In this article we will talk about the different theoretical formulas derived on basis of thin and thick lens equations, and the promise of ‘ray tracing’ optics for improved refractive results. In the process, we will talk about the schematic eye, without which no understanding of biometry formula will be complete. For a simple understanding of biometry formula, and for a beginner’s guide, you may refer to the article Biometry for Cataract - The art of applying Mathematics to Biology published at quickguide.org (https://www.quickguide.org/post/biometry-for-cataract-the-art-of-applying-mathematics-to-biology).
IOL formula began its journey during late 1960s and early 1970s. Fyodorov introduced his theoretical vergence formula for anterior chamber IOL power calculation based on 3 variables - Axial length, corneal reading , and the predicted post-operative ACD (anterior chamber depth). Since this formula was introduced, an array of IOL formula have come up for over half a century, that can be broadly classified into - regression based, theoretical -vergence, ray tracing ( also accounts theoretical calculation ), and artificial intelligence. In line with the objective of this article, we will cover the vergence and ray tracing based formulas. It would be proper to clarify here, that ray tracing is based on theoretical thick lens vergence formula too. All theoretical vergence based formula, namely - Fyodorov, Binkhorst, Collabander (all thin lens formula) as well as new age Barrett Universal formula (thick lens formula) are based on Gaussian optics. Gaussian optics is a framework for describing optical phenomena, which is based on geometrical optics (also called ray optics) and makes extensive use of the paraxial approxiamtion(rays that pass through the center of the lens, that is close to lens optical axis).
Developed by Carl Friedrich Gauss (1777-1855), a leading mathematician of his time, it is based on the following:
i) Light is described with light rays drawn geometrically (geometric optics). Thus wave optics or wave effects on optics as well as diffraction and dispersion in the material in which light is travelling is ignored.
ii) It is based on assumption of rotationally symmetric optics with a distinct optical axis.
(rotationally symmetric is the property of a shape that remains same when rotated around the center)
iii) Gaussian optics is based on paraxial rays, and rays that pass through the region outside the optical axis is ignored.
Paraxial rays may be defined as rays that make small angle of incidence to the optical axis of the system and lies close to the optical axis. Paraxial rays lie close to chief ray passing through the optical axis (fig1). Thus in Gaussian optics all marginal rays are ignored. The term geometric optics is often used interchangeably with ray optics. Here, do not get confused with Ray Optics IOL formula, which is one of the feature of Olsen (PhacoOPtics®) formula, and will be explained subsequently.
Gaussian optics helps us to arrive at both ‘thin’ lens, and 'thick lens' formula.
In Gaussian optics, a thin lens is such that its thickness is so negligible that all refraction is considered to happen only at one plane. The thin lens concept in physics do not consider optical aberrations or dispersion (chromatic aberration) of light. The thin lens has two refracting surfaces, however, the lens thickness is so negligible that all its refraction is considered to happen at one plane. Or in other words, the thickness of the lens is disproportionately smaller than its radius of curvature. Biometry formulas like Fyodorov (1975), Colenbrander (1973), Binkhorst (1975) were thin lens theoretical vergence formula.
In contrast, a thick lens formula considers the lens optic to have separate pairs of refracting plane - one anterior and the other posterior refracting plane. The plane at which the first refraction occurs is called the first principal plane, while the plane at which the second refraction happens as light comes out of the lens is called the second principal plane. For calculation of IOL power by thick lens formula, the second principal plane is more important and therefore taken into account by new age theoretical formula like Barrett ( The Barrett Universal formula has been explained at this site and can be accessed at https://www.quickguide.org/post/the-science-behind-barrett-universal-ii-formula ).
In Gaussian optics, a thick lens will have cardinal points, that is, three pair of points or planes located on the optical axis of a rotationally symmetric lens.
