Toroidal optical dipole grating

15 October 2001

Optics Communications 198 (2001) 101±105

www.elsevier.com/locate/optcom

Toroidal optical dipole grating A.A. Lipovka *, M.A. Cervantes Centro de Investigaci on en Fisica, Universidad de Sonora, Rosales y Boulevard Transversal, Col. Centro, Apartado Postal 5-088, Edif. 3-I, 83000 Hermosillo, Sonora, Mexico Received 30 May 2001; accepted 13 August 2001

Abstract A method for producing multiple coaxial toroidal optical dipole grating is presented. The analytical solution for beam forming the grating is obtained. This solution presents a structure of the toroidal optical grating. The possibility to change the con®guration of this grating in real time is stressed. Beams of this kind will have broad application in 2D atom lasers. Ó 2001 Elsevier Science B.V. All rights reserved. Keywords: Optical traps; Bose±Einstein condensation; Laser cooling and trapping; Optical lattices

1. Introduction As it is well known, beams forming optical dipole traps (ODT) are used to produce optical tweezers for particles and atoms [1,2]. By using the ODT, it is possible to achieve deeper threedimensional cooling of atoms with aid of a method based on expansion of an ensemble of cooled atoms from a small trap into a shallow one of much larger size [3]. In this, and other cases, it is very important to have ¯exibility to some extent in changing the shape of the ODT during the cooling process. More important is the main advantage of ODT compared with magnetic traps (in which the fermion's spin is locked to the ®eld direction) to achieve Bose±Einstein condensate (BEC) of some atomic species [4], that is due to the fact that ODT can store atoms in any sub-state of electronic

* Corresponding author. Tel.: +52-6259-2156; fax: +52-62126649. E-mail address: [email protected] (A.A. Lipovka).

ground state (see also Ref. [5] and references therein). For these reasons ODTs are applied in the manipulation of BEC and they have attracted the attention of researchers during the last few years. Lately, a number ODT have been proposed for very speci®c applications. For example the ODT suggested by Hammes et al. [4] is actually a gravito-optical surface trap and do not allow to manipulate atoms successfully. The optical bottle discussed in Ref. [6] was produced with Laguerre± Gaussian beam with aid of an amplitude hologram and cannot be implemented in real time. On the other hand, the use of multiple coaxial toroidal ODTs for BEC is very interesting from the point of view of the possible generation and investigation of quantum vortices on atomic mesoscopic rings. It is on the basis of these multiple rings that a feedback mechanism for a 2D atom laser is conceived. In this fashion, the radial tunneling between condensates would take place across toroidal dark regions (see Ref. [7] and references therein).

0030-4018/01/$ - see front matter Ó 2001 Elsevier Science B.V. All rights reserved. PII: S 0 0 3 0 - 4 0 1 8 ( 0 1 ) 0 1 4 9 2 - 4

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A.A. Lipovka, M.A. Cervantes / Optics Communications 198 (2001) 101±105

The ®rst step towards developing toroidal trap was made recently [7]. The toroidal ODT was obtained by the same method of Laguerre± Gaussian beam (see Ref. [6]). However, the authors recognize that there are two problems associated with this technique. One is connected with the decrease of mode purity from holographic generation that leads to substantial diculties in producing LG mode with high azimuthal index (it actually should be less than l6). The other is the inherent diculties of construction of the toroidal optical dipole grating (TODG) this way, which is very important for production of a circular grating for atoms and hence a 2D atom laser [7]. Besides, as it was mentioned, this approach cannot be implemented in real time. In this letter, we propose a new approach to producing a TODG which do not present these problems. In this report, we assume that the laser frequency is far from any resonance with the con®ned atomic sample, i.e. when the dipole potential is proportional to the laser intensity.

