![]() This seemed inadvisable as it might be misinterpreted as an improvement or refinement of the results obtained by Petzval and Rayleigh. It would of course be possible to compromise and adopt an intermediate best value of u. He mentioned that Petzval’s value, u = π, was quoted in a pamphlet on pinhole photography and remarked that the corresponding “ detail in a photograph … was not markedly short of that observable by direct vision.” At the same time, he stated that images obtained for u = 1.8π “ fully bore out expectations.” It appears that Rayleigh reached a similar conclusion. The same can be inferred from the observation made by writers on pinhole photography that, although the best aperture diameters are usually stated within 0.01 mm, deviations on the order of 0.1 mm have little effect on image quality. This ambiguity can undoubtedly be attributed to the fact that the various minima of image size found in this analysis are all shallow so that, on the whole, the difference between u = π and 1.8π is insignificant for practical purposes. We have verified Rayleigh’s value as a minimum of area width, confirmed Petzval’s value by computations of flux widths, and were unable to make a choice using the theory of resolution. On account of the complicated nature of the Lommel profiles it is no surprise that the above analysis gives no unequivocal answer regarding a best configuration parameter u for lensless imaging. 6, they are neatly clustered around a median value of The values of u at which these minima occur are all different. 5, showing that a shallow but discernible minimum ofį ( Δ v / u ). The results obtained are illustrated in Fig. The flux fractions used for these computations were 0.3333, 0.5, 0.6667, 0.75, and 0.8378, the latter being equivalent to the Airy disk in Table 1. (10) in a different manner so that it would directly yield the values of u for whichį ( Δ v / u ) is contained in the smallest possible width This was deemed insufficient for judging the profiles in their entirety. Δ v / u is a minimum, they are only on the order of 35 %. While approaching 50 % for small values of u, as should be expected from Table 1, these flux fractions decrease for larger values of u and, where These lists were used as lookup tables, and linear interpolation was used to obtain final results for given values of u andį ( Δ v / u ) obtained in this manner for the area widths computed earlier are plotted as the lower curve in Fig. Equation (10) was used to generate lists, accurate to six digits, ofį ( Δ v / u ) for consecutive upper limits N ≤ 500 and increments δ v = 0.01 u. (8) but v and | α ( u, v)| 2 are replaced by their arithmetic means for each element of summation. The latter has decidedly the higher resolving power, but the advantage is to some extent paid for in the greater diffusion of light outside the image proper.” He conducted visual and photographic experiments to settle this question and found that the sharpest images were obtained for The only question that can arise is between u = π and u = 2π. 3 and, without additional calculations, judged that “ u = ½π is too large and u = 3π is too great. Rayleigh plotted the diffraction profiles which are reproduced here as Fig. In the past, computations based on these equations were tedious but nonetheless Lommel provided numerical tables of E( u, v) and Rayleigh used these tables for a further analysis of pinhole imaging. 1, E geom( v) is the geometrical irradiance in the absence of diffraction, Φ 0 is the radiant flux admitted by the aperture, | α ( u, v)| 2 is the modification of E geom( v) by diffraction, and L( u, v) and M( u, v) are functions defined by Lommel as linear combinations of infinite series of Bessel functions. Where E( u, v) is the irradiance at the point P of Fig. The numerical values of other measures of image size are reported and compared to equivalent parameters of the Fraunhofer-Airy profile that governs imaging with lenses. The smallest discernible detail (pixel) in a composite image is defined by an expression found by Rayleigh on applying the half-wave criterion and is shown to be consistent with the Sparrow criterion of resolution. The numerical result ( u = π) obtained for the best configuration parameter u which defines the optical setup is consistent with the quarter-wave criterion, and is the same as the value reported in a classical paper by Petzval but smaller than the value ( u = 1.8π) found by Lord Rayleigh. The diffraction limit for lensless imaging, defined as the sharpest possible point image obtainable with a pinhole aperture, is analyzed and compared to the corresponding limit for imaging with lenses by means of theoretical considerations and numerical computations using the Fresnel-Lommel diffraction theory for circular apertures.
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