The Change of Spherical Aberration During Accommodation and Its Effect on the Accommodation Response

The change of spherical aberration during accommodationand its effect on the accommodation response

Grupo de Investigacin de Ciencias de la Visin,Universidad de Murcia, Murcia, SpainNorberto Lpez-Gil

Grupo de Investigacin de Ciencias de la Visin,Universidad de Murcia, Murcia, SpainVicente Fernndez-Snchez

Theoretical and ray-tracing calculations on an accommodative eye model based on published anatomical data, together withwave-front experimental results on 15 eyes, are computed to study the change of spherical aberration during accommodationand its influence on the accommodation response. The three methodologies show that primary spherical aberration shoulddecrease during accommodation, while secondary spherical aberration should increase. The hyperbolic shape of the lenssurfaces is the main factor responsible for the change of those aberrations during accommodation. Assuming that the eyeaccommodated to optimize image quality by minimizing the RMS of the wave front, it is shown that primary sphericalaberration decreases the accommodation response, while secondary spherical aberration slightly increases it. The total effectof the spherical aberration is a reduction of around 1/7 D per diopter of stimulus approximation, although that value depends onthe pupil size and its reduction during accommodation. The apparent accommodation error (lead and lag), typically present inthe accommodation/response curve, could then be explained as a consequence of the strategy used by the visual system,and the apparatus of measurement, to select the best image plane that can be affected by the change of the sphericalaberration during accommodation.

Keywords: accommodation, aberrations, primary spherical aberration, secondary spherical aberration,accommodation response

Citation: Lpez-Gil, N., & Fernndez-Snchez, V. (2010). The change of spherical aberration during accommodation and itseffect on the accommodation response. Journal of Vision, 10(13):12, 115, http://www.journalofvision.org/content/10/13/12,doi:10.1167/10.13.12.

Introduction

The fundamentals of the mechanism of human accom-modation were already studied in the 17th century byThomas Young, who showed that the eye is able to generatea change in focus mainly by changing the curvature of thefront surface of the lens (Young, 1801). Youngs findingshave been totally corroborated by many in vivo (Jones,Atchison, Meder, & Pope, 2005; Koretz, Bertasso, Neider,True-Gabelt, Kaufman, 1987; Neider, Crawford, Kaufman,& Bito, 1990; Strenk et al., 1999) and in vitro (Glasser &Campbell, 1998; Manns et al., 2007) measurements usingdifferent techniques that show, with great detail, thechanges of the lens during accommodation. Wave-fronttechnology has been mainly used during the last 15 yearsin the eye to get objective in vivo measurements of howthe power of the eye changes when the light passesthrough different parts of the pupil (Howland & Howland,1977; Liang, Grimm, Goelz, & Bille, 1994; Sminov,1961). This technology has also been applied to theaccommodated eye, showing a curious effect: the power ofthe eye, which is usually larger in the periphery of thepupil than at its center, varies during accommodation insuch a way that in the accommodated eye it is larger in the

center than in the periphery of the pupil (Atchison, Collins,Wildsoet, Christensen, & Waterworth, 1995; Buehren &Collins, 2006; Cheng et al., 2004; He, Burns, & Marcos,2000; Ivanoff, 1947; Kooman, Tousey, & Scolnik, 1949;Lopez-Gil et al., 2008; Ninomiya et al., 2002; Plainis,Ginis, & Pallikaris, 2005; Radhakrishnan & Charman,2007; Tscherning, 1900). That power distribution is relatedto the radial symmetrical high-order terms of the ocularwave front, usually known as spherical aberration (SA).For clarity, it is important to point out that there are several

mathematical expressions for SA depending on the poly-nomial expansion used and the degree of the expansion. Forsimplicity, in this article, we will use the expansion of thewave front in terms of Seidel and Schwarschild aberrations(Mahajan, 1991):

W > Ad rr0

2 As r

r0

4 Bs r

r0

6; 1

where r is the radial coordinate of the entrance pupil, andr0 is the actual entrance pupil radius. Ad corresponds to thecoefficient to the defocus, while As and Bs correspond toprimary (SA4) and secondary (SA6) spherical aberrations,respectively.

Journal of Vision (2010) 10(13):12, 115 http://www.journalofvision.org/content/10/13/12 1

doi: 10 .1167 /10 .13 .12 Received June 27, 2010; published November 12, 2010 ISSN 1534-7362 * ARVO
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Actually, Zernike polynomial expansion is also beingused broadly, for which the wave front in Equation 1would be written as (Thibos, Applegate, Schwiegerling,Webb, & VSIA Standards Taskforce Members, 2002):

W> a02ffiffiffi3

p2>2j1 a04

ffiffiffi5

p6>4j6>2 1

a06ffiffiffi7

p20>6j30>4 12>2j1; 2

where > = rr0, a20 is the defocus Zernike term, and a4

0 and a60

are the fourth- and sixth-order Zernike spherical aberrationterms, respectively. Equations 1 and 2 show that Seideldefocus (Ad) and Zernike defocus (a2

0) coefficients aredirectly proportional, as well as primary Seidel SA andfourth-order Zernike (a4

0) SA, and secondary Schwarschild(Bs) and sixth-order Zernike SA (a6

0). In particular, for aZernike expansion up to the sixth order, the relationshipbetween SeidelSchwarschild and Zernike SA coefficientscan be calculated from Equations 1 and 2, giving:

a04 As

6ffiffiffi5

p Bs4ffiffiffi5

p ; 3

a06 Bs

20ffiffiffi7

p : 4

There is a complete agreement among researchers aboutthe decrease (change in the negative direction) in As (or a4

