Abstract

We report the realization of a new iterative Fourier-transform algorithm for creating holograms that can diffract light into an arbitrary two-dimensional intensity profile. We show that the predicted intensity distributions are smooth with a fractional error from the target distribution at the percent level. We demonstrate that this new algorithm outperforms the most frequently used alternatives typically by one and two orders of magnitude in accuracy and roughness, respectively. The techniques described in this paper outline a path to creating arbitrary holographic atom traps in which the only remaining hurdle is physical implementation.

© 2008 Optical Society of America

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    [PubMed]
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    [CrossRef] [PubMed]
  28. We introduce several quantities in this manuscript which are functions of the input and output plane coordinates. For notational simplicity, we indicate this functional dependence only for the first use and when quantities are explicitly written as functions of the coordinates.
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    [CrossRef]
  38. H. Kim and B. Lee, "Diffractive optical element with apodized aperature for shaping vortex-free diffraction image," Jpn. J. App. Phys. 43, 1530-1533 (2004).
    [CrossRef]
  39. P. Senthilkumaran, F. Wyrowski, and H. Schimmel, "Vortex stagnation problem in iterative Fourier-transform algorithms," Opt. Lasers Eng. 43, 43-56 (2005).
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    [CrossRef] [PubMed]
  43. D. Jaksch, C. Bruder, J. I. Cirac, C. W. Gardiner, and P. Zoller, "Cold bosonic atoms in optical lattices," Phys. Rev. Lett. 81, 3108-3111 (1998).
    [CrossRef] [PubMed]
  44. B. DeMarco, C. Lannert, S. Vishveshwara, and T.-C. Wei, "Structure and stability of Mott-insulator shells of bosons trapped in an optical lattice," Phys. Rev. A 71, 063601 (2005).
    [CrossRef] [PubMed]
  45. A. Bezryadin, C. N. Lau, and M. Tinkham, "Quantum suppression of superconductivity in ultrathin nanowires," Nature 404, 971-974 (2000).
    [CrossRef] [PubMed]
  46. A 768×768 pixel kinoform and 256 discrete phase levels were used to match a commercially available, scientificgrade SLM: the Hamamatsu X8267.
    [PubMed]
  47. The code used to generate results for the MRAF algorithm in this paper is available at http://research.physics.uiuc.edu/DeMarco/publications.htm.
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    [CrossRef] [PubMed]

2007 (2)

B. T. Seaman, M. Krämer, D. Z. Anderson, and M. J. Holland, "Atomtronics: Ultracold-atom analogs of electronic devices," Phys. Rev. A 75, 023615 (2007).
[CrossRef] [PubMed]

D. R. Scherer, C. N. Weiler, T. W. Neely, and B. P. Anderson, "Vortex formation by merging of multiple trapped Bose-Einstein condensates," Phys. Rev. Lett. 98, 110402 (2007).
[CrossRef] [PubMed]

2006 (2)

T. Inoue, N. Matsumoto, N. Fukuchia, Y. Kobayashi, and T. Hara, "Highly stable wavefront control using a hybrid liquid-crystal spatial light modulator," Proc. SPIE 6306, 630603 (2006).
[CrossRef] [PubMed]

V. Boyer, R. M. Godun, G. Smirne, D. Cassettari, C. M. Chandrashekar, A. B. Deb, Z. J. Laczik, and C. J. Foot, "Dynamic manipulation of Bose-Einstein condensates with a spatial light modulator," Phys. Rev. A 73, 031402 (2006).
[CrossRef] [PubMed]

2005 (3)

J. Sebby-Strabley, R. T. R. Newell, J. O. Day, E. Brekke, and T. G. Walker, "High-density mesoscopic atom clouds in a holographic atom trap," Phys. Rev. A 71, 021401 (2005).
[CrossRef] [PubMed]

P. Senthilkumaran, F. Wyrowski, and H. Schimmel, "Vortex stagnation problem in iterative Fourier-transform algorithms," Opt. Lasers Eng. 43, 43-56 (2005).
[CrossRef] [PubMed]

