1. Introduction

Laser (optical) ceramics possesses a number of advantages as compared to single crystals, including large aperture, homogeneity and high concentration of doping ions, a possibility of controlling physicochemical and spectroscopic characteristics, and low cost. Ceramics had poor optical quality for quite a long time. Active investigations of laser ceramics were stimulated by the development at the end of the 1990s of the technology of its sintering by the method of isostatic pressing [1] and vacuum sintering [2]. In the 2000s the quality of ceramics became comparable to the quality of the best single crystals. Currently, optical ceramics is manufactured from diverse cubic crystals, including hard to grow ones, for example, Tb₂O₃ [3]. Ceramics is widely used not only as an active laser medium (when doped by Nd, Yb, Ho, Тm, Pr and other ions), but also in passive Q-switched modulators [4], optical windows [5], Faraday isolators [6], and Raman lasers [7]. For the high peak/average power laser applications, ceramics has two major advantages: i) large size (as glass) and high thermal conductivity (like crystal) and ii) a possibility to manufacture unique materials, which are very hard to grow as a single crystal, i.e. sesquioxides doped by two or three different ions. A large amount of works were devoted to measurement of the properties (material constants) of ceramics and to their comparison with the properties of single crystals (see, e.g., [8, 9] and references therein). It was shown that ceramics possesses better microhardness and a larger thermal destruction parameter, but the majority of ceramics and single crystal properties coincide. In particular, they include radiation and absorption spectra, transition cross-sections, upper level lifetime, thermal conductivity, temperature dependence of the index of refraction, laser damage threshold, and chemical stability.

Ceramics is an ensemble of densely packed grains having characteristic size l_g ranging from several to 100 microns. The thickness of the boundary between the grains is less than 1 nm. The crystallographic axes orientation in each grain is random. Crystals with cubic symmetry have a unitary linear tensor of dielectric permittivity like glasses. Consequently, the index of refraction is the same for any single crystal orientation, as well as for ceramics. The nature of ceramics manifests itself in the presence of the effects dependent on crystal orientation, for instance, at thermally induced birefringence determined by the piezooptical (or photoelasticity) tensor whose symmetry in cubic crystals is lower than in glasses. This difference is characterized by the optical anisotropy parameter equal to unity for glasses. Taking into consideration the photoelastic effect, the dielectric permittivity tensor ceases to be unitary, hence, the refractive index depends on crystal orientation and radiation polarization. Thermal effects in laser ceramics were studied in a number of works [10–12], where it was shown that thermally induced distortions, on the average over beam aperture, are the same as in glass with slightly different values of thermooptical constants [13, 14]. However, some effects inherent in ceramics have no analogs either in glass or in single crystals, namely, small-scale (of order grain size) spatial phase and polarization modulation of a laser beam transmitted through a ceramic element with thermal load. The physical explanation of this fact is the following: two arbitrary beams spaced apart by a distance larger than grain size pass through a statistically independent set of grains, thereby acquiring different polarizations and phase distortions.

Much like the piezoelasticity tensor, the tensor of cubic nonlinearity χ⁽³⁾ of isotropic (cubic) crystals has a lower symmetry than χ⁽³⁾ of glasses. In the frame of reference related to the crystallographic axes, the tensor χ⁽³⁾ for isotropic crystals is defined by χ_xxxx and the cubic nonlinear anisotropy parameter σ

Of the greatest interest are four χ⁽³⁾-effects: the generation of cross-polarized wave (XPW), self-phase modulation (SPM), large-scale whole-beam self-focusing (WBSF), and small-scale self-focusing (SSSF). The first three effects may be both, parasitic and useful: XPW is used for increasing the contrast of femtosecond pulses, SPM for pulse spectrum broadening aimed at pulse shortening, and WBSF for mode-locking. SSSF is always a parasitic effect as it leads to breakdown of optical elements.

The impact of single crystal orientation was studied in detail only for XPW [18–22], where it was shown that the XPW efficiency for the [110] orientation is higher than for the [001] orientation. However, optimal orientation was not found. Crystal orientations and radiation polarization at which the XPW efficiency is equal to zero were found in [22, 23]. The effective nonlinearity determining SPM and WBSF was calculated for these particular cases. SSSF was also studied only for three particular cases: for the [001], [110] and [111] orientations of a single crystal [24, 25]. To the best of our knowledge, none of these χ⁽³⁾-effects was discussed in the literature for ceramics.

