Refractive extended depth of focus lens design based on periodic power profile for presbyopia correction

Abstract

Purpose

Limitations of existing diffractive multifocal designs for presbyopia correction include discrete foci and photic phenomena such as halos and glare. This study aimed to explore a methodology for developing refractive extended depth of focus (EDoF) lenses based on a periodic power profile.

Methods

The proposed design technique employed an optical power profile that periodically alternated between far, intermediate and near distances across the pupil radius. To evaluate the lens designs, optical bench testing was conducted. The impact on visual performance was assessed using a spatial light modulator-based adaptive optics vision simulator with human subjects. Additionally, the effects of pupil size and lens decentration on retinal image quality were theoretically examined. A comparative performance analysis was carried out against a typical diffractive trifocal design and a monofocal lens.

Results

The proposed design method was found to be effective in uniformly distributing light energy across all object distances within the desired depth of focus. While trade-offs between overall image quality and depth of focus still exist, the EDoF lens design, when tested in human subjects, provided a continuous depth of focus spanning over 2.25 D. The results also revealed that the EDoF design had a slightly higher dependence on changes in pupil size and lens decentration compared with the diffractive trifocal design.

Conclusion

The proposed design method showed significant potential as an approach for developing refractive EDoF ophthalmic lenses. These lenses offer a continuous depth of focus but are slightly more susceptible to variations in pupil size and decentration compared with the diffractive trifocal design.

Introduction

Several multifocal designs for the correction of presbyopia have been proposed in the literature and applied to contact lenses² and intraocular lenses (IOLs).^3-5 Refractive type optical designs have been commonly used for contact lenses. Multifocal IOLs are based on diffractive as well as refractive designs.

A classic example of the diffractive type is the kinoform design, which induces discrete wavefront jumps to produce diffraction effects. By adjusting the locations and heights of the phase jumps, the number of foci and the distribution of light energy to individual foci can be controlled. Bifocal and trifocal diffractive designs are incorporated into commercially available IOLs, and complete mathematical descriptions of these lenses can be found in Zhang⁶ and Gatinel et al.⁷ Recently, an alternative multifocal diffractive design was proposed, characterised by sinusoidal, smooth wavefront profiles instead of the discrete phase jumps found in traditional kinoforms.^{8, 9} This design can achieve high diffraction efficiency and poses fewer manufacturability challenges due to its smooth shape. A relatively low incidence of severe dysphotopsia has also been reported with this design.¹⁰ Both diffractive designs described above are based on a periodic wavefront profile defined as a function of $r^2$, where $r$ represents the radial distance from the pupil centre. The advantages of these diffractive properties include (1) enhanced image quality at the designated foci and (2) performance that is independent of changes in pupil size. However, they are not without drawbacks, such as the photic phenomena of halo and glare, and significant degradation in image quality between the foci.¹¹

To overcome these limitations, some extended depth of focus (EDoF) IOL designs have been proposed.¹² Additionally, premium monofocal IOLs, based on refractive design concepts, have been developed and demonstrated to reduce photic phenomena but with limited depth of focus (DoF).^13-21

We propose a method for designing a refractive EDoF lens that modulates the shape of the power profile and repeats this pattern periodically. It is essential to distinguish this periodic smooth design from the sinusoidal diffractive design. The diffractive designs distribute light energy into different diffractive orders due to the periodic wavefront structure relative to $r^2$. When this periodicity relative to $r^2$ is altered (for instance, becoming relative to $r$), and the structure becomes less dense, the design may shift from being diffractive to refractive. Consequently, the distribution of light energy can be more precisely controlled throughout the DoF by manipulating the local refractive power.

The aim of this study was to assess the feasibility of creating such a wavefront profile and to evaluate its performance through focus simulations and empirical tests, including bench testing and human subject experiments.

Methods

In this section, we have undertaken a theoretical exploration of the proposed design method. Three designs are introduced to exemplify the design flexibility. Subsequently, building upon these examples, a final design is proposed. Following the theoretical groundwork, the characteristics of the proposed design were tested via a vision simulator and simulations.

