Ординатура / Офтальмология / Английские материалы / Seeing_De Valois_2000

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accommodative range is more than 10 D in a young eye, but this value declines steadily throughout adulthood (Duane, 1922) and eventually disappears (a condition called presbyopia) as about 55 years of age when the lens has become completely inflexible (Glasser & Campbell, 1998).

In addition to changes with age that occur in all eyes, there are other changes that occur in a significant minority of eyes. For example, many eyes grow excessively in the axial direction during childhood or early adulthood, which causes the retinal image of distant objects to be formed in front of the retina. As the target is brought closer to the eye, the optical image will move backwards towards the retina and eventually come into focus on the photoreceptors. Such eyes are called “nearsighted” (myopic), a condition which is easily corrected by weakening the optical power of the eye by wearing a spectacle or contact lens with negative power. The opposite condition of “far-sightedness” (hyperopia) results when the eye lacks su - cient power in the resting state to focus distant objects on the photoreceptor layer. The young eye easily overcomes this condition by exerting some accommodative e ort, but a presbyopic eye requires corrective lenses. Other eyes may have a cornea or lens which is not radially symmetric, which causes contours of di erent orientations to produce images at di erent distances. This condition of astigmatism is corrected by spectacle lenses which have the same asymmetry, but opposite sign. Although other defects in the eye’s optical system occur in the human population, the only optical defects which are routinely corrected by spectacle or contact lenses are myopia, hyperopia, presbyopia, and astigmatism.This traditional practice is likely to change in the near future, however, as new technologies for measuring and correcting the eye’s aberrations mature (Liang, Grimm, Goelz, & Bille, 1994). The fascinating potential for supernormal visual performance resulting from comprehensive optical correction of all the optical imperfections of the eye is but one intriguing spin-o from current research in this area (Liang, Williams, & Miller, 1997).

B. Physics of Image Formation

If an eye could be fitted with a perfect optical system, it would focus all rays of light from a distant point source into a single image point on the retina, as shown in Figure 2a. In this case the only optical question to be answered is, Where is the image point located? To answer this question we may apply the basic laws of paraxial (Gaussian) optical theory to describe the idealized case of perfect imaging. Real eyes, on the other hand, su er from three types of optical imperfection which are not treated by paraxial theory: aberrations, diffraction, and scattering. While the mechanism is di erent in each case, the common e ect of these imperfections is to spread light across the retina, as illustrated in Figure 2b. Thus a second, and more di cult, question to be answered is, What is the spatial distribution of light intensity in the image? To answer this question will require the concepts and computational tools of Fourier analysis.

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the image formed by a collapsing spherical wavefront is a single point at the center of curvature, determining image location is the same problem as locating the center of curvature of a refracted wavefront. Such problems are more easily solved if we describe wavefronts in terms of their curvature (i.e., inverse of radius of curvature), and for this reason we define the vergence L of a spherical wavefront as

where n is the refractive index of the medium in which the wavefront exists, and r is the radius of curvature of the wavefront.

Two rules illustrated in Figure 3 govern the imaging process: the transfer rule and the refraction rule. The transfer rule describes how the vergence of a wavefront changes as the wavefront propagates from a starting point A through the distance d to any later point X on route to the center of curvature C. To derive the transfer rule we simply express the linear relationship

In words, the transfer rule of Equation 3 says that the vergence “downstream” at some point X is given by the vergence “upstream” at A, divided by 1 minus the product of the upstream vergence and the “reduced” distance traveled by the wavefront. The term reduced distance denotes the physical distance traveled by a light ray, divided by the refractive index of the medium.

The refraction rule describes how the vergence of a wavefront changes when entering a new medium. The interface between the two media can be any shape, but in the paraxial domain the shape is approximated by a spherical surface with radius of curvature r. By definition, the refracting power F of the surface is

where n is the refractive index of the medium the wavefront is leaving, and n is the index of the medium the wavefront is entering. If the wavefront incident on such a surface has vergence L immediately prior to entering the new medium, then immediately after entering the new medium the wavefront has vergence L given by the refraction equation

In words, Equation 5 says that the wavefront vergence after refraction is equal to the vergence prior to refraction plus the power of the refracting surface.

