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

MTF is a constant function extended over the whole pupil (a window function).

Point Spread Function

If the object is a point at infinity, then instead of observing a point image on the retina - a bright narrow spot with brutal transition between brightness and opacity - we observe a spread spot, called "Point Spread Function (PSF)"[27-29]. The PSF is the spread function observed on the retina when the object is a point at infinity. Thus, the PSF is obtained merely by a FT of the aberration function AB(u,v), which is the OTF in the case of coherent illumination. Because the retina, like any optical detector, is sensitive to light intensity and not to phase, the PSF is defined as the complex square of the FT of the OTF. If we do not calculate the complex square and limit calculation to the FT of AB(u,v), then we call the resulting function ab(x,y) "Amplitude Point Spread Function (APSF)".

Image on the Retina

Let us now consider two points A and B at infinity as depicted in figure 5. The respective images, replay fields, observed on the retina are obtained by a double FT. At the entrance of the eye, each point gives rise to an oblique plane wave because propagation through the air comes mathematically down to a FT. The system of the eye operates a second FT which should produce again a point. However, because the pupil is of limited extent, we then obtain a diffraction spot depending on both size and shape of the pupil. For a circular pupil, the diffraction amplitude distribution has a Bessel form. For an axial object at infinity, the replay field is nothing else than the APSF. What we obtain on the retina for the two points A and B is the algebraic sum of the amplitude profiles of the two diffraction fields. However, for incoherent illumination, the intensity

Figure 5: Images of two points at infinity through the optical system of the eye. A diffraction spot of Bessel form results from the double FT if the pupil is circular.

profiles are summed rather than the complex amplitude profiles. In other words, what we obtain on the retina is the sum of two shifted versions of respectively the APSF for coherent illumination and the PSF for incoherent illumination. Thus because of possible destructive interference, we may see dark regions in the intersection zone of the two diffraction spots in the case of coherent illumination. However for incoherent illumination, interference fringes do not exist. Let us keep the analysis for incoherent illumination for the next section.

For an arbitrary object, the image is obtained by a convolution of the object with the APSF: ab(x,y). We remind the reader that the APSF is the FT of the OTF which is the aberration function AB(u,v) [29]:

obj(x,y) ab(x,y) ?Ft ♦ OBJ (u,v) [AB(u,v) (13)

where denotes convolution and capital letters stand for the FT.

Incoherent Illumination

In contrast to coherent illumination where the OTF and the aberration function AB(u,v) are identical, the OTF is the autocorrelation product of AB(u,v) for incoherent illumination. In other words, the OTF is the FT of the PSF. Since the PSF is a real function, the OTF is a hermetian function: OTF(-u,-v) = OTF(u,v)*, where * denotes the complex conjugate. The MTF is then an even real function. The image is obtained by a convolution of the object with the PSF = /ab(x,y)/2 rather than the APSF = ab(x,y). Instead of relation (13), we obtain the following Fourier pair [28,29]:

obj(x,y) /ab(x,y)/2 ?FT ♦ OBJ (u,v) (AB(u,v) (AB(u,v)

(14)

where denotes correlation.

Since the PSF is the complex square of the APSF, the first function is generally steeper (smaller extent) than the modulus of the second function. Then, the convolution product of (14) provides in general a less spread image compared to (13). However, for some special objects, the convolution product of (14) might possess a larger extent compared to the coherent case (13).

Optical Transfer Error (OTE)

For a given object, two wavefront aberrations with the same RMS error but with different types do not necessarily cause the same degradation of image quality [12]. This is mainly due to the fact that the RMS error does not provide sufficient information about the phase behavior of the optical field corresponding to the given wavefront. Two optical fields may have the same RMS error in terms of wavefront, but the phase is less varying in one field compared to the second. The idea of the Optical Transfer Error (OTE) is based on this interpretation, namely the influence of phase. In other words, it is based on the phase variability of the OTF.

The phase variability of any scalar function can be expressed by the gradient of this function. The

Section IV: Aberrations and Aberrometer Systems

gradient transforms a scalar function into a vector function as follows :

(15)

or in polar form :

(16)

where denotes the gradient.

A criterion "ote" of the phase variability of the OTF may be the intensity of the gradient of this function over its extent:

(17)

where 1/(2π) is an angular normalization factor.

According to relation (17), the "ote", which quantifies the variability of the OTF, is also identical to the RMS signal duration (extent) according to one of the approximation formulas used in signal processing [30]. This signal s(x,y) is, in our case, the APSF for coherent illumination and the PSF for incoherent illumination. The RMS duration of a signal is often approximated by the integral of the second moment of the complex square of this signal:

(18)

where for coherent illumination, s(x,y) = APSF(x,y)=ab(x,y) and OTF(u,v)=AB(u,v), and for incoherent illumination s(x,y)= PSF(x,y)=/ab(x,y)/2 and the OTF is the FT of the PSF. See appendix A for proof.

