Effect of pupil size on steady state accommodation

Cop)r!ghr

EFFECT

OF PUPIL SIZE ON STEADY ACCOMMODATION

(

oc-c-6989 35 53.00 t 0 00 1955 Prrg~mon Press Ltd

STATE

P. A. WARD and W. N. CHARMAS Department

of Ophthalmic

Optics. Sackville

The University of Manchester Street. .Manchester 4160 IQD. {Rrceitwi

29 Ocfoher

Institute England

of

Science & Technology.

I9g4)

Abstract-Experiments are described in which monocular accommodation response!stimulus curves were measured with Sneiien targets over the stimulus range from + I .O to - 5.0 D, using artificial pupils with diameters of 1.0. 1.5. 2.0. 2.5 and 3.0 mm and a constant retinal illuminance of 600 td. The results agree with those of earlier authors in showing a diminished response with smaller pupils. The slopes of the quasi-linear central portions of the response/stimulus curves are well described in terms of a geometrical optical approximation in which the accommodation system works to produce a retinal blur circle whose diameter is a linear function of the dioptric difference between the magnitudes of the stimulus and the accommodative resting state, this blur circle diameter being independent of the pupil diameter. Further consideration of diffractive effects suggests that contrast changes in the intermediate spatial frequency components (z 5 c,‘deg) of the retinal image may play a dominant role in guiding the response. Accommodation

Pupil

Ocular

modulation

transfer

INTRODUCTION

Following early work by Morgan (1944). there is now general agreement the that accommodation response/stimulus curve typically takes the form shown in Fig. I. Within the range of accommodation of the individual observer, errors or accommodation occur at both high and low stimulus levels, corresponding to near and distant objects. Under natural observing conditions, the slope of the quasi-linear central portion of the curve appears to depend upon the nature of the spatial frequency information available to the higher centres controlling the response. If the target itself is very blurred or only contains very low spatial frequencies (e.g. Heath, 1956; Charman and Tucker, 1977; Owens, 1980) the gradient of the response curve is low, becoming zero under “empty field” conditions (Whiteside, 1953, 1957; Nadell and Knoll, 1956). Similarly if the sensitivity of the visual system to spatial contrast is lowered by, for example, a reduction in luminance, the slope of the response/stimulus curve again falls, once more becoming essentially zero under conditions of very low or zero luminance (e.g. Johnson, 1976). The constant, non-zero level of accommodation response (typically + i .5 D) reached under degraded stimulus conditions (empty-field myopia, night myopia, dark focus etc.) is often termed the tonic accommodation or resting state (e.g. Leibowitz and Owens, 1975a, 1975b). It is inferred that it represents the equilibrium state of an accommodative system which depends upon the antagonistic action of parasympathetic and sympathetic innervation, increased parasympathetic activity being demanded for near work and increased sympathetic for far (e.g. Toates, 1972). If the details of the accommodation control system are to be successfully modelled, there is evidently a

need to understand more fully the parameters affecting the slope of the response/stimulus curve. One path to such increased understanding is through the use of small artificial pupils. Various authors have pointed out that, to a first approximation, such pupils have the primary effect of increasing the ocular depth-of-focus and hence allowing adequate vision with larger errors in accommodation. Two groups

--?-----

Stimulus

I’

(0)

Fig. I. Schematic diagram of a typical accommodation response/stimulus curve. In the present paper the widely used sign convention of visual optics is followed: an object x m in front of the eye has a vergence of - I/x D and demands for accurate focus a response of + I/x D. The 45’ dashed line represents the “ideal” response/stimulus line for an in-focus retinal image. R is the clinically determined amplitude of accommodation; S and P are the stimulus ranges which are thought to involve increased sympathetic and parasympathetic activity respectively. The value of response at which response and stimulus are equal, which corresponds to the “tonic” accommodation or resting state is typically c I.5 D. 1317

(Ripps ef al.. 196Z: Henncssy rr 01.. 3975) have subsequently demonstrated that. in consequence. the slope of the monocular response stimulus curve decreases as the pupil size is reduced below c 3 mm. Monocular observation is used to reduce complications due to vergence input to the accommodative system.

