## Abstract

The generation of cavitation bubbles in water by focused laser pulses has been thoroughly studied, particularly due to the wide range of applications in biomedical fields, in particular for ophthalmic laser surgery. The presence of optical aberrations in any optical system deteriorates the precision of light focusing and may reduce the laser interaction efficiency in transparent media. In this paper, we analyze the influence of several controlled optical geometrical aberrations in the formation and dynamics of cavitation bubbles in water caused by laser-induced optical breakdown using femtosecond laser pulses. The maximum cavitation bubble diameter reduces with increasing aberrations, showing a drop of the energy coupling efficiency, ultimately leading to the absence of cavitation bubble. The study reveals that the secondary astigmatism and spherical aberrations degrade more rapidly the beam concentration than the other aberrations studied here (namely astigmatism first and second, coma, spherical and trefoil). The cavitation energy threshold becomes equally increased. This observation relates very well to numerical simulations evaluating the laser fluence at and around the focusing plane in the presence of aberrations. These results open the door to more controlled ultrafast laser assisted surgeries and processing in transparent media.

© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

When an ultrashort laser pulse is focused in water, non linear absorption mechanisms such as multiphotonic and avalanche ionizations may take place leading to the formation of a local transient electronic plasma, generating a cavitation bubble [1]. In the last years, laser induced-optical breakdown (LIOB) using femtosecond laser has gained interest due to many applications particularly in the biomedical fields [2]. LIOB depends on different parameters of the laser radiation, such as the pulse duration, polarization [3], and energy [4,5]. Several authors have determined the LIOB thresholds for different media [6,7] with a particular attention payed to the effects of optical aberrations [1,8]. In particular, the plasma generation, the laser energy conversion into the bubble and the LIOB threshold have been studied, showing the beneficial effect of a stronger focusing angle to reach the optical breakdown in liquid. [9–11].

The performance of an optical system can be easily distorted by optical aberrations. They can cause undesirable effects such as a blur on an imaging system, spreading of the light [12] or unwanted effects on human vision [13]. Some of these aberrations are part of the system and cannot be easily corrected or compensated; for instance, aberrations found in human eyes [14–16] and animal eyes [17]. If remarkable progress has been achieved in the field of adaptive optics to counteract these aberrations, the implementation may significantly add to the optical system cost, complexity and response time. Moreover, unwanted effects caused by spherical aberrations can be found in ultrashort laser systems [18,19] when used to induce optical breakdown by nonlinear effects inside transparent materials [20]. It is also known that the interface between air and the transparent media usually associated with focusing inside a transparent media leads to the onset of depth-dependant spherical aberrations [21] that can be corrected using a wavefront correction [22]. Considering focusing a laser inside a transparent medium, aberrations cause a spatial spreading of the focal volume, leading to a drop of the maximal intensity level, a widening of the photo-irradiated zone thus decreasing the interaction efficacy when probing and/or modifying the medium in 3D. Having in mind the large amount of applications where ultrafast laser are focused inside transparent media, such as eye surgery, waveguide photowriting, 2-photon polymerization, it is of outmost importance to precisely evaluate the degradation effects of various aberrations, identifying the most problematic ones in order to eventually better address and counteract their detrimental effects.

Thus, we propose here a detailed study of the effects and dependence of classical geometrical aberrations, namely astigmatism, coma, spherical, trefoil and secondary astigmatism. In this paper, the formation and dynamics of cavitation bubbles in water as a result of LIOB by a purposely aberrated ultrafast laser beam are presented. The wavefront of a femtosecond laser beam was modulated using a phase-only spatial light modulator in order to generate specific aberrations in a controlled manner. The analysis of the bubble dynamics was done in-situ using a time-resolved photography optical technique. We observe that the maximum cavitation bubble diameter reduces with increasing aberrations eventually leading to the absence of cavitation bubble. We have also conducted numerical simulations evaluating the laser fluence at the focusing plane in the presence of aberrations. These calculations permit to predict the bubble generation that occurs when the laser fluence surpasses the fluence threshold. Thus, the drop of laser fluence due to aberration is thought to be the main cause of the reduction of the bubble generation efficacy. It is found that secondary astigmatism and spherical aberrations have a stronger detrimental effect on the laser focusing than the other aberrations considered here, notably due to their larger longitudinal spreading effect. Also, we verified that the ratio of the bubble maximum radius with the collapse time remains constant in all the conditions of this study. This confirms that the onset of aberrations has a predominant effect of reducing the fraction of laser energy well-coupled to the medium with no other altering effects of the bubble cavitation dynamics.

