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Abstract
In various applications, it is necessary to understand laser field dynamics during its propagation, especially at the focal position including the dispersed energy surrounding the main pulse, called the halo effect. For instance, the properties of electron beams produced by laser wakefield acceleration (LWFA) strongly depend on the laser energy distribution and its halo in the vicinity of the focus. Indeed, under certain conditions, this halo, or even its internal structures, can propagate and be self-focused independently of the main pulse in the plasma. This paper aims to provide sufficient tools to properly describe the behavior of a focused laser beam, including the halo. Subsequently, an optical description regarding the source of this halo is provided. A more accurate estimation of the input laser beam that should be used in simulations of high-power laser applications may now be obtained. Finally, one may also find ways to positively manipulate the laser beam. Using Fresnel diffraction theory, the propagation and focusing of an experimental high-power (sub-petawatt) aberrated beam is numerically calculated. The shape of the focused beam pattern within a few Rayleigh lengths is analysed as a function of main aberrations (up to the 14th term of Zernike polynomials). Furthermore, at the focus position, the spreading of the energy is compared to both the case of a perfect diffraction-limited Gaussian and a super-Gaussian beam.
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1. Introduction
Since the experimental demonstration of the laser by Maiman in 1960 [1], multiple devices and systems based on lasers were developed and now surround our lives. In the last 60 years, laser systems were continuously improved and one major breakthrough was made by Strickland and Mourou with the CPA (chirped pulse amplification) technique in 1985 [2], thus, opening the path for very intense lasers. Therefore, various applications are now accessible, and many more under development thanks to these powerful lasers, such as laser fusion or laser particle acceleration that might lead to very compact systems compared to km-scale current accelerators. A particular application, called Laser Wake Field Acceleration (LWFA), initially suggested in 1979 by Tajima et al. [3], proposed to use the laser as an electron beam generator, but also as a driver to accelerate these electrons at relativistic speed. Very recently, FEL (Free-Electron Laser) radiation sources based on laser accelerators were demonstrated [4,5]. However, the path towards industrialisation of such devices now highly requires a stability comparable to classical accelerators, hence a better understanding of laser beams instabilities and disturbances when being focused is required, which is the main purpose of this article.
Indeed, drawbacks are inherent with high-power lasers: as the beam must be expanded to avoid damaging optics while increasing in energy, the perfect paraxial theory can no longer be applied as incident angles on curved surfaces increase. Also, optics have to be larger and are more subjected to imperfections. Thus, the beam is more affected by aberrations. Although some solutions like deformable mirrors may be implemented, it remains important to understand how these aberrations may affect the beam during its propagation and focusing while interacting with the propagation medium [6,7]. As a consequence, focused high-power lasers often exhibit an imperfect shape with a non-negligible disturbance surrounding the main spot, thus reducing the effective energy and inducing unwanted effects [8]. In LWFA, this disturbance, commonly referred as halo, was numerically studied and detrimental consequences for the electron beam were underlined [9,10], even though the true nature of this effect was not explained. However, input beams used for simulations were only a mathematical model to reproduce the experimental intensity distribution at the focus, but the propagating behaviour, related to the beam phase, was not taken under consideration [11]. What is more, the effect of a purposefully imperfect laser beam was studied in [12]. The phase of the beam was retrieved through two measured intensity patterns using Gerchberg-Saxton algorithm (GSA [13]), and implemented into simulations. Despite the authors concluded that a real beam pattern gives numerical result closer to experimental ones regarding electron generation, still a clear description of how this disturbance around the beam is created, and thus may be handled, is missing. In [14], the beam wavefront was purposefully modified, and the effect on self-focusing and generated electrons was investigated and demonstrates that the wakefield may be tuned to change both the charge and the energy of electrons. Moreover, He et al. demonstrated that electrons may be controlled by using feedback on the laser wavefront [15], while Mangles et al. were able to control the X-rays spectrum emitted by electrons by tuning the wavefront [16]. Also, it has been established that comatic aberration can be self-healed and does not affect the properties of the electron beam [17]. This underlines the importance of the beam propagation inside the plasma, and thus of its phase component and halo structure, despite its origin has not been yet clarified.
One typical way to propagate and focus a given laser pattern is to calculate the field with Fresnel diffraction theory. Various algorithms were already developed and demonstrated great abilities [18–21]. Other methods, based on vectorial models, also exist to take into account the polarisation [22], and present very consistent results between simulations and experiments [23]. Very recently, Zemzemi used Fresnel integral in the near field to simulate the resulting focused beam when using a hole mirror, and studied how electron generation was affected [24]. On the other hand, it has already been shown that Gaussian beams and top-hat-like beams do not propagate the same, nor get affected similarly by aberrations [25]. Furthermore, investigations regarding the shape of PSFs (Point Spread Function - the focusing of a single point source) as function of aberrations were already conducted and underlined that all aberrations will not disturb similarly the focused shape [26,27]. In [28], the shape of the focused spot (PSF deformation) for primary aberration (defocus, spherical, astigmatism) is calculated for various strength of the aberration. Finally, in [29], multiple focused beam intensity patterns, with a great variety of patterns for the disturbance, are used to train a neural network to retrieve aberrations contained inside the initial beam.
It is clear that, as of now, the understanding of the focusing-induced halo, as its dependency with the initial beam aberrations and pattern is not complete. In the presented work, we aim at tackling current issues related to the halo, that may also be linked with laser instabilities. One main objective is to get a better knowledge about the halo source, thus drawing lines to potentially handle it in subsequent works. Consequently, these better insights on the halo should improve the global understanding of LWFA, but also help to perform more accurate PIC (Particle In Cell) simulations of such accelerators. Our investigations will be broad as the whole phase map of the beam will be considered, and each aberration may be treated independently. Other high-power laser-related applications, for which the focusing pattern matter, may also be impacted by the presented results.