0)during accommodation; in fact, this effect was alreadyknown by researchers in the first half of the last century.As a matter of fact, Young (1801), and later Tscherning(1900), using what can be assumed to be the first subjectiveaberroscope, showed that the eye had positive primaryspherical aberration (As), whose magnitude decreasesduring accommodation. Similar results using other tech-niques were obtained by Ames and Proctor (1921), Ivanoff(1947), Jenkins (1963), Kooman et al. (1949), and Sminov(1961). In the last two decades, several other studies, mostof which made use of aberrometric techniques, showedsimilar results both after in vivo measurements (Atchisonet al., 1995; Buehren & Collins, 2006; Cheng et al., 2004;He et al., 2000; Lopez-Gil et al., 2008; Ninomiya et al.,2002; Plainis et al., 2005; Radhakrishnan & Charman,2007) as well as after in vitro ones (carried out by stretchingthe lens; Glasser & Campbell, 1998; Manns al., 2007). Insome cases, the authors have found a decrease in a4

0 duringaccommodation, starting and finishing with positivevalues, but a4

0 goes usually from a positive value for theunaccommodated eye to a negative one for accommodationstates beyond 4 D. In other cases, the unaccommodated eyepresents negative a4

0, which becomes even more negativeduring accommodation. Thus, although the value of the a4

0

coefficient at the starting point (relaxed eye) could be

different, there is no question about the typical decrease(Atchison et al., 1995) of the coefficient of fourth-orderSA (As or a4

0) during accommodation in the young eye.However, after more than one century, there is at present

still no explanation of the mechanism for these changes inspherical aberrations during accommodation. The questionthat arises is whether or not those changes modified theaccommodation response or somehow benefit human vision.Although a few examples of the potential benefit of thedecrease of primary SA on the accommodation responsehas already been pointed out by some researches (Buehren& Collins, 2006; Plainis et al., 2005) following exper-imental in vivo measurements, little has been said abouthow much it really benefits the accommodation response.Moreover, as far as we know, except for three studiesVtwoin humans (Lopez-Gil, Lara, & Fernandez-Sanchez, 2006;Ninomiya et al., 2002) and another one in pigs (Roorda &Glasser, 2004)Vthere are no other studies that inves-tigated the possible increase or decrease in the secondarySA, and its influence on the accommodation response hasnever been investigated. In the following, we will give ananswer to these points by means of theoretical calculations,computer simulations, and experimental measurements.

Methods

Theoretical calculationsSpherical aberration change during accommodation

Thomas Young already demonstrated several centuriesago that accommodation is achieved by means of changesin the lens (Young, 1801), in particular, a reduction of theradius of curvature of both surfaces, together with anincrease in the lens thickness and a small displacement ofthe lens center toward the cornea. Classical and moderntechniques (Glasser & Campbell, 1998; Jones et al., 2005;Koretz et al., 1987; Neider et al., 1990; Strenk et al., 1999)have shown that the largest modification corresponds tothe reduction of the first radius of curvature. For instance,Le Grands eye model indicates that for 7 D of accom-modation, the first radius of curvature goes from 10.2 to6 mm, while the second radius only decreases from j6to j5.5 mm and the center of the lens moves forward0.15 mm. In vitro measurements of young subjects lensesalso confirmed that the lens front surface undergoes abigger curvature change than the posterior surface afterstretching (Glasser & Campbell, 1998; Manns et al., 2007).Since it is the physiological change of the lens that

makes accommodation possible, it should also be the mainfactor responsible for the decrease in spherical aberrationduring accommodation. Then, assuming radial symmetry,we can model the lens front and back surfaces using aconic surface of radii R1L and R2L and conic constants k1Land k2L, respectively. Appendix A shows the mathematical

Journal of Vision (2010) 10(13):12, 115 Lpez-Gil & Fernndez-Snchez 2

calculations used to find the dependency of the eyes wavefront with k1L and k2L (Equation A8), as well as thechange of As and Bs during accommodation (Equations A13and A14).Assuming that the pupil diameter does not change during

accommodation, Equation A10 indicates that As shouldchange in the negative direction (decrease) during accom-modation since both surfaces of the lens contribute to it. Inparticular, for the front surface R1L decreases (Dubbelman,van der Heidje, & Weeber, 2005), "1L (the magnificationbetween the plane of the lens front surface and the eyesentrance pupil plane) is positive and remains nearly constant,and k1L is negative (hyperbolic shape; Dubbelman et al.,2005). For the back surface, a similar situation occurs,because although the change of index of refraction isnegative, the radius of curvature (R2L) is also negative andso is k2L. (In fact, k1L and k2L seem to become even morenegative (Dubbelman et al., 2005), while "1L and "2Ldecrease only slightly during accommodation (see Computersimulations section).) On the other hand, Equation A11shows that Bs should change in the positive direction(increase) after accommodation, since its value dependson k1L

2 and k2L2 .

Refractive state change during accommodation

Let us study the effect of the wave-front variation duringaccommodation upon the effective change of the refractivestate defined as the inverse of the distance (vergence) of theretinal conjugate plane, that is, the target vergence requiredto maximize retinal image quality. Then, as hypothesis, weassume that the eye accommodates to a certain stimulusuntil its retinal image is maximized (Lopez-Gil, Fernandez-Sanchez, Thibos, & Montes-Mico, 2009; Tarrant, Roorda,& Wildsoet, 2010).Conversion from wave front (Equation A8) to refractive

change and accommodation can be calculated by takinginto account that the refractive change produced by adefocus, Ad, for a pupil radius r0 corresponds to j2Ad/r0