Relation (18) points out that the OTE is a quality criterion acting at two levels: system level and image level. The first equality of this relation expresses the optical quality of the system and the second equality expresses the image quality for a point object at infinity. Thus, the OTE measures the optical excellence in terms of the shape of the wavefront, like the RMS error, as well as the theoretical performance results, like the Strehl ratio.

For coherent illumination, relation (17) gives after some algebra:

(19)

or in polar form:

(20)

In the case of incoherent illumination, since the OTF is defined by the autocorrelation of AB(u,v), the variability of the OTF is generally smaller than that of the function AB(u,v) itself. This aspect can be seen in the signal plane. The object is convoluted with the complex square of ab(x,y), which is generally less narrow than the APSF itself: ab(x,y). To precisely determine the incoherent OTE, one can opt for a numerical solution. In other words, the integrals of relation (17) will be calculated as a finite sum where numerical derivatives are used [12,31].

For Zernike terms, the radial integral is bounded between 0 and 1. We then obtain for coherent illumination:

(21)

Because of relation (21), the OTE is proportional to the RMS value for a fixed aberration type. The OTE of each of the first 36 Zernike terms is given in Table 2. Reference [9] calculates the OTE for orders up to 20 (231 terms). All terms have an RMS value of 1 but different OTE values. Table 2 and reference [9] point out that the OTE increases with the order for a given aberration type even though the RMS error is fixed to 1. We also see that the OTE decreases with the angular velocity cn in the terms containing angular components Gn(θ)=cos(cnθ) or Gn(θ)=sin(cnθ) (see in Table 2 or [9]). We remind the reader that Zernike terms can be expressed by two separate functions: Zn(ρ,θ)=Rn(ρ)Rn(θ). In fact, the terms with high angular velocity possess low gradients for the radial functions Rn(ρ) and vise-versa. For instance, the reader can compare Z24 with Z27 or Z28 with Z31. Thus, the radial variation is the most determining for the phase variation of the OTF (OTE) and therefore for the image quality (for a given object).

We note that to be more precise in the calculation of the OTE, the tilt elements in the expressions of coma terms should be omitted for coherent illumination. Then, the tilt introduces no alteration of the image but merely a lateral shift.

In summary, wavefront aberrations with the same RMS value can have very different effects on the performance results of the system. For a given object, the degradation of image quality increases with the amount of phase variation of the OTF (OTE). For a given object, high order Zernike terms in general decreases image quality more than low order terms. However, depending on the respective

OTE value, some high order terms of may be less degrading than some low order terms [30]. Moreover, the OTE offers a global statistic for the PSF. It evaluates the extent of this function, which is a very useful information for analyzing the PSF.

Section IV: Aberrations and Aberrometer Systems

The term H is already analyzed in relations (19) to (20). Let us focus on the term G and replace in our case A(u,v) by the pupil function P(u,v):

(24)

DIFFRACTION EFFECT

Until now diffraction has been ignored. All what is said before is only valid if we consider the low-pass version obj’(x,y) of the object as the input signal instead of the object itself obj(x,y).

Diffraction Caused by the Pupil

Let us consider relation (17) and use the amplitude-phase representation of the coherent OTF: OTF(u,v) = AB(u,v) = A(u,v)exp(iϕ(u,v)). We than obtain:

(22)

Because of limited size of the pupil, the OTF is not a pure phase function (constant amplitude). In other words, the gradient of the amplitude function A(u,v) in Eq. (22) should not be ignored. The function A(u,v) can be divided with respect to three parts, namely the interior and the exterior of the pupil and the transition zone which is the contour of the pupil. Over the first two zones, the gradient of A(u,v) is zero whereas on the transition zone the gradient is represented by a Dirac delta function (unit impulse). Thus, taking into account the pupil size Eq.(22) becomes:

Relation (24) points out that G is the square of the extent of Fourier Transform p(x,y) of the pupil (p(ρ,θ) in polar form). If the pupil is of circular shape with radius r, we then obtain:

(25)

The square of p(ρ,θ) is a function composed of a central bright disk surrounded by alternative dark and bright disks. If we suppose that most of the energy (24) is condensed in the central bright disk of p(ρ,θ) according to relation (25), G can be approximated by the extent of this disk. Taking into diffraction effect through the pupil, the OTE expressed in relation (20) must be corrected by the term G.

For incoherent illumination, the diffractionfree OTF is already not a pure phase function: OFT(u,v)=A’(u,v)exp(iϕ(u,v)). Now if we take into account the effect of the pupil function P(u,v), we obtain: OTF(u,v)= A(u,v)exp(iϕ(u,v)), where A(u,v)= A’(u,v)P(u,v). We can then follow the previous analysis (22) to (25).

