Ординатура / Офтальмология / Английские материалы / Optics of the Human Eye_Atchison, Smith_2000

n

r d

E, E'

F, F'

P, pi

N,N'

V, V'

258 Appelldict',;

References

Emsley, H. H. (1952). Visual Optics, vol. 1, 5th edn, pp. 4Q.41, 346. Butterworths.

Gullstrand, A. (1909). Appendix 11: Procedure of rays in the eye. Imagery -laws of the first order. In Helmholtz's Handbucll der Pllysiologiscllell Opti«, vol. 1, 3rd edn (English translation edited by J. P. Southall, Optical Society of America, 1924).

Helmholtz, H. von (1909). Halldbllcll dcr Pllysiologischell Optik, vol. 1, 3rd edn (English translation edited by J. P. Southall, Optical Society of America, 1924).

Kooijrnan, A. C. (1983). Light distribution on the retina of a wide-angle theoretical eye. J. 01'1. Soc. Am., 73, 1544-50.

LeGrand, Y. and El Hage. S. G. (1980). Physiological Optics. Translation and update of Le Grand Y. (1968). La dioptriquc de l'oeil e! sa correction, vol. I of Optiquc pll)fSiologiql/e, pp. 65-7. Springer-Verlag.

Liou, H.-L. and Brennan, N. A. (1997). Anatomically

accurate, finite model eye for optical modeling. f. Opt. Soc. Am. A., 14, 1684-95.

Lotrnar, W. (1971). Theoretical eye model with aspheric surfaces. J. 01'1. Soc. Am., 61, 1522-9.

Navarro, R., Santamaria, J. and Bescos, J. (1985). Accommodation-dependent model of the human eye with aspherics. f. Opt. Soc. Am. A., 2, 1273-81.

Rabbetts, R. B. (1998). Benllettalld Rabbett,' Clinical Visual Optics, 3rd edn, pp. 209-13. Butterworth-Heinemann.

The point spread function (PSF)

We show how to calculate the PSF, taking into account diffraction, defocus, aberrations, polychromatic light, the photopic luminous efficiency function V(It) and the StilesCrawford effect. Scatter is more difficult to include in these calculations, so we will

neglect it.

The PSF can be calculated if the wave aberration in the pupil of the eye is known. From optical image formation theory (see

Smith and Atchison, 1997), the PSF is related to the wave aberration via a Fourier trans-

form. The background for the following equations is taken from the above reference.

Rather than express the PSF as a light distribution at the retina, we will express it as the

equivalent distribution projected back into object space. The major difference will be that,

in the first case spatial co-ordinates will be in linear quantities such as millimetres, and in

the second case they will be in angular units. Calculating the point spread function back in object space, taken as air, also avoids the need

to use the image space refractive index in the diffraction integral.

Before we present equations for calculating the PSF from the wave aberration in the pupil, we must first distinguish the amplitude PSF from the intensity PSF. The amplitude PSF is

the complex amplitude of the light distribution, whereas the intensity PSF is the actual

light distribution that we measure with a light meter. When we refer to the PSF, we mean the

intensity PSF unless otherwise indicated.

If we denote the amplitude PSF as ga(u,v), where u and v are the directions in object space, and the PSF as g(u,v), these two quantities are related by the equation

where * refers to the complex conjugate. The amplitude PSF is related to the wave aberration W(X,Y)by the equation

ga(u,v) =CJJEP(X,y)e-i2lt(UX+vY)dXdY (A4.2)

E implies integration over the pupil of radius 15, (u, v) are related to the actual angles (in

radians) Ox and 0y in the X and Y directions, respectively, by the equations

(X, Y) are the cartesian co-ordinates in the pupil, and P(X, Y) is the complex amplitude in

the pupil known as the pupil function, which is mathematically defined as

The constant k =2rrlIt, A(X, Y) is the amplitude transmittance at the point (X, Y) in the pupil, and is included to allow for the Stiles-Crawford effect. W(X, Y) is the wave aberration as an optical path difference, and is expressed in normal units of distance and not wavelength.

Equation (A4.2) is a Fourier transform of

260 Appel/dices

the pupil function P(X/ Y)/ and is zero outside the pupil. However, it would not be a Fourier

transform if the object space variables were the real angles Ox and 0y/instead of II and v. In other words, the use of II and v and not Ox and 0y allows the amplitude point spread function

to be expressed as a Fourier transform.

We will regard the Stiles-Crawford effect as rotationally symmetric in the pupil, and can write it as

where f3 is the Stiles-Crawford attenuation factor to base e and the //2/ factor is included because we must use an 'amplitude' Stiles-Crawford effect and not the normal 'intensity' effect (Krakau, 1974). Typical values for f3 are given in Chapter 13.

The wave aberration function

W(X, Y)

The wave aberration function W(X/ Y) is described in detail in Appendix 2. It is one way of quantifying the level of aberrations. The

function is a polynomial in X and Y/ and the different terms represent the different

aberrations - e.g. spherical aberration and coma. It can also incorporate defocus and

chromatic aberrations.

Defocus

If the eye is defocused by an amount L1F/ e.g. 0.5 0/ a defocus term W (X2 + y2) can be

ZO

added to the wave aberration polynomial,

Chromatic aberration

As discussed in Chapter 17/ longitudinal and transverse chromatic aberration arise because

of the dispersion of the ocular media. The dispersion leads to a chromatic change in

power L1F(,t) given by the equation

where F(A) is the power at the wavelength A and X is the reference wavelength for zero

chromatic aberration. Equations for F(,t) are given in Chapter 17.

For on-axis and off-axis point spread

functions, longitudinal chromatic aberration is taken into account with a wave aberration

term Wz,o(,t)(Xz + yZ). The co-efficient is related to the chromatic change in power L1F(A) by the equation

For off-axis point spread functions, transverse chromatic aberration is taken into account with a wave aberration term W11(,t)Y or W1,1(A)X. The co-efficient is related to the chromatic change in power L1F(,t) by the equation

where 0 is the angular distance off-axis, EN is

the distance from the entrance pupil to the front nodal point, and n(,t) is the refractive

index of the vitreous medium. Transverse chromatic aberration leads to a PSF having a transverse shift that is wavelength dependent.

Polychromatic sources

For polychromatic light sources, the PSF is

calculated at a number of wavelengths. The chromatic aberrations are included in the wave aberration function. Each point spread

function is weighted by both the relative sensitivity of the eye V(,t) (Chapter 11) and the

relative radiance of the source S(,t). The polychromatic point spread function is formed by adding the weighted individual point spread functions, but only after these have been expressed in terms of the real ?ngles Ox and 0y/ instead of 1I and v. That

1S,

(A4.11)

Computation checks

For any computation of the point spread function via equation (A4.2), it is good practice to have independent checks of the final result. The following three conditions