B. DeMarco, C. Lannert, S. Vishveshwara, and T.-C. Wei, "Structure and stability of Mott-insulator shells of bosons trapped in an optical lattice," Phys. Rev. A 71, 063601 (2005).
[CrossRef] [PubMed]

2004 (6)

N. Bertaux, Y. Frauel, P. Réfrégier, and B. Javidi, "Speckle removal using a maximum-likelihood technique with isoline gray-level regularization," J. Opt. Soc. Am. A 21, 2283-2291 (2004).
[CrossRef] [PubMed]

H. Kim and B. Lee, "Diffractive optical element with apodized aperature for shaping vortex-free diffraction image," Jpn. J. App. Phys. 43, 1530-1533 (2004).
[CrossRef]

H. Kim, B. Yang, and B. Lee, "Iterative Fourier transform algorithm with regularization for the optimal design of diffractive optical elements," J. Opt. Soc. Am. A 21, 2353-2365 (2004).
[CrossRef]

J. Estève, C. Aussibal, T. Schumm, C. Figl, D. Mailly, I. Bouchoule, C. I. Westbrook, and A. Aspect, "Role of wire imperfections in micromagnetic traps for atoms," Phys. Rev. A 70, 043629 (2004).
[CrossRef] [PubMed]

O. Ripoll, V. Kettunen, and H. P. Herzig, "Review of iterative Fourier-transform algorithms for beam shaping applications," Opt. Eng. 43, 2549-2556 (2004).
[CrossRef] [PubMed]

S. Bergamini, B. Durquié, M. Jones, L. Jacubowicz, A. Browaeys, and P. Grangier, "Holographic generation of microtrap arrays for single atoms by use of a programmable phase modulator," J. Opt. Soc. Am. B 21, 1889-1894 (2004).
[CrossRef] [PubMed]

2003 (4)

D. McGloin, G. C. Spalding, H. Melville, W. Sibbett, and K. Dholakia, "Applications of spatial light modulators in atom optics," Opt. Express 11, 158-166 (2003).
[CrossRef] [PubMed]

D. G. Grier, "A revolution in optical manipulation," Nature 424, 810-816 (2003).
[CrossRef] [PubMed]

A. E. Leanhardt, Y. Shin, A. P. Chikkatur, D. Kielpinski, W. Ketterle, and D. E. Pritchard, "Bose-Einstein condensates near a microfabricated surface," Phys. Rev. Lett. 90, 100404 (2003).
[CrossRef] [PubMed]

B. Damski, J. Zakrzewski, L. Santos, P. Zoller, and M. Lewenstein, "Atomic Bose and Anderson glasses in optical lattices," Phys. Rev. Lett. 91, 080403 (2003).
[CrossRef] [PubMed]

2002 (6)

An IFTA is most generically described using the block-projection algorithm formalism, which is not necessary for the relatively simple MRAF algorithm. For a description of the block-projection formalism as applied to IFTAs, see R. Piestun and J. Shamir, "Synthesis of Three-Dimensional Light Fields and Applications," Proc. IEEE 90, 222-244 (2002).
[CrossRef] [PubMed]

S. Bühling and F. Wyrowski, "Improved transmission design algorithms by utilizing variable-strength projections," J. Mod. Opt. 49, 1871-1892 (2002).
[CrossRef] [PubMed]

J. S. Liu and M. R. Taghizadeh, "Iterative algorithm for the design of diffractive phase elements for laser beam shaping," Opt. Lett. 27, 1463-1465 (2002).
[CrossRef] [PubMed]

A. E. Leanhardt, A. P. Chikkatur, D. Kielpinski, Y. Shin, T. L. Gustavson, W. Ketterle, and D. E. Pritchard, "Propagation of Bose-Einstein condensates in a magnetic waveguide," Phys. Rev. Lett. 89, 040401 (2002).
[CrossRef] [PubMed]