(23)

Diffraction Caused by the Variability of the OTF

In the XVIIth century, Grimaldi, using a simple experiment (a light source, an aperture and a screen), had observed the progressive transition between light and shadow and regarded the corpuscular theory, supposing the rectilinear propagation of light as insufficient to explain such an effect [32]. Although diffraction is historically linked to borders effects, it is wrong to limit this phenomenon to this effect (pupil borders)[33]. The aperture presents amplitude change: brutal amplitude variation from opacity to transparence or the opposite. Indeed brutal amplitude and phase changes are linked together as shown in Figures 6a and 6b. The grating of Figure 6a presents a periodic brutal phase change between 0 and π. To simplify comprehension, we intentionally have chosen the extreme case of brutal change. The function f(x) presents the transmittance of the binary phase grating of Figure 6a. For simplicity, we limit the analysis to the one-dimensional consideration. Only two values, -1 and 1, are possible for the function f(x). The grating of Figure 6a is a transparent diffraction optical element, such as a glass plate, that modules the phase between 0 and π [34]. In other words the thickness difference inside the glass plate provides a difference of optical path that is equal to an odd multiple of the half of the wavelength: (2k+1)λ/2. The amplitude grating of Figure 6b possesses a transmittance g(x) which can take either the value 0 or the value 1.

Both phase transmittance f(x) and amplitude transmittance g(x) are linearly linked together: g(x) = 0.5 f(x) + 0.5. The main difference between both functions is the DC term of 0.5 (or zeroth harmonic term)[35,36]. The multiplicative factor of 0.5 merely stands for energy conservation. Thus, both phase grating of Figure 6a and amplitude grating of Figure 6b show the same diffractive behavior. The only differences concern the zeroth diffraction order (Fig. 6b) and energy repartition. If we separately place the two gratings in the front focal plane of an aberration-free lens, we observe the same diffraction

pattern in the back focal plane of the lens (Fourier plane) except for the following two aspects (Same observation is valid if we consider the diffraction patterns at infinity – Figure 6). First because the amplitude grating is a blocking element and not a totally transparent element, half of the input light energy will be lost in the opaque zones of Figure 6b. This explains the role of the multiplicative factor in the expression of g(x). Second, the DC term (0.5) of g(x) produces a useless diffraction spot (called zeroth diffraction order : theoretically a Dirac) in the center of the Fourier plane. This lost light represents the half of the energy collected in the Fourier plane, therefore a quarter of the input energy. The multiplicative factor in the expression of g(x), which has the value 0.5, indicates that only a quarter of the input energy is useful in the output plane (1st and -1st order). Indeed, if we calculate the energy we obtain a ratio of 0.52=0.25 between useful information in g(x) and in f(x).

Except for energy loss and the appearance of an additional zeroth diffraction order, binary amplitude variation and phase variation cause identical diffraction effects. Since the OTE evaluates phase variation, it then presents an indication on the diffraction behavior of the optical system. The OTE measures how diffractive the system is. Because of this character, in addition to the Rayleigh resolution criterion, the OTE is a useful statistic to analyze the resolution power of the system. We remind the reader that the Rayleigh criterion is based on the diffractive behavior of the system and especially on the diffraction effect of the limited size of the pupil. To be more precise, this criterion states that maximum resolution, which theoretically can be achieved, is obtained as follows. The images of two points can be separated if the maximum of the first diffraction disc is located at the first minimum of the second disc (Fig. 5). This criterion only considers border effects and does not worry about the interior of the wavefront, which may be very diffractive and therefore seriously affects resolution. Diffractive behavior of the interior of the wavefront is however quantified by the OTE.

HOW TO MEASURE

ABERRATIONS

Monochromatic aberration of the human eye has been studied during almost a half century [37,38]. Several methods, such as subjective methods using successive determination of the aberration at different points in the pupil [39,40], the Foucault knife-edge method [41], aberroscope methods [42,43], and the Hartmann-Shack wavefront sensor [38], have been developed. Most of these methods are time consuming and not precise. However, the advantages of the

Hartmann-Shack method are that it is precise fast and objective [38]. The method has been primarily used in adaptive optics applied to astronomy [44].

Hartmann-Shack Technique

The most commonly used apparatus measuring aberrations of human eye objectively are based on the Hartmann Shack principle. At present, there is no optical detector able to directly measure phase variation. Thus phase, which presents whole information in the wavefront, should be transformed into

another type of information in order to be detected. The principle of the Hartmann-Shack sensor consists of transforming phase information at the wavefront level into lateral shift in the Fourier plane by operating a FT. To measure the phase in several elementary zones of the wavefront, we should operate a FT for each of these zones. This can be offered by a lenslet array which performs local FTs [45,46].