Periodic refractive extended depth of focus (PREDoF) lens design

The proposed lens design was based on local refractive power, a vital element affecting optical performance. The refractive power of a wavefront is defined by the radial slope and location. The relationship between local refractive power and wavefront can be expressed by the following equation.

$$\varphi (r)=\frac{1}{r}\frac{dW(r)}{dr}$$ (1)

where $W(r)$ denotes the wavefront and $\varphi (r)$ represents the local sagittal wavefront vergence (local refractive power) at a radial distance, $r$, from the pupil centre. It is worth noting that the term pupil in this study refers to the eye’s entrance pupil. The refractive power corresponds to the inverse of the distance between the lens and the point where the ray intersects the optical axis. This equation is distinct from the curvature description, utilising a different mathematical representation for dioptric power.²²

To achieve a broad DoF and ensure independence from pupil size, the power can be evenly distributed across the entire region of the lens. This uniform distribution can be achieved through periodic oscillation of the power structure along the lens radius;²³ a design method termed the PREDoF design. Within each cycle, the sagittal wavefront vergence $\varphi (r)$ (measured in dioptres, D) at a distance $r$ mm from the centre can be expressed by a set of subsequent equations.

$$\varphi (r)=P\times S(\rho (r))$$ (2)

$$\rho (r)=M{\left(\frac{r}{R}\right)}^N-\lfloor M{\left(\frac{r}{R}\right)}^N\rfloor$$ (3)

where $\rho (r)$ is an intermediate function of $r$ (in mm), $N$ is the exponential power, $M$ determines the number of cycles along the total design radius $R$ (in mm), the operator ⌊ ⌋ takes the floor value, $P$ is the total scanning power range (in D), which controls the overall DoF range and $S(\rho )$ is a shape function. As illustrated in Figure 1(a), for Design 1, the shape function $S(\rho )$ is chosen as $S(\rho )=1-\mid 2\rho -1\mid$. As M increases, the production of more cycles across the radius accelerates the iteration of the optical power and enhances pupil-size independence. However, this may potentially diminish the refractive quality of the lens while enhancing its diffractive characteristic.

Figure 1.

(a) Three representative periodic refractive extended depth of focus (PREDoF) power profiles are illustrated across a pupil radius of 3 mm. Design 1 (solid line), with parameters P = 3, M = 3, R = 3, and N = 1, focuses on emphasising intermediate and near visions. The definition of the parameters can be found in equations (2) and (3). $P$ is the total scanning power range (in D), which controls the overall DoF range, $M$ determines the number of cycles along the total design radius $R$ (in mm) and $N$ is the exponent. Designs 2 and 3 are adjusted empirically to favour far vision alone and both far and near visions, respectively. (b) Wavefront aberration translated from the power profile of Design 1. (c) Through-focus retinal image quality of the three designs as evaluated using the area under the modulation transfer function (areaMTF) metric with a central 4 mm pupil in a 6 mm lens design.

Subsequent to the power profile design, the wavefront map was numerically integrated. The slope of the wavefront, represented as $\frac{dW(r)}{dr}$, was calculated using equation (1), with the differential of $r(dr)$ set to 19.57 μm. Subsequently, the wavefront aberration map was obtained by adding up these increments. Figure 1(b) illustrates the wavefront map derived from the power profile represented by Design 1.

In practical terms, the shape function $S(\rho )$ can be replaced with other functions to fulfil specific design objectives; it can be vertically shifted, and its shape can be freely designed. The slope of $S(\rho )$ determines the light energy density. The flatter the local curve, the more energy is distributed across that specific dioptric value. Figure 1(a) shows two additional examples of power profiles. Designs 2 and 3 distinguish themselves from Design 1 by modulating the shape of function $S(\rho )$. In Design 2, the slope of the power profile at far vision is flatter than that for near vision. In Design 3, the slopes for both far and near visions are flatter than intermediate vision. Consequently, Design 2 is solely emphasised for far vision, while Design 3 is emphasised for both far and near visions.

The utilisation of the area under the modulation transfer function (areaMTF) is justified by its demonstrated strong correlation with visual acuity. Thus, areaMTF was utilised as the metric for assessing a range of target vergences from 0 to 3 D. A monochromatic light condition was assumed (λ = 579 nm). AreaMTF measures the area under the radially averaged modulation transfer function (rMTF) over a spatial frequency range of 0 to 15 cycles/degree (c/deg), as defined by the following equation.²⁴

$$areaMTF={\sum }_{n=0}^{\lfloor \frac{15}{\delta }\rfloor }\frac{\delta }{15}rMTF(n\delta )$$ (4)

where $\delta$ represents the sampling size of $rMTF$ in c/deg, and it was set at 0.59 c/deg; $n$ is an integer ranging from 0 to the floor value for $15∕\delta$.

