2.1.

Experimental Assays and Numerical Modeling

We performed two series of incisions in opaque corneas (prepared as described in the following subsection):

Due to the relative opacity of the cornea, the energy used was adjusted at a value corresponding to four times the threshold radiant exposure at the surface.

Using the ZEMAX software permitting numerical modeling of optical systems, we estimated the contribution of spherical aberrations to the decrease of the irradiance as a function of the numerical aperture at focusing depths ranging up to $1000\mu \mathrm{m}$ , which is the typical thickness of a pathological cornea.

2.2.

Setup and Sample Preparation

We performed experiments on human corneal grafts obtained from the Banque Française des Yeux (French Eye Bank, BFY). The study was conducted according to the tenets of the Declaration of Helsinki and the French legislation for scientific use of human corneas. The corneo-scleral buttons were recovered from one to two weeks after enucleation and stored in CorneaMax solution at room temperature until the experiments. The buttons made available to us for research purposes had previously been considered unsuitable for transplantation by the BFY. They generally presented a defective endothelium, which normally ensures the regulation of the corneal hydration. As a consequence, bathing them in the solution promotes swelling, thus reproducing corneal edema.

The cornea was mounted onto an artificial chamber specifically conceived to reproduce natural mechanical and pressure conditions and to maintain the natural curvature of the specimen. By filling the chamber with a bicarbonate-buffered saline (BBS) solution, a pressure is applied to the cornea and the hydration of the tissue is preserved. The sample could be positioned in three dimensions by step motors with submicrometric resolution, allowing a laser scan in three dimensions.

For cutting the corneal tissue, we used two laser systems:

In both configurations, a beam expander allowed overfilling the pupils of the objective, which focused the laser beam into the sample. Focusing optics were mounted either on an independent holder or onto an inverted optical microscope inserted into the beam path. In the latter case, in order to preserve the in situ configuration, a chamber in Plexiglas was used and a camera was mounted on the side of the microscope with a video port. The experimental setup is presented in Fig. 1 .

Following laser cutting, the samples were prepared for histological and ultrastructural analysis by light microscopy (Zeiss Phomi2) and transmission electron microscopy (TEM; Philips CM10), respectively. They were chemically fixed and embedded in a resin. Subsequent semithin and ultrathin sections were obtained using an ultramicrotome (Reichert OmU2). Semithin sections were stained with toluidine blue, while ultrathin sections were colored by uranyl acetate and lead citrate solutions for observation.

3.1.

Laser-Induced Streaks

Using light microscopy, we performed an a posteriori analysis of lamellar corneal incisions. Analysis showed that laser-induced side effects occurred on the optical axis well outside the point-spread function (PSF) of the focused beam. They correspond to darkening of the tissues as streaks, which form above and below the lamellar plane where the cut was obtained. This effect occurred when using focusing numerical apertures of 0.15, 0.3, 0.5, and 0.75. It was observed in all irradiated corneas, although not in all analyzed histological sections. Electron transmission microscopy of corneal ultrathin sections revealed that the permanent modifications produced in the cornea when using NAs ranging between 0.15 and 0.5 have a striking periodical structure as a consequence of laser irradiation. In some cases, the tissue is just stained (i.e., modified in a way that reduces its transparency for electrons); in others, collagen fibrils have been disrupted [Fig. 2 ]. Lamellar incisions performed at $0.75\mathrm{NA}$ showed no periodical features, and the extent of the out-of-focus modification is strongly reduced, reaching a maximum length of $2.0\pm 0.2\mu \mathrm{m}$ [Fig. 2]. We measured the period from TEM photographs. The corresponding values are reported in the graph in Fig. 3 , where we have classified them by the numerical aperture used. The measured average period ranges between $0.22\pm 0.05\mu \mathrm{m}$ and $0.47\pm 0.09\mu \mathrm{m}$ . The periodical structure can extend up to $30\mu \mathrm{m}$ , and there is no qualitative decrease of the effect produced by the laser (staining or disruption) as we move away from the lamellar plane of the incision.

Evidence of material modification in the form of streaks has been reported in previous studies for corneal tissue;¹⁹ however, a periodical feature was not observed, probably due to analysis at a lower resolution. To the best of our knowledge, we are first to show the occurrence of self-organized features into corneal tissue.