In the first part of this article, a brief description of the LWFA facility, from where the experimental beam (intensity distribution and phase) was taken and from where this halo issue arose, will be given. Then, the theory deployed to propagate and focus the beam (Fresnel diffraction) will be introduced in a second part, and Zernike formalism, used to describe aberrations, will be presented. Finally, our propagation algorithm and its associated simulations parameters will be detailed. Then, obtained results will be presented. The influence of the beam model (Gaussian, hyper-Gaussian (also called super-Gaussian) or experimental one), but also of aberration composition will be studied. Finally, we will discuss about limitations that exist in current optical beam quality criteria, and we will draw the consequences of our predicted results about the halo concerning LWFA applications.
2. Experimental facility, materials, and motivation of this work
For some simulations presented in this paper, a real laser beam (intensity and phase) is used. Data were obtained from the LAPLACIAN (Laser Acceleration Platform as a Coordinated Innovative Anchor) experimental facility located at RIKEN SPRING-8 Center. It consists in a Ti:Sapphire high-power CPA laser delivering 1 J of energy through a compressed pulse of about 25 fs (FWHM), centered around 800 nm, at a repetition rate of 1 Hz. For LWFA applications, an F/20 off-axis parabola (f = 1600 mm) is used to focus the laser inside a gas jet, where electrons are produced and injected inside the wakefield. Then, they get accelerated by a strong electric field that can achieve hundreds of GV/m.
A camera made by Basler (A2A5320, CMOS, 2.7 $\mu$m pixel size) was used to measure the laser beam profile in the near field, before the focusing optics. Furthermore, aberrations contained inside the beam were measured using an SID4 wavefront sensor from Phasics (182x136 sampling points, with an aperture size of around 5x3.6 mm$^2$). A high-quality beam splitter was used to send around 1% of the beam (reflection part) towards the SID4 sensor. A ND6 filter was used to attenuate the beam, and a beam reducer was also set to fit the beam size with the sensor size.
A second camera (Basler A2A3840-45umBAS, 2 $\mu$m pixel size) was used to measure the laser beam profile in the far field as shown in Fig. 1. For numerical calculations, a homemade Matlab code based on a discrete Fourier transform algorithm (FFT) was developed, as presented in section 3.3.
Fig. 1. Distribution of intensity (far field, on target, with a Strehl Ratio of around 0.83). The halo structure around the main focused Gaussian spot is clearly identifiable.
The main motivation of this work was taken from the necessity of achieving a high stability during electron beam generation. One obvious source of instability lies with the laser pulse used to drive the generation and acceleration of electrons. As shown in Fig. 1, a complex structure is surrounding the main Gaussian spot (obtained alone under a perfect case) and can reach non negligible intensities. This disturbance, called a halo, can fluctuate shot-by-shot, and interact non-linearly with the gas target, thus resulting in instabilities. Hence, it is crucial to understand the origin and the dynamics of this halo, nevertheless without any consideration about the absolute perfectness of the resulting focal spot that is another kettle of fish.
3. Fresnel diffraction method and simulation parameters
In the presented work, a scalar model of light is used and polarisation-dependent phenomena are not considered. The propagation of the laser beam is addressed using the diffraction theory developed by Fresnel.
Moreover, the work conducted within this paper investigates cases where the initial beam diameter $\Phi$ is around a twentieth of the focusing distance $f$ (1.6 m, at $\lambda$ = 800 nm), up to which the propagated beam is observed. Thus, the Fresnel number given by:
verifies the condition $F \gg 1$, allowing us to use the Fresnel diffraction theory in the near field [30]. Furthermore, as the angle between the initial beam and the propagated beam is small, the paraxial approximation can also be applied to simplify the equations.3.1 General scheme followed—Fresnel diffraction integral
We assume an initial 2D beam of amplitude $A (x,y)$, propagating in the $z$ direction, with an aberration induced phase $W (x,y)$ given by:
where $\varphi$ is the phase contained inside the beam and described using Zernike polynomials, in section 3.2 .Then, the beam goes through a focusing system of focal length $f$, which induces a phase:
with $k = \frac {2\pi }{\lambda }$ being the wavenumber. The source beam $E (x,y,0) = [A.W.F] (x,y,0)$ just after the focusing system is then propagated using Fresnel diffraction over a given distance $z$, close to the focal length, as illustrated in Fig. 2.Under the paraxial approximation ($\{(x-u)^2,(y-v)^2\} \ll z^2$, $(u,v)$ being the spatial coordinates after propagation), which allows a second-order expansion, the propagated field $E_2 (u,v,z)$ at a distance z from the source is given by:
Fig. 2. Considered geometry and method for the focusing of an aberrated beam using Fresnel diffraction.
3.2 Phase description using Zernike polynomials
The phase map applied over the entire beam size $\Phi$ is described using Zernike polynomials formalism, up to the 4$^{th}$ order [30–32]. This set of orthogonal polynomials is widely used in optics as they make it easier to describe a given phase as a linear combination of polynomials, each of them representing a particular type of aberration. In our phase description (using the first 14 terms of Zernike), considered aberrations are: tilt, coma, first and second order astigmatism, defocus, spherical aberration, trefoil and quadrafoil.
