Purpose: Our goal was to validate the accuracy, repeatability, sensitivity, and dynamic range of a Hartmann-Moiré (HM) wavefront sensor (PixelOptics, Inc.) designed for ophthalmic applications. Methods: Testing apparatus injected a 4 mm diameter monochromatic (532 nm) beam of light into the wavefront sensor for measurment. Controlled amounts of defocus and astigmatism were introduced into the beam with calibrated spherical (-20D to + 18D) and cylindrical (-8D to + 8D) lenses. Repeatability was assessed with three repeated measurements within a 2-minute period. Results: Correlation coefficients between mean wavefront measurements (n = 3) and expected wavefront vergence for both sphere and cylinder lenses were >0.999. For spherical lenses, the sensor was accurate to within 0.1D over the range from -20D to + 18D. For cylindrical lenses, the sensor was accurate to within 0.1D over the range from -8D to + 8D. The primary limitation to demonstrating an even larger dynamic range was the increasingly critical requirements for optical alignment. Sensitivity to small changes of vergence was constant over the instrument′s full dynamic range. Repeatability of measurements for fixed condition was within 0.01D. Conclusion: The Hartmann-Moiré wavefront sensor measures defocus and astigmatism accurately and repeatedly with good sensitivity over a large dynamic range required for ophthalmic applications.
© 2009 Optical Society of America
Wavefront sensing technology has many applications in vision science where objective measurements of ocular wavefront aberrations of human eyes are required. For example, the Shack-Hartmann (SH) wavefront sensor introduced by Liang  has been used in numerous laboratory studies of the aberrations of normal eyes [2–5] and corneal topography . Furthermore, this technique has also been widely used to measure the aberrations of abnormal eyes in cases of myopia [7–9], keratoconus [10–13], refractive surgery [8,14–17], contact lens [18,19], intraocular lens , and peripheral vision [20,21]. However, the SH sensor lacks the necessary dynamic range  to measure accommodating or highly aberrated eyes encountered in some applications (e.g., keratoconus [23,24] and peripheral vision ) without an additional focusing mechanism.
Recently, a new wavefront sensing technology called the Hartmann-Moiré (HM) wavefront sensor  (PixelOptics, Inc.) was developed for applications requiring a large dynamic range with good sensitivity. It consists of two high density Hartmann screens (pinhole array ) separated by a precisely calibrated distance. The first screen serves as a standard Hartmann screen that samples the wavefront to form a distorted spot pattern that encodes the wavefront gradient. Since the Hartmann screen lacks the lenslets found in a SH wavefront sensor , the spot pattern is available for analysis at a series of discrete planes over a relatively large range of axial distances. A large dynamic range is achieved by selecting a relatively small axial distance between the sampling plane and the analysis plane. However, the penalty for choosing a short axial distance is low sensitivity since the lateral motion of the spots is proportional to axial propagation distance for a wavefront of fixed slope. To recover adequate sensitivity, a second Hartmann screen rotated relative to the first screen intercepts the spot pattern at the analysis plane to produce a moiré deflectogram (Fig. 1), from which wavefront phase can be retrieved by various methods [28,29]. The purpose of our study was to test the dynamic range and sensitivity of this wavefront sensor for measuring defocus and astigmatism.
The single-pass test-bench shown in Fig. 2 was used to inject test wavefronts with known aberrations into the HM wavefront sensor. A laser beam (532 nm) collimated with a shear-plate interferometer passed through a circular aperture (4mm-diameter) that served as the entrance pupil (EP) of the system. The EP plane was optically conjugated to the first Hartmann screen via a pair of relay lenses (T1-T2). The second Hartmann screen was located downstream and rotated relative to the first Hartmann screen. The density of both screens was 500 holes per inch. The resulting moiré spot pattern was recorded by an 8-bit B/W CCD camera focused on the plane of the second Hartmann screen. From the recorded moiré deflectograms, the ‘peaks method’  was used to determine the spherical and cylindrical powers of the tested trial lens.
Wavefront aberrations were introduced into the light beam with Topcon ophthalmic (‘corrected curve’) trial lenses designed to minimize spherical aberrations. The lens set contained 77 spherical lenses ranging from -20 D to + 18 D and 16 cylindrical trial lenses ranging from -8 D to + 8 D. We calibrated all of the trial lenses with a clinical lensometer. The lenses were placed adjacent to the entrance aperture and thus were conjugate to the first Hartmann screen.
A selected trial lens was placed at the entrance aperture of the HM wavefront sensor and a series of three measurements were taken within two minutes without any adjustments to the measuring system. To determine dynamic range, this procedure was repeated for all of the test lenses in the set described above without changing anything else in the system. Since the sensor was not realigned to the beam between measurements, the standard deviation of the three repeated measurements indicates the level of camera noise and laser diode fluctuation during the experiment.
To determine sensitivity to small changes in wavefront aberrations, a large offset of defocus (-20D, -10D, or + 10D) was introduced into the beam with the corresponding spherical trial lens. For each of these offset values, a series of small changes in defocus was added by introducing an additional lens ranging from -0.5 to + 0.5 D in 1/8 D steps.
