We have used an adaptive optics system to study subjective blur limits of second-order and derived aberrations of the human eye.^{1–3} One aberration was manipulated whereas other aberrations, including higher-order aberrations, were held constant.

Blur limits for defocus were determined when second order astigmatism was corrected in white light.^{2} With a 0.1 logMAR letter size and without correcting higher order aberrations, mean “noticeable” blur limits were ±0.30 diopter (D), ±0.24 D, and ±0.23 D at 3 mm, 4 mm, and 6 mm pupils, respectively. When all monochromatic aberrations were corrected in white light, blur limits were reduced by a (non-significant) 8%, whereas changing the white light to monochromatic light reduced the limits by an additional 12%. Ratios of troublesome to noticeable and of objectionable to noticeable blur limits were 1.9 and 2.7, respectively.

In white light and 5 mm pupils, blur limits for (crossed-cylinder) astigmatism and cylinder were ∼0.9 and 1.5 times those for defocus blur limits but with meridional influence amounting to about 20%.^{1,3} These values are similar to the 1.0 and 1.4 times expected on theoretical grounds.^{4} As for defocus in this and the other study,^{2} correcting higher-order aberrations of the eye produced only minor reductions in these limits. The influences of blur criterion and letter size on blur limits were similar for the three aberration types.

In addition to the second-order aberrations, Atchison et al.^{1} considered blur limits for the third-order Zernike aberration of trefoil for two subjects only. With other ocular aberrations corrected, the ratio of just objectionable to just noticeable blur limits in these two subjects was much higher for trefoil (3.5) than for defocus (2.5) and astigmatism (2.2). This was supported with modeling of the effect of different aberrations on retinal image quality and by studies investigating the influence of aberrations on visual acuity.

The radial dependence of the aberrations may be the origin of the differences in the blur limit ratio. If this is the case, it might be expected that spherical aberration, with fourth-order dependence, may demonstrate higher blur limit ratios than shown for trefoil. However, Zernike spherical aberration has a second-order, defocus, component that may counteract this. In this study, we investigate the blur limits for the higher order aberrations of coma, trefoil, and spherical aberration. As well as Zernike spherical aberration, Seidel spherical aberration has been considered because this does not have a second-order component.

#### METHODS

##### Subjects

There were seven subjects in good ocular and general health. They were part of a bigger group of 14 subjects considered for cylinder blur limits.^{3} Age range was 18 to 55 years. Only right eyes were used. Subjects had ≤0.50 D cylinder by subjective refraction and corrected visual acuities of at least 6/6. Subjects were cyclopleged with 1% cyclopentolate. Pupils were dilated to at least 6 mm.

This experiment was completed for seven subjects (defocus), six subjects (coma), five subjects (trefoil), and six subjects (the spherical aberration conditions). Four subjects completed all parts. Three subjects did not complete one or more of the parts because their settings reached the safe limits for one or more actuators of the deformable mirror: one subject did not complete trefoil, one subject did not complete trefoil and coma, and one subject did not complete the Seidel spherical component of the spherical aberration (his results for both spherical aberration conditions were discounted). Because some subjects reached the safety limits for combinations of larger letters and less sensitive criteria, it can be concluded that the ratios of the higher aberration blur limits compared with those for defocus was underestimated for the group for at least these combinations.

##### Optical System and Procedures

The optical system and its operation has been described in detail elsewhere^{3} and only minimal details are given here. The eye's wavefront aberrations were measured with a super luminescent diode (830 nm, 25 nm FWHM). A deformable mirror (Imagine Eyes HASO 32) was used to correct an eye's aberrations and the system's residual aberrations (astigmatism only or all aberrations apart from defocus) and induce other aberrations. Aberrations were described by coefficients of Zernike polynomials, with 63 Zernike polynomials included to reconstruct the wavefront. A visual stimulus display was provided by an organic LED display and an aperture conjugate with the eye pupil. A collimated infrared laser diode (808 nm) was added to allow closed loop operation. This passed through the measuring system, bypassing the subject's eye and hence not being visible to it, and was used to control the system aberrations (eye's static aberration correction and dynamic aberration generation) with the help of customized deformable mirror control software.

