Cavity-enhanced dipole forces for dark-field seeking atoms and molecules Tim Freegarde Dipartimento di Fisica, Università di Trento 38050 Povo (TN), Italy J F Allen Physics Research Laboratories, University of St Andrews, Fife KY16 9SS, Scotland David McGloin, Kishan Dholakia R 2 R 1 even even odd Cartesi an cylindr ical Hermite- Gaussian Laguerre- Gaussian i + j 2p + |m | RAY OPTICS • 2 round trips before repeating • inverted image after 1 round trip • returning beam forward beam GAUSSIAN BEAMS CAVITY MODES • half modes simultaneously resonant • (anti-)symmetric image = superposition of even(odd) modes 0 1 0 f f 1 0 0 1 HALF TRIP ROUND TRIP q f q 2 L/ R 1 L/ R 2 0 1 1 confocal • high • low towards low intensity towards high intensity L 20 L 00 OPTICAL BOTTLE BEAM • Freegarde & Dholakia, Phys Rev A, in press • see Arlt & Padgett, Opt. Lett. 25 (2000) 191-193 • Laguerre-Gaussian superposition: CONFOCAL CAVITIES COAXIAL RING ARRAY • Freegarde & Dholakia, Opt. Commun. 201 99 (2002) • see Zemánek & Foot, Opt. Commun. 146 119 (1998) • use single Gaussian beam of waist w 1 larger than that of the fundamental cavity mode (w 0 = w 1 ) • counterpropagating beam smaller by same factor (w 2 = w 0 ) • beams of equal power cancel where nodal surfaces intersect 2 1 2 2 2 1 2 0 1 1 ln z w z w z w z w z r • intensity minima form a series of coaxial rings spaced by /2 • traps deepest when = 0.492 • r 0 ~ 0.7 w 0 (z) pm pm pm z r a z r , , L • with = 0.492, 99% of power in first 5 modes OPTICAL DIPOLE FORCE LAGUERRE-GAUSSIAN BEAMS • return beam larger than forward beam to avoid nodal surfaces • cancellation at cavity centre • constructive interference elsewhere thanks to different radial dependences and Gouy shifts • with = 0.5, the maximum modulation depth is 7%. 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 counter-propagating beams is apparent. Here, l = 100 mm, = 780nm, = 2. Left: depth of modulation due to interference between forward and return beams. Black=0, white=100%. MECHANICAL AMPLIFIER Amplitudes a p0 of mode components forming the complete five-component optical bottle beam with =2. 40 30 20 10 00 525 . 0 332 . 0 165 . 0 332 . 0 691 . 0 L L L L L E • five component superposition optimizes trap depth for given radius: COMPOSITION • 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 85 Rb at 0.2 nm detuning 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. Intensity distribution when the cavity mirrors are 0.1 mm from their confocal separation (l/l = 0.001), for r 2 = 0.99, t 2 = 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. 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, = 780 nm, = 0.492. • trapping of spectrally complex atoms and molecules • investigation of vortices in quantum degenerate gases 14 • coupling between adjacent microtraps 15 • cooling via coupling to cavity radiation field 16-18 Applicatio ns: SINGLE TOROID • in preparation • dissimilar forward/return waist sizes eliminate nodal planes • magnetic field free toroidal trap for study of vortices in condensates 14 REFERENCES 1 R. Grimm, M. Weidemüller, Y. B. Ovchinnikov, Adv. At. Mol. Opt. Phys. 42 (2000) 95-170 2 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 3 J. D. Miller, R. A. Cline, D. J. Heinzen, Phys. Rev. A 47 (1993) R4567-4570 4 M. D. Barrett, J. A. Sauer, M. S. Chapman, Phys. Rev. Lett. 87 (2001) 010404 5 T. Takekoshi, B. M. Patterson, R. J. Knize, Phys. Rev. Lett. 81 (1998) 5105-5108 6 N. Davidson, H. J. Lee, C. S. Adams, M. Kasevich, S. Chu, Phys. Rev. Lett. 74 (1995) 1311- 1314 7 P. Rudy, R. Ejnisman, A. Rahman, S. Lee, N. P. Bigelow, Optics Express 8 (2001) 159-165 8 S. A. Webster, G. Hechenblaikner, S. A. Hopkins, J. Arlt, C. J. Foot, J. Phys. B 33 (2000) 4149-4155 9 T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimuzu, H. Sasada, Phys. Rev. Lett. 78 (1997) 4713-4716 10 R. Ozeri, L. Khaykovich, N. Davidson, Phys. Rev. A 59 (1999) R1759-1753 11 J. Ye, D. W. Vernooy, H. J. Kimble, Phys. Rev. Lett. 83 (1999) 4987-4990 12 S. Jochim, Th. Elsässer, A. Mosk, M. Weidemüller, R. Grimm, Int. Conf. on At. Phys., Firenze, Italy, poster G.11 (2000) 13 P. W. H. Pinkse, T. Fischer, P. Maunz, T. Puppe, G. Rempe, J. Mod. Opt. 47 (2000) 2769-2787 14 E. M. Wright, J. Arlt, K. Dholakia, Phys. Rev. A 63 (2000) 013608 15 P. Münstermann, T. Fischer, P. Maunz, P. W. H. Pinkse, G. Rempe, Phys. Rev. Lett. 84 (2000) 4068-4071 16 T. Zaugg, M. Wilkens, P. Meystre, G. Lenz, Opt. Commun. 