Under Zernike formalism, a polynomial $Z_n^m$ (where $n$ and $m$ stand respectively for the radial and azimuthal index, with $n\ge m$) is used to represent each aberration. Their general expression is given by:
where each polynomial is normalized so that the standard deviation is equal to 1 and ($\rho$, $\theta$) are polar coordinates. The radial part $R_n^m$ is defined as:Then, the phase distribution $\varphi (\rho,\theta )$ is expressed as:
where $C_s$ is a coefficient that represents the RMS amplitude of the aberration of index $s$ over the whole the beam, expressed in $\lambda$ unit (as used in Fig. 3). Thus, the total RMS value for the phase $\varphi$ is $\sigma = \sqrt {\sum _{s=1}^{14} C_s^2 }$.Altogether, the Strehl ratio, which is given by [33]:
is often used to indicate whether the beam is close to a diffraction-limited one ($S_R = 1$) or of having a very poor quality ($S_R \approx 0$). Furthermore, the Maréchal condition ($S_R = 0.8$, which corresponds to a case where $\sigma = \frac {\lambda }{14}$ [34]) is widely used as a limit from where the beam may be assumed close enough from diffraction-limited.Fig. 3. Experimental distribution of aberration measured with a SID4 wavefront sensor, up to the 4$^{th}$ order of Zernike polynomials. The strength coefficient $C_s$ of each aberration is expressed in $\lambda$ unit. The corresponding Strehl ratio is 0.83.
Experimentally measured aberrations, and the associated phase map, are shown on Fig. 3. This set of aberrations is used as a starting point for our real beam profile. To observe the effect of either improvement or degradation of real aberrations, this set is modified both globally (all aberrations are changed) or independently (one aberration at a time in the set is modified). Modifications were ranging from 0.25 times to 12 times, in RMS value, to see how they can affect the focused beam.
3.3 Numerical computation using a Fourier transform
As Eq. (4) has no analytical solution for real beams, one may solve this problem through numerical calculations. Practically, this equation can be written as a 2D-Fourier transform of $E (x,y,0) \, \mathrm e^{i\frac {k}{2z}(x^2+v^2)}$ taken at the spatial frequencies of $f_x = \frac {u}{\lambda z}$ and $f_y = \frac {v}{\lambda z}$.
As said in the introduction, there are various possible methods to solve this problem, but the one based on an FFT algorithm has been shown to be faster for a high number of points [20].
The following steps are followed:
- • First, we have to choose the number of points n for sampling, and the windows size D (this gives an initial spatial sampling of $\delta x = \frac {D}{n}$). The corresponding sampled frequencies obtained after Fourier transform are $\delta f_x = \frac {1}{n \delta x}$, thus $\delta u = \frac {\lambda z}{D}$. To decide n and D, both the initial beam and the focused one should be correctly sampled. One may get an estimation of the focused beam waist size by considering a diffraction-limited laser: $w_{0,f} = \frac {2 \lambda f}{ \pi \Phi }$.
In our case, $n = 2^{14}$ points and $D = 8$ m were used, which corresponds to $\delta x =$ 0.5 mm and $\delta u =$ 0.16 $\mu$m compared to $\Phi \approx$ 80 mm and $w_{0,f} \approx$ 11 $\mu$m. The corresponding Rayleigh length is $z_R = \frac {\pi w_{0,f}^2 }{\lambda } \approx 0.5$ mm.
- • Then, in the case of an experimental beam (already sampled with the camera pixel size), an interpolation has to be performed to fit the beam into the windows of size D with n points. Furthermore, the beam must also be centered before applying the FFT.
In Fig. 4, the interpolated experimental beam is plotted with also the Gaussian beam and hyper-Gaussian beam (order 4) that were used in simulations. The waist size that best fits the experimental beam has been taken as 35 mm.
The mathematical description of the hyper-Gaussian beam of order $n_g$ is given by:
where $n_g\ge 1$ ($n_g = 1$ corresponds to the classic Gaussian function). - • The phase is calculated according to the 14 aberration coefficients, and integrated within the initial beam amplitude distribution.
- • The FFT algorithm is applied, and the propagated beam amplitude is obtained. To get the intensity, one may just take the squared modulus of the amplitude.
Fig. 4. Laser profiles used for simulation (left: 1D cut along x-axis; right: 2D intensity distribution). From top to bottom: Gaussian, hyper-Gaussian and experimental beam.
4. Results
4.1 Realistic experimental beam versus ideal Gaussian beams
In simulations involving high-power lasers, especially in LWFA with PIC simulations, implementing a real noisy beam is not very often done. Instead, a Gaussian beam approximation is usually taken, and sometimes a better estimation employs the hyper-Gaussian modeling.
In order to understand the differences during propagation between these 3 possible representations (real, Gaussian and hyper-Gaussian beams), we have run simulations while increasing the strength of aberrations (which corresponds to a decreasing in the Strehl ratio), and focused patterns were compared. The 14 considered aberrations (presented before in section 3.2) were used with the same weight, and the total RMS value for the phase $\sigma$ was ranging from $\lambda$/40 ($S_R = 0.98$) to $\lambda$/3 ($S_R = 0.01$).
First, we analyse how well the energy is focused inside the main spot, as function of the Strehl ratio: our criterion is to measure the radius ($R_{80\%}$, expressed in waist unit of a perfect beam ) up to which 80% of the energy is encircled, as shown in Fig. 5.