We note, however, that depth-of-focus is not the only factor influenced by pupil size. In practice, diffractive effects result in a substantial loss of visually-significant, high spatial frequency information from the in-focus retinal image when small pupils are used: only in a geometrical approximation is the in-focus image independent of pupil diameter. Moreover, the existing experimental studies of monocular response with small pupils can be criticised on the grounds that they relied on measuring the accommodation response of the fellow, rather than the observing eye, with some consequent uncertainty: Hennessy et al. (1976) only used the very small stimulus distance range of 0.5-6.0 m (- 2 to - 0. I7 D vergence). We have therefore repeated such studies, using apparatus which allows direct measurement of the response of the observing eye. More importantly, we have attempted to relate the observed responses to the corresponding optical characteristics of the retinal image in order to obtain a better understanding of the observed accommodation response/stimulus curves. METHODS

The apparatus used is shown in Fig. 2. The subject viewed monocularly one of a series of targets (T) placed at distances (I .0-O.? m) which gave an accommodative stimulus range from - I.0 to - 5.0 D in 0.5 D steps. The range was extended to + 1.0 D by the use of supplementary lenses (Ll), each placed close to the eye. Targets were transparencies of high-contrast arrays of illiterate E’s (Bailey and Lovie, 1976) and each target was photographically scaled for the corresponding viewing distance so that its overall subtense at the eye was constant and its letter sizes ranged from the equivalent of 6/3 to 6/60 Snellen. The targets were back-illuminated by tungsten light through a narrow passband green gelatine filter (F), to give a mean illuminating wavelength and halfwidth of about 540 and 60 nm respectively. Subjects viewed the targets with their right eyes through 1.0, 1.5, 2.0, 2.5 and 3.0mm diameter artificial pupils (AP) cut in red gelatine filter material, the left eye being occluded. The transmittance of the red filter material was effectively zero for the green wavelengths of the targets. Suitable values of neutral density filter (ND) were placed in the light path of the targets so that, with allowance for the subsequent light loss due to the beamsplitter (BS) and the area of each artificial pupil, the retinal illuminance was always 600 td. The pupils were carefully centred to

Fig. 2. Schematic diagram of apparatus. The projector P illuminates the opal perspex diffuser 0 through the green filter F. A target in one of the alternative positions T is viewed by the subject S through the artificial pupil AP. The lens LI allows the target vergence to be changed if required. Accommodative response is assessed with the laser optometer constituted by the laser L, the light chopper C, the beam expander L2, L3, the plane mirrors Ml, M2 and the slowly rotating drum Dr. The laser speckles are viewed superimposed on the targets via the supplementary lens L4 and the beamsplitter BS.

the eye, head position being maintained with a chinrest and brow-bar. The beamsplitter allowed the subject to see, superimposed on the target field, the red speckle pattern from a conventional laser optometer (Charman and Tucker, 1977). The chopper (C) gave a speckle presentation of 0.5 set every 5 sec. Movement of the optometer drum (Dr) along the observing path until the speckle pattern appeared stationary allowed the accommodation state of the eye to be calculated from the drum position, appropriate corrections being made for the position of the plane of stationarity with respect to the drum surface (Charman and Chapman, 1980) for the longitudinal chromatic aberration (Bedford and Wyszecki, 1957) of the eye between the green target and red (633 nm) laser wavelengths and for the effects of the supplementary lenses L I and IA. The use of the red filter for the artificial pupils meant that the speckle pattern was effectively seen through the full natural pupil of the viewing eye (-4 mm diameter under the conditions of the experiment) thus minimising ocular depth-of-focus and optimising precision during the estimate of accommodation, while maintaining the required small pupil for the target observation. The precision of the bracketing technique used for accommodation estimation was typically about 0. t 5 D. The targets were presented in sequence from + 1.0 to - 5.0 D. Subjects were instructed to always direct their attention to the smallest letters that they could see and to maintain best possible focus at all times. They were allowed to relax between successive target presentations. If a subject reported tiredness or