The refraction rule expresses in quantitative terms the net result of two factors acting to change the curvature of the wavefront. These two factors are the wavelength of light, which varies with refractive index of the medium, and the curvature of the refracting surface. This interaction is most easily understood for the case of an incident plane wave, as illustrated in Figure 3b. The incident plane wave is analogous to a marching band of musicians traveling across an open field.The musicians all march to the same beat and their cadence (i.e., temporal frequency of light oscillation) never varies. The curved surface separates the dry marching field where the musicians have a long stride (i.e., wavelength is relatively long) from a muddy field where the musicians must shorten their stride to avoid slipping (i.e., the wavelength of light is reduced). Since the musicians near the middle enter the muddy field first, that part of the line slows down first. As time progresses, more and more of the marching line enters the slower medium, but because the interface is curved, those musicians farthest from the centerline stay on dry land the longest and therefore travel the farthest before they too must enter the slower field. By the time the entire marching line has passed into the new medium, the end of the line has advanced farther than the center and so the line is curved. So it is with light. The incident wavefront is delayed more near the centerline than at the edges, and thus the wavefront becomes curved. This is the phenomenon of refraction. The alternative description of refraction as the bending of light rays by an amount which depends upon the angle of incidence relative to the surface normal is known as Snell’s Law.

Given the above rules and definitions, the paraxial location of the image of an axial point source formed by an arbitrary number of refracting surfaces may be found by repeated application of the refraction and transfer equations. To begin, one uses the definition of vergence (Equation 1) to determine the wavefront vergence of incident light immediately prior to refraction by the first surface. The refraction rule (eq. 5) then yields the wavefront vergence immediately after refraction. The transfer rule (Equation 3) is then applied to determine the new vergence of the wavefront after propagating from the first surface to the second surface. The “refract and transfer” process is repeated as many times as necessary until all surfaces have been encountered. After the final surface, the definition of vergence yields the center of curvature (i.e., the image point) of the exiting wavefront.

To determine the image of object points which are o axis but still within the paraxial image domain, the simplest method is to apply a magnification formula derived from Snell’s Law. Consider the ray of light shown in Figure 4 emanating from the tip of the arrow object and intersecting the refracting surface at the vertex (i.e., where the surface intersects the optical axis of symmetry). The incident ray makes an angle u with the optical axis, and after refraction the same ray makes angle u . Snell’s Law is a law of angular magnification

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FIGURE 4 The geometry of Snell’s law for refraction of the chief ray from an o -axis object point yields the paraxial formula for image magnification.

which, in the paraxial domain where the small angle approximation sin u u is appropriate, becomes

Replacing the angles of incidence and refraction with their trigonometrical equivalents, and applying the small angle approximation tan u u, Equation 7 becomes

and by the definition of vergence, this result simplifies to

The ratio of image height to object height is called the linear magnification of the optical system. Thus the lateral location of a paraxial object point is found by application of the linear magnification formula

2. Geometrical Optics in the Nonparaxial Domain

Ouside the paraxial domain of perfect imaging, aberrations are said to exist when refracted light rays from a single object point intersect the image plane at di erent locations, causing the light to spread out over an area. This distribution of light intensity in the image of a point source, called a point spread function (PSF), is a fundamental characterization of the system. The PSF also has great practical utility because it allows the computation of the image of an arbitrary object as the superposition of individual PSFs from every point in the object.

Two general methods for determining the PSF of any optical system are (a) direct observation of the image formed by the system for a point object, and (b) computational modeling of the system. Direct observation of the PSF of the human eye is complicated by the inaccessibility of the retinal image, which forces us to use the eye’s optical system a second time to make the observation. For this reason, the com-

putational approach using an optical model of the eye has played an especially important role in understanding the optical imperfections of the eye. Many such models have been proposed over the last century or more, and most modern schematic eye models have been implemented with computer programs. Given such programs, one common method for determining the PSF is to introduce a uniform bundle of rays that fills the model eye’s pupil and then to trace all the rays to their destination on the imaging screen (retina). Each ray intersects the screen at a single spot, so the image as a whole may be visualized by a diagram of spots for the entire bundle. With elementary trigonometry and Snell’s law of refraction, it is not di - cult to write a computer program to perform such ray tracing and so determine numerically a spot diagram rendition of the PSF for the eye.

