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

where n' and n are the object and image space refractive indices, L is the object vergence (= n/I) and L' is the image vergence (= n'/1'),

the distance I is measured from the front principal plane at P and the distance l' is measured from the back principal plane at P'. F is the equivalent power of the system, defined by equation (Al.18), and is the power of an equivalent thin lens placed at the principal planes (Figure A1.6).

In this situation, equation (A1.14) for the

transverse magnification is still applicable and, using the symbols shown in Figure A1.6, this equation can be written as

It can also be expressed in terms of the distances I and l' of the object and images from the respective principal planes by the equation

Gaussian optics

Sometimes it is useful to assume that rays that are beyond the paraxial region, according to the approximation in equation (A1.2), nevertheless behave as if they are paraxial rays (i.e, are aberration-free). The application of par-

axial approximations beyond the paraxial region is called Gaussian optics.

Summary of main symbols

j angle of incidence

u paraxial ray angle 11 paraxial ray height n refractive index

F (equivalent) power of a surface or lens

Csurface curvature (reciprocal of radius of curvature)

dseparation between adjacent refracting surfaces

For thin components, this is the distance from the vertex V of the component to the axial object point O. For thick components,

the distance is measured from the front principal plane P to the axial object point

o

L corresponding vergence (= n/ I) M transverse magnification

H optical invariant

11 object size F focal point N nodal point

P principal point

Q off-axis object point o on-axis object point V vertex point

A prime (') superscript following the above

symbols (except F, C, d, M, and H) means that a quantity occurs after refraction or in image space.

Quantification of aberrations

In the absence of aberrations, diffraction and scatter, all rays from any point in object space, that are refracted by an optical system, are focused to one point in the image plane - that is, they are concurrent. The position of this point can be predicted by using the paraxial equations described in Appendix 1. In the presence of aberrations, the rays are not

concurrent at the expected image point, but intersect the paraxial image plane in a spreadout pattern.

There are three common ways of quantifying the aberrations of an optical system; namely wave, transverse and longitudinal. Aberrations of a system are often determined by calculating the aberrations of representative rays. For each ray, there is always a wave and a transverse aberration, and sometimes a longitudinal aberration. Figure A2.I shows a schematic system and typical rays traversing the system. The wave, transverse and longitudinal aberrations of this ray are described as follows.

Rays from an axial point

Figure A2.Ia shows a particular real ray OBB'H in a beam from the point O. This ray should cross the optical axis and the paraxial image plane at the point 0' but, because of aberrations, it intersects the axis at the point G and the paraxial image plane at the point H.

The path OBO' indicates the route of the corresponding unaberrated ray. The aber-

ration may be specified in terms of any of the following quantities:

where the square brackets refer to optical path lengths, which are products of physical path lengths and refractive indices. The wave aberration is the difference in optical path length between that of the pupil ray (the central ray of the beam) and that of any other ray of the beam. In this case, it is assumed that

the pupil ray of the beam travels along the optical axis.

Rays from an off-axis point

Figure A2.Ib shows a general ray from an offaxis point Q. If such a ray does not intersect the optical axis or the pupil ray of the beam, longitudinal aberration is not applicable to the ray. The transverse aberration is Q'H. The wave aberration is similar to that given by equation (A2.Ia), but now the pupil ray follows the path QEE'Q' instead of the optical axis. In summary,

There are other methods of quantifying the level of aberrations of rays and beams that depend upon the particular aberration. For

example, the spherical aberration of the eye is often expressed as an equivalent power error.

