Theoretically formulated in the 1970s within the context of nonrelativistic quantum mechanics, Airy beams have been experimentally realized for the first time only recently, paving the way to innovative optical techniques. While their remarkable features, a non-diffracting property and a transverse shift of the intensity maximum during propagation, are currently theoretically described from the wave optics viewpoint, here their exact relation to rays and geometric wavefront aberrations is revealed using a wavefront family that includes two-dimensional Airy beams. Several members of this family are computationally and experimentally implemented here. The lateral shift of Airy beams during propagation is presented in the context of the three-dimensional caustic representation. This new description allows re-emphasizing the use of "Airy-like" beams in computational imaging for depth of focus extension.
We previously demonstrated that radial basis functions may be preferred as a descriptor of free-form shape for a single mirror magnifier when compared to other conventional descriptions such as polynomials [Opt. Express 16, 1583 (2008)]. A key contribution is the application of radial basis functions to describe and optimize the shape of a free-form mirror in a dual-element magnifier with the specific goal of optimizing the pupil size given a 20 degrees field of view. We demonstrate a 12 mm exit pupil, 20 degrees diagonal full field of view, 15.5 mm eye clearance, 1.5 arc min resolution catadioptric dual-element magnifier design operating across the photopic visual regime. A second contribution is the explanation of why it is possible to approximate any optical mirror shape using radial basis functions.
A distortion mapping and computational image unwarping method based on a network interpolation that uses radial basis functions is presented. The method is applied to correct distortion in an off-axis head-worn display (HWD) presenting up to 23% highly asymmetric distortion over a 27°x21° field of view. A 10(-5) mm absolute error of the mapping function over the field of view was achieved. The unwarping efficacy was assessed using the image-rendering feature of optical design software. Correlation coefficients between unwarped images seen through the HWD and the original images, as well as edge superimposition results, are presented. In an experiment, images are prewarped using radial basis functions for a recently built, off-axis HWD with a 20° diagonal field of view in a 4:3 ratio. Real-time video is generated by a custom application with 2 ms added latency and is demonstrated.
Abstract— Previously, it was demonstrated that radial basis functions may be preferred as a free‐form shape descriptor for a single‐mirror magnifier, justified by a performance increase measured by the MTF, when compared to other conventional descriptions such as multivariate polynomials (e.g., Zernike polynomials or x‐y polynomials). The benefit in performance increase can be used to expand the pupil diameter from 8 to 12 mm given a 20° field of view and a 15‐mm eye clearance or to increase the field of view. The main contribution in this paper is the investigation of the field‐of‐view limit in a dual‐element magnifier where the free‐form mirror is described with radial basis functions. Our main result in this paper is an estimate of the field‐of‐view limit of the dual‐element magnifier to be approximately 25° full‐field diagonal, given the specific geometry described in the paper. The impact of the astigmatic node placement in a rectangular image field on the modulation transfer function is also analyzed for the particular dual‐element magnifier geometry.
This paper studies the application of radial basis function networks to the description, design and optimization of free‐form optical surfaces, which find key applications in head‐worn displays.
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