Figure 1(c) demonstrates the performance of the three designs evaluated via the areaMTF metric. Notably, Design 2 prioritised the enhancement of far vision, compromising near vision. In contrast, Design 3 improved both far and near visions but made concessions on intermediate vision.

To achieve a more extensive DoF, Design 4 was conceived and tested for the rest of this study, based on adjustment in local slopes in the power profile. Its power profile, denoted by $S(\rho )$ was constructed with seven distinctive zones; the heights of Zones 1–7 adhered to a ratio of 1:2:1:2:1:2:1. By manipulating the slope of the power profile within each zone, the distribution of light energy across each zone could be effectively controlled, thereby producing an optimised power profile with a significant DoF. Figure 2(a) exhibits the shape of this profile, while Figures 2(b) and 2(c) display the colourmap representation of the wavefront and its cross-sectional profile, respectively.

Figure 2.

(a) Power profile for Design 4, conceived by dividing the entire dioptric range into seven zones, with the slope of each zone tailored for desired performance. (b) The resulting wavefront map from the optimisation. (c) Cross-sectional wavefront profile along the centre of the aperture of the wavefront depicted in (b).

Adaptive Optics Visual Simulator

The through-focus performance of Design 4 was verified using a custom-built adaptive optics vision simulator, which incorporated a spatial light modulator (SLM). Positioned at the pupil conjugate, the SLM (JD7554, Jasper Display Corp., jasperdisplay.com) corrected the eye’s aberration and induced the designed wavefront as depicted in Figure 3. A linear polariser was placed before the SLM to facilitate phase control. A laser beam (λ = 900 nm) from a super-luminescent diode was directed towards the retina, and the aberrated wavefront from the eye was measured by a custom-made Shack–Hartmann wavefront sensor. A digital micromirror device (DMD, model DLP 4710, Texas Instruments, ti.com) was employed to display visual stimuli, and a narrowband interference filter (λ = 579 ± 5 nm) created a monochromatic light condition. During optical bench testing, a doublet imaging lens and a charge-coupled device (CCD) camera (CXE-B013-U, Mightex Systems, mightexbio.com) substituted for the eye.

Figure 3.

Schematic of the spatial light modulator (SLM) based adaptive optics visual simulator's optical layout. The Shack–Hartmann wavefront sensor measures the eye's aberrations, and the SLM corrects these aberrations, implementing the lens design profile. Visual stimuli are rendered by the digital micromirror device (DMD) chip and filtered with a 579 ± 5 nm interference filter, maintaining a 4 mm entrance pupil diameter at the eye.

Subjects

The University of Houston Research Review Board approved this study, and all subjects provided informed consent before participation. All procedures involving human subjects adhered to the ethical standards of the Declaration of Helsinki. Five subjects, between 26 and 51 years of age, who had normal visual function with no history of ocular surgery, were recruited. Prior to the experiment, cycloplegia was achieved by instilling tropicamide ophthalmic solution (1%) and phenylephrine hydrochloride ophthalmic solution (2.5%) to each subject's left eye.²⁵

Through-focus image quality and visual performance measurement

In this study, a diffractive trifocal lens design was introduced, featuring three distinct foci at 0, 1.25 and 2.5 D, with the distribution of light energy being 42%, 15% and 26%, respectively. This trifocal lens was formulated by amalgamating two diffractive bifocal lens profiles.⁷ The specific phase profile employed in this study is depicted in the Appendices (Figure A1). The lens was designed without any apodisation. Moreover, an aberration-free monofocal lens design was evaluated as a control condition.

To assess the through-focus image quality of the three lens profiles (monofocal, trifocal and PREDoF Design 4), both bench testing and visual performance measurement were executed using an adaptive optics visual simulator. Bench testing encompassed capturing through-focus images of a high-contrast tumbling letter ‘E’ chart (with Snellen acuity progressing from top to bottom: 6/12, 6/9, 6/7.5, 6/6; as depicted in Figure 5). Various defocus magnitudes, ranging from 0 to 3 D in increments of 0.25 D, were incorporated into the lens profiles to emulate different object distances. The through-focus areaMTF values (shown in Figure 4) were drawn from the images utilising Fourier frequency analysis.

$$MTF=\mid \frac{FT[capturedimage]}{FT[displayedimage]}\mid$$ (5)

where FT denotes Fourier transform and MTF denotes the modulation transfer function.

