Ray-mapping based calculation of the intra-corneal wavefront and corneal ablation profile for ocular aberration corrections

Yiyu Li; Rong Zhu; Wei Huang; Hao Chen

doi:10.1364/OSAC.2.000253

1. Introduction

Wavefront-guided (WG) corneal ablation [1–3] has been proposed to correct the traditional sphere and cylindrical error of the eye and to reduce the ocular high-order aberrations (HOAs) [4–6]. The corneal ablation profile (CAP) is simply the inverse of the ocular wavefront map, divided by the difference of refractive index between cornea and air within the optical zone [7,8]. The reduction in ocular HOAs is expected to increase the retinal image resolution and contrast. However, patients undergoing this treatment may still have more HOAs postoperatively than they did preoperatively [9,10]. Several studies, which compared the results of WG ablation and the standard spherocylindrical treatment, do not support the superiority of WG treatments in terms of visual quality [11,12]. Most of the population who have a low magnitude of HOAs would probably not benefit from a WG treatment.

Several factors could lead to the induction of HOAs in a WG treatment, such as the precision of wavefront measurement, the decentration ablation [13], the loss in ablation energy due to the inclined incidence in the periphery [14]. Up to now, the accuracy of the WG algorithm which is responsible for the conversion from the ocular wavefront to the tissue ablation has been received little attention. The current used WG based algorithm assumes that the spatial distance from the exit pupil, which is the reference plane to measure the ocular aberration, to the corneal surface can be neglected. Hence, an approximate point-to-point coordinate mapping is used in the determination of the local tissue ablation for the phase compensation to correct the local wavefront error, without any consideration of the actual path of rays changed by the refraction on corneal surface. Since the HOAs of an optical system are related to the ray height and incident angle of the light rays according to the aberration theory, this approximation in GW based algorithm may be the source of the significant levels of HOAs even if the treatment is not affected by the other related factors.

In this paper, we address a novel ray-mapping (RM) based algorithm for the calculation of CAP, using the numerical simulation based on three-dimensional ray tracing. The measured ocular wavefront and the elevation data of the anterior corneal surface (ACS) are used in a reverse ray tracing process to acquire the intra-corneal wavefront. The desired postoperative ACS is obtained by solving the ray mapping problem between the intra-corneal wavefront and the target ocular wavefront. By comparing with the conventional WG based algorithm in aberration correction simulation, RM based algorithm shows a much higher accuracy both in the aberration-free correction and the aberration selective correction.

2. Intra-corneal wavefront

The ocular wavefront W_m shown in Fig. 1(a) is generally measured at the exit pupil plane of eye by aberrometers such as Hartmann-Shack wavefront sensor [15]. This wavefront originates from a focus point of illumination on the retina and has underwent the refractive system of the whole eye, so W_m is actually an emergent wavefront in the space with surrounding media of air. Here, we set a sign conversion by following a ray from left to right going out of the eye. The propagation distance going with the ray takes a positive value and that going against the ray takes a negative value. The wavefront W_m could propagate forward along the z axis in air from the exit pupil to any position beyond the cornea.

Fig. 1. Wavefront propagation simulated by ray tracing. (a) The measured ocular wavefront W_m propagates in air along the local normal directions to generate the propagated wavefront W_p; (b) reverse engineering of the intra-corneal wavefront W_ic by the reverse ray tracing of the measured ACS from the propagated wavefront W_p.

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Wavefront propagation can be simulated in two different ways, Fresnel diffraction integral method and geometrical ray tracing. We choose the ray tracing method because of the similarity between the propagated wavefront and the original wavefront due to the limited propagation distance of several millimeters. The diffraction effect at the wavefront boundary accounts for very limited degradation of visual quality which is mainly determined by the wavefront aberration within the large pupil aperture. Therefore, the diffraction effect is out of our concern in this context.

The propagating wavefront can be characterized as many rays travelling in different directions determined by the local normals of the wavefront surface. By knowing the measured Zernike coefficients [16,17] of the wavefront W_m, the direction cosine of normal vector can be calculated as

(1)$${\textbf{N}_\textbf{m}} = \frac{1}{{\sqrt {1 + {{\left( {\frac{{\partial {W_m}}}{{\partial x}}} \right)}^2} + {{\left( {\frac{{\partial {W_m}}}{{\partial y}}} \right)}^2}} }}\left( { - \frac{{\partial {W_m}}}{{\partial x}},\; - \frac{{\partial {W_m}}}{{\partial y}},\;1} \right)$$

It is easy to show that the coordinates of the point P in the propagated wavefront W_p are relate to the coordinates of point M in the original wavefront W_m as

(2)$$\mbox{P} = \mbox{M} + d{\textbf{N}_\textbf{m}}$$

where, d is the vertex distance set to be 4.5 mm that covers the distance d₁ = 3.5 mm from the exit pupil to the ACS and the distance d₂ = 1 mm from the corneal surface to the propagated wavefront W_p. The wavefront W_p appears to be roughly the same profile as the wavefront W_m, but the boundary of the wavefront aperture varies especially when low order aberrations, such as defocus or astigmatism, dominate in W_m.