This paper is organized as follows : after a description of the experimental details, we evaluate the effect of the aberration on the laser fluence at the focusing plane numerically and experimentally. Both studies reveal a drop of the laser mean fluence in the focusing plane for all aberrations with a stronger effect observed for secondary astigmatism and spherical aberrations. Then, the time-resolved results are presented showing the transient cavitation bubble dynamics with a sub-microsecond timescale. We observe the decrease of the maximum bubble diameter with increasing aberration and show that it can be clearly related to the aberration-induced drop of laser fluence. There also, the secondary astigmatism and spherical aberrations appear to have the strongest influence on the cavitation success. Finally, the ratio of the maximum bubble radius with the bubble collapse time is verified for all the conditions presented here.

## 2. Experimental details

The experimental apparatus is depicted on Fig. 1. This set-up permits to 1) add controlled aberrations to an ultrafast laser beam and analyse their effects on the focused laser spot and 2) image the cavitation bubble formation and dynamics in water in a time resolved manner as explained hereafter.

#### 2.1 Aberrations and beam analysis

Single pulses from a femtosecond laser (Satsuma - Amplitude) at $\lambda =1030$ nm with a pulse duration full width half maximum of $\tau _s=280$ fs and a narrow laser spectral bandwidth (below 2 nm), are directed onto a phase-only spatial light modulator LCoS (Hammamatsu). The laser beam is polarized using a half wave plate ($HWP$) to generate the polarization required on the LCoS in order to achieve phase-only modulation. The laser energy is controlled using another HWP with a polarizing cube (not shown). The input beam was expanded using a telescope, formed with lenses $L_1$ and $L_2$, to fit the LCoS dimensions. By displaying pre-calculated phase masks onto the LCoS, controlled aberrations are imprinted on the laser beam as pure wavefront modulation. The phase mask calculation is explained in Section 3. The controlled aberrated beam was focused inside a water container, consisting in a cubic polystyrene Petri dish filled with water, by an aspheric lens $L_4$ ($f_4= 18.4$ mm). The non-aberrated beam waist diameter measured 7.8 $\mu$m in the focusing plane, very close to the theoretical one with a M$^2$ of 1.2. The baseline aberrations of the set-up were found to be negligible as the LCoS was operated with its build-in wavefront correction and the ultrafast laser beam was guided by a hollow-core fiber (not shown) with a very clean output before reaching the setup.

The laser energy used during all the experiment was 2.37 $\mu$J, the energy was measured behind $L_4$. Using this laser energy and the beam parameters, the mean fluence used for cavitation bubble formation was 4.92 J/cm$^2$ at the focusing point $L_4$ in the water container.

The experimental analysis of the focused laser beam is performed by imaging the laser intensity on the CCD sensor (Thorlabs DCC3240M) through the microscope objective ($MO$) and $L_3$, replacing the bandpass filter $BPF$ by an optical density filter to avoid laser-induced damages on the CCD. The analysis of the beam was conducted at low energy with the filled water-container to obtain beam measurements as close as possible to the cavitation bubble experiment. The imaging plane is that of highest intensity with slight variations of the imaging plane position for all the conditions of this study.

#### 2.2 Time resolved photography of cavitation bubbles under aberration

Time resolved photography has been extensively used to analyze the evolution of the dynamics of cavitation bubbles [8,23,24]. The experimental apparatus used here completes the state of the art by adding the LCoS modulator on the pump beam, i.e. the ultrafast laser beam before focusing into water, thus enabling controlled wavefront modulation as explained above.