In the present work we undertook a detailed study of XPW, SPM and WBSF in ceramics. We revealed qualitative and quantitative differences of these effects in ceramics and in single crystals. As ceramics consists of single crystal grains with random orientation, it is reasonable to consider first the case of a single crystal of arbitrary orientation. This will be done in Section 2. The efficiency η of XPW for ceramics and nonlinear incursion of phase Φ characterizing SPM and WBSF will be addressed in Section 3.

2. χ⁽³⁾-effects in a single crystal of arbitrary orientation

Equations describing the propagation along the z-axis of a plane monochromatic wave in an isotropic crystal with cubic nonlinearity may be written for convenience in dimensionless variables [19]:

In the earlier studies the system of Eqs. (2) was solved numerically, but it allows a significant simplification, namely, it may be reduced to one Eq. for the variable К = E_y/E_x:

We will restrict our consideration to the linear input polarization that is assumed without restricting generalization to be parallel to the y-axis: E_x(z = 0) = 0, i.е. K(z = 0) = 0. The solution of Eq. (9) К(z = L) allows easy calculation of the XPW efficiency η

Before we pass over to ceramics, let us ascertain which single crystal orientations are regarded to be dedicated. For the [111] orientation (α = π/4, tg²β = 2), the XPW efficiency η = 0 for an arbitrary value of γ, i.e., for any linear polarization. For all other orientations there is an optimal angle γ_opt at which η takes on its maximum value η_max. The authors of [18, 19, 21] showed that η_max for the [110] orientation (α = 0, β = π/4) is higher than for the [001] orientation (α = 0, β = 0). Note, however, that other orientations have not been studied. For each orientation, i.e., for each pair of angles α and β, we solved Eq. (9) numerically and found γ_opt and η_max. The curves for η_max(α,β) and γ_opt(α,β) for the BaF₂ crystal (σ = −1.2 [19]) are plotted in Fig. 1. Both functions have a period of π/2 for both angles. It is clear from the Fig. 1 that, for small values of В-integral, the [110] orientation is optimal and for large values of В-integral there is another optimal orientation. Note that one orientation corresponds to six pairs of extreme values of angles α and β [see Fig. 1 and Table 1]. Hereinafter we will not distinguish between the physically equivalent orientations, e.g. [110], [101], [011], [1 ̅10] and so on.

 

Fig. 1 Maximum XPW efficiency η_m (upper row) and optimal angle γ_opt (lower row) of BaF₂ crystal (σ = −1.2) for different values of В-integral.

Download Full Size | PDF

Table 1. Optimal single crystal orientation of XPW.

View Table | View all tables in this article

From Eq. (8) it is clear that the linear polarization of radiation does not change, i.e., K(z) = K(0) = 0, if Γ₁ = 0, which coincides with the condition obtained in [23]. For the [111] orientation this condition holds true for arbitrary γ. For any other crystal orientation, there exist four linear polarizations (four values of the angle γ = γ₀), for which K(z) = K(0), two of which are stable and two unstable [22]. In these cases the anisotropic properties of the crystal vanish and it becomes equivalent to glass with effective value of nonlinearity χ_eff varying from χ_xxxx to (1-2σ/3)χ_xxxx, depending on orientation, i.e., χ_eff>χ_xxxx, if σ<0. Consequently, all χ⁽³⁾-effects (SPM, WBSF, and SSSF) are like in glass with χ_xxxx = χ_eff, for example, Φ = Вχ_eff/χ_xxxx. The values of γ₀ and χ_eff for some of these orientations are given in Table 2. In particular, if σ = 1.5 (we do not know crystals with such σ), then for the [110] orientation there exist two polarizations (γ₀ = 35.3 degrees and γ₀ = 144.7 degrees) for which χ_eff = 0, i.e., there is no cubic nonlinearity. The values of χ_eff in Table 2 coincide with the results of studies [24] of SSSF in crystals with [111], [001] and [110] orientations. For instance, for [111] SSSF will develop at the beam intensity (1-σ/2) times lower that for [001] at γ = 0.