We believe that the out-of-focus modifications induced in the tissue when performing lamellar incisions are due to local maxima in the laser intensity. This hypothesis is supported by the observation that in some corneal sections, optical breakdown occurred, thus disrupting the collagen fibrils. When the radiant exposure threshold for optical breakdown is not reached, the staining of the tissue is likely generated by low-density plasma oxidizing the fibrils. A previous work showed that by nonlinear absorption of NIR femtosecond pulses, photolysis of water molecules occurs producing reactive oxygen species (ROS), such as $\mathrm{O}{\mathrm{H}}^{-}$ and ${\mathrm{H}}_2{\mathrm{O}}_2$ (Ref. 19). In addition, cell damage induced by NIR irradiation and ascribed to photochemical effects has been previously reported.²⁰ In our studies, periodic tissue modifications might be produced by nonlinear effects related to an increase of the intensity-dependent refractive index, causing self-focusing and subsequent plasma defocusing, which likely occur at low and moderate numerical apertures.^{21, 22, 23} The combination of these two phenomena results in a successive focusing–defocusing effects, giving rise to filamentation. For ultrashort irradiation, self-focusing occurs when the pulse peak power exceeds a critical power $P_{\mathit{cr}}$ that in the case of a Gaussian beam is given by^{24, 25} :

where $n_0$ and $n_2$ stand for the linear and nonlinear refractive indexes of the material. Considering $n_{0\text{-}\mathit{\text{cornea}}}=1.38$ and $n_{2\text{-}\mathit{\text{cornea}}}\approx n_{2\text{-}\mathit{\text{water}}}=1.2\times {10}^{-6}{\mathrm{cm}}^2∕\mathrm{W}$ , using the value of water to model the nonlinear properties of corneal tissue, $P_{\mathit{cr}}$ corresponds to values of $10.2\mathrm{MW}$ at $\lambda =1.06\mu \mathrm{m}$ and $5.8\mathrm{MW}$ at $\lambda =800\mathrm{nm}$ . As the occurrence of self-focusing depends on laser power, whereas photodisruption is dependent on irradiance, the strength of the focusing will determine which mechanism will prevail. Because of the smaller focusing area, high NAs permit us to use lower pulse energies and to remain below the critical power. Based on our experimental conditions, we calculated the peak powers that we reached in order to induce optical breakdown. The values are reported in Table 1 . We recall that the incisions were performed at a depth of $150\pm 10\mu \mathrm{m}$ and at four times the surface threshold irradiance in order to ensure optical breakdown in the bulk of nontransparent samples. Although the beam experiences attenuation, which reduces the radiant exposure in the volume of the tissue, we estimate that the power is in a range where self-focusing is generated, for $\mathrm{NA}=0.15$ and $\mathrm{NA}=0.3$ . In fact, based on considerations on the laser propagation in opaque corneas reported in a previous investigation,¹⁸ typically for a cornea with mild edema, the $1∕e$ penetration depth is in the order of $300\mu \mathrm{m}$ . According to Eq. 4 (see the following), the energy at $150\mu \mathrm{m}$ is attenuated by a factor of 0.6, which would support our estimation.

Table 1

Values for the experimental threshold energy (10% measurement uncertainty), four times the power corresponding to the experimental threshold energy, and the critical power for self-focusing to be produced, as a function of different laser parameters.

	NA=0.15 , λ=1.6μm , τ=500fs	NA=0.3 , λ=800nm , τ=150fs	NA=0.5 , λ=800nm , τ=150fs	NA=0.75 , λ=800nm , τ=150fs
Experimental thresholdenergy $\left(E_{t\text{-}\mathit{exp}}\right)$	$2\mu \mathrm{J}$	$280\mathrm{nJ}$	$130\mathrm{nJ}$	$60\mathrm{nJ}$
$4\times \mathrm{P}$ correspondingto the $E_{t\text{-}\mathit{exp}}$	$16\mathrm{MW}$	$7.5\mathrm{MW}$	$3.4\mathrm{MW}$	$1.6\mathrm{MW}$
$P_{\mathit{cr}}$	$10.2\mathrm{MW}$	$5.8\mathrm{MW}$	$5.8\mathrm{MW}$	$5.8\mathrm{MW}$