2

(Mahajan, 1991).Therefore, following Equation A6, the refractive change

induced by the change of the defocus term generated byboth surfaces of the lens is

j nLj n 1"1L2 R1L

j1

"2L2 R2L

: 5

Equation 5 corresponds to the paraxial change ofaccommodation generated by the front surface of the lens,that is, the change of the refractive state during accom-modation caused by the front surface of the lens if high-orderaberrations were not been taking into account. However,Equation A8 showed that when the eye accommodatesthere is not only a change in defocus but also in primaryspherical aberration. It is well known that in any optical

systemwith certain amount of primary spherical aberration,As, the best image plane in terms of minimum variance ofthe wave-front aberration (RMS) will be observed in thepresence of a defocus AdV, such as AdV= jAs (Mahajan,1991). Appendix B shows an extension of this effect in thepresence of primary (As) and secondary (Bs) sphericalaberrations. In that case, the defocus necessary tominimize RMS is: AdV= jAs j (9/10)Bs. Therefore, fromEquations A9 and A10, the refractive effect of primaryand secondary SAs on the refractive state would corre-spond to

nLj nr204

k1L

"1L4 R1L

3j

k2L

"2L4 R2L

3

; 6

and

9nLj nr4080

k1L2

"1L6 R1L

5j

k2L2

"2L6 R2L

5

; 7

respectively.The term Wc(r) in Equation A8 represents the wave-

front variation induced by the cornea. Assuming that thereis no any change in the cornea during accommodation, thecontribution of the cornea to the change of the refractivestate during accommodation would come from the changeof the defocus originated by the change of corneas primaryand secondary SAs undergone when the pupil decreases itssize. Then, after Equations 57, and assuming that neitherthe indices of refraction (Hermans, Dubbelman, Van derHeidje, & Heethaar, 2008) nor "1L and "2L vary duringaccommodation (see Computer simulations section), theaccommodation (refractive change from a relaxed state toa given accommodation state) can be expressed as

A jnL j n 1"V21LRV1L

j1

"V22LRV2L

!j

1

"21LR1Lj

1

"22LR2L

!" #

nLj n4

kV1LrV2

"V41LRV31L

jkV2LrV

2

"V42LRV32L

!j

k1Lr2

"41LR31L

jk2Lr

2

"42LR32L

!" #

9nLj n80

kV21LrV4

"V61LRV51L

jkV22LrV

4

"V62LRV52L

!j

k21Lr4

"61LR51L

jk22Lr

4

"62LR52L

!" #;

8

where values without prime corresponds to the relaxedstate and those with prime corresponds to the accommo-dated eye. The first bracket in Equation 8 is directlyproportional to the curvature of the front and backsurfaces of the lens (1/R1L and 1/R2L; in agreement withexperimental measurements; Koretz, Cook, & Kaufman,2002) and is the main factor responsible for the refractivechange of the eye during accommodation. However, sinceboth surfaces of the lens have a negative asphericity, the


last two brackets also add some amount of accommoda-tion (third and fourth brackets). The bracket correspondingto the effect of the primary spherical aberration (secondbracket) produces a decrease in the accommodation, whilethe one corresponding to the secondary spherical aberration(third bracket) produces an increase in the accommodation.In total, the effect of spherical aberration results in areduction of the accommodation since the second brackethas a larger weight than the third one.The reduction of accommodation response produced by

the decreases of the primary spherical aberration duringaccommodation (second bracket in Equation 8) can beunderstood as follows. In the relaxed eye as the myopiceye shows in Figure 1 (top), there is usually a certainpositive total amount of SA4 since the cornea has a largerpositive value than the one of the lens (with negativesign). That positive SA4 is compensated by some negativedefocus (Figure 1, top) to minimize the RMS of the wavefront. However, in the accommodated eye (Figure 1,bottom), SA4 decreases becoming negative, and in orderto attain a good retinal image, it will need an extra defocusto minimize the RMS with opposite sign to the one neededin the relaxed eye. Then, for paraxial rays, the accommo-dation response of the eye should be larger than it really isbecause the paraxial object, OP, pass from a more hyper-metropic to a more myopic position with respect ORMSafter accommodation (Figure 1).To understand the effect of sixth-order spherical aberra-

tion, a similar analysis could be done, but in this case, theeffect of SA6, which pass from negative to positiveduring accommodation (see Results section), will create alarger accommodation in the case of non-paraxial rays.

However, its effect on the accommodation is muchsmaller than SA4 as will be seen later, and the totalchange of spherical aberration reduces the accommodationresponse of the eye.

Computer simulations

The exact change of spherical aberration and refractionduring accommodation can be calculated by means of ray-tracing software applied to a model eye. We use Zemax-EEand a model eye with the features summarized in Table 1 tocompute the change of both spherical aberrations as wellas the refraction change of the eye during accommodation.The data shown in Table 1 were measured by Dubbelman,Weeber, van der Heijde, and Volker

1.2945 j 0.0014D (R2 = 0.9987), for the front and backsurfaces of the lens, respectively. Thus, even when it comesto sixth-order spherical aberration, which includes a term in"j6 (see Equation 8), the value of that terms does notchange more than 1.6% across the abovementioned 7-Daccommodation range. That is the reason why in Table 1,"1L and "2L are considered to be constants.

