The Talbot Moir? interferometer has been previously used for measurement of wavefront error. It is based on the use of two Ronchi grids spaced a specific distance apart and the second grid is rotated relative to the first as shown in Figure 1. A monochromatic wavefront propagates through the first grid and is observed just after the second grid. This observed irradiance distribution is called an interferogram. Similar to the Shack-Hartmann wavefront sensor,1 the captured image from the Talbot Moir? wavefront sensor can be processed in the spatial or Fourier transform domain to extract wavefront information. In the spatial domain, the interferogram can be processed to directly extract fringes2 or fringe directions,3 which, in turn, are used to calculate the shape of the incident wavefront. In the Fourier domain,4 specific regions of the spectrum can be isolated to yield the wrapped phase derivatives in the ? and y direction. After unwrapping, the phase derivatives can be used to reconstruct the incident wavefront. These methods lead to a complete description of the incident wavefront.
However, in many cases, only the defocus and astigmatism components of the wavefront are needed. To recover this information, the analysis of the interferogram can be significantly simplified. In this article, we illustrate a Fourier transform domain method of extracting the astigmatic refraction from an incident wavefront by finding the location of two specific peaks in the Fourier transform of the interferogram. The theoretical basis for this recovery is shown and an example from a real eye is given.
MATERIALS AND METHODS
A Ronchi ruling is a transmission grating that consists of opaque parallel bars with a 50% duty cycle. A Ronchi grid is the equivalent of placing two Ronchi rulings at the same plane where the rulings are rotated perpendicular to each other, thus making an array of square apertures. The period of the resulting square apertures is given by the parameter p in both the x and y directions. The dimension of each square aperture is pi 2 × pi 2.
Figure 1. Basic layout for Talbot Moir? wavefront sensor. The first grid is aligned with the ? and y coordinate axes. The second grid is located d downstream from the first grid and is rotated with respect to the ? and y coordinate axes. The camera (not shown) would be positioned to the right of grid g2 and its sensor would be conjugate with grid g2. Typical grid spacing p for use as an ocular wavefront sensor is on the order of 1000 lines per inch.
The corresponding d location is referred to as the Talbot plane. Typically, ? in Eq.(4) is selected to be 1. We now place a second Ronchi grid g2 at the Talbot distance d downstream from gl and we rotate g2. This is illustrated in Figure 1. The superposition of the propagated gl grid (grid gl propagated to the plane of g2) and the g2 grid form a moir? pattern. The combination of the moir? effect with the Talbot distance requirement leads to the name Talbot Moir? interferometer. The primary peaks (ignoring the cosine squared term) for the Fourier transform Gl of the propagated gl grid are shown in the left side of Figure 2 as the solid circles. In Figures 1, 2, and 4 and the following text, Gl represents the Fourier transform of gl and G2 represents the Fourier transform of g2. The peaks for the rotated G2 grid are shown in the left side of Figure 2 as the hollow red circles. The field on the right side of the g2 grid is calculated as a multiplication in the spatial domain or convolution in the Fourier domain.
Two of the Fourier transform peaks from the resulting field are shown on the right side of Figure 2. These are the peaks that also appear in the Fourier transform of the irradiance distribution (the interferogram) that is imaged by the charge coupled device for the wavefront sensor. The captured image (the interferogram) used for processing is conjugate to the plane of grid g2. Note that the angle the peaks make with the spectral axes is half of the rotation angle of grid g2.
EFFECT OF DEFOCUSED INCIDENT WAVEFRONT
Figure 2. For an incident plane wave, we illustrate the main peaks in the Fourier transform domain for the individual grids (left) and for the captured interferogram (right). RO and Rl are locations of spectral peaks due to grid gl (propagated to the plane of g2); QO and Ql are corresponding locations of spectral peaks due to rotated grid g2; and DO and Dl are the locations of spectral peaks due to the convolution of the corresponding R and Q points.
Figure 3. Basic geometry for calculation of the period of the grid at the plane of gl and as it is propagated by a diverging wavefront a distance d downstream from grid gl.
