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Abstract
A method for enhancing the temporal contrast of high-power femtosecond laser pulses is proposed. The suppression of low-intensity radiation and the simultaneous 100% transmission of a pulse peak are attained due to the nonlinear phase difference π between the orthogonally polarized waves, leading to a 90-degree rotation of polarization. The polarization interferometer has an in-line geometry that does not demand spatial beam separation. The output pulse compression and power enhancement are implemented as a result of self-phase modulation in the interferometer and subsequent reflection from the chirping mirrors.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Many experiments on studying the behavior of matter in extreme light fields require both a high peak laser pulse power and a high temporal contrast. To increase the power the method of nonlinear compression has been actively employed in the recent years. As a follow-on to the theoretical predictions of Fisher, Kelley, and Gustafson [1], the recent effort in nonlinear compression relies on the use of a nonlinear plate that introduces self-phase modulation and chirping mirrors that induces negative dispersion. The method is called TFC (Thin Film Compression) [2], CafCA (Compression after Compressor Approach) [3], or post-compression [4]. For pulses with an energy of a few Joules [5,6] and even tens of Joules [7,8], multiple compression of femtosecond laser pulses was demonstrated with almost no energy loss; for more details, see the review [9].
Temporal contrast is the ratio of the intensity at the peak of the pulse to the intensity on its wings. For many experiments, of great importance is the far temporal contrast (1ps – 1ns from the peak of the main pulse) determined by the value of the pulse pedestal, which appears, as a rule, due to amplified spontaneous emission in laser amplifiers of CPA lasers (Chirped Pulse Amplification) [10] or due to amplified parametric emission in OPCPA lasers (Optical Parametric Chirped Pulse Amplification) [11]. Contrast can be enhanced making use of plasma mirrors [12], second harmonic generation [13], cross-polarized wave (XPW) generation [14], and spectral pulse filtering after self-phase modulation [15,16].
In [17], it was proposed to use a nonlinear Mach-Zehnder interferometer with symmetric arms, where both beam splitters have 50% intensity reflectance. Let ΔΨ be the phase difference without non-linearity. If ΔΨ=0, then the radiation of the pedestal does not pass to the dark port, and the contrast will be infinitely high. In practice, the increase in the Сout/Cin contrast is determined by inaccuracy of meeting the condition ΔΨ=0:
The main pulse acquires a nonlinear phase (B-integral)
where n2 and L are, respectively, the nonlinear index of refraction and the thickness of the nonlinear medium, k=2π/λ, λ is the wavelength in vacuum, and I is the intensity. The nonlinear phase difference ΔB is nonzero by virtue of the different lengths of the nonlinear elements in the interferometer arms. If ΔB=π, then the dark port becomes light. The efficiency η (the ratio of the intensity in this port to the input intensity) depends on ΔB asThe nonlinear phase of the input pulse is equal to the arithmetic mean between the nonlinear phases Bx and By in two interferometer arms:
On reflection from the chirping mirror, the pulse is compressed, which leads to an increase in its peak power Pout/Pin that may be assessed by the approximate formula [18]
The drawback of the Mach-Zehnder interferometer is the need to split the beam into two arms in space and then combine the beams with interferometric accuracy. This imposes strict requirements on the accuracy of the alignment. The in-line geometry is much more attractive in practice. In this paper, we propose to use a nonlinear polarization interferometer (NPI), in which the beams are not spatially separated.
2. Nonlinear polarization interferometer (NPI)
The NPI consists of two phase plates located between crossed polarizers (Fig. 1). The axes of the polarizers in Fig. 1 are shown by the vectors r (reflection) and t (transmission). The t axis of the first polarizer and, accordingly, the input field polarization Ein are tilted to the y axis by an angle π/4+δ. The phase plates are made of the same uniaxial crystal, have the same thickness L, the same angle ϑ (the angle between the optical axis and the wave vector), but different angles φ (the angle of rotation around the optical axis). The polarizations eo (ordinary wave) and ee (extraordinary wave) in the plates are oriented along the x and y axes and are rotated by 90 degrees relative to each other around the z axis: the x-polarization in the first crystal is an ordinary wave (refractive index no) and in the second crystal is an extraordinary wave (refractive index ne), and the y-polarization is vice versa (see Fig. 1).
