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Cavity-enhanced dipole forces for
dark-field seeking atoms and molecules
David McGloin, Kishan Dholakia
Tim Freegarde
Dipartimento di Fisica,
Università di Trento
38050 Povo (TN), Italy
OPTICAL BOTTLE BEAM
• Freegarde & Dholakia, Phys Rev A, in press
• see Arlt & Padgett, Opt. Lett. 25 (2000) 191-193
J F Allen Physics Research Laboratories,
University of St Andrews,
Fife KY16 9SS, Scotland
Dipole force traps for dark-field seeking states of
atoms and molecules require regions of low
intensity that are completely surrounded by a
bright optical field. Confocal cavities allow the
resonant enhancement of these interesting
transverse mode superpositions, and put deep offresonant dark-field seeking dipole traps within
reach of low-power diode lasers.
COAXIAL RING ARRAY
• Freegarde & Dholakia, Opt. Commun. 201 99 (2002)
• see Zemánek & Foot, Opt. Commun. 146 119 (1998)
• use single Gaussian beam of waist w1 larger than that of the
fundamental cavity mode (w0 = a w1)
• Laguerre-Gaussian superposition:
• cancellation at cavity centre
• constructive interference elsewhere
thanks to different radial
dependences and Gouy shifts
• return beam larger than forward
beam to avoid nodal surfaces
• with a = 0.5, the maximum
modulation depth is 7%.
• counterpropagating beam smaller by same factor (w2 = a w0)
OPTICAL DIPOLE FORCE
• beams of equal power cancel where nodal surfaces intersect
(
)
w
z
1
r0 (z )  ln
w2 (z )
J dipole traps eliminate the magnetic fields needed for MOTs1
COMPOSITION
• five component superposition optimizes
trap depth for given radius:
• high 
E  0.691L00  0.332 L10  0.165 L20  0.332 L30  0.525 L40
• trap intensity nearly half that at centre of simple Gaussian
beam with same waist and power as forward beam
• 99.99% mirrors with 100 mW at 780 nm would give 5 K trap
depth for 85Rb at 0.2 nm detuning
towards
high intensity
towards
low intensity
• low

FAR OFF RESONANCE2-5
J broadband interaction and
J minimal scattering, hence suitable for
spectrally complex atoms and molecules
L intense laser beam needed to
compensate for interaction weakness
• traps deepest when a = 0.492
• r0 ~ 0.7 w0(z)
• with a = 0.492, 99% of power
in first 5 modes
RESONANT CAVITIES11-13
J can greatly increase circulating intensity,
as optical absorption is low
L optical field not a single cavity mode
L/R2
• transverse mode degeneracy allows
enhancement of mode superpositions
for complex field geometries
1
confocal
0
1
L/R1
Intensity distribution around the centre of a confocal cavity.
Dashed and solid lines indicate the nodal and antinodal planes;
the dotted line shows where the lowest part of the trap wall is
maximum. Logarithmic contours (four per decade) refer to the
peak intensity on axis. l = 100 mm, l = 780 nm, a = 0.492.
• three different views of physics:
RAY OPTICS
• 2 round trips before repeating
• inverted image after 1 round trip
• returning beam  forward beam
R2
R1
GAUSSIAN BEAMS
 0

 1 f

f

0

f2
q 
q
HALF TRIP
even
Cartesian
cylindrical
odd
i+j
2 p + |m |
even
Hermite-Gaussian
Laguerre-Gaussian
 1

1



2
2
 w (z ) w (z ) 
 2

1
• intensity minima form a series of
coaxial rings spaced by l/2
BLUE-DETUNED6-10
J dark-field seeking to minimize residual
perturbations
L need isolated islands of low intensity for
closed trapping region
CONFOCAL CAVITIES
Intensity distribution within a perfectly confocal
resonator.
Above left: mean intensity shown for central
40% of the cavity. The solid lines show
where the forward beam has fallen to e-2 of
its on-axis intensity.
Above: viewed on a wavelength scale around
the cavity centre, the modulation due to
interference between the counterpropagating beams is apparent. Here,
l = 100 mm, l = 780nm, a = 2.
Left: depth of modulation due to interference
between forward and return beams.
Black=0, white=100%.
2
Intensity distribution when the cavity mirrors are 0.1 mm
from their confocal separation (Dl/l = 0.001), for r2 = 0.99,
t2 = 0.01. The nodal surfaces, shown dashed, are now curved,
reflecting the increase in Gouy phase with mode number.
Central and trapping intensities are reduced by about a third.
Applications: • trapping of spectrally complex atoms and
molecules
 1 0 


 0  1
• investigation of vortices in quantum
degenerate gases14
ROUND TRIP
CAVITY MODES
• half modes simultaneously resonant
• (anti-)symmetric image =
superposition of even(odd) modes
• coupling between adjacent microtraps15
• cooling via coupling to cavity radiation field16-18
Amplitudes ap0 of mode components forming
the complete five-component optical bottle
beam with a=2.
LAGUERRE-GAUSSIAN BEAMS
MECHANICAL AMPLIFIER
• the Laguerre-Gaussian cavity modes L pm are solutions to the
paraxial wave equation in cylindrical polar coordinates,
col intensity
• moving the mirrors from their
confocal separation causes an
amplified displacement of the
trap centre
exp (i(2 p  m  1) tan (z z R )) m  2r 

