1 Introduction
Small-scale self-focusing (SSSF) is, actually, filamentation instability of a plane wave propagating in a medium with cubic nonlinearity. The SSSF manifests itself as a fast amplification of wave components with high spatial frequencies, which ultimately leads to a beam splitting into filaments, thus deteriorating the beam quality and significantly increasing the risk of optical elements breakdown. SSSF was first theoretically predicted by Bespalov and Talanov[ Reference Bespalov and Talanov 1 ], experimentally confirmed in Refs. [Reference Chilingarian2–Reference Abbi and Mahr6], and the theory was further developed in Refs. [Reference Abbi and Kothari7–Reference Marburger12]. At the linear stage of development, SSSF is reduced to the energy transfer from a plane wave to spatial noise, that is, to the amplification of seed noise waves. The gain K depends on the angle θ = k ⊥/k between the wave vectors of the noise component k ⊥ and plane wave k = n 0 k 0 (where n 0 is the linear index of refraction). In the linear theory, K has a maximum value at the angle as follows:
$$\begin{align}{\theta}_{\mathrm{max}}=\sqrt{\frac{2{n}_2I}{n_0}},\end{align}$$
where I is the intensity of a plane wave and the refractive index of a nonlinear medium is defined as n = n 0 + n 2 I (where n 2 is the nonlinear index of refraction). In the linear approximation, the maximum gain is as follows:
$$\begin{align}K\left({\theta}_{\mathrm{max}}\right)=\cosh\ (2B),\end{align}$$
where B is the B-integral:
$$\begin{align}B={k}_0{n}_2 Il,\end{align}$$
and l is the length of the nonlinear medium.
In classical SSSF works, nanosecond pulses were considered, where the beam was usually split into filaments at B > 3. SSSF studies for femtosecond pulses[ Reference Mironov, Lozhkarev, Ginzburg, Yakovlev, Luchinin, Shaykin, Khazanov, Babin, Novikov, Fadeev, Sergeev and Mourou 13 – Reference Kochetkov, Martyanov, Ginzburg and Khazanov 16 ] showed that there is a significant difference between nano- and femtosecond pulses, which allows using beam self-filtering as a method for SSSF suppression. There are two mechanisms of beam self-filtering during propagation in free space. In the first (which we will call spatial self-filtering) the noise components propagate at an angle to the main beam and, if the beam passed a rather large length of free space, these components go outside the beam aperture. According to the second mechanism, called temporal self-filtering, the noise components lag behind the main beam, which, in the case of short pulses, allows reducing noise at the input to the nonlinear medium and, hence, suppressing the SSSF. The efficiency of both self-filtering methods depends on the angle θ max. For nanosecond pulses, a typical θ max value is about 1 mrad, which impedes self-filtering as it demands a too large propagation length. At the same time, for femtosecond pulses, the intensity is as a rule three orders of magnitude higher and, according to Equation (1), the typical values of θ max are tens of mrad, which makes self-filtering an efficient mechanism of noise reduction for θ of the order of θ max.
The linear theory of SSSF gives a correct result only at an early stage of instability development, when the power of the amplified noise is low. From a practical point of view, this stage is not so interesting, since because of small noise the beam retains an acceptable quality. Of major interest is the determination of the B-integral values at which hot spots leading to optical breakdown appear in the beam, or at which a significant part (e.g., 10%) of the plane wave energy is converted into noise. Thus, to quantify the efficiency of self-filtering for suppressing SSSF, the nonlinear regime should be considered. It is also worth mentioning that the experimental results demonstrated a significant suppression of SSSF, which cannot be explained by the existing theoretical concepts[ Reference Martyanov, Ginzburg, Balakin, Skobelev, Silin, Kochetkov, Yakovlev, Kuzmin, Mironov, Shaikin, Stukachev, Shaykin, Khazanov and Litvak 17 ]. One of the goals of our paper is to propose such a concept.
