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

80 Image jormation and refraction

ignore the effects of aberrations and diffraction/ the light distribution of Q/ on the retina is uniform and the area illuminated is the projection of the exit pupil at E', through Q/ onto the retina at Qb/. Therefore, the beam cross-section on the retina has the same shape as the pupils. It is assumed that the normal pupil of the eye is circular, and therefore the light distribution at Qb' is called the defocus blur disc for the object point Q. Similarly, the circle centred on R' is the defocus blur disc for

the object point O.

The centre of the defocus blur disc is at Qb' , and this is the intersection of the paraxial pupil ray (the line of sight) with the retina. We can see from Figure 9.1 that the retinal image of the object OQ is blurred or defocused, and it is a different size from the focused image O/Q/ formed in front of the retina. In this chapter, we explore the effect of such a defocus on the apparent size of the blurred retinal image and the size of the defocus blur disc. We start with the retinal image size of the object OQ.

Retinal image size

Consider the points 0 and Q shown in Figure

9.1 as the ends of a line object. Each point on this line is imaged as a defocus blur disc

(assuming a circular pupil), but only the blur discs for the points 0 and Q are shown. We could define the size of the blurred image of OQ as being measured from the bottom of the blur disc centred on Ob' to the top of the blur disc centred on Qb/. However, with this definition the de focused image size depends

'1

upon pupil size, and it is preferable to have a definition that is independent of pupil size. A definition that satisfies this criterion is the

image size measured from the centres of the defocused blur discs at Ob' and Qb', that is,

the distance Tlb' where the pupil rays from 0 and Q meet the retina.

In this case of a defocused retinal image we should not use the nodal ray to determine the image size, as we did in Chapter 6 for focused images. The nodal ray is not the central ray of the beam, and may not be part of the imageforming beam. It can be seen in Figure 9.1 that, if the pupil diameter is small enough, the

rays through the nodal points are blocked and do not reach the retina. The likelihood of this happening increases as Q moves further away from the axis. When we investigate the size of

a defocused image, we should use pupil rays only. We can find equations for this image size and its magnification relative to the focused image.

The size of the defocused image

Figure 9.2 shows the eye focused on the point O2/ but observing the scene at the plane at 0 1, From this figure, the size Tlb' of the defocused image is

Now the angles uz/and Uzshown in Figure 9.2 are connected by equation (5.7). Therefore

(9.2)

with the value of the constant mz depending upon the actual schematic eye used, in

particular the equivalent power and positions of the pupils and principal planes. Combining the above two equations gives

is the angular size of the object, measured at the entrance pupil. Equation (9.3) shows that

the retinal image size (whether focused or not)

is proportional to the angular size of the object measured at the entrance pupil at Ez.

More meaningful than the defocused image size 1]b' is its size relative to the size 1]' of the

image if it were in focus. This ratio is the magnification M

The size of focused images was discussed in Chapter 6, and equation (6.6) is relevant. Here

we have

where the subscript '1' refers to the quantities

measured for the eye focused on the point 01 instead of Oz. Equation (9.5) can now be written as

mzuzEz'R'

M =m E'R' (9.7a)

1ul 1

Using equation (9.4), we can express this

equation in terms of the distance dJ and dz of the focused planes from the corneal vertex as

As the eye changes its refractive state, there is

a change in entrance and exit pupil positions. This in turn leads to changes in most of the above quantities.

An eye focused at a finite distance, looking at an object at infinity

Before deriving an equation for the expected change in the image size, we will examine the

optics of this process. As the eye accommodates, the equivalent power increases and,

according to equation (6.11), the in-focus image size must decrease. However, because

the object is at infinity, this focused image is formed in front of the retina as shown in Figure 9.3. Therefore, we may expect that the

perceived image also decreases in size. However, in Figure 9.3, the height of the

retinal image is set by the position where the pupil ray intersects the retina, and this position is higher than the image point Q'.

To determine whether the observed image

actually decreases or increases with accommodation, it is possible to derive a

suitable equation directly from first principles using Figure 9.3. Equation (9.3) still applies,

and for the focused image we can use equation (6.11), with the angle (J replaced by

the angle u1 and the power here written as Fe' Thus

(9.8)

Since the object is at infinity, Ul = uz, and if we now substitute for 1]b' from equation (9.3) and

82 Image[ormation and refraction

for 1"1' from the equation (9.8), equation (9.5) becomes

Example 9.1: Compare the retinal image sizes of the moon for the two extreme states of accommodation of the Gullstrand number 1 eye.

