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

two cases was done by simply placing a negative or positive power lens in front of the eye, until both points were joined.

Now if the patient has a different disorder other than hyperopia or myopia (both cases are generally referred to as defocus), that is, astigmatism, a different procedure had to be done. In 1961, Smirnov described a method for measuring the astigmatism using Scheiner Disk. Smirnov used a fixed point source of light for the central hole, but a moveable source for the peripheral hole (see picture 2). In this way he could, for each meridian of the eye, change the angle of incidence of the peripheral ray and make it join the central ray at the retina. Because this could be done for each point at the entrance pupil, a more general measurement of the eye aberrations could be made. It is interesting to note that, because the power (Diopters) of a lens is proportional to the vergence of rays, the measurement of the displacements of the moveable light source (see parameters at figure 2) could be used to determine the correction needed at that point of the pupil. By doing this for different meridians Smirnov could also determine the patient’s astigmatism. This subjective instrument is still used today for vision research and became known as the Scheiner-Smirnov aberrometer.

Figure 2. Smirnov aberrometer

The methods of Scheiner and Smirnov were based on what physicists call ray optics, because they are based on geometric optics principles. Much before Smirnov, in 1884, Tscherning [18] constructed an instrument based on a different principle, which he named the "aberroscope". He used a square grid superimposed on a 4 diopter spherical lens and projected it’s image at the fundus of the eye (see figure 3).

Figure 3. Tscherning’s aberroscope.

Subjects were asked to look to a distant light source and then sketch the grid pattern that they were seeing on piece of paper. By analyzing these patterns Tscherning could have a subjective measure of the aberration.

In the turn of the century (1900), Hartmann[16] proposed what became known as the Hartmann screen, a surface with apertures in the form of circle that could be used to analyze optical aberrations.

In 1960 Bradford Howland [19] invented the cross cylinder aberroscope and used it to investigate the aberrations of camera lenses. Instead of using a spherical lens to shadow the grid on the retina, he used a crossed cylinder lens of 5 diopters with the negative cylinder axis at 45 degrees. In this way

Howland could obtain sharper images than those obtained by Tscherning’s aberroscope.

In 1971 Shack [15] improved Hartmann’s ideas for measuring wave fronts. Instead of using a screen with several holes, he used several microlenses instead, and in this way a sharper image at the focal plane was obtained for each bundle of rays.

In 1977 HC Howland [20] used a subjective aberroscope to investigate 55 eyes. The main result of that study was that coma-like aberrations dominate the aberration structure of the eye at all pupil sizes. This was also the first work to introduce Zernike [21] polynomials to describe wave-front aberrations, although these had been in use much before in other fields of optics [22].

In 1984 [23] the cross cylinder technique was improved by adapting photographic cameras and taking pictures of the grids on the retina (see picture 4). This was an important step because it made the aberroscope an objective technique, and allowed quantitative comparisons of aberrations.

Section IV: Aberrations and Aberrometer Systems

One of the greatest steps towards an objective and precise instrument for measuring eye aberrations was taken in 1994 at the University of Heidelberg. Liang and colleagues [14] reached for inspiration in the field of astronomy. They adapted a device called the Hartmann-Shack [15, 16, 17] wavefront sensor to an optical system and took pictures of eyes, calculating afterward their aberrations, also using Zernike polynomials. Up to that date, the HS sensor had been used mainly in observatory telescopes to measure and correct aberrations caused by the turbulence of our atmosphere [24]. These systems are based on wave-front sensors and deformable mirrors, a field generally called adaptive optics [42]. The differences in air temperature at the different levels of the atmosphere cause some rays of light to travel faster than others. Although these differences in speed are tremendously small, light rays from a distant star arrive at the telescope mirror at infinitesimally different times. This apparently neglectable effect causes considerable distortion in astronomical images, and many times even cause difficulties for the astronomers to recognize what they are looking at (see figure 5). Since 1994, the Hartmann-Shack sensor has become one of the most popular techniques for measuring eye aberrations. For this reason we will talk more about it and expose diagrams of it’s functioning and mathematical characteristics in more detail in the next section.

After Liang’s work in 1994 at Heidelberg, many authors have used the HS sensor as part of other optical systems with different objectives. One of the most distinguished applications was conducted by Liang again, David Williams and colleagues, this time at the University of Rochester in 1997 [26, 27]. Williams and his group used the HS sensor to measure the optical aberrations of the eye and sent this information to a deformable mirror. This mirror corrected the aberrations and allowed high-resolution images to be taken from the retina. This technique has been called "high resolution ophthalmoscopy" and it allows imaging single photoreceptor cells. One

Figure 5. Aberration in galaxies and stars, with and without correction, caused by atmospheric turbulence. From the Gemini telescope of the Astronomy Institute of Hawaii. The adaptive optics technique is also used in the Hubble telescope, although in this

case turbulence is caused by other factors [25].

of the most prominent applications is in the early diagnosis of retinal diseases such as macular degeneration and glaucoma.

