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Creating Dark Lines in Space with Linear Zone Plates

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Creating Dark Lines in Space with Linear Zone Plates Creating Dark Lines in Space with Linear Zone Plates Intel Science Talent Search (Physics) November 2010 Pradyoth Kukkapalli Charter School of Wilmington Wilmington, DE and Laser Teaching Center Department of Physics and Astronomy Stony Brook University 1 Introduction This project was inspired by a general fascination with zone plates and the interesting work on zone plates containing a π-phase jump done by Vinas et al. [1]. Such specialized optical devices create a series of dark focal points in space which can be used for precision alignment. Zone plates are optical elements which focus light by diffraction instead of refraction as in conventional lenses [8]. A Fresnel-type zone plate has many concentric circular regions which are alternately opaque and transparent. The widths and radii of the zones are such that light diffracted from the center of every transparent region reaches a focal point on the axis of the plate in phase [7]. The result is constructive interference, and the creation of a bright spot. Such Fresnel zone plates also have minor bright spots along the beam axis, which occur when the phase shift from adjacent transparent regions is an integer multiple of 2π. Linear zone plates have one-dimensional patterns and act like cylindrical lenses to create line foci [10]. Introducing a π-phase jump across the center of such a zone plate effectively inverts the focus, creating a \dark line in space" between two bright lines. A π-phase jump occurs when one half of the zone plate has transparent regions where the other half has opaque regions. The phase jump causes half of the light passing through the zone plate to be completely out of phase with the other half of the light, thus causing fully destructive interference along the dark line [1]. Sinusoidal zone plates have smooth transitions from completely transparent regions to completely opaque regions, unlike binary zone plates, which are strictly opaque or transparent. The smooth transitions lead to sharper focal lines, and eliminate the minor focal lines that are present with the use of binary zone plates [3, 9]. Zone plate patterns were generated by using Mathematica. Transmittance equations were derived for each of the four zone plates, and Mathematica's ContourPlot function was used to plot these equations which yielded the zone plate designs. Several 8 mm square patterns generated in this way were imaged on to 35 mm black and white film by photographer Gene Lewis [6]. 1 The goal of our project was to create and evaluate four linear zone plates. There were two binary zone plates, a conventional one and one with a π-phase jump. Likewise, there are two sinusoidal zone plates, a conventional one and one with a π-phase jump. The zone plates were illuminated by a HeNe laser. The interference patterns from the zone plates were projected directly onto a CCD camera without a lens. The greatest challenge throughout this project was making sure that the zone plate designs were of a high quality despite their complexity. 2 Background Zone plates were first invented by Lord Rayleigh in April 11, 1871 [7]. A zone plate is made of alternating rings of opaque and transparent rings. Traditional zone plates are made such that the alternating transparent rings only allow light that constructively interferes to pass through. This causes the light coming from the transparent zones to be out of phase by no more than π. Zone plates have two unique configurations, even and odd. Odd zone plates have a transparent zone at the center, whereas even zone plates have an opaque zone at the center [7]. Despite this difference, both even and odd zone plates produce the same interference pattern. But, the light coming from an odd zone plate is out of phase by π with the light coming from an even zone plate. This concept is especially important to the idea of π-phase jumps. 2.1 Zone Plate Basics In order to achieve constructive interference at the focus, the alternating zones must have specific widths such that the difference in path length from the distance and the center of the zone plate to the focus is an integer multiple of the wavelength of the light [7]. This ensures that the light coming from a zone is not out of phase by more than π with the light coming from another zone. Therefore, if one has an odd circular Fresnel zone plate, 2 then the path length of the light coming from the center of the zone plate is f. If the light coming from each transparent zone can be out of phase by no more than π, then the path difference between the light from the center of the first zone and the light from the edge of λ the first zone must be 2 . Therefore, the path length for the light from the edge of the first λ transparent zone to the focus is f + 2 . Furthermore, the light traveling from one edge must λ travel an additional 2 compared to the light coming from the previous edge [7]. Therefore, the path length of any light coming from the edge of any boundary, which is an n number of boundaries away from the center is nλ d = f + n 2 Furthermore, if the radius of the nth boundary is rn, then by using the Pythagorean theorem, p 2 2 dn = f + rn Then setting the two equations equal to each other and solving for rn, the distance of the zone boundaries from the center of the zone plate is described by the equation, r n2λ2 r = fnλ + n 4 2.2 The π-Phase Jump For the importance of analyzing the interference pattern of the zone plate, one must analyze the wave equation in relation to the zone plate. The wave equation is, A Φ(d; t) = ei(kd−!t) d Then by considering the light illuminating the zone plate to be uniform, one can ignore the 3 time dependence portion of the wave equation, thus yielding, A Φ(d) = eikd d It is important to note why two wavefronts that arrive at the same point, out of phase by π, destructively interfere. This can be accomplished by examining the sum of the wave equations of the two different wavefronts, A A Φ + Φ = eikd + ei(kd+π) 1 2 d d Using Euler's formula, this can be rewritten as, A Φ + Φ = (cos kd + i sin kd + cos kd + π + i sin kd + π) 1 2 d A = (cos kd + i sin kd − cos kd − i sin kd) d = 0 This concept is important to understand, because, the zone plate creates destructive interference by using this property of electromagnetic radiation. By combining an even zone plate and an odd zone plate it is possible to create destructive interference because the light that comes through the even side of the zone plate will be out of phase by π in relation to the odd zone plate. Creating a zone plate that produces two seperate wavefronts that are out of phase by π is known as introducing a π-phase jump to a zone plate. This destructive interference at the focus of the zone plate is what results in the dark focal spots. 4 2.3 Sinusoidal Zone Plates Sinusoidal zone plates differ from traditional binary zone plates in that the transparency of the zone plate varies sinusoidally from the center of the zone plate, unlike binary zone plates which are strictly transparent or opaque [3, 9]. The phase of the light coming from the transparent regions of the zone plate is not uniform, so binary zone plates produce extraneous background interference patterns. They also produce minor foci, which form at fractional distances of the focal length, because of their non-uniformity. However, sinusoidal zone plates reduce any such background interference as well as minor foci, which helps to further improve the quality of the dark lines, thus creating finer dark lines [3, 9]. The binary transmittance values for a zone plate which alternate from 0 to 1 are described by the equation, 1 ipπr2 2 1 1 X − 2 t(r ) = + ( ) e r1 2 iπ p=−∞ However, this has to be simplified to a sinusoidal variation, which then yields an equation of the form, 1 ± cos kr2 t(r) = ; 2 where k is a constant. The value of k can be determined by recognizing that although different, both sinusoidal and binray zone plates possess many of the same characteristics. Therefore, the equations that describe a binary zone plate can still be used to describe sinusoidal zone plates, which can then be used to find the constant k. It was important in this project to make the sinusoidal and binary zone plates as consistent as possible, so it became necessary to derive an equation describing the transmittance values of binary zone plates in a simple manner as well. This can easily be done, by recognizing that the transmittance values have to be either 0 or 1, thus resulting in the equation, 1 ± sgn(cos kr2) t(r) = 2 5 Figure 1: Zone plate designs: binary zone plate (top left), sinusoidal zone plate (top right), binary zone plate with π-phase jump (bottom left), and sinusoidal zone plate with π-phase jump (bottom right). By using the equation, r n2λ2 r = fnλ + n 4 it was possible to find the constant k, by recognizing that rn, indicates the distance at which a binary zone plate changes transmittance, which is the same as when t(r) changes 2 from 1 to 0 or vice versa.
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