Effect of pupil size on steady

1319

state accommodatxn

4 or

?uplI 0

d~amerer

;mm)

10

00 1.0

00

-1.0

-20 Stimulus

- 3.0

-4.0

-50

t D 1

Fig. 3. Monocular accommodation response/stimulus data for subject R.G. at the 5 pupil diameters indicated. Green light, retinal illuminance 600 td. The horizontal arrow indicates the measured dark focus

for this subject.

difficulty in maintaining accommodation a further rest period was allowed. A complete run with a single pupil typically took - 20 min. After data had been collected for all 5 pupil diameters, subjects were allowed 10 min dark adaptation and their accommodation in the absence of an illuminated target was assessed to yield the dark focus or tonic accommodation. Although measurements of dark focus obtained in this way might appear to be potentially susceptible to near adaptation effects (e.g. Ebenholtz, 1983). repeat measurements of the dark foci of some subjects in separate observing sessions showed that such effects were not significant, probably because the main study involved a sequence of relatively brief observations of targets at a wide range of vergence levels (+ I to -5 D) rather than prolonged observation of near targets. SUBJECTS

Five adult observers were used, aged between 20 and 30 years. All had clinical amplitudes of accommodation of at least 8 dioptres, as determined with a near-point rule (Duke-Elder, 1970). Where necessary, an accurate refractive correction was worn throughout the experiment. Two of the observers were experienced in such measurements, three were naive: the latter were allowed one practice session before data were recorded. RESULTS

A typical set of data for one subject is shown in Fig. 3. A larger accommodative lag or error is evident

for the smaller pupils when near targets are observed (i.e. in the ‘*parasympathetic range” between the dark focus and the near point) although with this subject the most accurate responses do not occur with the largest pupils. It is clear also that, with all diameters of pupil, accommodation response equals the magnitude of the accommodation stimulus at a value close to the dark focus of the eye concerned. For more distant targets, lying beyond the position corresponding to the dark focus, the pupil dependence of the response is much less marked. This might be expected, since if we initially assume that the far point of each eye lies at infinity (0 D), the “sympathetic” range of the accommodative system is much smaller than the “parasympathetic range” and any flattening of the response curve in this region is inevitably less striking. Figure 4 shows that these general features were reproduced in the other subjects: for clarity only the results for 1.0, 2.0 and 3.0 mm diameter pupils are shown. Table I emphasises the similarity between the values at which the response (AR) and stimulus (AS) become numerically equal (obtained by interpolation between neighbouring data points) and the dark focus for the same eye. The product moment correlation between the arithmetic mean values for the five eyes and the corresponding dark focus is r = 0.67. The magnitudes of the slopes of the individual response/stimulus curves in the stimulus region which lies closer than the dark focus value are given in Table 2 and further illustrate the decrease in slope of the “parasympathetic” branch at small pupil diameters.

...,

mm

S.W.

0

v

/

I /

-.-

2 mm

-

3mm

’

/

/‘/,

/ t

0

I -4

-2

Stimulus

IO)

Fig. 4. Monocular response:stimulus data for four further subjects. For clarity, only results for 1, 2 and 3 mm diameter pupils are shown. ..’ .. I mm pupil; -‘-2 mm pupil; 3 mm pupil. Horizontal arrows represent dark focus responses.

Table I. Positions of accurilte focus (IARt - I AS/) for the five subjects at five pupil diameters, together with the dark focus values for the subjects. Ail Ihe response values, given in dioprres. are very similar for an individual subject Response of subject (D) H.W. P.A.W. S.W.