For a lens, the basic imaging properties such as the image size, focal length, object distance are with reference to these three pair of cardinal points. Without the cardinal points, it would be difficult to understand the focal length of a thick lens, as we are left to assumptions to draw the focal length of the lens with respect to different alternate surfaces of the lens, like, lens vertex, centre of the lens, etc. Cardinal points are therefore theoretical points that optical engineers use to construct a lens with a given power. The Barrett Universal formula (https://www.quickguide.org/post/the-science-behind-barrett-universal-ii-formula) is a ‘thick lens’ formula, which use the cardinal points to understand the IOL power and Effective Lens Position (ELP) of the lens.
Cardinal Points in thick lens Gaussian optics consists of:
1. Object and Image focal points.
2. Prinicpal points/principal planes - front and back
3. Front and back nodal points.
Principal planes - The back principal plane (which is of our interest in IOL calculation formula) can be constructed by tracing a single ray that comes from infinity from the left and gets refracted after crossing the lens anterior surface, passes through the lens material and gets refracted again as it emerges from the posterior plane of the optic before it comes to focus.
In Fig2, a ray of light coming from Point A falls at the anterior lens surface (point B), refracts and further bends/refracts to fall on the image focal point I. If you draw a line backwards such that line IC meets the extension of the ray of light AB at point P’’, then the Point P” is second principal point. Thus you can draw multiple rays being refracted backwards to get multiple principal points. The line that joins all such principal points is the second principal plane, which interacts with the optical axis of the lens at point P (fig 3). Similarly, the first principal plane on the anterior side of the lens can be created, but the second principal plane is more important with respect to thick lens vergence based theoretical formula like Barrett Universal, as this is the plane that accounts for the calculation of the Effective Lens Position of the Lens (ELP). Accordingly, in Barrett Universal, which is based on Gaussian thick lens formula, the Lens Factor is the distance from the Iris plane to the second principal plane of the lens, accounting for determination of the ELP of the lens.
The focal point of a thick lens is the distance from the point where all light converges to the second principal plane, and not to the lens vertex or any other place.
Nodal Points: The nodal points in Gaussian optics, is referenced to the principal planes. Like principal planes, there are two nodal points of the thick lens - the front and back nodal points. The nodal point is described as the incident ray passing through the front nodal point and emerging through the second nodal point, all leaving at a same angle (fig 4). That is, though the incoming rays are refracted, yet the outgoing rays from the lens do not change the direction of path in relation to the incident ray.
Having known all the three pairs of cardinal points of a thick lens, it is easier to calculate the power of the lens
Applying the three pairs of cardinal points, we can arrive at the lens maker’s formula.
------------ (1)
Where o is the object distance, i is the image distance, and f is the focal length.
Lens maker's formula is the relation between the focal length of a lens to the refractive index of its material and the radii of curvature of its two surfaces. It is used by lens manufacturers to make the lenses of a particular power from the material (glass) of a given refractive index.
The above equation can be further broken down in the thin lens formula:
where f is the focal length of lens, n is the refractive index of lens medium, R1 is the anterior radius of curvature, and R2 the posterior radius of curvature. This formula is based on thin lens and therefore does not take into consideration the thickness of the lens, since it assumes the thickness of the lens is infinitely small than the radius of curvature of the lens.
The refraction of light by the eye is a complex process because there are more than one refracting surfaces, that is the cornea and the lens, and the distances between these refracting surfaces need to be considered also. Light has to pass through varying optical medium with different refractive indices. Keep in mind, the human crystalline lens itself has multiple refractive indices. Thus Gaussian optics and thick or thin lens formula may not be easily applied to get an IOL power with such complex dynamics in place. Therefore models of eye that help to apply Gaussian optics after taking into account the complexities of eye have been developed for well over two centuries.
The first accurate schematic eye is attributed to Listing, who in 1851 developed an eye with a single refracting surface cornea and a homogeneous lens refracting surface. However, it is Allvar Gullstrand (1862-1930) who is credited to have developed the most comprehensive schematic eye that used a four lens surface (two refracting surfaces of the cornea and lens) which included lens refractive index complexities. This is also known as Gullstrand exact eye.