trap behavior. Consider the situation depicted in Fig. 1. Let a quasi-monochromatic point source of wavelength k illuminate a Fabry±Perot (F±P) etalon characterized by a thickness `, refractive index n0 ; and re¯ectance R. With aid of a positive lens, light is brought to a focus a distance f away. An opaque di€racting screen is positioned at this plane with a circular aperture concentric with the fringes. The observation plane is located at a distance z to the right. Coordinate systems …x1 ; y1 †…q1 ; /1 † and …x0 ; y0 †…q0 ; /0 † are attached to the focal plane and the observation plane respectively. In a previous report [8], we used the Rayleigh±Sommerfeld di€raction formula to analyze the di€raction of multiple interfering beams by the circular aperture. There, the far ®eld was particularly analyzed asymptotically. In this work, we focus our attention in the ®eld distribution not so far from the aperture. Well known Rayleigh±Sommerfeld di€raction formula gives for ®eld distribution at point of observation …q0 ; z†:

2. Theory

U …q0 ; /0 jz† ˆ Z Z exp…ikz† 2p a U …q; /1 † ikz 0 0  k  exp i ‰q20 ‡ q21 2q0 q1 cos…/0 2z

The beam under discussion was obtained the same way as it was described before [8]. Our system encloses three sub-systems: the FP, the focusing lens and the diaphragm. Each of them is susceptible of being electronically controlled in a prescribed manner so as to conform with a desired

 /1 †Š

 q1 dq1 d/1 :

…1†

Integration over /1 give us Bessel function and we obtain the expression: Z b A J0 …kq0 q1 =z† exp…ib1 †q1 dq1 ; U …q0 ; /0 jz† ˆ kz 0 1 Reid …2† where the ®eld amplitude distribution produced by the F±P was used [9]: U …q1 † ˆ Ai

1

T ; Re…id†

dˆ

4pn` cos h ‡ /; k

…3†

where R ‡ T ˆ 1, and Fig. 1. Schematic of the array for di€raction of F±P fringes. Dotted line is the intensity distribution of the incident ®eld across the di€racting aperture (one on-axis point-like fringe and one o€-axis ring-like fringe).

s  0 2 n q21 cos h ˆ 1 : n q21 ‡ f 2

…4†

A.A. Lipovka, M.A. Cervantes / Optics Communications 198 (2001) 101±105

The term Reid < 1 and for this reason the factor 1 …1 Reid † can be expanded in a series. By the substitution q1 ˆ bx we obtain the following expression for our integral: U …q0 † ˆ

1 2pb2 X Rm expfi2mn0 kl ‡ i/g kz mˆ0   Z 1 i um x2 x dx:  J0 …vx† exp 2 0

…5†

Here the following variables were introduced: v ˆ kq0 b=z;

um ˆ

kb2 z



2mzl n0 f 2

 1 ;

u0 ˆ

kb2 ; z …6†

and b is radius of the aperture. In order to obtain a convergent series near the aperture, we should expand this integral of the parameter v=um P in series s (more exactly …v=u † J ), because um  1 and m s s v  1 in this region. For this purpose, it is convenient to use the series representation with Lommel functions: 1

an ˆ n…2kn0 l ‡ u†

(  i U ˆb R exp…im…2kn l ‡ u†† exp um mˆ0 )   s 1  u X iv m exp i Js …v† : 2 sˆ0 um 1 X

m

0

…8†

v2 ; 2un

bn ˆ n…2kn0 l ‡ u† ‡

un : 2

…9† …7†

for …v=u† < 1 and we obtain for the ®eld U the following expression: 2

1 X m X Rm‡n cos…am an † um un mˆ0 nˆ0  X  1 X m  dmn dmn Rm‡n 1 ‡ 2 2 u m un mˆ0 nˆ0 "   s 1 X v2  cos…bm bn † Js2 u u m n sˆ0  2 s  l 1 X s X v um l ‡ … 1† um un un sˆ1 lˆ1 #  l ! 1 m 1 X X Rm‡n un ‡ Js‡l Js l um um un mˆ1 nˆ0  s 1 X s 1 X v2 l  sin …bm bn † … 1† um un sˆ1 lˆ0 "  #   l l um um un un  Js‡l Js l 1 un v um v  l 1 X 1 X 1 X Rm‡n v um un un mˆ0 nˆ0 lˆ0 !  p  Jl cos am bn l 2