This work is devoted to a detailed numerical simulation of the nonlinear stage of SSSF, taking into account various self-filtering parameters of the input pulse. The conditions under which self-filtering leads to significant suppression of the filamentation instability are found, and a new type of instability is discovered.
2 Problem statement
The propagation of the field in a medium with Kerr nonlinearity will be studied using the nonlinear Schrödinger equation (NSE)[ Reference Khazanov, Mironov and Mourou 18 ] of the following form:
$$\begin{align}\frac{\partial A}{\partial z}+\frac{i}{2k}{\Delta }_{\perp }A+i\frac{3\pi {\omega}_0{\chi}^{(3)}}{2n\left({\omega}_0\right)c}{\left|A\right|}^2A=0,\end{align}$$
where A is the complex amplitude of the field, z is the longitudinal component, c is the speed of light in vacuum,
${{\omega}_0=c{k}_0}$
, Δ⊥ is the transverse Laplace operator and χ(3) is the cubic nonlinear susceptibility. To simplify the computations, we omitted in the NSE the time dependence of the complex field amplitude and neglected the finite transverse size of the beam. The simulation was made for a square 5 mm × 5 mm beam region with periodic boundary conditions.
At the entrance to the nonlinear medium (z = 0), the field is represented as a plane wave with superimposed noise. The noise is determined by the manufacturing quality of the optical elements through which the beam has passed, and is usually characterized by a 1D power spectral density PSD(k x ). The PSD(k x ) function typical of the optical elements is well approximated by the power dependence[ Reference Spaeth, Manes, Widmayer, Williams, Whitman, Henesian, Stowers and Honig 19 – Reference Liu, Jin, Zhou, Bai, Zhao, Yi and Shao 21 ]:
$$\begin{align}\mathrm{PSD}\left({k}_x\right)=\frac{\Phi_1}{k_x^{\beta}},\end{align}$$
where Φ1 and β are constants. The constant β takes on various values from 1 to 2 in different works and spatial frequency ranges, where β = 1.55 is the most frequent one. As shown in Ref. [Reference Ginzburg, Kochetkov, Mironov, Potemkin, Silin and Khazanov22], the
$\mathrm{PSD}$
in Equation (5) corresponds to the 2D power spectral density of the noise PSD2(k
⊥) in the following form:
$$\begin{align}\mathrm{PSD}2\left({k}_{\perp}\right)=\frac{\Phi_2}{k_{\perp}^{\beta +1}},\end{align}$$
where Φ2 is a constant. Thus, for PSD2(k ⊥) we adopt the dependence in Equation (6), where β = 1.55.
The beam self-filtering is modeled using the expression for temporal self-filtering[ Reference Khazanov 23 ] at which the free space for a Gaussian pulse in the time domain is the filter of spatial frequencies (angles θ) with transmittance:
$$\begin{align}{T}_{\mathrm{time}}\left(\theta \right)=\exp \left(-{\left(\ln (2)\frac{L}{\lambda}\frac{\theta^2}{N}\right)}^2\right)=\exp \left(-\frac{\theta^4}{\theta_{\mathrm{thr}}^4}\right),\end{align}$$
where L is the distance passed by the pulse in free space, λ = 2π/k 0, N is the ratio of the pulse duration to the period of the field and θ thr is the threshold value of the angle of self-filtering (the angle at which T time(θ) = 1/e). Further, for definiteness, we will assume θ thr = 4 mrad, which corresponds, for example, to L = 1.6 m for the pulse duration of 60 fs and wavelength of 910 nm.
Equation (7) was obtained for an ideal Gaussian pulse. In practice, the contrast of femtosecond pulses is limited, as a pedestal outside the main pulse restricts the filter contrast (Equation (7)), which was taken into consideration in the numerical simulations. It is worth noting that at spatial filtering the free space is a filter with transmittance T space(θ), the shape of which[ Reference Kochetkov, Martyanov, Ginzburg and Khazanov 16 ] is close to Equation (7), but the contrast may be much higher, as there is almost no spatial pedestal. An important feature of both T time(θ) and T space(θ) is a very sharp decrease when θ exceeds the threshold value θ thr.