97 (1993) 189-193 17 M. Gangl, H. Ritsch, Phys. Rev. A 61 (1999) 011402 18 V. Vuletic, S. Chu, Phys. Rev. Lett. 84 (2000) 3787-3790 dipole traps eliminate the magnetic fields needed for MOTs 1 FAR OFF RESONANCE 2-5 broadband interaction and minimal scattering, hence suitable for spectrally complex atoms and molecules intense laser beam needed to compensate for interaction weakness BLUE-DETUNED 6-10 dark-field seeking to minimize residual perturbations need isolated islands of low intensity for closed trapping region RESONANT CAVITIES 11-13 can greatly increase circulating intensity, as optical absorption is low optical field not a single cavity mode • transverse mode degeneracy allows enhancement of mode superpositions for complex field geometries • an arbitrary field may be written as a superposition pm L kz m z R kr z w r z w r z w r L z w z z m p m p p z r m m p R m pm i i 2 i exp 2 2 tan 1 2 i exp ! 1 ! 4 , , 2 2 2 2 2 2 1 0 L z z z z R R 2 • the Laguerre-Gaussian cavity modes are solutions to the paraxial wave equation in cylindrical polar coordinates, where are Laguerre polynomials and , , . x L m p • three different views of physics: 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 off- resonant dark-field seeking dipole traps within reach of low-power diode lasers. 2 1 0 R z z w z w 2 0 w z R • Laguerre-Gaussian beams , of non-resonant waist radius w 1 , correspond to superpositions of resonant L-G beams with the same azimuthal index m = s. The first three coefficients are: ) ( qm 1 L pmq a p s ps s p s p a sin ! ! ! cos 1 0 2 2 1 1 1 sin 1 cos sin ! 1 ! ! cos s p s p s p a p s ps 2 2 2 2 2 2 1 2 cos sin 2 cos sin 1 cos sin ! 2 ! ! 2 ! cos p s p s p s p s p a p s ps 0 1 1 0 0 1 1 0 sin w w w w w w w w col intensi ty trap centre intensity trap centre position • moving the mirrors from their confocal separation causes an amplified displacement of the trap centre • amplification by same factor as intensity enhancement LARGE PERIOD STANDING WAVE • in preparation • see D M Giltner et al, Opt. Commun. 107 227 (1994) • pattern period = /sin • 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 • amplification mechanism may be compared to Vernier scale between Gouy phases of different Laguerre-Gaussian components
Cavity-enhanced dipole forces for dark-field seeking atoms and molecules
Feb 11, 2016
L 20. L 00. 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. J F Allen Physics Research Laboratories, University of St Andrews, - PowerPoint PPT Presentation
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Cavity-enhanced dipole forces for dark-field seeking atoms and moleculesTim Freegarde
Dipartimento di Fisica,Università di Trento38050 Povo (TN), Italy
J F Allen Physics Research Laboratories,University of St Andrews,Fife KY16 9SS, Scotland
David McGloin, Kishan Dholakia
R2R1
even evenoddCartesiancylindrical
Hermite-GaussianLaguerre-Gaussian
i + j2p + |m |
RAY OPTICS• 2 round trips before repeating• inverted image after 1 round trip• returning beam forward beam
GAUSSIAN BEAMS
CAVITY MODES• half modes simultaneously
resonant• (anti-)symmetric image =
superposition of even(odd) modes
010
f
f
1001
HALF TRIP ROUND TRIP
qfq2
L/R1
L/R2
0 1
1confocal
• high
• low
towardslow intensity
towardshigh intensity
L20
L00
OPTICAL BOTTLE BEAM• Freegarde & Dholakia, Phys Rev A, in press• see Arlt & Padgett, Opt. Lett. 25 (2000) 191-193
• Laguerre-Gaussian superposition:
CONFOCAL CAVITIES
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 = w1)
• counterpropagating beam smaller by same factor (w2 = w0)
• beams of equal power cancel where nodal surfaces intersect
2
12
22
120
11lnzwzwzw
zwzr
• intensity minima form a series of coaxial rings spaced by /2
• traps deepest when = 0.492
• r0 ~ 0.7 w0(z)
pm
pmpm zrazr ,, L
• with = 0.492, 99% of power in first 5 modes
OPTICAL DIPOLE FORCE
LAGUERRE-GAUSSIAN BEAMS
• return beam larger than forward beam to avoid nodal surfaces
• cancellation at cavity centre• constructive interference
elsewhere thanks to different radial dependences and Gouy shifts
• with = 0.5, the maximum modulation depth is 7%.
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 counter-propagating beams is apparent. Here,l = 100 mm, = 780nm, = 2.
Left: depth of modulation due to interference between forward and return beams. Black=0, white=100%.
MECHANICAL AMPLIFIER
Amplitudes ap0 of mode components forming the complete five-component optical bottle beam with =2.
