Abstract

Two-dimensional spatial light modulators have been employed to create static and dynamic phase masks for embedding multiple vortices and exotic intensity-void structures in laser beams. A variety of patterns of singularities, producing dark longitudinal and transverse intensity channels, have been created. The uniformity, quality, and suitability of these patterns as elements for atom optics (e.g., atom-tunnel beam splitters) have been studied as a function of the phase quantization level and spatial resolution of the phase mask. Specifically, we show that (1) high-quality modes, those that propagate long distances and can be focused, can be generated when the number of phase steps between 0 and 2π on the phase mask exceed four and (2) atom confinement increases with the charge of the vortex.

© 2006 Optical Society of America

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2005 (1)

2004 (1)

S. Orlov and A. Stabinis, "Propagation of superpositions of coaxial optical Bessel beams carrying vortices," J. Opt. A, Pure Appl. Opt. 6, S259-S262 (2004) and references therein.
[CrossRef]

2003 (2)

2000 (1)

1999 (1)

1998 (1)

J. Yin, Y. Zhu, W. Jhe, and Z. Wang, "Atom guiding and cooling in a dark hollow laser beam," Phys. Rev. A 58, 509-513 (1998).
[CrossRef]

1997 (1)

1996 (1)

1993 (1)

G. Indebetouw, "Optical vortices and their propagation," J. Mod. Opt. 40, 73-87 (1993).
[CrossRef]

1987 (1)

1974 (1)

J. F. Nye and M. V. Berry, "Dislocations in wave trains," Proc. R. Soc. London, Ser. A 336, 165-190 (1974).
[CrossRef]

1970 (1)

J. W. Goodman and A. M. Silvestri, "Some effects of Fourier-domain phase quantization," IBM J. Res. Dev. 14, 478-484 (1970).
[CrossRef]

Allen, L.

L. Allen, S. M. Barnett, and M. J. Padgett, Optical Angular Momentum (Institute of Physics, 2003).
[CrossRef]

Alonzo, C. A.

Barnett, S. M.

L. Allen, S. M. Barnett, and M. J. Padgett, Optical Angular Momentum (Institute of Physics, 2003).
[CrossRef]

Berry, M. V.

J. F. Nye and M. V. Berry, "Dislocations in wave trains," Proc. R. Soc. London, Ser. A 336, 165-190 (1974).
[CrossRef]

Carcole, E.

Chattrapiban, N.

Cofield, D.

Cottrell, D. M.

Davis, J. A.

Durnin, J.

Gluckstad, J.

Goodman, J. W.

J. W. Goodman and A. M. Silvestri, "Some effects of Fourier-domain phase quantization," IBM J. Res. Dev. 14, 478-484 (1970).
[CrossRef]

Hill, W. T.

Indebetouw, G.

G. Indebetouw, "Optical vortices and their propagation," J. Mod. Opt. 40, 73-87 (1993).
[CrossRef]

Jhe, W.

J. Yin, Y. Zhu, W. Jhe, and Z. Wang, "Atom guiding and cooling in a dark hollow laser beam," Phys. Rev. A 58, 509-513 (1998).
[CrossRef]

Law, C. T.

Lee, W. M.

Milam, D.

Molina-Terriza, G.

Nye, J. F.

J. F. Nye and M. V. Berry, "Dislocations in wave trains," Proc. R. Soc. London, Ser. A 336, 165-190 (1974).
[CrossRef]

Orlov, S.

S. Orlov and A. Stabinis, "Propagation of superpositions of coaxial optical Bessel beams carrying vortices," J. Opt. A, Pure Appl. Opt. 6, S259-S262 (2004) and references therein.
[CrossRef]

Padgett, M. J.

L. Allen, S. M. Barnett, and M. J. Padgett, Optical Angular Momentum (Institute of Physics, 2003).
[CrossRef]

Recolons, J.

Rodrigo, P. J.

Rogers, E. A.

Roy, R.

Rozas, D.

Silvestri, A. M.

J. W. Goodman and A. M. Silvestri, "Some effects of Fourier-domain phase quantization," IBM J. Res. Dev. 14, 478-484 (1970).
[CrossRef]

Song, Y.

Stabinis, A.

S. Orlov and A. Stabinis, "Propagation of superpositions of coaxial optical Bessel beams carrying vortices," J. Opt. A, Pure Appl. Opt. 6, S259-S262 (2004) and references therein.
[CrossRef]

Swartzlander, G. A.

Tao, S. H.

Torner, L.

Wang, Z.

