Abstract
Purpose:
Controversial opinions exist regarding optimum laser beam characteristics for achieving smoother ablations in laserbased vision correction. The purpose of the study was to outline a rigorous simulation model for simulating shotbyshot ablation process. The impact of laser beam characteristics like super Gaussian order, truncation radius, spot geometry, spot overlap, and lattice geometry were tested on ablation smoothness.
Methods:
Given the super Gaussian order, the theoretical beam profile was determined following LambertBeer model. The intensity beam profile originating from an excimer laser was measured with a beam profiler camera. For both, the measured and theoretical beam profiles, two spot geometries (round and square spots) were considered, and two types of lattices (reticular and triangular) were simulated with varying spot overlaps and ablated material (cornea or polymethylmethacrylate [PMMA]). The roughness in ablation was determined by the rootmeansquare per square root of layer depth.
Results:
Truncating the beam profile increases the roughness in ablation, Gaussian profiles theoretically result in smoother ablations, round spot geometries produce lower roughness in ablation compared to square geometry, triangular lattices theoretically produce lower roughness in ablation compared to the reticular lattice, theoretically modeled beam profiles show lower roughness in ablation compared to the measured beam profile, and the simulated roughness in ablation on PMMA tends to be lower than on human cornea. For given input parameters, proper optimum parameters for minimizing the roughness have been found.
Conclusions:
Theoretically, the proposed model can be used for achieving smoothness with laser systems used for ablation processes at relatively low cost. This model may improve the quality of results and could be directly applied for improving postoperative surface quality.
The development of science and technology in the field of eye surgery has been rewarding. The past few decades have seen improving standards of refractive surgery in the human eye and an incremental growth in the effort to achieve perfect vision through surgical means. More precise lasers with small laser spots and high repetition rates are now widely used to manipulate the shape of the cornea to correct refractive errors. Corneal remodeling is essentially similar to any other form of micromachining. The lasers used in micromachining are normally pulsed excimer lasers, where the duration of the pulses is very short compared to the time period between the pulses. Although pulses contain little energy, given the small beam size, energy density can be nevertheless high. This high energy and the short pulse duration yield a high peak power.
The radiant exposure, measured by the pulse energy density, governs the amount of corneal tissue removed by a single pulse. In excimer laser refractive surgery, this energy density must exceed 40 to 50 mJ/cm^{2}. The depth of the produced ablation crater by a single impact relates to the radiant exposure, also affecting the thermal load per pulse increasing with increasing fluence.^{1}
Achieving accurate clinical outcomes and reducing the likelihood of a retreatment procedure are major goals of refractive surgery. Several parameters characterizing the laser beam are critical for an accurate and safe refractive surgery,^{2–5} and can influence the laser ablation process as well as its outcome. These parameters include (but are not limited to) laser wavelength, pulse duration, super Gaussian order, spot diameter, spot truncation, radiant exposure, placement of laser spots achieved with the laser scanners, and the debris removal mechanism.^{6}
Despite myriad technological advances in this field, laser corneal refractive surgery still presents some challenges in achieving higher ablation smoothness and minimizing the applied energy on the cornea. The temporal and spatial distribution of the laser spots (scan sequence) has shown to affect the surface quality and maximum ablation depth of the ablation profile. Smoothness of ablation may also vary with different excimer lasers systems.^{7} In a study, ablations were performed on polymethylmethacrylate (PMMA) plates, with four different excimer lasers: VISXStar, Coherent Schwind Keratom I/II, Chiron Technolas Keracor 117C, and the Nidek EC5000, to determine and compare the homogeneity and smoothness of the surface. It was concluded that the laser with Scanning spot technology produced smooth ablations even up to −9.00 diopters (D). Ablation smoothness is also influenced by the spot positioning algorithms. Dago et al.^{8} performed ablations on PMMA plates using four scanning excimer lasers, two with flying spot technology (ZeissMeditec MEL70, and a Russianmade unit, Microscan) and two Nidek models with scanning slit delivery systems and an expanding iris diaphragm (EC5000 and EC5000 CX). The smoothest surface was obtained in samples produced by ZeissMeditec MEL70 unit (root mean square [RMS] = 112 ± 23 nm), followed by the Nidek EC5000 CX (RMS = 153 ± 12 nm), and the Microscan (RMS = 181 ± 11 nm). It was concluded that scanning excimer lasers based on flying spot technology created smoother ablations on PMMA plates compared to the older Nidek EC5000 unit.
It has been theoretically shown that corneal laser surgery could benefit from smaller spot sizes and higher repetition rates.^{9,10} Furthermore, higher refractive settings correlate with decreasing surface smoothness. These results have been reproduced in PMMA by O'Donnell et al.,^{11} showing an increase of 25 nm roughness per micron of ablation in PMMA.
Deviations of LambertBeer law affect corneal refractive parameters after refractive surgery.^{12} Several mathematical models have been proposed in the recent times, particularly for laser tissue interaction in refractive surgery, in the form of modifications in the LambertBeer law.^{13–24} Despite the various modeling approaches comparing the overall predicted performance of laser platforms, an extensive analysis of the impact of individual laser beam characteristics like spot energy, spot diameter, super Gaussian order, truncation radius, spot geometry, spot overlap, and lattice geometry on ablation smoothness is not existing in the literature, to the best of our knowledge. These laser beam characteristics may individually affect the ablation smoothness; for example, truncating the flanks of the beam profile to avoid thermal loads, preferring flatter beams for a higher ablation volume per laser pulse, preferring smaller spot sizes to increase the resolution for ablating fine structure, preferring lower pulse energy for imparting lesser energy on the cornea (but being more sensitive to perturbations), or preferring higher pulse energy for achieving stability but at the expense of higher thermal load, may all impact the ablation smoothness either constructively or destructively.
An analytical model was proposed to optimize the ablation efficiency based on different metrics of ablation derived from the modification in the LambertBeer law.^{25} Extending this model, the main purpose of this paper is to describe a rigorous and generic simulation model for simulating the sequential shotbyshot ablation process based on a measured or modeled beam profile. Another aim is to test the impact of laser beam characteristics like spot energy, spot diameter, super Gaussian order, truncation radius, spot geometry, spot overlap, and lattice geometry on ablation smoothness, for both theoretically modeled super Gaussian beam profiles and measured intensity beam profiles acquired using a beam profiler camera. In order to account for the impact of deviations in real world settings on the methods, a rigorous perturbation analysis was also performed.
Methods
A simulation model was developed for simulating the sequential shotbyshot ablation process based on a measured or theoretically modeled beam profile, following the simulation methodology as:
Simulation Methodology
The following steps were implemented to simulate a sequential shotbyshot ablation process:

