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# Zernike polynomials

The Zernike polynomials are mathematical functions which are used to analyze and describe the wavefront error of the human eye.  Some of the Zernile polynomials carry the name of classic optical aberrations defocus, astigmatism, coma, trefoil… This semantic is also used by wavefront sensing instruments in their output. In addition, some other optical quality metrics are usually associated with Zernike wavefront decomposition; RMS Zernike coefficients, Strehl ratio, etc.

These maps are available within the software of the OPD Scan wavefront sensor. The Zernike chart display the list of the first 28 Zernike modes (6th order), and their coefficients, for a 6 mm zone. The WF (WaveFront) /OPD (Optical Path Difference) / HO (High Order) is a representation of the wavefront error of the measured eye, relative to an eye which would be « perfect », that is free of aberrations. The PSF (Point Spread Function) chart corresponds to the aspect of focal point on the retina formed by the measured eye when the eye has been corrrected with the best possible spectacle correction (the eye is free form « low order » aberrations). Some of these terms may seem esoteric to te clinician who is not familiar with wavefront sensing: hopefully, this page may be at rescue.

This page is a simplified introduction aimed at familiarizing the clinicians and eye-care practitioners with the basic features and role of Zernike polynomials in the wavefront sensing.

To understand the contribution of Zernike polynomials to wavefront sensing, and avoid too much of the mathematics, it is important to introduce some basic definitions and concepts. In particular, it is important to understand what the Zernike polynomials are used for, which requires the reader to move from the domain of geometrical optics (where light propagates a « rays ») to the domain of wavefront optics (where light propagates as « waves »). In Ophthalmology, Zernike polyomials are mainly used to sample the wavefront error of the human eye. It is also sometimes used to descripe the shape of the cornea.

Although the shape of the cornea dictates most of its optical properties, the Zernike polynomials cannot be interpreted interchangeably when it relates to the shape of an optical wavefront vs the shape of an optical surface. Basically, Zernike polynomials can fit any surface defined on a circular domain.

This page will focus on the use of Zernike polynomials for describing optical wavefront, and it is important to study first the main characters of wavefront sensing – in a simplified way.

## Light rays, photons and waves

First, let us consider a « perfect » optical system (top), and a « perfect » eye (bottom).

The perfect optical system 1 brings all the rays emitted by a distance source to a focal point. The rays can emanate from a point source located at infinity, like a star, which emits light in all directions. Our optical system captures a little portion of this emitted light. If the optical system is perfect, it must brings all the rays to intersect at the same point, which is the focal point. Because of the very long distance from the star to the eye gazing at it, the rays which are captured by the pupil of the eye are parallel. This can be also assumed if the light source is some 4 to 5 meters away (the pupil diameter being less than 8 mm in most eyes).

In fact, light rays do not exist, but allow to conveniently represent the assumed path of « light », which travels in straight line in the emptiness of space. It can be refracted at each optical interface (the surface of a lens, the cornea, etc.) and the change in the direction of the paths obeys to the « law of refraction » which we also call the « Snell Descartes law ».

Light rays can also be conceived as the path taken by photons, which are the light energetic particles, and are without mass. Photons can only be « seen » once they have interacted with some photosensitive membrane; there are of little help here.

Finally, light rays can represent the direction where “light waves” interfere in a constructive manner.  Light waves are representing the oscillation of the electromagnetic field. All this can be simplified and the trajectory of the oscillation of the electromagnetic field represented as a longitudinal sine wave, where crests and troughs alternate regularly. The distance between two consecutive crests (or troughs) is the wavelength of the emitted light. Most natural light sources are polychromatic. In contrary, laser light is very (not completely though) monochromatic. Wavefront sensing is performed in infrared light, and the following discussion will assume a monochromatic source of light.  Infrared is invisible to the human eye, but can travel through ocular medias, be reflected by the retina, and this makes wavefront sensing possible and at the same time comfortable to the patient.

## What is a wavefront?

A wavefront is an abstract construction which is formed by joining all the points which are « in phase » at the same time. A wavefront is also defined on a domain. The rays and wavefront are mutually perpendicular: light rays may be conceived as lines which materialize the local direction of the wavefront propagation.