In vivo ocular wave-front changeduring accommodation

To measure the real change of spherical aberrationduring accommodation and to see its effect in theaccommodation response, wave-front measurements werecarried out under natural pupil and monocular conditions in15 young subjects aged 2038 (24 T 6 years) having nopathologies, amblyopia, or accommodation dysfunctions.Their sphere ranged between 0.38 D and j3.06 D with acylinder ofj0.38 T 0.25 D, which was not corrected during

the experiment. The aberrometer of choice was the irx3(Imagine Eye, France), which had a Badal system embed-ded inside to be able to move a target from 0.5 D beyondthe subjects far point to 9.5 D closer than that point. Thestimulus varied its vergence at steps of 0.5 D, leaving 1.5 sfor the subject to accommodate. The stimulus accommo-dation range was always larger than the subjects accom-modative amplitude. The target consisted of a polychromaticimage of a balloon at the end of a road, containing a widerange of spatial frequencies in all directions, with a meanluminance of 50 cd/m2.The subject was instructed to keep the stimulus as clear

as possible during the experiment. We assume that therefractive state of the subject for any position of the stimuluswithin the interval of vision will be such that maximizesimage quality according to the minimum RMS metric.All the subjects were informed about the study and

signed an informed consent. The tenets of the Declarationof Helsinki were followed. All subjects ran at least 3 trialsto get used to the methodology prior to recording themeasurements.

Results

Theoretical calculations and computersimulations

Applying Equations A17 and A18 (with [As]T = 0.429 2mand [Bs]T = 0.081 2m for a 4.5-mm pupil) to the valuesshown in Table 1 permits us to obtain the theoreticalchange of As and Bs during accommodation. (Changes inZernike instead of Seidel during accommodation cansimply be calculated taking into account Equations 3and 4.)Figure 2 shows those changes for a constant pupil size

of 4 mm and also when it is assumed that the pupildecreases during accommodation in the way shown inTable 1. The results yielded by the computer simulationsare also presented for comparison purposes. Simulatedvalues of As and Bs were obtained, including in Equations 3and 4 the Zernike values calculated by the Zemax-EEsoftware for a wavelength of 0.587 2m (Equations 3 and 4).Assuming that in the relaxed eye accommodation

response is 0 D for a certain vergence of the stimulus (R),accommodation response to a certain vergence of theobject, X (corresponding to a stimulus accommodation ofR j X), is defined by the change of refractive state of theeye. For instance, for a myope with a refraction R = j2 D,which changes the refractive state fromj2 D (relaxed eye)to j4 D when seeing an object placed at 20 cm (j5 D ofvergence), it will be said that for a stimulus accommodationof 3 D (= j2 D j (j5 D)), the eye is presenting anaccommodation response of 2 D (= j2 D j (j4 D)).

Radius of the front surfaceof the cornea

7.87

Asphericity of the front surfaceof the cornea

0.85

Corneal refractive index 1.376Corneal thickness 0.574Radius of the back surfaceof the cornea

6.4

Asphericity of the back surfaceof the cornea

0.82

Refractive index of the aqueoushumor (na)

1.336

Anterior chamber depth 2.996 j 0.036DEntrance pupil radius 2 or 2.25 j 0.05 * DRadius of the front surfaceof the lens (RL1)

1 / (0.0894 + 0.0067D)

Asphericity of the front surfaceof the lens (kL1)

j4.5 j 0.5D

Magnification between EPand L1 ("L1)

1.1269

Refractive index of the lens (nL) 1.4293Lens thickness 3.638 + 0.043DRadius of the back surfaceof the lens (RL2)

j1 / (0.1712 + 0.0037D)

Asphericity of the back surfaceof the lens (kL2)

j1.43

Magnification between EPand L2 ("L2)

1.2945

Refractive index of the vitreoushumor (nv)

1.336

Axial length 24

Table 1. Parameters of the eye model used for theoretical andcomputer simulations. Note: D indicates the stimulus accommo-dation in diopters. Distances are in millimeters.


Figure 3 shows the resulting theoretical accommodationresponse, obtained based on the data of Table 1 andEquation 8. The contribution of each surface of the lens isshown separately, and so are the different effects on theaccommodation response of defocus and fourth- and sixth-order spherical aberrations.

In vivo ocular wave-front changesduring accommodationSpherical aberration

Figure 4 shows the values of As, Bs, and pupil diameterduring accommodation for the 15 subjects included in thestudy. (Although the computer simulations and in vivo

measurements have been obtained discretely by using astimulus that approaches to the eye by steps of 0.5 D, acontinuous line has been used to connect the results in thefigures, for clarity.)

Accommodation response

Figure 5 shows the mean accommodation responsefor all subjects using Equation 8. To show the effect ofboth spherical aberrations, we have computed the accom-modation response both including and excluding theircontribution.If we define the accommodation amplitude (AA) as the

maximum range of accommodation (maximum minusminimum accommodation response), we can calculatethe effect of SA4 and SA6 on the AA. Figure 6 shows theresults for the 15 subjects included in the study, as well asthe mean value.

Discussion

Decrease in spherical aberration duringaccommodation

Figures 2A and 2B show that Equations A11 and A12correctly predict the trend suggested by the simulated raytracing of SA4 and SA6. Nevertheless, there is no exactmatch, especially in the case of SA6 (Figure 2B). Thereare three main reasons for this small discrepancy. First,the modification undergone by the wave front after goingthrough any of the lens surfaces has been approximatedby the shape of the front surface multiplied by therefraction index difference before and after the surface(Equations A6 and A7). That approximation overestimatesthe effect of that surface on the eyes spherical aberration,above all for high aberration values (i.e., the edges of thepupil). Second, in our theoretical calculations, we haveassumed a constant value for "1L and "2L, but in reality,

Figure 2. Theoretical (continuous lines) and computer-simulated (discontinuous lines) changes of (A) primary and (B) secondary sphericalaberrations during accommodation for a fixed 4-mm pupil (gray lines) and for a variable pupil size (which decreases duringaccommodation; black lines).

