## Ⅰ. Introduction

Refractive error in the human eye means a state in which the incident parallel light cannot be focused on the fovea of the retina. Refractive anomalies are divided into myopia, hyperopia, and astigmatism.^{1,2)} To correct this, wear eyeglasses and contact lenses or perform refractive surgery.^{3-6)}

Conventional corneal refractive surgery has been classified into two categories. LASIK (laserassisted in situ keratomileusis) incision the corneal surface to make flaps and removing some corneal stroma, and LASEK (laser epithelial keratomileusis) Cutting from the epithelium, the outermost layer of the cornea, to the anterior part of the corneal stroma.^{7-9)} However, LASIK has the disadvantage that the flap boundary is weak and can be peeled off by impact after surgery. And LASEK has the disadvantage of slow recovery of vision because it removes the epithelium.^{10-15)}

SMILE (small incision lenticule extraction) surgery is a surgical method created to make up for the weaknesses of LASIK and LASEK. Figure 1 shows the SMILE surgical procedure using a femtosecond laser to excision lenticule in the stroma of the cornea and then removes it into micro incision gap. SMILE has the advantage of fast recovery of vision due to less damage to the epithelium and a small incision area.^{16,17)}

Among corneal refractive surgery, SMILE surgery is gaining popularity. Despite good visual acuity, visual performance is affected by aberrations caused by changes in corneal curvature after surgery. Therefore, in this study, the change in corneal curvature and corrected vision acuity according to the amount of lenticule cut was confirmed. Also, by analyzing the change of aberration according to the corrected vision acuity, the effect of the visual performance after surgery was theoretically confirmed.

## Ⅱ. Methods

Normally, in SMILE surgery, an excision a lenticule is made approximately 0.12 mm behind from the anterior surface of the cornea using a femtosecond laser. The ablation amount varies depending on the degree of refraction anomaly, but the change in curvature of the posterior surface of the cornea is small because it is cut at a position close to the anterior surface of the cornea.

In order to simplify a theoretical analysis, in this study, we analyzed based on Listing’s reduced eye. Ray tracing was performed to analyze the change of focal length and aberration according to SMILE surgery. We used Snell’s law to calculate the focal length along the height of the incident light. In addition, the difference in the distance between the focal point and the retina position was defined as aberration. It was assumed that the change in refractive power following SMILE surgery occurred only in the anterior surface of the cornea, and the changes in curvature and refractive power were calculated.

SMILE surgery changes the curvature of the cornea by removing a lens-shaped incision in the cornea. To simplify the analysis, assume that the curvature of the posterior surface of cornea remains constant and only the anterior surface changes.

## Ⅲ. Results

### 1. Changes in Corneal Curvature and Refractive Power Following Surgery

Figure 2 shows the changes in the anterior curvature of the cornea by lenticule removal by SMILE surgery. If the lenticule center thickness to be removed is *d*, then the positional change of the apex of the anterior surface of cornea is also assumed to be *d*. And also, let’s the diameter of the lenticule to be cut is *D* .

Figure 3 shows the parameters for analyzing the SMILE operation effect, numerically. The radius of curvature of the anterior surface of cornea before and after surgery are *r* and *r*′, respectively. After surgery, the refractive dioptric power of the anterior surface changes according to the radius of curvature and the vision is corrected accordingly. In the figure 3, the distance between the preoperative cornea and the postoperative corneal center is *d* equal to the thickness of the incision lens.

Since the anterior surface of cornea is a circle with a radius *r* around center *C* , the equation for this is

Assume that the anterior of the cornea after surgery is also a circle. In other words, the equation of a circle passing *P* , *O* ′, *Q* is

Where the new circle passes three points$P\left(-\sqrt{{r}^{2}-{\left(D/2\right)}^{2}},D/2\right),{O}^{\prime}\left(-r+d,0\right)$, and $Q\left(-\sqrt{{r}^{2}-{\left(D/2\right)}^{2}},-D/2\right)$. Substituting the coordinates of these three points into the circle equation

A combination of the above two equations yields

Expand this expression and solve for the axis coordinate of the center *C* ′, center of the new circle

Substituting these results into a new circle equation, a new radius of curvature can be obtained.

Here, the radiuses of curvature and will be treated only with absolute values without sign.

Figure 4 shows the result of calculating the anterior curvature radius (*r* ′) of cornea according to the ablation amount (*d*) using equation 6. As expected, the radius of curvature increases as the ablation amount increases.

In the above result, if *d* = 0 , it can be seen that *r* ′ = *r*. In the case of *d* = *s* (where $s=r-\sqrt{{r}^{2}-{\left(D/2\right)}^{2}}$), the spherical surface becomes a straight line passing through points *P* , *S* , and *Q* , and the radius of curvature *r* ′ should be infinite. That is, in the above equation, we can see that the radius of curvature becomes infinite because the denominator becomes zero.