This figure depicts the relations between the wavefronts, rays, and light waves emitted by a point source. At some distance from the source, the rays within a small bundle are parallel, and the wavefront takes on a flat shape.

In ocular wavefront sensing, this domain is the entrance pupil of the eye.

Along the way from the star to the eye, the points which are « in phase » and which will be  captured by the eye within its pupil are located on a flat disc (the previous figures depicts the points all located at the top of the light waves crests) .

Light slows when travels in a «denser » transparent material, such as the corneal stroma. The higher the refractive index, the less the speed and the shorter the wavelength becomes.

Because of the gross geometry of the eyeball, and the fact that light waves shorten (the speed or celerity of light is reduced) in the ocular medias, the phase relations are modified: the convexity of the corneal dome and the variable thickness of the crystalline lens causes the peripheral light waves to be less retarded than the central ones. Peripheral waves travel slightly more in the air than central ones, which hit first the corneal apex, and are thus retarded relative to the peripheral ones.

Light waves are guided toward a point where they can arrive in phase, where the oscillations will add up to increase the amplitude of the electromagnetic field oscillations. The intensity of light (the energy) is proportional to the square of the amplitude of the oscillations.

Suppose that you could revert the path of the time, what would happen to the light waves emitted by a star? The light would travel back along the same rays, concentrically, and arrive in phase at the center of the star (where it was emitted « in phase » if the source is « coherent »). To form a bright image of a star, all what is needed is to bring some of the emitted light waves to interfere constructively at a point.. and make that point located on a photosensitive membrane (such as a screen, or the photoreceptor layer of the retina).

If the eye is a « perfect eye », then the location where the light waves  interferes constructively after their travel through the anterior segment is at the fovea . In the vitreous cavity, the envelope of the points which are in phase would form an exact portion of a sphere centered on the fovea.

## The optical path

From the star to the fovea, along the path of each « light ray » captured by the perfect eye,  the number of oscillations  (the number of light waves) is equal. This conditions ensures that there all the light waves interfere constructively at each end of the path.

This total number of light waves represent the optical path. Although le wavelength shorten in the eye media, the optical path is equal to the physical distance that light would had traveled if there had been no shortening of the light waves, for the same number of electromagnetic field oscillations which were accomplished along the whole path.

Representation of the optical path of a train of light waves between two time points, when lights travel from A to B, and B to C. The segment AB is in air (n=1), the segment BC is in a glass medium (n>1). The optical path corresponds to the number of oscillations multiplied by the wavelength in air. It can be obtained by multiplying the physical distance with the value of the refractive index of the medium where light propagates.

Thus, in the perfect eye, there are no differences in the optical path taken by all the light waves leaving the very distant light source, and impinging on the fovea. When this condition is realized, all the incoming light waves interfere constructively at the fovea.

However, even in a perfect eye, the law of physics prevent the light emitted by an even infinitesimal light point source to form a true image point. The phenomena of diffraction, which is inherent to the wave nature of light, accounts for this impossibility.

In geometrical optics, light rays are infinitely thin, and can intersect in an infenitesimal point. When lightwaves hit a small obstacle, they are dispersed « around » and beyond it. The edge of the pupil can be seen as a singular obstacle which scatters the light energy in all the directions behind it. However, in practice, most of the incident light energy will be scattered within a small angle, especially when the pupil is large. This is equivalent to consider that the margninal ray is slightly displaced « outward ». In a perfect eye, at the foveal plane, light energy will be spread, as some lightwaves will not interfere constructively in the very center of the formed figure. The point spread function describes the combined effect of aberration and diffraction on the repartition of the light energy at the foveal plane, which is the net result on how the light interferes at that plane.

In a « perfect eye », also called a « diffraction – limited eye », the point spread function (PSF) is not a point but a small patch bright disk surrounded by a very faint halo of light energy. Interestingly, the smaller achievable dimension of the central disk (1.5 microns) is about the same size of the smallest foveal cones ( the effect of the pupil diffraction is minimized when the pupil is dilated).