Given the super Gaussian order (
N) and Full Width Half Maximum (FWHM),
R_{0} (the beam size when the radiant exposure falls to 1/
e^{2} its peak value) was calculated for a theoretical super Gaussian beam using a modified equation based on the LambertBeer law:

For a fixed resolution of 6 μm per element, a square matrix was calculated with defined number of elements (hereafter called as pixels), tightly enclosing the foot print of one laser pulse, calculated as explained in Reference
24.

Two theoretically modeled Gaussian spot geometries were simulated. For the round spot geometry, the value of R_{0} (calculated in previous steps) was kept constant at each meridian. For the square spot geometry, the value of R_{0} was meridionally changed to match the size of the square matrix resulting in a Gaussian profile with a square base.

A normalized intensity distribution of the super Gaussian beam profile was calculated using the standard form of LambertBeer law at each pixel as
^{24}:
Where I represents the intensity calculated for each pixel position defined by the coordinates
n and
m,
I_{0} represents the peak radiant exposure (normalized to
I_{0} = 1), and
r represents the radial distance.

The intensity distribution was scaled for a given spot energy, using the normalized intensity distribution (
Equation 2) and resolution (6 μm) as:

From the calculated intensity distribution, the ablation profile (ablation depth [ds]) for a single laser pulse with normal incidence was calculated as explained in Reference
24, using the following relation:
Where
θ is the deviation from normal incidence (
θ = 0°).
Figure 1 presents the twodimensional and threedimensional simulated ablation profile for a single laser pulse with square and round spot geometry.

A square ablation matrix of given dimensions was defined to sequentially position several ablation profiles corresponding to a single laser pulse, simulating a larger ablation area.

Two types of lattices were implemented for sequential spot placement within the ablation matrix, reticular and triangular lattice.

The reticular lattice was realized by regularly arranging the ablation profiles within the rows and columns in the ablation matrix.

The triangular lattice was realized by shifting the ablation profiles in every consecutive row and column, such that each ablation profile had an equal radial distance to all the neighboring ablation profiles in its vicinity.

