Abstract

Holographic optical traps use the forces exerted by computer-generated holograms to trap, move and otherwise transform mesoscopically textured materials. This article introduces methods for optimizing holographic optical traps’ efficiency and accuracy, and an optimal statistical approach for characterizing their performance. This combination makes possible real-time adaptive optimization.

© 2005 Optical Society of America

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References

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    [CrossRef]
  15. A. Jesacher, S. Fürhapter, S. Bernet and M. Ritsch-Marte. �??Size selective trapping with optical �??cogwheel�?? tweezers.�?? Opt. Express 12, 4129�??4135 (2004).
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  24. L. P. Ghislain, N. A. Switz and W. W. Webb. �??Measurement of small forces using an optical trap.�?? Rev. Sci. Instr. 65, 2762�??2768 (1994).
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  25. F. Gittes, B. Schnurr, P. D. Olmsted, F. C. MacKintosh and C. F. Schmidt. �??Microscopic viscoelasticity: Shear moduli of soft materials determined from thermal fluctuations.�?? Phys. Rev. Lett. 79, 3286�??3289 (1997).
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  27. K. Berg-Sørensen and H. Flyvbjerg. �??Power spectrum analysis for optical tweezers.�?? Rev. Sci. Instr. 75, 594�??612 (2004).
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  28. F. Gittes and C. F. Schmidt. �??Interference model for back-focal-plane displacement detection in optical tweezers.�?? Opt. Lett. 23, 7�??9 (1998).
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    [CrossRef]
  32. M. Polin, D. G. Grier and S. Quake. �??Anomalous vibrational dispersion in holographically trapped colloidal arrays.�?? Phys. Rev. Lett. submitted for publication (2005).
    [PubMed]
  33. P. T. Korda, G. C. Spalding and D. G. Grier. �??Evolution of a colloidal critical state in an optical pinning potential.�?? Phys. Rev. B 66, 024504 (2002).
    [CrossRef]

Acta Cryst. (1)

J. L. Aragón, G. G. Naumis and M. Torres. �??A multigrid approach to the average lattices of quasicrystals.�?? Acta Cryst. A58, 352�??360 (2002).

Appl. Phys. A (1)

E.-L. Florin, A. Pralle, E. H. K. Stelzer and J. K. H. Hörber. �??Photonic force microscope calibration by thermal noise analysis.�?? Appl. Phys. A 66, S75�??S78 (1998).
[CrossRef]

J. Colloid Interface Sci. (1)

J. C. Crocker and D. G. Grier. �??Methods of digital video microscopy for colloidal studies.�?? J. Colloid Interface Sci. 179, 298�??310 (1996).
[CrossRef]

J. Mod. Opt. (2)

H. He, N. R. Heckenberg and H. Rubinsztein-Dunlop. �??Optical particle trapping with higher-order doughnut beams produced using high efficiency computer generated holograms.�?? J. Mod. Opt. 42, 217�??223 (1995).
[CrossRef]

N. B. Simpson, L. Allen and M. J. Padgett. �??Optical tweezers and optical spanners with Laguerre-Gaussian modes.�?? J. Mod. Opt. 43, 2485�??2491 (1996).
[CrossRef]

Nature (2)

K. Svoboda, C. F. Schmidt, B. J. Schnapp and S. M. Block. �??Direct observation of kinesin stepping by optical trapping interferometry.�?? Nature 365, 721�??727 (1993).
[CrossRef] [PubMed]

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

Opt. Commun. (3)

J. E. Curtis, B. A. Koss and D. G. Grier. �??Dynamic holographic optical tweezers.�?? Opt. Commun. 207, 169�??175 (2002).
[CrossRef]

J. Liesener, M. Reicherter, T. Haist and H. J. Tiziani. �??Multi-functional optical tweezers using computer-generated holograms.�?? Opt. Commun. 185, 77�??82 (2000).
[CrossRef]

M. Meister and R. J.Winfield. �??Novel approaches to direct search algorithms for the design of diffractive optical elements.�?? Opt. Commun. 203, 39�??49 (2002).
[CrossRef]

Opt. Express (2)

Opt. Lett. (4)

Phys. Rev. B (1)

P. T. Korda, G. C. Spalding and D. G. Grier. �??Evolution of a colloidal critical state in an optical pinning potential.�?? Phys. Rev. B 66, 024504 (2002).
[CrossRef]

Phys. Rev. E (3)

S.-H. Lee and D. G. Grier. �??Flux reversal in a two-state symmetric optical thermal ratchet.�?? Phys. Rev. E 71, 060102(R) (2005).
[CrossRef]

K. Ladavac, K. Kasza and D. G. Grier. �??Sorting by periodic potential energy landscapes: Optical fractionation.�?? Phys. Rev. E 70, 010901(R) (2004).

M. Pelton, K. Ladavac and D. G. Grier. �??Transport and fractionation in periodic potential-energy landscapes.�?? Phys. Rev. E 70, 031108 (2004).
[CrossRef]

Phys. Rev. Lett. (5)

S.-H. Lee, K. Ladavac, M. Polin and D. G. Grier. �??Observation of flux reversal in a symmetric optical thermal ratchet.�?? Phys. Rev. Lett. 94, 110601 (2005).
[CrossRef] [PubMed]

P. T. Korda, M. B. Taylor and D. G. Grier. �??Kinetically locked-in colloidal transport in an array of optical tweezers.�?? Phys. Rev. Lett. 89, 128301 (2002).
[CrossRef] [PubMed]

M. Polin, D. G. Grier and S. Quake. �??Anomalous vibrational dispersion in holographically trapped colloidal arrays.�?? Phys. Rev. Lett. submitted for publication (2005).
[PubMed]

