Ординатура / Офтальмология / Английские материалы / Wavefront Analysis Aberrometers and Corneal Topography_Boyd_2003

As depicted in Table 2, each term of this basis of order k is a combination of Taylor terms of order k or lower. The normalization factor expressed by a square root in Table 2 makes the basis orthonormal. The Cartesian and polar expressions of the

terms Zn (n=0 ... 230) are given in reference[9]. The

highest power of the normalized radius ρ(0≤ρ≤1), is called the order of the term Zn. Thus, there are N+1

terms for each order N.

Properties of Zernike Basis: Some properties of this basis are summarized in Table 3. Other properties are given in reference[10]. One of the most useful properties of the Zernike basis is the fact that individual terms in the wavefront W(x,y) can be related to common aberration behavior such as spherical aberration, coma and high order astigmatism. It means that the Zernike expansion classifies the deformations inside the wavefront. In addition, this basis has the advantage to be orthogonal in a continuous fashion over the interior of a unit circle : ρ2 = x2 + y2 < 1. We note that the basis is in general

Section IV: Aberrations and Aberrometer Systems

not orthogonal over a discrete set of data points within a unit circle. In Table 2, 36 Zernike terms (orders up to 7) are listed and presented in Cartesian and polar form. Each order N involves N+1 terms. The polar form is more convenient for two main reasons. First, this form represents the terms of the basis in form of a product of two separable functions in ρ and θ: Zn(ρ,θ)=Rn(ρ)Gn(θ), where the radial func-

tion Rn(ρ) is a Taylor polynomial in r and Gn(θ) is a periodic trigonometric function. Second, the polar form evidences the properties of the Zernike basis.

The list of terms in Table 2 points out that each term contains an appropriate amount of each lower order term to make it orthogonal to each lower order term. For example, Z8 contains an appropriate amount of Z2, whereas to Z31 an appropriate amounts of Z17, Z7 and Z1 have been algebraically added. In addition to orthogonality, the algebraic addition of amounts of lower order terms to high order terms offers some interesting properties. This addition makes each term minimize the RMS wavefront error (Eq. 1): the RMS of Zk(ρθ) is equal to 1. Moreover, except for the first term, Z0(ρθ) called piston, the average value of each term over the unit circle is zero. In other words, if we consider the volume under the curve of the wavefront function W(x,y), we note that half of this volume is over the reference plane (x,y) whereas the other half is under this plane.

Wavefront Error: To evaluate wavefront error, we generally opt for the standard deviation, namely the RMS error, which expresses how distant the wavefront W(x,y) is from the zero function. To be more precise, the RMS value is the normalized volume under the curve of the square of the function W2(x,y). Because the normalized Zernike basis is an orthonormal basis, the RMS wavefront error is the square root of the sum of the square of all coefficients ak of the Zernike terms:

(2)

We note that the wavefront W(x,y) must be normalized to the unit circle 0< ρ2 = x2 + y2 < 1 in order to apply relation (2).

Transfer of Basis

Moving from one basis to another is possible. To do this, we need to know two things. First we need to know the expression of the function W(x,y) in the first basis B. Second we need to know the expression of each element of the first basis B in the second basis B’. We need to know the coefficients an in the following equation:

W(x,y)=a0 B0(x,y)+ a1 B1(x,y) + a2 B2 (x,y) +...+ aN BN(x,y) (3)

or in a matrix form:

(4)

Where each element Bn(x,y) of the basis B can be decomposed in the basis B’ as follows:

Bn(x,y)=M0nB’0(x,y)+M1nB’1(x,y)+M2nB’2(x,y)+...+MNnB’N(x,y)

(5)

or in a matrix form:

(6)

Thus, we obtain:

(7)

or in a compact form:

W(x,y)=AT(MT (B’=A’T (B’ (8)

where M is the transfer matrix. By simple algebraic multiplication of M with the coefficients A of the function W(x,y) in basis B, we determine the coefficients A’ of the W(x,y) in the basis B’.

For the opposite situation, we need the inverse matrix M-1 which we apply according to following transfer equation:

A=M-1 (A’

The inverse matrix M-1 can be determined analytically or numerically by matrix inversion of M. If we opt for the analytical approach then we need to express each element of the basis B’ in a linear sum of elements of the basis B:

Section IV: Aberrations and Aberrometer Systems

B’n (x,y) = M-10nB0(x,y) + M-11nB1(x,y)+M-12nB2(x,y) + ... + M-1Nn

BN (x,y)

However, the numerical solution can be a serious source of error if the basis is composed of a large number N of elements. Error increases cumulatively during matrix calculus[9].

Let us now turn to wavefront aberration where two basis "Taylor" (B’) and "Zernike" (B) are brought into play. The elements of the basis B’ and B respectively are listed in Table 1 and 2. The analytical expression of the Zernike terms (elements of the basis B) is generally available in polar form. To determine the elements of the matrix M, we only need to express the Zernike terms in Cartesian form (Taylor) as performed in Table 4.