Table 2. Effective nonlinearity χ_eff for ceramics and some orientations for which XPW is not observed.

View Table | View all tables in this article

In a general case of varying polarization, i.e. K(z)≠const and η≠0, it is impossible to unambiguously introduce the concept of effective nonlinearity, as the problem does not reduce to the case of isotropic medium in principle. At the same time, for many applications it is natural to determine χ_eff by analogy with the previous paragraph on the basis of the nonlinear phase incursion in initial polarization Φ Eq. (11). Note that in this case χ_eff also changes from χ_xxxx to (1-2σ/3)χ_xxxx.

During propagation, circular polarization (K = ± i) does not change only for two orientations [001] and [111]. That was pointed to in [23] and can be readily proved by substituting K = ± i into Eq. (8). The effective nonlinearity χ_eff is identical for the left and right polarizations and is presented in Table 2.

For the laser media, one should include gain in Eq. (2), which makes the problem much more complicated. It is a subject for a future study. Here we just mention that laser amplification (cross-section) is isotropic in a cubic crystal, hence, gain impact is the same in crystals and ceramics.

3. χ⁽³⁾-effects in ceramics

3.1 Formulation of the problem

Ceramics is an ensemble of single crystal grains having characteristic size from several microns to a hundred microns with very thin (less than 1 nm) boundaries between them. The physical mechanism of the orientation dependent χ⁽³⁾-effects in ceramics is the same as in single crystals: anisotropy of tensor χ⁽³⁾ originated by cubic symmetry. The principal difference from a single crystal is that the crystallographic axes orientation in each grain is random. The problem of wave propagation in ceramics with cubic nonlinearity reduces to a consecutive solution of the problem for a single crystal of arbitrary orientation considered in the previous section (see also the Appendix).

The field polarization, i.e., the value of К at the output of the first grain was found by numerical solution of Eq. (8) and was the boundary condition for the second grain, and so on up to the last grain. Knowing К(z = L), one can readily calculate the XPW efficiency η by formula (10). Analogously, the nonlinear phase incursion was calculated by Eq. (11). The Euler angles α, β and γ for each grain were assumed to be random values with probability density

The calculations show that the results of averaging weakly depend on D_g; all the results that will be presented below were obtained for D_g = 0.2l_g. Consequently, <η>, <Φ>, D_η and D_Φ are functions of three variables: В-integral [see Eq. (3)] characterizing nonlinearity; σ [see Eq. (1)] that is a characteristic of nonlinearity anisotropy, namely, the difference between crystal and glass; and N that is the average number of grains on the beam path:

In practice, the value of В-integral is limited by SSSF leading, at a certain characteristic value of B₀, to optical element breakdown. Search for the exact value of B₀ for ceramics is beyond the scope of this paper, so we will restrict ourselves to the analogy with glasses for which SSSF has been well studied. The typical value of B₀ for glasses is 3. However, for high power femtosecond lasers this value may be increased significantly by spatial beam self-filtering [27]. Here we will consider the 0<В<5 range.

Crystal orientation is of no importance for glass (σ = 0), all its grains are identical. Ceramics does not differ from an isotropic medium. In this case, η≡0 and Φ≡B irrespective of random series. The higher |σ|, the stronger the difference between single crystal and ceramics from glass is. The value of |σ| is known only for a small number of crystals [21], one of the highest of which is in BaF₂: σ = −1.2. The results of calculations presented below were obtained for σ = −1.2.

3.2 XPW in ceramics

The XPW efficiency <η> and its standard deviation D_η as a function of В and N are plotted in Fig. 2(а). First of all, note that <η> is proportional to 1/N and is described well by the formula <η>≈(B²/32 + B⁴/112)/N up to <η> = 0.01. This formula permits easy averaging of the XPW efficiency with respect to pulse or beam. For example, assuming Gaussian pulse shape, the integration with respect to time t yields${\text{η}}_{\text{pulse}}\approx \left(B_m^2/\left(32\sqrt{3}\right)+B_m^4/\left(112\sqrt{5}\right)\right)/N$, where B_m = В(t = 0).

 

Fig. 2 Average XPW efficiency <η> and standard deviation D_η versus В-integral (а), probability density P(η/<η>) for N = 100 (b), and value of η for 2000 random series for N = 100, В = 3 (c). Figure (c) models the profile of the beam in cross-polarization; note a substantial η scatter from zero to 0.1.

