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We measure the surface ablation threshold fluence of porcine corneal stroma for 100 fs laser pulses, with wavelengths between 800 nm and 1450 nm, generated by a Ti:sapphire-pumped optical parametric amplifier. The ablation threshold was found to vary only slightly within this wavelength range, between 1.5 and 2.2 J/cm2. The data suggest a rapid increase of the ablation threshold for wavelengths up to about 1000 nm, followed by a plateau for longer wavelengths. This behavior is partly confirmed by a simple theoretical model of the ablation process. The influence of the wavelength on the physics of surface ablation is discussed on the basis of the model.
©2008 Optical Society of America
1. Introduction
Femtosecond lasers (FSL) currently find important applications in the creation of Laser In Situ Keratomileusis (LASIK) flap, where it has been shown to provide better visual outcome, less higher order aberrations, glare and haloes [1], compared with traditional microkeratomes [2,3]. FSL is also used to create intrastromal channels for implanting intracorneal ring segments for keratoconus [4]. There are also other promising applications of FSL currently under development, such as full thickness corneal transplantation [5], deep lamellar endothelial keratoplasty [6–8], and the photodisruption of the human trabecular meshwork in glaucoma patients [9]. To further develop the applications of FSL in ocular surgery, it is nevertheless essential to improve issues such as smoothness of the FSL ablation surface [7], postoperative inflammatory response [10], and fibrotic scarring [11].
A route toward improving FSL surgical technique consists in exploring the effects of the laser parameters (such as pulse duration, wavelength, and numerical aperture of the focusing optics) on the ablation of biological tissues. A key parameter for characterizing the laser-tissue interaction is the ablation threshold, defined as the minimum laser energy per unit surface (fluence) required to induce detectable changes in the material. A low ablation threshold is generally favorable for minimizing collateral damage near the ablation area. Although the relationship between the thresholds for ablation at surface and in the bulk (cavitation) of corneas has not been clearly established yet, it is reasonable to assume that both processes occur in similar conditions, namely when the water contained in tissues is vaporized. (See [12] for a discussion on cavitation in water.)
Ablation threshold dependence on pulse duration has been well-documented in common dielectric materials such as silica (SiO2) [13–16], and in corneas [16,17]. The wavelength dependence of the ablation threshold has been studied recently in fused silica and calcium fluoride (CaF2) between 250 nm and 1700 nm, for laser pulses of about 150 fs [18]. It was found that for both materials the ablation threshold increases as a function of the wavelength and reaches a plateau of about 3 J/cm2 near 1000 nm.
Several papers have been devoted to ablation of biological tissues (see [12] and references therein). In this study we investigate, to the best of our knowledge, for the first time the surface ablation threshold of corneal stroma as a function of wavelength for femtosecond laser pulses. Freshly enucleated porcine eyes were used for this study. The wavelength tunable laser pulses were generated from a Ti:sapphire-pumped optical parametric amplifier (OPA). The resulting pulses have a duration of 100 fs and wavelengths ranging between 800 nm (near infrared) and 1450 nm (infrared). The relatively high energy of the OPA output (which strongly depends on the wavelength) allowed us to perform reproducible ablation measurements within this wavelength range.
The experimental data are compared with the results of a theoretical model involving the physical mechanisms which are assumed to be the most significant in the ablation process. This model was used to predict the ablation threshold over a wavelength range wider than that investigated in our experiments, and to infer the laser intensity profiles in the target at the ablation threshold.
This paper is organized as follows. In the next section we describe the experimental set-up and methodology used to measure the dependence of ablation threshold on the wavelength. Experimental results are then presented and discussed. The model and its results are discussed in Section 4. Section 5 concludes the paper.
2. Experimental set-up
The experimental set-up and procedure are almost the same as those presented in Ref. [16]. Only the main details will be repeated here. A schematic diagram of the experimental set-up for ablation threshold measurement is shown in Fig. 1. In the present set-up the femtosecond laser is a Ti:sapphire-pumped OPA (HE-TOPAS-800-fs from Light Conversion).
Fig. 1. Experimental set-up : λ/2: half wave plate, P: Polarizer, BS: Beam Splitter, E: Energy Detector, M: Mirror, Ob: Microscope Objective, S: Sample, T: Translation Stage.
The laser pulse was focused on the sample surface with a microscope objective (NA=0.1), at an angle normal to the target surface. The sample was installed on an XYZ translation stage and moved to expose fresh tissue for each laser pulse. A low power He-Ne laser was aligned collinear to the incident laser beam to facilitate locating the targeted sites. A CCD camera (Hamamatsu C2741-03D, with spectral sensitivity between 400 nm and 2200 nm) was used to visualize the image of the surface of the sample by transmission and to position the sample surface at the beam focus. The transmission coefficients of the beam splitters BS1 and BS2, and of the microscope objective were measured for each wavelength, to know the fraction of the incident energy sent to the sample. Pulse durations of 100±20 fs have been measured for several wavelengths. We expect this uncertainty to be of relatively minor importance since a plateau in the ablation threshold as a function of pulse duration was observed near 100 fs at 800 nm [16].