R.G.

1.50 2.60 2.35 3.20 2.30

I 85 2.40 2.20 2.13 2.20

2.25 1.80 2.50 ?.?O 2.00

2.30 1.95 1.50 2.10 2.75

I.00 1.30 1.20 1.10 1.19

Meen (all pupils) (SD)

2.59 0.36

2.16 0.20

2.!5 0.26

2.12 0.46

I.19 0.1-l

Dark focus

2.25

2.57

I .83

I.65

t.14

Pupil diameter (mm)

K,M.W.

I.0 1.5 2.0 2.5 3.0

DISCUSSION

When interpreting data of the present type, it seems reasonable to search for a common factor capable of linking the observed response behaviour at the different pupil diameters. Since the retinal image and its changes provide the starting point for driving the overall accommodative system, we have chosen to concentrate on this aspect. Descriptim

in terms ofgeometrid

which was linearly dependent upon the dioptric difference between the magnitude of the stimulus vergence (AS) and the dark focus (ABIAS), where we follow the terminology used by Hung and Semmfow

Table 2. Magnitude of the slope (/m 1)of the quasilinear response curves for a stimulus interval -2.5 to - 5.0 D inclusive. Values calculated by regression using the method of least squares

optics

If we assume a geometrical optical model. then we might postulate that, within the quasi-linear “parasympathetic” portion of the response curve, and for all pupil diameters, the accommodation system worked to produce a retinal blur circle diameter (D)

Subject K.IM.W. H.W. P.A.W. SW. R.G.

1.0

Pupil diameter(mm) 1.5 2.0 2.5

3.0

0.29 0.19 0.48 0.19 0.49

0.3 I 0.39 0.56 0.70 0.79

0.82 0.37 0.79 0.06 0.95

0.82 0.55 0.70 O.SL 0.74

0.71 0.52

0.78 0.59 0.97

Effect of pupil size on steady state accommodation in rhcrr control system model (Hung and Semmiow, 1987). The diameter of this “permitted” blur circle for any particular stimulus level vvould be independent of the pupil diameter (d) but would vary with such factors as the luminance level and the characteristics of the individual observer i.e.

D =(I( /ASI - IABIASI)

(i)

where n is a constant for a fixed observer and observing condition. Elementary visual optics shows, however, that any blur circle diameter is itself dependent upon the pupil diameter. the accommodative error [i.e. the difference between the magnitudes of the stimulus vergence and the accommodative response (AR)], and the dioptric length of the eye, which is usually assumed to be 60 D i.e.

D =$(iASi

- IARl).

Lastly. the magnitude of the resultant slope of the accommodation response/stimulus line in the “parasympathetic” range will be given by (JAR/ Irn’ = (JASl Combining

/ABIAS])

(iii)

- /ABIASl)

(i) and (ii) we find:

d(lAS] :.

-

d((lASl

- IARl)=60a(lASl - IABIASl)--(IARI = 60a(lASl

or, introducing

- IABIASI) - IABIASI)) - IABIASI)

(iii) ({(I - lnz I) = 60n.

(iv)

Fig. 5. Logarithmic plots of (I - lm 1) against d, where tf is the pupil diameter and lm / the magnitude of the slope of the corresponding response/stimulus line. (a) Present data for the “parasympathetic” stimulus range -2.5 to -5.0 D. Green light. 600 td. Logarithmic mean and standard deviation for 5 subjects. (b) Data of Ripps er a/. (1962) for stimulus range approximately -2 to -8 dioptres. White light, 3400 td. Arithmetic mean and standard deviation for 5 subjects. (c) Data of Hennessy Ed ui. (1976) for stimulus range 0.17 to -2.0 dioptres. White light, 62.5 td. Arithmetic mean of 5 subjects. In each case the dashed line has been drawn with a slope of - 1 through the data point for the smallest pupil.

