/W,[100],λ=0.532μm
(21)
n2,KerrI(ω1;ω1)=4.61±0.801020m2/W,[010],λ=0.532μm
(22)
corresponding to c11 = 5.37 ± 0.89 · 10−22 m2/V2 and c11 = 4.58 ± 0.79 · 10−22 m2/V2, respectively. At 1064 nm we get
n2,KerrI(ω1;ω1)=7.01±1.011020m2/W,λ=1.064μm
(23)
corresponding to c11 = 6.79 ± 0.98 · 10−22 m2/V2. For this measurement the polarization direction is unknown. Instead for the femtosecond experiments we get [19

19. H. P. Li, C. H. Kam, Y. L. Lam, and W. Ji, “Femtosecond Z-scan measurements of nonlinear refraction in nonlinear optical crystals,” Opt. Mater. 15, 237–242 (2001). [CrossRef]

]
n2,KerrI(ω1;ω1)=5.30±0.511020m2/W,[100],λ=0.78μm
(24)
n2,KerrI(ω1;ω1)=4.50±0.511020m2/W,[010],λ=0.78μm
(25)
corresponding to c11 = 5.18 ± 0.50·10−22 m2/V2 and c11 = 4.39 ± 0.50·10−22 m2/V2, respectively. When reducing the repetition rate from 76 MHz to 760 kHz they saw similar results.

Ganeev et al. [20

20. R. Ganeev, I. Kulagin, A. Ryasnyanskii, R. Tugushev, and T. Usmanov, “The nonlinear refractive indices and nonlinear third-order susceptibilities of quadratic crystals,” Opt. Spectrosc. 94, 561–568 (2003). [Opt. Spektrosk. 94, 615–623 (2003)]. [CrossRef]

] 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 |n2,cascI|=n2,KerrI. 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 [44

44. R. A. Ganeev, private communication (2012).

]. Then the only cascading channel is ooe, with Δkooe = −655 mm−1, and using Eq. (1) we get n2,cascI=0.261020m2/W that is therefore self-focusing (as they also note themselves). Using the Z-scan method they measured n2I=7.4±2.21020m2/W with the uncertainty estimated to be 30%, so correcting for cascading
n2,KerrI(ω1;ω1)=7.14±2.221020m2/W,λ=1.064μm
(26)
corresponding to c11 = 6.92 ± 2.15 · 10−22 m2/V2. At 532 nm they measured n2I=8.0±2.41020m2/W. Here cascading is more significant as Δkooe = −194 mm−1, ρ = 4.8°, deff = 1.50 pm/V and n2,cascI=2.221020m2/W. Therefore the Kerr value corrected for cascading becomes
n2,KerrI(ω1;ω1)=5.24±2.411020m2/W,λ=0.532μm
(27)
which corresponds to c11 = 5.78 ± 2.41 · 10−22 m2/V2.

Moses et al. [15

15. J. Moses, B. A. Malomed, and F. W. Wise, “Self-steepening of ultrashort optical pulses without self-phase modulation,” Phys. Rev. A 76, 021802(R) (2007). [CrossRef]

] 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 ooe 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 n2,totI=0 then a reverse calculation though Eq. (1) gives n2,KerrI=n2,cascI. Using the value deff = 1.8 pm/V they inferred the value n2,KerrI=4.6±0.91020m2/W (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 [45

45. J. A. Moses, private communication (2010).

]. 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
n2,KerrI(ω1;ω1)=5.54±0.981020m2/W,λ=0.8μm
(28)
which corresponds to c11 = 5.41 ± 0.95 · 10−22 m2/V2.

Fig. 2 (a) Spectral bandwidth at 10 dB below the peak of the experimentally recorded spectra vs. Δk for two different intensities. (b) Similar data from numerical simulations of the coupled SEWA equations [9, 37] (using the anisotropic quadratic and cubic nonlinear coefficients in Appendix C and Table 1, respectively). The input conditions were similar to the experiment (Gaussian pulse with 39.3 nm bandwidth and a group-delay dispersion of −330 fs2). (c) and (d) show the spectra on a linear and log scale right at the transition (Δk = 88 mm−1), where the total SPM is near zero. (e) Various spectra for I = 200 GW/cm2.