There is a great difference between using an ideal Gaussian beam (the radius at $S_R =$ 0.1, 0.5 and 0.9 is respectively $R_{80\%} =$ 2.97, 1.09 and 0.92 in waist unit), the hyper-Gaussian (respectively $R_{80\%} =$ 3.78, 1.68 and 1.17) or the real beam profile (respectively $R_{80\%} =$ 3.83, 1.64 and 1.13). This underlines a relative difference around 20% in average between the real beam and the Gaussian one, while it is never more than 5% between the real and the hyper-Gaussian case.
Fig. 5. Radius (expressed in waist unit) that contains 80% of the beam energy as a function of the Strehl ratio. Comparison between ideal Gaussian beams and the experimental beam.
The initial difference between the Gaussian and hyper-Gaussian case lies with the fact that, in a first approximation, the pattern at focus is related to the Fourrier transform of the beam pattern before focusing. In such situation, a Gaussian beam is then focused as a Gaussian beam, while an hyper-Gaussian beam will be transformed into a Gaussian beam surrounded by some natural ring. However, this being said, one can observe that when aberrations are introduced, more energy starts to spread out of the main spot for the hyper-Gaussian case compared to the Gaussian one (the slope is higher). In other words, apart the natural ring structure of the hyper-Gaussian beam, it is expected for the noisy halo around the main spot to be more pronounced in presence of aberrations.
Furthermore, we have plotted in Fig. 6 (resp. Fig. 7) the beam intensity distribution at the waist (resp. 2 mm before the waist) for two values of the Strehl ratio (0.81 and 0.54). The trend here for the real beam and the hyper-Gaussian one is consistent with previous observations conducted at the focal point, as compared to the Gaussian case, the intensity spreading at the focal spot is clearly higher, and the halo more developed. However divergent observations are obtained at 2 mm before the focus. Indeed, at this position, intensity shapes are very similar between the Gaussian beam and the hyper-Gaussian, while they are slightly different with the experimental beam. Additionally, the real beam is also less homogeneous, which shows that a realistic beam behaves differently from the ideal Gaussian and hyper-Gaussian case when propagated. Moreover, for these Strehl ratios, Gaussian and hyper-Gaussian beams remain quite homogeneous and not so distorted during their propagation.
Fig. 6. Focused intensity profiles for two Strehl ratios (0.81 (top) and 0.54 (bottom)), in log scale. From left to right: Gaussian case, hyper-Gaussian case and experimental one.
Fig. 7. Intensity profiles at 2 mm before the focal point and for two Strehl ratios (0.81 (top) and 0.54 (bottom)). From left to right: Gaussian case, hyper-Gaussian case and experimental one.
From these simulation results, two important conclusions can be drawn. First, in presence of aberration, a Gaussian beam will be more resilient during focusing, and the development of the halo around the main focused spot will be more contained. Secondly, the beam pattern before the waist is more stable and homogeneous in the case of an ideal Gaussian beam. Thus, this confirms the necessity of using a realistic beam for simulations and calculations as its behaviour is highly distinct from the ideal Gaussian beam, though hyper-Gaussian beams remains, at least, a good approximation at the focal point.
4.2 Real beam profile
In this part, the experimental beam pattern is used as a starting point. First, we analyse the variation of each aberration independently to see how the beam is affected. Then, we apply the phase measured experimentally and we either increase it (up to 12 times) or decrease it (down to 0.25 times). Furthermore, these modifications of the experimental phase are also performed independently for each aberration.
4.2.1 Independent effect of each aberration
Primary aberrations (see section 3.2) have been applied separately to the beam, and the radius at 80% of the encircled energy is calculated as a function of the Strehl ratio, as shown in Fig. 8. It can be seen that each aberration, while they have identical weight (thus the same Strehl ratio), does not spread the energy at the focus spot in the same way. For very good quality beams ($S_R > 0.9$), there is almost no difference between each of them, except for the tilt. However, for example at $S_R = 0.81$ (Maréchal criterion), the considered radius ranges from 1.22 for the quadrafoil to 1.49 for the coma, and 1.34 for the defocus. At $S_R = 0.6$, these values are respectively 2.92, 2.6 and 2.31. Moreover, aberrations that belong to the same family (for example, first order astigmatism and second order one) also exhibit different behaviour, the radius being the same at $S_R = 0.81$ but being notably different at $S_R = 0.6$ (2.23 versus 2.92). Altogether, this behaviour remarkably depends on the considered value of the Strehl ratio: the more detrimental aberration (regarding the energy dispersion in the halo) at $S_R = 0.81$ is not at all the same than at $S_R = 0.6$ (coma and spherical aberration in these cases).
Fig. 8. Radius (expressed in waist unit) that contains 80% of the beam energy as a function of the Strehl ratio. Comparison between each aberration that was isolated and applied to the real beam.
Furthermore, in Figs. 9 and 10 the beam pattern during its propagation (from -3 mm to +3 mm, relatively from the focal position) has been plotted. In Fig. 9, all aberrations have been set to the same weight ($\lambda$/30) while in Fig. 10 they are all set to 0 except the first order astigmatism, set to $\sqrt {14}\lambda$/30. Thus, both cases correspond to the same Strehl ratio ($S_R = 0.54$).
We can observe that both the focal point and the intermediate focused beam are completely different while they stand for the same Strehl ratio. These two examples illustrate that even though the beam is affected by a same amount of aberration, but with a different distribution, then its resulting halo and its pattern during focusing is totally different. For instance, at 2 mm before the focus position, we may see a single beam not homogeneous for the first case (Fig. 9), while we get two separate and homogeneous beams for the second case (Fig. 10). Moreover, the halo at the focus is also disparate, and more pronounced in the second case.