J. Fortágh, H. Ott, S. Kraft, A. Günther, and C. Zimmermann, "Surface effects in magnetic microtraps," Phys. Rev. A 66, 041604 (2002).
[CrossRef] [PubMed]

P. Senthilkumaran and F. Wyrowski, "Phase synthesis in wave-optical engineering: mapping- and diffuser-type approaches," J. Mod. Opt. 49, 1831-1850 (2002).
[CrossRef] [PubMed]

2001 (2)

2000 (3)

1999 (1)

R. Ozeri, L. Khaykovich, and N. Davidson, "Long spin relaxation times in a single-beam blue-detuned optical trap," Phys. Rev. A 59, R1750-R1753 (1999).
[CrossRef] [PubMed]

1998 (2)

V. V. Kotlyar, P. G. Seraphimovich, and V. A. Soifer, "An iterative algorithm for designing diffractive optical elements with regularization," Opt. Laser Eng. 29, 261-268 (1998).
[CrossRef] [PubMed]

D. Jaksch, C. Bruder, J. I. Cirac, C. W. Gardiner, and P. Zoller, "Cold bosonic atoms in optical lattices," Phys. Rev. Lett. 81, 3108-3111 (1998).
[CrossRef] [PubMed]

1997 (1)

1996 (1)

H. Aagedal, M. Schmid, T. Beth, S. Tiewes, and F. Wyrowski, "Theory of speckles in diffractive optics and its application to beam shaping," J. Mod. Opt. 43, 1409-1421 (1996).
[CrossRef]

1995 (1)

1994 (1)

1990 (2)

1988 (1)

1987 (1)

1986 (1)

H. Akahori, "Spectrum leveling by an iterative algorithm with a dummy area for synthesizing the kinoform," App. Opt. 25, 802-811 (1986).
[CrossRef]

1980 (1)

J. R. Fienup, "Iterative method applied to image reconstruction and to computer-generated holograms," Opt. Eng. 19, 297-305 (1980).
[PubMed]

1975 (1)

App. Opt. (1)

H. Akahori, "Spectrum leveling by an iterative algorithm with a dummy area for synthesizing the kinoform," App. Opt. 25, 802-811 (1986).
[CrossRef]

Appl. Opt. (5)

J. Mod. Opt. (3)

S. Bühling and F. Wyrowski, "Improved transmission design algorithms by utilizing variable-strength projections," J. Mod. Opt. 49, 1871-1892 (2002).
[CrossRef] [PubMed]

P. Senthilkumaran and F. Wyrowski, "Phase synthesis in wave-optical engineering: mapping- and diffuser-type approaches," J. Mod. Opt. 49, 1831-1850 (2002).
[CrossRef] [PubMed]

H. Aagedal, M. Schmid, T. Beth, S. Tiewes, and F. Wyrowski, "Theory of speckles in diffractive optics and its application to beam shaping," J. Mod. Opt. 43, 1409-1421 (1996).
[CrossRef]

J. Opt. Soc. Am. A (6)

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

Jpn. J. App. Phys. (1)

H. Kim and B. Lee, "Diffractive optical element with apodized aperature for shaping vortex-free diffraction image," Jpn. J. App. Phys. 43, 1530-1533 (2004).
[CrossRef]

Nature (2)

D. G. Grier, "A revolution in optical manipulation," Nature 424, 810-816 (2003).
[CrossRef] [PubMed]

A. Bezryadin, C. N. Lau, and M. Tinkham, "Quantum suppression of superconductivity in ultrathin nanowires," Nature 404, 971-974 (2000).
[CrossRef] [PubMed]

Opt. Eng. (2)

J. R. Fienup, "Iterative method applied to image reconstruction and to computer-generated holograms," Opt. Eng. 19, 297-305 (1980).
[PubMed]

O. Ripoll, V. Kettunen, and H. P. Herzig, "Review of iterative Fourier-transform algorithms for beam shaping applications," Opt. Eng. 43, 2549-2556 (2004).
[CrossRef] [PubMed]

