**Purpose**:
Controversial opinions exist regarding optimum laser beam characteristics for achieving smoother ablations in laser-based vision correction. The purpose of the study was to outline a rigorous simulation model for simulating shot-by-shot 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 Lambert-Beer 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 root-mean-square 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.

^{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}

^{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}

^{7}In a study, ablations were performed on polymethylmethacrylate (PMMA) plates, with four different excimer lasers: VISX-Star, Coherent Schwind Keratom I/II, Chiron Technolas Keracor 117C, and the Nidek EC-5000, 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 (Zeiss-Meditec MEL-70, and a Russian-made unit, Microscan) and two Nidek models with scanning slit delivery systems and an expanding iris diaphragm (EC-5000 and EC-5000 CX). The smoothest surface was obtained in samples produced by Zeiss-Meditec MEL-70 unit (root mean square [RMS] = 112 ± 23 nm), followed by the Nidek EC-5000 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 EC-5000 unit.

^{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.

^{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 Lambert-Beer 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.

^{25}Extending this model, the main purpose of this paper is to describe a rigorous and generic simulation model for simulating the sequential shot-by-shot 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.

- 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*(calculated in previous steps) was kept constant at each meridian. For the square spot geometry, the value of_{0}*R*was meridionally changed to match the size of the square matrix resulting in a Gaussian profile with a square base._{0} - A normalized intensity distribution of the super Gaussian beam profile was calculated using the standard form of Lambert-Beer law at each pixel as
^{24}: Where I represents the intensity calculated for each pixel position defined by the coordinates*n*and*m*,*I*represents the peak radiant exposure (normalized to_{0}*I*= 1), and_{0}*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 two-dimensional and three-dimensional 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.
- 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 Lasercam-HR-UV and Coherent BeamView-Software, with a laser trigger frequency of 49 Hz from a distance of 150 cm).

^{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.

*R*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.

_{0}*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.

^{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.

^{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.

^{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.

*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.

^{27}Perez-Merino 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 epithelium-stroma 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.

^{29}To test the effect of radiant exposure on surface smoothness, Fantes et al.

^{30}ablated rabbit corneas with the 193-nm 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 peak-to-valley 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 temperature-induced 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.

^{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 granule-like features on the ablated surface of the specimens. The specimens on which the smoothing procedure was performed (root-mean-square [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).

^{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 patient-reported 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.

^{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.

^{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 re-epithelialisation.

^{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.

^{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 pulse-to-pulse 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 E-Abstract 2660) used a 1.0-mm 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 50-μm deep ablation and 0.65 to 1.12 μm for a 250-μm 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).

^{59}It has been shown that a top-hat laser beam of 1.0 mm (Gaussian with full-width 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*= 0.425 mm).

_{0}^{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.

^{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.

*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.

**S. Verma**, SCHWIND Eye-Tech-Solutions (E);

**J. Hesser**, None;

**S. Arba-Mosquera**, SCHWIND Eye-Tech-Solutions (E)

*J Biomed Opt*. 2015; 20: 78001.

*J Refract Surg*. 2006; 22: S969–S972.

*Appl Opt*. 2006; 45: 5118–5131.

*Lasers Med Sci*. 2004; 19: 112–118.

*Lasers Med Sci*. 2003; 18: 112–118.

*J Cataract Refract Surg*. 2014; 40: 477–484.

*J Refract Surg*. 2001; 17: 43–45.

*J Refract Surg*. 2004; 20 (suppl 5): S730–S733.

*J Cataract Refract Surg*. 2009; 35: 363–373.

*J Refract Surg*. 2003; 19: 623–628.

*J Refract Surg*. 1996; 12: 171–174.

*Opt Express*. 2006; 14: 5411–5417.

*Opt Express*. 2011; 19: 4231–4241.

*Lasers Surg Med*. 2008; 40: 483–493.

*J Opt Soc Am A Opt Image Sci Vis*. 2007; 24: 265–277.

*Invest Ophthalmol Vis Sci*. 2011; 52: 3713–3722.

*Ophthalmic Surg Lasers Imaging*. 2004; 35: 41–51.

*Lasers Surg Med*. 1994; 15: 107–111.

*Opt Express*. 2007; 15: 7243–7244.

*Appl Opt*. 2005; 44: 4528–4532.

*Opt Lett*. 2010; 35: 1789–1791.

*Appl Opt*. 2008; 47: 5354–5357.

*J Opt Soc Am A Opt Image Sci Vis*. 2004; 21: 926–936.

*Opt Express*. 2008; 16: 3877–3895.

*Biomed Opt Express*. 2013; 4: 1422–1433.

*Opt Express*. 2008; 16: 20955–20967.

*Optom Vis Sci*. 2010; 87: E469–E474.

*Invest Ophthalmol Vis Sci*. 2008; 49: 3910.

*Modern Tribology Handbook: Principles of Tribology*. Vol. 1. Boca Raton, FL: CRC Press; 2001: 49–114.

*Lasers Surg Med*. 1989; 9: 533–542.

*J Refract Surg*. 2005; 21: 260–268.

*Invest Ophthalmol Vis Sci*. 2012; 53: 1296–1305.

*Advances in Optical Technologies*. 2010; 2010: 538541.

*J Refract Surg*. 2005; 21: 469–475.

*Opt Express*. 2008; 16: 11808–11821.

*Opt Express*. 2009; 17: 15292–152307.

*J Refract Surg*. 1996; 12: 401–411.

*J Cataract Refract Surg*. 2004; 30: 2536–2542.

*J Refract Surg*. 2001; 17: 43–45.

*Opt Express*. 2007; 15: 7243–7244.

*J Cataract Refract Surg*. 1997; 23: 1042–1050.

*Semin Ophthalmol*. 1998; 13: 79–82.

*Arch Ophthalmol*. 2001; 119: 889–896.

*J Cataract Refract Surg*. 2005; 31: 48–60.

*Br J Ophthalmol*. 2001; 85: 345–349.

*Br J Ophthalmol*. 2008; 92: 1397–1402.

*Eye (Lond)*. 2005; 19: 584–588.

*J Refract Surg*. 2015; 31: 281–282.

*J Refract Surg*. 2014; 30: 202–207.

*Refract Corneal Surg*. 1992; 8: 54–59; discussion 60.

*Corneal Mean Curvature Mapping: Applications in Laser Refractive Surgery*[thesis]. Columbus, OH: The Ohio State University; 2004.

*Am J Ophthalmol*. 2003; 135: 267–278.

*J Refract Surg*. 2000; 16: 177–186.

*J Laser Micro/Nanoengineering*. 2016; 11: 100–103.

*29th International Congress on Applications of Lasers & Electro Optics*. Laser Institute of America; 2010: 707.

*J Manufacturing Processes*. 2012; 14: 425–434.

*Proc SPIE Int Soc Opt Eng*. 2005; 5687.

*J Refract Surg*. 2001; 17: S588–S591.

*J Refract Surg*. 2003; 19: 15–23.

*Opt Lett*. 2005; 30: 1336–1338.