1. Introduction

Processing of the cornea with fs-lasers has been established as a powerful surgery method in ophthalmology in the last years. Besides the most frequent application for the creation of flaps for refractive surgery (as in Laser in Situ Keratomileusis - LASIK), some other techniques like refractive correction (lenticle), anterior and posterior lamellar keratoplasty, or tunnel creation for intracorneal ring segments also employ ultrashort laser processing [1,2]. These methods make use of the fact that cornea is transparent in the near-IR wavelength range where the most common commercially available fs-lasers work. Thus, by focusing the laser beam with sufficiently high numerical aperture optics to a small spot (typically < 10 µm) inside the tissue, even at moderate laser energies of a few µJ precise cutting by help of photodisruption becomes possible. The necessary optical breakdown is initiated by multiphoton and tunnelling ionization already during the ultrashort pulse [3]. This constitutes the main advantage of fs-laser processing compared with the thermally induced ablation by ns-laser pulses: precise tissue cutting while exhibiting only minimal mechanical and thermal damage [4].

It is obvious that the high peak intensities needed to generate nonlinear absorption of radiation for intracorneal cutting also induce other nonlinear processes in the tissue, which might have crucial influence on the processing result. Nonlinear effects which have to be regarded include self-focusing and filamentation, self phase modulation, second harmonic generation or white light generation. The main nonlinear effect contributing to a deviation of the focus position is self-focusing [5]: above the so called critical power for self-focusing (P_c), the intensity dependent refractive index n = n₀ + n₂ ⋅I (n₀: linear; n₂: nonlinear refractive index of material; I: time averaged intensity of laser) causes a different focal position as the one predicted by linear Gaussian beam optics. Depending on whether n₂ is positive or negative, the focus moves backward or forward along the direction of the beam. If the n₂ of cornea turns out to be large enough to cause focal shifts of several µm, nonlinear refraction would be an important issue for the precision of fs-laser-based eye surgery. As the nonlinearity of cornea as a biological material may vary appreciably from eye to eye, knowledge of the corneal nonlinear refractive index appears to be crucial for precise cutting, in particular when, e.g., lenticle cutting is intended.

To the best of our knowledge however, no reliable values for P_c and n₂ of cornea have been published to date, although a lot of work has been dedicated to investigations concerning the processing of cornea with fs-lasers. For instance, theoretical models using water as a model substance for cornea were developed for plasma formation [6], as well as for nonlinear side effects during processing with low and high numerical apertures [7,8]. Also the observation of intrastromal streaks around the cut [9], sometimes accompanied by a periodic nanostructure [10], were mostly interpreted using the nonlinear refractive index of water. Very recently Poudel [11] measured the self-focusing of a fs-laser beam at 780 nm and a pulse duration of 130 fs in pig cornea and gelatine. He stated that for propagation depths lower than 250 µm a focal shift is not visible.

The present work is dedicated to the determination of the nonlinear parameters of porcine cornea and their influence on the cutting precision which can be achieved for instance with fs-LASIK techniques. To get reliable results despite the rather delicate preparation and conservation of corneal slices, we employ a combined approach: (i) the critical power for self-focusing P_c and the n₂ of pig cornea are studied using a z-scan experiment [12] based on a 300 fs-laser at 1030 nm wavelength. (ii) Using the same laser source, intrastromal cutting is performed with repetition rates up to 350 kHz; discussing, in terms of self-focusing theory, the differences of achieved and intended cutting depths obtained at various laser parameters, we arrive at an independent determination of n₂ of porcine cornea. The consistency of the values derived from the different methods allows us to discuss the implications of the nonlinear refractive index for fs corneal surgery.