The technique involves a double pass. Light enters the eye, and then diffusely reflects from the retina so that it fills the whole pupil. The wavefront exiting the eye is imaged onto a lenslet array, which transforms this wavefront into a matrix of spots. For an aberration-free eye, the wavefront at the output of the eye is a plane wave. Thus in this ideal case, one should observe a regular array of bright spots, uniformly separated in the back focal plane of the lenslet array. However in reality, a deformed wavefront comes out of the eye. This wavefront results in an irregular array of spots, where each one is laterally shifted, with respect to the corresponding lenslet center (fig. 6).

wavefront g(x,y) illuminates only (M+1)(N+1) lenslets of the lenslet array, we obtain:

(28) Let us define gm(x,y) as the mth segment of the wavefront g(x,y). This mth segment illuminates

the mth lenslet and extends over the interval: –D/2+mD <x<D/2+mD and –D/2+nD <y<D/2+nD. So, we have:

(29)

Using Gmn(u,v) the Fourier transform of gmn(x,y), we obtain:

(30)

Thus, the size of the diffraction spot and whether spots overlap depend on the bandwidth of gmn and therefore on the aberration type and

Without loss of generality, let us suppose that this signal is imaged on the lenslet array (x-y plane) without any lateral magnification. So, we obtain at the exit of the lenslet, the following optical signal:

where l(u,v) is the transmittance of the lenslet array with focal length f and lenslet pitch D. If the imaged

order [31]. Using this simulation, the user learns how the Hartmann-Shack pattern looks like for any type and amount of aberration and for any combination of types (Figure 7). The Web interactive program allows us also to see the effect of the order on spot size and lateral spot shift.

Derivation of the Curve of Aberrations

The local shift observed in the focal plane of a lenslet is expressed as follows:

(31)

where ƒ is the focal length of the lenslet array, dx' and dy' are the lateral shifts of the spot corresponding to one lenslet, and W(x,y) is the aberration at a point (x,y). We observe an irregular array of spots in the focal plane, at a distance f behind the lenslet array[38].

One can perform the numerical reconstruction of W(x,y) by a wavefront fit. The method consists of fitting the best possible expression of W(x,y), the partial derivatives of which will satisfy equation (34). The wavefront estimation can be made by first choosing a suitable polynomial expression for the wavefront and using a least squares estimating routine to estimate the best coefficients for each polynomial term[47]. A complete method for calculating

these coefficients has been proposed by Cubalchini [48]. We then obtain an explicit analytical form of the aberration. The polynomial coefficients calculated can then be converted to a finite sum of orthogonal functions. If Zernike polynomials are used then the aberration types are directly deduced [29]. The preponderant aberration types are determined by identifying the polynomials that contribute the most to the deformation of the wavefront.

Hartmann-Shack Patterns

The relationship between myopia and aberrations [49,50] has been investigated by using the Hartmann-Shack apparatus [51]. We observed that the more myopic the eye, the smaller the measured Hartmann-Shack pattern. As shown in Figure 8b, the measured pattern for an emmetropic eye with pupil dilation fills the extent of the pupil. However, for a –5.0D myopic eye, the measured pattern is smaller because of large aberration (fig. 8c). The lateral shift of outermost spots is an indication on the amount of aberration. The measured pattern is even smaller for a myopia of –9.25D (fig. 8d).

Recently the effect of photorefractive keratectomy [52-54] and laser in situ keratomileusis [55-60] has been the subject of active research. The Hartmann-Shack apparatus has been largely used for this purpose. Let us consider one case of photorefractive keratectomy [54]. The patient is a 26-year-old woman with an initial myopia of -4.0D for the right eye. Aberration is measured for the eye two years after surgery (visual acuity after surgery : 6/6, no correction is required). Figure 8 shows the detected pat-

terns before (7mm) and after dilation (9mm) of the eye. Both patterns have almost the same extent although the pupil sizes are different. Indeed, all spots are confined in a diameter smaller than 4.5mm given that the sampling interval is 0.5mm. The arrows in Figure 8b show the shifts of the outermost spots. Because the outer spots moved radially into the central zone one can state that positive spherical aberration is dominating. The outermost spot has moved at least across 4 adjacent spots.

APPENDIX A

For simplicity, let us consider the one dimensional to analyze the dynamics of a spectrum G(u)=A(u)exp(iϕ(u)) that is the Fourier transform of g(x). The derivative of G(u) is:

(A1)

and its complex square is:

(A2)

From the derivation properties of the Fourier transform, we know that dG(u)/du is the Fourier transform of (–ix)g(x). Thus, using Parseval’s theorem we obtain the following relation:

(A3)

The right hand integral is nothing else the expression of the RMS duration of the signal defined by the second moment.

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