We report the first observation of accelerating parabolic beams. These accelerating parabolic beams are similar to the Airy beams because they exhibit the unusual ability to remain diffraction-free while having a quadratic transverse shift during propagation. The amplitude and phase masks required to generate these beams are encoded onto a single liquid crystal display. Experimental results agree well with theory

Bandres, Miguel A.

Cottrell, Don  M.

Davis, Jeffrey A.

Mitry, Mark J.

Optics Express

English

Caltech Authors - Main

Observation of accelerating parabolic
beams
Jeffrey A. Davis,1 Mark J. Mitry,1 Miguel A. Bandres,2
and Don M. Cottrell1
1San Diego State University, Department of Physics, San Diego, CA 92182-1233
2California Institute of Technology, Pasadena, CA 91125
∗Corresponding author: jdavis@sciences.sdsu.edu
Abstract: We report the first observation of accelerating parabolic beams.
These accelerating parabolic beams are similar to the Airy beams because
they exhibit the unusual ability to remain diffraction-free while having a
quadratic transverse shift during propagation. The amplitude and phase
masks required to generate these beams are encoded onto a single liquid
crystal display. Experimental results agree well with theory.
© 2008 Optical Society of America
OCIS codes: 050.1970, 100.5090, 140.3300, 230.6120.
References and links
1. G. A. Siviloglou and D. N. Christodoulides, “Accelerating finite energy Airy beams,” Opt. Lett. 32, 979–981
(2007).
2. G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating Airy beams,”
Physical Review Letters 99, 213901 (2007).
3. I. M. Besieris and A. M. Shaarawi, “A note on an accelerating finite energy Airy beam,” Opt. Lett. 32, 2447–2449
(2007).
4. G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Ballistic dynamics of Airy beams,” Opt. Lett.
33, 207–209 (2008).
5. M. V. Berry and N. L. Balazs, “Nonspreading wave packets,” American Journal of Physics 47, 264–267 (1979).
6. M. A. Bandres, “Accelerating parabolic beams,” Opt. Lett., doc. ID 96139 (posted 20 June 2008, in press).
7. M. A. Bandres and J. C. Gutie´rrez-Vega, “Airy-Gauss beams and their transformation by paraxial optical sys-
tems,” Opt. Express 15, 16719–16728 (2007).
8. K. Banerjee, S. P. Bhatnagar, V. Choudhry, and S. S. Kanwal, “The anharmonic oscillator,” Proceedings of the
Royal Society of London. Series A, Mathematical and Physical Sciences 360, 575–586 (1978).
9. J. A. Davis, D. M. Cottrell, J. Campos, M. J. Yzuel, and I. Moreno, “Encoding amplitude information onto
phase-only filters,” Appl. Opt. 38, 5004–5013 (1999).
10. J. B. Bentley, J. A. Davis, M. A. Bandres, and J. C. Gutie´rrez-Vega, “Generation of helical Ince-Gaussian beams
with a liquid-crystal display,” Opt. Lett. 31, 649–651 (2006).
11. J. A. Davis, C. S. Tuvey, O. Lo´pez-Coronado, J. Campos, M. J. Yzuel, and C. Iemmi, “Tailoring the depth of
focus for optical imaging systems using a Fourier transform approach,” Opt. Lett. 32, 844–846 (2007).
12. J. A. Davis, P. Tsai, D. M. Cottrell, T. Sonehara, and J. Amako, “Transmission variations in liquid crystal spatial
light modulators caused by interference and diffraction effects,” Optical Engineering 38, 1051–1057 (1999).
1. Introduction
Recently Siviloglou et al. [1, 2, 4, 3] theoretically and experimentally demonstrated the ex-
istence of the accelerating Airy beams. These are finite-energy solutions to the paraxial wave
equation that have the unusual ability to remain almost diffraction-free while having a quadratic
transverse shift during propagation over long distances. These Airy beams are exponentially
decaying versions of the pure Airy beams studied by Berry and Balazs [5] in 1979. While the
#97187 - $15.00 USD Received 10 Jun 2008; revised 9 Jul 2008; accepted 13 Jul 2008; published 8 Aug 2008
(C) 2008 OSA 18 August 2008 / Vol. 16,  No. 17 / OPTICS EXPRESS  12866
pure Airy beams are perfectly diffraction-less and have a quadratic transverse shift during prop-
agation, they cannot be physically realizable because they have infinite energy. The Airy beams
have finite-energy and retain these unusual properties over long distances. Recently, Bandres [6]
introduced the accelerating parabolic beams that are also exact solutions of the two-dimensional