Principles of Eye Aberration Measurements with the HartmannShack Sensor

In this section we’ll describe in more details the optical and mathematical principals of eye aberration measurements with the use of the HartmannShack (HS) sensor. In the year of 1971 Shack [15] proposed the use of micro-lens arrays instead of regular Hartmann screens. To understand the basic principle of the HS sensor let’s go back to the ScheinerSmirnov aberroscope described in section (1). Remember that the subject had to tell whether the

Figure 6. A dot of light reflecting at the fovea and leaving the eyes of three subjects with myopia, hyperopia and a normal eye (emetropic).

We may notice that a point of light scattered at the retina of a normal eye generates at the exit pupil what we call in physics a plane wave front. Although we have mentioned this term in the previous section, this is a good opportunity do define what it means. As we all know, light may be described as rays, such as in geometrical optics, or as waves, in physical optics. When describing light as wave phenomena, it has, as any other wave in physics, a wavelength, a velocity, amplitude and a phase (see these parameters in figure 7).

The phase of a wave is determined by the position of the wave crest. The wave front of a bundle of rays is determined by the connection of crests of neighboring waves. In figure 8 we may see two kinds of wave-fronts, one

that is said to be "in phase" and other that is said to be "out of phase", that is, with aberrations.

Now looking back at figure 6 we may see that the wave-front that leaves a perfectly emetropic eye is a plane; for a myopic the optical system is too strong so the wave-front is convergent, and for a hyperopic it is diverging. Notice then that by looking at the light that leaves the eye one can determine the same abnormalities that Smirnov did by shining light towards the eye. It is at exactly this point that the HS sensor makes a big difference, because it allows us to quantify objectively how converging, or diverging, or plane, or whatever wave-front format, is leaving the eye. In this way it’s possible to measure refraction at localized points over the pupil.

Imagine that we put a 15 by 15 square array of very small spherical lenses (with 0.5 mm in diameter each) in front of the eyes in figure 6. Since all lenses have the same focal distance ƒ we may put some kind of light

sensor behind them, like a photographic film or, even better, a CCD camera (see diagram on figure 9). For the first eye, because light has a plane wave front, when they hit the HS lenses a symmetric group of equally space spots will form at the CCD plane. For the myopic eye a matrix of spots more gathered towards the center will be formed and for the hyperopic, a matrix of more separated spots.

The most interesting aspect about the HS sensor is that, by comparing the dot pattern of a distorted wave-front with those of a plane wave front, one may precisely determine the exact shape of the distorted wave-front. This is so because the amount of displacement of each dot is directly proportional to the distortion of the wave front (see figure 9 and equation (1)).

Liang at. Al. used the optical diagram depicted in figure 10 to measure optical aberrations of two subjects [14].

Figure 9. A plane wave front focuses light at a point that lies over the optical axis of the lens, but a distorted wave front focuses light at a displaced point. The amount of displacement determines the wave front distortion.

In figure 11 we may see an example of the type of images that are obtained with the optical setup in figure 10. The center of "mass" of each spot is detected using image-processing algorithms[28]. The χ and coordinates of each spot are then compared with coordinates of corresponding spots of a calibration image. The calibration image is obtained from an aberration free eye (emetropic), or more often from an artificial eye.

Figure 11. Example of image formed at the CCD plane using the optical diagram of figure 10.

The mathematics to calculate this distortion is not very trivial because it involves differential calculus and other advanced topics in mathematics. Nevertheless we’ll make a brief description of these calculations here, and one may skip this discussion without compromising the comprehension of the remaining sections of the chapter. We’ll following describe how the distorted wave front is calculated by using Zernike polynomials.

In figure 9 the local slope of the wave front, in the vertical and horizontal directions of the CCD image plane, may be written as

Section IV: Aberrations and Aberrometer Systems

(1a)

(1b)

where ƒW and ƒW denote the partial derivatives of

ƒx ƒy the wave aberration function; x e n

is the position of the n’th spot for the calibration

emetropic eye (or calibration eye) and x a n

is the n’th position for the measured ametropic eye. So our goal is to find the aberration function (W). There are several techniques to find a function from a set of it’s derivatives, but the most popular method in this case has become the approximation using Zernike polynomials[21] and the Least Square method[29, 30]. We will not get into the mathematical aspects of why this technique became so popular for describing wave aberrations. What we may state is that one of the strongest reasons is that Zernike polynomials form an orthogonal basis and that it has ideal symmetries for describing optical aberrations. We’ll see later that each Zernike polynomial is associated with a typical optical aberration.