L pm (r , z , J ) 
L p 
2
(1  d 0 m)p (p  m )!
w(z )
 w(z ) 
4 p!
trap centre
intensity
trap centre position
• amplification by same factor
as intensity enhancement
Variation of trap col (dotted) and trap centre (dashed)
intensities – in units of the well depth at zero mirror
displacement – and trap centre position (right hand scale)
as mirrors are displaced from their confocal separation.
L00
•
amplification mechanism may be
compared to Vernier scale between
Gouy phases of different LaguerreGaussian components
SINGLE TOROID
e (r , z )   a pm L pm (r , z )
pm
• Laguerre-Gaussian beams Lqm, of non-resonant
waist radius w1, correspond to superpositions of
resonant L-G beams with the same azimuthal
index m = s. The first three coefficients a pmq are:
a ps 0  cos
• magnetic field free toroidal trap for study of vortices in
condensates14
j

a ps 2  cos s 1 j
• in preparation
• see D M Giltner et al, Opt. Commun. 107 227 (1994)
J
s 1

a ps1  cos s 1 j
• pattern period = l/sinq
• dissimilar forward/return waist sizes eliminate nodal planes
2
2
 2r 


r
i
kr




 w(z )2  exp  w(z )2  2 R(z )  imJ  ikz 




m
2
LARGE PERIOD STANDING WAVE
• in preparation
2
2
pw(0 )
z
2
m
R
(
)
(
)
(
)
(
)


(
)



L
x
where p are Laguerre polynomials and R z z
, w z w 0 1 z zR , zR
.
l
z
• an arbitrary field may be written as a superposition
(1)
L20
1
• 2-D Hermite-Gaussian analysis;
astigmatism renders out-of-plane
direction non-confocal
• high Q: all (odd) even modes
give (anti-)symmetric
field pattern
finite Q: half-axial modes
contribute
(p  s )!
p! s!
sin j
p
w0 w1  w1 w0
j

sin
w0 w1  w1 w0
(p  s )! p 1 j [ 2 j  (  ) 2 j ]
sin
p cos
s 1 sin
p! (s 1 )!
(p  s )!

sin p 2 j{[p cos 2 j  (s  1)sin 2j][p cos 2 j  (s  2)sin 2 j] p cos 2 j }
2! p! (s  2 )!
REFERENCES
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
R. Grimm, M. Weidemüller, Y. B. Ovchinnikov, Adv. At. Mol. Opt. Phys. 42 (2000) 95-170
S. L. Rolston, C. Gerz, K. Helmerson, P. S. Jessen, P. D. Lett, W. D. Phillips, R. J. Spreeuw, C. I. Westbrook, Proc. SPIE 1726
(1992) 205-211
J. D. Miller, R. A. Cline, D. J. Heinzen, Phys. Rev. A 47 (1993) R4567-4570
M. D. Barrett, J. A. Sauer, M. S. Chapman, Phys. Rev. Lett. 87 (2001) 010404
T. Takekoshi, B. M. Patterson, R. J. Knize, Phys. Rev. Lett. 81 (1998) 5105-5108
N. Davidson, H. J. Lee, C. S. Adams, M. Kasevich, S. Chu, Phys. Rev. Lett. 74 (1995) 1311-1314
P. Rudy, R. Ejnisman, A. Rahman, S. Lee, N. P. Bigelow, Optics Express 8 (2001) 159-165
S. A. Webster, G. Hechenblaikner, S. A. Hopkins, J. Arlt, C. J. Foot, J. Phys. B 33 (2000) 4149-4155
T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimuzu, H. Sasada, Phys. Rev. Lett. 78 (1997) 4713-4716
R. Ozeri, L. Khaykovich, N. Davidson, Phys. Rev. A 59 (1999) R1759-1753
J. Ye, D. W. Vernooy, H. J. Kimble, Phys. Rev. Lett. 83 (1999) 4987-4990
S. Jochim, Th. Elsässer, A. Mosk, M. Weidemüller, R. Grimm, Int. Conf. on At. Phys., Firenze, Italy, poster G.11 (2000)
P. W. H. Pinkse, T. Fischer, P. Maunz, T. Puppe, G. Rempe, J. Mod. Opt. 47 (2000) 2769-2787
E. M. Wright, J. Arlt, K. Dholakia, Phys. Rev. A 63 (2000) 013608
P. Münstermann, T. Fischer, P. Maunz, P. W. H. Pinkse, G. Rempe, Phys. Rev. Lett. 84 (2000) 4068-4071
T. Zaugg, M. Wilkens, P. Meystre, G. Lenz, Opt. Commun. 97 (1993) 189-193
M. Gangl, H. Ritsch, Phys. Rev. A 61 (1999) 011402
V. Vuletic, S. Chu, Phys. Rev. Lett. 84 (2000) 3787-3790