The wave aberration function

The aberration of a particular ray depends upon the co-ordinates of the point B, where

the ray passes through the entrance pupil. If we set up a (X, Y) co-ordinate system in the entrance pupil, as shown in Figure A2.1b, we can express the wave aberration as a function of X and Y. The wave aberration for the ray through B is also a function of the position 11 of the object point Q in the field. The wave aberration W is thus a function of X, Y, and 11

and can be expressed as a power series in these variables. For a rotationally symmetric

system, it has the form

W(1];X, Y) =0W~O(X2 + yz)z + 1W3,1 11(XZ

+ Y.')y + zWzof (X2 + y2)

+ zWz,zfy2';' 3Wl,1~Y

+ higher order aberrations (A2.2)

The first five terms in this expansion are the

244 Al'l'c/ldices

primary aberrations or third order aberrations, and are known as

It is often more convenient to express this function in terms of polar co-ordinates (r,q,) where

and q, is the angle between the line EBand the Y axis, as shown in Figure A2.1b. The above polynomial can then be expressed in the form

W(11;r,q,) = oW400 + 1W3 l11rJ cos(q,)

+ 2W2,o112r2 +'2W2,2112r2 cos2(q,) + 3W 1113r cos(q,)

+ higher order aberrations (A2.4)

The level of the wave aberration need only be as large as a wavelength to have a significant effect on image quality.

Units of aberrations

All the aberrations defined above are

distances or differences in distances, and therefore have units of length. Because even very small values can be significant, it is

common practice to express wave aberrations in units of the wavelength. For example, a

distance of 0.001 mm is equivalent to two wavelengths if the wavelength is 500 nm.

Defocus and wave aberration

The wave aberration polynomial is very useful in studying the effects of a defocus as well as aberrations on an image. If a defocus is present, we can add the following term:

oW2,Or 2

to the wave aberration polynomial given° by

equation (A2.4), and the value of W2 0 is related to the defocus M by the equation'

where M is the change in power required to produce that level of defocus.

Calculation of the wave aberration function

The definition of wave aberration given by equations (A2.1a and b) is ideal for the

theoretical analysis of, and deriving explicit equations for, the wave aberration, but only for relatively simple systems. For complex systems, it is easier in practice to determine

the wave aberration by numerically ray-trac- ing specific rays. Such ray-traces do not give the path QBQ' but the path QBB'H, so we cannot determine the wave aberration using equation (A2.1a or b). Therefore, in practice, we need a different and more practical definition of wave aberration.

In the ideal optical system, a point source can be thought of as emitting light with

spherical wavefronts as shown in Figure A2.2. These wavefronts enter the system through the entrance pupil. If the imagery is perfect, these wavefronts exit the system also with a

spherical shape. However, if aberrations are present, the wavefronts are distorted. The wave aberration is a measure of the distortion.

Figure A2.2 shows a distorted or aberrated wavefront leaving the exit pupil, and also shows an undistorted wavefront (the refer-

ence sphere). The wave aberration of any ray is the optical distance along the ray between the distorted wavefront and the reference

sphere. Arbitrarily, the two wavefronts are defined to coincide at the centre of the exit

pupil.

Since for any specific optical system the position and shapes of all the surfaces are known, the ray intersection points with each surface can be found by conventional geo-

metrical and trigonometrical rules and the above path differences can be calculated for

any specific ray.

The two definitions of wave aberration

used (equations (A2.1 and A2.7)) are slightly different, but for low levels of aberrations will lead to very similar results. They will give different results for large levels of aberrations,

but in this case, some other equations used in the chapters covering aberrations will also be

in error; for example, the relationship between wave, transverse and longitudinal aberrations used in Chapter 15.

Seidel aberrations

There are seven Seidel aberrations, and these are listed in Table A2.1 along with some of

their properties. Only five of these are monochromatic aberrations, and these are related to

the primary wave aberrations given in equation (A2.2). The Seidel aberrations can be

calculated from the paths of the paraxial marginal and paraxial pupil rays.