Only a single layer of ablation matrix was simulated for both the types of lattices, without superimposing one complete ablation matrix on top of another ablation matrix.

A smaller ablation matrix of given dimensions was calculated after cropping the complete ablation matrix. This was done to avoid the artifacts arising from uneven ablation spot placement at the edges of the ablation matrix.
Figure 2 compares the reticle and triangular lattice in a cropped ablation matrix.

The roughness in ablation was estimated using the metric
Where
RMS is the root mean square error, calculated as the standard deviation of the cropped ablation matrix, and Layer depth is the average ablation depth in the cropped ablation matrix.

All the steps were repeated for different values of overlap distances starting from 6 μm (equivalent to interpulse distance of 1 pixel) to 888 μm (equivalent to interpulse distance of 148 pixels [foot print of one spot]), with an increment of 6 μm. The increment of 6 μm represents the resolution limit of the simulation model.

Following similar methodology (from step 6 to 14), the roughness in ablation was calculated for a measured spot profile from an excimer laser system (1050 Hz repetition rate with spot energy 1 mJ), measured using a beam profiler camera (Coherent LasercamHRUV and Coherent BeamViewSoftware, with a laser trigger frequency of 49 Hz from a distance of 150 cm).
In all the simulations, the area of square ablation matrix was set to 2.56 mm
^{2} (with side 1.6 mm). The standard test setting with the values of various input parameters is presented in
Table 1. Most of these values were retained in all the simulations unless stated otherwise. In the tables, the spot depth and spot volume were calculated using the methods presented in Reference
25.
Table 1 Standard Test Settings: The Values of Various Input Parameters Used in the Simulation Methodology
Table 1 Standard Test Settings: The Values of Various Input Parameters Used in the Simulation Methodology
Analysis methodology included analyzing single parameters characterizing the beam profile individually, and estimating their optimum value by comparing the simulated roughness in ablation for different values of the parameters in an exploratory manner. A proper optimum configuration was found for the given input parameters, for minimizing the roughness in ablation, under the assumption that the combination of all optima also represents an optimum. The simulation program was developed in SCILAB (Scilab Enterprises, Versailles, France), and the analysis was performed using Microsoft Excel (Microsoft Corporation, Redmond, WA, USA). The optimum condition for the following parameters characterizing the beam profile was explored.
Optimum Truncation Size
Roughness in ablation was calculated at different overlap distances, for a super Gaussian beam profile (
N = 1) with three truncation fractions (100%, 70%, 50%). The truncation fraction is hereby determined by the ratio,
The complete beam profile is represented with a truncation fraction of 100%. Smaller truncation fractions represent more cutting off the flanks of the beam profile. Different truncation fractions were realized by changing the values of
R_{0} while keeping the truncation diameter (PH) constant. These test settings are summarized in
Table 2. Other input parameters including the spot energy were kept constant as presented in
Table 1.
Table 2 Optimum Truncation Size: Test Settings for Finding the Optimum Truncation Size for a Lower Simulated Roughness in Ablation
Table 2 Optimum Truncation Size: Test Settings for Finding the Optimum Truncation Size for a Lower Simulated Roughness in Ablation
Optimum Super Gaussian Order
Roughness in ablation was calculated at different overlap distances, for three super Gaussian beam profiles (
N = 1,
N = 2, Flat Top). For the three beam profiles, spot energy, truncation diameter, and other parameters were kept constant as presented in
Table 1; however, some parameters changed as shown in
Table 3.
Table 3 Optimum Super Gaussian Order: Test Settings for Finding the Optimum Super Gaussian Order (N) for a Lower Simulated Roughness in Ablation
Table 3 Optimum Super Gaussian Order: Test Settings for Finding the Optimum Super Gaussian Order (N) for a Lower Simulated Roughness in Ablation
Optimum Spot Geometry
Roughness in ablation was calculated at different overlap distances, for two spot geometries (round and square geometries). The input parameters were retained for both the spot geometries as shown in
Table 1.
Optimum Lattice Geometry
Roughness in ablation was calculated at different overlap distances, for two lattice geometries (reticular and triangular geometries). The input parameters were retained for both the lattice geometries as shown in
Table 1.
Measured Versus Theoretically Modeled Beam Profile
The roughness in ablation was calculated for a measured spot profile from an excimer laser system (as explained in the step 15 of the simulation methodology). The outcomes (roughness in ablation) of the theoretically modeled beam profile (input parameters defined in
Table 1) were compared with the outcomes of the measured beam profile.
Impact of Ablation Material
For the input parameters specific to the human cornea (in