Figure 1(b) shows the revers ray tracing starting at the wavefront W_p and going backwards through the ACS. The space on the left side of the ACS is supposed to be semi-infinite and named as intra-corneal space, since the posterior corneal surface (PCS) is not involved in the simulation. The challenge in ray tracing is to determine the intersection of the ray with the freeform ACS. This problem can be solved with an iterative calculation as shown in Fig. 2:

Fig. 2. Iterative ray tracing algorithm of the measured ACS

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1. Based on the measured data of the corneal topography, the ACS is represented using Zernike polynomials [18] as $(3)$$S({x,y} )= \sum\limits_{n,m} {a_n^mZ_n^m({x,\;y} )}$$$ where $a_n^m$ is the Zernike coefficient with order n and meridional frequency m.
2. The point C₁ is found by computing the intersection of the ray with a reference spherical surface which shares the common vertex of ACS and has a radius of curvature of 7.2 mm according to the data of the schematic eye models.
3. The coordinates of a point C₂ on ACS are calculated with the same x and y coordinates as C₁. Then a tangent plane through C₂ is constructed to find the intersection point C₃ with the ray, which is generally closer to the ACS than the point C₁. The construction of the tangent plane requires the normal direction N_c of ACS which can be calculated using the same formula as Eq. (1): $(4)$${\textbf{N}_\textbf{c}} = \frac{1}{{\sqrt {1 + {{\left( {\frac{{\partial S({x,y} )}}{{\partial x}}} \right)}^2} + {{\left( {\frac{{\partial S({x,y} )}}{{\partial y}}} \right)}^2}} }}\left( { - \frac{{\partial S({x,y} )}}{{\partial x}},\; - \frac{{\partial S({x,y} )}}{{\partial y}},\;1} \right)$$$
4. The step 3 is repeated until the point C₃ found on the ray is close enough to the ACS. For practical purposes, the iterative process is terminated when the change in point coordinates between successive approximations is less than 1.0 × 10⁻¹² mm which commonly can be achieved in four iterations.

The incident angle of the backwards traveling ray at refraction on the ACS is calculated by means of the following vector dot product

(5)$$\cos {\theta _1} = {\textbf{N}_\textbf{c}} \cdot {\textbf{N}_\textbf{m}}$$

The refractive angle θ₂ is given by Snell’s law with the refractive index of corneal epithelium n_c = 1.401 and air n_a = 1

(6)$${n_\textrm{a}}\sin {\theta _1} = {n_\textrm{c}}\sin {\theta _2}$$

The direction cosine of the refracted ray traveling in the corneal epithelium is given by

(7)$${\textbf{N}_\textbf{i}} = \frac{{{n_\textrm{a}}{\textbf{N}_\textbf{m}} - {\textbf{N}_\textbf{c}}({{n_\textrm{a}}\cos {\theta_1} - {n_\textrm{c}}\cos {\theta_2}} )}}{{{n_\textrm{c}}}}$$

This reverse ray tracing is repeated at corneal stroma interface with refractive index of n_s = 1.376, then terminated at point I as shown in Fig. 1(b) according to the predetermined constant optical path length (OPL). The coordinates of the ending point of each ray are solved in the same way and finally give the surface profile of the intra-corneal wavefront W_ic.

From the viewpoint of ocular wavefront measurement, the calculated intra-corneal wavefront actually originates from the same focus point of illumination on the retina and is related to the refraction of the crystal lens and the PCS. Thus, the intra-corneal wavefront already contains the aberration components caused by the crystal lens and the PCS. Since corneal ablation does not change the topographies of the crystal lens and the PCS significantly, we can assume that this calculated intra-corneal wavefront remains unchanged postoperatively. Thus, with knowledge of the intra-corneal wavefront W_ic, we are able to calculate the CAP for the desired postoperative ocular wavefront.

Note that, in all the simulations the central thickness of corneal epithelium is set to be 0.05 mm. The thickness distribution of corneal epithelium is calculated according to the estimated dioptric power (−1.0 Diopters) of corneal epithelium and the measured data of ACS. For simplicity, we assume that the refractive surgery, such as LASIK and SMILE, dose not change the thickness distribution of corneal epithelium significantly. Additionally, the refractive index 1.333 of the pre-corneal tear film with uniform thickness of 5 μm is also employed in the simulations.

3. RM based calculation of CAPs

Aberration corrections with corneal ablation assisted by RM based CAP calculation can be categorized as aberration-free correction and aberration selective correction as shown in Fig. 3, according to whether an iterative operation is involved in ray mapping calculation.

Fig. 3. RM based calculation of the desired postoperative ACS. (a) Aberration-free correction; (b) aberration selective correction

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3.1 Aberration-free correction

The simplest case is that all the ocular aberrations are corrected to produce the emergent planar wavefront W_f as shown in Fig. 3(a), which means that the postoperative ACS must be able to collimate the divergent rays starting from the intra-corneal wavefront. The ray mapping can be easily found by following the refracted rays parallel to the z axis from the ablated ACS to the wavefront W_f.

The constant OPL rules all the rays traveling from the intra-corneal wavefront W_ic to the emergent wavefront W_f. Therefore, the coordinates of point C′ on the ablated ACS can be solved easily as the point C′ and the point F share the common x and y coordinates according to the ray mapping relation.

An initial ablation depth of zero can be set at the center of the preoperative ACS, which is suitable for the treatment of hyperopic eyes. In the case of myopic eyes, the ray mapping calculation must be accompanied by the optimization of ablation depth increased from zero to its optimum value that allows for the continuous laser ablation across the whole optical zone with the lest tissue consumption.