1321

Hence a plot of logj I - jm /) against log ci should have a slope of - I. Figure 5 shows the average results for our subjects when plotted in these terms for the “parasympathetic” range -2.5 to -5.0 D inclusive. i.e. for stimuli closer than the dark focus values. Also shown are similar plots for data from Ripps er al. (1962) for the stimulus range - 1.0 to -3.0 D and from Hennessy et al. (1976) in the range -0.17 to -2.0 D (i.e. stimuli generally farther than the dark focus value). Evidently, if allowance is made for the difIerent experimental conditions. the present data and those of Ripps ef nl. agree quite well with the geometrical predictions. Those of Hennessy et al. do not, presumably because accommodation performance in the “sympathetic” range farther than the dark focus behaves differently: one would, in fact, expect the slope of the accommodation response/stimulus curve to fall to zero at the far point (0 D stimulus) and to rise steadily as stimulus AS comes nearer to ABIAS, making a linearly-fitted slope in this range of doubtful validity. We note. further, that the clinically .‘emmetropic” observer whether with or without the aid of correction is in fact usually slightty myopic in his true refraction. since conventional techniques specify the use of “least minus” or “most plus” corrections. Response to a zero stimulus is, then, typically -0.5 D. a target at optical infinity only being seen clearly with the aid of ocular depth-offocus. Hence, the magnitude of the extent of “sympathetic” range of response is smaller than the magnitude of the dark focus response. In the present study, the response at zero stimulus ievel for the clinically emmetropic or corrected subjects was typically - 1.0 D with the largest diameter pupil (3 mm). We attribute this higher than expected “lead” in accommodation partly to the increased depth-offocus resulting from the use of an artificial pupil which was somewhat smaller than the natural pupil normally associated with the retinal illuminance employed, and partly to our use of a green (z 540 nm) target wavelength. The longitudinal chromatic aberration of the eye between this wavelength and the mean effective wavelength which is thought to apply during clinical refraction with white tungsten light (- 580 nm, Charman, 1975) is about 0.2 D. so that the subjects would be expected to display an additional “myopia” of this amount during the observations. The values of the constant a deduced from the mean data of Fig. 5(a) and (b) are about 1OT5m* for the present study and 5 x 10-6m’ for the results of Ripps et nl. Thus a “parasympathetic” stimulus ( 1AS I - 1ABIAS I ) of I .O D would lead to accommodative errors yielding corresponding retinal blur circle diameters of about 10 or 5 pm. At this level of approximation, this seems a plausible order of magnitude in relation to a typical central fovea1 inter-cone separation of c 2 pm. If the data of Fig. 5 are examined in more detail,

approximation that the it becomes clear log( 1 - Inr 1) = -log~f - constant may start to break down for pupil diameters approaching 3 mm. This could be attributable to the fact that optical while negligible for pupil diameters aberration. <‘mm (Smirnov. 1962; Berny and Slansky. 1969; Walsh et al.. 1984). starts to become increasingly important at larger pupil diameters: studies of the pupil dependence of ocular depth-of-focus show very similar effects (Tucker and Charman. 1975). It is of interest to compare the geometrical optical description of the data with the response/stimulus relation predicted by current control theories (Hung and Semmlow, 1982). Rearrangement of equation (iv) yields

(n2 ( = I --.

600 ti

Hence, from (iii) (IAS

(AR(-(ABIASI=

- (ABIASl).

The linear part of the “parasympathetic” thus described by lAR/ =

l-7 (

IAS/ +T

range is

IABiASl.