Figure 5 is a graph of the refractive power change according to the ablation amount for the Listing’s reduced eye of *r* = 5.5 mm and *n* ′ = 1.33. The theoretical value is calculated by substituting equation 8 into the refractive power change $\Delta D=\left(n-1\right)\left(\frac{1}{r}-\frac{1}{{r}^{\prime}}\right)$ of the reduced eye, and compared with the value provided by the laser manufacturer. As the amount of cutting increased, the corrected visual acuity (ie, the amount of change in refractive power) gradually increased. Comparing the theoretical value (inverted triangle) and the clinical reference value (square) of corrected visual acuity according to the ablation amount, the theoretical value was larger when the cutting amount was small and the reference value was larger when the cutting amount was large.

### 2. Aberration with Changes in Corneal Curvature

The Listing’s reduced eye has an axial distance of 22.2 mm and a refractive power of 60 D. There are many causes of ametropia, but for the clarity of this study, we analyzed the case of myopic eyes with more than 22.2 mm of axial length.

Figure 6 shows the basic structure of the reduced eye (dotted line) and myopia (solid line). In addition, it shows the state of light refraction in the anterior surface of cornea after SMILE surgery.

When parallel light is incident on the cornea, the angle of incidence is

According to Snell’s law of refraction in the anterior surface of cornea, the angle of refraction is

If *h* > *D* /2, it is outside the surgical area, so *r* ′ = *r*, which is equal to the radius of Listing’s reduced eye. In Fig. 7, the image distance of parallel rays is

Using the relationship between angles, *θ* ″ = *θ* - *θ* ′ and equations 9 and 10

When parallel light enters the eye with corrected vision, the rays focus on the retina. Therefore, the focal length by paraxial approximation is

As the corrected visual acuity increases, both the ablation mount and the radius of curvature of the cornea by surgery increase.

Let’s analyze the change of aberration according to the corrected visual acuity by defining the difference between the image length, *s* ′ and the corrected axial length, *f* ′. As the radius of curvature increases, the limit of the image length in equation 12 is

Thus, the aberration converges to zero. Where the Taylor series

was used.

Numerical analysis using equations 8, 12, and 13 yielded the black square graphs in figure 7(a). As the absolute value of corrected visual acuity increases, the aberration decreases. This means that the corrected visual acuity is large, and the corrected axial length is large. To correct this, the thickness of the lenticule must be large. Therefore, it is believed that the spherical aberration is reduced because the curvature of the cornea becomes smaller and spherical surface becomes flatter after the procedure.

The blue circle graph in figure 7(a) shows the opposite trend of the former, and its absolute value is also large. In other words, the greater the corrected visual acuity, the more spherical aberration after surgery. The result of this spherical aberration can be explained in the figure. In the surgical region (*h* ≤ *D* /2 ) around the optical axis, the curvature of the anterior surface of cornea is reduced, so that spherical aberration is reduced. On the other hand, outside the surgical area (*h* > *D* /2 ), the curvature of the cornea remains unchanged after surgery. Therefore, the greater the corrected visual acuity, the greater the change in curvature between the two regions, which causes larger spherical aberration.

Figure 7(b) shows the image point for the rays that enter at different heights. Light rays entering the surgical area form an image point away from the cornea and have a low aberration. On the other hand, light rays incident out of the surgical region form an image point at a point close to the cornea by a large refractive power and have a relatively large aberration.

In analyzing spherical aberration by SMILE surgery, it is necessary to distinguish between two areas. The spherical aberration caused by the light rays entering the surgical region and the spherical aberration caused by the light rays incident out of the surgical region have different causes. It is also because the magnitude of the aberration is significantly different.

## Ⅳ. Discussion and Conclusion

In corneal surgery for vision correction, including SMILE surgery, as the corrected visual acuity increases, the ablation amount of the lenticule increases and the anterior curvature change of the cornea, also, increases. Previous studies have shown that the corneal surface is flatter after surgery than before surgery with corrected corneal surgery. it is known that the aberration increases as the corrected visual acuity increases.^{18-23)} To confirm this phenomenon, we theoretically analyzed corneal curvature changes and aberrations following SMILE surgery.

We drove anterior curvature of the cornea following lenticule removal. Using this result, the change of refractive power and the corrected visual acuity according to the ablation amount were calculated. As the ablation amount increased, both the new radius of curvature and corrected visual acuity increased. In addition, in the case of myopic eyes with longer axial lengths from the reduce eye, the image length equation was driven by the curvature change by SMILE surgery. By defining the difference between the image length and the axial length as aberrations, we analyzed aberrations in and out of the surgical area.

Within the surgical area, the ablation amount and radius of curvature also increase as the corrected visual acuity increases. Therefore, the spherical surface is flattened so that the spherical aberration is reduced. On the other hand, the radius of curvature remains unchanged outside the surgical area. Thus, as the corrected visual acuity increases, the difference in curvature inside and outside the surgical area increases, so that spherical aberration increases.

The prognosis is known to be the best in myopia patients around -3.00 D. Through this study, it was confirmed that the increase of aberration according to the increase of myopia is one of the optical phenomena that is not the technique of ophthalmologist. In the follow-up study, we attempt to investigate the optical characteristics of SMILE surgery using various surgical cases. We expect to be able to theoretically analyze the causes of halo and glare phenomena that are common in patients after surgery.