## The Strehl ratio

For a specific pupil diameter, the height of the peak of the diffraction limited PSF can serve as a reference. The « Strehl ratio » correpsonds to the ratio between the maximal height of the eye’s PSF and the diffraction limited PSF. Because of the effect of the monochromatic aberrations which impair the optical quality of any biological system such as the human eye, monochromatic Strehl ratio of the « best » (less aberrated) eyes is usually not higher than 0.15.

In addition to diffraction (which explains alone the concentric ripples of the left PSF, which is also called a Airy disk), the PSF on the right is « altered » by some coma aberration. This results in an increase in the spread of light energy, and a reduction of the height of the « tallest » peak of the PSF.

## The wavefront error

To apprehend the concept of wavefront error, which Zernike polynomials are there to describe, let us consider two optical systems: optical system 1 and optical system 2.

Optical system 1 is a « perfect system », at least for imaging distant object points. The rays collected from an a point source located at infinity are all refracted in a single point. If we put a screen in the focal plane of the system, we may record a bright point, which is the « perfect » image of the point source (of course we have neglected the diffraction from the edge of the pupil) The wavefront error of the optical system 2 compared to the « diffraction limited » optical system 1 is characterized by local phase advances or retardations (optical path differences, which are represented as little arrows), which cause the incident light energy to spread more than what is imposed by the diffraction only.

In geometrical optics, a perfect system brings into the same focal point all the rays of an incident parallel bundle. When the system is free of optical aberrations, all the incident rays are refracted in a single focal point, which is called the « Point Spread Function (PSF) ». In the case of a non-aberrated system, the PSF is a point (of course, we have neglected again the effect of diffraction) and the envelope of the refracted wavefront (in green) is a portion of sphere ( or a circle in cross section).

Rays can be visualized as pointing to the local direction of the propagation of the wavefront envelope. In the case of a spherical wavefront, all the rays point toward the center of the sphere, and the center of that sphere coincides with the focal point.

The spherical wavefront can serve as a « reference wavefront surface », since it materializes the wavefront formed by a perfect optical system. When some of the incident rays are not focusing in the same location, due to optical aberrations, the refracted wavefront is no longer perfectly spherical. The departure between the shape of this aberrated wafefront and the « perfect spherical » wavefront corresponds to the wavefront error.

Aberrations can occur if the corneal curvature, and the lens or intraocular lens position and « optical powers » do not « match », that is do not to bring all the light waves to interfere constructively at the same location. If the light waves interfere at a same location, but if the retina is located in a different plane, the eye will suffer from pure defocus. Spectacles glasses can selectively modulate the optical path between central and peripheral rays to bring the constructive interference back at the foveal plane.

In general, even when the best spectacle power is selected, human eyes will suffer from some imperfections which will alter the optical path along some rays (the optical path along the rays located at the periphery of the entrance pupil is slightly different to the optical path of the central rays). These imperfections correspond to what is called the « high order aberrations » in wavefront optics.

## Outgoing aberrometry

It is important to note that wavefront aberrations of the human eye are generally measured using outgoing aberrometry, i.e for a wavefront leaving the eye. In such situation, the reference wavefront is not a portion of a sphere, but a flat disc.

The same aberrated optical system 2 is represented. Top : in outgoing aberrometry, the wavefront is collected outside of the optical system. In the case of the human eye, infrared light is emitted and focused on the foveal plane, through a limited pupil aperture (this incident bundle of rays is assumed to be imune to the aberrations of the eye). After foveal reflexion and « backward refraction » by the optical system (cornea and lens in the case of the eye), the wavefront is collected and analyzed (see next).

In a perfect optical system, the exiting wavefront would be a a flat disc (limited by the extent ot the entrance pupil diameter of te eye – the iris pupil). In an aberrated system, the wavefront is not flat, and the departures from that wavefront to its « mean » (which is by convention its  » zero level ») correspond to the optical aberrations.