J. Yin, Y. Zhu, W. Jhe, and Z. Wang, "Atom guiding and cooling in a dark hollow laser beam," Phys. Rev. A 58, 509-513 (1998).
[CrossRef]

Yin, J.

J. Yin, Y. Zhu, W. Jhe, and Z. Wang, "Atom guiding and cooling in a dark hollow laser beam," Phys. Rev. A 58, 509-513 (1998).
[CrossRef]

Yuan, X.-C.

Zhu, Y.

J. Yin, Y. Zhu, W. Jhe, and Z. Wang, "Atom guiding and cooling in a dark hollow laser beam," Phys. Rev. A 58, 509-513 (1998).
[CrossRef]

Appl. Opt. (1)

IBM J. Res. Dev. (1)

J. W. Goodman and A. M. Silvestri, "Some effects of Fourier-domain phase quantization," IBM J. Res. Dev. 14, 478-484 (1970).
[CrossRef]

J. Mod. Opt. (1)

G. Indebetouw, "Optical vortices and their propagation," J. Mod. Opt. 40, 73-87 (1993).
[CrossRef]

J. Opt. A, Pure Appl. Opt. (1)

S. Orlov and A. Stabinis, "Propagation of superpositions of coaxial optical Bessel beams carrying vortices," J. Opt. A, Pure Appl. Opt. 6, S259-S262 (2004) and references therein.
[CrossRef]

J. Opt. Soc. Am. A (1)

J. Opt. Soc. Am. B (1)

Opt. Express (1)

Opt. Lett. (4)

Phys. Rev. A (1)

J. Yin, Y. Zhu, W. Jhe, and Z. Wang, "Atom guiding and cooling in a dark hollow laser beam," Phys. Rev. A 58, 509-513 (1998).
[CrossRef]

Proc. R. Soc. London, Ser. A (1)

J. F. Nye and M. V. Berry, "Dislocations in wave trains," Proc. R. Soc. London, Ser. A 336, 165-190 (1974).
[CrossRef]

Other (1)

L. Allen, S. M. Barnett, and M. J. Padgett, Optical Angular Momentum (Institute of Physics, 2003).
[CrossRef]

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Figures (17)

Fig. 1
Fig. 1

Phase masks and a cross section of the converted beam for a charge 1 vortex embedded on a plane profile leading to a LG 10 mode (left column) and a Bessel profile leading to a J 1 mode (right column). The parameter Δ indicates the physical size of a 2 π phase range on the mask.

Fig. 2
Fig. 2

From left to right, a phase mask and the corresponding beam profile of a channel beam for a vortex with a half-integer charge embedded in a plane background [Eq. (3), with n = 7 2 , α = 0 , and δ = 0 ]. The mask has four sections, three of which have phase variation of 0 to 2 π (black to white). The fourth contains a phase variation of 0 to π (black to gray). The location of the π dislocation currently along the positive x axis can be rotated by changing δ.

Fig. 3
Fig. 3

Experimental arrangement consisting of (1) a Ti:sapphire laser, tuned near 780 nm , which is expanded and collimated by lenses L 1 and L 2 before being sent to the SLM; (2) neutral density filters to control the intensity of the modulated beam before being detected by a CCD camera, and (3) a PC to process the image and control the SLM.

Fig. 4
Fig. 4

Normalized power in the first Bessel ring, P fr ( N , G , n ) , for α = 5500 m 1 as N, G, and n are varied. (a) fixed spatial resolution, P fr ( N = 768 , G , n ) (top); (b) fixed phase quantization level, P fr ( N , G = 256 , n ) showing an approximate n trend, as discussed in the text (middle); and (c) fixed charge, P fr ( N , G , n = 1 ) (bottom).

Fig. 5
Fig. 5

Intensity within a small wedged area for n = 1 , 5, 10, and 15, with N = 768 , G = 256 , and α = 5500 m 1 , as a function of angle around the ring showing no holes in the profile. The area consisted of a 2° wedge bounded by the FWHM determined by the front and back slopes of the ring. The large periodic fluctuations are due to reflection between the input and generated beams, as discussed in the text.

Fig. 6
Fig. 6

Intensity profiles of the first Bessel ring generated with α = 5500 m 1 for n = 1 , 5, 10, and 15 as a function of normalized radial coordinate, where one unit on the axis corresponds to 330 μ m , showing the peak position and the width increasing with n. These curves correspond to N = 768 and G = 256 and were obtained by scanning two concentric circles approximately one CCD pixel width centered at the vortex radically across the first ring.