Figure 5.

Captured images of a letter ‘E’ acuity chart, utilising monofocal, trifocal and periodic refractive extended depth of focus (PREDoF) profiles across a 2.5 D focus range, with increments of 0.5 D (including 0.25 D and 1.25 D). Letter sizes, from top to bottom rows, correspond to Snellen acuity of 6/12, 6/9, 6/7.5 and 6/6 (or 0.30, 0.18, 0.10 and 0.00 LogMAR, respectively). Image intensity is individually normalised. A 4-mm entrance pupil diameter was used throughout. Note that the 6/6 (logMAR 0.00) line may not be well visualised due to the space limit. Magnified image will improve the visibility of the letters.

Figure 4.

Through-focus area under the modulation transfer function (areaMTF) comparison for three optical designs: periodic refractive extended depth of focus (PREDoF), trifocal and monofocal. A higher areaMTF value signifies enhanced retinal image quality. The entrance pupil diameter was 4 mm.

The evaluation of subjects’ visual performance involved measuring high-contrast visual acuity through focus under the three optical designs, following a randomised sequence. For more direct comparison, the eye's intrinsic aberrations were fully corrected for all conditions. The reported visual acuity represented the mean of three measurements at each specified target vergence. A tumbling Snellen letter ‘E’ was used for visual acuity testing, and an adaptive staircase method, employing QUEST²⁶ was utilised to adjust the letter size based on subjects' responses to the letter's orientation. Following data collection, the visual acuity was ascertained at the 62.5% threshold of the fitted cumulative Weibull function.

Effects of pupil size and decentration

Furthermore, the study probed the impact of pupil size on visual quality through theoretical calculation. The lens profile was truncated to diameters of 3, 3.5, 4, 4.5 and 5 mm. The areaMTF was subsequently calculated for each pupil size across the various target vergences. Lastly, the standard deviation of the areaMTF values across all the pupil sizes was computed, serving to illustrate the fluctuations in retinal image quality at each target vergence.

The effect of lens decentration on retinal image quality was examined through bench testing. The lens profile, generated by the SLM was laterally shifted while keeping the size and location of the pupil constant, to emulate lens decentration. Various decentration magnitudes, including 0.25, 0.5, 0.75 and 1 mm, were scrutinised. The images of the letter ‘E’ chart, captured by the CCD camera, were subsequently analysed to determine the MTF.

Data analysis

Statistical analyses were conducted on the subjective visual acuity results. Each analysis incorporated the Kolmogorov–Smirnov test to ascertain the normality of the variables. A one-way analysis of variance (ANOVA) was employed to scrutinise the differences among the three distinct optical conditions at each vergence level. For post-hoc analysis, the Tukey method was used to perform multiple comparisons within the three subgroups,.

Results

Through-focus retinal image quality

The simulated through-focus image quality under the three conditions is illustrated in Figure 4. The monofocal design manifested a high areaMTF value for far vision (0 D), yet its image quality decreased relatively quickly with target vergence. As anticipated, the trifocal design exhibited three foci at 0, 1.25 and 2.5 D, with the image quality diminishing between these focal points. Conversely, the PREDoF design slightly compromised the image quality at the far and near foci, but displayed a more sustained through-focus image quality.

Figure 5 presents the results of the through-focus bench tests, featuring images of the tumbling letter ‘E’ chart generated by the three lens profiles. The image quality across the focus range aligns with the results illustrated in Figure 4.

Through-focus visual acuity

Figure 6 summarises the through-focus visual acuity results in LogMAR, with the y-axis inverted for enhanced visualisation of through-focus performance. As anticipated, the monofocal condition demonstrated superior visual acuity (−0.27 ± 0.14 LogMAR) at 0 D, although its performance deteriorated significantly with an increase in target vergence beyond 1 D. The trifocal design also displayed improved visual acuity at 0 D (−0.20 ± 0.02 LogMAR) compared to the PREDoF design. Even though there was a noticeable improvement in visual acuity at 1.25 D (−0.02 ± 0.08 LogMAR) and 2.5 D (−0.10 ± 0.08 LogMAR), the visual acuity between the foci was significantly impaired. On the other hand, the PREDoF design maintained superior visual acuity (better than 0.00 LogMAR) over an approximate target vergence range of 2.25 D, dropping below 0.00 LogMAR between 2.25 and 3 D. The PREDoF design significantly outperformed both the monofocal and trifocal designs at 1 and 1.5 D (P < 0.05).