3.2 Aberration selective correction

In this case, a target ocular wavefront carrying the specified residual aberrations must be predetermined. For a given ray starting from the intra-corneal wavefront, there is no explicit ray mapping relation can be found since the intersection of the ray with the target ocular wavefront is unknown. The calculation of ray mapping requires an iteration process as follows:

1. Wavefront propagation simulation is first performed with the target ocular wavefront W_t, which is normally positioned at the exit pupil plane of eye, to obtain the propagated target wavefront W_pt as shown in Fig. 3(b).
2. An initial ray mapping relation, which could be the same coordinate-mapping pattern used in the aberration-free correction, is setup for the initial coordinate calculation of the point C₁ on a new ACS ${S_\textrm{1}}({x,y} )$. The initial ablation depth is set to be zero first.
3. To evaluate the deviation in ray mapping, the ray tracing method is used for the new ACS ${S_\textrm{1}}({x,y} )$ to calculate the exact ray path and the corresponding emergent wavefront. All the rays start at the intra-corneal wavefront W_ic, travel through the new CAS, and terminate at the ending points determined by a constant OPL, meanwhile generating the profile of the emergent wavefront W_e set at the same location of W_pt.
4. The wavefront deviation is defined by the difference between the simulated emergent wavefront W_e and the propagated target wavefront W_pt as $(8)$$\mbox{WD} = {\mbox{W}_\textrm{e}}({x,y} )- {\mbox{W}_{\mbox{pt}}}({x,y} )$$$ which can be transferred into the topography compensation of ACS ${S_\textrm{1}}({x,y} )$ as $(9)$$\Delta S = \frac{{\mbox{WD}}}{{{n_\textrm{s}} - {n_\textrm{a}}}}$$$ This compensation must be applied along the travelling direction N_i of each incident ray, so that the corresponding coordinates of the compensated ACS ${S^{\prime}_\textrm{1}}({x,y} )$ are given by $(10)$${\mbox{C}^{\prime}_1} = {\mbox{C}_1} + \Delta S \times {\textbf{N}_\textbf{i}}$$$
5. The ablation depth is determined temporarily by comparing the compensated ACS ${S^{\prime}_\textrm{1}}({x,y} )$ with the original ACS $S({x,y} )$.
6. The step 3 to 5 are repeated until the RMS(WD) is less than 1.0 × 10⁻⁸ mm, so that an ultra-high accuracy in ray mapping from the intra-corneal wavefront W_ic to the propagated target wavefront W_pt is achieved. Meanwhile, the optimal design of the postoperative ACS with the optimum ablation depth is obtained.

4. Numerical simulation results

4.1 Intra-corneal wavefront

An example of the elevation data of ACS and the ocular wavefront was measured simultaneously from an emmetropic eye of a 38-year-old male volunteer (YYL) with an ophthalmic biometer (KR-1W, Topcon Medical Systems, Oakland, NJ.). KR-1W uses Placido rings to map the corneal surface and provides the elevation data arranged in 19 discrete rings with the angular sampling step of 1 degree. This polar coordination system was converted to Cartesian coordinate system. Then, the corneal surface profile within an aperture of 6.4 mm in diameter was fitted using Zernike polynomial expansions up to order of 16 (see Fig. 4(c)). The achieved RMS fit error was less than 0.03 μm. Meanwhile, the measured ocular wavefront W_m at 6 mm pupil diameter was also represented as a Zernike expansion up to order of 6 (see Fig. 4(a)).

Fig. 4. Numerical calculation of the intra-corneal wavefront W_ic by reverse ray tracing. (a) The measured ocular wavefront W_m propagates in air along the local normal directions to generate (b) the propagated wavefront W_p; (c) measured ACS; (d) reverse ray tracing of the measured ACS with rays starting from the wavefront W_p; (e) calculated intra-corneal wavefront W_ic.

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In simulation of wavefront propagation from the measured wavefront W_m to the propagated wavefront W_p, the rays were uniformly spaced on the measured wavefront W_m with a grid pitch of 0.03 mm in the lateral directions, providing forty thousand rays in total. The different directions of rays are responsible for the irregular spacing of the traced rays when arriving at the propagated wavefront W_p. The same bundle of rays was employed in the following reverse ray tracing of the measured ACS to obtain the intra-corneal wavefront W_ic as shown in Fig. 4(d). The ACS was aligned with the pupil center so as to share the common coordinate system with the wavefront W_p. The rays were traced backwards numerically by setting the negative values of the travelling distances. The intra-corneal space is assumed to be semi-infinite temporarily and the distance between the wavefront W_ic and the ACS is set to be 2 mm.

4.2 Aberration-free correction

The objective wavefront refraction of a subject is related to the coefficient values of the second-order wavefront aberrations by the formula suggested by Thibos [19]. In this way, the refractive error, represented by the conventional sphero-cylindrical notation (in “negative-cylinder”) form, is calculated to be (−0.29 D − 0.29 D × 108°) for the emmetropic eye discussed in Section 4.1. In order to transform this emmetropic eye into an ametropic eye model, we can manually modify the second-order Zernike coefficients in the wavefront W_m intentionally and use the same high-order Zernike coefficients in the wavefront W_m and the same measured data of ACS for the ametropic eye model. Hence, we can have the myopic eye models with Zernike coefficient $a_2^0 > 0$ and the hyperopic eye models with Zernike coefficient $a_2^0 < 0$ as listed in Table 1. The refractive astigmatism (or cylinder error) is determined by the Zernike coefficient $a_2^{ - 2}$ and $a_2^{ + 2}$. Here, only the ametropia with refractive astigmatism less than − 2.50 Diopter is considered. Note that, all the ametropic eye models share the common preoperative $\textrm{RMS}\;\textrm{HOA} = \sqrt {\sum\limits_{n > 2,m} {{{({a_n^m} )}^2}} }$, which is calculated to be 3.37 × 10⁻⁴ mm based on the measured high-order Zernike coefficients in the wavefront W_m.