Zernike polynomials are useful to express the wavefront error, which is a departure from the flat disk, as a sum of elementary deformations, which adds up to reconstruct that  wavefront error. It is useful to keep in mind that the latter corresponds to a variation of from the expected optical path of the reference wavefront. It is generally expressed in ocular wavefront sensing as a distance unit, which is close to the dimensions of light waves : the unit is the microns(1 micron is 1/1000 of a millimeter). If you could measure the length of each arrow, take its square value, sum up all those squared values, and then take the square root of that total number, you would get the total RMS (Root Mean Square) value of the wavefront error. The RMS enables to quantify the importance of the wavefront error.

The RMS value

The RMS coefficient is an important concept. There is no one single RMS though. A RMS coefficient refers to « one » single aberration mode (a Zernike mode), or a group of Zernike modes (ex: coma-like aberration RMS). When the RMS of a group of aberration has to be computed, it is not achieved by adding the values of each RMS coefficient. Rather, each RMS coefficient is squared. The sum of these squared coefficient is then calculated. Finally, the global RMS is equal to the square root of the sum of the squared coefficients (remember Pythagore  calculation of the hypotenuse of a triangle). The value of the RMS coefficient is very sensitive to the value of the wavefront domain diameter (pupil diameter).

The RMS is expressed in microns, and corresponds to the « weight » of a specific mode, or a group of modes, in the wavefront error.

## Schack-Hartmann wavefront sensing

After emision by a laser diode of a bundle of infrared light focused at the foveal plane,  its reflexion forms an exiting wavefront which exits the entrance pupil of the eye and before  being sampled by a matrix of microlenselets.  The wavefront is tesselated (sampled) into small portions, each of which is focused on a CCD sensor. The wavefront shape can be deducted form the location of the centroids of the spots array collected by the CCD sensor. Zernike polyomials are useful to interpret the shape, as the first Zernike modes correspond to the classic aberrations which were coined first in optics and astronomy.

Schematic depiction of wavefront sensing with a Schack Hartmann sensor.

In the case of a myopic eye, the collected wavefront will exhibit a spherical shape, which would collapse at the punctum remotum:

Schematic depiction of the wavefront reconstruction of a myopic eye with a Shack Hartmann aberrometer. In this example, the eye is only affected by myopia, and does not exhibit any high order refractive error. Because the myopic eye presents some excess in axial length, the wavefront formed by the foveal reflexion of coherent incident infrared light exits the eye with some central retardation. Without the optical system of the aberrometer, this wavefront would collapse to the punctum remotum of the eye, which is locate at finite distance in the case of a myopic eye.

The wavefront shape of a myopic eye is dominated by a a spherical curvature, which central radius is equal to the punctum remotum (PR) of the eye, where most of the wavefront collapse (if myopia is the only aberration of the measured eye the wavefront entirely collapse at the PR). The Zernike defocus term also has a strong curvature. It is parabolic (a parabola is a curve obtained by plotting Y= X^2). and this shape is not exactly spherical, but the discrepancy can be somehow neglected at that scale.

When the exiting wavefront meets the surface of a “concave lens” with appropriate power, (due to the curvature and refractive index >1 of the lens), its propagation speed decreases (inside the lens material, the wavelength shortens). Since the peripheral portion of the exiting wavefront reaches the lens medium before its center, the latter, which still travels in the air, gains some phase advance. This causes the wavefront to take on a flat geometry, if the thickness of the concave lens is adjusted to cancel the phase shits.

The refraction of a purely flat wavefront forms a regular spots array after refraction by the microlenselet array of the Schack Hartmann sensor. The power of the “concave lens” which is required to flatten the wavefront corresponds to the myopic error in that plane. If the wavefront is not perfectly flat, due to what is called “high order aberrations”, the spots array is slightly irregular.

Schematic depiction of the wavefront reconstruction of a myopic eye with a Shack Hartmann aberrometer. In this example, the eye is not only affected by myopia, but some positive spherical aberration as well. Positive spherical aberration can be seen as an excessive myopic error close to the edge of the pupil. The spot array after refraction by the microlenselet array of the “flattest” possible wavefront is not regular but present some residual distorsion in the periphery.
Positive spherical aberration would be caused by the induction of excessive phase advance for the wavefront periphery; due to this optical path difference, the peripheral rays may not interfere in phase with the central ones at the lens focal plane. The calculation of the wavefront error can be performed by measuring the distances between the actual spots and their expected positions, as this distance is proportional to the local slope of the wavefront.