When a laser pulse propagates through optical elements, inaccuracy in manufacturing the shape of optical surfaces gives rise to phase noise (the imaginary part of the complex field amplitude) and various scratches and dust particles to amplitude noise (the real part of the complex field amplitude). During the propagation in free space, the phase noise and amplitude noise transform into each other; therefore, after passing through a large number of optical elements, it can be assumed with a high degree of certainty that the noise is half phase and half amplitude. It is this noise that was used in the simulation of the SSSF. In this work, we will neglect the noise arising from the refraction at the input surface of a nonlinear medium.
An example of intensity and phase distribution in the beam at the input of the nonlinear element is given in Figure 1. The noise in Figure 1 has been filtered according to Equation (7) with the self-filtering threshold θ thr = 4 mrad. The input noise power P noise is about 0.02% of the power P 0 of the principal wave. This noise gives the ratio of the maximum intensity in the beam I max to the mean intensity I 0 of about 1.07.
Figure 1 Intensity and phase distribution in the input beam within a 5 mm × 5 mm area.
Figure 2 SSSF for θ max = 1 mrad at noise filter contrast 108: (a) noise spectrum (solid curves), gain K(θ) in the linear regime for B = 5 (dashed curve); (b) intensity distribution in the beam for B = 5 within a 5 mm × 5 mm area.
The main goal of the simulation was to see the difference in the development of instability in two cases: θ max< θ thr = 4 mrad and θ max > θ thr. The first case is typical for nanosecond and picosecond pulses, for which θ max is about 1 mrad; therefore, self-filtering does not affect the noise components with maximal increment. The case with θ max > θ thr is typical for femtosecond pulses, for which θ max is usually tens of mrad and the noise components with maximal increment are filtered at the input to the nonlinear medium, and only components with a much lower gain remain. Hence, we performed numerical simulations for four θ max values: 1, 3, 10 and 30 mrad. If fused silica is used as the nonlinear medium, then the above values of θ max correspond to intensity I 0 = 3, 27, 300 and 2700 GW/cm2, respectively. The calculations were performed up to the values of the B-integrals, at which a significant power of the principal wave was converted into noise. With a further increase in the B-integral, the sizes of bright spots become smaller than the cell size of the computational grid, which varied from 2.5 to 20 μm in different calculations.
3 Results of numerical simulation
The calculated noise spectra S(θ) = PSD2(k ⊥ = kθ) are shown in Figures 2(a) and 3(a) for the original beam (B = 0) and the beam that has passed in a nonlinear medium distance corresponding to B = 3, 4, 5, 6; the self-filtering contrast is 108. Figure 2(a) is for θ max = 1 mrad and Figure 3(a) is for θ max = 3 mrad. The dashed curves correspond to the gain K(θ) in the linear (without plane wave depletion) regime[ Reference Khazanov, Mironov and Mourou 18 ]:
$$\begin{align}K=1+\frac{2}{\frac{2{\theta}^2}{\theta_{\mathrm{max}}^2}-\frac{\theta^4}{\theta_{\mathrm{max}}^4}}{\cdot \sinh}^2\left(B\sqrt{\frac{2{\theta}^2}{\theta_{\mathrm{max}}^2}-\frac{\theta^4}{\theta_{\mathrm{max}}^4}}\right).\end{align}$$
Figure 3 SSSF for θ max = 3 mrad at noise filter contrast 108: (a) noise spectrum (solid curves), K(θ) in the linear regime for B = 6 (dashed curve); (b) intensity distribution in the beam for B = 6 within a 2.5 mm × 2.5 mm area.
Figure 4 SSSF for θ max = 10 mrad at noise filter contrast 108: (a) noise spectrum (solid curves), K(θ) in the linear regime for B = 11 (dashed curve); (b) intensity distribution in the beam for B = 11 within 2.5 mm × 2.5 mm area (see Silin_supplementmovie1.avi for 0 ≤ B ≤ 11 within a 5 mm × 5 mm area).
