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# Novel Wavefront Decomposition Method

## Another publication has been issued on this important topic:

### Watch the videos:

NB : Some repetitions are inevitable and intended for didactic purpose

#### I wish to thank the co-authors who have contributed to this work: Guillaume Debellemanière (MD), Radhika Rampat (MD), and my PhD directors: Laurent Dumas (PhD), and Jacques Malet (PhD).

I also wish to thank the Nidek company and its RnD team to have implemented this new method in a beta-software now able to prcess wavefront data obtained with my OPD-Scan III wavefront sensor.

## INTRODUCTION

### We owe much of our current understanding of the optical aberrations of the human eye to Zernike and famous visual scientists who consensually adopted Zernike polynomials to aid in the standardized interpretation of ocular wavefront data whenever it is convenient to express it in polynomial form.

This paper has been published in the Journal of Refractive Surgery in 2002.

## THE PROBLEM

### Zernike polynomials are not exactly true polynomials in the strict mathematical sense, but the product between a normalization factor, a polynomial of a certain degree (n), and a trigonometric function (sine or cosine) of a certain azimuthal or angular frequency (m).

Analytical expression of the Zernike modes.

### The radial order of a Zernike mode is the « HIGHEST » order but not the only one which may be contained within that mode. One mode of order n may contain lower order terms eg r^(n-2), r^(n-4). Zernike spherical aberration (Z40) contains a lower order term in r^2, which corresponds to a defocus phase error. The Zernike coma mode contains a lower order term in r^1. Conversely, some other modes like trefoil or quadrafoil (all the mode located on the edge of the pyramid) are « pure » in their maximal radial order (ex: r^3 for trefoil).

This table represents the analytical expression of the first Zernike polynomials (up to the 4th order). The radial function may comprise of one term (eg: Z1+/-1 with a unique term in r^1), or of several terms (eg: Z40 with a term in r^4, a term in r^2, and a constant term in r^0). The azimuthal function is a sine or cosine function where the argument (multiplication function) is multiplied by m. Here, modes for which n=|m| are the only ones which do not contain a lower order term in their analytical expression.

### The first 28 Zernike polynomials are displayed in a pyramid where they are ordered vertically by radial degree and horizontally by azimuthal degree.

Zernike modes pyramid.

### – The higher degree component should predict the residual optical defects that persist after correcting the considered eye with the best spectacles correction.

Wavefront derived metrics

## So, the high order aberrations are not purely representing wavefront distortions that cannot be corrected by spectacles.

### Lower order aberrations are located in the first three rows of the Zernike pyramid.

Lower Zernike modes

### Defocus, being spherical (rotationally symmetrical) or astigmatism (azimuthal), is a pure quadratic phase error (r^2). In cross-section, the phase error can be superimposed onto a quadratic function.

Defocus spherical and cylindrical phase error.

### Higher order aberrations should correspond to phase errors which can be modeled with polynomials with a radial degree of 3 (coma), 4 (spherical aberration), and higher.

Higher order Zernike modes (coma and spherical aberration). Left: Irregular astigmatism is the hallmark of the optical properties of keratoconic corneas. Along the contour of the keratoconus apex, which is often displaced inferiorly, the distance travelled by the light waves is longer than along the superior contour – This causes the wavefront surface to be distorted asymmetrically. This distortion cannot be corrected by spectacles. The optical effect of such an aberration is to spread the light in a specific direction. The PSF (Point Spread Function exhibits a « comet » shape. (hence, the term « coma » to depict such aberration which is modelLed by a cubic function). Patients who complain of ghost images or monocular diplopia exhibit increased amounts of coma aberration.
Right: After refractive surgery, patients may complain of halos at night. These are typically caused by spherical aberration. The wavefront error is modeled by a pure r^4 phase error (similar to a Seidel spherical aberration mode) across the pupil. Depending on the sign of the coefficient, the aberration is either positive (typically post myopic LASIK surgery) or negative (post-hyperopic LASIK surgery).

### Most aberrated eyes suffer from a mixture of higher order aberrations. To quantify the degree of wavefront distortion, one cannot simply add the coefficient weighting each of the implied aberrations. Negative and positive coefficients would cancel each other out. Fortunately, using orthogonal modes allows the use of simpler mathematics. Fortunately (for this task), Zernike polynomials were conceived orthogonal:

Orthogonality and RMS calculation.

## However, there is a price to pay for orthogonality: some of the higher order Zernike modes have lower order terms embedded, and this will be the cause of major issues for human eye wavefront analysis.

### This excerpt is from the original paper of Zernike where the polynomials are defined, outside any ophthalmic context:

Classic paper from F. Zernike where his polynomials were defined.

### As you can see in the following list of their radial components,  the Zernike spherical aberration, as opposed to the Seidel spherical aberration, contains a defocus term, and the Zernike coma aberration contains a tilt term. Zernike secondary astigmatism contains a low order astigmatism term.

Analytical expression of the radial polynomial R(n,m) of some of the higher order Zernike modes

### – The orthogonality of the modes of the same azimuthal frequency m (seen as vertical columns of the Zernike pyramid) has to be ensured by the orthogonality of the radial components within the tensor products (a new vector space formed by interaction between two vectors).

Low order terms in higher order Zernike modes.

## THE  TILT IN ZERNIKE COMA

Decomposition of the Zernike coma mode.