Fig. 9. Initial beam pattern and intensity distribution after focusing (f=1.6 m) from -3 mm to +3 mm after the focal position. All aberrations values ($C_s$) are set to $\lambda$/30 (then $\sigma = \sqrt {14}\lambda$/30 and $S_R = 0.54$).
Fig. 10. Initial beam pattern and intensity distribution after focusing (f=1.6 m) from -3 mm to +3 mm after the focal position. Only first order astigmatism is used ($\sigma _{AS1} = \sqrt {14}\lambda$/30, then $S_R = 0.54$).
And not least, another case is interesting to be presented and concerns the effect of defocus aberration. In Fig. 11, all the aberration values ($C_s$) are set to 0 except the defocus set to $\lambda$/10, which corresponds to a Strehl ratio of around 0.67. Even though defocus is an aberration than cat be easily corrected, however it is also the most unpredictable when working with high-power lasers as some temperature fluctuations in all the laser rods can induce a lot of changes in the defocus aberration, as discussed in [35]. This aberration is often ignored but as shown in Fig. 11, even if the pattern at the focal plane is not drastically disturbed (an homogeneous ring structure appears), we can clearly see that the distribution after and before is very different from what one could expect. Therefore, for LWFA, a complex propagation inside the gas may be expected, and justifies to not ignore this aberration.
Fig. 11. Initial beam pattern and intensity distribution after focusing (f=1.6 m) from -3 mm to +3 mm after the focal position. Only the defocus is applied ($\sigma _{d} = \lambda$/10, then $S_R = 0.67$).
4.2.2 Variation of the initial set of aberrations
The experimental phase (see section 3.2) taken from LWFA experiment is used as a typical reference for aberrated beam. To tailor this phase from a highly corrected to a poorly corrected one, a numerical coefficient has been applied to the phase coefficients $C_s$. The multiplicative factor chosen ranges from 0.25 times to 12 times. The resulting radius at 80% of the encircled energy, as a function of the Strehl ratio, is plotted in Fig. 12. As shown in Fig. 12, sometimes this typical experimental phase was modified globally (the coefficients for all aberrations are increased by the same amount (from 0.25 to 5 times), and the radius at 80% is represented by the red points on the green line), and sometimes only one kind of aberration was changed, while keeping the others constant. In this later case, only one coefficient was increased from 0.25 to 12 times, which corresponds to the blue points; this process has been applied to all of the same family of aberration (astigmatism first or second order, trefoil, quadrafoil,…). The purpose of such modifications is to observe if, for a constant Strehl Ratio, a small increase of all aberrations in one time is worse than a stronger increase of one single aberration.
Fig. 12. Radius (expressed in waist unit) that contains 80% of the beam energy as a function of the Strehl ratio after focusing of the experimental aberrated beam. The initial set of measured aberrations (Fig. 3) is increased by 0.25 to 12 times. Points with line stand for a uniform change in all aberrations while the individual points corresponds to a change of only one aberration type at a time.
For example, without any global multiplicative factor, the Strehl ratio equals 0.83 while it changes to 0.48 (resp. 0.92) if this factor becomes 2 (resp. 0.5). On the other hand, the Strehl ratio changes to 0.79 (resp. 0.54 ) if only the astimatism of first order (0 and 45 degres) is multiplied by 2 (resp. 12).
This is consistent with the former analysed result, where aberrations were isolated before being applied to the beam (see previous section). Clearly, we can see that a regular increase in all aberrations does not produce the same energy spread inside the focused beam than when only one aberration is changed, while the Strehl Ratio is fixed. Indeed, at $S_R = 0.55$ and 0.81, we have $R_{80\%} =2.35$ and 1.27 (resp. $R_{80\%} =3.18$ and 1.20) for the uniform modification (resp. the isolated modification). As a consequence, for the one aiming at correcting its beam aberration, it may be wiser to target in priority the more detrimental aberrations than trying to improve the global quality with all possible aberrations.
Moreover, this trend is highly non-linear and fluctuating, though above a given beam quality (which may be a slightly higher than Maréchal criterion, around $S_R = 0.9$) there is almost no more impact due to the considered aberration set.
5. Discussions—halo issue in LWFA
5.1 Limitation in using only the Strehl ratio
As it has been presented before (see Fig. 8, 9, 10 and 12), the beam halo may be very different depending on the used set of aberrations, even with the same Strehl ratio. Similarly, the beam shape (homogeneity, size, number of hot spots,…) before reaching the focal point will also be highly depending on the chosen aberrations for a same Strehl ratio. Thus, this underlines that the Strehl ratio is not necessarily the best criterion to adopt concerning the beam quality in LWFA, despite its current extensive use. Initially, the Strehl ratio was developed in optics for astronomy to help define when a system is able to resolve two separate points (e.g.: two stars observed through a telescope), and is still very suitable for this purpose. However, for applications involving high-power fs laser like LWFA, it seems like that this criterion is not distinctive enough and does not help to decide if a critical point has been reached or not. In other words, it does not give sufficient information about the energy concentration and the beam pattern anomalies during propagation. Indeed, LWFA applications require to concentrate the energy as best as possible and to avoid disturbance such as the halo that may change the shape and the propagation path of the wakefield, but this halo pattern is not correlated with the Strehl ratio.