Figure 5 SSSF for θ max = 10 mrad, no self-filtering: (a) noise spectrum (solid curves), K(θ) in the linear regime for B = 6 (dashed curve); (b) intensity distribution in the beam for B = 6 within a 1 mm × 1 mm area (see Silin_supplementmovie2.avi for 0 ≤ B ≤ 6 within a 5 mm × 5 mm area).
For B = 3, 4, maximum gain is seen at θ = θ
max, which is quite expected. For B = 5, 6, the radiation collapses and the noise spectrum approaches white noise, with the noise power increasing by many orders of magnitude, even outside the instability region, that is, at θ >
$\sqrt{2}$
θ
max. The intensity distribution in the beam is illustrated in Figure 2(b) (for θ
max = 1 mrad, В = 5) and in Figure 3(b) (for θ
max = 3 mrad, В = 6). The collapse is seen in Figures 2(b) and 3(b). The radiation is focused into a large number of filaments of huge intensity: I
max/I
0 = 274 for θ
max = 1 mrad and I
max/I
0 = 187 for θ
max = 3 mrad. In this case p = 0.28 and 0.45, where p = P
noise /P
0, that is, 28% and 45% of the beam power has been converted into noise, respectively. The power in each filament is P
fil ≈ 2.25P
cr, where the following applies:
$$\begin{align}{P}_{\mathrm{cr}}=\frac{0.174{\lambda}^2}{n_2{n}_0}\end{align}$$
is the critical power of self-focusing[ Reference Gol’dberg, Talanov and Erm 24 ], and a typical distance between the filaments is λ/(n 0 θ max). Thus, for θ max < θ thr we have obtained a classical filamentation instability. The maximum values of I/I 0 on the color palettes in Figures 2(b) and 3(b) are determined by excluding 0.05% of points with maximum intensity. Therefore, the maximum values of I/I 0 on the palettes are less than those of I max/I 0.
The effect of beam self-filtering is expected to manifest itself at θ max > θ thr. The noise spectra S(θ) for θ max = 10 mrad are shown in Figures 4(a) and 5(a) for various values of the B-integral. Figure 4(a) corresponds to the case with self-filtering of the beam before it enters the nonlinear medium with a contrast of 108, and Figure 5(a) corresponds to the case without self-filtering. In the latter case, the noise gain maximum is at θ = θ max. There is no such gain maximum in the noise spectra after self-filtering, and the nonlinear stage looks qualitatively different, as is clearly demonstrated in Figure 4(b) for B = 11 (p = 0.32). It can be seen that, instead of a large number of bright dots, a random honeycomb structure has appeared. A typical cell size of this structure corresponds to λ/(n 0θthr). To the best of our knowledge, such an instability has never been observed before either in numerical simulation or in experiment. We will refer to it as honeycomb instability. The honeycomb structure is preserved with an increase in the B-integral, the intensity in the walls of the honeycombs increases and their thickness decreases until it becomes less than the cell size of the computational grid. For comparison, the intensity distribution without self-filtering is shown in Figure 5(b) for B = 6, that is, for a value significantly smaller than that in Figure 4(b). Here, 41% of the beam power was converted into noise (p = 0.41), and a classical collapse of the radiation into a large number of bright points with a typical transverse scale λ/(n 0θmax) is visible. There are two videos of SSSF for θ max = 10 mrad (the time scale corresponds to the beam propagation in a nonlinear medium and, accordingly, to the growth of the B-integral) in the supplementary materials for this paper. In one of the videos (Silin_supplementmovie1.avi), the calculation was performed taking into account beam self-filtering, which leads to the development of honeycomb instability. In the second video (Silin_supplementmovie2.avi), there is no beam self-filtering, and filamentation instability develops.
Figure 6 SSSF for θ max = 30 mrad at noise filter contrast 108: (a) noise spectrum (solid curves), K(θ) in the linear regime for B = 16 (dashed curve); (b) intensity distribution in the beam for B = 16 within a 1 mm × 1 mm area.