### A unit coefficient z31 = 1 micron AND « contains » the equivalent amount of tilt: z11 = 3 microns! The addition of this tilt to the r^3 phase error results in a rotation of the most advanced lobe-like deformation compared to the most delayed lobe-like deformation! It will also cause an artefactual reduction in the RMS of the Zernike coma mode compared to a « pure Seidel-like r^3 » mode by a factor OF 3 !!

Zernike coma vs pure wavefront coma.

### However, the use of a coma Zernike mode to reconstruct such wavefront adds the need for a non-null tilt coefficient, to compensate for the tilt present within the Zernike coma mode.

Zernike decomposition of pure wavefront coma.

### In this theoretical example, we consider a pure cubic wavefront (trefoil and coma-like – without tilt). The functions are normalized (the normalizing coefficients are not shown here). All coefficients are set to 1.

Conversion of pure r^3 (cubic) wavefront coefficients into Zernike coefficients.

### Here is an example of a wavefront decomposition of an eye with keratoconus due to the presence of r^1 terms in the coma modes, artifactual tilt can appear in the Zernike decomposition. This is why, in eyes with increased levels of odd order coefficients such as coma, some concomitant tilt elevation is always observed. This is a problem as wavefront sensing being performed coaxially shouldn’t measure significant tilt. This tilt is an artifact caused by the necessity to cancel the tilt introduced by the coma mode.

Tilt aberration in an eye with keratoconus.

## THE  DEFOCUS IN ZERNIKE SPHERICAL ABERRATION

### A cross-sectional depiction of the decomposition of a Z40 mode into pure r^2 and r^4 phase errors (equivalent to Seidel modes) is shown.

Decomposition of a Z40 mode into its pure constitutive r2 and r4 wavefront errors.

### Spherical aberration should specifically refer to an error in r^4 in the wavefront decomposition. In some emmetropic eyes with high levels of spherical aberration (e.g. post LASIK), the central portion of the wavefront is flat. However, the reconstruction of the spherical aberration with a Zernike SA mode will bring some unwanted defocus, which has to be compensated for.

Zernike spherical aberration vs pure (e.g. Seidel) spherical aberration.

### This compensation will, however, appear in the Zernike decomposition as Zernike defocus; the degree of induction of defocus from the Zernike SA will change with the pupil diameter, as can be inferred from the profile of the aberration, in which the central portion is dominated by the influence of the r^2 term.

Zernike decomposition of pure wavefront spherical aberration.

## The presence of defocus in the Zernike spherical aberration mode is very detrimental to the wavefront interpretation.

Reconstruction of pure (e.g. Seidel) spherical aberration.

### This is a pure r^4 wavefront (rotationally symmetrical spherical aberration – in the Seidel sense, along with 4th order non-rotationally symmetrical modes such as secondary astigmatism and quadrafoil). The functions are Normalized (coefficients not shown here). The coefficients are set to 1:

Conversion of pure r^4 wavefront coefficients into Zernike coefficients.

## CLINICAL EXAMPLE: ZERNIKE DEFOCUS SIGN INVERSION

### This clinical example highlights the consequences of the spurious relationship between defocus and spherical aberration in the Zernike classification:

Clinical example: defocus sign inversion with pupil dilation.

## CLINICAL EXAMPLE: NARROW FUNCTIONAL OPTICAL ZONE

### The patient is emmetropic after LASIK with myopic correction delivered on a small optical zone, but the Zernike predicted refraction is myopic:

Example of the discrepancy between 1) the objective and predicted Zernike refraction, 2) The perceived vs Zernike predicted retinal image.

## CONSEQUENCES

### Because of the contamination of some HOA modes by LO terms, the ability of Zernike modes to predict refraction is significantly altered in aberrated eyes. As HO modes contain low order phase errors, some « compensation » in the low order coefficients is expected.

Low vs High Zernike split.

## We have created new orthogonal and Normalized modes G(n,m) to replace the high order Zernike « impure » modes. These modes are represented in cross-section and in their pyramid; note that their central portion is flat. Their analytical expression doesn’t contain low order terms (2 or less):

New higher order modes G(n,m)

LDHD (Low Degree High Degree) new modes pyramid.

## 1) COMA:

Zernike coma mode vs new coma mode

## 2) SPHERICAL ABERRATION

Zernike spherical aberration vs new spherical aberration

### The image plane metrics are different – this is expected. The defocus present in the Z40 mode accounts for the extra undulations of the PSF:

Image plane metrics.

## 3) SECONDARY ASTIGMATISM

Zernike secondary astigmatism vs new secondary astigmatism. Note the tormented aspect of the higher order Zernike modes (bottom right) which is caused by the presence of lower order astigmatism within their analytical expression (to satisfy the orthogonality property within these modes).

## CLINICAL APPLICATIONS:

### Truly allow envisioning the conception of methods to achieve high-quality visual function / ‘SUPER-NORMAL’ vision

#### See other relevant examples and material :

Pre and Post LASIK aberrations

PresbyLASIK

Other examples

Wavefront sensing

Zernike polynomials

### 3 réponses à “Novel Wavefront Decomposition Method”

1. Roberto Bellucci dit :

Congratulations Damien, an attractive (for us non-scientists) explanation of many doubts and discrepancies found in our daily practice.

2. Dr Damien Gatinel dit :

Thanks Norberto, my pleasure !

3. Nice work and explanation, Dr. Gatinel (and the whole team)! I have some questions, but I prefer to ask you directly next time I see you.

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