Table 1. Second order Zernike coefficients in the wavefront W_m and the corresponding wavefront refraction

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Since the second-order aberrations in wavefront W_m has been changed to be different for each ametropic eye model, it is necessary to perform the reverse ray tracing again to update the corresponding intra-corneal wavefront in each case to be the input data for the following calculation of CAP.

As an example, Fig. 5 shows the results of RM based calculation of CAP in case 3. Since the corneal astigmatism estimated from the measured ACS topography is quite limited, the large refractive astigmatism (−2.50 Diopter) is supposed to come from the internal ocular optics (with the crystalline lens as the main component). Thus, the astigmatic feature can be observed in the calculated intra-corneal wavefront W_ic (see Fig. 5(b)). As proposed in Section 3.1, by utilizing the explicit ray mapping from the wavefront W_ic to the planar wavefront W_f, the desired postoperative ACS (see Fig. 5(d)) as well as the optimized ablation depth is obtained. The ablation depth combined with the sag difference between the measured ACS (see Fig. 4(c)) and the calculated postoperative ACS gives the RM based CAP_RM (see Fig. 5(e)).

Fig. 5. RM based calculation of CAP for aberration-free correction in case 3. (a) Emergent planar wavefront W_f; (b) intra-corneal wavefront W_ic; (c) calculation of the postoperative ACS ruled by the explicit ray mapping from the wavefront W_ic to the wavefront W_f; (d) calculated postoperative ACS; (e) calculated RM based CAP_RM.

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On the other hand, by knowing the wavefront W_m of each case, the CAP in the optical zone for the conventional WG corneal ablation is given by

(11)$$\mbox{CA}{\mbox{P}_{\mbox{WG}}} = \frac{{\max ({{\mbox{W}_\textrm{m}}({x,y} )} )- {\mbox{W}_\textrm{m}}({x,y} )}}{{{n_\textrm{s}} - {n_\textrm{a}}}}$$

Thereby, the ablation parameters including ablation depth, optical zone and tissue consumption can be compared between the RM based algorithm and the WG based algorithm as shown in Table 2. As compared with the constant aperture size of the optical zone in WG based method, RM based method requires smaller aperture size of the optical zone for myopic cases and larger aperture size for hyperopic cases. Only in the myopic cases can the tissue consumption be saved with RM based method.

Table 2. Ablation parameters and residual wavefront error in aberration-free correction

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In addition, the sequential ray tracing is performed with the calculated postoperative ACS to evaluate the optical quality both for RM based method and WG based method. In each case, all the rays starting from the intra-corneal wavefront W_ic travel through the ablated ACS by refraction, and finally give the emergent wavefront ${\mbox{W}_\textrm{e}}({x,y} )$ whose deviation from the ideal planar wavefront W_f is estimated by the parameter called RMS wavefront error. In all the cases, RM based corneal ablation method has superior performance in RMS wavefront error. We have also calculated the RMS HOA, which is not listed here, finding that GW based method induced more HOAs than preoperation both in case 3 and case 6.

We also computed the retinal image of an extended object and the modulation transfer functions (MTFs) based on the simulated postoperative emergent wavefront ${\mbox{W}_\textrm{e}}({x,y} )$ [20]. The generalized pupil function is constructed as:

(12)$$\mbox{PF}({x,y} )= A({x,y} )\mbox{exp}\left[ {i\frac{{2\pi }}{\lambda }{\mbox{W}_\textrm{e}}({x,y} )} \right]$$

where $A({x,y} )$ denotes a pupil aperture. The squared module of the Fourier transform of $\mbox{PF}({x,y} )$ gives the point spread function (PSF) of the eye.

(13)$$\mbox{PSF}({x,y} )= {|{\mbox{FT}({\mbox{PF}} )} |^{2}}$$

The retinal image is then estimated by convolving the eye’s PSF with the object $\mbox{O}({u,v} )$.

(14)$$\mbox{I}({x,y})= \int\!\!\!\int {\mbox{O}}({u,v})\cdot {\mbox{PSF}} (u-x /M,v - y/ M)dudv$$

The MTF is the modulus of the optical transfer function (OTF) obtained as the Fourier transform of the PSF.

As shown in Fig. 6, the quality of the retinal image degrades rapidly in WG based method for the myopic cases with the increase of refractive astigmatism, while RM based method performs well in all the cases because of the much smaller RMS wavefront error achieved. Additionally, the calculated MTF in RM based method is very close to the diffraction limited performance of 6 mm pupil diameter as depicted in Fig. 7(b), so it surpasses the MTF obtained in WG based method obviously. Only in hyperopic cases can WG based method perform well in the quality of retinal image and MTF, because the required tissue consumption is much less than that in myopic cases.

Fig. 6. Retinal image of postoperative eye models in simulation of aberration-free correction with the CAP calculated by RM base algorithm or GW base algorithm.

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Fig. 7. Calculated MTFs of ametropic eye models. Defocus is not included in the wavefront error for MTF calculation. (a) Preoperative MTFs; (b) MTFs of postoperative eye models in simulation of aberration-free correction with the CAP calculated by GW base algorithm.

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Since the RM based method provides an almost perfect solution of corneal ablation for aberration-free correction, we can use the CAP_RM as a baseline to estimate the deviation of corneal ablation in WG based method, simply represented by

(15)$$\Delta \mbox{CAP} = \mbox{CA}{\mbox{P}_{\mbox{WG}}} - \mbox{CA}{\mbox{P}_{\mbox{RM}}}$$

As shown in Fig. 8, an over-ablation situation can be observed in most cases with WG based method. The insufficient ablation can be found in case 3 and the corneal periphery in hyperopic cases.