Distinction has to be made between two different “pupils.” Pupil P_{2} was the correction pupil within whom wavefront aberrations of the eye were corrected to target a predesignated wavefront (e.g., an ideal plane wavefront or a spherical wavefront). The imposed extra aberrations that formed the target wavefront to the eye were also generated in this pupil. This pupil was 5.7 mm diameter. Pupil P_{3} was the aberration measurement pupil and it was the same as the pupil used for visual display (5.0 mm).

Subjects were given a control box with a rotating knob (to control mirror shape) and a button. Subjects rotated the knob in clockwise and anticlockwise directions to identify different blur criterion limits (just noticeable blur, just troublesome blur, and just objectionable blur), and pressed the button for each limit. It was permissible to rotate the knob backward and forward to help determine limits. The customized software included a voice function to remind subjects of the direction to turn the knob and which blur limit was being determined.

Visual displays were presented on a black-and-white OLED display (eMargin, 800 × 600, 15 × 15 μm pixel size). Three Optotype letters, randomly selected from 10 letters (D, E, F, H, N, P, R, T, U, V, Z), were displayed with highest contrast black font on white background (∼100 cd/m^{2}). Target detail size was 0.1, 0.35, and 0.6 logMAR and five repetitions were made for each of these.

Subjects were cyclopleged with a drop of 1% cyclopentolate at least 20 min before commencement of the experiment, with an additional drop applied every 90 min. We checked with a hand optometer that subjects had minimal residual accommodation before proceeding with measurements. Instructions were given about determining just noticeable blur, just troublesome, and just objectionable blur limits.

Repeated measures analyses of variance were conducted on blur limits with subjects as repeated measures, and with adaptive optics condition, meridian, blur criterion, blur direction, and letter size as within-subject factors. Greenhouse-Geisser correction was used for F-tests where Mauchly test of sphericity was significant for the within-subjects factors.

##### The Aberrations

The required aberrations were combinations of Zernike aberrations. These were produced by custom software as described by Guo and Atchison.^{3}

##### Defocus

The form of defocus as a wave aberration is

where (ρ, θ) are polar co-ordinates in the pupil, Z_{2}^{0} is the Zernike defocus polynomial and C_{2}^{0} is its coefficient. The wave aberration can be converted into a longitudinal aberration as a mean sphere (or spherical equivalent)

where R is the pupil semi-diameter.

The step size was 0.05 μm.

##### Coma

There are two Zernike aberration polynomials for coma: vertical coma Z_{3}^{−1} with coefficient C_{3}^{−1} and horizontal coma Z_{3}^{1} with coefficient C_{3}^{1}. We considered the Zernike aberration coefficients in magnitude and axis format^{5} using

Here, n = 3, m = 1, and so we have

The Zernike coma aberrations are combined in the wave aberration

Substituting C_{3}^{−1} for C_{3}^{1} tan α_{31} in Eq. 4 into Eq. 5 gives

Equation 4 Image Tools |
Equation 5 Image Tools |

with C_{3}^{1} = A_{coma}, C_{3}^{−1} = A_{coma} tan α_{31} for 0 ≤ α_{31} < π, α_{31}≠π/2. For the special case of π/2 radians, it is necessary to set C_{3}^{−1} = A_{coma}, C_{3}^{1} = 0.

In experiments, α_{31} was set to 0, 45, 90, and 135° and A_{coma} was adjusted positively and negatively by rotating the control knob anticlockwise and clockwise, respectively.

Introducing coma induced tilt in the system, partly because of the difference between the two pupil sizes P_{2} and P_{3}, but probably mainly because of coupling using the Zernike modal wavefront construction method. The problem with this is that the target appears to move, thus, resulting in eye rotation which in turn misaligns the pupil and the line of sight with the measuring system. Thus, a compensatory tilt was included to prevent apparent movement of the target. The appropriate compensation was done in a trial-and-error method. For a given subject, we would run a sequence of mirror settings for coma and notice the tilt that appeared (Fig. 1, left). We adjusted the compensatory tilt in the software and repeated the sequence of mirror settings. This process was repeated until tilt was small (Fig. 1, right). The precompensations varied between subjects and depended on individual aberrations.

The step size A_{coma}was 0.05 μm for four subjects and 0.075 μm for a fifth subject.

