A general method to analyze the loss of ablation efficiency at non-normal incidence in a geometrical way is provided. The model is comprehensive and directly considers curvature, system geometry, applied correction, and astigmatism as model parameters, and indirectly laser beam characteristics and ablative spot properties. The model replaces the direct dependency on the fluence by a direct dependence on the nominal spot volume and on considerations about the area illuminated by the beam, reducing the analysis to pure geometry of impact. Compensation of the loss of ablation efficiency at non-normal incidence can be made at relatively low cost and would directly improve the quality of results.
© 2008 Optical Society of America
Since the introduction of laser refractive surgery, technology has evolved significantly. Today’s technology uses sophisticated algorithms, optimized tools in the planning, and proposes the challenge of improving surgery outcomes in terms of visual acuity and night vision. At the same time, patients have a better understanding and are better informed with regard to the potential of laser refractive surgery, raising quality requirements demanded by patients.
The available methods allow for the correction of refractive defects such as myopia , hyperopia , or astigmatism . One of the unintended effects induced by laser surgery is the induction of spherical aberration , which causes halos and reduced contrast sensitivity. The loss of ablation efficiency at non-normal incidence can explain, in part, many of these unwanted effects, such as induction of spherical aberrations or high order astigmatism and consequently the extreme oblateness of postoperative corneas after myopic surgery .
Probably the earliest references related to the loss of ablation efficiency in laser refractive surgery refer to the observation of hyperopic postoperative refractions (hyperopic shifts) after negative cylinder ablation of the cornea . This hyperopic postoperative refraction had not been planned and depended on various factors, such as the laser system used, the amount of negative cylinder corrected, or the presence or absence of spherical terms in the ablation profile.
For the surgeons, it was difficult to adequately compensate this effect in their nomograms in order to achieve the desired refractive correction. According to some surgeons, some manufacturers introduced the concept of coupling factor, defined as the average sphere resulting from the application of one diopter of negative cylinder. Despite its empirical nature, this coupling factor enabled surgeons to plan their treatments with a reasonable degree of success.
One clue that relates this coupling factor to the loss of efficiency is analysis of the effect in the correction of simple negative astigmatisms. As shown in Fig. 1, these cases revealed that the neutral axis became refractive, being less ablated in the periphery as compared to the center. Similar experiences [7,8] were observed using phototherapeutic keratectomy (PTK), but the results were not as conclusive, as cases where PTK is performed in large diameters are rare.
1.1 The simple model 
The simple model of ablation efficiency due to non-normal incidence is based on several assumptions, including that the cornea can be shaped spherically, that the energy profile of the beam is flat, or that reflection losses are negligible. Also, the spot overlap is not considered. The efficiency is defined as the ratio between the depth of impact on each point and the nominal impact depth (at normal incidence),
where Eff is the ablation efficiency at a radial distance r of the optical axis of the ablation center, and d is the depth of the impact at a radial distance r of the optical axis of the ablation.
This model provides an analytical expression for an adjustment factor to be used in photorefractive treatments that includes both compensation for reflection and for geometric distortion. Lateron, the authors refined the model by incorporation of non-linear deviations with regard to Lambert-Beer’s law.
1.3 The model by Dorronsoro-Cano-Merayo-Marcos 
This model provides a new approach to the problem. Flat and spherical polymethylmetacrylate (PMMA) substrates were ablated with a commercial excimer laser system. The relationship between the profiles obtained in spherical and flat sheets of PMMA was used for estimating the ablation efficiency depending on the distance from the optical axis of the lens. The predicted changes in efficiency were reasonably well correlated with the changes in asphericity and spherical aberration observed clinically using the same laser system, so that a correction factor valid for a given algorithm and a laser in particular could be derived.
2. Materials and methods
2.1 Calculation of the depth per shot
Corneal remodeling is essentially similar to any other form of micro-machining. The lasers used in micro-machining are normally pulsed excimer lasers, where the time length of the pulses is very short compared to the time period between the pulses. Although the pulses contain little energy, given the small size of the beams, energy density can be high for this reason; and given the short pulse duration, the peak power provided can be high.