Download Full Size | PDF

It is clear from Fig. 2(а) that D_η≈1.4<η>, i.e., the standard deviation is larger than the average value. This means that the probability density P(η) has a wide maximum, its right wing is much larger than the left one, and the most probable value is much smaller than the average value. As an example, plots for P(η/<η>) are given in Fig. 2(b) for N = 100. The curves almost coincide for different values of B, if the horizontal axis is normalized to <η>. Such a large value of standard deviation results in strong spatial modulation of the laser beam in cross-polarization [see Fig. 2(c)].

If N ⇒ ∞, then from the point of view of XPW, ceramics is very much like glass – cross-polarization is not generated (η ⇒ 0). At a finite value of N, XPW is generated but with much lower efficiency than in a single crystal of optimal orientation [see Fig. 1]. The primary difference of ceramics and single crystal from glass is that due to the large value of D_η the XPW effect manifests itself in small-scale (about grain size) polarization modulation induced by cubic nonlinearity, rather than in the change of the polarization of the whole beam. In a single crystal, polarization is also different at different points of the cross-section because of different beam intensities and, hence, B-integral. However, these changes are large-scale (of order beam diameter), and are absent for a flat-top beam. Apparently, when the beam has passed the polarizer, polarization modulation transforms to intensity modulation.

It is worthy of note that the cross-polarization with the intensity proportional to η is modulated almost to zero independent of В or N, as D_η is of order <η> [see Fig. 2(c)]. At the same time, the depth of small-scale modulation of initial polarization the intensity of which is proportional to (1-η) is approximately equal to <η> and is always much less than unity.

The calculations show that, if the incident polarization is circular, then the efficiency of generating cross-polarization is much less and is described well by the formula <η>≈B²/(200N).

Measurement of XPW efficiency seems to be the simplest way for experimental verification of the theory. A ceramic sample should be placed between two polarizers and <η>(B) dependence should be measured and compared with Fig. 2(a).

3.3 Nonlinear phase incursion in ceramics

As was mentioned above, small-scale intensity modulation of initial polarization is insignificant at large N. Consequently, like in the isotropic case, SPM and WBSF are determined by the dependence of the nonlinear phase incursion Φ on intensity (i.e. on В-integral). The time dependence of Φ leads to SPM and pulse spectrum broadening, the dependence of Φ on radius results in wave front distortion and subsequent WBSF. The calculated curves of the nonlinear phase average over random series <Φ> and its standard deviation D_Φ are plotted in Fig. 3. Note that <Φ>≈1.45B for arbitrary N – the curves in the Fig. 3(a) almost coincide for all N. For comparison, Fig. 3(а) shows Φ(B) curves for single crystals of some orientations at which no XPW occurs [see also Table 2]. One can see in the Fig. 3(a) that ceramics has an intermediate position between different single crystal orientations. The calculations for σ = [-1.5,1.5] yielded

 

Fig. 3 Average nonlinear phase <Φ> and D_ΦN^1/2 (dashed line) for ceramics (the curves almost coincide for all N) and nonlinear phase Φ for some single crystal orientations versus В-integral (а), probability density P(Φ) for N = 100 (b), and value of Φ for 2000 random series for N = 100, В = 3 (c). Figure (c) models the profile of beam wave front in initial polarization; note a small Φ scatter from 4.15 to 4.6 rad.

Download Full Size | PDF

One can see in Fig. 3(a) the standard deviation D_Φ≈0.22ВN^-1/2. As N>>1, D_Φ is much smaller than the average value of <Φ>. In particular, if N ⇒ ∞, the nonlinear incursion of the phase <Φ> is identical at all points of the cross-section (D_Φ ⇒ 0), like in single crystals. The probability density P(Φ) differs appreciably from P(η): it is symmetric and has a narrow maximum. Curves for P(Φ) are plotted in Fig. 3(b) for N = 100 and different values of B. The spatial modulation of laser beam phase is relatively small because of the small value of D_Φ [see Fig. 3(c)]. Even for В = 5, the beam phase modulation (D_Φ) is typically about 0.11 rad (for N = 100), which will deteriorate beam quality only slightly and the impact on laser pulse compression after SPM will be negligible. At the same time, phase modulation in ceramics may enhance the SSSF effect substantially, as its typical transverse scale is close to the transverse scale at which self-focusing instability is maximal [29].

