When the eye is in-focus, the retinal image plane is always optically conjugate with the object of regard. A certain range closer or farther away from the initial object position causes an imperceptible blurring of the retinal image. This is defined as the depth-of-field and is dioptrically analogous to the depth-of-focus (DoF) on the retina. Both the geometrical and the physical properties of an optical system can be used to predict these phenomena. The former method employs principles of paraxial optics, like controlled growth of spot sizes, while the latter accounts for diffraction and can be derived from the modulation transfer function (MTF) calculations. Since it is well agreed that the eye, though robust, is optically imperfect due to the presence of aberrations,1 it seems more appropriate to derive these measures from the physical-optics standpoint.

The DoF of the human eye is not only a function of optical parameters such as pupil size and aberrations, but is also affected by retinal, neural and more complex psychophysical factors.2-4 Among these, the Stiles-Crawford effect (SCE) is one of the most important retinal factor. This is a consequence of the directional sensitivity of the foveal photoreceptors whereby they elicit a visual response more efficiently when the incoming pencil of light lies close to the optical axis (from centre of the pupil) rather than when it enters obliquely. This effect has often been used to explain why the DoF does not decrease as rapidly as predicted by the geometrical model when the pupil size increases. The underlying idea is that the reduced impact of marginal rays due to the SCE reduces the effective pupil size and, therefore, increases the DoF.

It is known that the MTF of an unaberrated eye shows greater sensitivity to defocus than an optical system with aberrations. Although, these aberrations compromise the MTF at best-focus, they increase its relative measure with defocus and, therefore, result in a higher DoF.5 Thus, the DoF of an aberrated system is always higher than the equivalent unaberrated, or diffraction-limited, system. Additionally, monochromatic aberrations such as spherical aberration (SA) introduce blur to the visual system, rendering it less susceptible to the influence of chromatic aberration.6,7 These are examples of how aberrations can make a positive contribution to the optics of the eye. Clinically, this principle is utilized as a passive approach to correcting presbyopia with multifocal contact and intra-ocular lenses.

To understand how DoF might be optimally utilized in a corrective device, it is of value to consider how it is affected by changes in various aspects of the ocular system. For instance, its decrease with increasing pupil size, especially in aberration-free systems is a well-known fact. Moreover, studies have shown that the DoF is inversely related to spatial frequency.8,9 While in ideal eyes this effect is evident across the full range of relevant spatial frequencies, in real eyes it is only of importance up to about 8 cycles/degree; above that the DoF becomes virtually constant.10 Nevertheless, lower spatial-frequency targets produce retinal contrast profiles that are more tolerant to defocus.11

The contribution of spherical aberration (SA) to the enhancement of presbyopic vision and post-cataract (IOL) surgery has been a topic of interest and debate among the vision scientists in the recent past.5,7,10,12-17 Although most agree that the multifocal effect of low-level SA increases the eye's depth of focus and leads to pseudo-accommodation, 5,7,8,10,13,17 there are a few others who also demonstrated an improvement in the subjects’ spatial vision at the bestfocus position without compromising the subjective tolerance to defocus, despite a complete correction of the ocular SA.12,15,16 These contradictory findings leave the clinicians and surgeons doubtful, whether or not to leave the patient with the residual amounts of SA for the benefit of DoF.

Adding further to the complexity, the exact association between the sign of SA and the effective DoF has also not been well comprehended. Some researchers proposed and proved that the inherent blur induced by positive SA would leave the patient less susceptible to the effects of defocus and chromatic aberrations.5,7,10,17 The patent issued to Somani and Yee18 claims that negative SA provides significantly larger DoF than a positive SA of the same magnitude, which would be sufficient to mitigate presbyopic symptoms. However, there has been no additional evidence supporting this position. The accuracy of this proposal has considerable clinical ramifications as it could influence the design of multifocal corrective devices (e.g., IOLs), contact lenses and, potentially, presbyopic corneal ablations. This diversity of opinions amongst researchers and clinicians supports the need for further clarification. Accordingly, in the present work we sought to explore the influence of SA's sign on the optical DoF by using a ray tracing simulation. In addition, we also investigated the influence of the SCE, modeled as pupil apodization, on the spatial performance of centered optical systems.