The image path to observe the cavitation bubble begins at the flash lamp (High-Speed Photo Systeme, Wedel, Germany) that is collimated by $L_3$ and focused in the water-filled cubic Petri box of 20 mm size using $L_4$. The light from the laser and the flash lamp are both collected by an infinity corrected microscope objective $MO$ ($20x$, $NA=0.2$), and projected to the CCD triggered camera with the lens $L_5$ ($f_5$= 200 mm). A bandpass filter $BPF$ was used to protect the CCD from the laser radiation and let only the radiation from the flash lamp through the imaging path. Synchronization between flash lamp pulses and laser pulses was done using an FPGA (Xilinx Kinetex-7), enabling time-resolved pictures from a single laser pulse. The temporal resolution given by the flash lamp is around 100 ns with a jitter below $50\,ns$. The imaging setup has a lateral spatial resolution of 1.3 $\mu$m. In order to measure the bubble diameter on each photograph, the Otsu method was used to threshold the image [25] and automatically find the bubble borders using the Matlab programming software.

## 3. Effect of aberrations on the laser fluence

#### 3.1 Aberrations calculation

The complex amplitude of the laser beam impinging on the LCoS can be expressed as follows:

where $r$ is the radial distance to the optical axis normalized on the unit pupil $0 \leq r \leq 1$ , $\theta$ is the azimuthal angle, $U_i$ is the initial laser complex amplitude, and $W_n^m (r,\theta )$ the controlled aberrated wavefront modulation given by the LCoS, assuming that the optical set up aberration and intrinsic laser beam aberrations and are comparatively negligible. This wavefront was modeled using positive Zernike’s polynomials.Here $n$ and $m$ are both non-negative integers $(n\geq m)$, called radial degree and azimuthal degree respectively, and $c_n^m$ is the amplitude of the aberration coefficient, in rad. Even though the aberration catalog that can be reproduced with this representation is almost infinite, the study of the effect of aberrations on the laser induced optical breakdown, has been conducted for the most representative ones often met in the human eye such as spherical, coma, astigmatism and trefoil aberrations [26] and are listed in Table 1. A comprehensive study of their effect on the laser focusing in water can thus be easily related to eye surgery situations when "non-perfect" cornea and/or crystalline lens are on the laser path before the focusing plane. Lower-order aberrations such as phase piston, tilt and defocus are not studied here as their effects are at most a spatial displacement of the focal spot (along the optical axis for defocus, perpendicularly to the optical axis for tilt while the piston aberration only adds a constant to the absolute phase).

#### 3.2 Laser fluence numerical and experimental evaluation

The initial laser complex amplitude $U_i(r,\theta )$ was chosen to match a perfect Gaussian beam distribution with a flat wavefront, which is close to the experimental beam delivered by the femtosecond laser.

where $w$ corresponds to the radius of the Gaussian beam at $1/e^2$.Within the paraxial approximation and neglecting spectral effects, the electric field on the focal plane of a focusing lens $U_f (\rho,\vartheta )$, can be expressed using the Fourier transform:

Here, $(\rho,\vartheta )$ are the radial coordinates on the frequency plane, $k$ is the wave number, $\lambda$ the wavelength of the electric field, $f$ is the focal length of the lens, and $\mathcal {F}\{.\}$ corresponds to the Fourier transform. We used a classic Fresnel propagation code under the Matlab software to conduct the calculations [27].

The numerical mean fluence $NMF_n^m$ was computed at the focusing plane of the lens for the different aberrations following:

where $E_{pulse}$ is the pulse energy in J, $\tau _{Hz}$ is the laser repetition rate in Hz, $\left \langle \cdot \right \rangle _S$ corresponds to the arithmetic mean over the surface equal to the Gaussian beam (radius at $1/e^2$) and $I_f^{m,n} (\rho,\vartheta ) \propto U_f^{m,n} (\rho,\vartheta ) U_f^{*m,n} (\rho, \vartheta )$ represents the normalized laser intensity ($*$ is the complex conjugate of the electric field).The experimental mean fluence $EMF_n^m$ was calculated likewise i.e by using Eq. (5) as well, in which $I_f^{m,n}$ was replaced by the experimentally measured intensity profile $I_{fexp}^{m,n}$ from the beam analysis.