Solution: Equation (9.9) is used, with the following data from Table 5.1,

For the relaxed eye: Fe = 58.636100 0 For the

accommodated eye: m2 = 0.795850

and E2'R' =

21.173464 mm

Substituting these values into equation

(9.9) gives

M = 0.795850 x 0.058636100 x 21.173464 =0.98807

The value of 0.98807 correspond to a 1.2 % decrease in image size.

Thus, if the eye accommodated by 10 0 while looking at the moon, the moon's image would decrease by only 1.2%, showing that the retinal image size is remarkably stable with focus error.

The use of artificial pupils

The results of this section show that the effect

of a focus error on retinal image size depends upon the position of the entrance pupil of the

eye. In some experimental situations, the eye's natural pupil is dilated and artificial pupils are placed close to the spectacle plane about 12-15 mm in front of the eye. The artificial pupil becomes the effective aperture stop and the entrance pupil of the eye, and its position and size affect the size of defocused images.

Size of the defocus blur disc

The geometrical aberration-free defocus blur disc

Calculations of the diameter of the defocus blur disc, whether by physical or geometric optics, have traditionally used schematic eye models (e.g. van Meeteren, 1974; Charman

and Jennings, 1976; Obstfeld, 1982). These calculations use a schematic eye with a given refractive power (often that of Gullstrand's

number 1 model eye) and a specific pupil

diameter. Appropriate rays are traced to the retina to determine the diameter of the

defocus blur circle on the retina. The expected or perceived angular diameter of this blur disc is then found by calculating the angular diameter of the retinal image subtended at the back nodal point. This method has the disadvantage that schematic eyes are only specified in the relaxed or in a greatly accommodated state, usually about 10 0, and thus defocus calculations at other levels of accommodation are not readily found.

If the retinal image diameter is not required, the expected perceived angular diameter can be found by a very simple equation which,

although approximate, may be accurate enough for many circumstances. Smith (1982)

showed two derivations of this equation, the

second and simpler of which is presented here.

In Figure 9.4, an eye with its front principal plane at P is focused on a plane at R a distance lR from P. If the eye views a point 0 at distance 1while still focused on R, the point 0 appears as a blur circle or disc superimposed

on the plane at R. The longitudinal focus error

01 is

and the physical linear diameter 1/1, of the blur circle using similar triangles can be shown to be

where Dp is the diameter of the beam at the

principal planes. The expected perceived or visual angular diameter (/>of this disc must be

measured at the front nodal point N, as shown in the figure. Thus,

(/> =iPt,/(lN -IR) that is

In terms of vergences, this equation can be written

(9.13)

The sign is not important since the angle tP

corresponds to a diameter.

The beam diameter Dp is not in general the

diameter D of the entrance pupil. The relation between these two quantities can be found by referring to Figure 9.5. From this figure, the

two quantities are related by the equation

D =o.u:T)II

that is,

where T is the distance between the front principal plane and the entrance pupil. This

distance depends upon the level of accommodation, and is 1.7 mm in the relaxed

Gullstrand number 1 eye and 0.9 mm in the accommodated version. If Dp in equation (9.13) is replaced by D using equation (9.14), then

(9.15)

For low to medium levels of refractive error,

we can make some useful approximations as follows:

1.If T is neglected the fractional error is Til, and its value depends upon the level of accommodation and the distance I of the out-of-focus object. However, if this distance is infinite, the error is zero.

2.The quantity IN can be neglected if the quantity LR/N« 1. While IN depends upon the equivalent power of the eye and hence accommodation level, its value is about 5 mm. For a value of LR of 100, the error induced in neglecting this term is about 5%,but for LR =0, the error is zero.

If we accept these two approximations, equation (9.15) reduces to

If (LR - L) is replaced by the focus error L1L, the above equation can be re-expressed as

o

,-

Figure 9.5. The difference between beam diameter at the front principal plane and the entrance pupil.

which expresses the visual angular subtense tP of the defocus blur disc in terms of the

dioptric level of defocus L1L and the pupil diameter D. Thus, within the limits of the approximations made in deriving this equation, the angular diameter of the defocus blur disc is independent of the constructional

parameters of the schema tic eye and the acccommodation level.

If we express the pupil diameter in millimetres and defocus in dioptres and tP in minutes of arc, we have

84 Image[ormation and refraction

Example 9.2: Calculate the defocus blur disc size for an object at infinity viewed by aID uncorrected myopic eye with a 4 mm diameter pupil.