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) [24

24. M. Sheik-Bahae, D. Hutchings, D. Hagan, and E. Van Stryland, “Dispersion of bound electron nonlinear refraction in solids,” IEEE J. Quantum Electron. 27, 1296 –1309 (1991). [CrossRef]

]. 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 [16

16. R. DeSalvo, A. A. Said, D. 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, 1324–1333 (1996). [CrossRef]

], and by using the BBO band-gap value Eg = 6.2 eV [16

16. R. DeSalvo, A. A. Said, D. 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, 1324–1333 (1996). [CrossRef]

]. 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. [16

16. R. DeSalvo, A. A. Said, D. 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, 1324–1333 (1996). [CrossRef]

] 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 n2,KerrI near-IR values actually agree extremely well with the 2BM n2,KerrI 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

35. J. J. Wynne, “Optical third-order mixing in GaAs, Ge, Si, and InAs,” Phys. Rev. 178, 1295–1303 (1969). [CrossRef]

, 36

36. W. Ettoumi, Y. Petit, J. Kasparian, and J.-P. Wolf, “Generalized Miller formulæ,” Opt. Express 18, 6613–6620 (2010). [CrossRef] [PubMed]

]
Δijkl=χijkl(3)χi(1)χj(1)χk(1)χl(1)=χ(3)(ωi;ωj,ωk,ωl)χ(1)(ωi)χ(1)(ωj)χ(1)(ωk)χ(1)(ωl)
(30)
where χj(1)(ω0)=nj2(ω0)1 is the linear susceptibility. We here excluded the UV measurements below 400 nm as they obviously are not represented well by Miller’s scaling. Since Miller’s delta should be independent of wavelength this should allow us to calculate an average value based on the measurements at different wavelengths. We confirmed this hypothesis by checking that that Miller’s delta vs. measurement wavelength could be fitted to a line with a near-zero slope, before we calculated the weighted average (with a weight given by σj2 where σj is the estimated error of the jth measurement) as
Δ1111=52.3±7.7×1024m2/V2
(31)
where the standard deviation was calculated using an unbiased weighted average. This value corresponds to c11 = 5.0 · 10−22 m2/V2 and n2,KerrI(ω1;ω1)=5.11020m2/W at 800 nm. We see in 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.

Fig. 3 Summary of the experimental data for n2,KerrI-values from the literature corresponding to the c11 nonlinear susceptibility coefficient ( n2,KerrI=3c11/4n12ε0c). The plotted values are the ones reported in Sec. 4.2, i.e. the data values do not necessarily correspond to the ones reported in the literature. References: Tan et al. 1993: [13]; Hache et al. 1995: [14]; DeSalvo et al. 1996 [16]; Li et al. 1997 [18]; Li et al. 2001 [19]; Ganeev et al. 2003 [20]; Moses et al. 2007 [15]. The theoretically predicted electronic nonlinearity is calculated with the 2-band model [24]. The average value curve was calculated through a weighted mean of the Miller’s delta from all data, except the UV measurements below 400 nm, and the shaded areas denoted “σ” and “2σ” represent one and two standard deviations, respectively.

4.3. Other tensor components

Banks et al. [21

21. P. S. Banks, M. D. Feit, and M. D. Perry, “High-intensity third-harmonic generation,” J. Opt. Soc. Am. B 19, 102–118 (2002). [CrossRef]

] 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. [42

42. I. Shoji, H. Nakamura, K. Ohdaira, T. Kondo, R. Ito, T. Okamoto, K. Tatsuki, and S. Kubota, “Absolute measurement of second-order nonlinear-optical coefficients of β-BaB2O4 for visible to ultraviolet second-harmonic wavelengths,” J. Opt. Soc. Am. B 16, 620–624 (1999). [CrossRef]

]. (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 [21

21. P. S. Banks, M. D. Feit, and M. D. Perry, “High-intensity third-harmonic generation,” J. Opt. Soc. Am. B 19, 102–118 (2002). [CrossRef]

] define Cμm = χ(3)/4, we have Cμm = cμm/4, so
c10(THG)=0.24±0.041022m2/V2,λ=1.053μm
(32)
The uncertainty of the measurement was not reported, but for a similar fit it was reported to be 15%, which is what we used above. They also measured a three times larger value by using different quadratic nonlinearities, namely d15 = 0.16 pm/V (as opposed to d15 = 0.03 pm/V).

