Fluctuations in accommodation have been shown to be correlated in the two eyes of the same subject. However, the dynamic correlation of higher-order aberrations in the frequency domain has not been studied previously. A binocular Shack-Hartmann wavefront sensor is used to measure the ocular wavefront aberrations concurrently in both eyes of six subjects at a sampling rate of 20.5 Hz. Coherence function analysis shows that the inter-ocular correlation between aberrations depends on subject, Zernike mode and frequency. For each subject, the coherence values are generally low across the resolvable frequency range (mean 0.11), indicating poor dynamic correlation between the aberrations of the two eyes. Further analysis showed that phase consistency dominates the coherence values. Monocular and binocular viewing conditions showed similar power spectral density functions.
© 2008 Optical Society of America
When a subject fixates steadily on a target, their focus is not static but displays small temporal instabilities about a mean level of accommodation. This is known as the microfluctuations of accommodation. These fluctuations have been found to have an amplitude of a few tenths of a dioptre and a frequency spectrum extending out to a few Hertz, see for example [1, 2]. These fluctuations can be represented by two dominant regions of activity in the frequency domain, namely the low frequency component (LFC < 0.6 Hz) and the high frequency component (1 ≤ HFC ≤ 2.3 Hz). In addition, aberrations beyond defocus, although comprised of much smaller magnitudes, have also been shown to exhibit dynamic behavior with frequency components as high as 70 Hz [3, 4].
Some of the potential sources of the ocular wavefront aberration dynamics, for example the physiological fluctuations in pupil size  and the ocular microtremor , have been shown to be correlated between the two eyes. As a result, the dynamics of the ocular wavefront aberrations between the two eyes are expected to be correlated, especially at the LFC of accommodation microfluctuations since it is considered to be under neurological control . A few studies with binocular infrared optometers that allow simultaneous measurement of steady-state accommodation in both eyes suggest significant correlation between the defocus term in the right and left eyes of the same subject [8, 9]. However, these conclusions are based on qualitative analysis of the time traces of focus variations during steady-state accommodation. Comparison of the coefficients of higher-order aberrations of the whole eye have also revealed mirror symmetry between the Zernike modes in both eyes, i.e. modes that are symmetrical about the vertical axis have similar magnitudes and signs, while modes that are asymmetrical about the vertical axis have similar magnitudes but are opposite in sign [10–16]. Aforementioned studies compared the static monocular measurements of their subjects. Currently, little if anything is known about the dynamic correlation of focus, as well as higher-order aberrations, in the frequency domain.
In natural viewing conditions, the world is seen with both eyes. It is therefore important for the measurement of ocular wavefront aberrations to be performed under binocular viewing conditions to imitate the actual visual process that takes place in real life. The choice of monocular versus binocular viewing in experimental studies has been an issue of debate over many years. This is especially of concern since pupil diameter has been shown to increase from binocular to monocular viewing [17–19]. It is well established that root-mean-square (rms) wavefront errors of the higher-order aberrations increase with larger pupil diameters [10, 13, 14, 20–23]. Campbell first questioned the use of monocular optometers where he suggested that the amplitudes of accommodation microfluctuations could be smaller under binocular conditions, possibly due to the presence of the convergence-fixation reflex . Since then, binocular optometers have been produced to simultaneously measure the focus changes in two eyes [8, 9, 24–26]. Recently, Seidel et al. observed a trend for greater fluctuations in accommodation among late-onset myopes as compared to emmetropes and early-onset myopes under monocular viewing conditions, although this did not reach statistical significance . When the targets were seen binocularly, however, no difference was found between the three refractive groups. It is unclear whether the amplitudes of the higher-order aberrations vary depending on monocular or binocular viewing. This issue is important during refractive surgery because the pre-operative measurement of the ocular aberrations conducted monocularly may not reveal the actual imperfections of the eye under binocular viewing. If there is a significant difference between the monocular and binocular conditions, subsequent correction of the higher-order aberrations may not be precise enough to achieve best-corrected vision during daily life.