The impact of optical imperfections on the visual performance of the human eye has been of keen interest to vision scientists for most of the last century. While tremendous improvements have been made regarding the ability to objectively measure these defects, inadequate knowledge of their interaction with one another has generally limited the understanding of their visual effects. The obvious exception to this is defocus. For all kinds of refractive error, this is known to be the major ocular aberration and its effect on spatial visual performance metrics such as acuity and contrast sensitivity has been studied with great care. Visual performance as a function of defocus for an object having a given spatial frequency will be a reflection of the DoF that exists in the system. DoF becomes an important means of obtaining optimized vision in cases where significant defocus is evident as may occur, for example, at near vergences when accommodation is reduced (as in the case in presbyopic patients), or when it is entirely absent (for instance, after monofocal intraocular lens implantation).

The visual impact of changes in DoF can be assessed in a number of ways. It is well known that measuring contrast thresholds for sinusoidally modulated gratings for a range of spatial frequencies is an effective way of evaluating spatial vision.25 Hence, contrast sensitivity functions over a range of defocus values are commonly used as a measure of the DoF.8 Alternatively, the DoF could also be defined as the range within which there is virtually no reduction in visual acuity.26 Unfortunately, neither of these strategies is amenable to simulation using ray-tracing calculations. Therefore, it was necessary to take a different line in interpreting the results of our modeling. Recalling that at a given spatial frequency the contrast sensitivity function is the product of the optical and the neural modulation transfer functions and knowing that defocus decreases the former but has no effect on the latter, we considered that the optical MTF should bear a direct linear relationship with the contrast sensitivity function. Since it is not feasible to incorporate neural factors into the ray tracing schema, we assumed these to be constant over all our models and selected the optical MTF as a predictor of visual performance. Hence the customary definition “peak width at half-maximum of MTF value” was considered as the adopted measure of DoF.

In the Zemax software, such MTF computations can be easily obtained by using the native settings of Fast-Fourier- Transform through-focus MTF (FFT MTF). This method slides the image surface, the retina in this case, moving it axially to produce the required amount of defocus. The MTF of the active configuration is then calculated for each point in the described range, while the spatial frequency in the image space is held constant. This is hardly physiological in the case of a real eye. Moreover, the dioptric differences that occurred due to the shifting of the focal plane in different directions could result in a slight asymmetry of the observed DoF measures. Additionally, spherical aberration could also be slightly affected by the displacement of the image position. Such factors complicate the findings obtained by the standard in-built methods. Hence, we have adopted the more robust through-focus MTF method, which consists of iterative paraxial lens corrections placed in front of the eye model.

In this experiment we have attempted to investigate the influence of the sign of the SA on the eye's DoF. Although the +SA and –SA results were close to being mirror images of one another for low-SF targets, there are some minor asymmetries in the DoF at higher SFs. Reviewing the DoF measures both from the through-focus MTF calculations as well as from the phase profile viewpoint, we noted that when SA was below (more negative than) -0.15μm, the models showed marginally higher DoF than their positive-SA counterparts. An alternative explanation for this result could be given by considering the caustic envelope formed by the spherical aberration in any optical system. By definition, this envelope is different on either side of the best focus. The zone around the envelope's centre of least confusion is called the caustic core, and it is the portion that contributes to an increased depth-of-focus. This also switches from one side of focus to the other when the sign of spherical aberration is changed and lies closer to the exit pupil of the optical system with positive spherical aberration. As a result, there is a marginal increase in the numerical aperture of the system. Exactly the opposite happens in the negative-spherical-aberration case. Taking into account that image quality does not degrade as rapidly with defocus when numerical aperture is decreased, this fact might explain the slightly greater depth-of-focus found in the case of negative spherical aberration.

The results reinforce the theory that spherical aberration, be it positive or negative, increases the DoF. Moreover, certain higher levels of negative SA, either intrinsic or induced by means of external optical components, do occasionally result in marginally higher levels of DoF than when the same magnitude of positive spherical aberration is present. This does not necessarily mean that image contrast will always be higher over a range of defocus. More importantly, such a small difference in DoF would not be sufficient to outweigh the effects of positive SA in mitigating presbyopic symptoms.

The explanation for the asymmetry in the through-focus MTF trends for positive and negative defocus might be well attributable to the strong Zernike-mode interaction between defocus and higher-order aberrations. This can probably be best comprehended by taking into account the geometrical optics of these two scenarios. In the positive SA model, when the retinal plane is shifted towards the hyperopic side, both SA and defocus add dioptrically and make the marginal rays to land farther away from the ideal location. On the other hand, when the retinal plane is moved towards the myopic side, the defocus now counterbalances the effect of SA, thereby improving the retinal image quality. The behavior of negative SA models and defocus could also be explained on a similar basis. This asymmetry around the optimum focus occurs particularly for mid and high spatial frequencies, i.e. around 10-30 cycles/mm, and for large pupil sizes, which is in good agreement with previous studies.27-30 In agreement with Zhang et al.,4 we also found that the SCE apodization has an impact on the quality of the defocused image only when defocus and spherical aberration have the same sign.

