1. Introduction

Since the advent of lasers to this day, the two most important goals of research have been generating the shortest possible (single cycle) [1] and ultra-high-power [2] pulses. The duration of ultra-high-power pulses is limited by the amplifier and compressor bandwidths and is about 10 cycles. The compression of such pulses not only increases their power, but also opens up opportunities for important applications such as terahertz emission from relativistic laser-driven plasma [3] or laser-wakefield electron acceleration by mixing spatio-temporal couplings and quasi-Bessel beams [4]. Pulse duration at the output of femtosecond lasers is usually a little higher than the Fourier limit, therefore for compressing a pulse by several times it is necessary to increase the spectrum width by at least the same number of times. For this, back in 1969 [5] it was proposed to use self-phase modulation of a laser pulse propagating in the medium with cubic (Kerr) nonlinearity, where the index of refraction n depends on intensity I: n = n₀+ n₂I (where n₀ is a linear index of refraction and n₂ is a nonlinear index of refraction). At the output of the nonlinear element the pulse is chirped. Next, the chirped mirrors make the pulse Fourier-limited again, but now it is much shorter than the initial pulse. This method of nonlinear compression of laser pulses is usually called post-compression.

Post-compression was considered in detail in several review papers [1,6–10] focused on different methods of its implementation and the corresponding ranges of pulse intensity. In recent years this method has been actively developing for ultra-high-power lasers and is known in the literature as Thin Film Compression (TFC) [11] or Compression after Compressor Approach (CafCA) [2,10]. Results of the related studies can be found in the reviews [9,10], as well in the most recent publications [12–14]. Another important for ultra-short pulse lasers technique is the frequency doubling [15–17] for high contrast interactions.

One of the main limitations of post-compression and frequency doubling is small-scale self-focusing (SSSF), that is spatial instability of a plane wave – the growth of the amplitude of spatial harmonic perturbations [18]. A great number of theoretical and experimental works demonstrated that, in the stationary approximation, the instability increment is determined by the B-integral (B = kLn₂I, where k = 2π/λ and L is the length of the nonlinear medium), the threshold value of which is approximately equal to 3. The assertion that for B > 3 the beam is inevitably divided into filaments is true for nanosecond pulses but is erroneous for the femtosecond ones. A consequence of this misconception is the opinion (see, for example, [7,8,19–22]) that large compressibility factors that require large values of the B-integral are impossible. The recent studies [9,10] contradict this statement. In particular, SSSF was not observed for the B-integral equal to 19 [23,24]. Besides, it was revealed experimentally [12] that fluence fluctuations at the output of a nonlinear element do not grow, despite B = 8. This effect is very surprising, since an increase in spatial harmonic perturbations should lead to an increase in fluence fluctuations. It was also found in [12] that fluence fluctuations start to grow during beam propagation after the nonlinear element. Both these facts were not explained theoretically. In the presented work we performed detailed spectral measurements of fluence fluctuations at the output of a nonlinear element (Section 2) and proposed a theoretical explanation for the mentioned effects (Section 3).

2. Experimental results

The schematic of the experiment is shown in Fig. 1. A beam from the laser PEARL (PEtawatt pARametric Laser [25]) with central wavelength 910 nm, pulse energy up to 17 J, FWHM-duration 65 fs and diameter 18 cm passed through an aperture (AP) with a diameter of 10 cm located in vacuum at a distance of 8 m from the last diffraction grating of the compressor. Further, the beam was directed to the nonlinear element (NE). The distance from the AP to the NE was about 5 cm and the diffraction did not lead to significant changes in the beam distribution in the NE plane. The average over beam cross section intensity was 0.8 TW/cm². A 4-mm thick KDP crystal or a 7-mm K8 (BK7) glass was used as the NE. The B-integral values in these two cases were 10 and 14, respectively. Immediately in front of the NE there was a depolished glass plate 0.5 mm thick that served as a noise source (NS). The distance between the NS and the NE was chosen to be small enough (3 cm) to exclude spatial [26] and temporal [9] self-filtering.

 

Fig. 1. Schematic of the experiment. AP – aperture, NS – noise source (depolished glass plate), NE – nonlinear element, W – attenuating wedges.

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The NS quality was such that most of the noise was introduced by the NS, whereas the noise from other optical elements located both before and after the NE could be neglected. After the NE, the beam on reflection from two wedges was attenuated by about 1000 times and escaped from the vacuum chamber. The B-integral in the vacuum window did not exceed 0.05. The telescope relayed the image from the NE output onto the CDD camera with a 15-fold reduction in size. This is how we measured the fluence ${w_{nf}}(\boldsymbol{r} )= \smallint I({t,\boldsymbol{r}} )dt$.