Figure 3. Theoretical accommodation response for a natural pupilcondition computed by both surfaces of the lens. Dark lines:contribution of the lens front surface; gray lines: contribution ofthe lens back surface. See legend for details about the differentcontributions from defocus, primary (1SA), and secondary (2SA)spherical aberrations. The steepest line represents the 1:1response.


they are not constant (see Computer simulations section).Third, the modification of the wave front after itspropagation from the second surface of the lens to thefirst one has been ignored, since it is expected to play avery small role.From Equation A8, it can be seen that the main factor

responsible for the decreases of SA4 and the increase inSA6 during accommodation is the front surface of thelens. In particular, for a 4-mm pupil, out of the total dropof SA4, 0.06 2m (86%) is due to the lens front surfaceand 0.01 2m (14%) to its back surface.From the values of Table 1 for a fixed pupil diameter of

4 mm and a stimulus accommodation of 4 D, we obtain atheoretical decrease in As equal to 0.94 2m, that is, $As =j0.99 2m (which corresponds to $a4

0 = j0.056 2m;Figure 2A). That value is a little bit higher than the oneyielded by the simulations: $As = j0.87 2m ($a4

0 =j0.052 2m) for the eye model based on the data ofTable 1 (Figure 2A), or $a4

0 = j0.05 2m when usingNavarros eye model (Navarro, Santamara, & Bescos,1985; Lopez

studies performed by various authors: Cheng et al. (2004),$a4

0 = j0.07 2m; Plainis et al. (2005), about $a40 = j0.054

2m; or Lopez-Gil et al. (2008), $a40 = j0.044 2m.

Atchison et al. (1995) found a change of fourth-orderspherical aberration in terms of defocus of j0.34 D, whichcorresponds to a $a4

0 = j0.051 2m for a 4-mm pupil.Radhakrishnan and Charman (2007) also found a value of$a4

0 = j0.048 2m, although he used a natural pupildiameter.To our knowledge, the only published experimental data

that show a significant increase in a60 during accommodation

for a fixed pupil size correspond to Ninomiya et al. (2002).Their data, although it cannot be directly compared withour calculations (Figure 2B) because the pupil diametervalues are different, show a similar trend. Figure 4Bclearly shows that, on average, there is an increase in a6

0,which goes from a negative value to a positive one. It isinteresting to point out that the effect of SA6 on theaccommodation response is not negligible, as shown inFigure 6B for subjects #6, #8, and #13.A close examination of Equations A19 and A20 reveals

that when the pupil decreases (rVG r), we can expect asmaller change of As and Bs during accommodation,relative to the situation when the pupil size remains

constant (rV= r). This fact is reflected in Figure 2A, wherethe slope of $a4

0 is bigger for the natural pupil conditionthan for the 4-mm-diameter pupil until about 5 D ofstimulus accommodation, for which the natural pupil sizeis about 4 mm. Then, Equations A21 and A22 indicatethat the change of a4

0 and a60 during accommodation will

Figure 6. (A) Amplitude of accommodation for each subject: dark blue bars show the AA without the effect of fourth- and sixth-order SAs;light blue bars show the real response including their effect. (B) Contribution of each spherical aberration to the amplitude ofaccommodation: fourth-order SA (light blue bars); sixth-order SA (dark blue bars). Last two bars show the mean value between subjects.

Figure 7. Experimental results for subject #4 regarding the changeof pupil size (dotted line, right scale), SA4 (black line, left scale),and SA6 (gray line, right scale) during accommodation.


be bigger for larger pupils, which is in good agreementwith the experimental data shown by Lopez-Gil et al.(2008) and Radhakrishnan and Charman (2007). As anexample, Figure 7 shows that when the pupil does notdecrease during accommodation (stimulus accommodationbetween 0 and 2.5 D) or even when it increases (around4 D of stimulus accommodation) the value of SA4 decreasesrapidly.Figure 7 also shows the typical decrease in SA4 going

from positive to negative values. For this subject, beyond8 D, SA4 reaches its lowest value and starts to increaseagain approaching zero. This behavior can be explainedby examining Equation A9: for 8 D of stimulus accom-modation, the curvature of the lens reaches a maximum,whereas the pupil diameter can still decrease. This effectcould indicate that the ciliary muscle could still contractfor a stimulus accommodation larger than 8 D, but that thelens cannot modify its shape any further (i.e., it cannotbecome more convex). A similar behavior can be seen forSA6 (Figure 4): there is an initial increase followed by anapproach to zero for a stimulus accommodation above 7 D.Although there is a large intersubject variability, the

mean values shown in Figures 4A and 4B confirm thetrend predicted by our calculations and previous results(Buehren & Collins, 2006; Ivanoff, 1947; Kooman et al.,1949; Plainis et al., 2005); that is, a decrease in As and anincrease in Bs. As indicated by other researches (Chenget al., 2004; Lopez-Gil et al., 2008; Plainis et al., 2005),most of the subjects have a positive As (a4

0 9 0) in therelaxed state, and a negative one beyond 34 D ofstimulus accommodation. However, there are also somesubjects for whom As (so a4

0) always remains positive (blueline in Figure 4A), and others for whom As has a negativevalue across the whole accommodation range, as shownby the brown line in Figure 4A, corresponding to subject#6. In these cases where As G 0 in the relaxed eye, theminimum SA4 was usually more negative than in the caseswhere the relaxed eye presented a positive SA4 value.