>

Since, in a geometrical approximation. the ocular depth-of-focus (DSP) is inversely related to the pupil diameter DSP=h

n

where the constant b varies with the observer and observing conditions and depends upon both optical and higher-level effects. Thus the response/stimulus line is given by IARJ=

l-

,AS,

60.0bDSP

i +

> 60.a .DSP b

. IABIASI

(v)

It is likely that the constants a and b have an approximately linear relationship to one another as conditions are changed. If this is true, any factor (in addition to pupil size) which increases DSP, such as reduced visual acuity or luminance (Tucker and Charman, 1975; Oshima, 1958) would be expected to reduce the slope of the accommodation response/stimulus curve, in agreement with experimental observations (Wood and Tomlinson, 1975; Johnson, 1976; Hung er al., 1983). Expression (v) may be compared with the results of the control system model of Hung and Semmlow (1982). This yields, under similar conditions ]ARl =

ACG [ 1 +ACG

1

IASI

-DSP[l+A~G]-tlAB~ASl[,+~CG]

or. reorganising that of (VJ

the equation

into a form slmiiar

to

I ’ (I + ACG) I ‘i\s’ + (I + XCG) .( IABIASl + DSP) - DSP. (vi) Equations(v) and (vi) both show a linear relationship between AR and AS. However. in the purely descriptive equation (v). the ocular depth-of-focus, DSP. influences the slope of the response/stimulus line, whereas in equation (vi) it does not, largely because Hung and Semmlow assumed that DSP was essentially constant for normal pupil diameters. Thus we would suggest that the parameter ACG in equation (vi) must be dependent upon DSP. Banks (1980) has, in fact, already persuasively argued the case for the significance of ocular depth-of-focus in relation to the slope of accommodation response/stimulus curves when considering the development of accommodation in early infancy. He points out that the increase in slope that is observed with increasing age is exactly that which would be expected on the basis of the reduction in depth-of-focus associated with the optical and neural changes in the developing visual system. It may be noted that equation (vi) does not yield the expected equality between AR and ABIAS when 1AR l = 1AS 1. In an attempt to overcome the limitations of (vi) in explaining why response can often be very close to the stimulus level, Hung er (11. (1982) have recently revived suggestions that accommodation fluctuations may have an important role in making the response more accurate.

Physical optical considerarions So far we have considered the data in terms of geometrical optical approximations. Hopkins (1957) suggests that such approximations describe the defocused image reasonably well when the relevant wavefront aberration coefficient, ~~~~~ satisfies the condition bczO > X. This corresponds to errors of focus greater than 8.6 D and 1.O D for I and 3 mm diameter ocular entrance pupils respectively (Smith, 1981). Clearly, then, since in the present data the error of focus ([AS ( - [AR I) is always smaller than such values, it is necessary to consider the effects of physical optics if the results are to be properly interpreted. The most direct approach to the changes in retinal image quality that occurred during the study is through the corresponding modulation transfer functions (MTFs). Since ocular aberration is known to be negligible for pupil diameters <2mm, and nearmonochromatic light (j. = 540 nm) was used, it is reasonable to assume that performance during the experiments was almost diffraction-limited, although, as already noted, some slight additional degradation due to aberration might have occurred at the largest pupil sizes used (2.5 and 3 mm).

Effect of pupil size on steady state accommodation

Spatial

[ c/deq)

frequency

Fig. 6. Theoreticai modulation transfer functions for an in-focus. aberration-free eye with the entrance pupil diameters indicated. working in monochromatic light of wavelength 540 nm. It is assumed that ocular performance in the absence of accommodative error during the experiments approximated closely to this level.

Figure 6 shows the theoretical, diffraction-limited, “in focus” MTF for each pupil size (i.e. when /AR1 = 1ASI = IABIASI) as derived from the numerical tables of Levi (1973) using conversion factors appropriate to the eye (Charman, 1983). Clearly, rather than all the “in-focus” retinaf images being perfectly sharp, corresponding to unit modulation transfer at all spatial frequencies. there is a considerable variation in modulation transfer with spatial frequency and pupil size: optical diffraction heavily degrades the information content of the retinal images at high spatial frequencies when small pupils are used. Knowing the typical slopes of the response stimulus curves in the “parasympathetic” range at each pupil size, and hence the error in focus (/AS / - /AR 1) as a function of stimulus AS, we can