Opt. Express (1)

Opt. Laser Eng. (1)

V. V. Kotlyar, P. G. Seraphimovich, and V. A. Soifer, "An iterative algorithm for designing diffractive optical elements with regularization," Opt. Laser Eng. 29, 261-268 (1998).
[CrossRef] [PubMed]

Opt. Lasers Eng. (1)

P. Senthilkumaran, F. Wyrowski, and H. Schimmel, "Vortex stagnation problem in iterative Fourier-transform algorithms," Opt. Lasers Eng. 43, 43-56 (2005).
[CrossRef] [PubMed]

Opt. Lett. (4)

Phys. Rev. A (7)

B. DeMarco, C. Lannert, S. Vishveshwara, and T.-C. Wei, "Structure and stability of Mott-insulator shells of bosons trapped in an optical lattice," Phys. Rev. A 71, 063601 (2005).
[CrossRef] [PubMed]

R. Ozeri, L. Khaykovich, and N. Davidson, "Long spin relaxation times in a single-beam blue-detuned optical trap," Phys. Rev. A 59, R1750-R1753 (1999).
[CrossRef] [PubMed]

B. T. Seaman, M. Krämer, D. Z. Anderson, and M. J. Holland, "Atomtronics: Ultracold-atom analogs of electronic devices," Phys. Rev. A 75, 023615 (2007).
[CrossRef] [PubMed]

J. Sebby-Strabley, R. T. R. Newell, J. O. Day, E. Brekke, and T. G. Walker, "High-density mesoscopic atom clouds in a holographic atom trap," Phys. Rev. A 71, 021401 (2005).
[CrossRef] [PubMed]

V. Boyer, R. M. Godun, G. Smirne, D. Cassettari, C. M. Chandrashekar, A. B. Deb, Z. J. Laczik, and C. J. Foot, "Dynamic manipulation of Bose-Einstein condensates with a spatial light modulator," Phys. Rev. A 73, 031402 (2006).
[CrossRef] [PubMed]

J. Fortágh, H. Ott, S. Kraft, A. Günther, and C. Zimmermann, "Surface effects in magnetic microtraps," Phys. Rev. A 66, 041604 (2002).
[CrossRef] [PubMed]

J. Estève, C. Aussibal, T. Schumm, C. Figl, D. Mailly, I. Bouchoule, C. I. Westbrook, and A. Aspect, "Role of wire imperfections in micromagnetic traps for atoms," Phys. Rev. A 70, 043629 (2004).
[CrossRef] [PubMed]

Phys. Rev. Lett. (5)

B. Damski, J. Zakrzewski, L. Santos, P. Zoller, and M. Lewenstein, "Atomic Bose and Anderson glasses in optical lattices," Phys. Rev. Lett. 91, 080403 (2003).
[CrossRef] [PubMed]

A. E. Leanhardt, Y. Shin, A. P. Chikkatur, D. Kielpinski, W. Ketterle, and D. E. Pritchard, "Bose-Einstein condensates near a microfabricated surface," Phys. Rev. Lett. 90, 100404 (2003).
[CrossRef] [PubMed]

A. E. Leanhardt, A. P. Chikkatur, D. Kielpinski, Y. Shin, T. L. Gustavson, W. Ketterle, and D. E. Pritchard, "Propagation of Bose-Einstein condensates in a magnetic waveguide," Phys. Rev. Lett. 89, 040401 (2002).
[CrossRef] [PubMed]

D. R. Scherer, C. N. Weiler, T. W. Neely, and B. P. Anderson, "Vortex formation by merging of multiple trapped Bose-Einstein condensates," Phys. Rev. Lett. 98, 110402 (2007).
[CrossRef] [PubMed]

D. Jaksch, C. Bruder, J. I. Cirac, C. W. Gardiner, and P. Zoller, "Cold bosonic atoms in optical lattices," Phys. Rev. Lett. 81, 3108-3111 (1998).
[CrossRef] [PubMed]

