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

Conclusion

Phakonit with an ultrathin 5 mm optic rollable IOL implantation is a safe and effective technique of cataract extraction, the greatest advantage of this technique being virtual astigmatic neutrality.

REFERENCES

1.Apple DJ, Auffarth GU, Peng Q, Visessook N. Foldable Intraocular Lenses. Evolution, Clinicopathologic Correlations, Complications. Thorofare, NJ, Slack , Inc., 2000

2.Agarwal A, Agarwal A, Agarwal S, et al. Phakonit:Phacoemulsification through a 0.9 mm corneal incision. J Cataract Refract Surg 2001; 27:1548-1552

3.Agarwal A, Agarwal A, Agarwal A, et al. Phakonit: Lens removal through a 0.9 mm incision. (Letter). J Cataract Refract Surg 2001; 27:1531-1532

4.Agarwal A, Agarwal S, Agarwal A. Phakonit and laser phaconit: Lens removal through a

0.9mm incision. In: Agarwal S, Agarwal A, Sachdev MS, Fine IH, Agarwal A, editors, Phacoemulsification, laser cataract surgery and foldable iols. New Delhi, India, Jaypee, 2000; 204-216

5.Tsuneoka H, Shiba T, Takahashi Y. Feasibility of ultrasound cataract surgery with a 1.4 mm incision. J Cataract Refract Surg 2001; 27:934940

6.Tsuneoka H, Shiba T, Takahashi Y. Ultrasonic phacoemulsification using a 1.4 mm incision: Clinical results. J Cataract Refract Surg 2002;28:81-86

7.Kanellpoupolos AJ. A prospective clinical evaluation of 100 consecutive laser cataract procedures using the Dodick photolysis neodymium: Yittrium-aluminum-garnet system. Ophthalmology 2001;108:1-6

8.Dodick JM. Laser phacolysis of the human cataractous lens. Dev Ophthalmol 1991; 22:58-64

Dr. Amar Agarwal, MS,FRCS, FRCOpth Dr. Soosan Jacob, MS, DNB, FERC

Dr. Athiya Agarwal, MD,FRSH, DO

Dr. Sunita Agarwal, MS, FSVH,FRSH,DO

Authors’ Institution and Affiliation:

Dr. Agarwal’s Eye Hospital &

Eye Research Centre, 19, Cathedral Road,

Chennai-600 086. Tamil Nadu, India.

Financial Support:

There was no financial support for the performance of this study

Proprietary Interest:

NIL

Address for Reprint:

Dr. Amar Agarwal, MS.,FRCS.,FRCOpth.,

Dr. Agarwal’s Eye Hospital & Eye Research Centre, 19, Cathedral Road,

Chennai-600 086. Tamilnadu, India.

Ph:+91-44-8116233,8113704.

Fax:+91-44-8115871

Habib Hamam, MD.

INTRODUCTION

Since antiquity, the Greeks attempted to decode the enigma of light and considered it a continuous phenomenon propagating in the form of a substance current called the "visual ray"[1]. Aristotle [2], interested in the sensation in general, refused to admit the existence of the visual ray and believed in the analogy between light and sound whose vibratory nature was already known[3]. In the XIth century, the thesis of the visual ray was definitively abandoned in favor of (Iraqi) Ibn al-Haytham’s waork[4] which revolutionized the optics[5,6]. This geometro-optician explained vision as follows and

supported his claims with experimental proofs4. Objects in darkness cannot be seen merely because no light comes from them that the eye can detect. However, an object, illuminated by a light source, reflects a part of this light that the eye collects and the brain interprets. Each point of the object emits an infinity of rays a part of which enters the optical system of the eye. This system modifies them to reconstruct an image. Ideally this image is a point, called image point, which all rays entering the system of the eye converge to.

Stated in wave optics, the system of the eye should transform the input wavefront into a perfect convergent spherical wavefront that has the image point as center (Fig. 1a). We note that an optical

wavefront represents a continuous surface composed of points of equal phase. Thus all image-forming rays, which travel normal to the exit spherical wavefront, meet in the focal point in phase (differences between the optical paths are multiples of the wavelength), resulting in maximum radiant energy being delivered to that point. In reality, this situation never occurs.

The rays modified by the optical system do not converge entirely to a common point image (Fig. 1b). For one object point correspond several image points that form a blurred image. This devia-

Section IV: Aberrations and Aberrometer Systems

tion from the ideal case is called "aberration" and is a measure of the optical quality of the system. Aberration can be quantified either with respect to the expected image point or to the wavefront corresponding to this ideal point. If we compare the real output wavefront to the ideal one, we call the difference between them "wavefront aberration" and denote it "W" (Fig. 2). In general, the more the wavefront aberration defers from zero, the more the real image differs from the ideal image. Therefore, the poorer is the quality of image formed on the retina.

In this chapter, we discuss wavefront aberrations and the ways to describe them. Our analysis will be simplified to allow people with no strong background in optics to follow it. In this framework, we introduce the Taylor and Zernike basis. We also introduce the reader to the notions of Optical Transfer Function (OTF), of Point Spread Function (PSF) and of Modulation Transfer Function (MTF). We discuss the criteria to quantify optical performance and define the often used statistical measures, namely the Root Mean Square (RMS) error, the Peak to Valley (PV) error and the Strehl error. A section will be devoted to a new measure called the "Optical Transfer Error (OTE)" or "Admittance Dynamics" which evaluates the variability of the OTF. Distinction will be made between coherent and incoherent illumination.

