**Purpose**:
Amplitude of accommodation (AA) is reportedly greater for myopic eyes than for hyperopic eyes. We investigated potential explanations for this difference.

**Methods**:
Analytical analysis and computer ray tracing were performed on two schematic eye models of axial ametropia. Using paraxial and nonparaxial approaches, AA was specified for the naked and the corrected eye using the anterior corneal surface as the reference plane.

**Results**:
Assuming that axial myopia is due entirely to an increase in vitreous chamber depth, AA increases with the amount of myopia for two reasons that have not always been taken into account. First is the choice of reference location for specifying refractive error and AA in diopters. When specified relative to the cornea, AA increases with the degree of myopia more than when specified relative to the eye's first Gaussian principal plane. The second factor is movement of the eye's second Gaussian principal plane toward the retina during accommodation, which has a larger dioptric effect in shorter eyes.

**Conclusions**:
Using the corneal plane (placed at the corneal vertex) as the reference plane for specifying accommodation, AA depends slightly on the axial length of the eye's vitreous chamber. This dependency can be reduced significantly by using a reference plane located 4 mm posterior to the corneal plane. A simple formula is provided to help clinicians and researchers obtain a value of AA that closely reflects power changes of the crystalline lens, independent of axial ametropia and its correction with lenses.

^{1}Ametropia is a potential factor influencing AA but studies have produced conflicting results. Some studies have reported greater AA in myopes than in emmetropes and hyperopes,

^{2–6}while others found the opposite

^{7,8}or no differences.

^{9}Table 1 summarizes the results found in those studies. Although most studies show that AA is larger in myopic eyes, results are not fully comparable because of differences in methodologies between studies. Different choices of reference plane for specifying vergence of the far and near points are of particular importance for our investigation of potential optical explanations for variability in results reported in the literature.

**Table 1**

^{10,11}and is normally taken into account.

^{12,13}

^{10}assuming that the distance between principal planes of the cornea and the lens remains constant during accommodation, a 10-diopter (D) change in refractive state is produced by a 12.95-D increase in equivalent power of the crystalline lens, which changes the equivalent power of the whole eye by 11.12 D. Although all three values are equally valid, they have different interpretations. A potential for confusion arises if a researcher or clinician uses one definition for measurement but a different definition for application (or interpretation) of that measurement, and consequently comes to an erroneous conclusion.

*R*−

*P*diopters. However, the use of such a practical definition can be difficult to connect with the more theoretical definitions of AA because of movement of the principal planes relative to the fixed anatomic location of the cornea. To take such movement into account requires knowledge of other parameters of the eye that are usually unknown (e.g., corneal power, anterior chamber depth, axial length). Thus, measurements of AA in two eyes with crystalline lenses that change the same way during accommodation may be different if, for instance, the axial lengths of the two eyes are different.

*AA*, using the corneal plane as reference plane for eyes with different axial refractive errors, we performed computer ray tracing simulations as well as analytical calculations under the assumption of paraxial optics. Ametropia was introduced into these models by changing the axial location of the retina relative to the second surface of the lens without changing the eye's optics.

_{N}^{14}). Paraxial analysis was performed for two different eye models, and nonparaxial analysis for one of them. This strategy enabled an assessment of how much the results depend on the specific eye model used and the Gaussian approximations in paraxial optics. We chose the eye models of Le Grand

^{15}and Navarro et al.

^{16}because they are parameterized for both the accommodated and the relaxed states. The value of

*AA*in the Le Grand eye model is approximately 7 D, so we selected that value as the accommodation value for the Navarro eye model as well. Axial ametropia was introduced into the models by changing the axial position of the retina. All simulations were carried out using ray tracing with a 5-mm entrance pupil diameter and a wavelength of 550 nm.

_{N}*AA*for the Gaussian approximation, we took into account the axial movement of principal planes of the eye's optical system that occurs when the lens accommodates. Formulas used for all calculations can be found in Appendix A. The analytical equations were verified by application to the two eye models, the results of which agreed with paraxial ray tracing results from Radiant Zemax.

_{N}*R*

^{2}> 0.999) with a negative slope indicating larger

*AA*in myopes than in emmetropes, and these in turn larger than in hyperopes. The unitless slope of the fitted lines ranged from −0.053 to −0.059.