It was important to minimize the experiment duration, to limit the dehydration of the corneal samples. Therefore, verification and required adjustments of sample position were performed every 9 shots. A typical experiment involved about 50 laser shots and lasted 15 minutes. Based on the measurements of Fisher et al. [19], and on our pre-test hydration control protocol, tissue dehydration during the experiments was expected to remain less than 10 %.
The CCD camera was also used to record the transverse beam profile and to determine the beam size. The laser fluence was calculated by using the laser beam energy EL and radius R at 1/e 2 of the maximum intensity on the sample surface, using the formula F=EL/πR 2. With this set-up, series of shots were performed at different fluences on the samples, each shot hitting a different location on the sample. Ablation threshold measurements were started from lowest fluence, progressively increasing the laser energy (0.1-10 µJ, measured by means of the microjoulemeter Model PE10 with Nova II display from Ophir Electronics) until damage became visible on the surface of the sample. After laser exposure, the ablation threshold of the sample was deduced from the visible damage area for each laser shot, which was observed using a 20× reflection mode optical microscope (Reichert, MEF4M). Examples of laser shots on stroma samples are shown in [16].
Fresh porcine eyes were obtained from a slaughterhouse and preserved at 4 °C in plastic bags, each cornea covered by its own conjunctiva, tenon and lids. Experiments on corneal tissues were performed within 8 hours after death. Just before starting the femtosecond laser ablation experiments, the corneal epithelium was gently removed (blade #15, Albion surgicals, Sheffield England), the corneoscleral button dissected and full thickness corneal layers (3 mm wide, from limbus to limbus) were prepared and gently laid flat on a glass slide, endothelial side down. During tissue preparation, hydration was maintained by instillation of balanced salt solution (BSS) every 3 minutes. BSS was not used during laser experiments to avoid any excess liquid that could interfere with the measurements. Sample preparation did not exceed 10 minutes. After the laser experimentation, the cornea specimens were immediately examined under the microscope. To minimize experimental uncertainties and verify the reproducibility of our results, three measurements were made for each tested wavelength.
3. Experimental results
We show in Fig. 2 the mean ablation thresholds of the porcine corneal stroma measured as a function of wavelength. Apart from the two data points near 1150 and 1200 nm, one observes low variations – between 1.5 and 2.2 J/cm2 – of the ablation threshold in the range of wavelengths investigated. Although the data points at 1150 and 1200 nm were reproducible, they raise doubts since we found similar sudden drops in the ablation threshold in different kinds of silica (standard microscope slides, Fused Quartz GE 124, and UV Grade Fused Silica) at the same wavelengths. (Note that, in Fig. 3 of [18], the data near 1000 nm is lacking for a different reason: the pulse energy at this wavelength was too small to ablate the samples [20].) This behavior may be related to a degradation of the transverse beam quality which is likely due to the fact that 1150 and 1200 nm lie at the edge of a functioning regime of our OPA.
Apart from the points at 1150 and 1200 nm, the data suggest that the ablation threshold increases with the wavelength up to ~ 1000 nm and then saturates (or even decrease), as suggested by the two dashed lines. These general trends are quite similar to those recently described for SiO2 and CaF2 [18]. For these two materials, a change is also observed near 1000 nm and the maximum ablation threshold is about 3 J/cm2. This similarity between cornea and dielectric materials on the behavior of ablation threshold as a function of wavelength confirms the universal picture of the FSL ablation process for non-metallic materials, which will be discussed in the next section. The same observation was made in [16], in a study of the ablation threshold as a function of the laser pulse duration.
It is worth pointing out that the general behavior of the ablation threshold observed in Fig. 2 does not correlate with transmittance of the cornea as a function of wavelength for low intensity radiation (i.e., radiation not causing any damage). As a matter of fact, transmittance in cornea (and in pure water) remains rather constant between 800 and 1300 nm and then drops rapidly to reach a minimum near 1450 nm [21]. In Fig. 2, however, no significant change in the ablation threshold can be seen near 1450 nm. This is a strong indication that ablation is essentially a nonlinear process (that is, which depends on the laser intensity) and that linear (low intensity) effects are negligible. One also notes that the low intensity transmission coefficient in cornea cannot explain the low thresholds at 1150 nm and 1200 nm seen in Fig. 2.