It is obvious that without non-linearity after the second crystal, the phase difference between the x- and y-polarized waves ΔΨ is equal to zero, since no and ne do not depend on the angle φ in any uniaxial crystal. Therefore, the polarization of the wave will be the same as at the input, and the pulse is completely reflected by the second polarizer, i.e. the transmission of the entire system is zero. It is worth noting that it is true for any frequency and, hence, the contrast enhancement does not depend on the pedestal bandwidth. For the NPI, instead of (1, 3) we have
Equations (4,5) for the NPI remain unchanged. As is seen from (7), the difference between the nonlinear phases ΔB = Bx-By in the NPI must be nonzero and the efficiency is maximal at ΔB=π and δ=0. For estimating ΔB, we assume for simplicity that the difference of group velocities of the o- and e-pulses having duration τ is rather strong, so that the length l of their separation in space is much less than the crystal length L, i.e l=сτ/|n0-ne|<<L. In other words, the x- and y-polarized pulses pass nearly all the path in the first and second crystals without mutual nonlinear cross-modulation. Then
Hereinafter the subscripts I and II denote the first and second crystal, and o and e the ordinary and extraordinary waves. From (8) it is clear that for ΔB≠0, Bo,II≠Bo,I or Be,II≠Be,I should be met. From (2) it follows that this can be attained (i) at anisotropic cubic nonlinearity (n2,I≠n2,II, as n2 depends on φ), or (ii) at different wave intensities in the x- and y-polarizations (Ix≠Iy, i.е. δ≠0), (iii) when both these cases occur simultaneously. Note that 100% efficiency is possible only in the first variant with symmetric arms (see Eq. (7). Below we will consider all the three variants.
2.1 NPI with symmetric arms and crystals of different orientations
Since in the NPI with symmetric arms δ=0, ΔB≠0 can be obtained only if n2 depends on the angle φ (L and ϑ in crystals I and II are the same). The expressions for n2,o(φ) and n2,e(ϑ,φ) for uniaxial crystals of any symmetry are available, for example, in [19]. Here we omitted the argument ϑ for n2,o, as n2,o does not depend on ϑ for a uniaxial crystal of any syngony. For the hexagonal syngony, neither n2,o nor n2,e depend on φ and, hence, for crystals of this syngony ΔB=0. At the same time, for the tetragonal syngony, n2,o depends on φ and n2,e is φ-independent, and for the triclinic syngony, vice versa, n2,e depends on φ and n2,o does not depend on it. From the analysis of the n2,o(φ) and n2,e(ϑ,φ) functions, it is easy to find optimal values of ϑ=Θ, φI=ΦI, and φII=ΦII for the NPI with symmetric arms, i.e. the values at which ΔB has a maximum (see Table 1).