8p2 I0 k2 z 2   1

where an and bn are:



 i J0 …vx† exp  ux2 x dx 2 0 "    1 v2 u ˆ  i exp  i  i exp  i u 2 2n # 1  s X s v  …  i† Js …v† u sˆ0

Z

Iˆ

103

i

v2 2um



Finally the intensity distribution at the point of observation can be straightforwardly obtained as I ˆ UU  :

In this expression the Bessel functions Jl …v† are denoted as Jl and dmn is the symbol of Kronecker. This is the exact analytical solution of the problem, which allows us to investigate in more detail the shape of the TODG, stability of the BECs in the traps and the capability of variation of the grating. In Fig. 2, we present radial plots of the intensity patterns so obtained. 3. Results of calculations One can see part of the TODG obtained under the conditions that the phase u ˆ 0 and size of the aperture b ˆ 0:212 cm. In this way, the intensity at the aperture consist of an on-axis fringe surrounded by a ring-like one (shown in Fig. 1 by

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A.A. Lipovka, M.A. Cervantes / Optics Communications 198 (2001) 101±105

Fig. 2. Coaxial toroidal grating (intensity of the di€racted beam) formed by array shown in Fig. 1. Only part of total picture is presented. Axis Z is the distance from aperture (45±50 cm), q is radius from Z-axis.

dotted line). Fig. 2 shows the structure of the toroidal traps which form the ordered coaxial toroidal grating. By use of a Fourier transforming lens the region of interest would be produced near the focal plane of said lens, the dimensions being determined by the focal length of the lens. Fig. 3 shows the e€ect of changing the diameter of the di€racting aperture to vary the number of enclosed fringes, an operation that can be accomplished, in principle, in real time by either changing the etalon spacing, the diaphragm diameter, or the lens focal length. To conclude, we would like to emphasize that, we presented the TODG based on the circular aperture di€raction of multiple interfering beams. The system can be implemented in real time and thus it o€ers the operator added ¯exibility in manipulating the trap. We would like to stress the point that, in principle, all the optical components of our system are susceptible of being functionally synchronized by using present time electronic technology and, therefore, we can anticipate that a better control on the dynamics of the condensate can be obtained with the present approach. The TODG for BECs discussed in the present report can constitute a step forward to obtain the feedback mechanism necessary for the construction of a 2D atom laser.

Fig. 3. Changing the geometry of the coaxial toroidal grating. The curves shown were obtained with values of (a) b ˆ 0:212, (b) b ˆ 0:245 and (c) b ˆ 0:274. Axis Z and q are the same as in Fig. 2. Variable parameter is bÐradius of aperture.

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A.A. Lipovka, M.A. Cervantes / Optics Communications 198 (2001) 101±105 [3] S. Chu, J.E. Bjorkholm, A. Ashkin, J.P. Gordon, L.W. Hollberg, Opt. Lett. 11 (1986) 73. [4] M. Hammes, D. Rychtarik, V. Druzhinina, U. Moslener, I. Manek-Honninger, R. Grimm, J. Mod. Opt. 47 (2000) 2755. [5] D.M. Stamper-Kurn, M.R. Andrews, A.P. Chikkatur, S. Inouye, H.J. Miesner, J. Stenger, W. Ketterle, Phys. Rev. Lett. 80 (1998) 2027±2030.

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[6] J. Arlt, M.J. Padgett, Opt. Lett. 25 (2000) 191. [7] E.M. Wright, J. Arlt, K. Dholakia, Phys. Rev. A 63 (2001) 013608. [8] M.A. Cervantes, A.A. Lipovka, M.M. Tavares, Opt. Commun. 185 (2000) 233±237. [9] M. Born, E. Wolf, Principles of Optics, Pergamon Press, New York, 1975.