##### Trefoil

The development for trefoil follows that for coma. There are two Zernike aberration polynomials for trefoil: oblique trefoil Z_{3}^{−3} with coefficient C_{3}^{−3} and horizontal trefoil Z_{3}^{3} with coefficient C_{3}^{3}. In magnitude and axis format (Eq. 3) with n = 3, m = 3, we have

The Zernike trefoil aberrations are combined in the wave aberration

Substituting C_{3}^{−3} for C_{3}^{3} tan 3α_{33} in Eq. 7 into Eq. 8 gives

Equation 7 Image Tools |
Equation 8 Image Tools |

with C_{3}^{3} = A_{trf}, C_{3}^{−3} = A_{trf}tan 3α_{33} for 0 ≤ α_{33}<π/3, α_{31} ≠ π/6. For the special case of π/6 radians, it is necessary to set C_{3}^{−3} = A_{trf}, C_{3}^{3} = 0.

Measurements were made in clockwise and anticlockwise directions for meridians 0°/60°, 15°/75°, 30°/90°, and 45°/105°. The step size A_{trf} was 0.05 μm for five subjects and 0.075 μm for one subject.

##### Zernike Spherical Aberration

This is given by

where the Zernike spherical aberration polynomial is give by

with coefficient C_{4}^{0}. Theoretically, reducing pupil size from 5.7 mm to 5.0 mm reduces the Zernike spherical aberration coefficient by 41%. At the same time, defocus with a coefficient −0.68 times the original spherical aberration coefficient is produced. Theoretically, the addition of defocus for the 5.7 mm pupil with a coefficient +0.89 times that of the original spherical aberration coefficient will compensate for the induced defocus for a 5 mm pupil. As for coma where we eliminated tilt, in practice this did not work well, and we used a trial-and-error method to determine an appropriate defocus precompensation for each subject (Fig. 2). As mentioned for coma above, this is probably a coupling issue. Precompensating for defocus induced some secondary spherical aberration, amounting to 16% of the primary spherical aberration in a subject with a high level of spherical aberration (Fig. 2, right). Of the aberrations requiring precompensation (coma and the two spherical aberrations), Zernike spherical aberration had the lowest signal-to-noise ratio, e.g., the subject in Fig. 2 had signal-to-noise ratio of 6.4 ± 0.6 dB (Fig. 3) compared with coma (20.1 ± 1.0 dB) and Seidel spherical aberration (17.0 ± 1.1 dB). The step size used was 0.04 μm for five subjects and 0.03 μm for one subject.

Equation (Uncited) Image Tools |
Figure 2 Image Tools |
Figure 3 Image Tools |

##### Primary (Seidel) Spherical Aberration and Longitudinal Spherical Aberration

Primary spherical aberration, sometimes called Seidel spherical aberration in visual optics (although this is not strictly correct) is given as W_{SSA}(ρ,θ) = W_{4,0}ρ^{4}, where W_{4,0} is the coefficient. This can be constructed from Zernike spherical aberration, piston and defocus as

Because Z_{4}^{0} = √5(6ρ^{4}−6ρ^{2}+1), Z_{2}^{0} = √3(2ρ^{2}−1), Z_{0}^{0} = 1, we can eliminate the zero-order and second-order powers in ρ by setting

and

to give

An alternative way of expressing the Seidel spherical aberration is as

A meaningful comparison between the influences of primary spherical aberration and Zernike spherical aberration is to divide the coefficient W_{4,0} by 6√5. We refer to W_{4,0}

as the modified Seidel spherical aberration coefficient. The root mean square (RMS) aberration contributed by Seidel spherical aberration is

A comparison with other aberrations on the basis of rms is thus given by using 2W_{4,0}/3√5).

Either the Zernike or primary spherical aberration can be used to determine longitudinal spherical aberration (LSA). By using

where r is the distance from the pupil center given by Rρ, we can show that the difference between the edge and center of the pupil is

As for Zernike spherical aberration, precompensatory defocus was also used. The Seidel spherical aberration step size was 0.20 μm for five subjects and 0.25 μm for one subject.

#### RESULTS

Table 1 includes results for the experiment. Each of the coma, trefoil, and spherical aberration sessions included measures for defocus, which are also included.