Table 1), roughness in ablation was calculated at different overlap distances. The various simulation steps were repeated for threshold ablation fluence (67 mJ/cm
^{2}) and absorption coefficient (5.2 μm
^{−1}) specific to PMMA.
^{26} The other input parameters were retained from
Table 1. The outcomes (roughness in ablation) for the two ablation materials (human cornea and PMMA) were compared.
Impact of Spot Energy and Spot Diameter
The analysis and optimization of various parameters was performed for a fix spot energy of ∼1 mJ and spot size (presented as the truncation diameter/foot print in
Table 1) of ∼0.9 mm. This represents a realistic limit commonly seen in laser vision correction.
^{25} We simulated the impact of other extreme values of spot energy (0.5 mJ and 1.4 mJ) on the metric roughness in ablation.
Table 4 presents a comparison of the three tested energy settings.
Table 4 Optimum Spot Energy: Test Settings for Finding the Optimum Spot Energy for a Lower Simulated Roughness in Ablation
Table 4 Optimum Spot Energy: Test Settings for Finding the Optimum Spot Energy for a Lower Simulated Roughness in Ablation
Furthermore, we simulated the impact of other extreme values of spot sizes (0.6 and 1.2 mm) on the metric roughness in ablation.
Table 5 presents a comparison of the three tested spot sizes.
Table 5 Optimum Spot Diameter: Test Settings for Finding the Optimum Spot Diameter for a Lower Simulated Roughness in Ablation
Table 5 Optimum Spot Diameter: Test Settings for Finding the Optimum Spot Diameter for a Lower Simulated Roughness in Ablation
Perturbation Analysis
The values of various laser beam characteristics in
Table 1 were simultaneously and randomly varied for each ablation spot in the ablation matrix. The impact of these perturbations was evaluated on the roughness in ablation. The following parameters were varied for the perturbation analysis: spot overlap, threshold ablation fluence, absorption coefficient for human cornea, spot energy, super Gaussian order, matrix size, and beam size. The spot overlap was varied between the ranges ± 6 pixels of the original value. The rest of the parameters were varied between the ranges ± 10% of the original value. The outcomes (roughness in ablation) with the standard test setting (
Table 1) were compared to the outcomes with the perturbed input parameters.
Results
In general, the layer depth progressed as 1/overlap
^{2}, decreasing as the overlap distance increased. For the input parameters presented in
Table 1, the progression of layer depth is shown in
Figure 3. In
Figure 3 and subsequent figures, the horizontal axis (overlap [μm]) represents the overlap distance (or interspot distance) between two consecutive ablation spots in micrometers, where a lower overlap represents a tighter spot placement.
The optimum condition for the following parameters characterizing the beam profile was found.
Optimum Truncation Size
The results of the simulations are presented in
Figure 4. The complete beam profile represented with a truncation fraction of 100% resulted in the smoothest ablations. A further truncation of the beam profile increased the roughness in ablation for overlap distances less than 600 μm. These results suggest that application of a pinhole to truncate the beam profile may marginally increase the ablation efficiency but dramatically deteriorates the ablation smoothness for the entire range of overlap distances below ∼600 μm (corresponding nearly to the size of a single spot). An exception was seen at overlap distances ∼400 μm, where three profiles collapsed.
Optimum Super Gaussian Order
The results of the simulations are presented in
Figure 5. The roughness increased for larger values of
N. Gaussian profiles theoretically result in smoother ablations, particularly for overlap distances smaller than ∼620 μm, corresponding nearly to the size of a single spot.
Optimum Spot Geometry
The results of the simulations are presented in
Figure 6. Theoretically, round spot geometries produced lower roughness in ablation compared to the square geometry, particularly for overlap distances smaller than ∼600 μm, corresponding nearly to the size of a single spot.
Optimum Lattice Geometry
The results of the simulations are presented in
Figure 7. For lower overlap distances (<200 μm), triangular lattices shall theoretically produce lower roughness in ablation compared to the reticular lattice. As the overlap distances increase, a shift was seen in the corresponding peaks and valleys in the curve for triangular and reticular lattice, with triangular lattice achieving its corresponding peaks and valleys (in terms of the roughness in ablation), at larger overlap distances compared to the reticular lattice. This suggests that at higher overlap distances, triangular lattice results in lower roughness in ablation compared to the reticular lattice.
Measured Versus Theoretically Modeled Beam Profile
The simulated roughness in ablation for the theoretically modeled beam profile was compared to the measured beam profile. This comparison is presented in
Figure 8. The measured beam profile corresponded well with the theoretically modeled beam for the entire range of overlap distances; however, the modeled beam profile showed comparatively lower roughness in ablation in general.
Impact of Ablation Material
The results of the simulations are presented in