Figure 6.

Visual acuity (LogMAR) of all subjects (N = 5) across the focus range under the three optical conditions. The vertical axis is inverted for better representation of performance differences. Error bars represent ±1 standard deviation. The entrance pupil diameter was 4 mm.

Effects of pupil size and decentration

Figures 7a and 7b show the through-focus areaMTF values for the trifocal and PREDoF profiles across different pupil sizes, respectively. The trifocal design preserved its characteristic pattern (three areaMTF peaks at the foci) across target vergences, while the PREDoF design revealed alterations in through-focus shapes. Figure 7c illustrates the variability in through-focus areaMTF values for different pupil sizes, represented by the standard deviation across the pupil sizes. The PREDoF design displayed larger variations at 0 and 1 D, whereas other target vergences revealed similar variations for both designs.

Figure 7.

Through-focus area under the modulation transfer function (areaMTF) comparison of the optical profiles for various pupil sizes. (a) Trifocal design. (b) periodic refractive extended depth of focus (PREDoF) design. (c) Standard deviation of the areaMTF values across pupil sizes for the two designs.

Figure 8 summarises the results of bench testing investigating the effect of lens decentration. The PREDoF design exhibits greater variability compared with the trifocal design. Specifically, standard deviations of variability at 0.25 D and 1.75 D are 0.014 and 0.023, respectively, for the PREDoF design, whereas they are 0.007 and 0.011, respectively, for the trifocal design.

Figure 8.

Effect of lens decentration on image quality through focus, represented by area under the modulation transfer function (areaMTF), analysed from images captured during bench testing. Different colours indicate various magnitudes of pupil decentration. The entrance pupil diameter was 4 mm. PREDoF, periodic refractive extended depth of focus.

Discussion

This research work proposes a method for designing radially symmetric periodic refractive EDoF optics, referred to as PREDoF, based on a periodic power profile. Using this method, a representative design was developed, which effectively achieved a continuous and wide DoF. The periodic power profile was expected to enhance tolerance to changes in pupil size and lens decentration, but was still shown to be slightly inferior to the diffractive design. A series of experiments, including bench testing and visual performance evaluation using an adaptive optics visual simulator, demonstrated the potential effectiveness of PREDoF for correcting presbyopia.

Periodic structures as a function of $r^2$ are commonly observed in diffractive lens designs. In this study, we correlated the periodicity with $r$ instead of $r^2$, resulting in the lens displaying refractive characteristics. As depicted in Figure 1, the power distribution control allowed light energy to be distributed to desired target vergences rather than fixed foci, enabling continuous DoF. Although the proposed PREDoF design achieved a large DoF, as shown in Figures 4 and 6, it slightly degraded far vision at 0 D compared with 0.25 D. This was due to a slight defocus shift towards intermediate vision, which was intentionally implemented to optimise the overall DoF. To enhance far vision, the original power profile could be altered slightly. However, such an adjustment may lead to a degradation in near visual quality, resulting in a reduction in the DoF. This trade-off between overall image quality and DoF represents an intrinsic limitation of any multifocal or EDoF design.

Many investigators frequently employ theoretical modelling to predict visual performance using optical designs. One intriguing question is the degree of correlation between our predictions and the actual measured visual performance. To achieve this, we utilised the valuable work of Alarcon et al., which offers an excellent tool for such investigations.²⁴ Using the proposed method, we transformed the areaMTF curves illustrated in Figure 4 to predict visual acuity. As shown in Figure 9, a comparison between the predicted visual acuity and the averaged measured results reveals a strong correlation, though the predicted visual acuity appears to be somewhat inferior. This discrepancy may be attributed to differences in the study conditions between the present word and that of Alarcon et al.²⁴ Specifically, our visual acuity measurements were obtained with a fixed (4 mm) pupil and without the eye’s native aberrations under monochromatic light, while their measurements were conducted with a natural pupil and native aberrations under white light.

Figure 9.

Averaged measured visual acuity against predicted visual acuity, using the method proposed by Alarcon et al.²⁴ Linear regression lines are dotted. PREDoF, periodic refractive extended depth of focus.