#### 3.3 Numerical and experimental results

Figure 2 i)-vi) presents the wavefront modulation applied to the LCoS associated with the different aberrations studied here. The corresponding experimental laser intensity distributions were imaged at the focusing plane in water using the beam analyzer for the different aberrated wavefronts as depicted in Fig. 2 a)-f). Optical aberrations were programmed into phase holograms using Eq. (2), for different coefficients $c_n^m$, and displayed on the LCoS. The ’aberration-free’ (flat wavefront on the LCoS) intensity distribution of $I_{fexp} (\rho,\vartheta )$ where $(W_n^m (r,\theta )=0)$ in the focusing plane of the lens, is shown in Fig. 2(a).

It can be clearly verified here that the onset of optical aberrations tends to spread the laser intensity. However, it is difficult to evaluate which aberration has the strongest spreading effect from these experimental pictures, even though the spherical aberration seems take over in terms of energy distribution widening (see Fig. 2(b)). Thus, in order to classify the aberrations with respect to their intensity spreading, the numerical and experimental mean fluence, respectively $NMF_n^m$ and $EMF_n^m$ were calculated following Eq. (5) as explained earlier.

The variations of the mean fluence $NMF_n^m$ with the aberration coefficients for different aberrations are shown in Fig. 3(a). where the numerical results of Eq. (5) are plotted. The aberration coefficient is directly related to the peak-valley amplitude of the wavefront modulation, thus it permits to compare the effect of the different aberrations while keeping an equal amplitude of phase modulation for all of them. As anticipated from the previous figure, a general trend can be observed that the onset of aberrations reduces $NMF_n^m$ for all cases. Moreover, the calculation of the focused mean fluence allows for a more precise analysis. The stronger effects with respect to the aberration coefficient are found for spherical and secondary astigmatism (star and dash curves, respectively) for which the mean fluence drops very quickly as the aberration coefficient increases. As an indication, the observed laser-induced optical breakdown (OB) threshold is also shown on the graph. This threshold fluence of 3.28 J/cm$^2$ was experimentally measured as the minimum fluence to induce a detectable cavitation bubble on the set-up. We mention here that the limited spatial and temporal resolution of the bubble characterization set-up could not allow to detect cavitation with either submicrometric dimensions [28] and/or below 10 ns duration [29]. Thus, this value may overestimate the actual minimum fluence to trigger a cavitation. This empirical value permits however to better visualize the detrimental spreading effect of optical aberration. It can be seen that for relatively small aberration coefficients of 0.5-1.5 rad, the spherical aberration and secondary astigmatism deteriorate the beam focusing to the point that the detectable OB is not reached. Figure 3(b)) permits to compare these numerical results with experimental measurements. The behavior of the experimentally measured mean fluence $EMF_n^m$ with respect to the variation of aberrations seen in Fig. 3(b) is remarkably similar to the computed results. The stronger effects of spherical aberration followed by secondary astigmatism is also observed on the experimental beam profile. Also, the ranking and ratio of the various aberrations with respect to their ’efficiency’ in spreading the laser spot is very similar on both numerical and experimental graphs.

Thus, it is clear both numerically and experimentally that the onset of optical aberrations spreads the laser beam and consequently reduces the laser fluence in the focusing plane. Spherical aberration and astigmatism clearly have the strongest effect on the focused laser spot spreading compared to the other aberrations at identical phase modulation range. One could argue that a simple increase of laser energy can compensate for this lack of beam concentration. However, three points are to be carefully considered. First, as the laser intensity distribution is altered by the optical aberrations, an increase of energy on such a spread laser spot reduces the precision of energy deposition, possibly leading to unwanted cavitation sites before and after the main focusing spot, as observed in [27]. It is obvious that this can’t be tolerated for any in-volume laser processing application, especially for eye surgery or photopolymerization. Second, the increase of energy when using ultrafast laser pulses enhances the occurrence of nonlinear propagation effects before reaching the main focusing spot [30]. Self-focusing may alter the position of the main focusing point and can also lead to non-controlled filamentation when local plasma are generated along the laser propagation axis. Self phase modulation also alters the pulse duration and temporal shape, adding to the risk of additional non desired cavitation sites. Third, safety issues may limit the amount laser energy available, particularly for eye surgery [31]. Thus, a thorough evaluation of the aberrations is indispensable for a well-controlled and localized laser interaction.