The shape of the wavefront can be measured from the departure of each sport with its expected position, that is the position that the spot would occupy in the case of purely flat wavefront (see next). Depending on the asphericity and refractive index values of the lens, the wavefront may take on a spherical curvature and converge to a sharp focal point (no spherical aberration). Positive spherical aberration would be caused by the induction of excessive phase advance for the wavefront periphery; due to this optical path difference, the peripheral rays may not interfere in phase with the central ones at the lens focal plane.

We have enough information to focus on the Zernike polynomials, and their role in wavefront analysis.

## The Zernike polynomials

Once we understand what the wavefront error is, we can focus on the Zernike polynomials.

The Zernike polynomials correspond to a kind of « taxonomy » for the optical aberrations that explain a wavefront error.

A wavefront error can be plotted with a color scale (micron steps units), on a domain which is the pupil disk:

In the polar coordinates system representation (ANSI recommendation), the wavefront error depicts the optical path difference (in microns) with a reference surface. The above representation corresponds to the optical path difference with a flat wavefront (green level). Each color step represent a phase shift of 1 microns. The zero level is the « mean » of the wavefront, which is the plane that separates in two equal components the relative phase advance and phase retardation. The « height » of that mean with regards to the lowest point (the most retarded point of the wavefront error) corresponds to the RMS value of the first Zernike term, named the piston.

### The Zernike pyramid

The Zernike modes are usually presented as a pyramid.

Representation of the Zernike pyramid formed by the first 28 modes (the Zernike pyramid has an infinite number of modes). The mean value of each mode (execept for the piston) is zero. Think of each mode (or harmonic) as a kind of particular wavefront error within the eye’s pupil. Some modes, such as defocus (Z 20), are familiar to the clinician, whereas other (ex: tetrafoil : Z 4-4) are not.

If you are inclined to more abstract thinking, you can think of each mode as a unit vector. In a 2 dimensional (2D) Cartesian domain, you could plot two unit vectors along the X and Y axis, which would be orthogonal and with a norm (length) equal to one. In a 3D domain, you could add a third unit vector along the Z axis, which is orthogonal to the X and Y axis. In such 3D space, any vector can be decomposed on a weighted sum of the X, Y and Z axis unit vectors.

In the « Zernike polynomial domain », you can extend such concept to mode than 3 « dimensions ». Each of these dimensions as its own unit vector, represented by a particular Zernike mode (each Zernike mode is normalized). Each of these dimensions would be perpendicular (orhogonal) to all the other ones. In such multi-dimensional space,  a wavefront error would correspond to a particular vector, which could be « broken down » in a sum of weighted unit vectors (i.e Zernike mode). The weight of each mode corresponds to the RMS coefficient of a Zernike wavefront decomposition.

You can try to relate the spatial shape of some Zernike modes to some ocular affection. For example, keratoconus, which is characterized by some corneal vertical asyemtry, will easily induce vertical coma Z 3+/-1, in which the wavefront error is displayed vertically and introduce asymmetry between the superior and inferior half of the pupil domain. The coefficient of the coma term would be negative in most cases, as the wavefront error would show relative inferior retardation (due to the low decentered apex) and superior relative advance.  It may not be surprising to find high levels of trefoil Z 3+/-3 or tetrafoil Z 4+/-4 , or any « higher-foils » Z n+/-n=m in eyes which have been operated with techniques such as radial keratotomy.  Horizontal coma Z 3/1  and trefoil Z 3/3 are frequently seen in advanced pterygium, due to the nasal corneal tear film distortion.

### The Zernike polynomials analytical expression

The anaytical expression of the Zernike polynomials seem complex, not to say ugly to the non mathematician.