Figure 7 SSSF for θ max = 30 mrad at noise filter contrast 1024: (a) noise spectrum (solid curves), K(θ) in the linear regime for B = 27 (dashed curve); (b) intensity distribution in the beam for B = 27 within a 2.5 mm × 2.5 mm area.
Now consider the case of θ max = 30 mrad that is typical of femtosecond pulses. The noise spectra S(θ) for different values of the B-integral are shown in Figure 6(a). Despite the beam self-filtering one can see gain maxima at θ = θ max. We attribute this to the fact that θ max = 30 mrad is far from the filtering threshold θ thr = 4 mrad; therefore, the amplification of the noise components at θ < 4 mrad is very low at the linear stage of the SSSF, and the noise components in the θ∼θ max region are in an advantageous situation despite the self-filtering. This is what differentiates the case with θ max = 30 mrad from the case with θ max = 10 mrad, where the components with θ < θ thr are advantageous, so the honeycomb instability develops faster than the filamentation one. The intensity distribution in the beam for B = 16 is shown in Figure 6(b). It can be seen that the beam collapsed into a large number of random points and there is no honeycomb structure (here p = 0.51). For the honeycomb structure to appear in the beam cross-section, the development of filamentation instability should be slowed down even more. The numerical simulations have shown that this can be achieved by increasing the noise filter contrast by up to 20 orders of magnitude or more. Naturally, such a level of noise suppression cannot be achieved in practice, even with spatial self-filtering. However, to demonstrate the physical aspect of the problem, we have performed the corresponding numerical simulations.
The noise spectra for θ max = 30 mrad at a noise filter contrast of the 24th order are plotted in Figure 7(a), and the corresponding intensity distribution in the beam for B = 27 is presented in Figure 7(b) (here p = 0.59). In this case, the filamentation instability is inferior to the honeycomb instability. A honeycomb structure with characteristic size λ/(n 0θthr) is well seen in the beam.
At the end of this section, we will consider a benchmark to provide some verification of our numerical simulation. Let us calculate the noise gain K(θ) in the linear mode (there is no depletion of the principal wave). In this case, the gain must coincide with the analytical formula (Equation (8)). Letting θ max = 10 mrad and the filter contrast be 108, we observed the appearance of a honeycomb structure with these parameters (see Figure 4). However, the input noise level we used produces a discrepancy between the gain K(θ) obtained from the numerical simulation and Equation (8) even at B = 1. Therefore, we have reduced the input noise level by two orders of amplitude. In this case, numerical simulation also leads to the appearance of a honeycomb structure; however, at small values of B, the gain coincides with Equation (8) with good accuracy. This can be seen in Figure 8, which shows the noise gain for two values of the B-integral, B = 5 and B = 8, obtained from numerical simulation and Equation (8).
Figure 8 Comparison of noise gain K(θ) in the linear mode obtained using numerical simulation and Equation (8).
4 The influence of beam self-filtering on permissible values of the B-integral
Beam self-filtering allows suppressing the noise components possessing maximum gain at the SSSF, thus increasing the value of the B-integral, at which the beam is not fit for most applications, that is, an appreciable part of the energy goes into noise, and/or I max becomes significantly larger than I 0. To understand how beam self-filtering affects the permissible values of the B-integral, we will construct various beam parameters as a function of B.