## 4. Time-resolved cavitation bubble results

#### 4.1 Reduction of $D_{max}$ with aberrations

The in-situ analysis of the cavitation bubbles evolution was done using the optical setup shown in Fig. 1. By adjusting the delay between the flash lamp and the femtosecond laser pulse, the cavitation dynamics were recorded with a sub-microsecond resolution. Several sequences of images were obtained depicting the formation and collapse of the cavitation bubble for various types of aberrations with different coefficients in order to evaluate the effect of increasing aberration on the cavitation phenomenon and compare with the fluence degradation observed above. An example of a typical image sequence of cavitation bubble dynamics is shown in Fig. 4(a). It corresponds to the ’aberration-free’ irradiation, which is used as a baseline to make the comparison with the controlled aberration cases. Recording such image sequences permits to extract relevant data such as the bubble collapse time as well as its time-evolving diameter $D$.

Figure 4(b) presents the evolution of the cavitation bubble diameter, calculated for each delay of 30 image sequences similar to the one shown in Fig. 4(a). The time-resolved bubble diameter was measured on each photograph over the 30 different capture sequences. The graph shows the corresponding time-dependent averaged diameter values as well as the standard deviations as vertical error bars. This permits to reduce the effect of the inevitable deviation of bubble size and lifetime from each optical breakdown. We note here that the imaging axis is parallel to the laser propagation meaning that the laser optical axis is perpendicular to the pictures. For a mean fluence of 4.9 J/cm$^2$, the cavitation bubble reaches its maximum diameter $D_{max}$ at 2 $\mu$s with a maximum diameter of 30 $\mu$m. The collapse time of the bubble is also around 2 $\mu$s. These results are in good agreement to what is usually reported in the literature [32].

Figure 5(a), shows how $D_{max}$ drops when the aberration coefficient is increased for each of the aberrations studied here. Each point of the graph is an average among 6 diameter measurements from time-resolved photography, with the vertical error bars showing the standard deviation. We observe a decrease of $D_{max}$ with increasing aberration, with a strong similarity to the decrease of the laser mean fluence as observed numerically and experimentally (see Fig. 3). A closer look to the likelihood between $D_{max}$ and the experimental mean fluence $EMF_n^m$ can be obtained by comparing Fig. 5(a)) with Fig. 5(b)), where the $EMF_n^m$ versus the aberration coefficients is depicted (values under the OB detection threshold are excluded). It can be seen that $D_{max}$ and $EMF_n^m$ know very similar variations with respect to the aberration coefficients. Spherical aberration and secondary astigmatism are also appearing to have the strongest influence on reducing the maximum bubble diameter. Similar trends between the mean laser fluence and $D_{max}$ are also observed for the other aberrations.

When operating above the LIOB fluence threshold, the laser energy coupled to the medium can be considered proportional to $EMF_n^m$ in a first approximation [28]. The similarities between $EMF_n^m$ and $D_{max}$ can thus be explained. It is therefore reasonable to consider $D_{max}$ as a reliable indicator of the success of laser energy coupling. This can be verified on the $D_{mas}$ versus $EMF_n^m$ plot showed in 5(c)) where a linear relationship is observable especially at high enough fluence.

As our set-up does not allow for side imaging of the cavitation, we numerically investigated the intensity spreading along the laser propagation ($z$ axis) due to aberrations. We have calculated the laser intensity distribution for several planes before and after the focal plane for each of the aberrations discussed here, using the Fourier propagation formalism described above. The results are presented in Fig. 6. The laser intensity distribution is plotted along the laser propagation $z$ axis, revealing the aberration spreading effect in a plane parallel to $z$. A quick look permits to realize that secondary astigmatism and spherical aberration have the strongest effect compared to coma, astigmatism and trefoil. Not only their maximum laser intensity is significantly more dropped, to 35-45% of the non aberrated case (not shown), but also the intensity spatial extent along $z$ surpasses that of the other aberrated beams. It is interesting to consider these results with the study of Vogel et al. [9] on the detrimental effect of spherical aberrations on the ns-laser plasma formation in water. Not only an increase of the LIOB in presence of spherical aberrations is reported with an axial spreading of the plasma, but the authors also anticipate that other aberrations have a similarly deleterious effect. The results presented here add well to this knowledge by considering additional aberrations and ranking them with respect to their degradation effect on fs-laser focusing and cavitation success, the latter being is directly evaluated.