Table 1 Image Tools |
Table 1 Image Tools |

##### Coma

Fig. 4 shows results for coma and defocus. For coma, the magnitudes are values of C_{31} and α_{31} as in Eq. 4. For defocus, magnitudes are averages of positive and negative limits. Mean blur limits and their 95% confidence limits were similar for coma and defocus (second and third columns of Table 1). Meridian did not have a significant effect on the subject group blur limits for coma (fourth column). The influence of blur limit criterion was much greater for coma than for defocus (fifth and sixth columns). The influence of letter size on blur limits was similar for coma and defocus (seventh and eighth columns; For coma, the influence appeared to be dependent on blur criterion although the interaction between blur criterion and letter size was not quite significant (F_{1.0,5.1} = 4.0, p = 0.10). There was a considerable range of sensitivity between subjects, with ratios of blur limits of the subjects (blur limits of subjects divided by limits for the most sensitive subject) showing much more variability between subjects for coma than for defocus (ninth and tenth columns).

##### Trefoil

Fig. 5 shows results for trefoil and defocus. For defocus, magnitudes are averages of positive and negative limits. For trefoil, the magnitudes and meridians are values of C_{33} and α_{33} as in Eq. 7. Mean blur limits and their 95% confidence limits were similar for trefoil and defocus (second and third columns of Table 1). Meridian did not have a significant effect on the subject group blur limits for trefoil (fourth column). The influence of blur limit criterion was much greater for trefoil than for defocus (fifth and sixth columns). The influence of letter size on blur limits was slightly less for trefoil than for defocus (seventh and eighth columns); for trefoil, the influence was dependent on blur criterion (F_{1.5,5.9} = 40.5, p < 0.001). There was similar variability between subjects for both trefoil and defocus (ninth and tenth columns).

##### Spherical Aberration

Figs. 6 and 7 show results for defocus, Zernike spherical aberration, and Seidel spherical aberration. Fig. 6 shows wave aberration coefficients, with Seidel spherical aberration given as the modified coefficient (from text after Eq. 14).

Equation 14 Image Tools |
Figure 6 Image Tools |
Figure 7 Image Tools |

The overall mean blur limits for Zernike spherical aberration were 0.5 times those for defocus and the limits for Seidel spherical aberration were 0.4 times (modified coefficient) and 1.7 times (RMS aberration) those for defocus (second and third columns of Table 1). The influence of blur limit criterion was greater for both spherical aberrations than for defocus (fifth and sixth columns). The influence of letter size on blur limits was similar for the spherical aberrations and defocus (seventh and eighth columns). The variability between subjects was higher for the spherical aberrations than for defocus, being three times greater for Seidel spherical aberration than for defocus (ninth and tenth columns).

Fig. 7 shows longitudinal aberrations. For both Zernike and Seidel aberrations, this is the difference between the pupil center and pupil edge LSAs. The mean Zernike and Seidel aberration blur limits were 4.5 and 3.3 times higher, respectively, than those for defocus.

#### DISCUSSION

When expressed as wave aberration coefficients, the just noticeable blur limits for coma and trefoil were similar to those for defocus, whereas the just noticeable limits for Zernike spherical aberration and Seidel spherical aberration (the latter given as an “rms equivalent”) were considerably smaller and larger, respectively, than the defocus limits. The limits for the higher aberration terms increased more quickly than those for defocus because the criterion changed to just troublesome and to just objectionable. The influence of letter size was similar, or possibly slightly less, for the higher order aberrations than for defocus. The comparative blur limits for Seidel spherical aberration and defocus depended on the way in which the former was expressed.

In the analyses, limits for aberration types were determined on different subsets of subjects, and this may give an incorrect impression of the relative effects of the blur criteria when comparing different aberrations. Fig. 8 shows the mean ratios of just objectionable to just noticeable blur criteria for 0.1 and 0.6 logMAR letters, both for the four subjects common to all aberrations and for all subjects used for a particular aberration. Although the ratios are sometimes quite different for the four subjects (solid lines) and for larger groups (dotted lines), the differences between 0.1 logMAR (circles) and 0.6 logMAR (squares) letters are generally similar. Ratios for defocus are affected little by letter size, whereas ratios for the higher order aberrations are greater for the larger than for the smaller letters by about 0.6: the relationships between the two blur criteria limits are considerably different for higher order aberrations than for defocus.

In a previous study,^{1} the relationships between the blur criterion limits were compared between trefoil and coma for two subjects. In that study, the ratio of just objectionable to just noticeable blur limits increased from 2.3 for 0.1 logMAR letters to 4.6 for 0.6 logMAR letters. Although confirming the trend, the differences were less marked in this study for five subjects (3.1 for 0.1 logMAR letters to 3.6 for 0.6 logMAR letters). One of the subjects was excluded for trefoil because he was out of range at the largest letter size and the just objectionable criteria. The just objectionable to just noticeable ratio was 2.7 times higher for 0.6 logMAR letters than for 0.1 logMAR letters in the only two meridians that could be assessed, and it is likely that if this subject could have been included that the discrepancies in blur ratios between the two letter sizes would have increased.