Many parameters have to be considered in designing an efficient laser ablation. One is the selection of the appropriate wavelength (193.3 ± 0.8 nm for ArF) with optimum depth of absorption in tissue, which results in a high-energy deposition in a small volume for a speedy and complete ablation. The second parameter is a short pulse duration to maximize peak power and minimize thermal conductivity to the adjacent tissue (ArF excimer based τ < 20 ns).
The radiant exposure is a measure of the density of energy that governs the amount of corneal tissue removed by a single pulse. In excimer laser refractive surgery, this energy density must exceed 40–50 mJ/cm2. The depth of a single impact relates to the fluence, and also the thermal load per pulse increases with increasing fluence. Knowing the fluence and details of the energy profile of the beam (size, profile, and symmetry), we can estimate the depth, diameter and volume of the ablation impact. Assuming a super-Gaussian beam energy profile, the following equation applies:
where I is the radiant exposure at a radial distance r of the axis of the laser beam, I0 is the peak radiant exposure (at the axis of the laser beam), R0 is the beam size when the radiant exposure falls to 1/e2 its peak value, and N is the super-Gaussian order of the beam profile (where N = 1 represents a pure Gaussian beam profile, and N→∞ represents a flat-top beam profile).
Applying Lambert-Beer’s law (blow-off model), the footprint (diameter) of the impact is:
where FP is the footprint (diameter) of the ablative spot and ITh is the ablation threshold for radiant exposure for the irradiated tissue or material below which no ablation occurs.
From these data (and the beam symmetry: square, hexagonal, circular), we can calculate the volume of ablation impact:
where VS is the volume of a single spot, and α the absorption coefficient of the irradiated tissue or material.
If the profile is symmetry square:
If the profile is symmetry square with rounded corners:
For human corneal tissue, the ablation threshold takes values of about 40-50 mJ/cm2 [13,14], and the absorption coefficient is about 3.33–3.99 µm−1 [13,14]. We chose values of 46 mJ/cm2 for the ablation threshold and 3.49 µm−1 as absorption coefficient of the human corneal tissue. For PMMA, the ablation threshold takes values of about 70–80 mJ/cm2 , and the absorption coefficient is about 3.7–4.4 µm−1 . We chose values of 76 mJ/cm2 for the ablation threshold and 4.0 µm−1 as absorption coefficient for PMMA. Calculating the volume of a single spot for the cornea, and dividing it by the volume of a single impact on PMMA, we get the so-called “cornea-to-PMMA-ratio”.
Another method to calculate the volume of a single impact is direct simulation. The volume of a PTK ablation corresponds to that of a truncated cone limited by the optical zone and the ablation zone (OZ and TZ) and by the depth of ablation, whereas the theoretical volume per pulse corresponds to the ablation volume divided by the number of pulses:
The volume of a PTK ablation without transitional zone corresponds to a cylinder of the ablation depth, limited by the optical zone:
2.2 Determination of the ablation efficiency at non-normal incidence
The introduction of the concept of aberration-free profiles made it necessary to compensate for the induction of aberrations originating from deterministic and repeatable causes, thus minimizing the induction of aberrations to noise levels, so that a “new” model had to be developed. The aim in developing this model was to understand the mechanisms that govern the loss of ablation efficiency and to be able to predict their effect under different working conditions.
1 .- Considering the preoperative corneal curvature and asphericity as well as the intended refractive correction, the radius of curvature and asphericity the cornea will have after 50% of the treatment are estimated. (As the radius of corneal curvature changes during treatment, the efficiency also varies over treatment. The value at 50% of the treatment was chosen as a compromise to consider both the correction applied and the preoperative curvature).
2 .- Considering the offset of the galvoscanners’ neutral position compared to the system axis, the angle of incidence of the beam onto a flat surface perpendicular to the axis of the laser is calculated:
where α is the angle of incidence on a “flat” surface, XG, YG the position of the galvoscanners, x, y the radial positions of the incident beam, and dG the vertical distance from the last galvoscanner to the central point of the ablation.