paraxial wave equation. These beams have the same unusual properties as the Airy beams, but
have an inherent parabolic geometry. In addition, these beams are more localized compared
with the Airy beams (i.e., pure accelerating parabolic beams decay as x−1/2 while pure Airy
beams as x−1/4 as x→−∞ ) and might be more suited for practical applications.
The accelerating Airy beams have been receiving a great deal of attention because they ap-
pear to propagate in curved line. Actually this claim must be viewed with a great deal of caution.
These beams exhibit a distance-dependent diffraction phenomenon where the interference max-
imum propagates without diffraction in a parabolic curved path. However the center of mass of
the beams remains constant during propagation as expected from geometric optics [3, 4, 7].
In this work, we report the first observation of accelerating parabolic beams [6]. The ac-
celerating parabolic beams were generated by encoding both the amplitude and phase of the
Fourier transform onto a single liquid crystal display (LCD). Our experiments demonstrate that
the accelerating parabolic beams can resist diffraction and experience a quadratic transverse
translation during propagation over long distances.
2. Accelerating parabolic beams
The finite-energy accelerating parabolic beams [6] are given by
φn(η ,ξ ,z) = ei(z/2kκ2−ia)(η2−ξ 2)/2ei(z/2kκ2−ia)
3
/3Θn (η)Θn (iξ ) , (1)
where (η ,ξ ) are parabolic coordinates defined as
(
x/κ− (z/2kκ2)2 + iaz/kκ2,y/κ
)
=
(
η2/2−ξ 2/2,ηξ) . (2)
Here, κ is a transverse scale and k is the wave number. The functions Θn (η), where n =
0,1,2, . . ., correspond to square integrable eigensolutions of the quartic oscillator equation [8]
and are depicted in Fig. 1(b). These functions are orthogonal, have n zeros and definite parity
given by the parity of n. At z = 0 the parameter a controls the exponential aperture function,
exp(ax/κ), that ensures the containment of the accelerating parabolic beams. Therefore for
finite-energy beams it is necessary that Re(a) > 0.
For a ≪ 1 the accelerating parabolic beams have a quasidiffraction-less behavior over long
distances that increase as a decreases. The case a = 0 corresponds to the pure accelerating
parabolic beams that are perfectly diffraction-free but have infinite energy. From Eq. (2) we can
see that during propagation the accelerating parabolic beams exhibit a transverse shift given by
xs =
1
κ3
( z
2k
)2
. (3)
It is difficult to experimentally create these beams because their width is small compared with
the pixel sizes of spatial light modulators. Consequently, it is easier to generate the Fourier
transforms of these beams and then optically create the desired beam in the focal plane of a
lens. The Fourier spectrum of the finite-energy accelerating parabolic beams at the plane z = 0
is given by [6]
F [φn] (kx,ky) ∝ exp
[
iκ3(k3x/3−a2kx/κ2 + kxk2y)
] (4)
×
[
exp
(−aκ2 (k2x + k2y
))
Θn
(√
2kyκ
)]
.
#97187 - $15.00 USD Received 10 Jun 2008; revised 9 Jul 2008; accepted 13 Jul 2008; published 8 Aug 2008
(C) 2008 OSA 18 August 2008 / Vol. 16,  No. 17 / OPTICS EXPRESS  12867
Fig. 1. (a) Theoretical intensity (top row) and phase (bottom row) of the Fourier spectrum
of several accelerating parabolic beams with n = {0,3,6} and a = 0.02. (b) Few quartic
oscillator eigenfunctions.
Fig. 2. Experimental results comparing the Airy and parabolic beams with n = 0, a = 0.02
and κ = 68.5 µm.
where (kx,ky) are the spatial frequency coordinates. The parameter κ controls the cubic phase
that characterizes the Fourier spectrum as well as the Gaussian amplitude factor. Note that this
parameter also controls the transverse shift as seen in Eq. (3).
Figure 1(a) compares the theoretical patterns for the intensity (top row) and phase (bottom
row) of the Fourier spectrum for the cases of n = {0,3,6} with a = 0.02. There are two critical
differences between the Fourier spectrum of the accelerating parabolic beams and the two-
dimensional Airy beams. First, the phase patterns are much more complicated for large values
of n. More importantly, the amplitude modulation is not a simple Gaussian as with the Airy
beams. Not only it is not circularly symmetric, but there is an extra modulation in the ky-
