Image formation of a one dimensional polynomial mirror: Numbers and parities of images Wei-Liang Hsu n , Stanley Pau University of Arizona, College of Optical Sciences, 1630 East University Boulevard, Tucson, AZ 85721, USA article info Article history: Received 21 April 2012 Received in revised form 9 July 2012 Accepted 24 February 2013 Available online 21 March 2013 Keywords: Polynomial mirror Real image Virtual image Image formation abstract This paper addresses the image formation, including both real images and virtual images, of the polynomial mirrors when analyzed by using ray optics. The imaging properties of polynomial mirrors, including mirrors with quadratic, cubic, and quartic polynomial shapes, are reviewed and generalized to higher order polynomial mirrors by being approximated as a piecewise set of displaced parabolic mirrors. The conditions in which multiple virtual images can be formed and the parities of the virtual images are investigated both theoretically and experimentally. & 2013 Elsevier Ltd. All rights reserved. 1. Introduction Image formation using aspherical mirrors is an important issue in optical engineering. The availability of low loss coatings that cover a broad spectrum and the lack of optical dispersion from reflection make mirrors an indispensable component in the design and construction of optical instruments and imaging systems. Curved mirrors of arbitrary shapes, such as deformable mirrors, have applications in adaptive optics [1], optical testing [2], astron- omy [3], remote sensing [4,5], communication [6] and medicine [7]. Liquid curve mirrors using electrowetting-controlled liquid have been demonstrated by Bucaro et al. [8] and large liquid metal based mirror can possess many unique properties compared to conventional glass mirrors [9]. For images reflected from an arbitrary smooth surface, Berry introduced the caustic-touching theorem to analyze the topologies of the virtual images of the extended light source, for example in the case of the Sun’s disk seen in rippled water [10]. Roman-Hernandez and Silva-Ortigoza improved the Ronchi test by applying the caustic-touching theorem on the parabolic mirrors [11,12]. In this paper, both the real images and the virtual images of a one- dimensional arbitrary shaped mirror, called a polynomial mirror, are analyzed by using ray optics, in order to evaluate the numbers and parities of the images. This polynomial mirror has a flexible pre- distorted surface defined mathematically by a polynomial of only one variable. Our method is very general and is not restricted to the topologies of virtual images and the Ronchi test. In addition, it can be applied to the design of an omnidirectional camera system using polynomial mirrors [13]. In Section 2, the imaging properties of quadratic mirrors are briefly reviewed, and the concepts are applied to the analysis of a cubic mirror. In Section 3, both the real and virtual image formations from polynomial mirrors, including a quadratic polynomial mirror, a quartic polynomial mirror, and other higher order polynomial mirrors, are analyzed. Section 4 presents the results of several virtual image formation experiments and investigates the conditions in which multiple virtual images of the same object can be formed. Finally, the unique properties of the polynomial mirrors are summarized. 2. Quadratic aspheric mirrors The polynomial mirror, sometimes called a funny mirror or carnival mirror, produces caricatures of human faces and can often be found at carnivals, museums and festivals. Typical dimensions of the mirror range from tens of centimeters to a few meters. Virtual images produced by such a mirror are purposefully designed to be highly distorted with spatially varying magnifications, i.e. a thin person’s reflection may appear fat and with an exaggerated nose. Fig. 1 shows a painting by Norman Rockwell which depicts the characteristics of a polynomial mirror. In order to understand the imaging properties of a polynomial mirror, a simple one- dimensional parabolic mirror with a fixed curvature R is considered. The surface is described by a quadratic polynomial of the form yðxÞ¼ ax 2 þ bx þ c ¼ ðxx o Þ 2 4f þ y 0 , ð1Þ Contents lists available at SciVerse ScienceDirect journal homepage: www.elsevier.com/locate/optlaseng Optics and Lasers in Engineering 0143-8166/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.optlaseng.2013.02.020 n Corresponding author. Tel.: þ1 520 621 2997. E-mail addresses: whsu@optics.arizona.edu, whsu@email.arizona.edu (W.-L. Hsu), spau@optics.arizona.edu (S. Pau). Optics and Lasers in Engineering 51 (2013) 986–993