The nonlinear diffractive index n2 depends not only on the angles ϑ and φ but also on the elements of the cubic nonlinearity tensor χijkl (i,j,k,l = a,b,c) in the intrinsic coordinate system abc; the с axis coincides with the optical axis. For uniaxial crystals of all symmetries there exist seven independent tensor components that are readily normalized to χaaaa :
Sets of independent hi for different symmetry groups are shown in Table 1. The substitution of (2) into (8) for different optimal values of Θ, ΦI, and ΦII yields
where the expressions for D for different symmetries are given in Table 1 and where IΣ=Ix+Iy is the intensity of the input pulse. For all types of symmetries, n2,o(φ=0)=χaaaa/(n0)2. Assuming that ΔB=π, from (9, 12) we find
Table 1. Expressions for Θ, ΦI, ΦII, D, and A for different crystal classes
The expressions for A are presented in Table 1. The values of all hi required for calculating A and D were found from the literature only for two crystals: KDP [20] and BBO [21]. Using these data, we obtained from (13) (for λ=910 nm and IΣ = 2 TW/cm2) <B>=14, L=3.4 mm for KDP and <B>=38, L=3.8 mm for BBO for λ=910 nm and IΣ = 2 TW/cm2. Such intensity is typical for high power lasers. Note that in this case large-scale self-focusing is not a problem for laser beams with a diameter of 1 cm and above. The estimation of the self-focusing distance Lsf using the formula from [22,23] shows that Lsf=9cm for a beam diameter of 1cm and Lsf=90cm for a beam diameter of 10cm. It is much more than a typical crystal thickness, hence, large-scale self-focusing may be neglected.
The value of the B-integral is limited by small-scale self-focusing [24]. Due to the effect of beam self-filtering during propagation in free space [9,25], self-focusing can be avoided for pulses with an intensity of about 1012 TW/cm2 at large values of B. This was experimentally demonstrated up to B=20-25 [26] with a beam of small diameter and B=8 for a beam of a subpetawatt laser with a diameter of 18 cm [8]. Thus, it seems hard to employ BBO in practice due to the too large value of <B>, whereas KDP may well be used. From (5) we find that due to compression the power will increase by a factor of 8. According to more precise calculations [9], taking into account the influence of the dispersion in KDP, a pulse with a duration of 50 fs will be compressed to 8 fs.
Thus, the use of the NPI with symmetric arms is limited by both the small value of anisotropy of cubic nonlinearity (BBO crystal) and the lack of information about this anisotropy (it is known only for two crystals). In addition, crystals of hexagonal syngony (for example, LiIO3) cannot be used in principle. The NPI with nonsymmetric arms significantly expands the possibilities of contrast enhancement.
2.2 NPI with nonsymmetric arms and identical crystals
It can be seen from (8) that for identical crystals, ΔB≠0, only if δ≠0, i.e. the input pulse polarization differs from the 45-degree one (Fig. 1). The obvious drawback of such an NPI is the unavoidable loss, since the efficiency (7) cannot be 100% in principle. We will show that by “sacrificing” the 100% efficiency, the capabilities of the NPI may be expanded substantially. For identical crystals φI=φII=φ, from (2, 8) we obtain
Assuming ΔB=π, from (9, 12) we find
Equations (14, 15) are valid for any symmetry. <B> depends only on δ and does not depend in any way either on the nonlinearity of the chosen crystal, or on its orientation. The curves for <B> and losses (1-η) as a function of δ are plotted in Fig. 2. The green curve is plotted using Eq. (15) and shows the δ-dependence of <B> for any crystal in NPI with nonsymmetric arms and identical crystals. The red (for KDP) and blue (BBO) curves are plotted using Eq. (13) and Table 1 and show the dependences for NPI with symmetric arms and crystals of different orientations.
Thus, any uniaxial crystal can be used, including crystals of hexagonal symmetry. At δ→0, the NPI, according to (7), has zero losses (1-η) = 0, but <B>→∞ (15). Accordingly, the choice of δ is a compromise. However, <B> takes on suitable values at practically acceptable losses, for example, when <B>= 9, losses account for 3%, while <B>= 14 for 1.2% only. For calculating L by the formula (15) for a-cut crystals (θ=π/2, φ=0), it is sufficient to know only h1 and n2,o(φ=0). These data are known for several crystals. For some of them, the curves for L(δ) are plotted in Fig. 3 at IΣ = 2 TW/cm2.
Note that, in contrast to the NPI with symmetric arms, in the NPI with nonsymmetric arms, the value of ΔB may be controlled by changing the angle δ (cf. (14) and (11)). It can be seen from (6), that for small δ, contrast enhancement practically does not depend on δ.