Solution: We substitute L!L = 1 and Dm m = 4 into equation (9.17a), to give

eP =3.483 x 1 x 4 =13.8 min. arc

Experimentally determined angular diameter of the blur discs

The accuracy of equation (9.17) relative to the more accurate equation (9.15) is shown in Figure 9.6, where we have plotted the values

for a 4.2 mm diameter pupil and an eye focused at different distances but viewing a scene at infinity - i.e. L =O. The values of T

and IN are taken from the Gullstrand relaxed number 1 eye data in Appendix 3. The approximate equation gives values that are too high, but by only a few per cent.

Smith (1982) suggested that the perceived

angular diameters of the blur discs can be measured using two small light sources, such as illuminated optical fibres, whose separation is adjustable. When these sources are defocused, they are seen as blur discs. The

Defocus (D)

Figure 9.6. The diameter (/>of defocused blur discs measured by Chan et al. (1985),corresponding values predicted by the approximate equation (9.17a),and values for a 4.2 mrn diameter pupil using the exact equation (9.15),with L =O.

separation between the sources is varied until the two blur discs appear to just touch, and the angular diameter of one blur disc is then equivalent to the angular separation of the sources. At low defocus levels the blur disc

takes on an irregular star-shaped pattern, which is due to ocular aberrations, making it difficult to locate the edge of the blur discs. At high levels of defocus, where the defocus dominates over the ocular aberrations, the

defocus blur discs are more regular.

Chan et al. (1985) used this technique to test the accuracy of equation (9.17). Six observers were used, and since the differences between observers were small, the mean data are shown in Figure 9.6. The spectacle plane lens

power was converted to an equivalent power at the front principal plane of the eye. There is good agreement between theory and measurements at intermediate levels of refractive error (3-9 D). However, at lower refractive errors the theoretical values are too

low, while at higher refractive errors the theoretical values are too high. At the lower levels of refractive error, the discrepancy is because of the effects of ocular aberrations and diffraction, which give a wider spread of light than expected by aberration-free geometrical optics. At the higher levels of refractive error, the deviations can be explained by equation (9.17) being only approximate. The more accurate equation (9.15) agrees wen with the measured values at these levels.

Defocus ratio

The level of defocus can be quantified by the diameter t/Jt,' of the defocus blur disc in the image, the diameter t/Jt, of the defocus blur disc in object space, or its corresponding angular size eP. Alternatively, if we want to study the effect of a defocus on the visibility of an object

of a certain size eP, a more appropriate quantity is the defocus ratio, which we define

by the equation

In angular terms in object or visual space we have

where ()is the angular size of the object.

Example 9.3: Calculate the defocus blur ratio for the situation given in Example 9.2, if the eye is observing a 6/6 letter, i.e. one that subtends 5 min arc.

Solution: Substitute cP = 13.8min arc and (J= 5 min arc into equation (9.19) to give

Defocus ratio = 13.8/5 = 2.76

Other effects of defocus

Alignment of two targets at different distances

In some visual experiments, the eye is aligned

by asking the subject to subjectively align two targets at different distances. Figure 9.7 shows points Q1 and Q2 aligned along the visual axis. If these points are so aligned, do they

appear superimposed in the visual field? In other. words, are their retinal images superimposed? By analysing the situation shown in Figure 9.7, we can deduce that the targets do not appear to be aligned. This

shows ~e point Q~ imaged, in focus, at Q2' on the retina. The other point Q1 is imaged in

front of the retina at QI' and is therefore

def<?cused at the retina. The retinal image of Q1 is the defocus blur disc centred on the

point Qlb', which is laterally displaced with respect to the in-focus image Q{ Therefore, while the two points are aligned along the visual axis, their retinal images are transversely displaced.

If the points are to appear superimposed,

the two targets must lie along the paraxial pupil ray (i.e. the line of sight) rather than the visual axis.