Effect of spherical aberrationon the accommodation response

Figure 3 shows that, for an eye having similar featuresto those presented in Table 1, its accommodation responsedepends mainly on the defocus change produced by thefront surface of the lens, which accounts for 82% of thetotal response. Defocus produced by the back surface ofthe lens accounts for 33%, while the change in SA4produced by the front surface of the lens accounts forj19%. The variation of SA6 due to the front surface ofthe lens accounts for 4%, while the variation of SA4 andSA6 associated to the back surface of the lens has anegligible effect on accommodation.Figure 3 also shows that the accommodation response

for 7 D of stimulus accommodation of our model eye with

a 4-mm pupil is j1.2 D lower when the effect of SA4 istaken into account. The effect of SA6, however, increasesthe accommodation response but smaller (around 0.3 D).So total effect of SA in the accommodation response was0.9 D for 7 D of stimulus accommodation, whichrepresents a decrease of 0.13 D per diopter of stimulusaccommodation.Experimental results also confirm that tendency. Figures 5

and 6 show that for all the subjects included in the studythe change of both types of spherical aberrations duringaccommodation results in a decrease in the accommoda-tion response. The average decrease was 0.9 D, althoughdecreases up to 1.3 D were observed (subject #1). Onaverage, the effect of SA4 was about 3 times larger thanthat of SA6. This means that SA4 accounts for j1.2 Dand SA6 for 0.3 D of the total AA (6.46 D). However, it isinteresting to see that in some subjects (such as #1, #6, #8,and #13) the effect of SA6 could be larger than half of adiopter, resulting in increases of AA that are far frombeing negligible. In other subjects, such as #9 SA6 did notplay any role in the accommodation response. Thesedifferences across subjects can be explained by having inmind the different geometries of the cornea and the lensand the differences in pupil size, since it decreases duringaccommodation. The larger the pupil, the larger the effectof SA4 and SA6 that is expected (Equations A10, A11,A19, and A20). Subject #6 is particularly interesting. Hehas a negative a4

0 for the relaxed eye, and this value barelydecreases during accommodation (see brown line at thebottom of Figure 5A). However SA4 has a clear negativeimpact on his accommodation, as shown in Figure 6B.However, the same subject shows a large increase in a6

0

Figure 8. Theoretical accommodative response when accommo-dative miosis is included (black lines) and when the pupil size isassumed to be constant and equal to 4 mm (gray lines). The thinlines correspond to the effect of SA6, dotted lines to the effect ofSA4, and dash-dotted lines to the effect of defocus (independentof the pupil size). Thick lines correspond to the total response.The steepest line (the one at the top) represents the 1:1 response.


(see brown line at the bottom of Figure 5B) resulting in abenefit on the AA of about 0.5 D (Figure 6B).As an example of the dependency of the accommoda-

tion response on pupil size and on its variation duringaccommodation, we can use the same theoretical eye(Table 1) under two different pupil conditions. The first onecorresponds to a 6.5-mm pupil, whose diameter decreasesat a rate of 0.2 mm per diopter of stimulus accommodation(resulting in a 5.1-mm pupil diameter at the end of the 7-Drange of stimulus accommodation). The second conditioncorresponds to a constant pupil size of 4 mm. Figure 8shows the results.In this case, the joint effect on refraction of SA4 and

SA6 is higher for the first condition (accommodativemiosis) than for the second one (fixed pupil), due to thefact that across the whole accommodation range the pupilis always larger in the first condition. In particular, in thefirst condition, the effects of SA4 and SA6 on refractioncause a total decrease in the accommodation response of1.5 D at 7 D of stimulus vergence, producing a decrease inthe accommodation response of 0.21 D per diopter ofstimulus accommodation. However, in the second con-dition, SA4 and SA6 account for about 0.9 D of theaccommodation response at 7 D of stimulus accommoda-tion (0.13 D per diopter of stimulus accommodation).Furthermore, in the first condition, the joint effect of SA4and SA6 was similar but in opposite sign to the effect ofdefocus produced by the second surface of the lens. Thetotal accommodation response is far from the 1:1 response.The same tendency was found in the experimental results(Figure 5), although the effects of SA4 and SA6 are not soimportant as in Figure 8.In Figures 2 and 8, we have assumed that accommo-

dation response was zero for an object at the subjects farpoint, so there is no accommodation lead. However, ifwe plot the accommodation response obtained using theminimum RMS using the origin of the stimulus to accom-modation axis (X-axis) corresponding to the paraxial farpoint (maybe preferred subjectively), the accommodationresponse would present a lead of accommodation if therelaxed eye has a positive SA4 (as is the case in most ofthe subjects analyzed, see Figure 4A). Then, Figure 5shows not only a lag but also a lead of accommodation.Figure 1 also shows schematically the presence of a leadof accommodation, which corresponds to the absolutedifference in diopters between the vergences of OP andORMS in the relaxed eye, while the lag of accommodationcorrespond to the same difference but in the accommo-dated eye (with an opposite sign than the lead). However,the presence of a lead or lag of accommodation woulddepend on the sign of SA4 in the relaxed eye. Thus, forSA4 = 0, no lead could be expected, while for SA4 G 0 alag of accommodation could be expected in the wholerange of accommodation, being much larger in theaccommodated eye than in the relaxed eye as predictedby Equation 8.

Depending on the subjects equivalent sphere correction(spectacles or contact lenses), accommodation responsecould also be modified if the refractive effect of thedistance between the spectacle and the eye is not takinginto account to calculate the exact value of the stimulus toaccommodation.The accommodation response caused by defocus alone

is also larger in the experiment (Figure 5) than in thesimulation (Figure 8). We explain this discrepancy interms of depth of focus, different mean age between theexperimental and theoretical model eyes, the approxima-tions used in Equations A6 and A7, and probably theeffect of the lens gradient index, not included in ourtheoretical and simulated studies (a recent investigation(Maceo et al., 2010) has pointed out the large effect on theaccommodation response that the lens gradient indexmight have).Contrary to what most people think, when a relatively

large value of positive SA4 is added to the eye, the eyebecomes more hyperopic, while adding a negative SA4 tothe eye, its refraction will be shifted in the myopic direction.This effect has been observed by many researchers afteradding some amount of Zernike SA4 by means of contactlenses or deformable mirrors (Applegate, Marsack, Ramos,& Sarver, 2003; Benard, Lopez-Gil, & Legras, submittedfor publication; Rocha, Vabre, Chateau, & Krueger,2009). This is because the refraction change induced bySA4 is mainly due to the amount of defocus (the term inr2) that the polynomial includes to balance the term in r4