Proc. IEEE (1)

An IFTA is most generically described using the block-projection algorithm formalism, which is not necessary for the relatively simple MRAF algorithm. For a description of the block-projection formalism as applied to IFTAs, see R. Piestun and J. Shamir, "Synthesis of Three-Dimensional Light Fields and Applications," Proc. IEEE 90, 222-244 (2002).
[CrossRef] [PubMed]

Proc. SPIE (1)

T. Inoue, N. Matsumoto, N. Fukuchia, Y. Kobayashi, and T. Hara, "Highly stable wavefront control using a hybrid liquid-crystal spatial light modulator," Proc. SPIE 6306, 630603 (2006).
[CrossRef] [PubMed]

Other (8)

A 768×768 pixel kinoform and 256 discrete phase levels were used to match a commercially available, scientificgrade SLM: the Hamamatsu X8267.
[PubMed]

The code used to generate results for the MRAF algorithm in this paper is available at http://research.physics.uiuc.edu/DeMarco/publications.htm.

T. Willmore, Riemannian Geometry (Oxford University Press, 1997).

We introduce several quantities in this manuscript which are functions of the input and output plane coordinates. For notational simplicity, we indicate this functional dependence only for the first use and when quantities are explicitly written as functions of the coordinates.

G. C. Spalding, J. Courtial, and R. DiLeonardo, "Holographic Optical Trapping," in Structured Light and its Applications: An Introduction to Phase-Structured Beams and Nanoscale Optical Forces, D. L. Andrews, ed., (Elsevier Press) (to be published).
[PubMed]

V. Soifer, Methods for Computer Design of Diffractive Optical Elements (John Wiley and Sons, NY, 2002).
[PubMed]

We assume in this paper that only the center of a Gaussian beam interacts with the CGH, making the effect of intrinsic phase curvature negligible. Any effect of the intrisic phase curvature can be removed in the final kinoform using a compensating phase profile.
[PubMed]

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1998).
[PubMed]

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Figures (8)

Fig. 1.
Fig. 1.

Schematic geometry for an IFTA. The optical field that propagates from the input to the output plane through a focusing objective is shown in green. The field is discretized using coordinates (x,y) in the input plane and (x′,y′) in the output plane. The dashed lines represent the clear aperture of the focusing optics. The matrix used to computationally represent the input field must be enlarged beyond this region and filled with zero intensity points (dark gray) to fully resolve the output plane. The physical size of the matrix used to represent the input plane is d.

Fig. 2.
Fig. 2.

Block diagram of an IFTA.

Fig. 3.
Fig. 3.

Target intensity profiles I 0 used to characterize the performance of the MRAF algorithm. The field-of-view for images (a), (b), (c), and (e) is a 200×200 pixel and for (d) and (f) is a 400×400 pixel subset of the output plane centered on I 0. The grayscale represents intensity, with black corresponding to the regions of zero intensity. The radius of the ring in (a) is 53 pixels and the waist for each Gaussian beam and the Gaussian cross-section of the ring is 14 pixels. The maximum intensity of the Gaussian beams is three times that of the ring. Each “tip” of the star-shaped pattern in (b) is 20 pixels from the center of the star; the two lines that intersect to form each “tip” subtend a 28° angle. To create the profile shown in (b), a uniform intensity profile with a star shape was convolved with a Gaussian with a 5 pixel waist. The profile in (c) was created by convolving a 58 pixel on edge square profile with a 3 pixel averaging filter. The overall dimensions of the profile in (d) are 288 pixels wide and 325 pixels high, and the intensity in the “base” regions is increased by 33%. The Gaussian ring in (e) has a 53 pixel radius and a 7 pixel r.m.s. width. The intensity in the 10 pixel wide gaps in (e) is suppressed by a factor of 2, and the “leads” in (e) are 185 pixels from end-to-end. The 264 pixel wide Gaussian “wire” in (f) has a 3.5 pixel r.m.s. width, and the Gaussian reservoirs in (f) have a 17.6 pixels r.m.s. radius. The center of each profile is displaced from the center of the output plane by (b) 37 pixels and (c) 63 pixels; the profiles in (a), (d), (e), and (f) are centered on the output plane.