Seidel aberrations are usually calculated as wave aberrations, although they are

occasionally calculated in the transverse or longitudinal aberration forms. Equations for their calculations can be found in texts by

Welford (1986) and Smith and Atchison (1997). They are calculated from the system constructional parameters (refractive indices, surface curvatures and shapes and surface separations), and depend upon the beam width and the field-of-view, which in turn are

defined by the paraxial marginal and paraxial pupil rays. Values are given in Appendix 3 for selected schematic eyes, and are calculated for

a beam width limited by the paraxial marginal ray passing through the edge of an 8 mm diameter entrance pupil (that is, through the edge of pupil of radius p=4 mm), and for a field width specified by the paraxial pupil ray

inclined at angle of 5° to the optical axis. Figure A2.3 shows a beam arising from the point Q, which is at the edge of the °nominal

field of view and subtends an angle 1 at the entrance pupil at E. The pupil ray subtends a paraxial angle ul to the axis, and this paraxial angle is the tangent of the real angle according

to

The combination of beam width and field size is embodied in the optical invariant H, which we introduced in Appendix 1. Using the paraxial marginal and pupil rays, we have

where u and h are the marginal ray angle and height at any plane in the system, uand 1i are

the corresponding pupil ray values, and 11 is the refractive index on the side of the plane where the above paraxial angles are measured. At the entrance pupil plane,

Therefore, for the data in Appendix 3,

For a beam with a different width and from a

different field point, the aberrations are different and can be found from

New aberration

value = old aberration value

x (fractional pupil radius)P

x (fractional field position r)q (A2.11)

where the values of p and q are given in Table A2.1.

The fractional value r can be used to specify the position of some other field point, say the point Q r shown in Figure A2.3, and this point is off-axis by an angle Of' measured at the entrance pupil. The fractional field value r is defined as

where Or is an off-axis angle point inside or outside the 5° field.

Seidel aberrations and the primary wave aberration co-efficients

The five monochromatic Seidel aberrations are directly related to the wave aberration co-

efficients of equations (A2.2) and (A2.4) as follows:

be represented by the corresponding vergences L and Lt. This is particularly useful if the imag~ plane is at infinity, because then the image surfaces cease to have any meaning. Using equations in Smith and Atchison (1997), for ray-tracing out of the eye and for the object (instead of the image) at infinity, we can show that the vergences are related to the off-axis

angle 8by the equations

where 8 is the off-axis angle in radians. These vergences are the same as peripheral power errors as used in Chapters 15 and 16.

In visual optics, the peripheral power errors Ls(8) and Lt( 8) are more meaningful than the Seidel or wave aberration values. However, if we wish to calculate the point spread or optical transfer functions, we need to know

the corresponding wave aberration coefficients W2,oand W2,2' We have

Modifications for a curved retina

The use of Seidel aberrations assumes usually that the image surface is flat. Some of the

above equations can be modified by taking into account the radius of curvature of the retina. The equivalent of equations (A2.16a and b), for the sagittal and tangential surface vergences, now become

whereretina. rR is the radius of curvature of the In the Seidel approximation, the wave

Seidel aberrations of a gradient index medium

Sands (1970) presented a set of equations for determining the Seidel aberrations of gradient index media. To make these equations consistent with the Seidel aberrations of Hopkins (1950) and Welford (1986), Sands' aberration values must be multiplied by a factor of two and the signs changed. The aberrations are broken up into two types: the refractive contribution arising at the surfaces, and the transfer contribution arising from the passage of the rays through the lens from the anterior to the posterior surface.

Refractive contribution

where Ct and C2are the front and back surface curvatures of the lens, d is its thickness,

dNo(Z)/dZ is the differential of No(Z) with respect to Z, h is the paraxial marginal ray

height at the surface, and h is the paraxial pupil ray height at the surface.