Studies have shown that aberration fluctuations can help to guide the accuracy of closed-loop accommodation responses [28, 29]. Since accommodation is controlled centrally, resulting in a concerted oculomotor response, highly correlated higher-order aberration dynamics may provide a more robust cue than uncorrelated aberration dynamics. Measurement of aberration correlations under binocular conditions may be a useful tool for the clinical evaluation of patients with accommodative dysfunction, e.g. lag of accommodation seen in progressive myopes . In the present study, a real-time binocular Shack-Hartmann sensor was constructed to allow the simultaneous measurement of ocular aberrations in both eyes. The high accuracy and repeatability of this type of sensor in the measurements of the ocular aberrations have been demonstrated in a number of studies [10, 31, 32], which makes it the device of choice for this study. We investigated the dynamic correlations of ocular aberrations in the right and left eyes of six subjects.
An infrared laser diode with central wavelength of 785 nm is used to create a point source on each retina. The output power of the laser source is controlled by aperture A 1. The average power of the laser at the corneae is 40 µW which is several orders of magnitude less than the maximum permissible exposure for up to 8 hours of continuous viewing at this wavelength . One drawback with the use of a coherent laser source is the manifestation of laser speckle that affects the measurement of Shack-Hartmann spot centroids . A rotating diffuser is therefore introduced at the focal point of lens L 1, i.e. at a plane conjugate to the retina, to introduce random phase variations. The diffuser consists of a diffusion filter (Super Gel filter 132, Rosco Laboratories, UK) and a motor rotating at 5000 rpm. The laser beam is then re-collimated with L 2.
In order to create a laser beacon on each retina of the two eyes with a single laser source, the laser beam is split into two by pellicle beamsplitter PBS. Since there are two ingoing beams at cube beamsplitter CBS, two undesirable outgoing beams will be produced from the simultaneous reflection-transmission property of the beamsplitter. Black plastic tubes are used to absorb and dissipate these unwanted beams. The ingoing beams are angled such that they are slightly off-axis at the corneae so that back reflections can be removed by careful positioning of aperture A 2 at a plane conjugate to the retinae . The incident beams are 1 mm in diameter at the corneae.
Laser beacons hit the retinae, reflect and propagate out of the eyes where they become aberrated due to optical imperfections. Plane mirrors PM 2 to PM 6 are used to balance the two optical paths so that the path length from both eyes to L 3 is equivalent to 500 mm. The system is designed so that the separation of both outgoing beams at CBS is 6 mm, enabling capture by a single camera, thus reducing cost and system complexity. This also avoids the problem of having to synchronize two cameras to capture the Shack-Hartmann spots at the same time.
Lenses L 3 and L 4 are used to relay the emerging wavefront at the pupils to the Shack-Hartmann wavefront sensor placed in a plane conjugate to the pupils. The ratio of the focal lengths between L 3 and L 4 determine the magnification factor of the pupil on the Shack-Hartmann sensor, which is equivalent to 0.4 for this system. Thus, a 6 mm pupil at the cornea is equivalent to a pupil size of 2.4 mm at the sensor. The sensor has a regular array of square lenslets, each with a focal length of 7mmand a width of 0.2 mm. The pupil is therefore sampled at an interval of 0.5 mm by the lenslet array. Typically, for a 6 mm pupil, there are 112 useable Shack-Hartmann spots. The Shack-Hartmann spots produced by the lenslets are captured by a CCD camera (Retiga Exi Fast 1394, QImaging, Canada) placed at the focal length of the lenslet array. The CCD chip resolution is 1392 × 1040 pixels and each pixel is equal to 6.45 µm × 6.45 µm (i.e. the chip size is 8.98 mm × 6.71 mm). The sampling frequency of the camera is 20.5 Hz.
Mirrors PM 7 to PM 8 are used to redirect the light paths so that all optical components can be fitted onto a 600 mm × 600 mm breadboard.