Now to perform the transfer from the Taylor to the Zernike basis, which is the most frequent situation, we should determine the inverse matrix M-1. In general, this transfer matrix is numerically calculated through matrix inversion. To avoid any source of numerical error (discussed in reference[9]), we developed a method that analytically determines the matrix where each element of the matrix M-1 is exactly represented by a rational number [9].

Types of Aberration

It is important to use the Zernike basis which decomposes the wavefront W(x,y) into terms with different optical behavior such as the two well known aberration types namely spherical aberration and coma. In other words, the Zernike expansion classifies the deformations inside the wavefront. Zernike terms with lower orders (smaller than 3) are generally not considered as aberration types because they can be corrected by conventional optics. One of the most important lower order terms is the defocus term (Z4 in Table 2). This defect arises when the image plane is longitudinally shifted with respect to the output plane. For the human eye, this means that the optical system focuses in a wrong location. Therefore the focus term merely represents myopia or hyperopia. Similarly, the tilt (Z1 and Z2) is not considered as aberration because it can be compensated by a conventional optical element namely the prism.

Theoretically, defocus can be considered as low order spherical aberration (Z12, Z24,…,

Z2n(n+1) , …). We see from Table 2, that these symmetrical aberrations include some amount of defocus. Spherical aberration arises when the optical system does not focus all parallel input rays in one single point. The focal point is longitudinally shifted depending on how far the ray is from the center. In other words, spherical aberration can be considered as radius-dependent defocus. Parallel rays entering the optical system near its center will be focused less

or more than parallel rays entering near the edges of the pupil. What we observe on the retina is a symmetrical blur surrounding the paraxial retinal image.

Coma is represented by asymmetrical Zernike terms (Z7, Z8, …, Z2n2-1, Z2n2, …). We see from Table 2, that these asymmetrical aberrations contain some tilt, that can be considered as a lower order coma (n=1). Like spherical aberration, coma shows some defocus of the image, but the blur is asymmetrical. The image of a bright point will have the shape of a comet. Coma can occur in the human eye generally for two main reasons. Firstly, it can appear when ocular components are not coaxial. Secondly, it can result from pupil decentration [11].

For a given object, reference [12] illustrates the behavior of the image when the amount of spherical aberration or coma increases or decreases.

HOW TO EVALUATE

ABERRATION

Aberrations can be evaluated at two levels. They can be evaluated 1) by directly analyzing them according to a certain reference or 2) by analyzing their impact at the output plane. In the first case we speak about the optical quality of the system because we analyze the system itself. In the second case we analyze the image plane and therefore we should speak about image quality. The question is whether a certain link exists between the two qualities.

Optical Quality

To measure the optical quality of the system, a reference is required. The reference should correspond to the ideal situation. If the system does its job in an ideal fashion, it should produce an image that is identical to the object in shape (not necessarily with the same size and orientation). Given that we are interested to what we observe at the output, the reference should be linked to the image. Thus the refer-

ence is the wavefront that produces the ideal image. The question is: which object’s image should we consider? Given that the object and the image are sets of points, we generally choose a single point as a reference object and in particular the axial point situated at infinity. Therefore, the reference is the spherical wavefront centered at the point image corresponding to an infinitely far axial point object.

Several statistics are used to evaluate the quality of optical systems [13-17]. The most commonly used statistic is the Root Mean Square (RMS) error. However, the criterion of Rayleigh to define perfect optical systems uses another measure based on the worst case error. In fact, Rayleigh suggested that if light arriving at the image point is never more than π/2 radians out-of-phase, the image would differ negligibly from that formed by a perfect system [18]. In other words, perfect diffraction limited images are formed by an optical system when all light rays entering the system converge at the focal plane with an optical path error of a quarter of wavelength of light or less. This criterion is based on the Peak-to- Valley (PV) error which is a worst case error statistic. This wavefront error is a measure of the distance from the highest to the lowest point on the deformed wavefront relative to the reference wavefront. Since this only compares two points on the surface, it is possible for two very different wavefronts to have the same PV error. This error does have the advantage that it is very easily estimated visually from the wavefront profile. It has the disadvantage that one small dig or a narrow crest in the wavefront can cause the PV error to be very large even though the optic may perform quite well.

In contrast to the PV error, the Root Mean Square (RMS) error is an area weighted statistic. As explained above (Eq. 1), the RMS is calculated as the standard deviation of the height (depth) of the wavefront relative to the reference at all the points in the wavefront. Due to its statistical nature, the RMS

Section IV: Aberrations and Aberrometer Systems

wavefront error is a very useful measure of optical quality [19]. Because the normalized Zernike basis is an orthonormal basis, the RMS wavefront error is the square root of the sum of the square of all coefficients of the Zernike terms.