Verification of the large dynamic range expected from the HM wavefront sensor is shown in Fig. 3. The wavefront defocus measured by the sensor varied linearly with trial lens power over the entire range from -20 D to + 18 D. The correlation coefficient between the two sets of the measurements was greater than 0.999. The discrepancy between HM wavefront sensor measurement (mean of the three consecutive measurements) and lens power was less than 0.1 D in all cases and the mean absolute discrepancy was 0.03D. The standard deviation of the three consecutive measurements was less than 0.007D in every case, which indicates good repeatability of the HM wavefront sensor during the experiment.
To confirm that the large dynamic range shown in Fig. 3 is not confined to defocus, the experiment was repeated for 16 cylindrical lenses ranging from -8 D to + 8 D as test cases. As shown in Fig. 4, the correlation coefficient between the HM wavefront sensor measurements and lens power was greater than 0.999 and the mean absolute discrepancy was smaller than 0.07D. The maximum absolute difference was 0.17 D at the -7D tested cylindrical lens. This agreement confirmed that the HM wavefront sensor is also capable of measuring astigmatism over a large dynamic range.
Sensitivity to small changes in spherical defocus was confirmed initially by the data in Fig. 3. The mean of the absolute discrepancy between measured and expected defocus for eight spherical trial lenses ranging from -0.5 D to 0.5D was 0.014D. When one outlier (the 0.25D lens) was excluded, the mean of the absolute difference was 0.0085D. This indicates that the HM wavefront sensor has good sensitivity for small levels of defocus of a collimated beam. Additional results obtained for small changes in defocus around a mean value of -20 D, -10 D and + 10 D of defocus are shown in Fig. 5. In every case the correlation coefficients of the linear regression were larger than 0.999 and the data were well fit by a line, the slope of which is close to one. This indicates that the HM wavefront sensor has good sensitivity over its full dynamic range.
Our experimental results confirmed that the HM wavefront sensor is accurate, repeatable, and sensitive to small changes of focus over the tested dynamic range from -20 D to + 18 D. For a 4mm pupil, -20 D defocus is equivalent to 12 micrometers of RMS wavefront error. This dynamic range is far greater than for SH wavefront sensors . To achieve a useful dynamic range, the typical ophthalmic wavefront aberrometer employs a focusing mechanism (e.g. Badal system ) that removes the bulk of the defocus before the wavefront enters the wavefront sensor. By contrast, the large dynamic range reported above is an inherent property of the HM wavefront sensor itself since no auxiliary focusing mechanism was employed in our tests. A similar result was demonstrated also for astigmatism, which confirms that the large dynamic range of the HM sensor is not specific to defocus. In this study to verify the instrument's dynamic range, we chose to use the ‘Peaks method’  for data analysis because it was adequate to obtain the sphere and cylindrical powers of the test cases. However, we anticipate that the dynamic range will also be large for the higher-order aberrations if a full Zernike expansion of the wavefront slope data is performed . Validation of that expectation is left for future study.
A wavefront sensor with large dynamic range but poor sensitivity would lack the ability to distinguish relatively small wavefront errors due to astigmatism or higher-order aberrations in the presence of very large levels of defocus associated with myopia or hyperopia. Unlike the traditional SH wavefront sensor, for which sensitivity and dynamic range are inversely related [30,31], the HM sensor maintained good sensitivity in the presence of large amounts of defocus (Fig. 5).
The source of the small discrepancies between the HM sensor measurements and the corresponding lens power is not necessarily a limitation of the wavefront sensor. Many factors might have contributed to this discrepancy. Firstly, the calibration of the trial lenses was limited by the resolution of our lensometer (0.1D). Secondly, although several alignment indicators and techniques (e.g. alignment ring, precise translation table, and feedback from the HM sensor) were adopted to ensure the accurate alignment of the test beam to the sensor, this experiment is still subject to alignment errors that grow larger as the power of the trial lens increases. For example, a 250 microns axial shift of a -20D spherical trial lens corresponds to 0.1 D of focusing error. Lastly the difference in entrance pupil diameters between the HM sensor and the lensometer may have contributed to the experimental discrepancy. Nevertheless, in spite of these many potential sources of error, the discrepancy reported in the experiment is still far smaller than the level of clinical significance (0.1 D) which suggests the accuracy of the HM is greater than we were able to measure.
Laser safety is essential when developing clinical instrumentation. Since the HM wavefront sensor consists of two Hartmann screens, its light efficiency is lower than the SH wavefront sensor. Nevertheless, with the optimal configuration a HM wavefront sensor is efficient enough to detect the wavefront aberrations of human eyes according to our preliminary experience with human subjects.
In summary, we have demonstrated experimentally that the Hartmann-Moiré wavefront sensor has a large inherent dynamic range that is especially well suited for the clinically abnormal, highly-aberrated eye. The sensor also maintains good sensitivity over its full dynamic range. Remaining issues relevant to the double-pass configuration required for a clinical aberrometer such as alignment repeatability, decrease in contrast , and light efficiency of the HM wavefront sensor remain to be investigated. Other potential ophthalmic applications for a wavefront sensor with large dynamic range include corneal topography  and the measurement of contact and intraocular lenses.
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