2.2. Target presentation
To permit open-view observation that stimulates accurate accommodation control, two hot mirrors HM 1 and HM 2 are placed in front of the eyes. These mirrors reflect infrared radiation but transmit visible light, thereby separating the infrared measuring rays from the visible light originating from the target. The distance between the hot mirrors is adjusted to match the inter-pupillary distance of the observer. The target was a Maltese cross presented on a LCD monitor (Sony Trinitron Multiscan 200 GS) placed at 2.7 m from the eyes, subtending 11.32 minutes of arc at the cornea. The luminance of the target was 255 cd/m 2.
2.3. Validation of both channels
Since we are interested in the dynamic changes of the ocular aberrations of the two eyes, a known aberration should induce the same changes in both channels. To demonstrate this, spherical trial lenses of equal power were inserted in the right and left channels to study the resultant change in the measurements registered by both channels. Results obtained are shown in Fig. 2(a). These measurements are significantly correlated with the actual power of trial lenses, where the correlation coefficient, r is 0.999 (p < 0.0001) and 0.997 (p < 0.0001) for the right and left channels, respectively. To study the agreement between the two channels, we used a Bland and Altman plot which is commonly used in clinical studies to compare two different methods of measurement . The Bland and Altman plot can be used to show the level of agreement between two different methods of measurement, and whether they can be used interchangeably. We used the same principle to compare the two channels, where the differences in the defocus term obtained with the two channels is plotted against their mean, as shown in Fig. 2(b). The mean of their differences is -0.005 D indicating negligible systematic bias, and the differences are within the 95% limits of agreement (mean ± 1.96 SD).
Cylindrical lenses of various powers (range +0.75 to -0.75 D) and axes (10 to 170 degrees) were also added to the artificial eyes with known spherical and cylindrical powers. The resultant spherocylindrical changes were compared with the ideal values calculated with power vector analysis , which allows the combination of two refractive components (the eye and the trial lens) that have different axis orientations. The r was found to be 0.997 (p < 0.0001) and 0.998 (p < 0.0001) for M, 0.991 (p < 0.0001) and 0.997 (p < 0.0001) for J 0 and 0.996 (p < 0.0001) and 0.939 (p < 0.002) for J 45, for the right and left channels respectively. These results shows that both channels can be used interchangeably, indicating the equivalence of both channels.
2.4. Experimental procedure
Six healthy subjects with no ocular pathology participated in this study. Their age ranged between 23 and 34 years with a mean age of 27 years. Two subjects were emmetropes, the rest were myopes with refractive errors ranging from -1.25 to -6.00 D with less than 1.50 D of astigmatism. We performed the measurements over the natural pupil size of each subject, with a mean of 5.1 ± 0.7 mm. All subjects had natural pupil sizes greater than 4 mm, hence dilation with mydriatics could be avoided. This is advantageous because previous studies have reported different wavefront aberration measurements when pupil dilations were achieved naturally and pharmacologically [37–39]. subjects wore their spectacles during the experiment. A bite bar was used to stabilize the heads of the subjects. The subjects positioned themselves until both laser beams appeared equally bright in both eyes.
During the experiment, subjects were instructed to remain stationary while maintaining focus on the target. They were allowed to blink naturally. The experiment was first conducted binocularly and then with each eye occluded in turn. For the monocular condition, only the uncovered eye was measured. Each measurement run lasted 24 s. Five repeated measurements were taken for each condition. Following each measurement, subjects removed themselves from the bite bar and were re-aligned before the subsequent measurement.
3. Data analysis
3.1. Wavefront reconstruction
All Zernike coefficients (excluding tip and tilt) up to and including the eighth radial order were calculated and reported based on the standard single indexing scheme recommended by the OSA/VSIA Taskforce . To obtain the Zernike mode number, j, the following equation is used:
where n is the radial order and m is the angular frequency of the standard double indexing scheme.
3.2. Removal of blink artefacts
Blinks lead to abrupt changes in the calculation of Zernike coefficients and hence result in higher magnitude of certain frequencies in the power spectrum analysis [41, 42]. The occurence of blinks was identified from the sharp spikes in the time course of the Zernike prismatic term (i.e. tip) . The time span of a typical blink is roughly 250 ms . According to our sampling rate, a total of 6 data points were deleted from the beginning of each blink. After that, a cubic spline function was used to interpolate the data points before and after the blink. This procedure was repeated for the same points in each Zernike coefficient of the two eyes.
