On the refractive properties of media with poles or zeros in the index of refraction

Abstract

The refractive properties of inhomogeneous fibers are examined with emphasis being placed on the limiting situation where the index of refraction possesses poles or zeros. If, e.g., the index of refraction is <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n(\xi) = \xi^{m}, \xi = r/a</tex> , where <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">a</tex> is the radius of the fiber and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</tex> an arbitrary constant, it is found that energy integrability is satisfied if <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m &gt; -1</tex> . When <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m \leq -1</tex> energy infinities occur. The ray behavior of such media is examined in terms of geometrical optics, and corrections to geometrical optics are obtained by an asymptotic analysis of the exact solution. For <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m &gt; 0</tex> , the lens is of the diverging type, and when the angle of incidence is <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\alpha = 0</tex> , geometrical optics predicts that rays "reflect" at various angles from the origin (depending on the value of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</tex> ). When <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m &lt; 0</tex> , rays "wrap" around the origin several times with a zero radius of curvature before they leave the lens ( <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\alpha = 0</tex> ). For <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-1 &lt; m &lt; -\frac{1}{2}</tex> , it is found that when <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\alpha= [(2m + 1)/2m] \pi</tex> caustics occur ( <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\alpha = 0, \pi/2</tex> excluded). Pictorial diagrams show the behavior of these caustics and the correction coefficients to geometrical optics are obtained.