In the introduction, we suggested that the dependence of the blur limit ratio on letter size might increase with the radial dependence on relative pupil diameter. If this is the case, then Seidel spherical aberration with only fourth-order dependence on relative pupil diameter might show a more pronounced effect than that of Zernike spherical aberration that has both second-and fourth-order pupil dependence, but this is not obviously the case (Fig. 8).

The wave aberration coefficients are the signed weightings of the contributions of aberration polynomials to the RMS aberrations. The RMS aberration is not a good quality metric. For example, the subjective refraction is better predicted by combining Zernike even-order terms to get a “paraxial” refraction rather than using the second-order terms alone which give the minimum RMS aberration,^{6} and the more meridionally dependent aberrations with a particular coefficient have less effect on visual acuity than less meridionally dependent aberrations with the same coefficient.^{7,8} Applegate et al.^{7,8} found that trefoil had less deleterious effects on visual acuity than coma of the same amount, but this was not apparent for blur limits in our study (compare Figs. 4 and 5).

Rocha et al.^{9} investigated the influence of different aberrations on visual acuity with an adaptive optics system that initially compensated for the aberrations of the subjects' eyes. For 5 mm pupils, individual second-, third-, and fourth-order Zernike rms aberrations of 0.1, 0.3, and 0.9 μm were introduced. At the lower aberration levels, the reduction in visual acuity was similar for defocus, coma, and trefoil (∼0.15 log unit for 0.3 μm), whereas at the highest aberration level, the reduction was greater for defocus (0.64 log unit) than for coma (0.34 log unit) and trefoil (0.23 log unit). Our results for blur limits are broadly in line with these visual acuity results. However, Rocha et al. found similar reduction in visual acuity for Zernike spherical aberration as for defocus at all blur levels, whereas we found more than a two-fold difference (defocus larger) in the blur limits at all criterion-letter size combinations.

To compare the effects of the two spherical aberrations, it is likely that the central part of the pupil is more important to vision than is predicted by the RMS aberration, which emphasizes the peripheral pupil contributions. This suggestion is supported by the second-order coefficients predicting refraction poorly with large pupils compared with a paraxial estimate that emphases the central contribution.^{6} For a particular coefficient value, the central part of the pupil contributes more to the wave aberration variation for Zernike spherical aberration than it does for defocus. However, the central part of the pupil contributes less to the wave aberration variation for the Seidel spherical aberration rms equivalent (Eq. 15) than for the same level of defocus coefficient. Our finding that the coefficient blur limits for Zernike spherical aberration are much smaller than those for Seidel spherical aberration, when the latter is given as a RMS equivalent (Table 1, second column), is in line with a recent finding that the majority of the visual impact of high levels of fourth-order Zernike aberrations is due to the second-order terms within the polynomials.^{10}

There will be both optical and neural factors contributing to the large range of blur limits found in this and our previous studies.^{1–3} As well as the aberrations, the Stiles-Crawford effect (which can be classified as either) is likely to be only a minor contributor.^{11,12} Concerning cortical effects, possibly some people might experience more rapid neural adaptation to changing levels of aberrations, and there is probably a big influence of personality type.^{13}

The experimental conditions were not naturalistic because subjects were cyclopleged. In the natural situation where accommodation, pupil size, and pupil center are variable, aberrations will change over time. This is likely to increase the tolerance to blur, because blur itself will be fluctuating and information can be integrated over time, e.g., a letter R may be difficult to distinguish from a letter P, but as aberrations change the tail of the R may at times become obvious.

Although this study used closed loop adaptive optics to manipulate aberrations, corrections were not applied to changing aberrations (either real changes or due to movement) of eyes. This would be the ideal and it is possible that, had this been able to be achieved, significant effects of full adaptive optics correction might have been found.

#### ACKNOWLEDGMENTS

This work was supported by Carl Zeiss Vision through a collaborative grant to QUT and provided the major elements of the optical system.

David A. Atchison

School of Optometry

Queensland University of Technology

60 Musk Avenue

Kelvin Grove, QLD 4059, Australia

e-mail: d.atchison@qut.edu.au