3.3. Coherence function analysis
The coherence function is a valuable tool for the investigation of the synergy between two concurrent time series [33, 45–47]. The main advantage of the coherence function is its ability to reveal the degree of correlation for each individual frequency component of the two signals. the coherence function of two signals is given by
where Gxy is the cross-spectral density (CSD) and Gxx and Gyy are the power-spectral density functions (PSDs) . A detailed explanation of coherence function analysis can be found in Hampson et al. . The coherence function value varies between 0 and 1, where 0 indicates complete lack of correlation and values close to 1 means the two signals are highly correlated at a particular frequency.
Each of the five 24 s measurement signals were divided into two 12 s signals, resulting in 10 data segments in total, hence 20 degrees of freedom. The frequencies that can be reliably resolved ranged from 0.08 to 10.25 Hz. Detrending was applied to each data segment to factor out the influence of sampling time on the PSDs by removing contributions from signals with periods longer than the segment length . A Hanning window was then used to minimize the bias due to spectral leakage. We used the method proposed by Wang and Tang to obtain the 95% confidence intervals for the coherence function . Modeling with twice the amount of the actual pupil movements shows that synchronized pupil translation due to instabilities in head position has minimal effect on the coherence values .
The wavefront map for the right and left eyes of subjects YP (5 mm pupils) and JC (6 mm pupils) are shown in Fig. 3. The wavefront maps are different for each subject. The rms wavefront errors were found to be similar in the two eyes of the same subject, comparable to the values found in the study by Marcos and Burns .
Figure 4 plots the correlation between the Zernike coefficients (up to and including the fifth radial order) of the right and left eyes for four subjects. To avoid the confounding effect of dissimilar refractive errors on the absolute aberration measured, we disregard subjects CS and CV from this part of the analysis (however, this does not affect the coherence values due to the normalization of the CSD by the PSDs). The sign of Zernike modes with odd symmetry about the y-axis in the right eye have been inverted to allow for the comparison of mirror symmetry . Correlation coefficients for each subject vary from -0.22 to 0.94. Two subjects (YP and EM) show high degree of mirror symmetry, where the correlation of the Zernike coefficients between both eyes is statistically significant (p < 0.001). The solid black line represents a linear fit to all the data for the four subjects (r = 0.39).
Power spectral density (PSD) functions of the rms wavefront error of the right and left eyes are very similar within the same subject. The mean slope for the right and left eyes of the six subjects is 1.14 ± 0.21 (mean ± 1 SD). Figure 5 shows the typical PSDs for subjects KH and EM.
Figure 6 plots the coherence of the rms wavefront error for all subjects. Generally, the coherence values are fairly low, with the mean across frequency and subject equal to 0.11±0.02 (mean±1 SD).
Figure 7 shows the coherence between each individual aberration of the two eyes of subject EM. The plots show minor differences for each individual subject but all of them display low coherence values (typically <0.1), except for the LFC region of defocus, where the mean coherence value is equal to 0.43. These plots vary for each individual.
The mean coherence values across subjects for each Zernike mode in the LFC region and across the resolvable frequency range are plotted in Fig. 8. The plot for HFC region is very similar to the one for the resolvable frequency range, hence it is omitted for clarity. In general, for most Zernike coefficients, the mean coherence values are slightly higher in the LFC region (mean 0.13) than the HFC region (mean 0.11) and the measurable frequency range (mean 0.11). Defocus (Z4) shows the highest coherence between both eyes (coherence value 0.30 for LFC region, 0.15 for HFC region and 0.13 across the resolvable frequency range).
PSDs of the accommodation fluctuations (specified by defocus Z4) for subject KH under monocular and binocular viewing conditions are shown in Fig. 9(a) and 9(c). PSDs of the higher-order aberrations (radial order third and above) are plotted in Fig. 9(b) and 9(d). There was no significant difference between the two viewing conditions for all six subjects in terms of accommodation microfluctuations and the higher-order aberration fluctuations, both at the LFC region and across the measurable frequency range (paired t-test, p > 0.05).
