![]() For numerical computations involving Zernike radial polynomials of n ≥ 40, Janssen and Dirksen suggested an alternate form of Eq. (1) with advantages in computation time, accuracy and ease of implementation. If the pupil function is rather roughly behaved, it may be necessary to include Zernike polynomials of very high orders. Table 1 lists explicitly the Zernike polynomials according to this indexing scheme. Since the power of every radial polynomial is n = m + 2 k and since ( m + 2 k ) + m = 2 ( m + k ) is fixed for every row, the rightmost entry of every row, with m = 0, has the highest power. As one can see, rows are arranged by the ascending order of m + k. The indexing scheme used by Zeiss and ASML is shown in Fig. 1. It is therefore a gross misnomer that the Zernike polynomials we lithographers use are often referred to as Fringe Zernike polynomials, as if there are various sets of such polynomials it is the “Fringe” indexing scheme of the one and only set of Zernike polynomials! Hence we believe that it was John Loomis who invented this indexing scheme in conjunction with the wavefront-fitting program called FRINGE, originally written by Jim Rancourt. Loomis, FRINGE User’s Manual, Optical Sciences Center, University of Arizona, Tucson, AZ, November 1976. 11),… Later, Loomis, PhD 1980, wrote a FRINGE MANUAL, and updated the program to output the 37 “FRINGE” Zernike polynomials, and the beginning of the confusion about whose numbering of the polynomials one might be using.” Citation in their article is: Optical Sciences Center, “FRINGE Software Program,” OSC Newsletter 8(12), 29 (1974). Parks’ article: 7 “The first program for analyzing interferograms was written by Jim Rancourt, PhD 1974 (Fig. We can learn the origin of this Fringe indexing scheme from Katherine Creath and Robert E. The ordering sequence of the Zernike polynomials used by Zeiss and ASML is a modified version of the indexing scheme originated at the University of Arizona. One advantage is that k is independent of m. Incidentally, using k instead of n to index the Zernike polynomials is not a bad thing. Defining n = m + 2 k brings us to Eq. (1) put forth originally by Frits Zernike. Therefore, the first thing to do is to obtain these orthogonal radial polynomials (actually the Zernike radial polynomials) by orthogonalizing the set. Once this is accomplished, both summations over k can be expressed as linear combinations of these polynomials. Our third and last step involves expressing r 2 k as a linear combination of orthogonal polynomials satisfying the orthogonal relation over the interval. In reaching the above expression, no requirement of rotational symmetry about an axis had to be imposed. (4) W ( r, θ ) = ∑ m = 0 ∞ cos m θ r m ( A m + A m ′ r 2 + A m ′ r 4 + ⋯ ) + ∑ m = 0 ∞ sin m θ r m ( B m + B m ′ r 2 + B m ′ r 4 + ⋯ ) = ∑ m = 0 ∞ cos m θ r m ∑ k = 0 ∞ A m k r 2 k + ∑ m = 0 ∞ sin m θ r m ∑ k = 0 ∞ B m k r 2 k. Although RGP lenses correct the irregular astigmatism, smaller comet-like retinal images in the opposite direction remain due to residual vertical coma.Eq. In addition to the larger amount of trefoil, coma, tetrafoil, and secondary astigmatism, keratoconic eyes tend to have a reverse coma pattern and reverse trefoil aberrations compared with normal eyes. Although the total HOAs were significantly (keratoconus and keratoconus suspect, P < 0.001 and P = 0.012, respectively) reduced with an RGP lens, the patterns of the axes of coma and trefoil were reversed with the lens. The mean axes of trefoil in patients with keratoconus (93.8 degrees ) and keratoconus suspect (100.6 degrees ) differed from that in normal subjects (35.4 degrees ), indicating that keratoconus has a reverse trefoil pattern from that of normal eyes. Zernike vector analysis showed prominent vertical coma with an inferior slow pattern, with mean axes of 82.5 degrees or 91.0 degrees in the patients with keratoconus or keratoconus suspect, respectively. Ocular higher-order aberrations (HOAs) were measured with a wavefront sensor for a 4-mm-diameter pupil, and the magnitudes, axes of trefoil, and coma were calculated by vector analysis. To determine the effect of RGP lenses, 19 eyes with keratoconus, 9 eyes with keratoconus suspect, and 17 normal eyes, with and without an RGP lenses, were compared. To measure the magnitude and orientation of the Zernike terms in keratoconic eyes, with and without rigid gas-permeable (RGP) contact lenses.Ī total of 76 eyes with keratoconus, 58 eyes with keratoconus suspect, and 105 normal eyes were studied.
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