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.

Table 1. Summary of the literature measurements of the cubic nonlinearities. The column (A) reports the original data and (B) our updated values, if any.

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Table 2. Proposed nonlinear susceptibilities for the BBO anisotropic Kerr nonlinearity. Note the c16 value is deduced from the c11 and c10 coefficients, and the c33 value is deduced from the c11, c10 and c16 coefficients. The Miller’s delta Δijkl are calculated from Eq. (30).

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4.5. Implications for cascaded pulse compression in BBO

Fig. 4 (a) Compression diagram for BBO type I cascaded SHG. In order to excite solitons Δk must be kept below the Kerr limit (red line). Optimal compression occurs when the cascaded nonlinearities dominate over GVM effects (Δk > Δksr, above the black line). Note that Δksr is calculated for the full dispersion case, and that for λ1 > 1.49 μm the FW GVD becomes anomalous. We have also indicated the operation wavelengths of Cr:forsterite, Yb and Ti:Sapphire based amplifiers. The Kerr limit employs Miller’s rule to calculate the nonlinear quadratic and cubic susceptibilities at other wavelengths, case (1) corresponds to the ‘old’ Kerr value n2,KerrI(ω1;ω1)=3.651020m2/W@850nm (taken from [17]) and case (2) corresponds to the Kerr value proposed in this work Δ1111 = 52.3 m2/V2, corresponding to n2,KerrI(ω1;ω1)=5.061020m2/W@850nm (b) Numerical simulation of the case marked with ‘x’ in (a): a 50 fs@1030 nm Iin = 500 GW/cm2 pulse propagating in a 25 mm BBO crystal with θ = 19.1°, ρ = 2.8° and ϕ = −90° (Δk = 80 mm−1). The simulations used the plane-wave SVEA equations [(Eqs. (50)(51)] including full dispersion and extended to include self-steepening, and case (1) assumes an isotropic Kerr nonlinearity and n2,KerrI(ω1;ω1)=3.651020m2/W@850nm, while (2) uses the anisotropic coefficients of Table 2, and Eq. (30) to calculate the nonlinear coefficients at 1030 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 Neff2|n2,tot|Iin (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

5. X. Liu, L.-J. Qian, and F. W. Wise, “High-energy pulse compression by use of negative phase shifts produced by the cascaded χ(2) : χ(2) nonlinearity,” Opt. Lett. 24, 1777–1779 (1999). [CrossRef]

, 7

7. S. Ashihara, J. Nishina, T. Shimura, and K. Kuroda, “Soliton compression of femtosecond pulses in quadratic media,” J. Opt. Soc. Am. B 19, 2505–2510 (2002). [CrossRef]

] 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 [7

7. S. Ashihara, J. Nishina, T. Shimura, and K. Kuroda, “Soliton compression of femtosecond pulses in quadratic media,” J. Opt. Soc. Am. B 19, 2505–2510 (2002). [CrossRef]

], as the intensity had to be kept low in order to avoid severe GVM-induced self-steepening. (Note that the simulations in [7

7. S. Ashihara, J. Nishina, T. Shimura, and K. Kuroda, “Soliton compression of femtosecond pulses in quadratic media,” J. Opt. Soc. Am. B 19, 2505–2510 (2002). [CrossRef]

] 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 [10

10. J. Moses and F. W. Wise, “Controllable self-steepening of ultrashort pulses in quadratic nonlinear media,” Phys. Rev. Lett. 97, 073903 (2006). [CrossRef] [PubMed]

] was done in the stationary regime, and indeed a few-cycle soliton was observed.