PuDil

z t;

1Omm -

06

ii G4

=

dlamefer

5

-

15mm

- -

20mm

---

25mm

-

30mm

FJ 02 7; :: I 0 -021 0

, 10 Spotiat

I 20 frequency

J 40

30 lc/deg

I

Fig. 7. Theoretical modulation transfer functions for outof-focus aberration-free eyes with the entrance pupil diameters indicated, working in monochromatic light of wavelength 54Onm. The errors of focus correspond to the accommodation lags for the “parasympathetic” stimulus 1AS 1- JABIAS = 3 D. Response/stimulus slope magnitudes of 0.42,0.62,0.72. 0.77 and 0.81 are assumed for the I .O, 1.5. 3.0. 2.5 and 3.0 mm diameter pupils respectively, these slopes being derived from the best-fitting straight line of slope - 1 to the raw data points of Fig. 5(a).

1373

similarly evaluate the representative MTFs at an> chosen stimulus level, Figure 7 shows such ocular MTFs for the various pupil5 for errors of accommodation corresponding to i AS 1 - IABIAS I = 3 D. The different .LlTFs are now very similar at relativeI> low spatial frequencies (< IO c!deg) but remain different at higher spatial frequencies, although the modulation transfer is always very low in this region. It must be remembered that the Snellen target arm) contained information over the full spatial frequency spectrum (although not. of course, at unit modulation) so that in searching for a common factor to explain accommodation performance at all pupil sizes. it seems reasonable to hypothesise that the subjects were utilising the spatial frequency region where the MTFs at all pupil sizes are similar, that is, they used primarily low. rather than high, spatial frequencies to guide their accommodation response. This is broadly in accord with the findings of Owens (1980) and Bour (1981) regarding the importance of -5 c,‘deg in controlling the spatial frequencies accommodation response. It does not appear that the response was “refined” by the utilisation of high spatial frequency information from the targets in the way suggested by Charman and Tucker (1977). Further insight into the possible value of different portions of the spatial frequency spectrum can be gained by considering the way in which the theoretical modulation transfer to the retina under “steady-state” conditions varies at fixed spatial frequencies, as a function of the “parasympathetic” stimulus ( j AS 1 - 1ABIAS I) and the pupil size. The appropriate focus errors (/AR 1 - 1AS 1) are, of course, known from the slopes of the response/stimulus lines and Fig. 8 shows the stimulus dependence of the modulation transfer to the retina at the three spatial frequencies I, 5 and 25c/deg, which typify the low, intermediate and high spatial frequency regions of the visually significant spectrum. In general terms, it can be seen that at I c:‘deg, although the behaviour is very similar at all pupil sizes, the accommodative errors have little effect on modulation transfer, since low spatial frequencies are very robust against defocus. Hence it is unlikely that such frequencies can play any important role in controlling the final steady-state accommodative response. On the other hand, at 25 c/deg, very disparate behaviour occurs in the modulation transfer variation for different sizes of pupil, making it improbable that this or other high spatial frequencies have any important general function in guiding the responses for our subjects. Only at 5 c/deg does the modulation transfer to the retina vary with stimulus level in a very similar way for all pupil sizes, while also showing substantial sensitivity to stimulus level. This supports the conjecture that optical performance at intermediate spatial frequencies - 5 c/deg may be the common factor linking accommodative responses at different pupil sizes. Such spatial frequencies not only possess the reasonable,

30

20

10

0

‘Parasympothetlc

snmulus

1 Omm.- -15~7;

40

I1 AS1 - IABIASl IlDl

-2omm.---2

51171n.- 3,omm

Theoretical dependence of modulation transfer at fixed spatial frequency of “parasympathetic” stimulus vergence ([AS1 - IABIASI)at a wavelength of 540nm. (a) I c/deg; (b) 5c/deg; (c) 25c/deg. I mm pupil; -. .I .5mm pupil; - ‘- 2 mm pupil; --- 2.5mm pupil; 3 mm pupil. For clarity, the curves for 1.5 and 2.5mm pupils at I c/deg have been omitted. It is assumed that the response/stimulus slopes are the same as those used in Fig. I.