Each Zernike mode, Z(n,m) is in fact a mathematical function with a normalizing constant (so that the square root of the sum of the squared residuals –RMS- of each function is one on a pupil of radius=1), a polynomial in r, and for some of the modes, a trigonometric function in t.

n is a positive integer called the « radial degree », m is a negative or positive integer called the « azimutal frequency ».

To obtain a particular Zernike mode of « n » radial degree and « m » azimutal frequency, just insert the actual values of n and m in the equations. There is an important relation between n and m, which is conveyed by another variable « k », wich is an integer comprised between zero and half of the sum or half of the difference between n and m. Because of these relations, the difference between the absolute values of n and m cannot be 1, or any odd number. Hence, there are no Z (1,0) or Z (4,-3) Zernike polynomials. When m is equal to zero, there is no angular dependancy. The mode is « rotationally symetrical ». Defocus and spherical aberration are such modes. Of course, regular astigmatism is azimutally variable (m =2). The normalization factor is relatively easy to calculate, it just depends on n. It is a scaling factor, which makes the RMS of each mode equal to one on the unit pupil (radius = 1).

Let us focus on a Zernike mode named trefoil : Z (3,-3). It may help to connect the equations with the shape of the modes.

Representation of triangular astigmatism (trefoil) with the Z3-3 polynomial on the normalized unit pupil disk (green ring). The enveloppe of this polynomial is equal to the product of a 3rd order polyomial radial term (r3 ) where r is the radial distance from the pupil center, and a trigonometric function with a azimuthal frequency of 3 (sin3t), where t correspond to the angle formed with the horizontal line (as for the common astigmatism axis angle plot). The normalization factor is a scaling number wich makes each mode having a total RMS of unity. If you would like to visualize each mode as a vector, the normalization factor makes this vector a « unit vector », having a norm equal to one.

This is another Zernike mode called « secondary astigmatism »:

Example of a non-radially symmetrical Zernike mode (secondary astigmatism). This mode is formed by the product of a normalization factor, a radial polynomial (function of r) and an azimutal function (function of t).

The primary spherical aberration, Z 4/0 is a famous Zernike mode:

Example of a non-radially symmetrical Zernike mode (secondary astigmatism). This mode is formed by the product of a normalization factor, a radial polynomial (function of r) but no azimutal function; it is rotationally symmetrical

The structure of the Zernike pyramid is based on the values of n and m:

In the Zernike pyramid, the position of each polynomial depends on its radial order and azimutal frequency.

In the central column, the modes are rotationally symmetrical (m=0), or invariant by rotation. This central column can be seen as an axial symmetry axis. On each line (same n value), the Zernike modes of opposite azimutal frequency value have the same overall shape, but a different orientation. These « pairs » are required to enable any mode to be freely oriented around 360°, by selectively adjusting the weight of each mode to obtain the desired orientation. For astigmatism, a purely « with a rule (WTR)» or « against the rule (ATR)» orientation would result in a null value of the « oblique » astigmatism component Z(2,-2) and some positive (ATR) or negative (WTR) RMS coefficient value of the Z(2,2) mode.
The first 6 Zernike polynomials correspond to « low order » aberrations (the highest radial term value is equal to 2). These aberrations can be corrected by spectacles. From n=3 and beyond, the remaining modes correspond to « high order » aberrations.

Here follows the analytical expression of the first Zernike mode in radial coordinates:

### Sampling the wavefront error into Zernike modes

A wavefront error can be sampled, or decomposed, into Zernike modes; each RMS coefficient corresponds to the weight of the mode.

The total wavefront error (left) is equal to the sum of all the non zero RMS coefficients  Zernike mode

The values of the coefficients are the direct consequence of the « shape » of the wavefront, and of the diameter of the domain where it is measured.  The larger the pupil diameter, the higher the absolute values of the Zernike coefficients. This increase is exponential (the higher the radial degree, the larger the amplitude of the variation with the pupil diameter).

### Examples

This is an example of a wavefront reconstruction obtained with the wavefront sensor OPDscan III (Nidek)

The wavefront sensor acquires optical data using a rotative slit and an automated skiascopy.

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