Ocular wavefront aberrations display mirror symmetry between the Zernike coefficients of both eyes in two out of four subjects. This is consistent with previous studies where mirror symmetry has been found in some but not all subjects [11, 12].
As shown in Fig. 5, the magnitude of the PSDs of the rms wavefront errors were very similar between the two eyes of each subject. The mean slope value is in agreement with the results of previous studies [3, 50]. Based on this similarity, one might expect the aberration dynamics to be highly coherent in both eyes. However, coherence function analysis showed that the coherence of the rms wavefront errors were generally quite weak for all subjects (Fig. 6). In order to achieve high coherence values, the signals from the two eyes need to have constant phase difference and also fixed amplitude ratio between the data segments. If either the phase difference or amplitude ratio or both factors change over time, the resultant coherence value will be low. In contrast, the PSDs are based on the average powers of the ten data segments, which is less sensitive to the variation in phase and/or amplitude ratio. Hence the use of PSDs to interpret the coherence of the aberrations between the two eyes can be misleading. This demonstrates that coherence function analysis is a more robust quantitative tool for the study of the correlation between two concomitant signals.
Phase consistency is important because a high phase correlation may be indicative of a central origin for aberration dynamics. In order to investigate the contribution of phase variations to the coherence values, we calculated the phase consistency . The spectrum S(f,t) of a signal s(t) is given by
where |S(f,t)| is the amplitude and φ(f,t) is the phase value. The coherence function from Eq. (2) can be rewritten as
where N is the number of segments. The numerator of the coherence function consists of amplitude and phase information. In order to study the impact of phase alone on the coherence values, the contribution of the amplitudes are removed by setting it to a constant. Coherence from Eq. (4) becomes
where φX,n(f,t) and φY,n(f,t) represent the phases of signal X and Y, respectively. This is known as phase consistency . Figure 10 shows the coherence values for the rms wavefront errors between the two eyes for two subjects based on the coherence function and phase consistency calculations.
Paired t-test shows that for each observer, there is no significant difference between the coherence function and phase consistency (p > 0.05), indicating the coherence values are dominated by phase.
RMS wavefront error represents the effect of a combination of all the aberration modes. This method of data representation might mask an individual aberration which could have shown a high coherence value. To avoid the masking effect, coherence values for each individual Zernike mode between the two eyes of subject EM are plotted in Fig. 7. The overall coherence is fairly weak for all aberrations and none of these Zernike modes displays high coherence, apart from defocus in the low frequency region.
In our previous publication, we have shown that the time-course records of the accommodation microfluctuations were qualitatively similar in the two eyes . This agrees with previous studies [8, 9]. This study further extended the work by looking at the dynamic correlation of defocus as well as higher-order aberrations. Averaging the coherence values across frequencies and subjects for Zernike modes up to and including the eighth radial order shows that defocus (Z4) displays the highest coherence among all the aberrations, which is greater in the LFC region than across the measurable frequency range (Fig. 8). Paired t-test of the average coherence value across the measurable frequency range shows that out of the 41 Zernike modes (up to and including the eighth radial order), there is no significant difference between the coherence function and phase consistency for 36 modes. For the other five modes (namely Z4, Z5, Z7, Z16 and Z24), phase consistency is significantly lower than the coherence function (p < 0.05). From this, we deduce that it is the phase consistency that dominates the coherence values.
Using a Shack-Hartmann sensor it has been shown that the LFC and HFC of the microfluctuations in higher-order aberrations as well as defocus show some correlation with components of the cardiopulmonary system [46, 52]. The LFC is related to heart rate variability and the HFC is related to the average heart rate. Hence one may expect the coherence values for the aberrations between the two eyes to be similar to those obtained in the aforementioned studies. The values obtained in this study for the LFC and HFC regions are consistent with the correlations found in the study by Hampson and colleagues . Zhu et al., however, found that by averaging across subjects, the average coherence values between Zernike modes (from second to fourth radial orders) and pulse rate was 0.51, and 0.55 for instantaneous heart rate . The discrepancies between the two studies warrant further discussion.