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 [37

37. M. Bache, J. Moses, and F. W. Wise, “Scaling laws for soliton pulse compression by cascaded quadratic nonlinearities,” J. Opt. Soc. Am. B 24, 2752–2762 (2007). [CrossRef]

]) criterion for this to happen is Δk > 0 as well as Δk low enough for |n2,cascI|>n2,KerrI(ω1;ω1). 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 χeff(3)(ω1;ω2)=c11/3, dashed grey line). Instead the SH SPM term lies well below the isotropic value [found taking χeff(3)(ω2;ω2)=c11, 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. [30

30. M. Bache and F. W. Wise, “Type-I cascaded quadratic soliton compression in lithium niobate: Compressing femtosecond pulses from high-power fiber lasers,” Phys. Rev. A 81, 053815 (2010). [CrossRef]

]. 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 [46

46. H. Guo, X. Zeng, B. Zhou, and M. Bache, “Electric field modeling and self-steepening counterbalance of cascading nonlinear soliton pulse compression,” (submitted to J. Opt. Soc. Am. B ), arXiv:1210.5903.

].

Fig. 5 Estimating the operating parameters vs. FW wavelength for cascaded SHG in BBO. (a) The θ-range for which Δk > 0 is achieved as well as |n2,cascI|>n2,KerrI(ω1;ω1). (b) The XPM term χeff(3)(ω1;ω2) and SH SPM term χeff(3)(ω2;ω2), both normalized to the FW SPM term χeff(3)(ω1;ω1). The range indicated corresponds to the θ range in (a). The dashed lines indicate the isotropic limit. All nonlinear coefficients are scaled to other wavelengths using Miller’s rule, and we used the SHG nonlinearities at 1064 nm from Appendix C as well as the cubic nonlinear parameters listed in Sec. 5. We also took ϕ = −π/2.

5. Summary

We have analyzed the Kerr nonlinear index in BBO from a number of experiments [13

13. H. Tan, G. P. Banfi, and A. Tomaselli, “Optical frequency mixing through cascaded second-order processes in beta-barium borate,” Appl. Phys. Lett. 63, 2472–2474 (1993). [CrossRef]

21

21. P. S. Banks, M. D. Feit, and M. D. Perry, “High-intensity third-harmonic generation,” J. Opt. Soc. Am. B 19, 102–118 (2002). [CrossRef]

], 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 n2,KerrI=5.11020m2/W, 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 [24

24. M. Sheik-Bahae, D. Hutchings, D. Hagan, and E. Van Stryland, “Dispersion of bound electron nonlinear refraction in solids,” IEEE J. Quantum Electron. 27, 1296 –1309 (1991). [CrossRef]

], 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.

Appendices

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 [30

30. M. Bache and F. W. Wise, “Type-I cascaded quadratic soliton compression in lithium niobate: Compressing femtosecond pulses from high-power fiber lasers,” Phys. Rev. A 81, 053815 (2010). [CrossRef]

], 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
2E(r,t)=μ02t2D(r,t)
(36)
where D = ε0E + P is the displacement field, and P is the induced polarization. In frequency domain, where E(r,ω)=dteiωtE(r,t) and similarly for P(ω), we may separate the linear and nonlinear polarization response as
P(r,ω)=ε0χ_(1)(ω)E(r,ω)+PNL(r,ω),
(37)
where χ̱(1)(ω) is the linear susceptibility tensor. The nonlinear part is usually expanded in a power series as
ε01PNL(r,ω)=χ_(2)(ω):E(r,ω)E(r,ω)+χ_(3)(ω)E(r,ω)E(r,ω)E(r,ω)+
(38)
The tensor products are calculated by summation over the indices ε01PNL,i=jkχijk(2)EjEk+jklχijkl(3)EjEkEl+. By assuming plane waves (no diffraction), we have 2=2z2 so
2z2E(z,ω)+ω2c2ε_(ω)E(z,ω)=ω2c2ε01PNL(z,ω)
(39)
where ε̱(ω) = 1 + χ̱(1)(ω) is the relative permittivity tensor, and we used ε0μ0 = 1/c2.