4030201000 525.0332.0165.0332.0691.0 LLLLLE
• five component superposition optimizes trap depth for given radius:
COMPOSITION
• 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
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.
Intensity distribution when the cavity mirrors are 0.1 mm from their confocal separation (l/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.
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, = 780 nm, = 0.492.
• trapping of spectrally complex atoms and molecules
• investigation of vortices in quantum degenerate gases14
• coupling between adjacent microtraps15
• cooling via coupling to cavity radiation field16-18
Applications:
SINGLE TOROID• in preparation
• dissimilar forward/return waist sizes eliminate nodal planes
• magnetic field free toroidal trap for study of vortices in condensates14
REFERENCES1 R. Grimm, M. Weidemüller, Y. B. Ovchinnikov, Adv. At. Mol. Opt. Phys. 42 (2000) 95-1702 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) 205-2113 J. D. Miller, R. A. Cline, D. J. Heinzen, Phys. Rev. A 47 (1993) R4567-45704 M. D. Barrett, J. A. Sauer, M. S. Chapman, Phys. Rev. Lett. 87 (2001) 0104045 T. Takekoshi, B. M. Patterson, R. J. Knize, Phys. Rev. Lett. 81 (1998) 5105-51086 N. Davidson, H. J. Lee, C. S. Adams, M. Kasevich, S. Chu, Phys. Rev. Lett. 74 (1995) 1311-13147 P. Rudy, R. Ejnisman, A. Rahman, S. Lee, N. P. Bigelow, Optics Express 8 (2001) 159-1658 S. A. Webster, G. Hechenblaikner, S. A. Hopkins, J. Arlt, C. J. Foot, J. Phys. B 33 (2000) 4149-41559 T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimuzu, H. Sasada, Phys. Rev. Lett. 78 (1997) 4713-471610 R. Ozeri, L. Khaykovich, N. Davidson, Phys. Rev. A 59 (1999) R1759-175311 J. Ye, D. W. Vernooy, H. J. Kimble, Phys. Rev. Lett. 83 (1999) 4987-499012 S. Jochim, Th. Elsässer, A. Mosk, M. Weidemüller, R. Grimm, Int. Conf. on At. Phys., Firenze, Italy, poster G.11
(2000)13 P. W. H. Pinkse, T. Fischer, P. Maunz, T. Puppe, G. Rempe, J. Mod. Opt. 47 (2000) 2769-278714 E. M. Wright, J. Arlt, K. Dholakia, Phys. Rev. A 63 (2000) 01360815 P. Münstermann, T. Fischer, P. Maunz, P. W. H. Pinkse, G. Rempe, Phys. Rev. Lett. 84 (2000) 4068-407116 T. Zaugg, M. Wilkens, P. Meystre, G. Lenz, Opt. Commun. 97 (1993) 189-19317 M. Gangl, H. Ritsch, Phys. Rev. A 61 (1999) 01140218 V. Vuletic, S. Chu, Phys. Rev. Lett. 84 (2000) 3787-3790
dipole traps eliminate the magnetic fields needed for MOTs1
FAR OFF RESONANCE2-5
broadband interaction andminimal scattering, hence suitable for
spectrally complex atoms and molecules
intense laser beam needed to compensate for interaction weakness
BLUE-DETUNED6-10
dark-field seeking to minimize residual perturbations
need isolated islands of low intensity for closed trapping region
RESONANT CAVITIES11-13
can greatly increase circulating intensity, as optical absorption is low
optical field not a single cavity mode
• transverse mode degeneracy allows enhancement of mode superpositions for complex field geometries
• an arbitrary field may be written as a superposition
pmL
kzmzR
krzwr
zwr
zwrL
zwzzmp
mppzr
m
mp
R
mpm ii
2iexp22tan12iexp
!1!4,,
2
2
2
22
21
0
L
zzzzR R2
• the Laguerre-Gaussian cavity modes are solutions to the paraxial wave equation in cylindrical polar coordinates,
where are Laguerre polynomials and , , . xLmp
•three different views of physics:
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 off-resonant dark-field seeking dipole traps within reach of low-power diode lasers.
210 Rzzwzw
20wzR
• Laguerre-Gaussian beams , 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 are:
)(qm1L
pmqa
psps sp
spa sin!!!cos 1
0
2211
1 sin1cossin!1!!cos
spspspa ps
ps
2222221
2 cossin2cossin1cossin!2!!2!cos pspsp
spspa ps
ps
0110
0110sinwwwwwwww
col intensity
trap centreintensity
trap centre position
• moving the mirrors from their confocal separation causes an amplified displacement of the trap centre
• amplification by same factor as intensity enhancement
LARGE PERIOD STANDING WAVE • in preparation
• see D M Giltner et al, Opt. Commun. 107 227 (1994)
• pattern period = /sin• 2-D Hermite-Gaussian
analysis; astigmatism renders out-of-plane direction non-confocal
• high Q: all (odd) even modes
give (anti-)symmetric
field patternfinite Q: half-axial modes
contribute
• amplification mechanism may be compared to Vernier scale between Gouy phases of different Laguerre-Gaussian components
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