Fig. 8. Deviation of corneal ablation in WG based method for aberration-free correction as compared with the baseline of CAP_RM from (a) case 1 to (i) case 9.

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4.3 Aberration selective correction

The ability to control the postoperative HOAs is of great significance to the custom corneal ablation especially for the purpose to produce a carefully specified amount of selected aberrations. As we know, the optical quality of the human eye varies across the visual field. The selected residual aberration for a given field point may generate an optimum compensation in the other field regions [21]. Additionally, aberrations interaction may optimize the visual performance [22]. So, this section is arranged to investigate the feasibility of CAP calculation for the generation of the desired residual wavefront aberrations.

The same ametropic eye models listed in Table 1 and their corresponding intra-corneal wavefronts are employed here again for the simulation of aberration selective correction by corneal ablation. We aim to correct the low-order aberrations including defocus and astigmatism, ensuring that the postoperative eye and the preoperative ametropic eye share the common HOAs. This requirement forces us to construct the target ocular wavefront W_t with the Zernike coefficients of the measured HOAs in wavefront W_m, then simulate the wavefront propagation to generate the propagated target ocular wavefront W_pt (see Fig. 9(a)) for the following iterative optimization of ray mapping as proposed in Section 3.2.

Fig. 9. RM based calculation of CAP for aberration selective correction in case 3. (a) Propagated target ocular wavefront W_pt; (b) intra-corneal wavefront W_ic; (c) calculation of the postoperative ACS ruled by the ray mapping from the wavefront W_ic to the wavefront W_pt; (d) calculated postoperative ACS; (e) calculated RM based CAP_RM.

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As an example, Fig. 9 shows the final results of the optimized ray mapping after 5 iterations for the required accuracy in case 3. It is hard to distinguish the calculated results including the ray trajectory and the CAP between Fig. 5 and Fig. 9 by their appearances, as the light-weight HOAs in the wavefront W_pt can hardly cause dramatic fluctuation to the numerical simulation in the macroscopic aspect but the microscopic aspect.

To deal with the WG based corneal ablation, the CAP in the optical zone is calculated by

(16)$${\mbox{CAP}}_{\textrm{WG}} = \frac{\max ({\mbox{W}}^{\prime}_{\textrm{m}}(x,y))- {\mbox{W}}^{\prime}_{\textrm{m}}(x,y)}{n_{\textrm{s}} - n_{\textrm{a}}}$$

where, ${\mbox{W}}^{\prime}_{\textrm{m}}(x,y)$ is composed of the low-order wavefront aberrations to be corrected in the wavefront W_m.

The comparison of ablation parameters between RM based method and GW based method can be found in Table 3. Similar performance can be observed in ablation depth and tissue consumption except for the optical zone diameter. The numerical simulation with ray tracing reveals more difference in the optical performance as shown in Table 4. RM based method does well in reproducing the desired HOAs with perfectly corrected low-order aberrations. By contrast, GW based method is unable to fully correct the low-order aberrations, leading to the residual spherical error and astigmatism although within the tolerance of normal eyes.

Table 3. Ablation parameters for aberration selective correction

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Table 4. Residual wavefront refraction and wavefront aberration in aberration selective correction

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GW based CAP calculation is unable to reproduce the desired HOAs, as additional HOAs are induced in the myopic cases but over-corrected in the hyperopic cases. Generally, the more the corneal tissue to be ablated in GW based method, the larger the deviation of RMS HOA from the target value of 3.37 × 10⁻⁴ mm. This significant change of HOAs can be found especially in $Z_4^0$, $Z_4^2$, $Z_6^0$ and $Z_6^2$ as shown in Fig. 10. As a consequence, the quality of the simulated retinal images obtained in GW based method (as shown in Fig. 11) is normally worse than that of RM based method, and degrades as the increase of refractive astigmatism in myopic cases. Note that, only HOAs in the calculated emergent wavefront have been considered in the retinal image simulation.

Fig. 10. Zernike coefficients of HOAs of postoperative eye models in simulation of aberration selective correction with the CAP calculated by GW base algorithm.

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Fig. 11. Retinal image of postoperative eye models in simulation of aberration selective correction with the CAP calculated by RM base algorithm or GW base algorithm. Only HOAs have been considered.

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The difference in CAP calculation between RM based method and WG based method has been shown in Fig. 12 which uses CAP_RM as baseline. It can be seen that WG based method generally provides an over-ablation solution in the corneal periphery for the myopic cases but an insufficient-ablation solution in the corneal periphery for the hyperopic cases.

Fig. 12. Deviation of corneal ablation in WG based method for aberration selective correction as compared with the baseline of CAP_RM from (a) case 1 to (f) case 6.

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5. Discussion and conclusion

By using geometrical ray tracing, accurate ray mapping can be achieved from the input wavefront to the output wavefront and vice versa through the refraction of the optimized corneal surface. The proposed CAP calculation requires using the ray mapping process twice. The first time is the reverse ray tracing from the measured ocular wavefront to the intra-corneal wavefront. The second time is the sequential ray tracing from the intra-corneal wavefront to the target emergent wavefront. Ruled by the Snell’ law and the constant OPL condition, ray mapping providing a geometrical approximation of wavefront propagation mainly has two advantages for CAP calculation. First, dynamic change of wavefront feature including amplitude, aperture size and vergence during the wavefront propagation can be taken into account. As a consequence, the aperture of optical zone which is different from the aperture of the measured ocular wavefront can be determined accurately for ametropic eyes. Second, tissue ablation can be calculated along the ray travelling direction not the surface sag direction along z axis in the cornea, therefore providing an effective way to compensate the wavefront error or to generate the desired wavefront of arbitrary HOA components.

