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Cavity-enhanced dipole forces for dark-field seeking atoms and molecules Tim Freegarde Dipartimento di Fisica, Università di Trento Povo (TN), Italy J F Allen Physics Research Laboratories, University of St Andrews, Fife KY16 9SS, Scotland David McGloin, Kishan Dholakia R2R2 R1R1 even odd Cartesian cylindrical 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 HALF TRIPROUND TRIP L/R 1 L/R 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) Laguerre-Gaussian superposition: CONFOCAL CAVITIES COAXIAL RING ARRAY Freegarde & Dholakia, Opt. Commun (2002) see Zemánek & Foot, Opt. Commun (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 ln zwzw zw zw zr intensity minima form a series of coaxial rings spaced by /2 traps deepest when = r 0 ~ 0.7 w 0 (z) pm 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 a p0 of mode components forming the complete five-component optical bottle beam with = 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 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 = 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, = 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 Applications: 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 1R. Grimm, M. Weidemüller, Y. B. Ovchinnikov, Adv. At. Mol. Opt. Phys. 42 (2000) 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) J. D. Miller, R. A. Cline, D. J. Heinzen, Phys. Rev. A 47 (1993) R M. D. Barrett, J. A. Sauer, M. S. Chapman, Phys. Rev. Lett. 87 (2001) T. Takekoshi, B. M. Patterson, R. J. Knize, Phys. Rev. Lett. 81 (1998) N. Davidson, H. J. Lee, C. S. Adams, M. Kasevich, S. Chu, Phys. Rev. Lett. 74 (1995) P. Rudy, R. Ejnisman, A. Rahman, S. Lee, N. P. Bigelow, Optics Express 8 (2001) S. A. Webster, G. Hechenblaikner, S. A. Hopkins, J. Arlt, C. J. Foot, J. Phys. B 33 (2000) T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimuzu, H. Sasada, Phys. Rev. Lett. 78 (1997) R. Ozeri, L. Khaykovich, N. Davidson, Phys. Rev. A 59 (1999) R J. Ye, D. W. Vernooy, H. J. Kimble, Phys. Rev. Lett. 83 (1999) S. Jochim, Th. Elsässer, A. Mosk, M. Weidemüller, R. Grimm, Int. Conf. on At. Phys., Firenze, Italy, poster G.11 (2000) 13P. W. H. Pinkse, T. Fischer, P. Maunz, T. Puppe, G. Rempe, J. Mod. Opt. 47 (2000) E. M. Wright, J. Arlt, K. Dholakia, Phys. Rev. A 63 (2000) P. Münstermann, T. Fischer, P. Maunz, P. W. H. Pinkse, G. Rempe, Phys. Rev. Lett. 84 (2000) T. Zaugg, M. Wilkens, P. Meystre, G. Lenz, Opt. Commun. 97 (1993) M. Gangl, H. Ritsch, Phys. Rev. A 61 (1999) V. Vuletic, S. Chu, Phys. Rev. Lett. 84 (2000) 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 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 kzm zR kr zw r zw r zw r L zw zzmp mp p zr m m p R m pm ii 2 i exp 22 tan12iexp !1 !4,, L z z zzR R 2 the Laguerre-Gaussian cavity modes are solutions to the paraxial wave equation in cylindrical polar coordinates, where are Laguerre polynomials and,,. xL 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 10 R zzwzw 2 0w 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 ps ps sp sp asin !! ! cos 1 0 sin1cossin !1! ! cos sp sp sp a ps ps cossin2cossin1cossin !2!!2 ! cospspsp sp sp a ps ps sin wwww wwww col intensity 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 (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

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