(Equation 2), thus minimizing the RMS. However, thatonly happens for relatively large amounts of induced a4

0.For small values of a4

0 (below approximately 0.1 2m), thevisual system does not have its refraction modified, as hasbeen pointed out recently by Cheng, Bradley, Ravikumar,and Thibos (2010). When Seidels primary sphericalaberration is added (Seidel SA, which is similar to anunbalanced Zernike spherical aberration), the oppositeeffect is expected. That is, low values of Seidel SA willmodify the refraction, while large values will hardlymodify it. In particular, positive small amounts of SeidelSA induce a negative defocus (hyperopic eye) while anegative small amount of Seidel SA induces a positivedefocus (myopic eye).Our theoretical results (Equation A8) suggest that

there is a relative small decrease in Seidel SA duringaccommodation. The average change of SA4 was 0.32 2m(0.06 2m for SA6), which is larger than 0.1 2m, indicatingthat probably the refraction change produced by the effectof SA4 (Figures 3 and 8) could be overestimated. That isprobably the main discrepancy between the experimentalaccommodation response (Figure 5) and the calculatedone (Figure 8). However, even in the case that only half ofthat SA4 was really used to change the refraction state ofthe eye, it also causes the decrease in AA by about 0.7 D(for a mean pupil size of 5 mm), which is a large value(10% of the 1:1 response). Moreover, we should keep in


mind that the change of SA4 during accommodation iscontinuous, so its effect could be different from the casewhere SA4 is added instantaneously, as in the studiespreviously (Benard et al., submitted for publication; Rochaet al., 2009) undertaken.Our main hypothesis maintains that the eye is accom-

modating to maximize the image quality. Assuming thatimage quality is maximized by minimizing the RMS ofthe wave front, Equation 8 predicts a decrease in theaccommodation response with respect the case where theeye would use only the paraxial rays (insensitive to SAeffects). This explains the presence of errors in theaccommodation response as shown in Figures 3 and 8.However, we do not really know if the visual systemminimizes the RMS during accommodation when thestimulus light is traveling from the stimulus toward theretina (first pass). Nevertheless, the light traveling fromthe retina to the apparatus of measurement (second pass)is also affected by the change of the spherical aberrationduring accommodation. Thus, depending on the method-ology used to measure the refractive state of the eye bythe apparatus, accommodation response could also show alead and/or lag of accommodation due to the second pass.The measured amount of accommodation lead or lag willthen depend on the depth of field, the amount of thespherical aberration presented in the relaxed eye, itschanges during accommodation, and the image qualitymetric used by the visual system and the apparatus used inthe measurements.

Conclusions

A decrease in fourth-order and an increase in sixth-order spherical aberration during accommodation are dueto the hyperbolic shape of the surfaces of the human lens.Depending on the objective method used to measure theaccommodation response, this variation of sphericalaberrationVand above all the one undergone by thefourth-order spherical aberration at the front surface ofthe lensVhas an important effect on the retinal imagequality, modifying the position of the image quality plane.In case that minimum RMS is used as the image qualitycriteria, the changes of fourth-order spherical aberrationproduce a decrease while the changes of sixth-orderspherical aberration increase the accommodation response.Total contribution of fourth- and sixth-order sphericalaberrations reduces the accommodation response about1/7 of diopter per diopter of proximity of the stimulus.Thus, excluding the small depth of field of the eye, the

apparent accommodation error in the accommodationresponse of the object in the interval of vision where theeye is getting a good (eventually the best) image is not

really an error but a consequence of the strategy used bythe visual system and the apparatus of measurement toselect the best image plane that can be affected by thepresence of spherical aberration in the relaxed eye and itschanges during accommodation.

Appendix A

The front surface of the lens can be described using aconic surface that, in Cartesian coordinates, can beexpressed as

xjx02 yj y02 zj z02 R1L2 jk1Lj 1z2;A1

where R1L and k1L are the radius of curvature and theconic constant of the surface, respectively (Figure A1).If the center of the conic surface has coordinates x0 = y0

and z0 = R1L, we can rewrite Equation A1 in polarcoordinates as

r21i k1Lz2 j 2R1Lz 0; A2

where r1i represents the radial coordinate at the iris plane.

Figure A1. Front surface of the lens represented as a conic curvewith its apex at the origin of coordinates.


Solving Equation A2 for z, we have

z R1L 1j

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1j k1L

r1iR1L

2r" #

k1L; A3

where only the negative root has been taken into accountin order to have the vertex of the surface at the originof coordinates (see Figure 1). Now we can rewriteEquation A3 as a Taylor-series expansion around theorigin, taking into account that

ffiffiffiffiffiffiffiffiffiffiffi1 xp 1 x

2j

x2

8 x

2

16jI: A4

As a result, following an expansion up to the sixth order inr1i, Equation A3 can then be expressed as (see Roorda &Glasser, 2004 for a similar result for a calculation up tofourth order)

z r1i r1i2

2 I R1L k1Lr1i

4

8R31L k1L

2r1i6

16R51L: A5

We can now think of a point light source located on thefovea that produces light. To exit the eye, this light willhave to go through the eye, as many of the aberrometerswork today. When the light is passing through the frontsurface of the lens, it will modify the wave front byadding