Our calculations demonstrate that in ceramics χ_eff≈(2/3-σ/4)χ_xxxx for circular polarization, which is 1.5 times less that for linear polarization. The same ratio (1.5) of effective nonlinearities for linear and circular polarization is true for the [111] crystal orientation and glass [Table 2].

4. Conclusion

To conclude, the following results have been obtained:

Appendix Dynamics of system (9)

The variable φ of system (9) is cyclic, so we will consider its dynamics in cylindrical phase space$G=\{\phi ,X:\phi \in S^1,X\in R^{+}\}$. The spatial coordinate ζ plays the role of time in the dynamical system (9).

Single crystal. System (9) is conservative and partitioning of G into trajectories is determined by the equilibrium states that may be centers or saddles [30]. The number of equilibrium states and their stability depend on parameters Г_i (i = 0…4) which, in turn, are dependent of angles α, β and γ. Typical phase portraits for some orientations of a single crystal: (a) [111] (α=π/4, tg²β=2), (b) [001] (α=0, β=0, γ=0), (c) [001] (α=0, β=0, γ=10°), (d) [110] (α=0, β=π/4, γ=π/6), (e) [110] (α=0, β=π/4, γ=π/3), (f) [348] (α=36.8°, β=32.7°, γ=42.5°) are presented in Fig. 4. Saddle equilibrium states are not shown for the sake of simplicity; their role reduces to the separation of the flow of trajectories in phase plane. The case of linear polarization at the crystal input corresponds to the initial conditions in the (φ, X) plane chosen on the φ axis (i.e. X(0)=0) or near it (in numerical experiments X(0)=10⁻⁴). Note that system (9) has a singularity at X=0 that vanishes at certain values of Г_i, for example, for the cases depicted in Figs. 4(a) and 4(b). In other cases at X=0 system (9) is defined only at the points φ =±π/2+2πk, k=0,1,2,…, and at X>0 the trajectory in the neighborhood of the φ axis moves with high speed, so that starting from an arbitrary value of φ any trajectory rapidly reaches the neighborhood of the points φ =±π/2. Further motions occur in the neighborhood of the curve connecting the points φ =π/2 and φ =-π/2 [heavy curves in Figs. 4(c)-4(f)]. Thus, for the majority of values of angles α, β and γ, the motion of the trajectories starting at the initial conditions X(0)=10⁻⁴ are determined by these curves.

 

Fig. 4 Phase portraits in (φ, X) plane for some single crystal orientations: (a) [111] (α = π/4, tg2β = 2), (b) [001] (α = 0, β = 0, γ = 0), (c) [001] (α = 0, β = 0, γ = 10°), (d) [110] (α = 0, β = π/4, γ = π/6), (e) [110] (α = 0, β = π/4, γ = π/3), (f) [348] (α = 36.8°, β = 32.7°, γ = 42.5°).

Download Full Size | PDF

Analysis of the phase portraits allows explaining the XPW efficiency for different crystal orientations. For example, in the cases depicted in Figs. 4(a) and 4(b), the trajectories with the initial conditions X(0)=0 remain on the straight line X=0, which corresponds to the XPW efficiency η=0. In the cases shown in Figs. 4(c)-4(f), the initial conditions for all the trajectories may be regarded to be identical φ (0)=±π/2, X(0)=10⁻⁴. Then the efficiency η is determined by the curves connecting the points φ =π/2 and φ =-π/2. The higher the curve, the higher the XPW efficiency is. For example, comparison of Figs. 4(d) and 4(e) shows that η is higher for the case depicted in Fig. 4(d).