Fig. 4.
Fig. 4.

Initial phase profiles K 0 (top row) and predicted initial intensity profiles |E (1) out |2 (bottom row) chosen for the target intensity profiles in Fig. 3. The phase profiles are 768×768 pixels and are shown in grayscale modulo 2π; white corresponds to a 2π phase. Conical phase profiles with B=117 mrad/px are employed in (a) and (e). Linear gradients of 136 mrad/px and 260 mrad/px with µ=0 and π/4 are used for (b) and (c), respectively. Quadratic phase profiles with R=0.31 mrad/px 2, R=0.3 mrad/px 2, R=0.34 mrad/px 2, R=1.4 mrad/px 2, R=0.5 mrad/px 2, and R=1.6 mrad/px 2;α=0.29 are applied in (a), (b),(c), (d), (e), and (f) respectively.

Fig. 5.
Fig. 5.

Intensity profiles If for the MRAF, GS, and AA (left, middle, right) algorithms for the test target profiles. Only the intensity in the signal region is shown, and the profiles are scaled so that the total power in the signal region is the same for each.

Fig. 6.
Fig. 6.

Final kinoforms Kf (top row) and predicted intensity profiles If (bottom row) produced by the MRAF algorithm for targets (a), (b), and (c). The mixing parameters used to generate these results are: (a) 0.40, (b) 0.35, (c) 0.40, (d) 0.30, (e) 0.35, and (f) 0.30. The signal region (red) for (a) is an annulus with with inner and outer radii 25 and 81 pixels; in (b) is a circle with a 40 pixel radius; for (c) is a square 75 pixels on edge; in (d) is a region 10 pixels from the edge of the target profile; for (e) a region 10 pixels away from where the intensity is 10% of the maximum intensity; and for (f) consists of two 53 pixel radius circles separated by 264 pixels and a connective region 25 pixels wide. The measure region (yellow) in (a) is an annulus with inner and outer radii 44 and 62 pixels; in (b), (e), and (f) is defined by a region in which the intensity of the target is greater than 10% of the maximum target intensity; in (c) is a square 57 pixels on edge; and for (d) is the edge of the target profile.

Fig. 7.
Fig. 7.

Variation of measures characterizing the MRAF algorithm performance as the mixing parameter m is varied. The efficiency ξ (red), roughness ρ (black), and error η (blue) are shown for target (a) for different values of the mixing parameter m. The inset shows detail around the globally-optimized value of m. The mixing parameter that minimizes η approximately coincides with an minimum in roughness ρ for the MRAF algorithm.

Fig. 8.
Fig. 8.

Histogram of the fractional error at each pixel evaluated for If for the MRAF algorithm used on target (a). The fraction of pixels in the signal region are binned with respect to the fractional error I ~ f 2 I ~ 0 2 I ~ 0 . The width of each bin is equivalent to a 1% fractional error. The solid black, blue dotted, and red dashed lines are the result for the MRAF, GS, and AA algorithms, respectively.

Tables (1)

Tables Icon

Table 1. Table comparing the performance of the MRAF to the GS and AA algorithms. The mixing parameters used for the AA algorithm are (a) 1.9, (b) 2.0, (c) 1.9, (d) 2.0, (e) 2.2, and (f) 2.5. The GS and AA algorithms converged in 100 iterations for the results in this table.

Equations (3)

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G ( n ) = { m I 0 | SR + ( 1 m ) E out ( n ) NR } e i arg [ E out ( n ) ] .
η = 1 N MR ( x , y ) MR [ I ~ f ( x , y ) I ~ 0 ( x , y ) ] 2 I ~ 0 ( x , y ) 2 .
ρ = ( x , y ) MR { H [ I ~ f ( x , y ) I ~ 0 ( x , y ) ] } 2 N MR

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