3 .- Considering the calculated curvature and asphericity at 50% of the treatment, the local angle of the cornea is calculated. Assuming the cornea as an ellipsoid, which satisfies Baker’s equation , the following equation results:
where θ is the angle of the local tilt of a corneal location, and RHT and QHT are the predicted radius of curvature and asphericity quotient at 50% of treatment progress.
4 .- To calculate the angle of incidence for each point on the corneal surface
applies, where β is angle of incidence.
5 .- The ablation efficiency is calculated by consideration of geometric distortions, reflections losses, and spot overlapping:
where the factor corresponds to the geometric distortions, the factor (1- R(x, y)) corresponds to the reflections losses, and y is the radial direction along which angular projection occurs.
The sums represent the overlap and extent along the size of the impact.
where Δx0 and Δy0 are the spot overlapping distances (i.e. the distance between two adjacent pulses) and x0,i,j and y0,i,j are the respective centers of the different spots contributing to the overlap at one corneal location.
If the galvoscanners are coaxial with the laser system:
If the distance from the last mirror to the ablation plane is large:
Eq. (16) further simplifies to:
If the spot overlapping is very tight and many pulses contribute to the ablation at each corneal location (i.e. Δx 0,Δy 0≪FP), Eq. 22 further simplifies to:
In this way, we removed the direct dependency on the fluence and replaced it by a direct dependence on the nominal spot volume and on considerations about the area illuminated by the beam, reducing the analysis to pure geometry of impact.
There are two opposing effects: the beam is compressed due to reflection and at the same time expands due to its projection angle.
6.- The compensation would be the inverse of efficiency:
7 .- We can develop the efficiency (or the compensation) in power series:
Therefore, instead of using the radius at half of the treatment, we can calculate the overall effect of the variation in efficiency over treatment.
8 .- Returning to the concept of radius and asphericity at half of the treatment, we can further simplify the model by defining an averaged spot depth as if the energy profile of the beam were flat and apply the simple model:
If this model is applied to a spherical surface (Q = 0) and the depth per layer equals the depth per pulse or the spots do not overlap, it simplifies to the simple model.
9 .- Losses due to reflection are generally negligible. This is so because the highest reflection contribution occurs for normal incidence, as well, and this component is already renormalized in Eq. (1) and (15). Therefore, we can further simplify the model as below:
10 .- In the case of a strong astigmatic component, we can continue to calculate the 50% of treatment:
As the radius of corneal curvature changes during treatment, the efficiency varies over treatment, as well. This change is shown in Fig. 6 for both corneal and PMMA ablations. The graph demonstrates that the ablation efficiency decreases steadily with increasing curvature, thus resulting in improvement of ablation efficiency during myopic corrections and increasing loss of ablation efficiency during hyperopic corrections.
The model considers curvature based upon radius and asphericity, the effect of the asphericity quotient is shown in Fig. 7. As expected, a parabolic surface provides higher peripheral ablation efficiency (due to prolate peripheral flattening) compared to an oblate surface (with peripheral steepening).
The model considers efficiency losses due to reflection losses, geometric distortions, and spot overlapping. Ablation efficiency effects due to reflection losses and geometric distortions are shown in Fig. 8. Note that the reflection losses already exist for normal incidence and decrease by a very small amounts towards the periphery. Although normal reflection losses approximately amount to 5%, they do not increase excessively for non-normal incidence. As our calculation defined ablation efficiency for a general incidence as the ratio between the spot volume for general incidence and the spot volume for normal incidence, it is evident that the so-defined efficiency equals 1 for normal incidences.
We removed the direct dependency on the fluence and replaced it by a direct dependence on the nominal spot volume and on considerations about the area illuminated by the beam, thus reducing the analysis to pure geometry of impact. However, the influence of radiant exposure can be seen in Fig. 9. Note that efficiency is very poor close to the ablation threshold and steadily increases with increasing radiant exposure approaching 100% ablation efficiency. It should also be noted that the difference between efficiencies for cornea and PMMA increases with lowering radiant exposure.