Ceramics. In this case, within a grain having length l_g, the dynamics of the φ and X variables is determined by system (9) with a set of parameters Г_i fixed for this length. Over the l_g interval the dynamics is described by one of the phase portraits analogous to the ones presented in Fig. 4, and motion occurs along one of the trajectories. An analogous process occurs in the next grain but in conformity with the trajectory of system (9) with different values of parameters Г_i, i.e., following a trajectory belonging to another phase portrait [see Fig. 4]. Further, the procedure is repeated and the resulting “glued” trajectory consists of fragments of different trajectories of system (9) having length l_g formed by ceramics grains. The length of the grains l_g and the set of parameters Г_i are random values in this case. Note that the variables of system (9) at the points of “joining” are continuous. By repeating construction of such “glued” trajectories we obtain a family of trajectories presented in Figures 5(a) and 5(b) for B=5, N=10 and N=100, respectively. These results show the existence of some average values of φ and X and their fluctuations which determine XPW efficiency.

 

Fig. 5 Trajectories originate at different initial conditions in (φ, X) phase plane corresponding to ceramics with B = 5, (a) N = 10, and (b) N = 100.

Download Full Size | PDF

Funding

FASO Russia (007-02-1225/2, 0035-2014-0007); Ministry of Education and Science of Russia (14.Z50.31.0007).

References and links

1. A. Ikesue, T. Kinoshita, K. Kamata, and K. Yoshida, “Fabrication and optical properties of high-performance polycrystalline Nd:YAG ceramics for solid-state lasers,” J. Am. Ceram. Soc. 78, 1033–1040 (1995).

2. J. Lu, M. Prabhu, J. Xu, K. Ueda, H. Yagi, T. Yanagitani, and A. A. Kaminskii, “Highly efficient 2% Nd:yttrium aluminum garnet ceramic laser,” Appl. Phys. Lett. 77(23), 3707–3709 (2000).

3. I. L. Snetkov, D. A. Permin, S. S. Balabanov, and O. V. Palashov, “Wavelength dependence of Verdet constant of Tb3+:Y2O3 ceramics,” Appl. Phys. Lett. 108(16), 161905 (2016).

4. Y. Feng, J. Lu, K. Takaichi, K. Ueda, H. Yagi, T. Yanagitani, and A. A. Kaminskii, “Passively Q-switched ceramic Nd³⁺:YAG/Cr⁴⁺:YAG lasers,” Appl. Opt. 43(14), 2944–2947 (2004). [PubMed]  

5. W. Kim, G. Villalobos, C. Baker, J. Frantz, B. Shaw, S. Bayya, S. Bowman, B. Sadowski, M. Hunt, B. Rock, I. Aggarwal, and J. Sanghera, “Overview of transparent optical ceramics for high-energy lasers at NRL,” Appl. Opt. 54(31), F210–F221 (2015). [PubMed]  

6. E. A. Khazanov, “Thermooptics of magnetoactive medium: Faraday isolators for high average power lasers,” Phys. Uspekhi 59, 886–909 (2016).

7. A. A. Kaminskii, S. N. Bagaev, H. J. Eichler, H. Rhee, K. Ueda, K. Takaichi, A. Shirakawa, H. Yagi, and T. Yanagitani, “Observation of high-order Stokes and anti-Stokes χ⁽³⁾-generation in highly transparent laser-host Lu₂O₃ ceramics,” Laser Phys. Lett. 3(6), 310–313 (2006).

8. A. A. Kaminskii, “Laser crystals and ceramics: Recent advances,” Laser Photonics Rev. 1(2), 93–177 (2007).

9. B. Denker and E. Shklovsky, Handbook of Solid-State Lasers: Materials, Systems and Applications (Woodhead Publishing, 2013).

10. E. A. Khazanov, “Thermally induced birefringence in Nd:YAG ceramics,” Opt. Lett. 27(9), 716–718 (2002). [PubMed]  

11. M. A. Kagan and E. A. Khazanov, “Compensation for thermally induced birefringence in polycrystalline ceramic active elements,” Quantum Electron. 33(10), 876–882 (2003).

12. I. L. Snetkov, I. B. Mukhin, O. V. Palashov, and E. A. Khazanov, “Properties of a thermal lens in laser ceramics,” Quantum Electron. 37(7), 633–638 (2007).

13. A. G. Vyatkin and A. E. Khazanov, “Thermally induced scattering of radiation in laser ceramics with arbitrary grain size,” J. Opt. Soc. Am. B 29(12), 3307–3316 (2012).