Unfortunately, ray tracing fails to account for di raction, the second major source of image degradation in the eye. Di raction is the name given to any deviation of light rays from straight lines which cannot be interpreted as reflection or refraction. If the phenomenon of di raction did not exist, then when a wavefront of light passes through an aperture (such as the eye’s pupil), the marginal rays would define a sharp boundary between light and shadow, as suggested by the dashed lines in Figure 5. In fact, light does penetrate the shadow and the e ect is not insignifi- cant. For an eye with a pupil diameter about 2 mm or less, di raction is the primary factor which prevents the formation of a perfect point image of a point source (Campbell & Green, 1965; Liang & Williams, 1997). Since the pupil diameter of the normal eye spans the full range from di raction-dominated performance to aberration-dominated performance, we require an optical theory of image formation which takes account of both factors. For this reason we will now set aside the ray theory of image formation in favor of the wave theory.

3. Wave Theory of Image Formation

According to the wave theory of image formation, the perfect image of a point object is formed by a collapsing hemispherical wavefront with center of curvature

FIGURE 5 Di raction by an aperture is the propagation of light into the geometrical shadow region shown by broken lines.

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located in the image plane. An aberrated wavefront, therefore, is characterized by its deviation from this perfect wavefront. Even without defects of this kind, however, the image of a point source through a circular pupil will still be spread out across the retina as a di raction pattern consisting of a bright central disk surrounded by unevenly spaced concentric circles known as the Airy pattern (after G. B. Airy, who first derived its analytical form). The underlying reason for this di ractive imperfection in the image is that the wavefront emerging from the eye’s pupil is only a small part of the complete wavefront needed to form a perfect point image.

The necessity of having a complete wavefront for perfect imaging follows logically from Helmholtz’s principle of reversibility: what is true for waves diverging from a point is also true for waves converging towards the point. If an ideal point source produces spherical waves expanding symmetrically in all directions of threedimensional space, then to produce an ideal point image requires spherical waves collapsing symmetrically from all directions of three-dimensional space. By a similar argument (C.Williams & Becklund, 1989), to mimic the e ect of an aperture which transmits only a segment of a diverging hemispherical wavefront propagating towards the plane of the aperture (Figure 6a) would require an extended source in order to cancel the wavefront everywhere except where the segment persists (Figure 6b).Therefore, by the principle of reversibility, a converging segment of a spherical wavefront passed by an aperture must produce an extended image. This is the phenomenon of di raction.

To take the next step and quantitatively describe the nature of the di raction pattern produced by a wavefront after passing through an aperture, we require some help from di raction theory. Di raction theory is not for the faint-hearted, as di raction problems have a reputation for being amongst the most di cult ones encountered in optics. Nevertheless the basic ideas are not di cult to understand if we leave rigorous mathematical proofs for the standard reference works (Born & Wolf, 1999; Goodman, 1968). As a bonus reward for this e ort we will have laid the foundation for Fourier optics, which is the primary computational method for modern image analysis.

FIGURE 6 To replicate the creation of a wavefront segment by an aperture (a) would require an extended source in the absence of the aperture (b).

FIGURE 7 Huygens’s theory of wavefront propagation.

A framework for thinking quantitatively about the di raction of propagating waves is illustrated in Figure 7.To understand how an expanding or collapsing wave propagates, Huygens suggested in 1678 that if every point on the wavefront were the center of a new, secondary disturbance which emanates spherical wavelets in the direction of propagation, then the wavefront at any later instant would be the envelope of these secondary wavelets. In 1818 Fresnel advanced Huygens’s intuitive notions by incorporating Young’s discovery in 1801 of the phenomenon of interference to propose a quantitative theory now known as the Huygens–Fresnel principle of light propagation. By this principle, Huygens’s secondary wavelets are directed (i.e., the amplitude is strongest in a direction perpendicular to the primary wavefront) and they mutually interfere to re-create the advancing wavefront. Consequently, when a wavefront encounters an opaque aperture, the points on the wavefront near the obstruction are no longer counter-balanced by the missing Huygen wavelets, and so light is able to propagate behind the aperture. Subsequently, Kircho demonstrated in 1882 that critical assumptions made by Fresnel (including the nonisotropic nature of Huygens’s wavelets) were valid consequences of the wave nature of light. In 1894 Sommerfeld rescued Kircho ’s theory from mutually inconsistent assumptions regarding boundary values of light at the aperture. In so doing Sommerfeld placed the scalar theory of di raction in its modern form by establishing that the phenomenon of di raction can be accurately accounted for by linearly adding the amplitudes of the infinity of Huygens’s wavelets located within the aperture itself. Mathematically this accounting takes the form of a superposition integral known as the Rayleigh-Sommerfeld di raction formula (Goodman, 1968, Equations 3 –26, p. 45). For a fuller account of the history of the wave theory of light, see Goodman, p. 32, or Born & Wolf, 1970, pp. xxv–xxxiii).