Other criteria exist and have been compared [36] in case of aberrations, like measuring the overlapping integral or the well-known M$^2$ parameter that describes how much a laser beam is far from a diffraction-limited Gaussian beam (single transverse mode). However, there is still a need to establish a better criterion to address these specific applications. For example, one could try to track the position of the mass center, the average peak intensity and their numbers, or the smoothness of the focus pattern as function of the aberration coefficient strength and of the aberration type. But, such a tough work of developing new metrics suitable for the halo, which probably depend on the targeted application, goes beyond the purpose of our article
5.2 Chromatic aberration
Another related issue lies with chromatic aberration. Indeed, especially for fs lasers, the spectrum is quite wide (up to 100 nm FWHM). In such a case, during the focusing, essentially axial chromatic aberration will arise. That will mainly induce a dependency of the focal position with the wavelength, and it may add more complexity to the beam shape around the focal plane.
However, it is difficult to make a clear model concerning the wavefront modification induced by chromatic aberration (it is not described by Zernike polynomials), and it is also not measurable with standard wavefront sensors. As a consequence, such a work goes beyond the scope of this article and the calculation model implemented. But, intuitively, the effective Rayleigh length of the focused laser may become longer and the beam pattern may also be slightly changed when considering all these wavelengths. Despite that this effect was not considered in this paper (only one central wavelength $\lambda = 800$ nm was taken), it should be considered in future works for improvements, as it may impact the halo.
Nevertheless, few previous existing articles have been studying this issue [37,38]. The wavefront dependency as function of the wavelength was either modeled or simulated (with an optical design software), and then the focus pattern dependency with the wavelength was simulated. But the possible relation with the halo is still missing.
5.3 Halo issue and focusing quality in LWFA
As already described in the introduction, the halo around the focused laser spot is a thorny question, which has already been partially addressed although it is not yet totally clarified.
In this paper, we have shown that the halo is a consequence of an aberrated beam that can not be perfectly focused. However, we also demonstrated that the halo shape is highly depending on the beam aberration distribution.
Additionally, the beam shape before the focusing also strongly depends on these aberrations: there are a lot of possible configurations regarding the intensity homogeneity, as also the number of individual beams inside the whole pattern. As a consequence, and considering the high-intensity involved in LWFA applications, even before reaching the focus point the beam may trigger locally some nonlinear effect inside the gas jet and plasma, first and foremost the self-focusing. Thus, the expected focal spot (experimentally measured or even numerically calculated from the beam pattern and phase distribution), which uses propagation under vacuum, may drastically change in reality.
Indeed, the relativistic self-focusing which depends on the laser pulse power P, occurs when this power exceeds the critical power $P_{c}$ given by [39]:
where $n_e$ is the plasma electron density and the critical density is $n_{c} = \frac {m_e \varepsilon _0 \omega ^2}{e^2}$ with $\omega$ the laser pulse angular frequency, $m_e$ and $e$ the electron mass and charge and $\varepsilon _0$ the permittivity. A typical value for $P$ in LWFA is around $100$ TW while $P_c$ is around $10$ TW. In other words, if even 10 % of the average laser power of the central part is reached inside the halo, it will start to self focus and generate other wakefields thay may disrupt the main one, which will be a huge issue for LWFA.Therefore, the halo structure should be a high concern for LWFA as it can trigger non-linear effect by itself, independently of the main pulse.
In Fig. 13 we have summarized the pattern of the beam at 2 mm before the focus (typical starting position of a gas jet in LWFA) and in the focal plane, for all aberrations. In all case, the strength of the aberration was set to $C_s = \lambda$/11 ($S_R =$ 0.72). A small comment is provided for each situation.
As said before, we can observe that for some aberrations (ex: astigmatism, defocus or trefoil) the energy will be more distributed inside the halo than in the main pulse, which should result in the self focusing of this part of the beam instead of the central part. This can lead to very complicated wakefields and change the properties of the electron beam.
Fig. 13. Patterns obtained 2 mm before the waist and at the waist (log scale for the intensity) for each aberration with an amplitude set to $C_s = \lambda$/11. Comments are provided regarding the homogeneity and possible issues in LWFA.
Besides, no temporal considerations were involved in our analysis. However, it may affect the beam propagation in two ways:
- • Depending on the beam pattern, some non-linear effects such as self-phase modulation or induced spatial dispersion may arise. In that case, depending on the halo distribution and its consequences, the beam duration and spectrum may be spatially changed.
- • A priori, aberration and induced halos are not depending on the beam duration (e.g: optical elements will always add the same phase modification to the full temporal beam); however, it may be slightly different between the ns pre-pulse and the true fs pulse, during pulse creation inside the laser cavity. This question should be at least studied as, if the halo has a temporal dependency, then the plasma channel dynamics will also be affected. To the best of our knowledge, temporal resolved wavefront measurements for fs high-power laser have not yet been investigated.
Lastly, forthcoming works and especially PIC simulations for LWFA should use more realistic laser profiles, both intensity and phase distribution, instead of ideal Gaussian models. This may broaden the understanding on the induced wakefield and hence on electron generation. As our propagation results showed, the laser distribution is complex before reaching the focus and the interaction with a plasma or a gas may highly change the expected behaviour. Reciprocally, it may also be possible to find a specific laser profile, with a given aberrations distribution, that gives rise to a better propagation and focusing inside the gas/plasma. In other words, there might be a way to purposefully deteriorate the wavefront and the beam profile so that the resulting focused beam (and its associated wakefield) becomes better for the targeted application ; this question is at least worth of being investigated in future works.