Finally, Fig. 10 to 12 compare the ablation efficiencies for cornea and PMMA obtained with the proposed model using the values reported in the study of Dorronsoro et al. In Fig. 10, the ablation efficiency for the spherical shapes prior to receiving any laser shot was evaluated, whereas Fig. 11 and 12 evaluate the average ablation efficiencies for the surfaces during a -12 D and a +6 D correction, respectively. Again, note that the ablation efficiency decreases steadily with increasing curvature, resulting in an improvement of ablation efficiency during the − 12 D correction as opposed to an increased loss of ablation efficiency during the +6 D correction.
The loss of efficiency is an effect that should be offset in commercial laser systems using sophisticated algorithms that cover most of the possible variables. However, an unambiguous definition of the meaning of an optimal ablation profile for corneal refractive surgery is not available to date. To try to compensate for existing patients’ aberrations, customized treatments were created. This customization can be achieved based on the wavefront aberration of the eye  (using Hartmann-Shack systems, for example) or by estimating the wavefront aberration of the cornea using topographic data [18,19]. Treatments guided by topography , guided by aberrometry , “wavefront-optimized” treatments , or treatments guided by asphericity  have all been proposed as solutions to the problem. However, considerations such as duration of treatment, amount of tissue removed , tissue remodeling, or the controversial results obtained so far difficult the selection of a unique type of profile. Parallelly to the clinical developments, increasingly capable, reliable, and safer laser systems with better resolution and accuracy are required.
This study provides an analytical expression for calculation of the ablation efficiency at non-normal incidence. The method results in a geometrical analysis of the volume per shot and of the area illuminated by the beam. The model directly considers curvature, toricity, asphericity, applied correction including astigmatism and system geometry as model parameters, and indirectly laser beam characteristics and ablative spot properties. Separate analysis of the effect of each parameter was performed.
Our approach reduces all calculations to geometrical analysis of the impact, the ablation efficiency does not primarily depend on the radiant exposure, but rather on the volume per single shot for the specific material and also on overlap and geometric considerations of the irradiated area per shot, supported by radiant exposure data. Different effects interact, the beam is compressed due to the loss of efficiency, but at the same time expands due to the angular “projection”. Using this model for ablation efficiency at non-normal incidence in refractive surgery, up to 42% of the reported increase in spherical aberrations can be explained.
Applying this comprehensive loss of efficiency model to a pure myopia profile in order to get the achieved profile etched into the cornea, we observed that the profile “shrinks”, steepening the average slope and then slightly increasing the myopic power of the profile as well as inducing spherical aberrations. The net effect can be expressed as an unintended positive spherical aberration and a small overcorrection of the spherical component. Applying this model to a pure hyperopia profile, we observed that the profile “softens”, flattening the average slope and then decreasing the hyperopic power of the profile as well as inducing spherical aberrations. The net effect can be expressed as an undercorrection of the spherical component and a small amount of induced negative spherical aberration. Applying this model to a PTK profile, we observed that the flat profile becomes myopic due to the loss of efficiency, resulting in an unintended myopic ablation (hyperopic shift).
Corneal curvature and applied correction play an important role in the determination of the ablation efficiency and are taken into account for accurate results. As a compromise between accuracy and simplicity, we decided to use the predicted radius of corneal curvature after 50% of the treatment as curvature metric for determination the ablation efficiency. However, corneal toricity and applied astigmatism, even though easily computed using the comprehensive model, do not have a relevant impact as long as their values correspond to those of normal corneas. Only when toricity or astigmatism exceed 3 D, their effects on ablation efficiency start to be significant.
System geometry is considered in this model using the offset of the galvoscanners’ neutral position compared to the system axis as well as the distance from the last galvoscanner to the central point of the ablation. Nevertheless, usually the galvoscanners are coaxial with (or determine the axis of) the laser system, and the distance from the last galvo-mirror to the ablation plane used to be large. Both conditions further simplify Eq. (16) to Eq. (22) explained in section 2 of this article.