Effect on visual acuity

Defocus reduces the quality of the retinal image and hence various aspects of visual performance. Uncorrected visual acuity correlates with refractive error. However, the correlation is not perfect, even after pupil size is taken into account (Atchison et al., 1979). The relationship has been quantified from time to time by various investigators fitting

regression equations to clinical data. Perhaps the most common equation used is of the form

w~e.re A is the (uncorrected) visual acuity in rrurumum angle of resolution in min arc, L1L is the refractive error (say in dioptres), and aand b are constants. This equation is not based

upon any theoretical optics justification. In contrast, an argument based upon the defocus

blu~ circle model described in the preceding section suggests that the minimum angle of resolution should be linearly related to

refractive error, except for low levels of refractive errors. Smith (1991) used the defocused blur disc concept and the optical

tran~fer .function to show that, for typical pupil diameters and for refractive errors greater than about 1 0, the expected relationship is a simple one of the form

o~

86 lmagt;[ormation and refraction

where D is the pupil diameter and k is a constant whose value depends on the structure of the acuity target and the recognition rate (e.g. 50 per cent, 75 per cent, etc). For letters of the alphabet and a 50 per cent recognition rate, the value of k is approximately 650 for A in minutes of arc, D in metres and.1L in dioptres (Smith, 1996).

For low levels of defocus, aberrations and diffraction dominate the quality of the retinal image, but these factors are beyond the scope of the paraxial optics discussion of this chapter. The influence of these factors is considered in Chapter 18.

The value of k and the corresponding defocus ratio

From the above value of k, we can estimate the defocus ratio of alphabetical characters at the 50 per cent threshold of visibility.

The letter size is regarded as five times the minimum angle or detail in the letter. Using this value, transforming equation (9.21) to the threshold letter size H in radians gives

H= 5A = 5 x 650 x D.1L x tr/(180 x 60)

=0.945 D.1L radians

and since cp= D.1L,

from equation (9.17)/it follows that

which indicates that a letter may be recog-

nized at a 50 per cent success level when the defocus blur disc is about the same diameter

as the letter height.

Summary of main symbols

I, l' object and image distances from front and back principal planes, respectively

Lcorresponding vergence of the distance 1

refractive error

distance of nodal point from principal plane

distance of object, conjugate with the retina, from principal plane

vergence corresponding to 1

distance of entrance pupil from front principal plane

defocused image size at retina object and image sizes angular size of object

diameter of defocus blur disc on retina and its conjugate in object space corresponding perceived angular diameter of defocus blur disc, measured at the nodal points (always positive)

E, E' position of entrance and exit pupils N,N' position of front and back nodal points 0/0/ general object point and correspond-

R/R' axial retinal point and corresponding conjugate in object space

References

Atchison, D. A, Smith, G. and Efron, N. (1979).The effect of pupil size on visual acuity in uncorrected and corrected myopia. Alii. J. Optom. Physiol. Opt., 56,

315-23.

Chan, C, Smith, G. and Jacobs, R. J. (1985). Simulating refractive errors: source and observer methods. Am. J.

OptOIll. Physiol. Opt., 62, 207-16.

Charrnan, W. N. and Jennings, J. A M. (1976). The optical quality of the monochromatic retinal image as a function of focus. Br. J. Physiol. Opt., 31,119-34.

Marsh, J. S. and Temme, L. A (1990). Optical factors in judgements of size through an aperture. HumanFactors, 32,109-18.

Obstfeld, H. (1982). Optics in Vision, 2nd edn. Butterworths,

Pardhan, S. and Gilchrist, J. (1990). The effect of

monocular defocus on binocular contrast sensitivity.

Ophthal. Physiol. Opt., 10, 33-6.

Acad. Optom.,33, 353-73.

Rosenfield, M., Ciuffreda, K. J., Hung, G. K. and Gilmartin, B. (1993). Tonic accommodation: a review. I. Basicaspects. Ophthal. Physiol. Opt., 13, 266-84.

Simpson, T. L., Barbeito, R. and Bedell, H. E. (1986).The

effect of optical blur on visual acuity for targets of different luminances. Ophtha/. Physiol. Opt., 6, 279--81.

Smith, G. (1982). Angular diameter of defocus blur discs.

Am. J. Optom. Physiol. Opt., 59, 885-9.

Smith, G. (1991). Relation between spherical refractive error and visual acuity. Optom. Vis. Sci., 68, 591-8.

Smith, G. (1996). Visual acuity and refractive error. Is

Image[ormation: thedefocused paraxial image 87

there a mathematical relationship? Optometry Today, 36 (17),22-7.

Smith, G., Meehan, J. W. and Day, R. H. (1992). The effect

of accommodation on retinal image size. Human Factors, 34, 289-301.

van Meeteren, A. (1974). Calculations on the optical modulation transfer function of the human eye for white light. Optica Acta,21, 395-412.