6. Conclusion
The source of the disturbance around the main Gaussian spot when focusing a high-power laser, and called a halo, has been identified as induced by wavefront aberrations. This halo structure is one potential candidate for the instabilities of generated electron beams in LWFA, and was explored for this reason. The importance of both the aberrations and the intensity distribution contained inside the laser pulse, regarding the impact on this halo, has also been evidenced. Owing to Fresnel diffraction theory, numerical computations were performed to investigate how beam patterns will evolve during either propagation or focusing. Using Zernike polynomials decomposition, the wavefront has been modified as desired and a great influence on both the beam propagation and its focusing shape, containing the halo disturbance, was observed. Inhomogeneities contained inside the beam pattern and the energy concentration, compared to a diffraction-limited beam and measured as a percentage of the total energy within a given radius, were studied. This work highlights the importance of taking into account both the considered beam intensity distribution and the wavefront error. Indeed, simulations, often using a perfect-Gaussian beam without aberration, will not be able to reproduce experimental results neither the halo effect as long as a real aberrated beam is not implemented.
Altogether, future works should take care of this issue in order to improve the understanding of high-power laser applications. In particular, LWFA of electrons may be greatly influenced by this halo, and the way the beam may propagate and interact within the gas or plasma medium. Knowing the beam initial pattern and phase, one should now calculate properly its focused shape and phase before introducing this beam as a driver for LWFA PIC simulations. Finally, one might manipulate the beam properties to obtain an optimal phase and pattern for a given gas or plasma medium, and it should be investigated as interaction with the beam remains intricate.
Funding
JST-Mirai Program (jP-MjMI17A1).
Acknowledgements
The authors are grateful to Prof. Yuji Sano and Dr. Naveen Pathak for their encouragements and helpful discussions. We would also like to thanks Dr. Zhan Jin, Dr. Izuru Daito and Dr. Takafumi Kondo for their support with the experimental facility.
This work was funded by the JST-Mirai Program, grant No. jP-MjMI17A1 and was partially supported by the ImPACT R&D Program of Council for Science, Technology and Innovation (Cabinet Office, Government of Japan). This work is also partially supported by ‘Dynamic Alliance for Open Innovation Bridgin Human, Environment and Materials’ from the Ministry of Education, Culture, Sports, Science and Technology of JAPAN (MEXT).''
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are available from the corresponding author upon reasonable request.
References
1. T. Maiman, “Stimulated optical radiation in ruby,” Nature 187(4736), 493–494 (1960). [CrossRef]
2. D. Strickland and G. Mourou, “Compression of amplified chirped optical pulses,” Opt. Commun. 56(3), 219–221 (1985). [CrossRef]
3. T. Tajima and J. Dawson, “Laser electron accelerator,” Phys. Rev. Lett. 43(4), 267–270 (1979). [CrossRef]
4. W. Wang, K. Feng, L. Ke, et al., “Free-electron lasing at 27 nanometres based on a laser wakefield accelerator,” Nature 595(7868), 516–520 (2021). [CrossRef]
5. M. Labat, J. C. Cadabag, A. Ghaith, et al., “Seeded free-electron laser driven by a compact laser plasma accelerator,” Nat. Photonics 17, 150–156 (2022). [CrossRef]
6. J. Alda, J. Alonso, and E. Bernabeu, “Characterization of aberrated laser beams,” J. Opt. Soc. Am. A 14(10), 2737–2747 (1997). [CrossRef]
7. V. Dev, A. N. K. Reddy, A. V. Ustinov, S. N. Khonina, and V. Pal, “Autofocusing and self-healing properties of aberration laser beams in a turbulent media,” Phys. Rev. Appl. 16(1), 014061 (2021). [CrossRef]
8. A. Thomas, S. Mangles, C. Murphy, A. Dangor, P. Foster, J. Gallacher, D. Jaroszynski, C. Kamperidis, K. Krushelnick, K. Lancaster, P. Norreys, R. Viskup, and Z. Najmudin, “Ultrashort pulse filamentation and monoenergetic electron beam production in lwfas,” Plasma Phys. Controlled Fusion 51(2), 024010 (2009). [CrossRef]
9. N. Nakanii, T. Hosokai, K. Iwasa, N. Pathak, S. Masuda, A. Zhidkov, H. Nakahara, N. Takeguchi, Y. Mizuta, T. Otsuka, K. Sueda, H. Nakamura, and R. Kodama, “Effect of halo on the stability of electron bunches in laser wakefield acceleration,” EPL 113(3), 34002 (2016). [CrossRef]
10. N. Pathak, A. Zhidkov, S. Masuda, T. Hosokai, and R. Kodama, “Effect of halo on high power laser pulse wake in underdense plasma,” Phys. Plasmas 23(11), 113106 (2016). [CrossRef]
11. I. Zemzemi, F. Massimo, and A. Beck, “Azimuthal decomposition study of a realistic laser profile for efficient modeling of laser wakefield acceleration,” in Journal of Physics: Conference Series, vol. 1596 (IOP Publishing, 2020), p. 012054.
12. J. Ferri, X. Davoine, S. Fourmaux, J. Kieffer, S. Corde, K. Phuoc, and A. Lifschitz, “Effect of experimental laser imperfections on laser wakefield acceleration and betatron source,” Sci. Rep. 6(1), 1 (2016). [CrossRef]
13. R. W. Gerchberg and W. Saxton, “A practical algorithm for the determination of plane from image and diffraction pictures,” Optik 35, 237–246 (1972).
