1. Introduction

Cross polarized wave generation [1–5] is a degenerate four wave mixing mechanism in which a single input wave is partly converted to a wave polarized perpendicularly to the input plane polarization. We have shown previously that this process has a direct application for contrast enhancement of femtosecond pulses in petawatt systems [6, 7]; in this respect its efficiency is of prime importance. Other arrangements that lead to an increase in contrast ratio in ultra intense laser pulses include implementation of non linear plasma mirrors [8] or the use of elliptic light [9] in polarization selective systems.

We have previously developed a model [10] that describes the evolution of amplitudes of the fundamental wave (A) and of the cross polarized generated wave (B) using the following system of equations:

where _{${\gamma }_1={\gamma }_0\left[1-\frac{\sigma }{2}{\sin }^2\left(2\beta \right)\right],{\gamma }_2=-{\gamma }_0\frac{\sigma }{4}\sin \left(4\beta \right)\mathrm{and}{\gamma }_3={\gamma }_0\left[\frac{\sigma }{2}{\sin }^2\left(2\beta \right)+\frac{1-\sigma }{3}\right].$} σ is the anisotropy of the χ ⁽³⁾ tensor and β is the angle between the input polarization direction and the [100] axis of the non linear crystal. γ ₀ is linked to the ${\chi }_{\mathit{\text{xxxx}}}^{\left(3\right)}$ value of the non linear medium by γ ₀ = 6${\pi \chi }_{\mathit{\text{xxxx}}}^{\left(3\right)}$/(8λn ₀). Initial conditions are as follows: A(0) = A ₀ and B(0) = 0, where |A ₀|² = 2I ₀/cε ₀ n with I ₀ the input intensity. The system (1) describes the interaction of two orthogonally polarized plane waves. However in the case of relatively short crystals (shorter than the input confocal parameter) and in the case of group velocity matching that perfectly holds in our case, the system (1) can be used to find pulse and beam distributions as B(t,r) and A(t,r) with r the transverse coordinate. The efficiency of the XPW process is then η = ${\int }_0^{\infty }$ r ${\int }_{-\infty }^{\infty }$ B(r)|² dtdr/${\int }_0^{\infty }$ r${\int }_{-\infty }^{\infty }$|A ₀(t,r)|² dtdr.

Theoretical predictions are illustrated on figure 1. XPW efficiency is drawn with respect to the parameter S = γ ₀|A ₀|² L, for various values of angle β. These numerical results correspond to an extension from [10] in which we use Gaussian spatial and temporal shapes for the input beam. The value of the anisotropy of the χ ⁽³⁾ tensor for BaF₂ σ = -1.2 is taken from Ref [11]. As a matter of fact, we have to refer to experimental conditions in order to give an estimation of the maximum value for S parameter. In typical experiments, a few millimeter long barium fluoride crystal is used, a material whose ${\chi }_{\mathit{\text{xxxx}}}^{\left(3\right)}$ value is around 1.59×10^-22 m²V² [11]. By considering an input pulse with a maximum intensity I₀ of 10¹² Wcm^-2 [6], we can estimate that the greatest experimental S value is around 5. As indicated on the Fig. 1, the maximum XPW efficiency in single crystal that corresponds to this value of S is limited to ~25 %.

 

Fig. 1. Theoretical prediction for the evolution of the XPW conversion efficiency as a function of parameter S = γ₀|A ₀|² L for different β angles between the input direction of polarization plane and crystal [100] axis.

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Published experimental results [2,4,6,7,10,12] indicate that the efficiency for cross polarized wave (XPW) generation in a single crystal is limited to around 10–12%, whatever are the length of the nonlinear media, the intensity used and the β angle value. In fact extensive experimental data indicate that the maximum deviation from low intensity optimal value (22.5°) is about -5⁰ (see Fig. 5(a) from Ref. [10]). These levels of XPW conversion correspond in fact to 15–17% internal efficiency of the process. As a matter of fact, it has been evidenced that long interaction lengths lead at least to a decrease in the XPW conversion efficiency due to the contribution from side effects. Among these side effects we note: self-focusing of the fundamental beam inside the crystal (that generates poorer spatial overlap between the FW and XPW beams and additional Gouy phase shift), and subsequent continuum generation [13] from refocused fundamental beam in the nonlinear crystal.