Gullstrand exact eye give us the measurements of the eye in both accommodated and non-accommodated states.
In the above image you can see the details of the schematic eye or model eye. The values of the refractive indices of the various refracting surfaces are illustrated. The difference in the refractive index of medium on both sides of the refractive surface as well as their radius of curvature determine the degree of refraction. The refractive index of the air is 1.0 and that of the anterior cornea is 1.376. The big difference in the refractive index of these two mediums explains why the anterior cornea has a higher power than the posterior cornea. The refractive index of aqueous is 1.336, which is of little difference from the corneal refractive index. Hence the posterior cornea accounts for less bending of light compared to the anterior cornea.
It is because of this that a swimmer without googles cannot see well under the water. Since the refractive index of water is 1.33, there is little bending of light at the water-cornea interface, as the swimmer dips his/her head in water. That is why you will find swimmers wearing a covered googles in water.
Cross-sectional image of Gullstrand exact eye with all three pairs of cardinal points.
Image credit: Clinical and Experimental Optometry 78.5 September-October 1995
So, in Gullstrand exact schematic eyes an attempt is made to represent the eye with all its measurements as exactly as possible.
Using the corneal refractive indices and radius of curvature as given by Gullstrand, the net corneal power can be derived by applying Gaussian thick lens formula optics:
Where, Da is the anterior corneal power; Dp is the posterior corneal power; ra is anterior radius of curvature of cornea; rp is posterior radius of curvature of cornea; Dnet is the total/net power of cornea.
Thus Gausian formula for total or net corneal power if you have to put in one single equation with all details would be
Dnet = 1.376 - 1.0000/ra + 1.336 - 1.376/rp - d/1.376(1.376 - 1.000/ra x 1.336 - 1.376/rp)
In reduced schematic eyes, many of the ocular complexities are ignored with the cornea and the lens regarded as having single refracting surfaces. The thin lens theoretical formulas like Fyodorov, Colabrander, etc follow such a reduced eye.
The below is a calculator that will help you to calculate the power of the IOL (lens) with the help of Gaussian thick lens formula if you know the radius of curvature of the anterior and posterior of IOL, the refractive index of the IOL, and the thickness of the IOL. The same formula can be used to calculate the corneal power if you know the radius of curvature of anterior and posterior sides of the cornea, input the refractive index of cornea (1.376), the thickness of the cornea, and also input the refractive index of air (1.0) and the refractive index of aqueous humor (1.336)
A calculator that helps you calculate the IOL power based on thick lens formula. Note, you can calculate the power of the cornea or any refractive medium, if you know the radius of curvature of the medium, the refractive indices, and thickness. (quickquide.org)
Limitations of Gullstrand eye and Gaussian optics formula:
A Limitation of Gullstrand eye is that it assumes the eye to be rotationally symmetric. The word rotationally symmetric means that the lens has an optical axis. Thus when the lens is rotated on its optical axis, the physical dimensions do not change, that is, the lens look similar even after being rotated on its axis. We understand that the eye is not rotationally symmetric. The real eye is thus aspheric, it is not a spherical surface. In Gullstrand eye, another limitation is that the fovea is positioned in line with the optical axis. In real eyes, the fovea is offset and there are natural decentrations of the eye - angle kappa and angle alpha.
On the other hand, Gaussian optics is limited by the fact that it takes in account only the paraxial rays of light and does not take into account marginal rays of light. However, we know in the eye, spherical aberration plays an important role and therefore paraxial ray- based lens calculation has a room for improvement. Thus, the retinal image cannot be well described by rays close to the optical axis alone.
Herein comes the importance of ray tracing in IOL power calculation. Before I go on to describe ray tracing in IOL power calculation formula, a quick review of ray tracing in optics will help set the right perspective. Ray tracing is a method used by optical engineers to determine optical system performance. Ray tracing is the act of tracing all rays of light through a system and calculating the angle of refraction at each surface of the lens. Thus in physics, ray tracing is a method to calculate the path of waves or rays that pass through the entire surface of the lens.