EFFECT OF ASTIGMATIC INCIDENT WAVEFRONT
In the case of an astigmatic incident wavefront where the principal axes are aligned with the axes of the coordinate system, the primary peaks in the Fourier transform of the propagated Gl pattern are arranged at the boundary corners and midpoints of an axis-aligned rectangle. By locating these peaks, we can readily determine the astigmatic wavefront powers using Eq. (5) and the axis of the cylinder is aligned with either the ? or y axis depending on the desired refraction notation (plus or minus cylinder).
Figure 4. Effect of defocuseci incident wavefront on the location of Fourier transform peaks in propagated grid Gl (left) and peaks after convolution by the Fourier transform of G2 (right).
The elements in the M matrix and b vector of Eq. (11) are obtained from the calculated Fourier transform peaks associated with Gl. We arrive at this system of equations by using RO and Rl on the right side of Eq. (7) and RO' and Rl' on the left side, and then rearranging so as to solve for the A, B, and C elements. We directly measure the location of peaks DO and Dl from the Fourier transform of the interferogram. We then use Eq. (6) to calculate the RO' and Rl ' location values required for the b vector. The QO and Ql location values required for Eq. (6) are obtained from system parameters or a calibration procedure. Likewise, the RO and Rl peak locations required for the M matrix are calculated from system parameters or obtained from a calibration procedure. See Figure 2 for identification of peaks D, R, and Q. Once we have the elements A, B, and C of the 2X2 matrix in Eq. (7), we find the principal scale factors by solving for the eigenvalues of the matrix.
Figure 5. A) Sample interferogram and B) corresponding Fourier transform of a human eye captured by Talbot Moir? wavefront system. Fourier transform has primary peaks of interest circled. Using the calculation method described in the text, the resulting mean spectacle correction for a series of 10 such examinations was -0.09 -1.45 × 69.1°.
The algorithm steps for calculating the astigmatic wavefront are:
1. Compute RO' and Rl' from the detected DO and Dl peaks.
2. Use Eqs.(10) and (11) to solve for the matrix elements A, B, and G.
3. Solve for the eigenvalues and the axis using Eqs.(12) and (14).
4. Recover principal powers using Eq.(13).
5. Report the astigmatic refractive values using Eq.(15).
A custom Talbot Moir? wavefront system as described above was developed to collect examination images from an astigmatic model eye and human eyes. Custom software was developed to compute the Fourier transform from the captured image, find the peaks of interest to sub-pixel accuracy, and apply the peaks algorithm described above. The peak locator algorithm looks for the highest value in a given region of interest. Then we fit the neighborhood of the maximum value with a two-dimensional polynomial. The maximum value of the polynomial is then taken as the sub-pixel location of the Fourier transform peak. We have found this algorithm to be significantly more precise than a simpler centroid calculation. We applied our Fourier transform calculation procedure to captured interferogram images and tabulated the results.
Five examinations were done of the astigmatic model eye. The mean sphere and cylinder were ?4.26 D (SD = 0.004) and -2.90 D (SD = 0.008), respectively. The axis for all measurements was 92?. These values were consistent with the model eye design parameters. For the human eye, the patient was a 46-year-old man with best spectacle-corrected visual acuity of 20/20. Ten examinations were done of the patient's eye. One of the captured interferogram images is shown in Figure 5 A. The corresponding Fourier transform with the main peaks circled in red is shown in Figure 5B. The mean sphere and cylinder were ?0.09 D (SD = 0.11) and -1.45 D (SD = 0.14), respectively. The mean axis was 69.1° (SD = 1.29).
The calculation time was 46 ms per image running on a 3.6-GHz personal computer using 512×512 fast Fourier transforms (FFTs). The majority of this time is spent calculating the FFT of the interferogram. This processing time is fast enough for real-time acquisition and display of a patient's refractive state.
The Fourier transform calculation procedure provided a fast and simple method of determining the spectacle correction in an aberrated ocular wavefront represented in the interferogram of a Talbot Moir? wavefront sensor. It may be especially useful for realtime calculation and display of ocular refraction. The technique may also be helpful in augmenting or providing a quality control check for a full wavefront reconstruction.
As is apparent from the text, the accuracy and range of the method are primarily limited by the ability to determine the location of the Fourier transform peaks. Calibration of the system must also be taken into account to remove any potential bias from the reported measurements. Beneficial future work in this area would include a detailed comparison of peak finding algorithms to determine their relationship to the overall accuracy and precision of the method, especially in the presence of higher order aberrations.
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