Design of Holographic Lightfields for Manipulation of Quantum Degenerate Gases
S. Kreppel1, C. Hamner2, J.J. Chang2, P. Engels2
1Carthage College, Kenosha, WI
2Dept. of Physics and Astronomy, WSU, Pullman, WA
Introduction
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.
3. Computer Generated Binary Holograms
1. Periodic Lattices
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.
Absorption
Mask
Optical
Test
Mathematica
Calculation
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.
a.
b.
Dipole Force
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.
The light is “blue detuned” in this case. The
E
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
Ó
Figure 2: Sample binary mask
in optical set-up
c.
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.
1st telescoping lens
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.
or
irr
m
Absorption
Mask
He-Ne Laser
f = 30mm
f = 200mm
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.
focusing lens
mask
focusing lens
f = 75mm
magnification lens
f = 30mm
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.
Figure 3: Fourier Transforms
of aperiodic absorption masks
imaging screen
Binary
Hologram
Holographic
Image
Conclusions and Future Plans
2. General Fourier Optics of aperiodic shapes
2nd telescoping lens
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
a.
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
b.
behind the second lens.
Original
Image
Figure 4: Computed binary holograms and the image
output
V V
V = −ρ • E
Experimental Setup
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.
a.
Optical Mathematica
Test
Calculation
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 turnaround times.
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.
8
References
b.
X
c.
T
• 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.