WG based CAP calculation, conventionally suggested to reduce preexisting aberrations and reduce the induction of new aberrations for customized refractive surgery, actually fails to address the HOAs in some cases according to our theoretical simulations. WG based algorithm may correct the low-order aberrations to a certain degree, however induces more HOAs simultaneously especially in the treatment of myopic eyes with significant level of refractive astigmatism. Generally, the more the calculated corneal tissue consumption is required, the worse the performance of WG based method in the manipulation of HOAs according to our simulation results. The suggestion can be made to replace the WG based CAP calculation with the new RM based algorithm which has been proved to support both the aberration-free correction mode and the aberration selective correction mode in theory. However, future clinical research is needed to verify the visual quality after RM based treatment.

Klein [23] proposed a similar theoretical approach to calculate the ablation profiles based on Hartmann-Shack wavefront slope data only, however limited to the application of aberration-free correction. Compared to Klein’s algorithm, the main advantage of our RM based algorithm is the ability to support unlimited patterns of the desired target wavefront.

The intra-corneal wavefront can hardly be measured directly by existing wavefront sensors. Ray optics simulation is the only way to derive the intra-corneal wavefront which is the foundation of RM base algorithm for the CAP calculation. The optical power distribution of the ACS also can be estimated by comparing the vergence between the intra-corneal wavefront and the measured ocular wavefront. To extend this research, the intra-corneal wavefront can propagate further through the PCS numerically to provide the intra-ocular wavefront in the anterior chamber, which makes it possible to evaluate the optical quality of crystalline lens in vivo. Thus, it is time to develop the new ophthalmic biometer which is able to measure the ocular wavefront and the topographies of both corneal surfaces simultaneously.

Some specific reasons for the induced HOAs are difficult to predict before the refractive surgery. In the short term, immediately after treatment, tissue ablation should lead to the flattening or steepening of ACS. However, the associated mechanical weakening due to tissue ablation and flap separation will cause the ACS reshaping in the long term under the intraocular pressure. Meanwhile, the postoperative PCS tends to show signs of central flattening and peripheral steepening [24], inducing fluctuation to the intra-corneal wavefront and ocular wavefront [25]. Wound healing which is responsible for tissue stiffening may also result in some reversal of the corneal shape changes. Therefore, the refinement of RM based algorithm for the CAP calculation and the further improvements of the surgical technique may help to reduce the side effects of these factors.

Funding

National Natural Science Foundation of China (NSFC) (61775171); Natural Science Foundation of Zhejiang Province (LY14F050009, LY16H120007); National Key Research and Development Program of China (2016YFC0100200).

Acknowledgment

We thank Fangjun Bao for valuable discussion.

References

1. M. Mrochen, M. Kaemmerer, and T. Seiler, “Wavefront-guided laser in situ keratomileusis: early results in three eyes,” J. Refract. Surg. 16(2), 116–121 (2000). [CrossRef]

2. M. Mrochen, M. Kaemmerer, and T. Seiler, “Clinical results of wavefront-guided laser in situ keratomileusis 3 months after surgery,” J. Cataract Refractive Surg. 27(2), 201–207 (2001). [CrossRef]

3. F. Manns, A. Ho, J. M. Parel, and W. Culbertson, “Ablation profiles for wavefront-guided correction of myopia and primary spherical aberration,” J. Cataract Refractive Surg. 28(5), 766–774 (2002). [CrossRef]

4. M. S. Shaheen, B. A. Bhalaby, D. P. Piñero, H. Ezzeldin, M. El-Kateb, H. Helaly, and M. A. Khalifa, “Wave Front-Guided Photorefractive Keratectomy Using a High-Resolution Aberrometer After Corneal Collagen Cross-Linking in Keratoconus,” Cornea 35(7), 946–953 (2016). [CrossRef]

5. J. A. Durán, E. Gutiérrez, R. Atienza, and D. P. Piñero, “Vector analysis of astigmatic changes and optical quality outcomes after wavefront-guided laser in situ keratomileusis using a high-resolution aberrometer,” J. Cataract Refractive Surg. 43(12), 1515–1522 (2017). [CrossRef]

6. M. Daniel, M. T. Leucci, V. Anand, L. Fernandez-Vega, S. A. Mosquera, and B. D. Allan, “Combined wavefront-guided transepithelial photorefractive keratectomy and corneal crosslinking for visual rehabilitation in moderate keratoconus,” J. Cataract Refractive Surg. 44(5), 571–580 (2018). [CrossRef]

7. V. V. Atezhev, B. V. Barchunov, S. K. Vartapetov, A. S. Zav’yalov, K. E. Lapshin, V. G. Movshev, and I. A. Shcherbakov, “Laser technologies in ophthalmic surgery,” Laser Phys. 26(8), 084010 (2016). [CrossRef]

8. R. R. Krueger, R. A. Applegate, and S. M. MacRae, Wavefront Customized Visual Correction: The Quest for Super Vision II, 2nd ed. (Slack Incorporated, 2003).