Here we present a first detailed description of a method [14] that enables a noticeable enhancement of the measured cross polarized wave generation efficiency. It consists in the use of two thin nonlinear crystals situated at a specific distance one to the other. Some preliminary results have been submitted in Ref. [15, 16]. We have improved by a 1.7 times factor the XPW generation efficiency for three different laser systems, at 620 nm, 800 nm and 1063 nm. We clearly prove that the scheme with two crystals (each L long) separated by optimal distance gives a better XPW generation efficiency than the one that can be obtained using a single crystal of length 2L. We have also developed a model that allows giving details upon advantages of the two crystals scheme. Using two nonlinear BaF₂ crystals, an internal efficiency that reaches an amount of 30% for cross polarized wave generation has been demonstrated. This result is extremely promising for future application of XPW generation process as a temporal and spatial filter of femtosecond pulses.

2. Experiment

The experimental scheme for the XPW generator consists of two nonlinear cubic crystals sandwiched between two crossed polarizers [11]. The arrangement is drawn in Fig. 2. Lenses of focal f = 1m or f = 0.3m are used as a way to reach specific energy density and confocal parameter on the first crystal. The moving distance between the two crystals is defined as d and the crystals may be rotated at optimum angle β. The experiments have been performed using a single beam that propagates along z axis of [001] cut barium fluoride (BaF₂) crystals. The faces of BaF₂ crystals and polarizers were uncoated. In the two crystal scheme, the position of the second crystal was optimized for each input energy value. Presented results correspond to β ₁ ~ β ₂ in the range 18 – 22.5° depending on input intensity. Experiments with one crystal performed for comparison with two crystals experiments have been performed by suppressing the second crystal and leaving the first one on the focus. The laser system currently used (results in Fig. 3(b) and Fig. 3(c)) is a colliding pulse mode-locked (CPM) dye laser (used maximum energy 200 μJ, duration 100 fs, frequency 10 Hz, wavelength 620 nm). Alternatively (results in Fig. 3(a)) we used a mode-locked Ti:Al₂O₃ laser at a wavelength of 800 nm, maximum energy 1.2 mJ and duration 45 fs. The signals from two photodiodes that measure input and output energies are digitalized for signal averaging and processing. For absolute energy measurements “OPHIR” energy meter was used. Essential results are gathered in Fig. 3.

 

Fig. 2. Schematic of the two crystals scheme for efficient cross polarized wave generation. Both crystals can be independently rotated around z axis for optimization of β ₁ and β ₂ angles and moved one from the other along z axis.

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We first comment on results from curves on Fig 3(a). We notice that at low input energy (~30 μJ) the measured efficiency is proportional to the square of the crystal length, independently of the number of crystals. At high intensities, the single BaF₂ crystal experiments, with respectively 2 mm and 6 mm thicknesses, are giving almost the same maximum conversion efficiency of about 10% that is in strong contrast with the theoretical prediction for efficiency vs. crystal length dependence. Among side effects described in Introduction, this one is clearly assigned to processes that depend on the propagation length. Using multiple crystals is therefore a way to overcome the early saturation of thick crystals for the XPW process. As seen on Fig. 3(a) with two BaF₂ crystals scheme a 1.7 times increased efficiency is achieved compared to single BaF₂ 2 mm crystal scheme. The optimal distance between the nonlinear crystals in this experiment was ~ 50 mm.

 

Fig. 3. Experimental dependence of the XPW conversion efficiency as a function of input pulse energy for different BaF₂ crystal lengths (a, b) and as a function of two crystals separation (c) for different focusing lenses and intensities; (b, c): experiments at 620 nm; (a): experiments at 800 nm; note that curves in (c) have been normalized to the efficiency at zero separation distance.