W1Lr1i nLj nazr1i; A6

where nL and na represent the index of refraction of theaqueous humor and the lens, respectively, already takinginto account the sign of the wave front when it is gettingout of the eye (Rocha et al., 2009).The wave front generated by the back surface of the

lens can also be analyzed using a similar theoreticalcalculation. In that case, we will obtain

W2Lr2i nvj nLzr2i; A7

where nv represent the index of refraction of the vitreous.The front surface of the lens is located at the stop plane

of the eye. The back surface is not located exactly at thestop plane of the eye, but the change in the wave front afterpropagation from the second surface to the front surfaceshould be small and may be neglected. Then, taking intoaccount that there is certain magnification between bothsurfaces of the lens and the entrance pupil plane (where

aberrations are measured), which correspond to "1L = r/r1iand "2L = r/r2i for the front and back surfaces, respec-tively, the total radial dependence of the wave front at theeyes entrance pupil would be, after Equation A5:

W r WC r W1L r W2L r WC r nLjnr

2

2

1

"1L2 R1L

j1

"2L2 R2L

nLj nr4

8

k1L

"1L4 R1L

3j

k2L

"2L4 R2L

3

" #

nLjnr6

16

k1L2

"1L6 R1L

5j

k2L2

"2L6 R2L

5

" #; A8

where WC(r) represents the radial contribution of thecornea, up to the sixth order (n = 2, 4, or 6), and where wehave assumed that na = nv = n. The term related to the firstbracket corresponds to defocus:

Ad L nLj nr20

2

1

"1L2 R1L

j1

"2L2 R2L

: A9

The terms associated with the second and third bracketsin Equation A8 are the primary (As) and secondary (Bs)spherical aberrations generated by the lens, respectively:

As L nLjnr40

8

k1L

"1L4 R1L

3j

k2L

"2L4 R2L

3

" #; A10

Bs L nLjnr60

16

k1L2

"1L6 R1L

5j

k2L2

"2L6 R2L

5

" #: A11

In Zernike expansion, aberration coefficients depends onthe pupil radius (see Equation 2), thus the variation of a4

0

would also be affected by the term WC (see Equation A8).In that case,

As TV As CV As LV As CrV4

r4

As LV

As Tj As L rV4

r4

As LV; A12

where the prime indicates the values after accommo-dation, and [As]T and [As]C are the values of the fourth-order spherical aberration of the whole eye and that


corresponding to the cornea, respectively. Thus, the changeof the total fourth-order spherical aberration during accom-modation is given by

$ As T As TVj As T As TrV4

r4j1

As LVj As LrV4

r4

: A13

Similarly, the variation in the eyes sixth-order spher-ical aberration, [Bs]T, would be

$ Bs T Bs TVj Bs T Bs T

rV6

r6j1

Bs LVj Bs L

rV6

r6

: A14

If we do not take into account pupil miosis duringaccommodation (rV= r; for instance, by using an artificialpupil in front of the eye), the last equations can berewritten as

$AsRT As0 RL j AsL; A15

$BsRT Bs0 RL j BsL; A16

where the symbol R indicates that the values corre-spond to a constant pupil size.However, when pupil miosis (natural pupil) is taken

into account, Equations A13 and A14 can be rewritten as afunction of the change undergone by the spherical aberra-tion for a constant pupil size (Equations A15 and A16), asfollows:

$ As T $ As RTrVr

4 As T

r04

r4j1

; A17

$ Bs T $ Bs RTrVr

6 BsT

r06

r6j1

: A18

Assuming that "1L and "2L do not change during accom-modation, this leads to

As 0RL As LVr

r0 4

; A19

Bs 0RL Bs LVr

r0 6

: A20

In terms of a Zernike polynomial expansion,Equations A10 and A11 would be (Mahajan, 1991)

a04

L nLj nr

40

48ffiffiffi5

p k1L"1L4 R1L

3j

k2L

"2L4 R2L

3

" #; A21

a06

L nLjnr

60

320ffiffiffi7

p k1L2

"1L6 R1L

5j

k2L2

"2L6 R2L

5

" #; A22

and Equations A13A20 remain the same except changingAs by a4

0 and Bs by a60.

Appendix B

The defocus Ad necessary to balance the effect of thepresence of primary and secondary spherical aberrations, interms of minimum RMS, will be obtained for an Ad valuethat minimizes the function bW(>)2 j bW(>)2 with

bW r n 1:

Z 10

Z 2:0

Ad>2 As>4 Bs>6n>d>dF:

B1

Thus,

bW > 2j bW > 2

1:

Z 10

Z 2:0

Ad>2 As>4 Bs>6

2>d>dF

j1

:

Z 10

Z 2:0

Ad>2 As>4 Bs>6

>d>dF

2

9112

B2s

4

45

A2s

1

12

A2d

1

6

BsAs

320

BsAd 1

6

AsAd: B2

The value Ad that minimized Equation B2 is obtainedusing

bW>2jbW>2Ad

16

Ad 3

20

Bs 1

6

As 0:

B3


So the defocus necessary to balance the primary andsecondary spherical aberrations generated by the eye duringaccommodation will be

Ad j 910

BsjAs: B4

Acknowledgments

This work has been supported by the FundacionSeneca (Region de Murcia), Spain (Grant 05832/PI/07)and by the Red Nacional de Optometra, Ministerio deCiencia e Innovacion (ACTION: SAF2008-01114-E).The authors want to thank the anonymous reviewers fortheir suggestions.

Commercial relationships: none.Corresponding author: Dr. Norberto Lopez-Gil.Email: [email protected]: Facultad de Optica y Optometra, Campus deEspinardo, Murcia 30071, Spain.

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The Change of Spherical Aberration During Accommodation and Its Effect on the Accommodation Response

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