HOW TO DESCRIBE

ABERRATIONS

As mentioned in the introduction, two concepts are possible to describe aberrations, namely ray and wave optics. An optical system provides a perfect formation of a point image if all image-forming rays meet in a single point. However, such an ideal condition is never fulfilled in practice because of the presence of "aberrations" [7]. Output rays corresponding to one point object do not meet in only one point. We can also express ideal image formation by means of waves. Indeed, a perfect optical system should provide a spherical converging wave centered at the ideal point image. In practice, output waves are different from this ideal wave and the deviation of the real wavefront with respect to the reference wavefront is called the "wavefront aberration" [8]. Given that wavefront aberration is most commonly used to express the optical performance of the eye, we intend to focus on wave concept in this chapter.

Like any object, phenomenon or event, wavefront aberration, as a physical quantity, should be identified by means of a decomposition in a con-

ventional system. For example, the event of birth can be identified by decomposing the date of birth in the commonly used dating basis, namely the 3-element system: day - solar month - solar year.

Basis

The system is called a basis if the decomposition is unique (to be rigorous this condition is not enough). For example, the day - solar month - solar year system is a basis. Your date of birth can be expressed only in one way. However, the system used for money is not a basis. If you tell me that you possess 58$, then I can not identify what you have in your pocket. You may have:

58$ = 2*20$ + 1*10$ + 1*5$ + 1*2$ + 1*1$

or 58$ = 1*20 + 2*10$ + 3*5$ + 3*1$

or ...

There are a lot of possible decompositions (combinations). Each decomposition is represented by a series of coefficients (bold).

Instead of solar calculus, one can use lunar calculus. The day of birth can be expressed in the following 3-element system: day - lunar month - lunar year. This system is also a basis. Because the representation (series of coefficients) inside a basis is unique, the transfer from one basis to another one is also unique. If each basis is composed of N elements, then a unique NxN matrix exists that allows us to move from one basis to the other. We need a 3x3 matrix to transfer the expression of the date of birth from the day - solar month - solar year system to the day - lunar month - lunar year system. However, there is no 3x3 matrix that allows us to make this transfer. We need in practice a long and complicated calculus to express the date of birth in the day - lunar month - lunar year system instead of the day - solar month - solar year system. The reason is that none of the two systems is rigorously a basis because some

elements depend on each others. The number of days of the second month (February) depends on the year in solar calculus. In lunar calculus, this kind of dependence is even deeper. Depending on the year (moon orbit), each lunar month (not only one month) can have 29 or 30 days. Thus, another necessary condition for a basis imposes that its elements are independent from each others.

The importance of the decomposition into a basis is such that a basis wisely chosen can give us specific information about the quantity to be examined. For example, by intelligently choosing the basis in which we decompose wavefront aberrations, we can identify the aberration types that the optical system suffers from.

Orthogonality

Let us consider a special basis namely the locality basis. The basis deals with this question: how can we localize a point in space (x, y, z)? For simplicity, let us limit our attention on a two-dimen- siononal plane (x, y) instead of a three-dimensional space (fig. 3).

Two basis are considered in (fig. 3). The point P1 is represented by two different combinations of coefficients (a1, b1). The distance between the origin O and the point P1 is identical in both cases (fig. 3a and 3b) : d1. In other words, if we go from the origin to P1 in a straight way, we cover the same distance in both cases. However, if we follow the path a1 and b1, the non-orthogonal basis (fig. 3a) is advantageous and corresponds to the shortest way. In other words, the point P1 seems to be nearer in the nonorthogonal basis.

Let us now turn to the second point P2. It is in the same position in both systems. However, if we go from the origin to this point following the nonorthogonal basis (a2 and b2), P2 seems to be very far although. In reality, it is very close to the origin (distance is short). The series of coefficients, or the

Section IV: Aberrations and Aberrometer Systems

Figure 3: Representation of two points P1 and P2 in two different basis, a1, b1, a2 and b2: respective coordinates (components) of P1 and P2, d1 and d2 : respective distances from the origin O. a) non orthogonal basis, distance may be smaller than the length of one the components. b) Orthogonal basis: distance is never smaller than the lengths of the components.

representation (a2, b2), is not a trustful representation of the real distance. Depending on where the point is (left, right, up, down), the coefficients can be big or small.

However, in the orthogonal basis, the coefficients (a2, b2) gives us a real idea (trustful information) on the distance which is expressed only by these two coefficients in a unique way:

The Cartesian and polar expressions of the terms Tn (n=0 ... 230) are given in reference [9]. The first 36 terms are listed in Table 1. The sum of the powers of x and y is called the order of the term Tn. Thus, there are N+1 terms for each order N.

This basis is however not orthogonal. Let’s consider for example a wavefront function defined by only four components (terms) with big coefficients, namely:

W(x,y) = 5 x2 - 5 x4 – 10 y3 + 10 y5.

To evaluate how distant (different) a function W(x,y) is from the zero function zero(x,y)=0, one can calculate the standard deviation, namely: the square root of the normalized volume under the curve of the square of the function W(x,y). This square root of the normalized volume is a kind of distance for curves. If we calculate the square root of the

normalized volume under the curve of W2(x,y) over the interior of a unit circle for the example given above, we obtain:

(1)

where the normalization factor S is the surface of the unit circle: S=π.

The amount d expresses how the function W(x,y) is distant from the zero function over the interior of the unit circle. We see that this distance is smaller than any one of the four Taylor coefficients. This is due to the lack of orthogonality. The Taylor coefficients do not give the remotest idea on the distance.