For SHG we define the electric field envelopes j for the FW (frequency ω1) and the SH (frequency ω2 = 2ω1), as well as envelopes for the nonlinear polarization response 𝒫NL,j
E(z,t)=12[u11(z,t)eik1(ω1)ziω1t+u22(z,t)eik2(ω2)ziω2t+c.c.]
(40)
PNL(z,t)=12[u1𝒫NL,1(z,t)eik1(ω1)ziω1t+u2𝒫NL,2(z,t)eik2(ω2)ziω2t+c.c.]
(41)
where the wave numbers are defined as kj2(ω)=ω2c2εj(ω) with εj(ω) = uj·χ̱(1)(ω), and where u is a unit polarization vector. In a lossless medium εj is real, and the linear refractive index is given by nj(ω)=εj(ω). The envelope wave equations are therefore
2j(z,ω)z2+2ikj(ωj)j(z,ω)z+[kj2(ω)kj2(ωj)]j(z,ω)=ω2c2ε01𝒫NL,j(z,ω).
(42)
The SVEA assumes |jz|kj(ωj)|j| giving
ij(z,ω)z+[kj(ω)kj(ωj)]j(z,ω)=ωj2nj(ωj)cε01𝒫NL,j(z,ω)
(43)
where we used the approximation kj2(ω)kj2(ωj)2kj(ωj)[kj(ω)kj(ωj)][54, p. 34]. (A more accurate approach is to first go into the co-moving reference frame and then perform this expansion [55, p. 14], but eventually the SVEA will discard terms so the result will be the same.) At this time, we have also used that the assumption of a slow envelope compared to the fast oscillating carrier term ejt means that the nonlinear polarization term on the right-hand side can be approximated as ω22kj(ωj)c2ε01PNL,j(ω)ωj22kj(ωj)c2ε01PNL,j(ω). We can now perform the Taylor expansion of the wave number as kj(ω)=m=0m!1kj(m)(ωm)(ωωj), where the dispersion coefficients are defined as kj(m)(ω)=dmkj(ω)/dωm. Following this, we multiply both sides with eiΩt where the frequency detuning is Ω = ωωj, and perform an inverse Fourier transform over Ω to time domain. This gives
[iz+ikj(1)(ωj)t+D^j]j(z,t)=ωj2nj(ωj)cε01𝒫NL,j(z,t)
(44)
This is the SVEA equations in the stationary laboratory frame, describing a pulse envelope propagating with group velocity 1/kj(1)(ωj). The time-domain dispersion operator is
D^j=m=2kj(m)(ωj)imm!mtm
(45)
and it can be truncated at any order to study effects higher-order dispersion.

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

30. M. Bache and F. W. Wise, “Type-I cascaded quadratic soliton compression in lithium niobate: Compressing femtosecond pulses from high-power fiber lasers,” Phys. Rev. A 81, 053815 (2010). [CrossRef]

, 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 PNL(2)(z,t)=χ_(2):E(r,t)E(r,t), 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: ε01ujPNL(2)(z,t). The only relevant terms are those that oscillate with the fast carrier frequency oscillations of the field ejt, as this defines the polarization envelope 𝒫. Doing this we get for the FW and SH fields
ε01𝒫NL,1(2)(z,t)=χeff(2)1*(z,t)2(z,t)eiΔkz,ε01𝒫NL,2(2)(z,t)=χeff(2)1212(z,t)eiΔkz
(46)
where the phase mismatch is Δk = k2(ω2) − 2k1(ω1). The effective susceptibility is calculated from the combination χeff(2)=ujχ_(2):ukul that appears from isolating the ejt term. The χ̱(2) is a rank 3 tensor with 27 different elements, but due to permutation symmetry in the the crystal axes indices there are only 18 nonzero elements [33

33. B. Boulanger and J. Zyss, International Tables for Crystallography (Springer, 2006), Vol. D: Physical Properties of Crystals, Chap. 1.7: Nonlinear optical properties, pp. 178–219. [CrossRef]

]. 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 12 in front of our envelope definition.