axis given by Θn
(√
2kyκ
)
. The Gaussian factor exp
[−aκ3 (k2x + k2y
)]
in the Fourier spectrum
assures that the beam carries finite energy if Re(a) > 0.
3. Experiment
Accelerated Airy beams were experimentally generated [2, 4] using a phase-only LCD to en-
code the phase of the Fourier spectrum of the desired Airy beam. The necessary amplitude
distribution is a simple two-dimensional Gaussian function that was obtained by illuminating
the LCD with a Gaussian beam whose diameter had been carefully prepared. They then per-
formed the optical Fourier transform in the focal plane of an external lens. This approach is
not feasible with the accelerating parabolic beams because the amplitude distributions of their
Fourier spectra are more complicated as mentioned above. Therefore another approach must be
used to encode the amplitude and phase distributions.
In this work, we generate the accelerating parabolic beams by combining both the ampli-
#97187 - $15.00 USD Received 10 Jun 2008; revised 9 Jul 2008; accepted 13 Jul 2008; published 8 Aug 2008
(C) 2008 OSA 18 August 2008 / Vol. 16,  No. 17 / OPTICS EXPRESS  12868
Fig. 3. Experimental results for accelerating parabolic beams with n = {0,3,6}, a = 0.02
and κ = 68.5 µm at several transverse z-planes.
tude and phase patterns onto a single LCD using a previously reported technique [9, 10, 11].
We begin with the Fourier spectrum of the desired accelerating parabolic beam function hav-
ing magnitude Mp (x,y) and phase φp (x,y) and written as Mp (x,y)exp [iφp (x,y)]. In this ap-
proach, we combine the phase pattern with a vertically oriented linear phase grating φG (y) =
2piy/d with period d as Mp (x,y)exp [iφp (x,y)+ iφG (y)]. The total phase is the sum of the
parabolic phase term with the grating phase. Amplitude information is then encoded by spa-
tially modulating the phase pattern with the amplitude portion of the desired pattern [9, 10, 11],
exp
{
iM′p (x,y) [φp (x,y)+φG (y)]
}
. The intensity that is diffracted into the first order varies spa-
tially and reproduces the desired amplitude and phase distribution. There is a slight distortion
that is corrected by modifying the amplitude term M′p (x,y) as described in Ref. [9]. Here the
total phase is defined φ = φp (x,y)+φG (y) in the range [−pi,pi] while the amplitude is defined
in the range
[
0≤M′p (x,y)≤ 1
]
. The zero order diffraction produces an undesired term that can
be spatially filtered.
In our experiments, linearly polarized light from an Argon laser is spatially filtered, ex-
panded, and collimated. The optical elements are encoded onto a parallel-aligned nematic liquid
crystal display (LCD) manufactured by Seiko Epson with 640×480 pixels with pixel spacing
of ∆ = 42 µm [12]. Each pixel acts as an electrically controllable phase plate where the total
phase shift exceeds 2pi radians as a function of gray level at the Argon laser wavelength of
514.5 nm. The linear phase grating period d was 8 pixels≈ 328 µm. The Fourier transform of
the mask was formed in the focal plane of a 1-meter lens to create the parabolic beam. As in
#97187 - $15.00 USD Received 10 Jun 2008; revised 9 Jul 2008; accepted 13 Jul 2008; published 8 Aug 2008
(C) 2008 OSA 18 August 2008 / Vol. 16,  No. 17 / OPTICS EXPRESS  12869
Fig. 4. Experimental results for acelerating parabolic beams with n = 4, a =
{0.01,0.05,0.1} and κ = 68.5 µm at several transverse z-planes.
Ref. [4], the LCD is placed in the front focal plane of the Fourier transform lens and the Fourier
transform is formed in the back focal plane. It is possible to give an initial launch angle [4] to
the accelerating parabolic beams by translating the center of the mask pattern by m pixels rela-
tive to the center of the LCD. The launch angle is related to this linear shift δ = m∆ by θ = δ/ f
where f is the focal length of the Fourier lens. We found this to be simpler than moving the
transverse position of the Fourier lens [4]. Consequently, Eq. (3) for the transverse position of
the beam is given by
xs = θz+
1
κ3
( z
2k
)2
. (5)
As stated earlier, our approach for encoding amplitude and phase information also generates
a strong zero-order beam that was spatially filtered. The output was then recorded with a CCD