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Figure 3(c) (f = 1m, E = 100 μJ and 50 μJ) illustrates more precisely the distance behavior as a function of energy density for the two crystal scheme. There is an optimal distance d (tuning d can lead at 100 μJ to an increase or a decrease in the efficiency) that furthermore depends on E value. These specific data have been normalized with respect to the efficiency at minimum distance between the crystals (d ~ 0). So we can conclude that a two crystal scheme (2 times L length) gives higher efficiency than a one crystal scheme using a 2L length. Increase in efficiency due to optimization of distance d can reach a factor of 2.5. On the same Fig. 3(c) we have plot the dependence of XPW efficiency against crystals separation for two experiments with approximately equal intensities but different confocal parameters: input energy E = E₀/10 and f = 0.3 m compared to E = E₀ and f = 1m). The measured optimal separation is 12 times smaller for smaller confocal parameter. All these findings clearly indicate the built up of a Kerr lens in the first crystal, which focal length is proportional to input confocal parameter [17].

Figure 3(b) focus once more on the distance critical behavior and also on the general characteristics of the multiple crystal setups. It is seen that with a “2 mm” crystal maximum achievable efficiency is 12 % at 150 μJ. The highest efficiency obtained with the two “1 mm” crystals scheme is 18.5 % at an optimal distance d of 75 mm. As the two “1 mm” crystals experiment has d optimized, in this specific case for highest intensity, it is appearing to be less efficient than a “2 mm” crystal for twice lower energy as the Kerr lens at this energy level does no more match the d distance between the 1 mm crystals. Moreover, using a two “2 mm” scheme the efficiency saturates at 20% at a distance of 65 mm. This efficiency value seems to be the maximum achievable in a two crystal scheme. It is important however to notice that 20% efficiency is obtained at almost 1.5 times smaller input energy (100 μJ) than it is required for maximum efficiency with the double “1 mm” scheme.

In conclusion, it is worth noticing that taking into account all losses from the faces of the two BaF₂ crystals and the output Glan the achieved 20% efficiency with the 2 crystals scheme would correspond to 30 % internal efficiency if crystals and analyzer have adequate coatings and have been designed for minima losses. This value breaks the theoretical “single crystal” maximum XPW efficiency limit presented in the Introduction.

3. Theoretical analysis

We present very briefly hereafter the theoretical approach used to describe the XPW generation with a two crystal scheme. The hypotheses used in our calculations are as follows. We remain in the approximation of a spectrally degenerate four wave mixing - the model does not include terms that account for generation of new spectral components (as in the experiment we take care to do not create any white light continuum). Moreover, for sake of simplicity, the first nonlinear crystal is situated at the waist of the focused fundamental beam. The model takes now into account that in the first crystal the two waves, FW and XPW, form independently Kerr lenses that result in their refocusing at a given discrete position from the first nonlinear crystal. The positions of the new focus planes are calculated and positioned at different locations for the FW and the XPW. In the second crystal the fundamental and cross polarized beams interact again continuing the process of XPW generation.

The effect of XPW generation in cubic (m3m) crystals in the slowly varying envelope approximation is described by the system of equations (1). Let us assume that the input linear polarized beam with Gaussian spatial shape enters the cubic crystal along its four-fold axis z. To describe the spatial profile in the perpendicular direction r, we denote the input linearly polarized beam by A = A(r, z) and the generated orthogonally polarized light by B = B(r,z). The initial conditions for the first crystal situated in the focus of a long-focal-length lens (at z = 0) are:

where ρ _0,A is the beam radius of the input FW beam. Numerical solution of the system of equations (1) gives in numerical form the spatial distributions of amplitudes [A ₁(r),B ₁(r)] and phases [ϕ_A (r),ϕ_B (r)] of the FW and XPW after the first crystal. We consider the crystal as being a thin Kerr lens and neglect the beam-size changes inside the nonlinear crystal. A thin linear lens of focal length f_L would introduce (in paraxial approximation) radial parabolic phase distribution of the beam (2) of the form ϕ(r) = ϕ(r = 0) + πr ²/λf_L , where λ is the wavelength. We then approximate the numerical data of non-linear phase shifts (NPS) ϕ_A and ϕ_B (r) with parabolas ϕ_A,B (r)= ϕ(0) + Δϕ_A,B r ² to find focal lengths of induced Kerr lenses: f_NL,A,B = aπ/(nΔ_A,B λ). In this expression, a is a dimensionless aberration factor, which is introduced to account that parabolic fit of NPS-distribution is a relatively accurate approximation. The value of a is usually taken to be between 3.77 and 6.4 [17]; for our calculations we used a = 4.5 for both beams. Using ABCD rules and in the approximation of Gaussian beams we find the positions of the new waists z _0,A,B of FW and XPW beams counted from the first crystal and their sizes ρ _1,A,B to be:

Since XPW-generation is a third-order non-linear process we assume that the size of the XPW-beam exiting the first crystal is ${\rho }_{0,B}=\frac{{\rho }_{0,A}}{\sqrt{3}}.$ This assumption is reasonable providing a not too high FW to XPW conversion, a case for which the XPW-beam will remain nearly Gaussian.

Once we know _z0,A,B and _ρ1,A,B we can find wave front and beam size values at every transverse plane after the first non-linear crystal. We consider that beams after the first crystal propagate like Gaussian beams with respect to wave fronts, which allows knowing the phase distributions. At the same time, intensity distributions after the first crystal (from z = 0), which are proportional to |A ₁(r)|² and |B ₁(r)², are rescaled in the transverse direction in concordance with beam-size changes.

If the second non-linear crystal is located at distance z = d behind the first crystal then the following expressions for the complex amplitudes of the fundamental wave and XPW are found to be:

where ${\xi }_{A,B}\left(z\right)=\frac{\lambda \left(z-z_{0,A,B}\right)}{{\mathit{\pi \rho }}_{1,A,B}^2}\mathrm{and}{y}_{A,B}=r\sqrt{\frac{1+{\xi }_A^2\left(0\right)}{1+{\xi }_A^2\left(z\right)}}.$ Moreover, Φ_A,B = Φ_A,B(r = 0)- arctgξ _A,B(0), an expression that accounts for NPS relations after the first crystal as well as for the Gouy phase accumulation when moving from plane z = 0 to plane z = d. Non-linear effects in the air gap between the crystals are assumed to be negligible.

To obtain the output XPW characteristics after the second crystal, the same system of differential equations (1) is solved again but with initial conditions at the front of the second crystal given in Eqs. (4). Finally, XPW power and conversion efficiency are found by numerical integration of the XPW and FW intensity over the transverse coordinate r. At this stage for simplicity we did not considered time integration, which means that the obtained final dependencies are valid for spatially transverse Gaussian and temporally rectangular shapes.

The above presented simple theoretical model could be easily extended to a more general frame where the waist of input fundamental beam does not coincide with the position of the first crystal.

One of the main outputs of the calculation, the existence of a spatially varying non linear phase shift, is a direct consequence of the formal introduction in coupled equations (1) of a spatial Gaussian shape. This nonlinear phase shift leads to evidence a Kerr lensing effect, which may drastically modify the characteristics for A(r,z) and B(r,z) beams and the overall efficiency. Several parameters have been evidenced to play a prime role: the confocal parameter, the position of the new focal plane (especially for the FW beam) and its characteristics (waist and energy density), and finally the optimal position of the second nonlinear crystal. These results are compared to the experimental findings in the next part.

4. Results and comparison with the experiment

The developed model explains qualitatively the XPW efficiency dependences as a function of the two crystals distance obtained experimentally (Fig. 3(c)). Such theoretical plots are shown on Fig. 4(a) for input parameters S = 1 and S = 2, input radius ρ ₀ =140 μm on the first crystal and fundamental wavelength λ_fund = 620 nm. There is a smooth variation of the efficiency that possesses a maximum at 4.9 cm and 6.5 cm in these specific cases. As expected from one crystal scheme [4, 8], we predict a higher efficiency at high intensities when β < 22°5. However, at S = 1 and even with β = 22°5, we calculate a noticeable increase in efficiency for the two crystal scheme. Figures 4(b) and 4(c) allow having a better insight into the modifications of the beams geometry, estimated from the exit of the first crystal. First of all, XPW beam waist has been dramatically reduced by a factor of √3 due to its creation by third order process; it does not produce strong Kerr lens effect due to its low intensity. At the opposite, FW that exit the first crystal is more or less re-focused at a given distance z _0,A,B that is function of S and ρ ₀ values. At small intensities (see Fig. 4(b)) the optimal distance d is very close to the new waist position of the fundamental beam. From Fig. 4(c) we see that, at higher intensities, the optimal distance is not the smallest diameter position of the fundamental beam, but rather shifted in direction to a wider beam size away from the first crystal. This is an indication that increased efficiency depends on two factors. Increased fundamental intensity is only one of these factors. The second important factor is that in the second crystal, thanks to Gouy phase shift effect on both waves, XPW generated in the second crystal may be in phase with the XPW generated in the first crystal. Therefore two XPW signals interfere constructively, increasing this way the intensity level for which quadratic dependence (efficiency vs. γ ₀|A ₀|² L) is valid. This process can be compared to quasi phase match (QPM) geometries [18] that allow coherent growing of a process output all along the length of the device. Accordance between experiment and model is qualitative. We cannot expect a better accordance since the model assumes that both fundamental and XPW beams have Gaussian transversal shapes between the crystals. However, the fundamental beam has a distorted Gaussian shape at high intensities and different values of the aberration coefficients for the two beams have to be used.