A key factor influencing the potential usefulness of these effects was pupil size. The DoF effects mentioned above were apparent in the initial modeling with a 6mm pupil, which provided a wide range of SA, from -0.20 to +0.20μm. However, with the 4mm pupil, all the models showed more or less similar DoF values for every spatial frequency under study. The reason behind this result is that the smaller pupil size effectively eliminates from the calculations the SA associated with larger apertures. It is also important to recall that one of the operands of the merit function, as defined, is the emmetropic 2nd-order refraction taken over a 6mm pupil. As the pupil size decreases, this non-zero refraction term brings down the central peak of the MTF. Understanding this point provides an insight into the design of aspheric multifocal contact and intraocular lenses, most of which have an optic diameter of about 6 to 8mm. Thus, there is a wide zone in which the power can change, allowing control of SA across the whole optic of the lens. Light rays passing through the lens periphery can thus be utilized to promote the passive increase in DoF. Unfortunately, with the physiological pupil size usually varying between 3-5mm, it is unlikely that this situation is achieved once the lens is in place on a real eye. Thus, it is likely that the actual DoF achieved would always be lower than the theoretical value.

The DoF was identical for both far and near vergences, a situation that can plausibly be attributed to the assumption that the aberration profiles of the unaccommodated and accommodated eyes were similar. This is certainly not the case among young adults, where SA is driven in the negative direction with accommodation, but it is closer to what happens among older people.31 The change of the aberration pattern with accommodation is strongly age-dependent and is a complex function involving multiple variables. As this was not taken into account for the current isolated ray tracing experiment, comprehension of the effects of SA on DoF for a range of accommodative targets may be partial. This theoretical study is also arguably limited, as the subjective effects of viewing objects in the presence of spherical aberration have not been considered. Future work will be needed to remedy this.

We also considered whether the SCE, if exists, would contribute to a further increase in DoF. While theoretically this appeared to be the case, smaller pupil sizes, such as the typical values observed (3-4mm) during everyday viewing conditions, substantially eliminate any effect. Moreover, the SCE was found to be beneficial only for the low-SF stimulus, with a mixed effect being evident for the mid and high SF's, commonly associated with reading and near tasks. In the light of this result, it is likely that the SCE will not usefully increase DoF in practical presbyopic correction situations.

As pointed out by Vohnsen,32 the pupil apodization technique used to model the SCE is subject to the aberration effects from the optics between the pupil plane and the retina. Unfortunately, due to the limitations of our modeling software we were not able to include such effects. As the aberrations incorporated into the present model appear to be less than those considered problematic by Vohnsen and, moreover, all previous experiments had been conducted using SCE neutralizing filters mounted in the pupil plane of the eye,22,27-30,33 we considered this approach to be reasonable. Nonetheless, readers should be aware, that the impact of the SCE on DoF in the presence of higher magnitudes of spherical aberration may differ slightly from those indicated here.

When it comes to considering the presbyopic-correction philosophy of using deliberately induced aberrations to increase the depth of focus, the current study indicates that certain amounts of positive or negative SA could have equal potential. However, for mid-to-high spatial-frequency tasks, which are typical of near and reading work, the experiment suggests that negative spherical aberration might be the preferred choice, although this will not result in all cases in higher DoF values. It should also be borne in mind that the amount of SA required to alleviate presbyopic symptoms is a function of pupil diameter and that, for close viewing distances, any residual accommodation will alter the effective ocular aberrations. These factors mean that, in practice, it would be a complex task to produce the calculated levels of DoF, since the combined spherical aberration of the corrective device and the eye would effectively need to be the same as that designed for the distant target.

A final point to be considered is that while certain high levels of negative SA do marginally improve the image contrast with defocus and effectively increase DoF, this increase is not symmetric about the best focus but, instead, it is limited to the myopic (positive defocus) side and, moreover, it is to a great extent spatial-frequency dependent. Therefore, it is likely that while manipulation of negative SA to raise the DoF might be achievable in some circumstances, these may not be sufficiently accessible to provide a useful improvement in near work in practical applications.