Design of Holographic Lightfields for Manipulation of Quantum Degenerate Gases

Experiments with ultracold quantum degenerate gases such as Bose-Einstein Condensates or degenerate Fermi gases are at the forefront of modern atomic physics. These systems allow the investigation of complicated quantum-mechanical dynamics of many-body systems in a well controlled environment. Since these gases require temperatures near absolute zero, they must be well isolated from any room temperature environment. Mechanical forces that laser beams exert on atoms are well suited for trapping and manipulating these atoms. The goal of our studies is the investigation of holographic techniques to generate nearly arbitrary lightfields for the manipulation of ultracold quantum gases. The project has three major components: periodic lattices, aperiodic patterns, and computer generated binary holography.

Introduction

S. Kreppel1, C. Hamner2, J.J. Chang2, P. Engels21Carthage College, Kenosha, WI2Dept. of Physics and Astronomy, WSU, Pullman, WA

focusing lens

Experimental Setup

The optical set-up for testing absorption masks is based on a 633 nm He-Ne laser beam. The laser is sent through telescoping lenses, an absorption mask, a focusing lens, a magnification lens and then is finally projected onto a screen and photographed with a CCD camera. When the absorption mask is illuminated by the coherent light from the laser, the interference pattern produces the Fourier transform of the mask at the focus of the 75 mm lens. Arbitrary patterns can be created by using computer generated binary holograms as absorption masks.

He-Ne Laser

f = 30mm

mirror

f = 200mm

mask

f = 75mm

f = 30mm

imaging screen

1st telescoping lens 2nd telescoping lens

magnification lens

The interaction between an atom and light can be understood by modeling the atom classically as a harmonic oscillator. When the atom is placed in laser light, the electric field E of the light induces a dipole moment p on the atom as seen in the figure below. The interaction energy between the induced dipole moment and the electric field is given by V= - p•E. When the frequency of the laser light is less than that of the atomic resonant frequency, the force on the atom is attractive. The laser light is referred to as being “red detuned”. The force on the atom is repulsive if the laser light frequency is greater than the atomic resonance.

Dipole Force

The light is “blue detuned” in this case. The force on the atom is proportional to the intensity of the laser light. This force is called the dipole force and can be used to trap and manipulate ultracold atoms.

induced dipole moment

EÓEVVV•−= ρ

1. Periodic Lattices

Figure 1: Several examples of tested absorption masks. The first column is the absorption mask tested while the second column is the photographed optical diffraction pattern from that mask. The third column is the theoretical Fourier transform generated from a Mathematica program written as part of this project. There is a very good agreement between the observed and calculated patterns.

According to the laws of propagation and diffraction of light, a convex lens can produce the Fourier transform of an input light field entering the lens. The Fourier transformed image appears in the focal plane of the lens. The first step of this project tested this relation on simple periodic shapes as seen in Figure 1.

All masks for this project were generated by programs written in Mathematica. The masks were then printed on a transparency and used in the optical set-up. An example of this set-up is displayed in Figure 2.

Figure 2: Sample binary mask in optical set-up

2. General Fourier Optics of aperiodic shapes

Figure 3: Fourier Transforms of aperiodic absorption masks

T

X

Absorption Mask

Optical Test

Mathematica Calculation

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c.

a.8

The aperiodic masks were tested in the same manner as in the periodic case. However, aperiodic masks provide more interesting transforms which lead to new possibilities for the manipulation of ultracold atoms. From both Figure 1 and Figure 3, one can see that the optical set-up produces a fairly accurate Fourier transform.

The rules that govern Fourier transforms allow for not only periodic shapes but aperiodic shapes as well. Part two of this project tested aperiodic masks, examples of which can be seen in Figure 3.

References• Brown, B., and A. Lohmann. "Complex Spatial Filtering with Binary Masks." Applied Optics IP 5.6

(1966): 967.• Chhetri, B., S. Serikawa, S. Yang, and T. Shimomura. "Binary phase difference diffuser: Design

and application in digital holography." Optical Engineering 41.2 (2002). • Heckenberg, N., R. McDuff, C. Smith, and A. Whit. "Generation of optical phase singularities by

computer-generated holograms." Optics Letters 17.3 (1992): 221-23.• Meschede, Dieter. Optics, Light and Lasers The Practical Approach to Modern Aspects of

Photonics and Laser Physics. New York: Wiley-VCH, 2007. • Walker, T. "Holography without photography." American Journal of Physics 67.9 (1999): 783-85.

Conclusions and Future Plans

The future plans include: - Write an optimization function to reduce the amount of noise and lost information for the binary hologram program. - Use of an aperture in optical set up in order to block out the central intensity. - Test a binary hologram on a BEC and Fermi degenerate gas. - Create a deep well lattice which can effectively trap a BEC for imaging purposes.

Through the course of this project methods to create periodic and aperiodic absorption masks as well as binary holograms for the formation of holographic lightfields have been tested. Several Mathematica programs were written throughout the course of the project as a means to create the masks and calculate their proficiency. An optics setup was constructed to test the actual performance. These programs and the experimental set-up have proven to be very effective. Printing binary holograms on transparencies using an inkjet printer has turned out to be a viable and efficient way to test calculated holograms with rapid turn-around times.

3. Computer Generated Binary Holograms

A modified optical set-up was used to optimize the photographs of the holographic output. The 633nm He-Ne laser was sent through two lenses of f = 30 mm and f = 1000 mm. The distance between the lenses was larger than that for a telescope (the sum of the focal lengths), so that the beam was converging behind the second lens. The binary hologram was placed directly behind the second lens.

Binary holograms provide a powerful technique to form nearly arbitrary lightfields. As part of this project, a program in Mathematica was developed for creating these binary holograms. First a random phase factor is multiplied to each pixel of the desired light pattern in order to spread the information of each pixel over a large area of the hologram. Then the Fourier transform is calculated, its imaginary part is discarded, and the real part is binarized. The resulting hologram reproduces the desired image, a symmetric twin image and a central bright spot due to use of only the real part of the Fourier transformed original image. In the photographs in Figure 4(b), the central maximum of intensity was removed for the purpose providing a better image quality.

Binary Hologram

Figure 4: Computed binary holograms and the image output

Original Image

Holographic Image

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b.

c.

AbsorptionMask

Optical Test

Mathematica Calculation

focusing lens

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a.