This investigation found that through-focus retinal image quality and VA in presbyopic eyes were significantly impacted by the combination of high levels of pupil transmission apodization and SA. Furthermore, the interaction of the transmission profile of the pupil with the eye's wavefront aberrations may be beneficial or detrimental to visual performance, depending on the type of ocular aberrations present. We found that apodization was beneficial for through-focus retinal image quality and VA for presbyopic eyes with monofocal and negative SA-induced multifocal corrections. In contrast, apodization had a negative impact on through-focus retinal image quality and VA in presbyopic eyes with large magnitudes of positive SA.
To understand the mechanism behind this study's observations, it is beneficial to consider the distribution of refractive power,
Φ(
r), across a distance-optimized pupil with Zernike primary SA. It is well known that induction of Zernike primary SA causes a defocus shift in peak image quality.
35,39–43 For example, induction of positive Zernike primary SA causes a hyperopic shift in best focus, whereas an induction of negative Zernike primary SA causes a myopic shift. The Zernike defocus coefficient C
20 required to optimize peak image quality may be determined computationally
35,39,43 or psychophysically.
40,41 In this study, C
20 was determined by optimizing the retinal image quality metric at distance (0 D). The power distribution is obtained from the expression for the distance-optimized wavefront with Zernike primary SA, as shown below:
where
where
r is the radial pupil coordinate,
R is the radius of the maximum pupil size,
C40 and
C20 are the coefficients of Zernike primary SA and defocus (in μm), respectively, and
Z40(ρ) and
Z20(ρ) are the Zernike polynomials for primary SA and defocus, respectively.
The expression for the refractive power distribution,
44,45 Φ(
r), is shown below:
Substituting
equation 4 into
equation 5, we obtain the final expression for the refractive power distribution,
Φ(
r), across a pupil with SA and defocus:
The defocus shift caused by induction of SA, referred to above, is represented by the constant C
2 0 term in
Equation 6.
Figure 7 illustrates the distance-optimized power distribution,
Φ(
r), for a 4-mm pupil with C
4 0 = ±0.2 μm. The sign convention is such that positive values of defocus along the
y-axis denote near-add powers.
SA has a parabolic distribution of power across the pupil,
43,45 as illustrated in
Figure 7. Positive SA is defined as rays in the pupil periphery having a larger refractive power than central rays (i.e., periphery contributing to near vision). Alternatively, negative SA is defined as rays in the pupil periphery having less refractive power than central rays. Hence the eye's properties of pupil miosis and a negative shift in SA during accommodative effort
46,47 work together to improve near image quality.
48
By apodizing the pupil with a Gaussian-type function, the pupil's peripheral rays have lower transmission relative to the central pupil. Therefore, the peripheral rays are penalized and have a reduced contribution to the retinal image. As shown in
Figure 7, the pupil's periphery contains the near-add powers for positive SA, similar to center-distance multifocal presbyopic corrections. As a result, through-focus retinal image quality and VA with positive SA worsens with apodization, particularly at intermediate and near target vergences. In contrast, negative SA (comparable to center-near multifocal presbyopic corrections) undergoes an attenuation of rays contributing to distance with apodization, thereby improving intermediate and near retinal image quality.
Although SA is predominantly positive in the normal, phakic population,
33,49,50 there are circumstances that can cause the eye's SA to reverse sign. For example, hyperopic refractive surgery,
51,52 contact lenses,
53 aspheric
54,55 and multifocal intraocular lenses,
2,56 and more recently, some refractive intracorneal inlays have been shown to induce significant magnitudes of negative SA. Although SA is known to increase the eye's depth of focus
35,54,57 (
Fig. 6), it also reduces contrast sensitivity
58,59 and leads to photic phenomena, such as halos and glare.
60 Photic phenomena constitute a major reason for patient dissatisfaction with extended depth-of-focus intraocular lenses.
61,62 As shown in this investigation, pupil transmission apodization yields a visual benefit for through-focus retinal image quality, and therefore may aid in overcoming limitations of current presbyopic corrections. The impact of pupil transmission apodization on glare sensitivity will be important to address in future studies.
Our findings are in agreement with previous studies.
26–28,63 Zhang et al.,
26 Atchison et al.,
27 and Mino and Okano
63 reported apodization to significantly increase the through-focus range of non–phase-reversed spatial frequencies. Phase effects are particularly important to visual tasks relying on stimuli with a broad spatial frequency bandwidth,
64 as in normal viewing conditions. Atchison et al.
28 illustrated the improvement in blurred image quality due to apodization with images convolved with the point spread function. Correcting the phase effects (i.e., phase-rectification) caused by optical defocus significantly improves VA.
65,66 Although apodization does not phase-rectify the retinal image, apodization does improve through-focus retinal image quality by shifting the first phase reversal of the optical transfer function to higher spatial frequencies.
The accurate prediction of through-focus visual performance from theoretical retinal image quality metrics is important for the design and optimization of ophthalmic lenses with extended depth of focus, such as presbyopic corrections and myopia control techniques.
67 Similar to previous studies,
35,68 through-focus VA was well predicted by the image convolution–based image-quality metric (
R 2 = 0.85). For comparison, other common retinal image quality metrics were also evaluated, such as the log of the visual Strehl ratio (VSOTF), log of the area under the modulation transfer function (aMTF), and wavefront RMS. The coefficients of determination (
R 2) of through-focus VA with all test conditions and log(VSOTF), log(aMTF), and wavefront RMS were 0.63, 0.74, and 0.71, respectively.
Quantifying the eye's depth of focus is subject to numerous factors that may influence the final outcome (for a detailed review, see Wang and Ciuffreda
69 ). For example, previous studies
40,70 that defined depth of focus using both positive and negative defocus values (e.g., full-width half-maximum of an image quality metric) found that SA had a similar impact on depth of focus regardless of its sign. In this study, image quality was optimized at distance and only positive defocus values (i.e., near objects) were tested to recreate realistic conditions. Under these circumstances, positive SA led to a larger increase in depth of focus as compared with negative SA. However, as shown in
Figure 6, apodizing the pupil's transmission function had a significant impact on depth of focus.
A limitation of this study was that all optical conditions were matched in target luminance with the purpose of isolating retinal sensitivity from optical effects. Had the conditions not been luminance-matched, apodization would have reduced the retinal illuminance by approximately 60% (
Fig. 1). Despite the retinal cone photoreceptors' more than four orders of magnitude of gain control for photopic vision,
71 the reduction in retinal illuminance is well known to affect visual performance
72,73 and may be clinically significant, especially at low light levels. However, further investigation is needed to determine if pupil transmission apodization is safe and acceptable.
For future study, it will be of interest to examine the effect of pupil transmission apodization on diffractive multifocal intraocular lenses' through-focus image quality. Diffractive multifocals have become a widely accepted option for clinicians aiming to restore functional near vision in presbyopes. Based on the results of this study, we can expect pupil transmission apodization to improve through-focus image quality with diffractive multifocal corrections.