Westheimer, G. and McKee, S. P. (1980). Stereoscopic

acuity with defocused and spatially filtered retinal images. J. Opt. Soc. Am., 70, 772--8.

Introduction

When an eye is corrected by an ophthalmic lens in the form of spectacles or contact lenses, aspects of the retinal image are changed as well as the image being focused. Most noticeably, the lens affects the size of the retinal image. This change in image size can be measured in two different ways. One of these is the spectacle magnification, which is the

ratio of the image sizes after and before correction. The second, which is of limited

use, is called relative spectacle magnification. This is the corrected or focused retinal

image size compared with that of a 'standard'

eye. These magnifications are associated with other effects. For example, a positive power

lens produces a magnified image and a blank part of the visual field called a scotoma. Image magnification of spectacle lenses is associated with prismatic effects and changes in the required eye rotation to look at an object which is not on the lens optical axis. There are also effects on binocular vision. In anisometropia the two eyes are corrected by

different lens powers, and differential effects between eyes, for example different amounts

of eye rotations, may produce eyestrain.

In this chapter we investigate some of these

racies, the equations are simple and readily show trends. Aberrations associated with the lenses are not considered. We refer readers who want a full thick lens treatment, or to

consider effects of aberrations, to texts such as [alie (1984) and Fannin and Grosvenor (1996).

Most of the equations in this chapter apply to both spectacle and contact lenses. Exceptions are the equations dealing with

rotational magnification and field-of-view in the section Effects on far and near points and accommodation demand. The equations assume

a rotating eye behind a fixed lens, which is not applicable for contact lenses, which rotate

with the eye. Most of the effects described

have much smaller magnitudes for contact lenses than for spectacle lenses, because of the

small distance between contact lenses and relevant ocular reference positions such as the

entrance pupil.

Spectacle magnification

Spectacle magnification SM is defined as

An equivalent definition is

of the defocused image (neglecting aberrations).

The following equation is derived for spectacle magnification using Figure 10.1. An object of height T/ is at a distance q from the entrance pupil of the eye at E (Figure 10.la). It subtends an angle ro at the entrance pupil of

the eye, given by

In Figure 10.lb, a correcting lens of power Fs has been placed in front of the eye. The object is a distance I from the lens, and the eye's entrance pupil is a distance he from the lens. The object subtends an angle t/J at the lens,

given by

The image of the object in the lens has height

T/'and is a distance I' from the lens. From the figure, T/'is given by

Substituting the right-hand side of equation (10.3) for t/J into equation (10.4) gives

The angle ro'subtended by the image at the entrance pupil E is given by

Substituting the right-hand side of equation (10.5) for T/'into equation (10.6) gives

From the definition of equation (10.la), SM is

(a)

(b)

Qr--hc-l/'---1

"- /--+--

Figure 10.1. The optics of spectacle magnification and the angular size of the images:

a.The uncorrected eye looking at an object.

b.The eye looking at the object through a correcting spectacle lens.

Substituting the right-hand sides of equations (10.2) and (10.7) for roand ro~ respectively, into equation (10.8) gives

If we replace I and I' by their corresponding vergences L (= 1/1) and L' (= 1/1'), we obtain

Given that L is known, L' is found simply by

the refraction equation L' = L + Fs' For the object at infinity, it can easily be shown that

Equations (10.8b) and (10.8c) can be extended to the thick lens case, with F now the

equivalent power of the lens and Land L' determined relative to the front and back principal planes, respectively, of the lens.

Examination of equation (10.8c) shows:

1.For positive lens powers, spectacle magnification is greater than 1, i.e, the observed object is enlarged. For negative power lenses, spectacle magnification is less than 1.

2.The departure of spectacle magnification from a value of 1 is increased with increase in distance between the lens and the eye.

In ophthalmic optics, it is more convenient to measure the back vertex power rather than the equivalent power of a lens. For the object at infinity, it can be shown (see, for example, Jalie, 1984)that

where t is lens thickness, n is lens refractive index, hve is the distance between the lens

back vertex and the eye's entrance pupil, F1is

the front surface power of the lens, and Fv is the back vertex power of the lens. This

equation shows how SM is dependent on the parameters»v»: of the lens. The expressions (1 -

and (l-hveF/ l-l on the right-hand side of the equation are referred to as the

shape factor and power factor, respectively.

Example 10.1: An object is 30 em in front of the corneal vertex of the eye. A

correcting thin lens of power -S 0 is