The SVEA equations can be extended to the slowly-evolving wave equations (SEWA) [9

9. J. Moses and F. W. Wise, “Soliton compression in quadratic media: high-energy few-cycle pulses with a frequency-doubling crystal,” Opt. Lett. 31, 1881–1883 (2006). [CrossRef] [PubMed]

, 37

37. M. Bache, J. Moses, and F. W. Wise, “Scaling laws for soliton pulse compression by cascaded quadratic nonlinearities,” J. Opt. Soc. Am. B 24, 2752–2762 (2007). [CrossRef]

] by including self-steepening effects, and also the non-instantaneous (delayed) Raman effect can straightforwardly be included [37

37. M. Bache, J. Moses, and F. W. Wise, “Scaling laws for soliton pulse compression by cascaded quadratic nonlinearities,” J. Opt. Soc. Am. B 24, 2752–2762 (2007). [CrossRef]

, 46

46. H. Guo, X. Zeng, B. Zhou, and M. Bache, “Electric field modeling and self-steepening counterbalance of cascading nonlinear soliton pulse compression,” (submitted to J. Opt. Soc. Am. B ), arXiv:1210.5903.

]. For simplicity such effects are neglected here.

B. Reduced nonlinear Schrödinger equation in the strong cascading limit

There is also a quintic nonlinearity from cascading mixed with the XPM term of the FW
χcasc(5)=12ω12χeff(3)(ω1;ω2)deff25n22c2Δk2,n4,cascI=4ω12nKerrI(ω1;ω2)deff2Δk2n12n2ε0c3
(56)
where the latter comes from the relationship n4I=5χ(5)/(4n13ε02c2), and the quintic nonlinear refractive index n4I was defined in Eq. (13). This term gives a nonlinear index change that is positive and ∝ I2, and becomes important for high pulse fluences [37

37. M. Bache, J. Moses, and F. W. Wise, “Scaling laws for soliton pulse compression by cascaded quadratic nonlinearities,” J. Opt. Soc. Am. B 24, 2752–2762 (2007). [CrossRef]

, 53

53. X. Zeng, H. Guo, B. Zhou, and M. Bache, “Soliton compression to few-cycle pulses with a high quality factor by engineering cascaded quadratic nonlinearities,” Opt. Express 20, 27071–27082 (2012). ArXiv:1210.5928. [CrossRef] [PubMed]

]. 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 [51

51. M. Bache, O. Bang, J. Moses, and F. W. Wise, “Nonlocal explanation of stationary and nonstationary regimes in cascaded soliton pulse compression,” Opt. Lett. 32, 2490–2492 (2007). [CrossRef] [PubMed]

]. 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 [3

3. R. DeSalvo, D. Hagan, M. Sheik-Bahae, G. Stegeman, E. W. Van Stryland, and H. Vanherzeele, “Self-focusing and self-defocusing by cascaded second-order effects in KTP,” Opt. Lett. 17, 28–30 (1992). [CrossRef] [PubMed]

] quadratic nonlinear strength parameter Γ=ω1deff|in|/(n1n2c), where |in| 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 [56

56. M. Bache, F. Eilenberger, and S. Minardi, “Higher-order Kerr effect and harmonic cascading in gases,” Opt. Lett. 37, 4612–4614 (2012). [PubMed]

]). 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

29. D. N. Nikogosyan, “Beta barium borate (BBO) - A review of its properties and applications,” Appl. Phys. A 52, 359–368 (1991). [CrossRef]

, 57

57. C. Chen, B. Wu, A. Jiang, and G. You, “A new-type ultraviolet SHG crystal - beta-BaB2O4,” Sci. Sin., Ser. B 28, 235–243 (1985).

]. The Sellmeier equations were taken from Zhang et al. [43

43. D. Zhang, Y. Kong, and J. Zhang, “Optical parametric properties of 532-nm-pumped beta-barium-borate near the infrared absorption edge,” Opt. Commun. 184, 485–491 (2000). [CrossRef]

]. We used the quadratic nonlinear tensor components measured by Shoji et al. [42

42. I. Shoji, H. Nakamura, K. Ohdaira, T. Kondo, R. Ito, T. Okamoto, K. Tatsuki, and S. Kubota, “Absolute measurement of second-order nonlinear-optical coefficients of β-BaB2O4 for visible to ultraviolet second-harmonic wavelengths,” J. Opt. Soc. Am. B 16, 620–624 (1999). [CrossRef]

]: 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 [58

58. R. S. Klein, G. E. Kugel, A. Maillard, A. Sifi, and K. Polgar, “Absolute non-linear optical coefficients measurements of BBO single crystal and determination of angular acceptance by second harmonic generation,” Opt. Mater. 22, 163–169 (2003). [CrossRef]

], 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 [59

59. R. C. Eckardt and G. C. Cattela, “Characterization techniques for second-order nonlinear optical materials,” Proc. SPIE 5337, 1–10 (2004). [CrossRef]

]; 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 [57

57. C. Chen, B. Wu, A. Jiang, and G. You, “A new-type ultraviolet SHG crystal - beta-BaB2O4,” Sci. Sin., Ser. B 28, 235–243 (1985).