Figure 9. The simulated roughness in ablation on PMMA tends to be lower than on the human cornea, although correspondingly, lower ablation volumes are also associated with PMMA. For a window of overlap distances (90–150 μm), the correspondence between PMMA and cornea was very good in terms of the roughness in ablation (indicated with a green window in
Fig. 9). As the overlap distances increase, a shift was seen in the corresponding peaks and valleys in the curve of PMMA and cornea.
Impact of Spot Energy and Spot Diameter
The results of the simulations are presented in
Figures 10 and
11. The roughness in ablation was lower for lower energy and larger spot sizes. This suggests that for smoother flat ablations, lower spot energy, and larger spot sizes should be preferred. However, this would limit the lateral resolution that is imperative to ablate the fine shapes needed in higher order correction in refractive surgery.
Perturbation Analysis
The results of the perturbation analysis are presented in
Figure 12. With respect to the standard test settings (
Table 1), the simultaneous perturbation in the input parameters, including the spot overlap distances, represents more realistic results expected in the real world. According to the expectations, the simulated roughness in ablation with the perturbed input parameters followed a similar but very jittery progression as the standard test settings. However, unlike the standard test settings, for the overlap distances below 300 μm, the simulated roughness remained closer to 0.1 μm and did not systematically reduce to zero roughness. This suggests that in the real world settings, minimum roughness in ablation shall be achieved at an overlap distance of 300 μm, and an overlap distance below 300 μm shall not detrimentally affect the roughness in ablation.
Proper Optimum for Minimum Roughness in Ablation
For the input parameters presented in
Table 1, a proper optimum configuration for minimizing the roughness is summarized in
Table 6. The optimum overlap distances correspond to the valleys in
Figure 7 (reticular lattice); that is, 96, 120, and 132 μm. However, considering the outcomes of the perturbation analysis, any spot overlap below 300 μm would result in minimum roughness in ablation given the variability in relevant parameters observed in a realistic setting.
Table 6 Proper Optimum: A Proper Optimum Configuration for Minimizing the Roughness in Ablation, for the Input Parameters Presented in
Table 1 Table 6 Proper Optimum: A Proper Optimum Configuration for Minimizing the Roughness in Ablation, for the Input Parameters Presented in
Table 1
Discussion
The roughness in ablation after refractive surgery is related to the transmission of light in the cornea.^{27} PerezMerino et al.^{28} analyzed the relationship among transmittance, scattering, and epithelial surface properties during wound healing after refractive surgery in hens operated using different refractive surgery techniques (Lasik, Lasek, PRK). Their results suggested that higher roughness in the epitheliumstroma interface causes a decrease of transmittance and an increase of scattering. Larger differences between internal and external roughness of epithelium contributes to produce a decrease of transmittance and an increase of scattered light.
Various measurement techniques have been developed for the measurements of surface roughness.^{29} To test the effect of radiant exposure on surface smoothness, Fantes et al.^{30} ablated rabbit corneas with the 193nm argon fluoride excimer laser at nine radiant exposures from 50 to 850 mJ/cm^{2}. They showed that the uniformity of the surface following laser ablation may play an important role in the rate of epithelial healing and amount and type of stromal scarring. It has been experimentally shown that high levels of surface roughness produced by some laser systems may be sufficient to degrade visual performance under some circumstances.^{31} O'Donnell et al.^{11} showed that surface irregularities in PMMA increase with ablation depth and proposed a unit of measure of roughness expressed as the peaktovalley distance in ablation. The excimer laser interacts with the nonablated residual stromal surface in a characteristic fashion not seen with isotropic, inorganic material. McCafferty et al.^{32} postulated that the surface changes demonstrated after excimer laser ablation may be indicative of temperatureinduced transverse collagen fibril contraction and stress redistribution, or the ablation threshold of the stromal surface may be altered. This phenomenon may be of increased importance using lasers with increased thermal load.
These researches substantiate the need for achieving a smoother surface in laser ablation for vision correction, for achieving higher fidelity in the postoperative outcomes. Smoothing agents and optimized energy distribution patterns have been explored to achieve smoother surfaces after laser ablation. ArbaMosquera et al.^{33} presented the dual fluence concept for the sequencing of laser shots in corneal ablation for achieving higher fidelity and avoiding vacancies and roughness of the cornea. Lombardo et al.^{34} examined the impact of smoothing agent (0.25% sodium hyaluronate) on postoperative roughness in porcine corneas subjected to Excimer laser photorefractive keratectomy, by means of atomic force microscopy. Images of the ablated stromal surface showed undulations and granulelike features on the ablated surface of the specimens. The specimens on which the smoothing procedure was performed (rootmeansquare [RMS] rough: 0.152 ± 0.014 μm) were more regular (P < 0.001) than those on which PRK alone was performed (RMS rough: 0.229 ± 0.018 μm).