To express the di raction formula requires a mathematical description of Huygens’s wavelets and a way of summing them up at an observation point R beyond the aperture as shown in Figure 8. Although a rigorous theory would describe light as a vector field with electric and magnetic components that are coupled according to Maxwell’s equations, it is su cient to use a much simpler theory that treats light

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FIGURE 8 Geometry for computing the contribution of Huygens’s wavelet at point S to the amplitude of light at point R.

as a scalar phenomenon, the strength of which varies sinusoidally in time. Thus, for a monochromatic wave, the field strength may be written as

where U(R) and (R) are the amplitude and phase, respectively, of the wave at position R, while is the optical temporal frequency. For mathematical reasons it is more convenient to express the trigonometric function as the real part of a complex exponential by writing Equation 11 as

where i 1 and U(R) is a complex valued function of position only:

The temporal oscillation of the field is not essential to the di raction problem, and for this reason we concentrate on the phasor U(R) as a description of a wavefront. Accordingly, we seek such a phasor description of Huygens’s wavelets. The required function for an arbitrary point S on a wavefront of unit amplitude is

where

H complex amplitude of light at R due to Huygen wavelet at S r radial distance from source point S to observation point R

k 2 / , wave number, converts r to phase shift in radianswavelength of light

angle between line RS and the normal to wavefront at S

Each of the three factors on the right-hand side of Equation 14 relates to an essential feature of Huygens’s wavelets. The middle factor is the standard expression for a spherical wavefront due to a point source. The numerator of this factor

accounts for the phase shift that results when the wavelet propagates from S to R, and the denominator accounts for the loss of amplitude needed to keep the total energy constant as the wavefront expands. This spherical wavelet is modified by the first factor, which says that the amplitude of the secondary source is smaller by the factor 1/ compared to the primary wave, and the phase of the secondary source leads the phase of the primary wave by 90 . The third factor in Equation 14 is the obliquity factor, which states that the amplitude of the secondary wavelet varies as the cosine of the angle between the normal to the wavefront at S and the direction of the observation point R relative to S.

Equation 14 describes the secondary wavelet produced by a primary wavefront of unit amplitude. Applying the actual wavefront amplitude U(S) as a weighting factor, the wavelet at point S is the product U(S)H(S,R). The total field at point R is then found by linearly superimposing the fields, due to all of the secondary wavelets inside the aperture A. The result is a superposition integral over the aperture

which is known as the Rayleigh–Sommerfeld di raction integral. (Goodman, 1968, p. 45, Equations 3–28).

4. Fourier Optics

Under certain restricted circumstances, the superposition integral of Equation 15 reduces to a convolution integral. This immediately suggests an application of the convolution theorem of Fourier analysis (Bracewell, 1969), which transforms the given quantities U and H into a corresponding pair of new quantities, which provides a complementary view of di raction and optical image formation. This is the domain of Fourier optics.

To begin, erect a coordinate reference frame centered on the aperture as shown in Figure 8, with the x, y plane coinciding with the plane of the aperture.The observation point at R has the coordinates (x, y, z), and an arbitrary point in the aperture plane has the coordinates (x , y , 0).The initial simplifying assumptions are that the observation point R is far from the aperture and close to the z-axis, in which case the obliquity factor cos in Equation 14 is approximately 1. Under these assumptions the distance r in the denominator of Equation 14 may be replaced by z. However, this is not a valid substitution in the numerator because any errors in this approximation are multiplied by a large number k. To deal with this problem we need to investigate in more detail how r depends on the coordinates of S and