Gaussian Optics Analysis for Human Eyes with Application for Vision Corrections

Aims: To derive formulae and analyze the roles of ocular components of human eye on the refractive power in various applications. Study Design: Gaussian optics analysis. Place and Duration of Study: Taipei, Taiwan, between May 2015 and August 2016. Methodology: An effective eye model is introduced by the ocular components of human eye including refractive indexes, surface radius (r1, r2, R1, R2) and thickness (t,T) of the cornea and lens, the anterior chamber depth(S1) and the vitreous length (S2). Gaussian optics is used to calculate the change rate of refractive error per unit amount of ocular components of a human eye. Results: For typical corneal and lens power of 42 and 21.9 diopters, the rate function defined by the change of refractive error (De) due to the change of ocular components, or Mj =dDe/dQj, with j=1 to 6 for Qj= r1, r2, R1, R2, t, T are calculated for a 1% change of Qj M1=+0.485, M2=-0.063, M3=+0.053, M4=+0.091, M5=+0.012,and M6=-0.021 diopters. For 1.0 mm increase of S1 and S2, the rate functions are: M7=+1.35, and M8=-2.67 diopter/mm. Conclusion: Using Gaussian optics, we have derived analytic formulas for the change of refractive power due to various ocular parameter changes. These formulas provide the amount of refractive error corrections in various applications including laser in situ keratomileusis (LASIK) surgery and scleral ablation for accommodation.


INTRODUCTION
Gaussian optics [1,2] has been used for the calculations of intraocular lens (IOL) power, accommodation amplitude in IOL and human natural lens and the refractive state of human eyes [1][2][3][4][5][6]. A complete optical description of a human eye should include its 12 ocular parameters, 4 refractive indexes, 4 surface radius and 2 thickness (for cornea and lens), the anterior chamber depth and the vitreous length (or axial length). The roles of these parameters on the refractive power of an eye have been reported only partially [2,3]. This study will derive analytic formulas for the change rate of refractive error per unit amount of selected ocular components.
A conversion function defined by the ratio between the refractive error change and the lens power change is used to calculate the rate functions. The roles of the surface radius of the cornea and lens on the refractive error, the competing factors of anterior chamber depth (resulting to hyperopic shift) and posterior chamber depth (resulting to myopic shift) are discussed. Finally, various new applications related to the formulas developed in this paper including laser in situ keratomileusis (LASIK) surgery, corneal cross linking (CXL) procedure, femtosecond laser surgery and laser scleral ablation for accommodation are presented in the manuscript. The analytic formulas developed in this paper provide useful clinical guidance for vision corrections in various applications.

The Effective Eye Model
By Gaussian optics theory (or paraxial ray approximation along the axial axis), the refractive error (De) as a function of the system effective focal length (EFL) (F), axial length (L) and position of the system second principal plane (L2) as follow [1,2].
where n1 is the refractive index of the aqueous humor. F is the system EFL defines the system total power D=1000n1/F (D in diopter, F in mm) which is determined by the corneal (D1) and lens power (D2) as follows [2,3]. where nj (j=1, 2, 3, 4) are the refractive index for the aqueous, vitreous, cornea and lens, respectively. The anterior and posterior radius of curvatures (in the unit of mm) of the cornea and lens are given by (r1, r2) and (R1, R2), respectively, where the only concave surface R2 is taken as its absolute value in this study. Finally, S is the effective anterior chamber depth, related to the anterior chamber depth (ACD), S1, by S=S1+P11+0.05 ( in mm), where P11 is the distance between the lens anterior surface and its first principal plane, and 0.05 mm is a correction amount to include the effect of corneal thickness (assumed to be 0.55 mm) [2,3]. The thickness terms in Eq.(2.b) and (2.c) are given by b=11.3/(r1r2), a=4.97/(R1R2) for refractive indexes of n1 = n2 = 1.336, n3 = 1.377 and n4 = 1.42; and t and T are the thickness of the cornea and lens, respectively.
As shown in Fig. 1, we have derived the equation [3] L-L2=X+ SF/f, with X=L-S-aT+0.05, where aT and 0.05 mm are the correction terms due to lens and cornea thickness. Eq.(1) may be rewritten in an effective eye model equation [3].
where f (in mm) is the EFL of the cornea given by f=1336/D1, and the nonlinear term k is about 0.003 calculated from the second-order approximation of SF/(1336f). The nonlinear term may also be derived from the IOL power formula [5]. We note that in Eq. (3), X, Z, S and f are in the unit of mm; D1, D2 and De are in the unit of diopter; and the 1336 is from 1000x1.366 in our converted units.