We removed the direct dependency on the fluence and replaced it by a direct dependence on the nominal spot volume and on considerations about the area illuminated by the beam, reducing the analysis to pure geometry of impact in this way. The influence of the radiant exposure is shown in Fig. 9. We found that the efficiency is very poor close to the ablation threshold and steadily increases with increasing radiant exposure approaching 100% ablation efficiency. Also, differences between the efficiencies for the cornea and PMMA were observed to increase with lowering radiant exposure. Actually, the key factor is not the peak radiant exposure of the beam, but rather the average spot depth (i.e. the ratio spot volume to spot area) (Eq. (23) and (33)).
The detailed model determines ablation efficiency considering geometric distortions, reflections losses, and spot overlapping. Geometric distortions are very important, because the angular projection expands the beam, thus spreading the beam energy over a wider area and flattening its radiant exposure. At the same time, spot overlapping is a major parameter, especially in flying-spot systems, where the spot spacing is small compared to the spot width and multiple spots overlap, all contributing to the ablation at each corneal location, whereas reflection losses can be neglected, because important reflection contribution already occurs in normal incidence and does not excessively increase in non-normal incidence. Based on these facts, further simplifications are possible (Eq. (22) to Eq. (35) in section 2 of this article).
Surface asphericity before ablation, and especially after completion of 50% of the treatment, refines this comprehensive approach. Simulations, based on cornea and PMMA, for extreme asphericity values (from asphericity quotient of -1 to +1) showed minor effects with differences in ablation efficiency of 1% in the cornea and 2% in PMMA even at distances of 4 mm radially from the axis. Hence, for corneas with normal curvature and asphericity spherical geometry seems to be a reasonably simple approach for calculating the ablation efficiency at non-normal incidence (Eq. (36) and (37)).
The loss of efficiency in the ablation and non-normal incidence are responsible for much of the induction of spherical aberrations observed in the treatments as well as the excessive oblateness of postoperative corneas observed after myopic corrections  (also part of some overcorrections observed in high myopias and many undercorrections observed in hyperopia) with major implications for treatment and optical outcome of the procedure. Compensation can be made at relatively low cost and directly affects the quality of results (after a correction of the profiles to avoid overcorrections or undercorrections in defocus and marginally in the cylinder).
Today, several approaches to import, visualize, and analyze high detailed diagnostic data of the eye (corneal or ocular wavefront data) are offered. At the same time, several systems are available to link diagnostic systems for measurement of corneal and ocular aberrations of the eye to refractive laser platforms. These systems are state-of-the-art with flying spot technology, high repetition rates, fast active eye trackers, and narrow beam profiles. As a consequence, these systems offer new and more advanced ablation capabilities, which may potentially suffer from new sources of “coupling” (different Zernike orders affecting each other with impact on the final result). The improper use of a model that overestimates or underestimates the loss of efficiency will overestimate or underestimate its compensation and will only mask the induction of aberrations under the appearance of other sources of error.
In coming years, the research and development of algorithms will continue on several fronts in the quest for zero aberration. This includes identification of sources for induction of aberrations, development and refinement of models describing the pre-, peri- and postoperative biomechanics of the cornea, development of aberration-free profiles leaving pre-existing aberrations of the eye unchanged, redevelopment of ablation profiles to compensate for symptomatic aberrated eyes in order to achieve an overall postoperative zero level of aberration (corneal or ocular). Finally, the optimal surgical technique (LASIK (Laser assisted in-situ Keratomileusis), LASEK (Laser Epithelial Keratomileusis), PRK (Photorefractive Keratectomy), Epi-LASIK …) to minimize the induction of aberrations to a noise level has not yet been determined.