Our analysis revealed that the lateral dimensions of the features obtained typically measure a fraction of the beam waist. This is in agreement with studies on filamentation in air²⁶ and in other transparent media (reviewed in Ref. 25). The high degree of nonlinearity of the process and different experimental parameters may explain the discrepancy between the subwavelength values of the periodical structure when focusing into the cornea and values of multiples of the wavelength obtained in these previous studies.²⁵

At an NA of 0.75, the power necessary to obtain photodisruption is well below the critical power for self-focusing, and the streak could be attributed to low plasma density produced in the Rayleigh range equal to $1.6\mu \mathrm{m}$ . An NA of 0.5 seems to correspond to a transitional regime for self-focusing to occur. At lower numerical apertures, other mechanisms are unlikely to explain the observed laser effect. Constructive interferences between the incident beam and the beam reflected by the plasma created in the focal plane cannot account for the effects observed below the focal plane (considering that the plasma is not reflecting in the forward direction). It does not seem probable either that local maxima in the axial light distribution in the focal volume (assumed to be an Airy pattern in our experiments) may cause the observed patterns. The first maxima of intensity of the Bessel function describing the irradiance distribution is at a distance of 130 $\lambda$ $(\mathrm{NA}=0.15)$ , 30 $\lambda$ $(\mathrm{NA}=0.3)$ , and 9 $\lambda$ $(\mathrm{NA}=0.5)$ from the focal plane, which is much greater than the period lengths we measured. Moreover, secondary maxima being less intense, an attenuation of the laser effects should be visible with increasing distance from the focal plane, which is in contrast with our results. Also, nanostructures have been reported in silicon attributed to nonlinear nanoplasmonic processes; however, the periodicity is produced in a plane parallel to the surface of the sample and the scanning plane²⁷ and not in the incident plane, as in our case.

3.2.

Influence of Spherical Aberrations on the Laser Focusing

According to previous studies (reviewed in Ref. 1) and in agreement with our experimental results presented earlier, increasing the focusing numerical aperture can reduce the extent of out-of-focus material modifications caused by nonlinear effects. However, at high NAs, spherical aberrations may become significant—in particular, when cutting opaque tissue—as they contribute to the broadening of the point-spread function and therefore raise the energy threshold and reduce the laser accuracy. In order to evaluate the beam focus degradation ascribed to spherical aberrations at different NAs as a function of the focusing depth, we performed numerical simulations using the ZEMAX software. Based on geometric ray tracing, ZEMAX allows the analysis of modeled optical systems. In particular, we computed wavefront aberrations in order to determine the Strehl ratio at different focusing depths. For our simple optical system consisting of optics focusing the beam into the cornea, the geometric approximation appears to be adequate. Design data for the doublet lens were available in the ZEMAX catalogue. The microscope objectives were modeled by a paraxial element corresponding to their numerical aperture, as we assumed that their point-spread function is reasonably close to the diffraction limit. The cornea was simulated as an isotropic material with a refractive index of 1.37. The merit function was minimized by optimizing the wavefront at $\lambda =1.06\mu \mathrm{m}$ , which is the typical operating wavelength of clinical systems, and the beam intensity throughout the entrance pupil of the system was considered uniform. ZEMAX enables calculating the peak-to-valley (P-V) and the root-mean-square (RMS) wavefront aberrations as well as the Strehl ratio. Peak-to-valley aberration refers to the maximum wavefront departure from the ideal value in both positive and negative directions, whereas the RMS wavefront error expresses the deviation averaged over the entire wavefront. According to the literature, the Strehl ratio $S$ can be approximated by the following relation:^{28, 29}

where $\sigma$ is the RMS value of the wavefront error. As Eq. 2 is no longer valid for Strehl ratios $<0.5$ , a different approximation is used by ZEMAX to calculate the Strehl ratio. It is given byand is valid for Strehl ratios as small as 0.10 (Ref. 30). Figure 4 shows the Strehl ratio computed with the RMS approach as a function of the focal plane depth into the sample. The $1∕e$ penetration depths measured from the Strehl ratio curves and peak-to-valley and RMS wavefront aberrations calculated at a depth of $500\mu \mathrm{m}$ are presented in Table 2 . We point out that these values underestimate the total aberration that the beam experiences during propagation: only spherical aberrations introduced by the working depth are considered here, and aberrations introduced by irregularities on the surface and local or low-frequency variations in the refractive index within the tissue are not accounted for.