but not excessive, sensitivity of modulation transfer to defocus that is required of any focusing system but also constitute powerful visual stimuli in general, since they lie close to the peak of the overall ocular contrast sensitivity functions (Owens, 1980). In the light of these findings, it is worth exploring the effect of changes in response on the contrast of the retinal image at these intermediate spatial frequencies. Charman and Tucker (1978) suggested that, in the maintenance of the nominally “steady state”

& g

response. the dbnumic changes III the contr,ijt ot’ the retinal image occasioned by small response change> plaq an important role. It was envisaged that ths response fluctuations -0.25 D that are kntiivn to occur at temporal frequencies up to ;L fsit Herz (,Arnulf and Dupuy. 1960; Campbell er uf.. 1959; Bour. 1981: Dcneuil, 1953) could produce small modulation changes A.\/ in the mean modulation ,\/ at any spatial frequency and that these changes generated the “signal” to drive the system. To see how a 5 c, deg spatial frequency component might fit into such a scheme, the data of Fig. 8(b) have been replotted in Fig. 9 to show the variation in modulation transfer with the error or lag in accommodation ( 1AS 1 - 1AR I). We see that, as would be expected, modulation transfer falls off more rapidly lvith accommodative error at the larger pupil sizes. The gradient is always an approximately linear function of the error up to the point where the modulation transfer first falls to zero. Any fixed level of accomodation fluctuation A(AR) will clearly produce a larger modulation change &C/ if it occurs about a mean position where there is a non-zero error, rather than zero error, as was noted by Charman and Tucker (1978). However, a difficulty arises when the effect of pupil diameter is considered. If the fluctuations are under “active” control, and are to work to produce some constant change in modulation transfer at any chosen mean value of modulation transfer, it is evident from Fig. 9 that smaller fluctuations in accommodation would be necessary with larger pupils, since the corresponding curve gradients are always steeper. In practice, experimental evidence (Campbell ct 01.. 1959) shows that the “fast” fluctuations, at -2 Hz, are larger with larger pupils, which suggests that they do not behave in the way demanded by the hypothesis. On the other hand. slower “drifts” (SO.3 Hz) are known to decrease as the pupil size increases (Campbell E! (I/.,

0.6

0

1.0

-02 0

1 IO

, 2.0 Error

Fig. 9. Theoretical (I AS1 - 1ARl) for

modulation a wavelength

m accomodalion

transfer at of 540 nm

5c/deg as a and the pupil

I 3.0

I 40

1 5.0

( D) function of error in accommodation sizes indicated. The curves have been

discontinued at the point where the valve of modulation transfer first becomes negative, i.e. “spurious resolution” first occurs (see. e.g. Smith, 1982).

Effect of pupil

size on steady

1959) and could therefore be acting to majn~ain modulation change in the way required by the hypothesis (see also Campbell er ol.. 19%). These accommodati.
CONCLUSiONS

The present results confirm those of earlier authors in showing that the magnitude of the slope of central region of the the accommodation response/stimulus curve decreases with pupil diameters below 3 mm. The linear region of the curves is well described by a simple geometrical optical model where the final steady-state blur circle diameter is a linear function of the difference between the accommodative stimulus and the accommodative resting state (1AS/ - / ABIAS/), and is independent of pupil diameter. Consideration of the effects of diffraction in relation to the observed results suggests that intermediate spatial frequencies ( - 5 cjdeg) in the retinal image were likely to have been of particular importance in guiding the responses, rlckrron,lec~~emenr-This Procurement Executive,

work

has been supported

by the

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