Fig. 7
Fig. 7

Modified intensity profiles of the profiles from Fig. 6. The higher-charge rings are focused so that all peaks coincide at a normalized radius of 1, where one unit on the axis corresponds to 330 μ m . Clearly, the peak intensities grow with n while the widths decrease. Compare with Fig. 6.

Fig. 8
Fig. 8

Intensity profiles of the first Bessel rings as a function of n but with different α values to make the peaks coincide at a normalized radius of 1, where one unit on the axis corresponds to 330 μ m . The units of α are inverse meters. Compare with Figs. 6, 7.

Fig. 9
Fig. 9

Viewed perpendicularly to the propagation direction of a tunnel (see Fig. 10), a cloud of cold Rb atoms expands and falls under gravity after being released from a MOT. The cloud takes the shape of the hollow Bessel beam, tuned to the blue of the Rb resonance line, which was generated with mask parameters n = 4 , N = 768 , and G = 256 . The power in the first ring is 60 mW . The Bessel beam is generated from Eq. (2) with α = 8300 m 1 . The time evolution increases from left to right with an increment of 1 ms between adjacent columns.

Fig. 10
Fig. 10

Arrangement to monitor Rb atoms confined to the optical potentials. Alignment of the probe beam for shadow imaging of the atoms along the tunnel (top) and perpendicular to the tunnel (bottom) is shown. The arrows labeled σ ± represent the MOT beams, whereas the loops represent the MOT magnetic coils. The two parallel lines represent the optical potential. The circular and elliptical clouds inside the parallel lines represent the Rb cloud before and after the MOT beams are turned off. The cube with the diagonal line represents a polarizing beam splitter, and the CCD cameras are labeled with the elements before them, representing the imaging lenses.

Fig. 11
Fig. 11

The phase mask and beam profile of tiled Bessel beams as described in the text. The mask is generated with α = 1800 m 1 , N = 768 , G = 256 , and n = 1 .

Fig. 12
Fig. 12

Left, the phase mask created from nesting seven vortices on a plane background; right, the output profile of the converted beam.

Fig. 13
Fig. 13

A sequence of pictures of a laser beam with two nested vortices where the distance between the singularities is increased systematically. The two vortices have the same charge ( n = 1 ) , and the Bessel background has α 7000 m 1 . The separation increases clockwise from top left in increments of 0.6 mm . When the two vortices are separated far enough that they encounter the neighboring ring (separation equals 4.2 mm in this case), they break up as shown in the lower left picture.

Fig. 14
Fig. 14

Left to right, a phase mask and a beam profile of a channel beam in a plane background. The origins of the ϕ j terms, i.e., ( x j , y j ) , are ( 0 , 0.40 ) , ( 0.20 , 0.35 ) , and ( 0.20 , 0.35 ) mm , where (0,0) is at the center of the mask. The parameter δ = 9 π 25 rad .

Fig. 15
Fig. 15

Possible beam splitter formed by intersecting two beams at a right angle containing line singularities (one with three channels and the other with one channel). The intersection would create a Y-shaped rectangular channel. The vertical channel divides into two nonorthogonal paths.

Fig. 16
Fig. 16

Possible beam splitter formed by intersecting two beams at a right angle containing line singularities (each with three channels). In this case, the vertical channel divides into four nonorthogonal paths.

Fig. 17
Fig. 17

Shadow images viewed along the beam of cold Rb atoms confined to several exotic optical potentials. From the left this figure shows: atoms freely expanding 3 ms after the MOT beams were turned off; atoms confined to the first ring of a J 0 beam generated with α = 8500 m 1 ; atoms confined within a J 4 beam generated with α = 10000 m 1 ; and atoms confined within the intensity voids of two nested vortices separated by 4 mm generated with α = 8000 m 1 . The three left images were taken 3 ms after the MOT beams were turned off, whereas the right image was taken 10 ms after the MOT was turned off.

Equations (6)

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Φ Vor ( ρ , ϕ ) = ( n ϕ + κ ρ ) mod 2 π ,
Φ Bes Vor ( ρ , ϕ ) = Φ Bes + Φ Vor ( κ ρ = 0 ) = ( α ρ + n ϕ ) mod 2 π .
Φ half = [ ( 2 m + 1 2 ) ϕ + δ ] mod 2 π ,
Φ nest = [ j = 1 M n j ϕ j ( x , y ) ] mod 2 π ,
Φ Bes nest = [ α ρ + j = 1 M n j Φ j ( x , y ) ] mod 2 π .
Φ Nest Half = { j = 1 M [ 1 2 ϕ j ( x , y ) ] + δ } mod π ,

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