Image Quality

In contrast to the RMS error, the Strehl ratio is more of a measure of optical excellence in terms of theoretical performance results rather than a quality criterion for the physical surface or the shape of the wavefront. Because the Strehl ratio is measured in the image plane, one can associate this ratio to image quality rather than optical quality. In fact, this measure includes indications on the image plane. It is defined as the ratio of the maximum intensity of the actual image to the maximum intensity of the fully diffraction limited image, both of them being normalized to the same integrated flux. In other words, The Strehl ratio is an expression of the amount of light contained within the Airy disk as a percentage of the theoretical maximum that would be contained within the disk with a perfect optical system. Although the Strehl ratio and the RMS are defined into different planes (system and image plane), they are linked together for small amounts of aberrations. In this case, the simple approximation formula, called Maréchal [20] is valid: Strehl Ratio = 1 - (2π RMS)2[21]. Some other approximations for the Strehl ratio have been suggested in literature[22, 23].

Optical and Image Quality

One might want to simultaneously combine the two qualities into a single measure. A criterion is required that evaluates the optical excellence in terms of the shape of the wavefront as well as the theoretical performance results. An example of such sta-

tistics has been recently proposed. It is referred to as the "Optical Transfer Error (OTE)" or the "Admittance dynamics", and it presents a measure of quality at both levels: system and image. Indeed, it links system quality to image quality. We will give more details for this measure later in a separate subsection.

Measures in Form of Functions

Some functions are used as measures for optical performance. As we will explain later when discussing the optical effects of aberration, the Point Spread Function (PSF) is a measure of how well one object point is imaged on the retina through the optical system of the eye. The Modulation Transfer Function (MTF) is a measurement of the system ability to transfer contrast from the object to the image plane at a specific resolution.

OPTICAL EFFECT OF

ABERRATION

A deformed wavefront provides a blurred image. However, the form of the blurred image depends on the type of aberration. For instance, symmetrical wavefront aberration produces a symmetrical blurred image. Now for a given object, if we increase the amount of aberration without changing its shape (multiplying the wavefront aberration by a constant bigger than 1), image quality will be further decreased. So we can say that the image quality decreases with the RMS value if the aberration form is maintained. However, it is not obvious that two wavefront aberrations with the same RMS error but with different types produce the same degradation of image quality[12]. Let us discuss these aspects in what follows. For a given system, the optical relationship between the image and its object is not identical for coherent and incoherent illumination. Because it depends on the illumination mode, dis-

tinction between coherent and incoherent illumination must be made when analyzing the optical performance of the system. In a first time, we will ignore diffraction effect caused by the finite size of the pupil.

Coherent Illumination

If the system is ideal (aberration-free), we should obtain a convergent spherical output wave. This ideal wave is considered as a reference. However in reality, we obtain a deformed wave, and the output wavefront differs by a wavefront aberration W(u,v) from the reference wavefront. Let AB(u,v) denote the optical field resulting from the wavefront deformation.

Transfer Functions

The image of point object at infinity is the response of the optical system when it is illuminated by AB(u,v) instead of the plane wave, that results from Fraunhofer diffraction field of the point object at infinity. Fraunhofer diffraction is mathematically formulated by a Fourier Transform[24-28]. Diffraction from infinity through the air results in a Fourier integral because of the infinite longitudinal succession of Hygens wavelets [28]. Fraunhofer diffraction can be observed at a finite distance if we use a spherical lens or diopter or mirror and observe the replay field in the back focal plane. As a consequence, the optical components of the eye also involve Fraunhofer diffraction. The response of an aberration-free system of the eye for an object at infinity can be obtained by a two successive Fourier Transforms. The first transform merely results from light propagation through the air and the second FT is performed by the nonaccommodating system of the eye. For a near object, the accommodating system of the eye behaves as an imaging system where the image plane is situated at the retina for an emmetropic eye. This imaging system can be modeled by two successive FTs. So the

flow diagram of figure 4 is valid for both near and far objects. The FT is exactly cyclic with order 4 and cyclic except of point symmetry with order 2 (output is a flipped version of the input). Because of this property, the image of any object through the aberra- tion-free system of the eye is a flipped version of the object itself but with a different scale. For a real eye, a modulation by the aberration function separates the two FTs as depicted in Figure 4. The pupil function performs a low-pass filtering [29]. In other words, we can ignore the pupil function if instead of the object, we use a low-pass version of it. We remind the reader that diffraction caused by the pupil is ignored.

Section IV: Aberrations and Aberrometer Systems

The aberration function AB(u,v) is also called the "Optical Transfer Function (OTF)" [29] or the "Optical Admittance". In the ideal case, the OTF transfers all frequencies without any modulation (modification). In reality the OTF is different from 1, and therefore it transfers the frequencies after some modulation (magnification or dumping). For this reason, the modulus of the OTF is called the "Modulation Transfer Function (MTF)"[27-29]. If the MTF is a wide function (large bandwidth – large frequency extent) with values close to 1, the frequencies are slightly modulated, which results in the image being closer in form to the object (low degradation of the image quality). However, if the MTF is a narrow function (narrow frequency extent) then the system acts as a low-pass filter and the high frequencies are

dumped. The optical system lets low frequencies go through and blocks high frequencies. As a consequence, the contour of the object is not reproduced in full in the image plane. In other words, the transition between high illumination and low illumination is no longer brutal. Thus, the contrast sensitivity is reduced. In our case of coherent illumination, the