We may write the wave aberration function

as

(2)

where are the coefficients of each polynomial and is each term of the Zernike polynomial. The objective is to find these coefficients based on the slopes, so we apply the partial derivatives to equation (2):

(3a)

(3b)

Ahora usamos el método del Menor Cuadrado para encontrar los mejores coeficientes de Zernike que interpolen las derivadas en (1) y para hacer eso sustituimos los valores en (3) por las derivadas en (1). Podemos entonces escribir los coeficientes como:

Donde el coeficiente de la matriz es de la forma:

[DER] es la columna de la matriz de derivados, y [MT] es la matriz de transformación obtenida por el método del menor cuadrado.

Como lo vemos en la tabla 1, cada polinomio de Zernike esta asociado a un tipo de aberración. Además de las aberraciones de alto orden, los valores de refracción de bajo orden pueden ser obtenidos por los coeficientes C3, C4e C5, asumiendo que el frente de onda del lente equivalente esferocilíndrico puede ser escrito como:

Present Technologies for

Optimizing Visual Acuity

Through Refractive Surgery

In this section we’ll describe the current state of the art technologies for refractive surgery that are being tested and that will probably be available in the very near future. We’ll explain the basics about the current elevation algorithms in corneal topography, how this data is being "plugged" into the laser computers together with de aberration data, and how the laser uses this information to guide the custom ablation using eye tracking and flying spot technology. Please notice that because this is an extensive and complex subject and out of the scope of this book, we will not get into very specific details of the ablation process. Nevertheless you will certainly acquire a "big picture" of the ultimate techniques for refractive surgery.

Corneal Topography and

Elevation Maps

In 1880 the Portuguese ophthalmologist Placido invented what today is known as the Placido Disk [1]. Since those days there has been an enormous evolution in equipments to measure the shape of the anterior corneal surface. After Placido’s invention Javal and others [31] adapted magnifying lenses to keratoscopes so reflected images could be seen in more detail, but instruments were still qualitative. Shortly after that, Gullstrand [31] adapted photographic techniques to keratoscopy. From the 1950’s

until the 70’s many authors developed quantitative instruments using Placido Disks and photographic cameras, and these instruments became generally known as Photokeratoscopes. Examination would take several days, if not weeks, because one would have to wait for pictures to be developed, and after that all curvature calculations would be done by hand, measuring radial distances of Placido Disks one by one, at different meridians, with a ruler. In the 1980’s this picture started to change because microcomputers were starting to spread and their prices were going down in a fast pace. From this point on, computer algorithms could do all measuring processes. The photographic camera was substituted by CCD cameras (from the words "Charge Coupling Device"), and cards called "frame grabbers", which grab images from the CCD into the computer memory. The manual and tiresome Placido image measurements could now be accomplished by image processing algorithms [28] and so the whole process would take only a few minutes. As computers grew more powerful, this process became faster and faster and with the popularity of colored monitors, the first high resolution corneal topography maps were plotted, originally suggested by Klyce [32]. These instruments are quite popular nowadays and became generally known as VKS (Videokeratoscopes) or Corneal Topographers.

Since the beginning of the computerized VKS, many authors have proposed different mathematical algorithms to calculate corneal features. It is important to notice that there are lots of parameters that may be calculated and that each of them have a different meaning. To make our point clear let’s look at the diagrams in figure 13:

Figure 13. Different descriptors for corneal surface, each one with it’s own advantages and disadvantages for corneal diagnosis. The curvature maps (A and B) and the refractive map (C) are particularly useful for pre and post-surgical evaluation, because they are proportional to corneal power. Descriptors D, E and F are all elevation maps. They measure the "true" topography of the cornea, relative to a plain (D), to a small sphere (E) and to a big sphere (F). A detailed analysis of advantages and disadvantages of each descriptor may be found in the works of Salmon [33] and Klein [34].

To understand the meaning of each descriptor we must first comprehend the words from optics and geometry "refractive power" and "curvature". Refractive power is related to how "strong" an optical component is. Refractive power of a lens, for example, is a measure of how strong it can "bend" light’s path, changing its direction inwards or outwards relative to the optical axis. Two factors determine the refractive power of a lens: it’s curvature and

Where D is the refractive power in diopters when the radius of curvature, r, is measured in meters n; is the index of refraction of the medium, and for the cornea the value 1.3376 is usually assumed [35]. The concept of index of refraction is associated to how much a medium decelerates light’s speed, so it’s value is given by the velocity of light in vacuum (c 300,000 km/s) divided by that of the medium in consideration.

Given these basic concepts we may state that descriptors A, B and C are good measures of the optical quality of the cornea. They differ from one another, but they will all be useful in indicating to the refractive surgeon how much change in power was obtained after surgery. A is called the Axial map and it measures power using the radius relative to the geometric axis of the cornea; B is the Tangential map and it measures power using the instantaneous radius of curvature; D, E and F are variations of true elevation maps and they are the descriptors which are interesting for customized corneal ablations, simply because they carry "true" topography data (just as in the topography of a landscape) that may be used by the laser to calculate ablation depth. We’ll talk more about this in the following paragraphs.