Derivation of the Rate Functions
To find the change of refractive error (De) due to the change of Qj, we further define Qj=(r1, r2, R1, R2, t, T, S1, S2) with j=(1 to 8), respectively. The ACD (S1) and vitreous length (S2) are related to the axial length by L=S1+S2+T. The derivative of the refractive error (De) with respect to these ocular parameter change (Qj) given by Mj=dDe/dQj, defines the rate function, or the change of De per unit amount change of Qj, where the standard notation "d" for "derivative" is used in this study.
In general, under the second-order approximation including the contributions from both n1/(L-L2) and (n1/F) in Eq.(1), one shall rigorously calculate the derivative dDe=Mj(dQj) based on Eq.(1). The complexity of this method is due to the nonlinear dependence of L2 on the ocular parameters [1][2][3]. In this study, an alternative method is described as follows. For cornea related ocular parameters Qj (with j=1,2,5), the rate function Mj may be calculated by the corneal conversion function C 1 = (dDe/dD1), such that On the other hand, the lens related parameters Qj (with j=3,4,6) , the rate function may be calculated by a lens conversion function C 2 = (dDe/dD2), It may be calculated, from Eq. (3.a) that C 1 =1.0, for initial De=0. In other words, the corneal power change is 100% converted to the system power or refractive error change. Similarly the lens conversion function given by C 2 = Z 2 , can be derived by taking the derivative of De in Eq. (3.a).

The Rate Functions
The increase of S1 results in a hyperopia shift (HS), whereas S2 results in a myopia shift (MS), where M8 is about two times of M7 which has two competing terms as shown by Eq. (7.a). The rather high change rate M12=-2.67 (D/mm) has significant impact on the onset of emmetropization and myopia which are governed by the correlation among the growth of axial length (L=S1+S2+T) and the power decrease of the cornea and lens when an eye grows [3]. The change rate M7 having a lower value than M8 can be analyzed as follows.
The competing between the MS (due to the increase of ACD, S1) and the HS (due to the associate decrease of S2 for a fixed axial length L=S1+S2+T) results in a net hyperopic-shift, because the hyperopic component is always the dominant one, since the corneal power (D1) is always less than the total system power (D) or F<f in Eq. (5.a). This new finding based on the analytic formula of Eq. (5) has not been explored before, in addition to the newly introduced conversion function.
The hyperopic shift due to the increase of S1 is equivalent to a myopic-shift when S1 decreases, or a forward movement of the lens. This feature is important for presbyopia accommodation which is contributed by two components: the lens curvature decrease and the lens forward movement [3,4]. The lens forward movement is also the main feature in an accommodative IOL and our formulas, Eq. (7) for M7 and M8 provide the amount of accommodation.

Clinical Applications
Two examples of applications related to the formulas developed in this paper, including laser in situ keratomileusis (LASIK) surgery and scleral ablation for accommodation are presented as follows. More clinical applications will be presented else where.

LASIK surgery [8]
LASIK is a procedure where one diopter correction only requires an ablation depth about 8 to 11 microns of the corneal central thickness [6] or a corresponding change of r1 about 0.16 mm based on Eq. (6.a). It is important to know that the corneal power change is 100% converted to the system power or refractive error change, as demonstrated by our cornea conversion factor C 1 . We should also note that the refractive error (De) defined on the corneal plan is the same as that of a contact lens. However, a conversion formula is needed when it is translated to a spectacle power Ds, given by De= Ds/ [1 -V Ds], where V is a vertex distance about 12 mm.
The central ablation depth for a 3-zone myopic correction is given by [8]. where W is the diameter of the outer ablation zone having a typical value of 6.5 to 7.5 mm; C is a nonlinear correction term given by C= 0.19 (W/r1) 2 , r1 is the corneal anterior radius of curvature. For examples, for r1=7.8 mm, (or a Kreading of K=43.2 D), C = (11.2, 13.2, 16.5)% for W =(6.0, 6.5, 7.00 mm. The reduction factor R=(0.70 to 0.85) depending on the algorithms used. For example, comparing to a single zone with W=6.5 mm, a 3-zone depth will reduces to 71.6% (or R = 0.716) when an inner zone 5.5 mm and an outer zone 6.5 mm are used.

Presbyopia treatment [9]
Scleral laser ablation and band expansion have been used to increase the space of the ciliarybody and zonus such that accomondation is improved by two components [9]: the lens translation and the lens shaping which are given by, respectively, M7 and M3. For older and/or harder lens, the accommodation is mainly attributed by the lens translation (or S1 change), whereas lens shaping dominates the power change in young or soft lens. It was known that change of the rear surface of the lens is about one-third of the front surface during accommodation [10], our formulas Eq. (6.c) and (6.d) shows that the contribution from R2 is about the same as that of R1, because of R2 (6.0 mm) <R1(10.2 mm), and M4=2.9 M3, for the same change of curvature, dR1= dR2.

CONCLUSION
Using Gaussian optics, we have derived analytic formulae for the change of refractive power due to various ocular parameter changes. These formulae provide the amount of refractive error corrections in LASIK and scleral ablation for accommodation. More clinical applications such as corneal cross linking, femtosecond laser surgery and accommodative IOL will be presented else where.

CONSENT
It is not applicable.

ETHICAL APPROVAL
It is not applicable.