camera. To magnify the image, we mounted a 10× microscope objective at a fixed distance in
the front of the CCD camera [11]. By translating this assembly, we imaged various locations
along the axial.
Figure 2 compares experimental results for the two-dimensional Airy beam and the parabolic
beams with n = 0 and where κ = 68.5 µ m for both beams.
Figure 3 shows experimental results for different accelerating parabolic beams with n =
{0,3,6}, a = 0.02 and κ = 68.5 µ m, at propagation distances of z = {0,15,30,45,60} cm.
The width of the image plane corresponds to ≈ 4 mm. As expected, the beams remain almost
diffraction free for ≈ 45 cm and have a transverse quadratic right shift during propagation. The
direction of the transverse shift can be reversed using the complex conjugate of the phase mask.
The theoretical behavior is in good agreement with the experimental results and therefore is not
#97187 - $15.00 USD Received 10 Jun 2008; revised 9 Jul 2008; accepted 13 Jul 2008; published 8 Aug 2008
(C) 2008 OSA 18 August 2008 / Vol. 16,  No. 17 / OPTICS EXPRESS  12870
0 10 20 30 40 50 60 70
−1
0
1
2
3
4 Experiment Theory
θ = +2.1
θ = 0
θ = −4.2
Propagation Distance [cm]
Tr
an
sv
er
se
 T
ra
sl
at
io
n 
[m
m]
Fig. 5. Transverse deflection of an accelerating parabolic beam with n = 0, a = 0.01 and κ =
68.5 µm as a function of the propagation distance for launch angles of θ = 0,+2.1,−4.2
mrad.
shown.
Next we examine the effects of changing the parameter a of the beams. In Fig. 4 we show
the experimental results for the propagation of the accelerating parabolic beams with n = 4,
a = {0.01,0.05,0.1} and κ = 68.5 µm, at propagation distances z = {0,15,30,45,60} cm.
These experimental results show clearly that, as the parameter a decreases, the beams propagate
almost diffraction-free over longer distances. For example the beams with a = {0.01,0.05,0.1}
start diffracting significantly at z≈ {60,45,30} cm, respectively, as can be seen from Fig. 4.
Figure 5 shows experimental results for the quadratic transverse translation of the n = 0
accelerating parabolic beam with a = 0.01 and κ = 68.5 µm from z = 0 to 70 cm. We show
three curves with different initial launch angles of velocities using values of θ = 0,+2.1,−4.2
mrad.
To obtain an accurate measurement of the transverse translation of the beams, we compare
the transverse deflection using the accelerating parabolic phase mask with the opposite deflec-
tion using the complex conjugate of the phase mask. Using one mask, we adjusted the transverse
position of the camera until the beam was at one edge. We then applied the conjugate mask and
moved the camera until the beam was again at the edge of the camera. The experimental accu-
racy of this technique was quite good at about 75 µm. The data fits the quadratic depencence on
propagation distance from Eq. (5) using the parameter κexp = 68.6 µm in excellent agreement
with the theoretical value. Beams with the same parameters, κ and a, but different orders, n,
propagate in exactly the same way and therefore they are not shown.
4. Conclusion
In conclusion, we have reported the first observation of accelerating parabolic beams. As
demonstrated in our experiments, the finite-energy accelerating parabolic beams exhibit the un-
usual ability to remain almost diffraction-free while having a quadratic transverse shift during
propagation over long distances. Our experimental results agree with theory. These accelerat-
ing parabolic beams are more tightly concentrated and have lower sidelobes compared with the
Airy beams. Consequently they might be more useful for practical applications as trapping.
Acknowledgments
J. A. Davis would like to thank Tomio Sonehara of Seiko Epson Corporation for the use of the
LCD and Mark Hatay for helping with the laboratory. M. A. Bandres acknowledges support
from the Secretarı´a de Educacio´n Publica de Me´xico.
#97187 - $15.00 USD Received 10 Jun 2008; revised 9 Jul 2008; accepted 13 Jul 2008; published 8 Aug 2008
(C) 2008 OSA 18 August 2008 / Vol. 16,  No. 17 / OPTICS EXPRESS  12871


Observation of accelerating parabolic beams

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Observation of accelerating parabolic beams

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