Table 2

1∕e penetration depths measured form the Strehl ratio curves and peak-to-valley and RMS wavefront aberrations calculated at a depth of 500μm as a function of the numerical aperture.

Numerical aperture	0.3	0.4	0.5	0.6	0.75
$1∕e$ penetration depth $\left(\mu \mathrm{m}\right)$	$>1000$	$>1000$	620	300	90
Peak-to-valley wavefront aberration	$0.048\lambda$	$0.164\lambda$	$0.450\lambda$	$1.088\lambda$	$3.776\lambda$
RMS wavefront aberration	$0.014\lambda$	$0.047\lambda$	$0.127\lambda$	$0.298\lambda$	$0.953\lambda$

One can immediately notice that for low numerical apertures, spherical aberrations can be neglected. Considering a maximum thickness of about $1\mathrm{mm}$ for an edematous cornea, the contribution to the beam attenuation becomes significant for $\mathrm{NA}>0.5$ . Using specific microscope objectives with numerical apertures of 0.4 and 0.6, we performed posterior-to-anterior incisions by correcting the spherical aberrations at a given focusing depth in the volume of the cornea. The correction depth and the yielded incision lengths, measured by light microscopy, are reported in Table 3 . It can be noticed that increasing the correction depth increases the incision length up to a point beyond which this tendency is reversed. As expected, the behavior is more prominent when using the $0.6\mathrm{NA}$ focusing optics, as compared to $0.4\mathrm{NA}$ , which produces reduced spherical aberrations. We believe that as we correct deeper in the edematous tissue, the scattering becomes significant, and when associated to the degradation of the beam away from the correction point, it prevents us from cutting farther. As a result, the maximum length of a cut can be less than the total thickness of pathological corneas.

Table 3

Incision lengths obtained when focusing the beam into a cornea using microscope objectives with correction of spherical aberrations at a given depth.

$\mathbf{NA}=\mathbf{0.6}$	Correction depth $\left(\mu \mathrm{m}\right)$	Yielded incision length $\left(\mu \mathrm{m}\right)$ (4 corneas)
	0	265	250	490	490
	300	395	295	550	540
	450	470	320	375	360
	640	425	220	252	250
$\mathbf{NA}=\mathbf{0.4}$	Correction depth $\left(\mu \mathrm{m}\right)$	Yielded incision length $\left(\mu \mathrm{m}\right)$ (2 corneas)
	0	425	445
	200	470	475
	400	435	475
	600	360	390

Even though the irradiance attenuation by spherical aberration does not follow an exponential behavior, for simplicity, we here associate a $1∕e$ penetration depth with this phenomenon and make a semiquantitative estimation. Therefore, if we consider that to a first approximation, both the extinction of the irradiance $I\left(z\right)$ by scattering and spherical aberrations may be described by an exponential decay, we can write

where $I_0$ , represents the incident irradiance, $z_0$ the depth for which the spherical aberrations have been corrected, and $l_{\mathit{cs}}$ and $l_{\mathit{sa}}$ the penetration depths for attenuation due to scattering and spherical aberrations, respectively. As the optical breakdown occurs for values of the radiant exposure higher than the threshold, relation 4 becomeswhich is equivalent to

We have represented the spherical aberration penetration depth as function of the correction length, considering two sets of parameters:

In both cases We notice that the predicted theoretical behavior is in agreement with the experimental findings: by increasing the correction depth, at a fixed set of values for $l_{\mathit{sc}}$ , $l_{\mathit{sa}}$ , and $I_0$ , the incision length increases up to a maximum point, and then it decreases. This tendency is enhanced when $l_{\mathit{sa}}$ decreases, even though the variation of $l_{\mathit{sa}}$ does not influence the maximum incision length. At a fixed value of $l_{\mathit{sa}}$ , the maximum incision length obviously increases with increasing $l_{\mathit{sc}}$ .

The preceding results provide direct evidence that in pathological corneas, both scattering and spherical aberrations are limiting factors.