The share of radiation power converted into noise p = P noise /P 0 versus the B-integral is shown in Figures 9(a) and 9(b). The maximum intensity in the beam normalized to mean intensity I max/I 0 versus the B-integral is plotted in Figures 9(c) and 9(d). The root mean square (RMS) intensity in the beam where the intensity is normalized to the mean intensity RMS I = RMS(I/I 0–1) as a function of the B-integral is shown in Figure 9(e). Finally, the RMS phase in the beam relative to the mean phase in the beam RMSφ = RMS(φ−<φ>) as a function of the B-integral is plotted in Figure 9(f). Figures 9(a), 9(c), 9(e) and 9(f) show the curves for the input noise level of 0.02% (this noise level was used for obtaining the results described in Section 3), and Figures 9(b) and 9(d) correspond to the input noise level of 0.002%. The curves in Figure 9 correspond to the four values θ max = 1, 3, 10 and 30 mrad. The curves for θ max = 1, 3 mrad are plotted for the case of self-filtering. However, since θ max< θ thr = 4 mrad, the self-filtering does not affect the result, so the curves in the absence of self-filtering are the same. For θ max = 10 mrad, the curves are plotted both for self-filtering and without it at a contrast of 108. For θ max = 30 mrad, the curves are plotted for two values of noise filter contrast, 108 and 1024, as well as without self-filtering. The effect of collapse is clearly demonstrated in Figures 9(c) and 9(d): at a definite value of the B-integral, I max increases ‘almost vertically’, despite the logarithmic scale.
Figure 9 (a), (b) Fraction of radiation power converted into noise, (c), (d) maximum intensity in the beam normalized to mean intensity in the beam, (e) root mean square (RMS) intensity in the beam and (f) RMS phase in the beam as a function of the B-integral. Curves (a), (c), (e) and (f) correspond to the level of input noise of about 0.02% of the beam power, while curves (b) and (d) are for the level of input noise of about 0.002% of the beam power. Self-filtering threshold θ thr = 4 mrad.
From the plots presented in Figure 9, we can draw the following conclusions. Without noise filtering, an increase in θ max leads to an insignificant decrease in the parameters p, RMS I and RMSφ, while I max remains almost unchanged. Thus, from the point of view of the I max growth, the admissible B-integral does not depend on θ max. Beam self-filtering at small values of the B-integral (B < 4) practically does not affect p or RMSφ, while from the point of view of I max and RMS I , self-filtering suppresses the SSSF. This means that the amplified noise is phase noise.
With self-filtering, all curves in Figures 9(a)–9(d) are notably shifted to the right, which points to a significant increase in the permissible B-integral. The increase in the noise filter contrast increases the admissible B-integral still more. Another important fact is that the decrease in the input noise level affects the increase in the permissible B-integral in different ways. Without self-filtering, as well as for θ max = 30 mrad and filter contrast 108, when a honeycomb structure is not formed, a decrease in the input noise by a factor of 10 increases the admissible B-integral by approximately 1.5. At the same time, during self-filtering, when a honeycomb structure appears, the admissible B-integral with the same decrease in the input noise increases by about 2–3 for θ max = 10 mrad and filter contrast 108 and by about 6–14 for θ max = 30 mrad and filter contrast 1024. The permissible values of the B-integral at θ max = 30 mrad are given in Table 1 for different values of filter contrast and input noise level under the specified criteria I max/I 0 < 1.5, p < 10%. It can be seen from the table that without filtering, the criterion I max/I 0 < 1.5 is more stringent than p < 10%. However, with an increase in contrast, the allowable values of the B-integral in terms of the criterion I max/I 0 < 1.5 grow faster and, with a high filter contrast, the energy transfer from the principal beam to noise comes to the fore. Note that even with a contrast of 106 only, the admissible values of the B-integral become two-digit.
Table 1 Permissible values of the B-integral for θ max = 30 mrad.