_{N}**Figure 1**

**Figure 1**

*AA*determined by paraxial and nonparaxial analyses of the Navarro eye models is 0.60 D. Estimated

_{N}*AA*values for the nonparaxial eye model are lower because of the presence of positive spherical aberration in the relaxed eye that decreases during accommodation, thereby resulting in a decrease in the accommodation response.

_{N}^{14,17}For smaller pupil diameters, spherical aberration has less effect on the Zernike refraction, resulting in a shift in

*AA*values toward the paraxial result.

_{N}*AA*from the corneal plane,

_{N}*C′*, as a function of axial length: where:

*A*and

_{D}*A*are the vergences of the far and near points measured from the first principal plane (

_{A}*H*) of the eye; is the displacement of the first principal plane from relaxed to the accommodated state; and are the distances from the first principal plane to the corneal plane of unaccommodated and accommodated eyes, respectively;

*φ*and

_{A}*φ*represent the powers of the eye when accommodating and in the relaxed state, respectively; and

_{D}*c*is defined by Equation A9. Equation 1 has a dependency on the axial length expressed as the distance between the secondary principal plane and the image plane in the relaxed eye, .

*AA*is the sum of three contributions. The first term corresponds to the change in power of the eye due to the lens when accommodating, and takes into account the distance between the corneal and principal planes. The second term in the equation is due principally to the movement of the secondary principal plane during accommodation. This term is always negative and, in absolute value, is smaller for myopes, so its effect is to reduce the

_{N}*AA*more in hyperopes. The last term in the equation is related to the movement of the first principal plane with accommodation, and its change during accommodation for different axial lengths is negligible compared to the other two terms (Fig. 2). All terms in Equation 1 are affected by the choice of reference plane

_{N}*C′*for specifying vergences.

**Figure 2**

**Figure 2**

*H*, inside the eye) to the corneal plane,

*C′*, for the purpose of defining AA. The second reason is related to the displacement of the posterior principal plane of the eye,

*H′*, toward the retina when the eye accommodates (see Appendix A for supporting data). For the Le Grand eye model, the first reason accounts for approximately 52% of the change in the AA with axial refractive error shown in Figure 1, while the second reason accounts for most (45%) of the remaining 48%. While the two reasons have independent origins, they are related to each other since the distance changes during accommodation due to the displacement of

*H*toward the retina.

^{15}The values obtained by Le Grand are lower than those obtained in this work because he used the anterior principal point,

*H*, as the origin of vergences. In 1864, using Helmholtz's diagrammatic eye, Donders

^{8}calculated the near and far points of three eyes with different axial lengths. Contrary to our findings, Donders found that hyperopes accommodate more than myopes. Based on our calculations, Donders' results can be explained by the fact that he used the nodal point of the unaccommodated eye,

*N*, as origin of vergence, which is closer to the retina than

*H*. Thus an opposite change in AA with axial refractive error compared to the one shown in Figure 1 can be expected. These results reveal the importance of choosing the origin of measurement appropriately.

*H*and

*H′*toward the retina during accommodation, on the choice of eye model, and on the degree of accommodation. Paraxial analysis of the Le Grand model indicates that this null position is 3.68 mm behind the corneal plane. Paraxial analysis of the Navarro eye model indicates the plane to be 4.49 mm behind the corneal plane for 10 D of accommodation and 4.29 mm behind for just 1 D of accommodation. Thus, a reference plane located 4 mm behind the corneal plane for specifying accommodation will strongly reduce the dependency of estimated values of AA on the axial refractive error of the eye. A practical advantage of this proposed reference plane is its close proximity to the eye's entrance pupil, which is normally approximately 3 mm posterior to the corneal plane. Thus when a Badal optometer is used to measure accommodation, and the focal plane of the Badal lens is positioned to coincide with the eye's entrance pupil, then measurements of AA should be largely unaffected by the internal effects described above. That expectation is consistent with the findings of Fisher et al.,

^{9}who reported that AA was independent of refractive error when specified relative to the eye's entrance pupil. However, this advantage of a Badal optometer would be lost for other configurations that do not use the entrance pupil of the eye for specifying AA.