It has to be mentioned that for a finer understanding of the phenomena in play, nonlinear propagation effects should be considered because of their influence on the energy coupling in transparent media, especially in the presence of aberrations [33]. However, a detailed study of this phenomena is beyond the scope of this paper. The good agreement between $EMF_n^m$ and $D_{max}$ evolution with respect to aberration coefficients suffices to support our main observation that spherical aberration and secondary astigmatism are of outmost important to evaluate and counteract when ultrafast pulses are used to generate bulk cavitation in water and more generally in transparent media.

#### 4.2 Ratio $D_{max}$ collapse time $T_c$

In order to investigate whereas the aberrated intensity distribution also affects the bubble dynamics once the cavitation occurs, the collapse time $T_c$ was measured for all the conditions of this study. Figure 7 depicts the time evolution of the bubble diameter $D$ for the 5 aberrations types discussed here with various coefficients. In this graph series, aberration was induced with different values for its respective aberration coefficient $c_n^m$. For each coefficient $c_n^m$ time-resolved image sequences were captured and analyzed, in order to study the temporal dynamics. As a global trend, the reduction of $D$ associated with increasing aberration coefficient appears clearly for each type of aberrations with again a stronger effect for spherical aberration and secondary astigmatism. It can also be seen that the collapse time $T_c$ is not constant and depends on the aberration coefficient. More precisely, $T_c$ is also reduced when the aberration coefficient increases and this tendency is similar for all the aberrations considered here. Thus, $D$ and $T_c$ both follow the same trend i.e they decrease when aberrations are augmented. It has been shown for a spherical bubble that the maximum diameter of the cavitation bubble $D_{max}$ in water is proportional to the collapse time of the bubble $T_c$ [32,33] following the Rayleigh’s equation (for a spherical bubble):

where $\rho _0=998$ kg/m$^3$ corresponds to the water density, $p_\infty =101.5$ kPa is the pressure in the water at the bubble wall, and $p_v=2.6$ kPa is the vapor pressure [34]. With these values, the ratio $\frac {D_{max}}{T_c}$ is theoretically around 21 m/s. We calculated this ratio for the data represented in Fig. 7 where a cavitation bubble was generated. It was found that this ratio is around 15, which is slightly smaller than what could be expected from Eq.4.2. The discrepancy can be explained by the fact that the bubbles generated in our experiments may have not been perfectly spherical. Indeed the on axis imaging does not allow to evaluate a possible elongation of the bubble along the optical $z$ axis that occur for elongated beams [24,35]. For this, a side-imaging arrangement should be used. Nevertheless, the key point does not concern the value of the ratio $\frac {D_{max}}{T_c}$ but the fact that this ratio was found constant in all cases with very little variations (less than 10%). This means that, in the frame of this study, the onset of aberrations does not alter the bubble collapse dynamics. Rather, aberrations, especially spherical and secondary astigmatism, have mostly a reduction of laser energy coupling efficiency to water.## 5. Conclusion

We have reported on the influence of controlled optical aberration in the formation and dynamics of laser-induced cavitation bubbles in water caused by femtosecond laser pulses. The maximum cavitation bubble diameter reduces with increasing aberrations, which can be related to a reduction of the energy coupling efficiency, ultimately leading to the absence of cavitation bubble. Spherical aberration and secondary astigmatism were found to have the strongest effect in terms of reducing the laser fluence at the focusing point (i.e spreading the intensity distribution). Numerical simulations were also conducted to evaluate the laser fluence at the focusing plane in the presence of aberrations confirming the predominance of spherical aberration and secondary astigmatism. These results are of outmost importance for biomedical applications, especially for eye surgery applications where the laser energy cannot be blindly increased to overcome the aberration-driven intensity spreading, because of safety issues.

## Funding

Keranova.

## Acknowledgments

We thank F. Romano, S. Valla, P. Gain and G. Thuret for fruitful discussions; we also thank Keranova for financial support.

## Disclosures

The authors declare no conflicts of interest.

## Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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