Modeling approaches have been proposed in the past to study the ablation profiles and outcomes of the refractive surgery excimer lasers.^{23,35} The predicted postoperative corneal ablation shape, ablated volume, asphericity, and spherical aberration varies across commercial laser platforms, as well as the relative contribution of ablation pattern designs and efficiency losses to the increased asphericity.^{36} We tested the laser beam characteristics to define a set of parameters characterizing the laser beam profile that can optimize the roughness in ablation. The assessment of quality of vision is now an essential aspect of postoperative assessments following refractive procedures. Quality of vision is a subjective entity and the perception of quality of vision consists of various factors. Theoretically, this model may improve the quality of results and could be directly applied for improving postoperative surface quality; however, formal assessment of subjective quality of vision through patientreported outcome questionnaires is necessary to analyze the improvements. This model is generic in nature and can be applied to any material for which the absorption coefficient and the ablation threshold for the specific wavelength and laser characteristics are known. It must be noted that the presented methods optimize each parameter independently under the assumption that the combination of all optima also represents an optimum.
Plastic models have been used in refractive surgery research and calibration for a long time, especially for the assessment of roughness and calibration of fluence.
^{8,11,37–39} However, it has not been until recently that plastic models have been used to study in detail the changes in the shape of the flat or spherical surfaces, after refractive surgery.
^{40,41} Ablating plastic model corneas not affected by biomechanical or other biological effects with clinical lasers allows to directly measure the actual ablation pattern provided by the laser, avoiding the approximations and assumptions used in theoretical models. However, plastic models are not intended to mimic the response of the cornea but rather used to characterize the laser systems. The differences in ablation process between collagen and PMMA are well documented in the literature. Although purely based on our simulations (and not mimicking the corneal response), the differences in the surface quality and ablation depths in PMMA and human cornea were still evident in our results (
Fig. 9). This is of clinical relevance if PMMA materials are used for initial testing or calibration of the laser system preceding treatment procedures performed in humans. The results suggest a window of overlap distances (<90–150 μm), where the correspondence in terms roughness, between PMMA and cornea, was better (indicated with a green window in
Fig. 9). Additionally, unlike other groups who found more roughness for deeper ablations in plastic models,
^{11} our model is consistent with constant roughness for deeper ablations, if one could dynamically adapt the overlap to the particular ablation.
Several factors are associated with the epithelial response in refractive surgery.^{42–46} An important aspect for consideration in postoperative refractive outcomes is the epithelial masking that will finish the smoothing process after the corneal ablation.^{47,48} An optimum topography of the stroma facilitates reepithelialisation.^{49,50} Additionally, it is known that stromal topography affects overlying epithelial function including the differential expression of both cellular and extracellular substances.^{51} Attempts have been made in the past to develop mathematical models used as the basis to design ablation patterns that compensate in advance for the expected corneal surface smoothing response.^{52–54} However, the mathematical model presented here does not take the above mentioned aspects into account in order to determine the final effect upon the overall refracting surface and level of corneal clarity. An investigation in corneal remodeling should be considered to further explore the benefits of the presented mathematical approach, closer to the real world conditions.
Nevertheless, based on the presented theoretical results, it can be foreseen that smoother surfaces would be achieved immediately after the ablation, using the proposed laser beam characteristics. This means at least two related potential advantages: (1) shortterm outcomes may be better in the time period where the epithelium remodeling/smoothing/masking takes place, and (2) time for surface recovery may be shorter—since the surface is smoother to start with, epithelium may need less remodeling, which means less time for remodeling. Some other advantages of this model can be speculated or at least subjected to clinical assessment, namely, improving the smoothness seems a no risk condition, improvement in shortterm outcomes (without compromising longterm ones), shorter recovery time to reach final visual acuity goal, higher levels of final visual acuity, shorter reepithelization time, reduced levels of induced higher order aberrations, and less haze response.