] that gave d15 = 0.07d22[29

29. D. N. Nikogosyan, “Beta barium borate (BBO) - A review of its properties and applications,” Appl. Phys. A 52, 359–368 (1991). [CrossRef]

], 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 [42

42. I. Shoji, H. Nakamura, K. Ohdaira, T. Kondo, R. Ito, T. Okamoto, K. Tatsuki, and S. Kubota, “Absolute measurement of second-order nonlinear-optical coefficients of β-BaB2O4 for visible to ultraviolet second-harmonic wavelengths,” J. Opt. Soc. Am. B 16, 620–624 (1999). [CrossRef]

]), 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.

Acknowledgments

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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X. Liu, L.-J. Qian, and F. W. Wise, “High-energy pulse compression by use of negative phase shifts produced by the cascaded χ(2) : χ(2) nonlinearity,” Opt. Lett. 24, 1777–1779 (1999). [CrossRef]

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M. Bache and F. W. Wise, “Type-I cascaded quadratic soliton compression in lithium niobate: Compressing femtosecond pulses from high-power fiber lasers,” Phys. Rev. A 81, 053815 (2010). [CrossRef]

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52.

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53.

X. Zeng, H. Guo, B. Zhou, and M. Bache, “Soliton compression to few-cycle pulses with a high quality factor by engineering cascaded quadratic nonlinearities,” Opt. Express 20, 27071–27082 (2012). ArXiv:1210.5928. [CrossRef] [PubMed]

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57.

C. Chen, B. Wu, A. Jiang, and G. You, “A new-type ultraviolet SHG crystal - beta-BaB2O4,” Sci. Sin., Ser. B 28, 235–243 (1985).

58.

R. S. Klein, G. E. Kugel, A. Maillard, A. Sifi, and K. Polgar, “Absolute non-linear optical coefficients measurements of BBO single crystal and determination of angular acceptance by second harmonic generation,” Opt. Mater. 22, 163–169 (2003). [CrossRef]

59.

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OCIS Codes
(160.4330) Materials : Nonlinear optical materials
(190.3270) Nonlinear optics : Kerr effect
(190.4400) Nonlinear optics : Nonlinear optics, materials
(190.5530) Nonlinear optics : Pulse propagation and temporal solitons
(190.7110) Nonlinear optics : Ultrafast nonlinear optics

ToC Category:
Nonlinear Optical Materials

History
Original Manuscript: December 18, 2012
Manuscript Accepted: January 17, 2013
Published: February 4, 2013

Citation
Morten Bache, Hairun Guo, Binbin Zhou, and Xianglong Zeng, "The anisotropic Kerr nonlinear refractive index of the beta-barium borate (β-BaB2O4) nonlinear crystal," Opt. Mater. Express 3, 357-382 (2013)
http://www.opticsinfobase.org/ome/abstract.cfm?URI=ome-3-3-357


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References

  1. L. A. Ostrovskii, “Self-action of light in crystals,” Pisma Zh. Eksp. Teor. Fiz.5, 331–334 (1967). [JETP Lett.5, 272–275 (1967)].
  2. J. M. R. Thomas and J. P. E. Taran, “Pulse distortions in mismatched second harmonic generation,” Opt. Commun.4, 329–334 (1972). [CrossRef]
  3. R. DeSalvo, D. Hagan, M. Sheik-Bahae, G. Stegeman, E. W. Van Stryland, and H. Vanherzeele, “Self-focusing and self-defocusing by cascaded second-order effects in KTP,” Opt. Lett.17, 28–30 (1992). [CrossRef] [PubMed]
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