9. D. J. Tanzer, T. Brunstetter, R. Zeber, E. Hofmeister, S. Kaupp, N. Kelly, M. Mirzaoff, W. Sray, M. Brown, and S. Schallhorn, “Laser in situ keratomileusis in United States Naval aviators,” J. Cataract Refractive Surg. 39(7), 1047–1058 (2013). [CrossRef]

10. D. Smadja, T. De Castro, L. Tellouck, J. Tellouck, F. Lecomte, D. Touboul, C. Paya, and M. R. Santhiago, “Wavefront Analysis After Wavefront-Guided Myopic LASIK Using a New Generation Aberrometer,” J. Refract. Surg. 30(9), 610–615 (2014). [CrossRef]

11. K. G. Stonecipher and G. M. Kezirian, “Wavefront-optimized versus wavefront-guided LASIK for myopic astigmatism with the ALLEGRETTO WAVE: three-month results of a prospective FDA trial,” J. Refract. Surg. 24(4), S424–S430 (2008).

12. M. R. George, R. A. Shah, C. Hood, and R. R. Krueger, “Transitioning to optimized correction with the WaveLight ALLEGRETTO WAVE: case distribution, visual outcomes, and wavefront aberrations,” J. Refract. Surg. 26(10), S806–S813 (2010). [CrossRef]

13. M. Bueeler, M. Mrochen, and T. Seiler, “Maximum permissible lateral decentration in aberration-sensing and wavefront-guided corneal ablation,” J. Cataract Refractive Surg. 29(2), 257–263 (2003). [CrossRef]

14. R. E. Ang, W. K. Chan, T. L. Wee, H. M. Lee, P. Bunnapradist, and I. Cox, “Efficacy of an aspheric treatment algorithm in decreasing induced spherical aberration after laser in situ keratomileusis,” J. Cataract Refractive Surg. 35(8), 1348–1357 (2009). [CrossRef]

15. J. Liang, B. Grimm, S. Goelz, and J. F. Bille, “Objective measurement of wave aberrations of the human eye with the use of a Hartmann–Shack wave-front sensor,” J. Opt. Soc. Am. A 11(7), 1949–1957 (1994). [CrossRef]

16. V. Lakshminarayanan and A. Fleck, “Zernike polynomials: a guide,” J. Mod. Opt. 58(7), 545–561 (2011). [CrossRef]

17. ISO, “Ophthalmic optics and instruments - Reporting aberrations of the human eye,” in ISO 24157:2008.

18. D. R. Iskander, M. J. Collins, and B. Davis, “Optimal modeling of corneal surfaces with Zernike polynomials,” IEEE Trans. Biomed. Eng. 48(1), 87–95 (2001). [CrossRef]

19. L. N. Thibos, X. Hong, A. Bradley, and X. Cheng, “Statistical variation of aberration structure and image quality in a normal population of healthy eyes,” J. Opt. Soc. Am. A 19(12), 2329–2348 (2002). [CrossRef]

20. P. Artal, Handbook of Visual Optics, Volume Two: Instrumentation and Vision Correction, 1st ed. (CRC Press, 2017).

21. S. Bará and R. Navarro, “Wide-field compensation of monochromatic eye aberrations: expected performance and design trade-offs,” J. Opt. Soc. Am. A 20(1), 1–10 (2003). [CrossRef]

22. R. A. Applegate, J. D. Marsack, R. Ramos, and E. J. Sarver, “Interaction between aberrations to improve or reduce visual performance,” J. Cataract Refractive Surg. 29(8), 1487–1495 (2003). [CrossRef]

23. S. A. Klein, “Optimal corneal ablation for eyes with arbitrary Hartmann–Shack aberrations,” J. Opt. Soc. Am. A 15(9), 2580–2588 (1998). [CrossRef]

24. M. L. Dai, Q. M. Wang, Z. S. Lin, Y. Yu, J. H. Huang, G. Savini, J. Zhang, L. Wang, and C. C. Xu, “Posterior corneal surface differences between non-laser in situ keratomileusis (LASIK) and 10-year post-LASIK myopic eyes,” Acta Ophthalmol. 96(2), e127–e133 (2018). [CrossRef]

25. T. Yamaguchi, K. Ohnuma, D. Tomida, K. Konomi, Y. Satake, K. Negishi, K. Tsubota, and J. Shimazaki, “The Contribution of the Posterior Surface to the Corneal Aberrations in Eyes after Keratoplasty,” Invest. Ophthalmol. Visual Sci. 52(9), 6222–6229 (2011). [CrossRef]

	Original emmetropia	Case 1	Case 2	Case 3	Case 4	Case 5	Case 6	Case 7	Case 8	Case 9
$a_{2}^{- 2}$ (μm)	0.157	0	−0.001	−0.001	0	−0.001	−0.001	0	−0.001	−0.001
$a_{2}^{0}$ (μm)	0.564	3.897	4.871	5.521	7.794	8.769	9.418	−3.897	−2.923	−2.273
$a_{2}^{+ 2}$ (μm)	0.215	0	1.378	2.296	0	1.378	2.296	0	1.378	2.296
Sph (D)	−0.29	−3.00	−3.00	−3.00	−6.00	−6.00	−6.00	3.00	3.00	3.00
Cyl (D)	−0.29	0.00	−1.50	−2.50	0.00	−1.50	−2.50	0.00	−1.50	−2.50
Axis (°)	108	/	90.00	90.00	/	90.00	90.00	/	90.00	90.00

