Radius dependence of Zernike Polynomials 𝐙𝟐𝟎 and 𝐙𝟒

𝟎 and its impact on corneal topography

Andreas Berke PhD1,3, Christoph Berke2, Gustav Pöltner Dipl.Ing.3,4

1. Cologne College of Optometry, Köln Germany; 2. Technical University Dresden, Germany; 3. University of Applied Sciences for Health Professions Innsbruck /Austria; 4. HTL-Kolleg Optometrie Hall in Tirol, Austria

Zernike Polynomials represent a system of complete orthonormal functions. The completeness of this system is defined solely on the unit circle. Any minor radius is contradictory to the requirement of completeness and limits the use of Zernike Polynomials. Z2

0 and Z4

0 were chosen because they show no angle dependence and are solely radius dependent. A diminution of the radius decreases Z4

0 (“spherical aberration”), but causes an increase of Z2

0 (“defocus”). Parts of the aberration Z4

0 are shifted to Z20 . The depiction of

deviations of a real surface from a reference surface is critical, because the values of the individual Zernike coefficients depend on the radius of the particular corneal zone. Furthermore, the values of the Zernike coefficients critically depend on the reference body that was used. The general validity of the Zernike representation of the corneal topography is narrowed.

To assess the radius dependence of Zernike Polynomials Z20 (“defocus”)

and Z40 (“spherical aberration”) and their impact on corneal

topography. The impact of the type of reference body used upon the Zernike representation is determined.

1s

email: berke@hfak.de; gpoeltner@gmail.com

Table 1: Zernike Coefficients |𝑐𝑛𝑚| for the first 5 orders for a spherical and an elliptical reference body

Introduction

Purpose

Methods

Conclusions

Zernike polynomials are commonly used in optics, optometry and ophthalmology. The customization of Lasik, ophthalmic lenses, and contact lenses is partially based on the application of Zernike polynomials. They are used to analyze wave front aberrations of the eye. With corneal topography they are suitable to describe the deviation of the real corneal topography from an ideal reference body with spherical or elliptical shape. Their advantages are the simple analytical properties inherited from the simplicity of the radial functions and the factorization in radial and azimuthal functions. Mathematically, the Zernike polynomials are a sequence of polynomials that are orthogonal on the unit disk, thus they are strictly radius dependent.

A change of the radius from 𝜌 to 𝜌‘ results into a new aberration W’

𝑊′ = 𝑎4𝑐40 6𝜌4 − 6

𝜌2

𝑎2 +1

𝑎4

which can be distributed among the angle-independent Zernike polynomials Z0

0, Z20 and Z4

0.

𝑊′ = 𝑐40′

𝑍40 + 𝑐2

0′𝑍2

0 + 𝑐00′

𝑍00

The new Zernike coefficients in terms of the original coefficients are 𝑐4

0′ = 𝑎4𝑐40

𝑐20′ = 3𝑎2 𝑎2 − 1 𝑐4

0 𝑐0

0′ = (1 − 3𝑎2 + 2𝑎4)𝑐40

Theory predicts a diminution of Z40 of 28% and increase of Z2

0of 26% when the radius is changed from 9.0 mm to 6.5 mm, which means that a is equal to 0.722. The theoretical values of the changes of Z2

0 and Z4

0 are in good accordance with the experimental values. The particular choice of the reference body is reflected by the values of the angle-independent Zernike polynomials Z2

0 and Z40 . Using a

spherical reference body increases the Zernike coefficient 𝑐20 by a

factor of nearly 6 and 𝑐40 by a factor of more than 3. The angle-

dependent Zernike polynomials and their coefficients are not changed, but this seems to be due to mathematical reasons, because it is much easier to redistribute the altered aberrations to angle- independent terms than to angle dependent terms.

Discussion

Figure 1: Scleral lenses with different central apertures (left: 9 mm; right 6.5 mm)

ε c11 c02 c22 c13 c33 c04 c24 c44 c15 c35 c55 0.75 0.009251 0.000694 0.003635 0.002372 0.000292 0.000628 0.000578 0.000255 0.000304 0.000051 0.000087

0.00 0.009251 0.004055 0.003635 0.002372 0.000292 0.002055 0.000578 0.000225 0.000304 0.000051 0.000087

Three patients (aged between 28 and 52 years) without any corneal abnormalities were fitted with black scleral lenses with a central aperture of 6.5 mm and 9.0 mm respectively (Figure 1). The contact lenses were purpose-made for the present study (FALCO 7.9 mm / 0.0 D / 16.0 mm / x290 µm / ioz 10.0 mm / r scleral 13.0 mm / dm scleral 13.0 mm / apertures 6.5/9.0 mm to 13.0 mm black / material: Optimum extrem, weiß) The corneal topography was measured with the Keratograph 4 (Oculus, Dutenhofen Germany) and the Pentacam Scheimpflug Camera (Ouclus, Dutenhofen Germany). Each measurement was repeated ten times. The theoretical values of Z2

0 and Z4

0 were calculated for both radii and compared to the experimental data determined by the keratometer. Furthermore, the Zernike representation was determined for two different reference bodies with eccentricities 𝜺 = 0.00 and 𝜺 = 0.75 respectively.

Results

Figure 3: Corneal topography for two different reference bodies (left e = 0.75; right e = 0.00)

A decrease of the central aperture from 9.0 mm to 6.5 mm results in a diminution of Z4

0 from about 0.66 ± 0.27 µm to 0.45 ± 0.22 µm (32%) and an increase of Z2

0 from 2.29 ± 0.46 µm to 3.05 ± 0.52 µm (33%).

Figure 2: Changes of 𝑐20 and 𝑐4

0 for two different radii (9mm/6.5mm).

Figure 3 shows the deviations of the real cornea from two different reference bodies with eccentricities of 0.00 and 0.75 respectively. Table 1 specifies the Zernike coefficients |𝑐𝑛

𝑚| for the first 5 Zernike orders. It is easily seen that only the angle-independent polynomials Z2

0 and Z40 are influenced by the choice of the reference body.

The aberration W due to Zernike polynomial Z40 is given by:

W = c40(6𝜌4 - 6𝜌2 +1).

A factor a is introduced to quantify the change of the radius from 𝜌 to the new radius 𝜌‘ : a = 𝜌‘/𝜌

0,0

0,5

1,0

1,5

2,0

2,5

3,0

3,5

c02 c04

[µm

]

9.0 mm 6.5 mm