5 Discussion of the results
The obtained results may be physically interpreted as follows. Without self-filtering, the instability is developing primarily at the spatial frequencies k
⊥max= k
0
n
0
θ
max and on the spatial scales w
max = 2π/k
⊥max = λ/(n
0
θ
max) corresponding to the highest increment. From the known expression for the length of self-focusing (the length at which a beam having diameter w and power
${P}_1$
collapses)[
Reference Gol’dberg, Talanov and Erm
24
],
$$\begin{align}{L}_{\mathrm{sf}}=\frac{0.0925{k}_0{n}_0{w}^2}{\sqrt{{\left(\sqrt{P_1/{P}_{\mathrm{cr}}}-0.852\right)}^2-0.03}},\end{align}$$
it follows that for w = w max the beam collapses at B ≈ 2.3. The real values are somewhat higher, because for the instability to form inhomogeneities of sufficient amplitude on a plane wave the beam must pass a definite distance. However, self-filtering of high spatial frequencies may result in the concentration of maximum noise power at the spatial frequencies k 0 n 0 θ thr and on the spatial scales w thr = λ/(n 0 θ thr), since these scales, although they have a smaller increment, are not subject to filtering. Thus, during self-filtering, there is a competition between the scales w max and w thr. If the contrast is high and θ thr is a little less than θ max (i.e., the difference in increments is not so significant), then the maximum noise power will be concentrated on the w thr scale. Alternatively, if the contrast is low and θ thr is much less than θ max, the scale w max will be the winner. In this case, the SSSF has a classical filamentation character and self-filtering leads to an increase in the admissible value of the B-integral, as the input noise power on the w max scale is significantly reduced. If the scale w thr ‘wins’, then the picture changes qualitatively. The filamentation instability is completely suppressed and the honeycomb instability develops, which is radiation focusing into a line (interface between the cells) rather than into a point, that is, 1D self-focusing occurs instead of a 2D one. Since the problem is isotropic, the directions of these lines are random, which gives a honeycomb-like spatial pattern (Figures 4(b) and 7(b)) with a typical size w thr. Note that the cell size w thr depends neither on I nor on n 2; therefore, the power per a single cell Iw thr does not depend on n 2.
Let us try to explain the absence of 2D self-focusing. Shortly before the development of filamentation or honeycomb instability (B is 2–3 radians less than in the case of collapse), the intensity distribution in the beam looks similar in both cases. An example of such a distribution is demonstrated in Figure 10. For the sake of generality, the color scale is not shown in Figure 10, as it may be different, but I
max here exceeds I
0 by several tens of percent. The characteristic scale w (characteristic transverse sizes of the spots) corresponds to the scale w
thr if θ
max > θ
thr (at a sufficient level of self-filtering contrast) and w
max if θ
max < θ
thr, or there is no self-filtering. Here, the conditions used to obtain Equation (10) are not fully realized, since there are no isolated round spots for which the self-focusing length could be calculated. One can see in Figure 10 compact bright (red) spots, but they are located on the elongated spots with a lower intensity. Therefore, the nonlinear phase gradient on the bright spots varies in different directions, resulting in different self-focusing rates in different directions. Further SSSF development depends only on the ratio of the scales w and w
max (in other words, on the
$\theta_{\max} / \theta_{\mathrm{thr}}$
ratio). If there is no self-filtering (or θ
max < θ
thr), then on the w = w
max scale the 2D self-focusing increment significantly exceeds the 1D self-focusing increment, as a result of which the elongated spots break up into separate round ones and classical 2D self-focusing occurs with the formation of filaments. At a high enough level of contrast, self-filtering (θ
max > θ
thr) results in the characteristic scale w = w
thr > w
max on which 2D self-focusing does not have such an advantage over 1D self-focusing. So, due to a larger nonlinear phase gradient in the transverse direction of the elongated spots than along them, 1D self-focusing is faster than 2D self-focusing. As a result, the elongated spots turn into honeycomb borders.
Figure 10 Example of intensity distribution in a beam shortly before the development of either filamentation or honeycomb instability.
The honeycomb patterns in nonlinear optics were reported in 2002[ Reference Bennink, Wong, Marino, Aronstein, Boyd, Stroud and Lukishova 25 ]. There are fundamental differences in the honeycomb structure presented in Ref. [Reference Bennink, Wong, Marino, Aronstein, Boyd, Stroud and Lukishova25] and in our paper. Firstly, somewhat different equations are solved. We consider the propagation in a medium with cubic nonlinearity (corresponding to the term in the NSE ~|A|2 A), while in Ref. [Reference Bennink, Wong, Marino, Aronstein, Boyd, Stroud and Lukishova25] the propagation in a medium with saturation of the linear part of susceptibility is considered (the corresponding term in the equation ~A/(1+|A/A sat|2)). Secondly, and most importantly, is that in our work there occurs 1D SSSF into the honeycomb borders, so the intensity of the honeycomb borders becomes significantly higher than the intensity of the honeycomb interiors (see Figures 4(b) and 7(b)), while in Ref. [Reference Bennink, Wong, Marino, Aronstein, Boyd, Stroud and Lukishova25], in contrast, the honeycomb borders are darker than the honeycomb interior.