The surface roughness is influenced by the formation of random or almost periodic holes with a depth of several micrometers. For high quality surface ablation, the formation of periodic structures and random holes should be avoided. This can be achieved by orienting the scan direction perpendicular to the polarization.
^{55} In order to minimize the surface roughness, Neuenschwander et al.
^{56} suggested that the optimum ratio between pulse distance and spot radius should be >1.0, and the ratio between the line distance and spot radius should be ≈ 0.5. Domke et al.
^{55} explored the optimal combination of pulsetopulse distance and fluence in order to minimize the surface roughness for the ablation of silicon, irradiated using an ultrafast femtosecond laser. They concluded that the maximum specific ablation rate was achieved at fluence of approximately 2 J/cm
^{2}. At a fluence of 2.8 J/cm
^{2}, the global minimum of the surface roughness was determined to be approximately 220 nm at pulse distance = 0.67* spot radius. The influence of the furrows on the surface roughness seemed be negligible at this fluence. Their results suggest that the optimal pulse distance increases with fluence.
^{42} The local ablation frequency and spot overlap has also shown to affect the surface roughness in PMMA.
^{57,58} Bende et al. (Bende T, et al.
IOVS 2003;44:ARVO EAbstract 2660) used a 1.0mm Gaussian beam flying spot excimer laser to study the impact of spot overlap and ablation frequency on surface roughness in PMMA flat ablations (like PTK). They found that the surface roughness varies as a function of ablation depth, where the surface roughness for a PTK ablation in PMMA plates varied between 0.26 and 0.49 μm for a 50μm deep ablation and 0.65 to 1.12 μm for a 250μm deep ablation. In PMMA, the minimal surface roughness was found for an overlap of 72.5%. Although we simulated only sequential spot placement in our methods, we also used a similarly defined spot (∼0.9 mm spot diameter Gaussian spot). Their findings for optimum overlap correspond well with the optimum 300 μm overlap distance we found with the perturbation analysis. Similarly, they found that increased fluence was associated with decreased surface roughness as well as decreased surface waviness, as suggested in our results for the optimum spot energy (
Fig. 10).
There is a delicate balance between the spot energy and spot diameter in terms of the roughness in ablation. Correcting the higherorder aberrations of the eye requires lasers with smaller spots and finer resolution.^{59} It has been shown that a tophat laser beam of 1.0 mm (Gaussian with fullwidth half maximum of 0.76 mm) is small enough to produce custom ablation for typical human eyes.^{60} Our analysis of the optimum spot energy and spot sizes suggested that for smoother flat ablations, lower spot energy and larger spot sizes should be preferred. However, for achieving the limit of lateral resolution to ablate the fine shapes needed in higher order correction in refractive surgery, a compromise must be made between ablation smoothness and spot characteristics (including spot geometry and energy). This compromise is reflected in our standard test settings (Spot Energy = 0.95 mJ, R_{0} = 0.425 mm).
Several units for measuring the roughness in ablation have been proposed.
^{11} In our analysis, we used the RMS per square root of layer depth to define roughness in ablation due to the stability observed for the entire range of overlap distances compared to other metrics such as RMS and RMS/LayerDepth (
Fig. 13). Further to the stability for the entire range of overlap distances, we consider that the variance could well be linear with ablation depth (Layer Depth), so RMS would be linear with sqrt(Layer Depth). In the light of the nonnormalized RMS roughness, and provided that different overlaps produce different layer depths, there would exist an optimum overlap for different depths. Therefore, our methods can be implied to optimize the overlap distances in the laser systems used in refractive surgery based on (1) the typical depth of refractive surgery treatments, (2) dynamically adapted to the particular maximum depth in every refractive surgery treatment, and (3) dynamically adapted to the local depth of each position in every refractive surgery treatment.
The oscillating nature observed in our results can be explained with an analogy to the Gauss's circle problem for the reticular lattice. The solution of the Gauss's circle problem determines the number of lattice nodes inside the boundary of a circle with center at the origin. Assuming that the ablation matrix is based on an equidistant grid, where a circle defines the boundary of one ablative spot and the center of this circle represents a lattice node on this grid (like
Fig. 2, top left). The overlap distance can be assumed as the distance between two consecutive nodes (i.e., the centers of two such circles in the ablation matrix). When this distance is too small, there would be larger number of overlapping pulses (larger number of lattice nodes within the perimeter of a single spot). As this distance increases, the number of overlapping pulses will decrease. For a laser ablation process, this also affects the layer depth (