Measurements were made in two regimes: linear and nonlinear. In the case of the linear regime, the NE was removed from the beam. Fluence fluctuation spectrum is

It makes sense to speak about SSSF only for θ much larger than the diffraction angle θ_diff≈0.01 mrad, i.e., at θ>0.1 mrad. On the other hand, the angle θ=4.6 mrad corresponds to the transverse size equal to one pixel of the CCD camera; therefore, within the θ=2.3-4.6 mrad range the data are “noisy” with the noise of the CCD camera itself. Figure 2 clearly demonstrates that after the nonlinear element, the fluence fluctuations do not increase in the 0.1 mrad <θ<2.6 rad range. This effect, that was also observed in [12], is absolutely unexpected, as the SSSF-induced amplification of spatial harmonic perturbations is expected to increase fluence fluctuations. The theoretical analysis explaining this effect is given in Section 3.

 

Fig. 2. Spectra of fluence fluctuations |S_nf|² for linear case (blue) and for nonlinear case with BK7 (red) and KDP (green). Vertical dashed lines show 1/20 of θ_cr.

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3. Fluence fluctuation gain at the output of the nonlinear element

Let the noise at the input of the nonlinear element be time independent:

The noise X(r, t) and its spatial spectrum S(κ, t) are related by the Fourier transform

Fluctuations of the fluence ${w_{nf}} = \int {|E |^2}dt$ from the value w₀ to an accuracy of |X|² has the form:

By substituting (4) into (5) and the result into (1), after integration we obtain a relationship between two spatial spectra: the spectrum of the noise field S and the spectrum of fluence fluctuations S_nf:

Following the theoretical approach of the works [18,27], we will assume that S(κ,t)=S(-κ,t). Then, from (6) we will find the spectrum of fluctuations at the output of the nonlinear element:

As shown in [27], the input S_in and output S_out noise spectra are related by the matrix U = U(ξ,B):

Here, θ and θ_cr are the external angles; the angles in the nonlinear medium are n₀ times smaller. The value of θ_cr typical for high-power femtosecond lasers is 30 mrad, and the corresponding spatial scale is 30 microns. From (8) we obtain

The substitution of (10) into (7) and the result into (1), taking into consideration that the input noise X_in does not depend on time, i.e., |S_in| and φ_in are time-independent, yields

Hereinafter, the angle brackets denote averaging over the series of random quantity X, as well as over the polar angle in κ-space. From (11) we find gain for the spectrum of fluence fluctuations K_nf:

3.1 Stationary case

Let I₀(t)=const, i.e., nothing depends on time. Then from (12) we have

From (13),(14) we obtain

The physics of (13),(15) is interpreted as follows. As shown in [27], in all three particular cases, φ_out is almost independent of φ_in and B. Note that for 2B$\mathrm{\kappa }$/$\mathrm{\kappa }$_cr<<1, φ_out ≈-π/2 (purely phase noise), therefore, fluence fluctuations grow only slightly; for $\mathrm{\kappa } = {\mathrm{\kappa }_{cr}}/\sqrt 2 $, φ_out ≈-π/4, hence the growth of fluence fluctuations is equal exactly to noise power growth; finally, for $\mathrm{\kappa }$=$\mathrm{\kappa }$_cr and B>>1, φ_out ≈0 (purely amplitude in-phase noise), therefore, the growth of fluence fluctuations is three times higher than the growth of noise power. Note that for 2B$\mathrm{\kappa }$/$\mathrm{\kappa }$_cr<<1, we have a purely phase noise at the output of the nonlinear element; however, with further propagation in free space, the noise becomes amplitude-phase one and fluence fluctuations grow, which was observed experimentally in [12].

3.2 Quasi-stationary case

Let the pulse have an arbitrary shape f(t):

If 2B$\mathrm{\kappa }$/$\mathrm{\kappa }$_cr<<1, then K_nf≈1, i.e., fluence fluctuations are B-independent, like in the stationary case. It is interesting to compare K_nf with K_ff obtained by the integration of (14) with respect to time:

The nonlinear element is usually followed by chirped mirrors that correct the phase of the time spectrum. It is easy to show that the chirped mirrors do not change the fluence w_nf, and, consequently, all the formulas obtained above, including (12)–(15) and (18),(19). In other words, the chirped mirror affects neither the fluence fluctuation spectrum S_nf(κ) nor the noise field spectrum S_ff($\mathrm{\kappa }$).