 

Fig. 4. (a) Theoretical dependences for XPW generation efficiency as a function of two crystals separation for S = 1 and angles β ₁ = β ₂ = 22.5° and for S = 2 and angles β₁ = β ₂ = 18°. The curves are normalized to the theoretical efficiency at zero separation (12% and 33%, respectively). (b,c) Change of the beam sizes of the fundamental and XPW waves in the space between the two crystals: (b) S = 1; (c) S = 2. The positions of the waists and the positions of the second crystal for maximum efficiency are shown.

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The theoretical model leads also to the conclusion that the optimal distance is linearly proportional to the confocal parameter of the input fundamental beam in the plane of the first crystal. This is illustrated on Fig. 5 where the optimal distance between the crystals for maximum XPW efficiency is plotted against the confocal parameter of the input fundamental beam for S = 1. This theoretical result is in good agreement with the observation shown on Fig. 3(c), where the optimal distance dropped 12 times for 3.3 times stronger focusing that gives 10 times smaller confocal parameter. From Fig. 5 we learn that for one and the same intensity to crystal length product the results for a given input radius can be directly used to predict the results for any other input radius.

5. Conclusions

We have demonstrated that the use of two thin crystals can solve the saturation problems that accompany the process of cross polarized wave generation. The use of two thin crystals in our case is not just equivalent to an increase of the thickness of the nonlinear medium and re-use of non-depleted fundamental output; this arrangement is known to be proposed for increasing the efficiency in SHG process [19, 20]. At highest possible energy time’s crystal length product, one can benefit from Kerr lens effect to increase the efficiency of the two crystal scheme above the saturation value typical for a single crystal scheme. Furthermore a more appropriate phase shift between the fundamental and XPW waves can be achieved. Loss corrected efficiencies as high as 30% have been obtained at optimized distances between crystals. Such a discrete configuration allows constructive interference between the XPW signals generated in both crystals. For best efficiency the second crystal has to be optimized with respect to the distance d from the first crystal. This distance is dependent on the relative phase between the two generated XPW, as a consequence of Gouy phase effect. The used energy density maximum value is limited by the appearance of side effects such as self-focalization, continuum generation inside the crystals and multi-photon absorption. Inside this range, the optimal experimental angular position β ₂ between the input direction of polarization and crystal axis [100] remains relatively close to 22°5 (though lower values are predicted by the model at even higher intensities). The described model that is an extension from previously described calculation firstly explains the obtained behaviors, secondly allows predicting the optimal geometrical arrangement as a function of laser characteristics.

 

Fig. 5. Optimal two crystal separation as a function of the confocal parameter b = 2${\pi \rho }_o^2$/λ of the input fundamental beam at the plane of the first crystal for S = 1. The circle corresponds to the maximum efficiency point of the curve with the same S shown on fig. 4a.

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On a general point of view the use of multiple BaF₂ crystals can be compared to QPM geometries, but applied for the first time to our knowledge to third order processes. We believe that the two crystal approach, we demonstrate leading to increased efficiency for cross polarized wave generation, can be applied to other cubic nonlinear optical processes as for example four wave mixing based wavelength shifting.