2. Experimental setup

The experimental set-up employed for the determination of nonlinear refractive indexes of corneas is a classical z-scan setup [12]. A fs-laser system operating at 1030 nm with pulse duration of 300 fs (‘Pharos’ from Light Conversion) was used to produce the nonlinear effects in the cornea. The fs-laser allows for a maximum output power of 6 W at variable repetition rates ranging from 10 kHz to 350 kHz. Repetition rates lower than 10 kHz can be realized with an integrated pulse picker; this option was used for the z-scans in order to avoid thermal lensing (see below). The laser beam was focused in the corneal tissue samples, which were mounted on a movable table. The samples were moved along the laser beam and the intensity transmitted through the sample was measured simultaneously with two photodiodes corresponding to the open and closed aperture signals. An additional photodiode provided a reference signal. The beam was focused to a beam waist of 30 µm and the laser power was kept as small as possible in order to induce nonlinearities without irreversibly altering the sample. The corneas were either completely extracted (thickness up to 1000 µm) or thin slices of cornea (100-200 µm thick) were fixed on the sample holder and scanned.

The second method for studying nonlinear effects in the cornea was cutting flaps by fs tissue processing using the same laser system in a different experimental setup. The laser beam was directed into a xyz-scanner equipped with a telecentrical f-theta lens, which focused the beam into the cornea at a well-defined depth. The optical system had a numerical aperture of 0.086 leading to a focal spot of about 4 µm radius. A sketch of the experimental setup is presented in Fig. 1 .

 

Fig. 1 Experimental setup for pig cornea processing. In the inset is shown the beam propagation in the cornea.

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The precise positioning of the eye was accomplished using a fibre coupled laser diode at 980 nm aligned in a confocal system. The collimated laser diode is following the same optical path as the fs-laser and is focused by the f-theta lens onto the vertex of the eye. The reflection from the cornea follows the way back to a beam splitter and is then captured in a 6 µm diameter monomode optical fibre and detected with a photodiode. When the signal at the photodiode reached the maximum, the laser diode was focused on the eye vertex. The positioning system has a resolution in axial direction of about 5 µm, being limited by the numerical aperture of the system and the optical aberrations in the system. A difference of 300 µm between the fs-laser and the laser diode focal position appears due to slightly different divergences of the two beams and the chromatic aberrations throughout the whole optical setup.

The pig eyes were fixed in a suction ring and kept at a low-pressure of 400 mbar. The processing of the cornea was computer-controlled, and the 3D profiles followed the geometry of the surface of the eye. The eye was not applanated before the processing as in most of the fs-systems used in refractive surgery. A layer of liquid (physiological saline) with a refractive index close to that of the cornea was brought on top of the eye, in order to achieve a flat surface at the entrance of the fs-laser. The thickness of the liquid could also be precisely measured with the positioning system. As well, the correction of the focal position due to refraction at the air/water interface, which has been accounted for by the control software for cutting 3D profiles, could be verified by this system. Flaps with a radius of 4 mm were produced at different depths in the cornea and the flap thickness was measured after surgery with an ultrasonic device. For these flap cutting experiments, the (single pulse) energy of the laser has been varied between 0.8 µJ and 2.5 µJ at repetition rates between 10 and 350 kHz.

3. Results and discussions

Z-scans have been conducted on a large number of corneas; complete corneas with a thickness between 800 and 1000 µm have been studied as well as flaps of 100 – 200 µm thickness cut out by our own fs setup. Repetition rates lower than 10 kHz have been used for all z-scans to avoid problems due to thermal lensing: if the delay between two consecutive pulses is shorter than the characteristic thermal diffusion time (which was calculated to be 100 µs for cornea [13,14]), the propagation of later pulses will be modified by the not yet relaxed temperature distribution profile left by the preceding ones. Such effects of cumulative heating with highly repetitive laser systems may occur also in transparent samples due to nonlinear absorption. Thermal lensing in cornea has been observed with ns-lasers [15].