Analyzing the different models available, the simple model of ablation efficiency at non-normal incidence bases its success on its simplicity, which forms the reason why it is still used by some trading houses. The problems arising from the simple model directly derive from its simplicity, and consequently the limitations of application as required by their implicit assumptions. The simple model does not consider the calculation nor the asphericity of the cornea, or the energy profile of the beam, or the overlap of impacts, overestimating the ablation efficiency, underestimating its compensation.
The Jiménez-Anera model provides an analytical expression for an adjustment factor to be used in photorefractive treatments, which includes both compensation for reflection and for geometric distortion, incorporating non-linear deviations with regard to Lambert-Beer’s law. It eliminates some problems of the simple model, because it considers the energy profile of the beam, overlapping, and losses by reflection. However, it does not consider the calculation nor asphericity of the cornea, it assumes that the energy profile is a Gaussian beam, assumes unpolarized light, it does not address the size or shape of the impact, it does not consider that the radius of curvature changes locally throughout the treatment, and accordingly the angle of incidence. Therefore, it often slightly overestimates the ablation efficiency, partially underestimating its compensation.
The Dorronsoro-Cano-Merayo-Marcos model provides a completely new approach to the problem. It eliminates many of the problems of both the simple model and the model by Jiménez-Anera, reducing the number of assumptions and using an empirical approach. Even so, it assumes that the reflection losses on cornea and PMMA are identical, it does not consider the local radius of corneal curvature, its asphericity or applied correction, it does not consider that the radius of curvature changes locally throughout the treatment, and accordingly the angle of incidence, and it does not consider the effects for different values of fluence.
The model described here eliminates the direct dependence on fluence and replaces it by direct considerations on the nominal spot volume and on the area illuminated by the beam, reducing the analysis to pure geometry of impact. The proposed model provides results essentially identical to those obtained with the model by Dorronsoro-Cano-Merayo-Marcos. Additionally, it offers an analytical expression including some parameters that were ignored (or at least not directly addressed) in previous analytical approaches. The good agreement of the proposed model with results reported in Dorronsoro’s paper - to our knowledge the first study using an empirical approach to actually measure the ablation efficiency - may indicate that the used approach including the discussed simplifications is a reasonable description of the loss of efficiency effects. In so far, this model may complement previous analytical approaches to the efficiency problem and may sustain the observations reported by Dorronsoro et al.
Even though a large number of detailed parameters are considered, this model is still characterized by a relatively low degree of complexity. In particular, the model could be further refined by incorporating non-linear deviations according to Lambert-Beer’s law or by considering local corneal curvature directly from topographical measurements rather than modeling the best-fit surface elevation.
The loss of efficiency is an effect that should be offset in commercial laser systems using sophisticated algorithms that cover most of the possible variables. Parallelly, increasingly capable, reliable, and safer laser systems with better resolution and accuracy are required. The improper use of a model that overestimates or underestimates the loss of efficiency will overestimate or underestimate its compensation and will only mask the induction of aberrations under the appearance of other sources of error.
The model introduced in this study eliminates the direct dependence on fluence and replaces it by direct considerations on the nominal spot volume and on the area illuminated by the beam, thus reducing the analysis to pure geometry of impact and providing results essentially identical to those obtained by the model by Dorronsoro-Cano-Merayo-Marcos, however, also taking into account the influence of flying spot technology, where spot spacing is small compared to the spot width and multiple spots overlap contributing to the same target point and the correction to be applied, since the corneal curvature changes during treatment, so that also the ablation efficiency varies over the treatment.
Our model provides an analytical expression for corrections of laser efficiency losses that is in good agreement with recent experimental studies, both on PMMA and corneal tissue. The model incorporates several factors that were ignored in previous analytical models and is useful in the prediction of several clinical effects reported by other authors. Furthermore, due to its analytical approach, it is valid for different laser devices used in refractive surgery.
The development of more accurate models to improve emmetropization and the correction of ocular aberrations in an important issue. We hope that this model will be an interesting and useful contribution to refractive surgery and will take us one step closer to this goal.
We thank Dr. Jesús Merayo-Lloves for his time and expertise to criticize this work.
References and links
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