14. L. Dickson, C. Underwood, F. Filippi, R. Shalloo, J. B. Svensson, D. Guénot, K. Svendsen, I. Moulanier, S. D. Dufrénoy, C. Murphy, N. Lopes, P. Rajeev, Z. Najmudin, G. Cantono, A. Persson, O. Lundh, G. Maynard, M. Streeter, and B. Cross, “Mechanisms to control laser-plasma coupling in laser wakefield electron acceleration,” Phys. Rev. Accel. Beams 25(10), 101301 (2022). [CrossRef]
15. Z.-H. He, B. Hou, V. Lebailly, J. Nees, K. Krushelnick, and A. Thomas, “Coherent control of plasma dynamics,” Nat. Commun. 6(1), 7156 (2015). [CrossRef]
16. S. Mangles, G. Genoud, S. Kneip, M. Burza, K. Cassou, B. Cros, N. Dover, C. Kamperidis, Z. Najmudin, A. Persson, J. Schreiber, F. Wojda, and C.-G. Wahlstrom, “Controlling the spectrum of x-rays generated in a laser-plasma accelerator by tailoring the laser wavefront,” Appl. Phys. Lett. 95(18), 181106 (2009). [CrossRef]
17. P. Cummings and A. G. R. Thomas, “A computational investigation of the impact of aberrated Gaussian laser pulses on electron beam properties in laser-wakefield acceleration experiments,” Phys. Plasmas 18(5), 053110 (2011). [CrossRef]
18. W. Zhang, H. Zhang, and G. Jin, “Frequency sampling strategy for numerical diffraction calculations,” Opt. Express 28(26), 39916–39932 (2020). [CrossRef]
19. S. M. Azmayesh-Fard, “Gaussian beam propagation: comparison of the analytical closed-form Fresnel integral solution to the simulations of the Huygens, Fresnel, and Rayleigh–Sommerfeld I approximations,” J. Opt. Soc. Am. A 30(4), 640–644 (2013). [CrossRef]
20. S. Shakir, D. Friend, A. Pease, T. Brennan, and T. Dolash, “Efficient matrix approach to optical wave propagation and linear canonical transforms,” Opt. Express 23(20), 26853 (2015). [CrossRef]
21. D. Hofer and B. G. Zagar, “A numerical approximation of the Huygens-Fresnel integral–simulations of a rough wetting problem,” Measurement 46(8), 2828–2836 (2013). [CrossRef]
22. S. Khonina, A. Ustinov, and E. Pelevina, “Analysis of wave aberration influence on reducing focal spot size in a high-aperture focusing system,” J. Opt. 13(9), 095702 (2011). [CrossRef]
23. Q. Li, M. Chambonneau, M. Blothe, H. Gross, and S. Nolte, “Flexible, fast, and benchmarked vectorial model for focused laser beams,” Appl. Opt. 60(13), 3954–3963 (2021). [CrossRef]
24. I. Zemzemi, “High-performance computing and numerical simulation for laser wakefield acceleration with realistic laser profiles,” Ph.D. thesis, Ecole Polytechnique (2020).
25. V. Mahajan, “Uniform versus Gaussian beams: a comparison of the effects of diffraction, obscuration and aberrations,” J. Opt. Soc. Am. A 3(4), 470 (1986). [CrossRef]
26. Z. Cao and K. Wang, “Effects of astigmatism and coma on rotating point spread function,” Appl. Opt. 53(31), 7325–7330 (2014). [CrossRef]
27. Y. Li, H. Zhou, X. Liu, Y. Li, and L. Wang, “Effects of aberrations on effective point spread function in STED microscopy,” Appl. Opt. 57(15), 4164–4170 (2018). [CrossRef]
28. A. Dixit, S. K. Mishra, and A. K. Gupta, “Sensitivity of singular beams in the presence of Zernike aberrations,” Opt. Lasers Eng. 71, 74–84 (2015). [CrossRef]
29. Z. Zhou, J. Zhang, Q. Fu, and Y. Nie, “Phase-diversity wavefront sensing enhanced by a Fourier-based neural network,” Opt. Express 30(19), 34396 (2022). [CrossRef]
30. M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, (Cambridge University Press, 2005).
31. F. Zernike, “Beugungstheorie des schneidenver-fahrens und seiner verbesserten form, der phasenkontrastmethode,” Physica 1(7-12), 689–704 (1934). [CrossRef]
32. J. Wyant and K. Creath, Basic Wavefront Aberration Theory for Optical Metrology, (Academic, 1992).
33. V. Mahajan, “Strehl ratio for primary aberrations in terms of their aberration variance,” J. Opt. Soc. Am. 73(6), 860 (1983). [CrossRef]
34. C. Sheppard, “Maréchal condition and the effect of aberrations on strehl intensity,” Opt. Lett. 39(8), 2354 (2014). [CrossRef]
35. X. Zhang, Z. Deng, Z. Hu, and X. Zhang, “Analysis of defocus aberration characteristics on a typical passively confocal unstable resonator,” Optik 202, 163625 (2020). [CrossRef]
36. E. Perevezentsev, A. Poteomkin, and E. Khazanov, “Comparison of phase-aberrated laser beam quality criteria,” Appl. Opt. 46(5), 774–784 (2007). [CrossRef]
37. H.-M. Heuck, P. Neumayer, T. Kühl, and U. Wittrock, “Chromatic aberration in petawatt-class lasers,” Appl. Phys. B 84(3), 421–428 (2006). [CrossRef]
38. U. Fuchs, U. D. Zeitner, and A. Tünnermann, “Ultra-short pulse propagation in complex optical systems,” Opt. Express 13(10), 3852–3861 (2005). [CrossRef]
39. G.-Z. Sun, E. Ott, Y. Lee, and P. Guzdar, “Self-focusing of short intense pulses in plasmas,” Phys. Fluids 30(2), 526–532 (1987). [CrossRef]