In the Zhu et al. study, they note that their coherence function analysis has low reliability given that for each aberration they only used a data record of 128 measurements. In contrast, over 1500 data points were used by Hampson et al., and 2400 data points in this study for data analysis. We found that by calculating the coherence function based on less data segments and hence less data points considerably raises the coherence values. However, we believe it is necessary to use more data segments to reduce the variance of the estimate. A further difference between the two studies is that the coherence function values in the Zhu et al. study were an average of normal and controlled breathing condition, whereas Hampson et al. and our studies used normal breathing.
It is known that the fluctuation of the tear film thickness affects the aberration dynamics especially after a blink [53–56]. Although no study has investigated the variation in the tear break-up of the two eyes, it is highly unlikely for the tear film to break up with the same pattern in both eyes of the same subject. Therefore, we suspect this inter-ocular dissimilarity might lower the dynamic coherence between the eyes.
The absence of peaks in the HFC in the PSDs of accommodation microfluctuations as shown in Fig. 9(a) and 9(c), can potentially be explained by several factors. It should be noted that most of the earlier studies used an infrared optometer to measure accommodation changes, which has a different measurement principle to the Shack-Hartmann sensor. Unlike the Shack-Hartmann sensor, it cannot distinguish the difference between Zernike modes and may have misinterpreted the fluctuations in other aberrations as the microfluctuations in the defocus term. Also, the magnitudes of the HFC have been shown to increase with target vergence, for example it has been found to increase by 22 times when he stimulus is moved from infinity to 25 cm from the subject . Since our subjects fixated at a target vergence of 0.37 D, these powers might be low, hence the absence of an obvious peak at the the HFC. Hofer et al. claim that the peak at the HFC was only detected occasionally in their subjects . Careful inspection of the PSDs of accommodation microfluctuations of our subjects shows the presence of a HFC peak in some but not all of the data segments. Therefore, averaging across segments would probably have averaged this peak out. The variation of HFC with pupil size is controversial, where one study indicated the disappearance of HFC with reducing pupil size  and another found an increased in the HFC as pupil size decreased . The authors of the later study therefore suggest that HFC may be the result of instability in the accommodation system, although the reason behind this is currently unknown.
There is no significant difference between the PSDs of the rms wavefront errors under monocular and binocular viewing conditions for all subjects. This is in contrast to the prediction made by Campbell  where he suggested smaller fluctuations may be observed when the subject viewed with both eyes due to the presence of the convergence-fixation reflex. In a study carried out by Seidel et al., late-onset myopes demonstrated reduced fluctuations in accommodation under binocular observations, but the results were not significant . Our study supports their findings since no difference was found in the magnitude of the PSDs of the accommodation microfluctuations and higher-order aberrations for all six subjects.
6. Future work
The effect of tear film break-up can be eliminated by fitting subjects with scleral contact lenses with dry front surfaces. These lenses will preserve a layer of tear film in between the lens and cornea, hence the blink reflex can be avoided. This will remove the effect of uneven tear film dynamics on the coherence values.
We wish to study the effect of adaptive optics correction of higher-order aberrations on steady-state and dynamic accommodation accuracy. The trend for increased microfluctuations under monocular viewing condition observed by Seidel and colleagues suggest that higher-order aberrations may play a lesser role in accommodation accuracy under binocular conditions. This work will be applicable to the practical study of accommodation responses in individuals at risk of myopia progression, e.g. clinical microscopists . Differentiating the usefulness of aberration dynamics as a guide to accommodation response accuracy under monocular compared to binocular conditions is therefore of significant value.
This study investigated the dynamic correlation between the ocular wavefront aberrations of the two eyes with a binocular Shack-Hartmann wavefront sensor. Coherence function analysis showed that the coherence values vary according to the subject, aberration mode and frequency component. In general, inter-ocular correlations of the aberrations are fairly weak for all subjects. Phase consistency between each data segment dominates most of the coherence values as compared to the effect of amplitude ratio. Monocular and binocular observations result in similar rms wavefront error dynamics.
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