2.3 NPI with nonsymmetric arms and crystals of different orientations
In this case, the advantages of the two options discussed above are combined in one. For crystals of different orientations for the optimal values of ϑ=Θ, φI=ΦI, and φII=ΦII presented in Table 1, from (8, 9, 12) we obtain
Equations (11, 13) are particular cases of eqs. (16, 17) at δ=0. As the values of hi needed for the calculation of A and D are known only for KDP and BBO, the δ-dependences of <B> and L in Figs. 2 and 3 are plotted only for these crystals. From Fig. 2 it is clear that, although the pair of BBO crystals with different φ angles cannot be used at δ=0 due to a very large value of <B> (section 3.1), it can be well used at δ=2 degrees, i.e. at negligibly small losses – less than 1%.
3. Discussion of the results
The simplest analysis made above shows that the NPI is promising for both contrast and power enhancement. A more detailed study requires a separate publication; here we will restrict ourselves to a brief discussion. It is convenient to consider NPI separately in the linear mode for low-intensity radiation (pedestal) and in the nonlinear mode (for the main pulse). The linear mode solely is responsible for the contrast, whereas nonlinear effects are in charge of the energy parameters of the main pulse (the efficiency of the NPI and the subsequent compression).
3.1 Contrast enhancement
It follows from (6) that the contrast is almost entirely determined by the value of the linear phase difference ΔΨ and tends to infinity at ΔΨ=0. This condition demands exact orientation of the crystals I and II, which can be readily ensured in practice, as small inaccuracies of their cut can be eliminated by angular alignment. This is also true for eliminating the disparity in the crystal lengths L. However, as a result of the deviation of the surface from the plane, the condition ΔΨ=0 cannot be fulfilled for all points of the cross section. For the characteristic inhomogeneity of the surface profile δL=λ/N, the substitution of ΔΨ=k(no-ne)δL into (6) will yield
Thus, crystals with a small birefringence are more preferable. The Cout/Cin values for several crystals at N=7 are listed in Table 2. The dispersion does not restrict contrast growth, since Cout/Cin is practically independent of the wavelength; the allowance for the dispersion in (18) will only lead to a small correction. Note that for the Mach-Zehnder interferometer, instead of (no-ne) in (18) there will be (n−1), which imposes more stringent restrictions on both surface quality and dispersion.
Table 2. Values of contrast enhancement Cout/Cin for δL/λ=1/7, crystal lengths L, and length l of the o and e pulses separation in space.
Another limitation of Cout/Cin is the contrast of the polarizers used. This is a technological problem especially relevant for ultra-high-power lasers, in which the reflective polarizers must have a large aperture (just as an illustration, we presented in Fig. 1 the transmissive polarizers). However, the use of several consecutive reflections can solve this problem.
Equation (6) was obtained for a negligible intensity, for which the nonlinear phase incursion is ΔB=0, i.e. for the far contrast. The near contrast after the NPI will also increase, although not as much. For estimates, we can use (6), substituting ΔB instead of ΔΨ. For example, at an intensity of 10% of the peak one, ΔB=π/10 and the contrast will increase by 40 times, i.e. the output intensity will drop down to 0.25%.
Finally, Eq. (6) is valid for ΔB=π which is correct on the beam axis but is violated at the beam periphery, because the periphery intensity is less than on the axis. As a result, Cout/Cin is smaller at the periphery. It is easy to show that for the intensity equal to 1/e of the maximum one, i.e. for ΔB=π/e the Cout/Cin ratio reduces by a factor of 3, which is not crucial, especially for super-Gaussian beams.