Besides thermal lensing, the quality of corneal z-scans depends strongly on preparation and the resulting optical quality of the samples, as well as on their possible degradation (dehydration) during measurement. Therefore, the pig corneas were carefully extracted and then scanned quickly. Nonetheless, in some cases the data quality was suffering considerably from strong scattering effects. In addition, the cornea in general is not an optically uniform sample, and also features a native curvature. Quite different areas are being probed during a z-scan because of the necessarily changing beam diameter; the obtained signals will therefore only give reliable results if the material properties are constant over the illuminated region. As a consequence of these, sample related difficulties a lot of cornea z-scans (about 100) had to be carried out to acquire the necessary accuracy of the results.

Figure 2 presents an example for simultaneously measured open and closed aperture scans. Typically more than 70% of the z-scans yielded similar results, where at least the peak – valley region (|z-z₀| ≤ z_R) allowed a reliable evaluation of an n₂ value, while the other had to be discarded because of the above given reasons. Several scans of comparable quality have been analyzed by the method described in [16] and [12], using the complete z-scan curve with open and closed aperture curves as presented in Fig. 2. The result is a positive time averaged nonlinear refractive index of about n ₂ = 2·10⁻¹⁹ m²/W. Besides the complete data evaluation, we also used the setup for z-scan experiments with varying intensity to estimate directly the critical power for self-focusing. The peak laser power above which the z-scans gave the first indication of nonlinear refractivity in the cornea, was experimentally found to be P_c ≈1.2 MW. Since the flaps are very thin and the transmitted light can only accumulate a nonlinear phase shift over a short path length, this value can only be interpreted as an upper limit for the critical power. With λ being the laser wavelength, the relation between P_c and n₂ is given by [17]:

 

Fig. 2 Z –scan results in the pig cornea with open and closed aperture, z_R is the Rayleigh range of the beam, z₀ the focal position.

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Using Eq. (1), we arrive at a lower limit for the nonlinear refractive index at n₂ ≥ 1·10⁻¹⁹m²/W. This is nicely compatible with the above given results from the complete fitting of the z-scans, and proves independently that the nonlinear refractive index of cornea clearly exceeds that of water [18,19] by more than a factor of two.

The other approach to study the self-focusing effect was processing the flaps in the cornea using the experimental arrangement presented in Fig. 1. Mostly a repetition rate of 200 kHz was used in order to achieve a short processing time of the flap. At energies above 0.85 µJ and appropriate spot distances in axial and radial direction, the flaps could be removed. To compare this value to the ablation breakdown thresholds reported for porcine cornea at 300 fs, which vary between 0.4 J/cm² and 1.8 J/cm² [4,20], we have to consider the given spot radius of 4 µm yielding energies between 0.2 and 0.9 µJ. Thus the results indicate that the higher value of the breakdown threshold is the correct one.

For a first series of experiments we chose pulse energies considerably higher than the threshold (1.3µJ) to produce easily removable flaps of a well defined thickness. Applying the same laser power and keeping the liquid layer above the eye constant at 300 µm, flaps of several different thicknesses were produced. For this purpose the geometrical depth of the focus was calculated in the limit of linear optics. The thickness of the removed flaps was then measured with the ultrasound device; the results are presented in Fig. 3 .

 

Fig. 3 The measured flap thickness in pig cornea depending on the programmed depth. The line graph is a fit to the data. The cut parameters were 1.3 µJ, 200 kHz.

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It should be mentioned that at some depths of interest, many flaps were processed and measured for reproducibility. Also, a change of repetition rate in the range of 50 kHz to 200 kHz had no influence on the curve. Quite clearly, the obtained cutting depths are different from the programmed ones. For depths of 200 µm or more there is an almost constant difference of 100 µm.

From these data alone it is not really clear if the offset is due to self-focusing in water and/or cornea, or if some undetected malfunction of the positioning system occurred. The later suspicion however could be cleared up by a second series of experiments, where two different thicknesses of the flaps were programmed (250 µm and 500 µm), and then cutting was performed with several different laser intensities. The obtained flap thicknesses, as shown in Fig. 4 , demonstrate unambiguously that the positioning system was working correct, and in fact self-focusing causes the deviations: in the low-power limit the flaps are approaching the programmed depth, while towards higher laser power the created flaps are getting monotonously thinner.