	Ablation depth (mm)		Optical zone diameter (mm)		Tissue consumption (mm³)		RMS wavefront error (mm)
	RM	WG	RM	WG	RM	WG	RM	WG
Case 1	0.0387	0.0388	5.9552	6.00	0.6078	0.6124	3.93 × 10⁻⁸	7.76 × 10⁻⁵
Case 2	0.0567	0.0566	5.9494	6.00	0.9757	0.9812	5.41 × 10⁻⁸	2.87 × 10⁻⁴
Case 3	0.0689	0.0685	5.9481	6.00	1.2264	1.2298	6.54 × 10⁻⁸	4.89 × 10⁻⁴
Case 4	0.0739	0.0747	5.9078	6.00	1.0723	1.1100	5.81 × 10⁻⁸	9.23 × 10⁻⁵
Case 5	0.0919	0.0925	5.9020	6.00	1.4326	1.4787	6.57 × 10⁻⁸	2.27 × 10⁻⁴
Case 6	0.1041	0.1045	5.9007	6.00	1.6789	1.7273	7.17 × 10⁻⁸	3.63 × 10⁻⁴
Case 7	0.0000	0.0000	6.0469	6.00	0.5508	0.5327	4.64 × 10⁻⁸	1.87 × 10⁻⁴
Case 8	0.0000	0.0000	6.0411	6.00	0.4211	0.4088	5.04 × 10⁻⁸	1.67 × 10⁻⁴
Case 9	0.0001	0.0001	6.0400	6.00	0.3354	0.3264	6.69 × 10⁻⁸	1.53 × 10⁻⁴

	Ablation depth (mm)		Optical zone diameter (mm)		Tissue consumption (mm³)
	RM	WG	RM	WG	RM	WG
Case 1	0.0361	0.0359	5.9555	6.00	0.4989	0.4976
Case 2	0.0542	0.0539	5.9497	6.00	0.8721	0.8705
Case 3	0.0664	0.0658	5.9485	6.00	1.1231	1.1191
Case 4	0.0716	0.0718	5.9081	6.00	0.9728	0.9952
Case 5	0.0897	0.0898	5.9023	6.00	1.3380	1.3680
Case 6	0.1019	0.1017	5.9010	6.00	1.5846	1.6167
Case 7	0.0000	0.0000	6.0473	6.00	0.5145	0.4965
Case 8	0.0000	0.0000	6.0415	6.00	0.3846	0.3723
Case 9	0.0000	0.0000	6.0404	6.00	0.2985	0.2896

	Sph (D)		Cyl (D)		Axis (deg)		RMS(WD) (mm)		RMS HOA (mm)
	RM	WG	RM	WG	RM	WG	RM	WG	RM	WG
Case 1	−0.002	0.029	−0.002	−0.010	93.9	85.7	3.48 × 10⁻⁸	5.44 × 10⁻⁵	3.37 × 10⁻⁴	3.65 × 10⁻⁴
Case 2	−0.002	0.114	−0.002	−0.117	93.9	89.7	4.94 × 10⁻⁸	2.28 × 10⁻⁴	3.37 × 10⁻⁴	4.01 × 10⁻⁴
Case 3	−0.002	0.145	−0.002	−0.175	93.9	89.8	6.12 × 10⁻⁸	3.58 × 10⁻⁴	3.37 × 10⁻⁴	4.77 × 10⁻⁴
Case 4	−0.002	−0.07	−0.002	−0.018	93.9	83.9	5.51 × 10⁻⁸	9.47 × 10⁻⁵	3.37 × 10⁻⁴	3.92 × 10⁻⁴
Case 5	−0.002	0.014	−0.002	−0.155	93.9	89.4	6.27 × 10⁻⁸	2.04 × 10⁻⁴	3.37 × 10⁻⁴	4.09 × 10⁻⁴
Case 6	−0.002	0.047	−0.002	−0.233	93.9	89.6	6.89 × 10⁻⁸	3.34 × 10⁻⁴	3.37 × 10⁻⁴	4.80 × 10⁻⁴
Case 7	−0.002	0.138	−0.002	−0.004	93.9	161	5.45 × 10⁻⁸	1.86 × 10⁻⁴	3.37 × 10⁻⁴	3.08 × 10⁻⁴
Case 8	−0.002	0.128	−0.002	−0.012	93.9	94.4	5.09 × 10⁻⁸	1.66 × 10⁻⁴	3.37 × 10⁻⁴	3.19 × 10⁻⁴
Case 9	−0.002	0.123	−0.002	−0.027	93.9	91.6	6.48 × 10⁻⁸	1.52 × 10⁻⁴	3.37 × 10⁻⁴	3.27 × 10⁻⁴

	Original emmetropia	Case 1	Case 2	Case 3	Case 4	Case 5	Case 6	Case 7	Case 8	Case 9
$a_{2}^{- 2}$ (μm)	0.157	0	−0.001	−0.001	0	−0.001	−0.001	0	−0.001	−0.001
$a_{2}^{0}$ (μm)	0.564	3.897	4.871	5.521	7.794	8.769	9.418	−3.897	−2.923	−2.273
$a_{2}^{+ 2}$ (μm)	0.215	0	1.378	2.296	0	1.378	2.296	0	1.378	2.296
Sph (D)	−0.29	−3.00	−3.00	−3.00	−6.00	−6.00	−6.00	3.00	3.00	3.00
Cyl (D)	−0.29	0.00	−1.50	−2.50	0.00	−1.50	−2.50	0.00	−1.50	−2.50
Axis (°)	108	/	90.00	90.00	/	90.00	90.00	/	90.00	90.00

Ray-mapping based calculation of the intra-corneal wavefront and corneal ablation profile for ocular aberration corrections

Abstract

1. Introduction

2. Intra-corneal wavefront

3. RM based calculation of CAPs

3.1 Aberration-free correction

3.2 Aberration selective correction

4. Numerical simulation results

4.1 Intra-corneal wavefront

4.2 Aberration-free correction

4.3 Aberration selective correction

5. Discussion and conclusion

Funding

Acknowledgment

References

Cited By

Figures (12)

Tables (4)

Equations (16)

OSA Continuum