14. I. L. Snetkov, D. E. Silin, O. V. Palashov, E. A. Khazanov, H. Yagi, T. Yanagitani, H. Yoneda, A. Shirakawa, K. Ueda, and A. A. Kaminskii, “Study of the thermo-optical constants of Yb doped Y2O3, Lu2O3 and Sc2O3 ceramic materials,” Opt. Express 21(18), 21254–21263 (2013). [PubMed]  

15. P. Samuel, T. R. Ensley, H. Hu, D. J. Hagan, E. W. Van Stryland, and R. Gaume, “Nonlinear refractive index measurement on pure and Nd doped YAG ceramic by dual arm Z-scan technique,” AIP Conf. Proc. 1665, 060010 (2015).

16. Y. Senatsky, A. Shirakawa, Y. Sato, J. Hagiwara, J. Lu, K. Ueda, H. Yagi, and T. Yanagitani, “Nonlinear refractive index of ceramic laser media and perspectives of their usage in a high-power laser driver,” Laser Phys. Lett. 1(10), 500–506 (2004).

17. Y. Wang, Z. Jia, X. Su, B. Zhang, R. Zhao, X. Tao, and J. He, “Third-order nonlinearity and saturable absorbed performance of Cr4+:Gd3Ga5O12 crystal,” Opt. Mater. Express 6(8), 2600–2606 (2016).

18. L. Canova, S. Kourtev, N. Minkovski, A. Jullien, R. lopez-Martens, O. Albert, and S.M. Saltiel, “Efficient generation of cross-polarized femtosecond pulses in cubic crystals with holographic cut orientation,” Appl. Phys. Lett. 92(23), 231102 (2008).

19. S. Kourtev, N. Minkovski, L. Canova, A. Jullien, O. Albert, and S. M. Saltiel, “Improved nonlinear cross-polarized wave generation in cubic crystals by optimization of the crystal orientation,” J. Opt. Soc. Am. B 26(7), 1269–1275 (2009).

20. A. Jullien, O. Albert, G. Chériaux, J. Etchepare, S. Kourtev, N. Minkovski, and S. M. Saltiel, “Two crystal arrangement to fight efficiency saturation in cross-polarized wave generation,” Opt. Express 14(7), 2760–2769 (2006). [PubMed]  

21. M. S. Kuz’mina and E. A. Khazanov, “Enhancement of temporal contrast of high-power laser pulses in an anisotropic medium with cubic nonlinearity,” Quantum Electron. 45(5), 426–433 (2015).

22. D. C. Hutchings, J. S. Aitchison, and J. M. Arnold, “Nonlinear refractive coupling and vector solitons in anisotropic cubic media,” J. Opt. Soc. Am. B 14(4), 869–879 (1997).

23. D. C. Hutchings and B. S. Wherrett, “Theory of the anisotropy of ultrafast nonlinear refraction in zinc-blende semiconductors,” Phys. Rev. B Condens. Matter 52(11), 8150–8159 (1995). [PubMed]  

24. M. S. Kuz’mina and E. A. Khazanov, “Spatial instability of the linearly polarized plane wave in a cubic crystal,” Radiophys. Quantum Electron. 59(7), 596–604 (2016).

25. V. N. Ginzburg, A. A. Kochetkov, M. S. Kuz’mina, K. F. Burdonov, A. A. Shaykin, and E. A. Khazanov, “Influence of laser radiation polarisation on small-scale self-focusing in isotropic crystals,” Quantum Electron. 47(3), 248–251 (2017).

26. R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (North-Holland Publ. Co. 1977).

27. S. Y. Mironov, V. V. Lozhkarev, G. A. Luchinin, A. A. Shaykin, and E. A. Khazanov, “Suppression of small-scale self-focusing of high-intensity femtosecond radiation,” Appl. Phys. B 113(1), 147–151 (2013).

28. M. Dabbicco, A. M. Fox, J. F. Ryan, and G. von Plessen, “Role of X⁽³⁾ anisotropy in the generation of squeezed light in semiconductors,” Phys. Rev. B Condens. Matter 53(8), 4479–4487 (1996). [PubMed]  

29. V. I. Bespalov and V. I. Talanov, “Filamentary structure of light beams in nonlinear liquids,” JETP Lett. 3(12), 307–310 (1966).

30. V. I. Nekorkin, Introduction to Nonlinear Oscillations (Wiley – VCH, 2015).