3.3 Nonstationary case

The stationary (13),(14) and quasi-stationary (18),(19) graphs for K_nf($\xi$) and K_ff($\xi$) are shown in Fig. 3 by blue and red curves, respectively. Using analytical methods it is difficult to take into account nonstationary effects – linear dispersion leading to pulse spreading and nonlinear dispersion leading to self-steepening. Expression (8) does not hold in this case; instead, the nonlinear Schrödinger equation should be solved numerically. We performed numerical simulation for a Gaussian input pulse f(t )=exp(-t²/τ²). Two additional parameters appear in the problem. These are N = τ/T – the pulse duration normalized to the field period T = 2π/ω₀ –and Λ=L_n/L_d – the ratio of the nonlinear length L_n= 1/(kn₂I_in) to the dispersion length L_d= τ²/k₂, where k₂ is the group velocity dispersion. For a given B_m, the parameter Λ is in charge of pulse spreading and N is responsible for self-steepening. We compared the simulation results with (18),(19). If Λ<0.003 and N > 10, the results of simulating the nonlinear Schrödinger equation differ from the quasi-stationary theory only slightly. The simulation demonstrated that the parameters of output radiation strongly depend on Λ and much less on N (for given Λ).

As an example, curves for Λ=0.0029, N = 12.9 (black), Λ=0.011, N = 12.9 (green) and Λ=0.035, N = 5.2 (cyan) are plotted in Fig. 3. The first two sets of parameters correspond to the experiments described above with KDP and BK7, respectively, and the third set of parameters corresponds to the experiments described in [12]. As seen from Fig. 3, for Λ=0.0029 the differences from the quasi-stationary theory are not essential (cf. the black and the red curves). For larger Λ, they become significant. Compared to the quasi-stationary case, the values of K_nf and K_ff decrease and their maxima shift towards smaller ξ. This is explained by the pulse stretching and its intensity reduction during propagation. Hence, the effective value of the B-integral decreases, which leads to a decrease in K_nf and K_ff, and the decrease in the effective values of κ_cr and θ_cr (9) leads to the shift of the K_nf and K_ff maxima.

 

Fig. 3. Curves for K_nf(ξ) (a) and K_ff(ξ) (b) for stationary expressions (13),(14) (blue), for quasi-stationary expressions (18),(19) (red), and numerical solution of the nonlinear Schrödinger equation for Λ=0.0029, N = 12.9 (black), Λ=0.011, N = 12.9 (green) and Λ=0.035, N = 5.2 (cyan).

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In the ξ<0.01 region corresponding to our experiments (Section 2) as well as to the data presented in [12], the value of K_nf is only a little greater than unity. Thus, the presented theory explains absence of significant growth of fluence fluctuations (K_nf≈1), despite the large values of the B-integral.

The simulation of beam propagation after the nonlinear element demonstrated the growth of fluence fluctuations. As noted above, this occurs because there is primarily phase noise at the NE output, while with further propagation in free space the fraction of the amplitude noise increases. The simulated results are shown in Fig. 4 and are in a good agreement with the experimental data [12]. Thus, the probability of breakdown of the optical elements located after the NE is higher than the probability of NE breakdown.

 

Fig. 4. Results of numerical simulation of beam propagation after 1.5-mm thick plate of quartz glass, the beam parameters are taken from [12].

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4. Conclusion

Analytical expressions for the spectrum of fluence fluctuations during radiation propagation in the Kerr nonlinear medium have been obtained in the stationary and quasi-stationary approximations. Numerical simulation for the nonstationary case showed that the obtained formulas give fairly accurate results if the ratio of the nonlinear length to the dispersion length is less than 0.003 and the pulse duration is more than 10 laser field cycles.

We have experimentally verified the earlier revealed effect [12] of a very weak growth of fluence fluctuations during propagation in a nonlinear medium, even for large values of the B-integral. We have shown that this effect occurs only for the fluctuations with a spatial scale much larger that the critical scale of self-focusing. We have proposed a theoretical explanation of this effect, according to which the noise on such scales becomes a phase one, hence the fluence fluctuations decrease. This also explains the growth of fluence fluctuations during beam propagation in free space after the nonlinear element, when phase fluctuations transform to amplitude fluctuations. Thus, the probability of breakdown of the optical elements located after the nonlinear element is higher than that of the nonlinear element. Note that this conclusion concerns only noises with a spatial scale much larger than the critical self-focusing scale that is about 30 microns for femtosecond lasers.

The results obtained may be interesting for pulse post-compression, frequency doubling and other experiments using transmission optical elements, such as vacuum windows, λ/2 and λ/4 plates, polarizers, beamsplitters, and the like.