A traditional kinoform diffractive design without apodisation maintains energy distribution to the individual focal points at any point on the diffractive zone.²⁷ Consequently, the energy distribution in a diffractive design is in theory not significantly affected by changes in pupil size. However, as shown in Figure 7, through-focus image quality still varies, possibly due to increased blur of out-of-focus light with increased pupil sizes. In contrast, the PREDoF design relies on finite cycles of power profile modulation across the lens diameter. Consequently, the relative power distribution to different object distances varies with pupil size. Thus, the through-focus performance is more dependent on pupil size than the diffractive design. Nonetheless, the periodic nature of the power profile in the PREDoF design should offer advantages in terms of pupil-size independency when compared with other refractive designs based on spherical aberration and add power.²⁸

Moreover, we conducted a comprehensive comparison of lens decentration effects between the PREDoF and the diffractive trifocal designs. As anticipated, the through-focus areaMTF curves of the trifocal design exhibited minimal variation with decentration, whereas the PREDoF design showed significant variation across the tested amounts of decentration. This observation indicates that the diffractive design exhibits greater resistance to decentration, offering enhanced robustness due to its uniform energy distribution over the lens surface, as mentioned earlier. However, it is worth noting that a modification to the PREDoF design could potentially reduce its sensitivity to decentration. For example, we explored variations by adjusting the parameter N to 2 or 3, instead of 1, and observed a reduction in overall variability in image quality (see Figure A2 in Appendices).

Recently developed refractive EDoF IOLs (also referred to as premium or enhanced monofocal IOLs) have shown promise in mitigating photic phenomena. In a metrology study by Labuz et al.,²⁹ it was reported that the refractive EDoF halo profile closely resembled that of a monofocal IOL. Additionally, Baur et al.³⁰ reported that, in comparison to diffractive IOLs, refractive EDoF IOLs exhibited a lower intensity of light beyond the core of the point spread function. Various clinical trials have further supported the notion that refractive EDoF lenses generate less dysphotopsia when compared with diffractive multifocal lenses,^{13, 14, 31, 32} though with a reduced DoF. Our design significantly extended the DoF to closely match the add power of the trifocal lens. However, it is essential to investigate the halo and glare of this design in future studies.

Another direction for future research is to address real-life situations. The current study was conducted under well-controlled optical conditions, utilising monochromatic light and an adaptive optics visual simulator to correct the eye’s native aberrations. However, a future study should evaluate the PREDoF design under normal viewing conditions, where the eye's monochromatic and chromatic aberrations are left uncorrected.

In conclusion, the PREDoF design method demonstrated remarkable flexibility in creating refractive wavefronts that optimised through-focus retinal image quality by manipulating parameters associated with the periodic optical power profile. This adaptability allows for customised designs tailored to individual patient needs while exhibiting some level of tolerance to changes in pupil size and lens decentration. This design concept holds great promise for the development of refractive ophthalmic lenses to address presbyopia effectively.

Key Points.

• This study proposed a design method for refractive extended depth of focus lenses using periodic power profiles.

• The design method can achieve a large depth of focus while exhibiting some level of pupil-size independence and lens decentration independence.

• An adaptive optics vision simulator equipped with a spatial light modulator was used to evaluate the effectiveness of the design in human subjects.

Appendices

Figure A1.

Wavefront profile of the diffractive trifocal lens used for comparison in the present study. The trifocal design was made by linearly summing two bifocal diffractive lenses designed to have three foci at 0, 1.25 and 2.5 D at a wavelength of 579 nm.

Figure A2.

(a) Power distribution of two profiles with same parameters ($P=3,M=5,R=3$, and $S(\rho )=1-\mid 2\rho -1\mid$) but different $N$ values. The definition of the parameters can be found in equations (2) and (3). $P$ is the total scanning power range (in D), which controls the overall DoF range, $M$ determines the number of cycles along the total design radius $R$ (im mm), $N$ is the exponent and $S(\rho )$ is a function to define the shape of the power profile. (b) Simulated through-focus area under the modulation transfer function (areaMTF) of the profile with $N=1$ under a series of decentration values (0, 0.25, 0.5, 0.75 and 1 mm). (c) Simulated through-focus areaMTF of the profile with $N=2$ under the decentration values. (d) Simulated through-focus areaMTF of the profile with $N=3$ under the decentration values. (e) Standard deviation of the areaMTF across the decentration values. The pupil diameter was set as 4 mm.

Data availability

Data underlying the results presented in this paper may be obtained from the authors upon reasonable request.