It should be noted that honeycomb instability occurs over a wide range of initial parameters. The above figures present the results when the degree of PSD decay in Equation (5) is β = 1.55 and the spatial frequency filter has the form in Equation (7). However, numerical simulation has shown that the honeycomb structure in the beam appears for the β values at least in the 0 ≤ β ≤ 3 range, and the sharpness of the spatial frequency filter can vary in the range at least from
$\exp \left(\hbox{--} {\theta}^2/{\theta}_{\mathrm{thr}}^2\right)$
to a step function. A necessary condition for honeycomb instability is sufficient filter contrast, which differs for different parameters. The parameters considered in the paper (β = 1.55, filter Equation (7) at θ
thr = 4 mrad, input noise power about 0.02% of the power of the main wave) require a filter contrast of at least 106 for θ
max = 10 mrad, while for θ
max = 30 mrad a filter contrast of at least 1020 is required. In any case, honeycomb instability exists (or does not exist) irrespective of the random realization of spatial noise used as a boundary condition for Equation (4).
We have considered a stationary instability model, taking into account pulse duration only in the filter transmittance (Equation (7)). Obviously, allowance for nonstationarity in the NSE (Equation (1)) will lead to pulse spreading in the course of propagation and, consequently, to a decrease in the effective B-integral; see, for example, Ref. [Reference Martyanov, Ginzburg, Balakin, Skobelev, Silin, Kochetkov, Yakovlev, Kuzmin, Mironov, Shaikin, Stukachev, Shaykin, Khazanov and Litvak17]. This means that the values of admissible B-integrals listed in Table 1 will be even higher for fs pulses. A detailed investigation of this issue will be the subject of another publication.
6 Conclusion
The results of the performed numerical simulation lead to the following conclusions on the efficiency of beam self-filtering in terms of instability suppression.
(1) At θ max > θ thr, the classical filamentation instability leading to the appearance of bright points in the beam develops slower, that is, at larger values of the B-integral.
(2) With a high self-filtering contrast, the filamentation instability does not develop at all, since it is inferior to the earlier unknown honeycomb instability, resulting in a honeycomb structure in which all energy is accumulated within the boundaries between the cells.
(3) Beam self-filtering at θ max > θ thr significantly increases the admissible values of the B-integral, at which the beam retains acceptable quality: the maximum intensity exceeds the average one by no more than 50% and the fraction of power in the noise does not exceed 10%. The higher the filtering contrast, the larger the admissible values of the B-integral. In the absence of self-filtering, the mentioned intensity increase occurs earlier than 10% of the beam power converting into noise, whereas in the case of self-filtering with a sufficiently high contrast, the opposite is true.
(4) The reduction of the fraction of noise power in the input beam with self-filtering increases the admissible B-integral by a much larger value than in the absence of self-filtering.
(5) This study may be a route to explain experimental results[ Reference Martyanov, Ginzburg, Balakin, Skobelev, Silin, Kochetkov, Yakovlev, Kuzmin, Mironov, Shaikin, Stukachev, Shaykin, Khazanov and Litvak 17 ] that cannot be explained by the existing theoretical concepts.
Supplementary materials
The supplementary material for this article can be found at http://doi.org/10.1017/hpl.2024.9.
Acknowledgements
This work was supported by the Ministry of Science and Higher Education of the Russian Federation (075-15-2020-906, Center of Excellence ‘Center of Photonics’).
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