Transfer contribution

Spherical aberration (5t)

- INo(d)h(d)u3(d) - No(0)h(0)u3(0)}

- 12S[4N2(Z)h4(Z) + 2Nt(Z)h 2(Z)1I2(Z) o

- 0.5No(Z)u4(Z)jdZI

Coma (52)

- INo(d)h(d)1I2(d)u(d) - No(0)h(0)u2(0)U(0)1

d

- 12J[4N2(Z)h3(Z)h(Z) o

+ Nt(Z)h(Z)u(Z)[h(Z)u(Z) +h(Z)u(Z)j

-0.5No(Z)1I3(Z)U(Z)]dZI

Astigmatism (53)

- INo(d)h(d)u(d)u2(d) - No(0)h(0)u(0)u2(0)l

- 121[4N2(Z)h2(Z)1i2(Z) o

+ 2Nt(Z)h(Z)h(Z)u(Z)u(Z)

-0.5NO(Z)1I2(Z)u2(Z)]dZI

Distortion (55)

- INo(d)h(d)u3(d) - No(0)h(0)u3(0)l

-121[4N2(Z)h(Z)h3(Z)

o

+N;(Z)h(Z)u(Z)[h(Z)u(Z) + h(Z)II(Z)]

-0.5No(Z)II(Z)u3(Z)]dZI

Relevance of gradient index expressions to the eye

If a ray passing through the lens of the eye has

a shallow trajectory, the angle u(Z) is small. In the expressions above, the effects of the No(Z) and Nt(Z) co-efficients are attenuated by the value of II, and so their contributions are also

likely to be small. The N2(Z) co-efficients are likely to be the main gradient index contri-

butors to Seidel aberration of the lens. The equations show that the spherical aberration

has the opposite sign to that of the N2(Z) coefficients. Thus, if the gradient index medium

is to reduce the spherical aberration of the eye, the sign of these co-efficients must be positive.

Summary of main symbols

rrelative field position defined by

equation (A2.12)

Hoptical invariant defined by equation (A2.9)

Seidel aberrations

Sl spherical aberration

S2 coma

S3 astigmatism

S4 Petzval curvature

S5 distortion

CL aberrationlongitudinal chromatic

CT transverse chromatic aberration

W(r) wave aberration for ray passing through the pupil at a height r

oW4,o etc. wave aberration co-efficients

References

Hopkins, H. H. (1950). The Wave Theory of Aberrations.

Clarendon Press.

Sands, P. ]. (1970). Third order aberrations of inhomogeneous lenses. J. Opt. Soc. Am., 60, 1436--43.

Smith, G. and Atchison, D. A. (1997). In TheEyeandVisual Optical Instruments, Ch. 33. Cambridge University Press.

Welford, W. T. (1986). Aberrations ofOptical Systems. Adam Hilger.

Introduction

This appendix lists constructional data, Gaussian constants such as the pupil positions and sizes, the positions of the cardinal points,

and the Seidel aberrations of several paraxial and finite schematic eyes. The constructional

data were taken from references, but Gaussian and aberration values were determined by the

authors.

Seidel aberrations were evaluated for the schematic eyes using an 8 mm entrance pupil

diameter, a semi-field angle of 5° and a

reference wavelength of 589 nm. The optical invariant H has the value 0.349955. The V-

value of all ocular media is taken as 50.23, based on equations (17.15) and (17.17).

Units

Distances are in millimetres and are generally measured from the front surface vertex of the cornea.

Powers are expressed in units of dioptres (0 or m").

Accommodation levels are measured at the anterior corneal surface vertex.

The Seidel aberrations and the primary wave aberration co-efficients are given in units of wavelengths (A. =589 nm), except for S5and Cp which are given as percentages.

Paraxial schematic eyes

List of eyes

Gullstrand exact (Gullstrand, 1909) relaxed

accommodated 10.878 0

Le Grand theoretical (Le Grand and El Hage,1980)

relaxed

accommodated 7.053 0 Gullstrand-Emsley simplified (Emsley, 1952)

relaxed accommodated 8.599 0

Bennett-Rabbetts simplified (Rabbetts, 1998)

relaxed

Emsley reduced (Emsley, 1952).