Fig. 2. Measured ablation threshold as a function of wavelength for a 100 fs laser pulse. Dashed lines show the main trends of the data (excluding the two points at 1150 nm and 1200 nm).
4. Modeling
The ultrashort laser pulse ablation process of transparent materials is generally assumed to occur as follows [13]. The laser energy is first absorbed mostly by electrons in the sample via two main mechanisms: (i) photoionization, in which bound electrons are directly excited by light in the conduction band of the material, and (ii) collisional absorption (inverse Bremsstrahlung) in which the electrons in the conduction band gain more energy through collisions with the atoms and are eventually able to induce further ionization of the sample. Once the electrons recombine with the ions the ionization energy is released (on a timescale estimated as one picosecond in water [22]), causing a local elevation of the temperature of the sample. Ablation takes place when the energy density required for vaporization is reached at the surface.
We model our experiments by means of the following pair of equations for the electron density ne and electron thermal energy density χe:
In these expressions, WPI and Ũi are the photoionization rate per unit volume and the effective ionization potential, respectively, as given by the Keldysh model for solids [23]. Ũi includes the oscillation energy of free electrons in the laser field and is thus higher than the band gap Ui between the valence band and the conduction band. In Eq. (2), I=cε 0|E|2 is the laser intensity and
is the collisional absorption cross section. Here, τc is the electron-lattice collision time, η 0 is the linear index of refraction, and ω is the laser frequency.
We estimate the collisional ionization rate by [24]:
where na is the initial atom number density, U͌i≈1.5Ũi is the electron energy required for impact ionization [25], and x=kBTe/U͌i, kB being the Boltzmann constant. We define the electron temperature Te from the expression:
The laser electric field E in the sample as a function of the axial position z is obtained by solving the Helmholtz wave equation for normal incidence:
Here εrel is the relative dielectric function of the medium, which is given consistently with Eqs. (1) and (2), by:
where η 2 is the nonlinear index of refraction, ncr=meε 0 ω 2/e 2 is the plasma critical density and k 0=ω/c. Note that the imaginary part of εrel represents absorption of the laser radiation.
Assuming an incident laser intensity I(t) with a temporal Gaussian envelope of 100 fs at full width half maximum, Eqs. (1) to (7) provide the laser electric field profile as a function of position z and time t.
Table 1. Values of the Parameters Used in the Model
The parameters used in the model are given in Table 1. To our knowledge, only the linear index of refraction η0 has been measured in cornea [26]. The parameters η2 and Ui used here are those of water (Refs. [27] and [28], respectively), as well as the parameter na. The parameter τc was adjusted to obtain the best fit to our data and will be discussed below.
Another parameter which has to be defined in the framework of this model is the damage threshold Dth. The criterion used in this study was that the laser energy density absorbed at the surface of the cornea is approximately the energy density required for vaporization of water near 0 °C initially, i.e., about 3 kJ/cm3. In the framework of this model, ablation occurs when neUi+χe=Dth at the corneal surface, neUi+χe being the electron energy density at the end of the laser pulse. Note that this criterion differs from the criterion ne≈ncr (i.e., when the plasma becomes opaque to radiation) used in several investigations on the influence of the pulse duration on the ablation threshold [13–16]. In the present study, the plasma becomes overcritical (i.e., ne>n 2 0 ncr) about 100 fs after the peak of the laser pulse when λ>1100 nm. However, the plasma is never overcritical at the peak of the laser pulse.
A low initial electron density ne(t=0)=108 cm-3 in the samples was assumed in the model. Other choices for this parameter had negligible effects on the results, even by varying it over several orders of magnitude (from ~0 to ~1014 cm-3).
Equations (1), (2) and (6) were solved numerically on a mesh size of 10 nm, with time steps of 2 fs, by using a time splitting method involving an explicit scheme for Eqs. (1) and (2), and an implicit scheme for Eq. (3). Open boundary conditions were used at both ends of the z axis in solving Eq. (3).
Fig. 3. Calculated ablation threshold as a function of wavelength for a 100 fs laser pulse compared with experimental data.
The results of the model are plotted in Fig. 3. One sees that the model reproduces roughly the main trends of the measurements suggested in Fig. 2. In particular, the increase of the ablation threshold with the wavelength is slower for λ>1000 nm. The step-like behavior observed in the calculated values is due to the Keldysh photoionization model, which involves abrupt variations of the photoionization rate as the number of photons required to excite the electrons from the valence band to the conduction band changes by one unit. The model does not reproduce the variations of the measurements at 1150 and 1200 nm. This either indicates that the data points near 1200 nm are erroneous or that the model is too simple to reproduce the observed behavior.