The Olsen (PhacoOptics®) as well as OKULIX Ray-Tracing-Calculation for the Pseudophakic Eye (Panopsis GmbH, Mainz, Germany) are ray tracing formula that takes in account both paraxial rays of light as well as marginal rays of light entering the eye. The advantage of ray tracing is that spherical aberration of cornea is taken in account. This may be especially useful in Lasik patients. In the post-LASIK case the front surface of the cornea is abnormal and it is not possible to apply standard models for the corneal optics. If you have more detailed information on the corneal asphericity (corneal asphericity is described in the article https://www.quickguide.org/post/spherical-aberration-cause-effect-of-undesired-higher-order-aberration) as well as front and back curvature of the cornea, ray tracing formula like PhacoOptics (Olsen) or OKULIX® can use these values directly in the calculations for better refractive outcome.
Thus ray tracing is based on Snell’s law of refraction :
n1 sin(i) = n2 sin(u)
where i is the angle of incident ray on the surface of the lens, u is the refracted angle, n1 is the refractive index of the first medium and n2 is the refractive index of the second medium. To get the full advantage of ray tracing formula it would be important to enter the full details of the eye and the lens. These are the radius of curvature as well as asphericity of the anterior and posterior corneal surfaces, IOL thickness, refractive index of the IOL, etc.
The advantage of ray tracing formulas is that it accounts for not only paraxial rays of light per Gaussian optics but also accounts for each marginal ray of light. Preussner (JCRS 2002) et al states that in the pseudophakic eye, difference between paraxial rays calculated with Gaussian thick lens formula and off-axis rays of light can be as much as 3.0 diopter in a worst case scenario. Using ray tracing, Olsen found that standard keratometers that use a Keratometry index of 1.3375 measures the corneal power as one dioptre higher than when one measures with true indices of cornea with ray tracing methods. However, the exact refractive indices of cornea can not be used in any traditional IOL calculation formula, as most traditional IOL calculation formula are based on an assumed K index of 1.3375.
The ray tracing is not the only feature of the Olsen formula. It brings in an unique concept of C-constant to predict the Effective Lens Position ( ELP) of the IOL. The C constant can be thought of as a ratio by which the empty capsular bag will encapsulate and fixate an IOL following in-the-bag implantation. It is based on the observation that after standardized lens surgery and in-the-bag implantation, the IOL tends to locate itself in a defined manner that is predictable according to the formula
IOLc = ACDpre + C x LTpre
where IOLc is the center of the IOL, ACDpre is the preoperative anterior chamber depth (ACD) (including corneal thickness), LTpre is the preoperative thickness of the crystalline lens, and C is a constant related to the IOL type.
However, one must not get the impression that Ray Tracing formula are a magic stick, which would help overcome poor refractive outcome. At its best, Ray Tracing formula can only add small incremental benefit in a virgin average eye by taking into account Higher Order Aberration (HOA) like spherical aberration. Most studies on such formula however show a benefit in myopic lasik patients, because of the altered shape of the cornea.
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References :
PhacoOptics IOL Power Calculation and Data Management - November 2017
Ray-Tracing Analysis of intra ocular lens in sity - Thomas Olsen, Mikkel Funding;J Cataract Refract Surg 2012; 38:641–647
Schematic eyes: history, description and applications - George Smith; Clinical and Experimental Optometry 78.5 September-October 1995
C constant: New concept for ray tracing–assisted intraocular lens power calculation- Thomas Olsen, Peter Hoffman; May 2014 Journal of Cataract and Refractive Surgery 40(5):764–773.
Ray tracing for intraocular lens calculation Paul-Rolf Preussner, MD, Jochen Wahl, MD, Hedro Lahdo, MD, Burkhard Dick, MD, Oliver Findl, MD; J Cataract Refract Surg 2002; 28:1412–1419.
Disclaimer: Opinion are based on experience and research and are for explanatory and illustrative purpose only and in no way be construed as definitive.