 

Fig. 4 The flap thickness dependence on the applied laser peak power in two different programmed depths 250 µm and 500 µm

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In order to derive the nonlinear refractive index of pig cornea from these results, we calculate the expected focal position by help of established descriptions of self-focusing. According to [5], the position of the focus due to self-focusing z_sf for an externally focused Gaussian beam entering the medium is given by:

Here w_Kerr is the beam radius at the location of the Kerr medium, l is the Kerr medium length (in our case the liquid thickness), P is the power of the beam and a = 1.723 for a circular beam. The Kerr lens diminishes the beam radius at the entrance into the cornea increasing the convergence of the beam. By presence of the Kerr medium alone, the geometrical focus will be displaced by a distance z_Kerr, decreasing the thickness of the produced flaps. This effect will become stronger with increasing laser power. A schematic of the beam propagation is shown in the inset of Fig. 1.

The beam profile at the output of the scanner was measured with a CCD camera and the beam propagation was fitted following the Gaussian optics theory. A beam waist of 3.8 µm was obtained by the fit and this value was used for the theoretical calculation. For every experimental conditions (different depths and different laser powers), the beam radius in front of the liquid layer w_Kerr and the position of the new focus z_Kerr were first calculated using Eq. (3), then the beam radius in front of the cornea w was deduced from the lens transformation of Gaussian beams and it was used in Eq. (2) to calculate z_sf. The values calculated by this procedure using n ₂ = 2·10⁻¹⁹ m²/W for cornea are in good agreement with the experimentally measured values, as can bee seen in Fig. 3 and 4 (solid lines representing the theoretical values). The remaining, very small deviation of theoretical and experimental values may be due to imperfections of the laser beam, or due to the fact that Eq. (2) is an approximate solution for the self-focusing position.

4. Conclusions

The reported results show that self-focusing effect plays an important role for the precision of cornea processing. In order to perform precise cutting in a well defined depth, one has to understand in detail the response of the medium to the applied laser pulse. The self-focusing depends on the applied peak power; therefore one has to keep the power as low as possible. Through the increase of the numerical aperture of the optical system, focal shifts due to self-focusing can be avoided because of then generally shorter nonlinear refraction path lengths. The breakdown energy decreases as well as the energy necessary to create the flap.

A very important result is the determination of the cornea nonlinear refractive index (n ₂ = 2·10⁻¹⁹ m²/W). One can see that this value is four times larger than the one of water, showing that the cornea has a stronger third order nonlinear response. Since corneal tissue typically contains more than 70% water, it is an obvious conclusion that the main contribution to its nonlinear optical behaviour comes from the collagen fibrils. Turning around the argument, our investigation demonstrates that modelling optical nonlinearity of cornea with the nonlinear parameters of water is not a very good approximation.

The large amount of publications about the use of fs-lasers employed for the processing of the cornea are emphasizing the precision of these devices compared to the traditional microkeratome assisted LASIK. Fs-laser processing is more localized and thermal and mechanical damage of the surroundings is strongly reduced. On the other hand the reproducibility of the cutting depths is still mostly in the range from 5 to 20 µm. Our study shows that self-focusing in the cornea, in connection with the nonlinear refractive index being different from eye to eye, may be a reasonable explanation for such deviations. In principle, one should therefore now attempt a systematic study to look for the anticipated correlation of the nonlinear refractive index of corneas and its effect on the cutting depth during fs laser processing. On the other hand, our current findings imply that the absolute distance of the cut from the eye surface should be different from eye to eye, but quite stable within a single operation. Therefore, the relative precision of an individual surgery appears not to be severely affected by our results. Nonetheless, self-focusing should be considered for any fs-LASIK technique to achieve the optimum depth precision for refractive correction.