3.2 Power enhancement
In the analysis presented in section 2, the influence of cubic nonlinearity is limited by the appearance of a nonlinear phase accumulated independently for the o- and e-polarized pulses. This is true only if the dispersion of the nonlinear medium is neglected and the pulses with o- and e-polarization are assumed to propagate without interference. The last assumption is based on the fact that the length of space separation of the o- and e-pulses, l=сτ/|n0-ne|, is much shorter than L, i.e. the orthogonally polarized pulses propagate along almost all the path in both the first and the second crystals without overlapping in space. The values of l and L for several crystals are given in Table 2 (τ=30фс, λ=910нм, angle δ in all lines is chosen so that <B>= 14). It can be seen that in a number of cases the condition l<<L is satisfied, but it demands a large value of |n0-ne|, which imposes restrictions on contrast enhancement (see (19)). The allowance for the impact of the section of the path on which the pulses overlap, ΔB=π being maintained, will lead to an increase in <B>. The estimates reported in [27] for the fiber demonstrate that, if the pulses overlap completely (i.e. if l>>L) <B> will increase by a factor of 6 as compared to (15), which significantly limits the use of the NPI. Obviously, if l is shorter but commensurate with L, then <B> will be only slightly larger than (15). Also, within a short distance lphase at the first crystal entrance, XPW generation may take place. It is efficient while phase mismatch is negligible, so we can estimate lphase=λ/|n0-ne|, which is obviously much less than l. It is possible to accurately determine <B> by solving the equations taking into consideration the cross-interaction in a uniaxial crystal. This equation should include terms with are responsible for XPW generation, see for example
Both linear and nonlinear group velocity dispersions have little effect on ΔB, since this influence is “subtracted”, being almost the same in the two crystals. At the same time, both dispersions affect the compression efficiency. The case of completely nonoverlapping pulses was considered, for example, in [9], where it was shown that the compression efficiency decreases only slightly. However, for partially overlapping o- and e-polarized pulses, this issue has not been investigated.
It is also worthy of note that the pulse after the NPI (before the chirping mirror) will be shortened, because the sine in Eq. (7) is equal to unity only at the pulse maximum. At the time instant when the intensity is equal to 1/e of the maximum, ΔB=π/e and the NPI efficiency (i.e. the transmission) will be η=30%. Energy efficiency ηenergy may be easily found by integrating η over time:
Using Eq. (7) for η and assuming a Gaussian pulse with δ=0 we find ηenergy=73%, i.e. the energy losses are 27%. An analogous compression occurs at XPW. This effect can make an impact on compression. On the one hand, part of the energy is lost at the leading and trailing edges of the pulse, but on the other hand, the chirp introduced by self-phase modulation in such a pulse will be closer to a linear one, and therefore, the compressed pulse will be closer to the Fourier-limited one.
Above we have considered only a flat-top beam. Laser beams are generally Gaussian or super-Gaussian. Beam shape impacts both on compression efficiency and beam focusability. The theoretical study of this impact may be found in [9].
4. Conclusion
We have proposed a method for the simultaneous enhancement of the temporal contrast and power of powerful femtosecond laser pulses using a nonlinear polarization interferometer (Fig. 1). For low-intensity radiation, the crystals in NPI induce a zero phase delay, so that the transmission through crossed polarizers is zero. Due to the nonlinear phase incursion π between orthogonal polarizations, the phase plate at maximal intensity becomes a λ/2 plate, thanks to which the transmission of the NPI approaches unity. The nonlinear phase incursion π can be provided both due to the anisotropy of the cubic nonlinearity (n2 depends on the angle φ), and due to different wave intensities in orthogonal polarizations. For the first option, the KDP crystal has suitable properties and for the second option, a wide range of uniaxial crystals. The pulse that has passed through the second polarizer is self-phase modulated, which allows it to be compressed by reflection from the chirping mirrors, thereby increasing the peak power.
The NPI has an in-line geometry that does not require spatial beam separation and can be used at the output of any lasers with TW power and higher, as well as in lasers with double-CPA.
Funding
Ministry of Science and Higher Education of the Russian Federation (agreement No. 075-15-2020-906., Center of Excellence “Center of Photonics”).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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