The decrease of the ablation threshold as the wavelength decreases is essentially due to the higher efficiency of photoionization at shorter wavelengths. Photoionization provide the seed electrons required to trigger electron avalanche through collisional absorption. The step- like behavior observed for λ>1000 nm indicates that photoionization is still significant at higher wavelengths. For the value used for τc, one can see that (ωτc)2≪1 in the range of wavelengths considered, so that the collisional absorption cross-section, Eq. (3), does not depend significantly on wavelength. This explains the slower increase of the threshold fluence observed for λ>1000 nm. The slight increase is due to the fact that the ionization thresholds Ũi and U͌i increase with wavelength.
The parameter τc obtained here is consistent with the values calculated for silica for several-eV electron energy [29]. Such energetic electrons are coherent with our calculations since the electron temperature attains for example 2.8 eV at 500 nm and 5.8 eV at 1600 nm near the peak of the laser pulse. However, the collision time obtained here is one order of magnitude smaller than the value of 1.7×10-15 s inferred from experiments in silica [30] for electron density estimated as 5×1019 cm-3. The only physical explanations we can provide for the smaller collision time obtained here with respect to the value measured in [30] are that: (1) the material considered here is very different from silica since corneas are not homogeneous media, and (2) τc may depend strongly on plasma conditions and the value obtained here is representative, on average, of our specific plasma conditions.
The influence of the collision time τc can be understood by examining Eq. (3). One can see that σ increases with λ for all τc and that, as τc increases, σ decreases for small values of λ (λ≪2πcτc) and increases for larger values of λ. Hence, increasing τc would result in an increase of the ablation threshold at lower wavelengths and a decrease at higher wavelengths, producing a flatter profile which would not be in agreement with measurements.
Fig. 4. Calculated flux profiles as a function of retarded time at various positions inside the sample for (a) 500 nm and (b) 1600 nm. Incident fluences are the calculated ablation thresholds 1.07 J/cm2 and 2.13 J/cm2, respectively.
Figure 4 shows the flux n0I of the laser pulse as a function of the retarded time η=t-z/νg, where νg is the group velocity in the medium, for various positions in the sample. The incident fluences for the 500 nm and 1600 nm cases considered in Fig. 4 are the corresponding calculated ablation thresholds shown in Fig. 3. One observes that the leading edge of the pulse does not experience strong absorption inside the sample. However, the pulse intensity decreases significantly as ionization starts to take place. The pulses are not strongly absorbed or reflected close to the surface and propagate deeply inside the sample. Absorption at λ=500 nm is more efficient than at 1600 nm due to stronger photoionization. Most of the energy removed from the laser pulse is absorbed in the sample and a negligible fraction is reflected. (Reflection is responsible for the small difference between the incident flux and the flux just below the surface, at 0+µm.)
5. Conclusion
Laser ablation thresholds were measured in porcine corneal stroma for 100 fs laser pulses, with wavelengths ranging between 800 nm and 1450 nm. Measurements suggest that the ablation threshold increases with wavelength and reaches a plateau of ~ 2 J/cm2 above ~ 1000 nm. This general behavior is consistent with recent measurements made in common dielectric materials (fused silica and calcium fluoride). A drop of the ablation threshold was found at 1150 nm and 1200 nm in our corneal samples as well as various kinds of silica. This behavior may be related to a lesser beam quality due to the fact that these two wavelengths lie at the edge of a functioning regime of our OPA.
To gain more insight into the physics of laser ablation, a theoretical model was implemented. The model is based on the assumption that the two main mechanisms for laser energy absorption are photoionization and collisional absorption. The main unknown parameter, the electron-lattice collision time, was obtained by fitting the data and was found to be consistent with calculated values found in literature for fused silica for several-eV electrons.
The model confirms the main trends suggested by the experiment although saturation of the ablation threshold for λ>1000 nm is not as complete as expected. The decrease of the threshold for shorter wavelengths is essentially due to more efficient photoionization. For wavelengths longer than ~ 1000 nm, collisional absorption is the dominant mechanism for energy deposition. The drop in the ablation thresholds found at 1150 nm and 1200 nm is not reproduced by the model. The model suggests that, at ablation threshold, a small fraction of the incident laser energy is absorbed or reflected at the sample surface and that the laser can propagate deeply (at least several tens of microns) inside the sample. Beam attenuation is nevertheless stronger at smaller wavelengths due to the greater efficiency of photoionization. Experimental verification of these predictions would be helpful for improving our understanding of the physics of ionization in solid materials.
Acknowledgements
The authors would like to thank Stéphane Payeur and François Poitras for technical support. This research was supported by the Canadian Institutes of Health Research and the FRSQ Research in Vision Network. The research described in this paper was performed at the Advanced Laser Light Source facility of the INRS – Énergie, Matériaux et Télécommunications, which is supported by the Canadian Foundation for Innovation.
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