Creating Dark Lines in Space with Linear Zone Plates

Pradyoth Kukkapalli, Charter School of Wilmington, Delaware
Marty Cohen and John Noé, Laser Teaching Center, Stony Brook University

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. 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. 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 2 π.

Linear zone plates, which are the subject of this project, have one-dimensional patterns and act like cylinder lenses to create line foci. 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.

The goal of our project is to create and investigate a linear sinusoidal zone plate with a π phase jump. 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. Zone plate patterns were generated by using Mathematica to map the varying transparency across the zone plate with the function

where f is the focal length, x is the transverse distance from the centerline of the zone plate and λ is the wavelength of the light illuminating it. Several 8 mm square patterns generated in this way have been imaged on to 35 mm black and white film by photographer Gene Lewis (Darkroom Specialties LLC, Eugene, OR). We are currently observing and recording the diffraction patterns that result when the zone plates are illuminated with the expanded beam from a HeNe laser.

We would like to thank the Simons Foundation for funding this research, Prof. Michael Raymer for recommending a printing service, and Prof. Harold Metcalf for establishing and supporting the Laser Teaching Center.

References

[1] S.B. Vinas, Z. Jaroszewicz, A. Kojolodziejczyk, and M. Sypek, "Zone plates with black focal spots," Applied Optics 31, 192 &ndash 198 (1992).

[2] J. Ojeda-Castaneda, and G. Ramirez, "Zone plates for zero axial irradiace," Optics Letters 18, 87 &ndash 89 (1993).

[3] T.D. Beynon, I. Kirk, and T.R. Mathews, "Gabor zone plates with binary transmittance values." Optics Letters 17, 544 – 546 (1992).

[4] J.W. Goodman, Introduction to Fourier optics. Robert and Company, 2005.

[5] Nityan Nair, "Diffraction with a twist: creating optical vortices with a spiral Fresnel zone plate." Siemens competition report, Oct. 2008. http://laser.physics.sunysb.edu/~nityan/Siemens.pdf