Fig. 3).
Correspondingly, the increment/decrement of lattice nodes within the perimeter of a single spot occurs in discrete steps (and for a reticular lattice this is a multiple of four, for a triangular lattice a multiple of six). Therefore, for some overlap distances, some lattice nodes appear exactly at the boundary of a circle, increasing the resulting roughness (RMS) to its peak. As this lattice node comes within the circle, the roughness gradually decreases to its minimum, and again peaks as the next group of lattice nodes appears at the boundary of the circle. The frequency of this oscillation depends on the overlap distance, changing rapidly (smaller period) for smaller overlap distances and slowly (larger period) for larger overlap distances. This suggests that the amplitude of RMS in the ablation matrix shall remain constant with increasing overlap distances; however, the frequency would change. The RMS in ablation obtained in our simulations (
Fig. 13) shows good correspondence with this model, with oscillations of increasing frequency and barely increasing amplitude (0–0.2 μm) seen as the overlap distance increases. The other metrics (RMS/sqrt[Layer depth] and RMS/Layer depth), however, increase in amplitude (but follow similar frequency as the RMS) as the layer depth decreases with increasing overlap distances. Furthermore, this analogous model may also explain the abrupt changes seen at some overlap distances in our results (
Figs. 4,
5), as the discretization due to the overlap distance affects the roughness (amplitude and frequency) differently for different beam profiles (due to truncation, super Gaussian order). A similar model can be designed to explain the results with the triangular lattice.
There are few limitations associated with our methods. The simulation methods were developed for a normal incidence but did not include the angular dependence of ablation efficiency and the increase in corneal asphericity due to ablation. The impact of beam characteristics on the surface roughness and ablation efficiency was analyzed considering a flat ablation surface. In addition, only a single ablation layer was simulated in our methods without superimposing one complete ablation layer on top of another ablation layer. There might be an improvement in terms of the roughness in ablation for some other multilayer lattice geometries, which shift every layer in order to optimize the coverage of unablated regions (like a “5 in a dice” pattern), however, at the cost of increasing the layer depth. Further exploration is needed to test the impact of other spot sequences (flying spot), lattice geometries (radial lattices), and ablation depth on roughness in ablation.
It should be also noted that in the real case, having the theoretical optimum does not suffice to reach the optimum real world performance since there are several factors like spot positioning errors, energy fluctuations,
^{61} drifts of the divergence, and thermal drifts involved. We have simulated and evaluated the impact of these factors under the perturbation analysis. A wide range (±10%) was chosen for changing the input parameters of the standard test setting (
Table 1), allowing testing the methods under extreme conditions of variability/technical error. The result of this analysis shows an optimum that lies close to the real world conditions. Therefore, concerning the spot positioning errors, the theoretical results should be modified to account for these fluctuations; for example, a local minimum of the roughness shall be taken, such that the typical spot positioning errors would not be much detrimental to roughness. Alternatively, for the input standard test setting, any spot overlap below 300 μm can be considered as optimum, based on the outcomes of the perturbation analysis.
The results show that the beam characteristics used in a corneal laser procedure has a major impact on the surface quality. Low overlap distances result in higher surface roughness compared to high overlap distances. Deeper analysis and knowledge regarding the influence of spatial laser spot distribution on the expected clinical outcomes is essential for designing safer refractive procedures with higher fidelity. From the simulations, a theoretical proper optimum configuration for minimizing the roughness in ablation for defined input parameters (
Table 1) has been found—specifically, round spot geometry; no spot truncation (truncation fraction = 100%); super Gaussian order
N = 1; and triangular lattice, with overlap distances corresponding to the valleys in
Figure 7, that is, 96, 120, and 132 μm. The obtained theoretical results should be modified to account for the fluctuations seen in the real world. For the input standard test setting, an overlap distance below 300 μm shall not detrimentally affect the roughness in ablation, given the perturbations observed in the real world.
Acknowledgments
Disclosure: S. Verma, SCHWIND EyeTechSolutions (E); J. Hesser, None; S. ArbaMosquera, SCHWIND EyeTechSolutions (E)
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