^{9}) do not include sufficient methodological details to allow correction of reported results for the internal effects described above. Nevertheless, we estimate the order of magnitude that might be expected for such corrections (assuming typical values for missing parameters) to be the same order of magnitude as the differences reported between myopic and emmetropic eyes (see Table B1 at the end of Appendix B). Future studies of accommodation should take into account such methodological details that appear to have been overlooked previously in order to accurately estimate AAs.

*AA*from the value of the vergence to the near point (assuming that the vergence of the far point is 0), which is referred to the plane of correction (ophthalmic or contact lens): where

_{N}*d*is the distance between the correction plane and the corneal plane in meters,

*P*is the power of the correction lens in diopters, and

_{L}*P*is the vergence in diopters of the corrected eye's near point as measured from the correction plane (with positive sign for real objects). As an example of application of Equation 2, let us assume a 5-D myope (

_{C}*P*= −5 D), corrected with spectacle lenses placed at 14 mm of the corneal plane (

_{L}*d*= 0.014 m). If the near point of this subject is located 12 cm from the spectacle plane (

*P*= 1/(0.12 m) = 8.33 D), the AA will be 6.17 D.

_{C}**P. Bernal-Molina**, None;

**F. Vargas-Martín**, None;

**L.N. Thibos**, None;

**N. López-Gil**, None

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*C′*and principal planes

*H*and

_{D}*H*. Relative distances from the principal planes to the cornea and the retina are also shown. Eye refraction is irrelevant in this demonstration, and a myopic eye is chosen for clarity in the diagram. Figure A1 also shows schematically the displacement of the principal planes toward the retina in the accommodated eye. These displacements are real and result from the increase in power of the eye during accommodation due to the increase in power of the lens. Values of those displacements depend on the choice of eye model and on the amount of accommodation, but not on the axial length of the eye,

_{A}*L*.

**Figure A1**

**Figure A1**

*n*and

*n′*are the indices of refraction outside the eye and the vitreous humor, respectively, and and ( < ) are the image focal lengths of the relaxed and accommodated eye, respectively. and are positive values, while

*a*and

*a′*values are negative for distances in front of the eye and positive for virtual objects. Equations A1 and A2 can be also rewritten in terms of vergences by defining , , , and , as: and The distance in Equations A1 and A2 can be related to the axial length as: If

*n*= 1, we have from Figure A1: and By means of Equations A3, A4, A6, and A7, the AA can be expressed as: where

*H*and

*H′*for correcting the eye's myopia (bottom). Distance from

*H′*to the corneal plane,

*C′*, is

*d*and has a positive value.

**Figure B1**

**Figure B1**

*C*is the position of the object whose image is

*C′*(

*C*and

*C′*are conjugate through the OS), we can apply generalized Gaussian correspondence laws taking

*C*and

*C′*as reference planes to compute the positions of the images of the far and near points of the corrected eye. Assuming that the lateral magnification of the OS for

*C*and

*C′*is

*β′*, and

_{c}*f ′*the focal distance of the OS, we have for the far point: and for the near point: where

*n′*is referred to the image space of the OS. Subtracting Equation B1 from Equation B2 yields: Now if the accommodative amplitude of the naked eye, referred to

*C′*, is defined as the difference between near-point vergence and far-point vergence, and that of the corrected eye, referred to

*C*, is similarly defined as then we finally have

*C*but we want to obtain the relation with origin on C′, . In a practical case, usually a thin ophthalmic lens is used in air at a distance

*d*in front of the eye for its correction. In that case, taking into account that

*d′*is the distance between the spectacle lens and

*C*, and that

*R*represents the refractive error of the naked eye that is perfectly corrected with a lens with power

*P*, we can use the following relations: to obtain So, finally we obtain

_{L}*C*to the correction plane, an expression relating AA (as defined in B4) of the naked eye with that measured clinically, can be found: where

*P*is the vergence of the near point with respect to the correction plane in absolute value.

_{C}*d*should now correspond to the distance from the correction to the corneal plane plus 4 mm:

*d*= 0 m in Equation B11,

*P*is the power of the contact lens and

_{L}*P*must be the vergence of the near point to the corneal plane (which is the correction plane in this case) in absolute value.

_{C}**Table B1**