F. Gittes, B. Schnurr, P. D. Olmsted, F. C. MacKintosh and C. F. Schmidt. �??Microscopic viscoelasticity: Shear moduli of soft materials determined from thermal fluctuations.�?? Phys. Rev. Lett. 79, 3286�??3289 (1997).
[CrossRef]

H. He, M. E. J. Friese, N. R. Heckenberg and H. Rubinsztein-Dunlop. �??Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity.�?? Phys. Rev. Lett. 75, 826�??829 (1995).
[CrossRef] [PubMed]

Rev. Sci. Instr. (5)

L. P. Ghislain, N. A. Switz and W. W. Webb. �??Measurement of small forces using an optical trap.�?? Rev. Sci. Instr. 65, 2762�??2768 (1994).
[CrossRef]

K. Berg-Sørensen and H. Flyvbjerg. �??Power spectrum analysis for optical tweezers.�?? Rev. Sci. Instr. 75, 594�??612 (2004).
[CrossRef]

P. T. Korda, G. C. Spalding, E. R. Dufresne and D. G. Grier. �??Nanofabrication with holographic optical tweezers.�?? Rev. Sci. Instr. 73, 1956�??1957 (2002).
[CrossRef]

E. R. Dufresne and D. G. Grier. �??Optical tweezer arrays and optical substrates created with diffractive optical elements.�?? Rev. Sci. Instr. 69, 1974�??1977 (1998).
[CrossRef]

E. R. Dufresne, G. C. Spalding, M. T. Dearing, S. A. Sheets and D. G. Grier. �??Computer-generated holographic optical tweezer arrays.�?? Rev. Sci. Instr. 72, 1810�??1816 (2001).
[CrossRef]

Other (3)

V. Soifer, V. Kotlyar and L. Doskolovich. Iterative Methods for Diffractive Optical Elements Computation (Taylor & Francis, Bristol, PA, 1997).

G. E. P. Box and G. M. Jenkins. Time Series Analysis: Forecasting and Control (Holden-Day, San Francisco, 1976).

H. Risken. The Fokker-Planck Equation (Springer-Verlag, Berlin, 1989), 2nd ed.
[CrossRef]

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Figures (4)

Fig. 1.
Fig. 1.

Simplified schematic of a holographic optical tweezer optical train before and after modification. (a) A collimated beam is split into multiple beams by the DOE, each of which is shown here as being collimated. The diffracted beams pass through the input pupil of an objective lens and are focused into optical traps in the objective’s focal plane. The undiffracted portion of the beam, shown here with the darkest shading, also focuses into the focal plane. (b) The input beam is converging as it passes through the DOE. The DOE collimates the diffracted beams, so that they focus into the focal plane, as in (a). The undiffracted beam comes to a focus within the coverslip bounding the sample. (c) A beam block can eliminate the undiffracted beam without substantially degrading the optical traps.

Fig. 2.
Fig. 2.

(a) Design for 119 identical optical traps in a two-dimensional quasiperiodic array. (b) Trapping pattern projected without optimizations using the adaptive-additive algorithm. (c) Trapping pattern projected with optimized optics and adaptively corrected direct search algorithm. (d) Bright-field image of colloidal silica spheres 1.53 μm in diameter dispersed in water and organized in the optical trap array. The scale bar indicates 10 μm

Fig. 3.
Fig. 3.

A three-dimensional multifunctional holographic optical trap array created with the direct search algorithm. (a) Refined DOE phase pattern. (b), (c) and (d) The projected optical trap array at z = -10 μm, 0 μm and +10 μm. Traps are spaced by 1.2 μm in the plane, and the 12 traps in the middle plane consist of ℓ = 8 optical vortices. (e) Performance metrics for the hologram in (a) as a function of the number of accepted single-pixel changes. Data include the DOE’s overall diffraction efficiency as defined by Eq. (15), the projected pattern’s RMS error from Eq. (16), and its uniformity, 1 - u, where u is defined in Eq. (17).

Fig. 4.
Fig. 4.

Power dependence of (a) the trap stiffness, (b) the viscous drag coefficient and (c) the viscous relaxation time for a 1.53 μm diameter silica sphere trapped by an optical tweezer in water.

Equations (48)

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E ( r ) = u ( ρ ) exp ( i φ ( ρ ) ) exp ( i kr ρ f ) d 2 ρ .
E ( r ) = j = 1 N u j exp ( i φ j ) T j ( r ) ,
T j ( r ) = exp ( i kr ρ j f ) .
E ( r ) = m = 1 M E m δ ( r r m ) , with
E m = α m exp ( i ξ m ) ,
E m = j = 1 N K j , m 1 T j , m exp ( i φ j ) ,
φ z ( ρ , z ) = k ρ 2 z 2 f 2 .
K j , m z = exp ( i φ z ( ρ j , z m ) ) .
φ ( ρ , ) = θ ,
K j , m = exp ( i φ ( ρ j , m ) )
Δ E m = K j , m 1 T j , m exp ( i φ j ) [ exp ( i Δ φ j ) 1 ] .
C = I + f σ ,
σ = 1 M m = 1 M ( I m γ I m ( D ) ) 2
γ = m = 1 M I m I m ( D ) m = 1 M ( I m ( D ) ) 2
= 1 M m = 1 M I m I m ( D ) ,
e rms = σ max ( I m ) .
u = max ( I m I m ( D ) ) min ( I m I m ( D ) ) max ( I m I m ( D ) ) + min ( I m I m ( D ) ) .
P ( r ) exp ( β V ( r ) ) ,
V ( r ) = 1 2 i = 1 3 κ i r i 2 ,
x ˙ ( t ) = x ( t ) τ + ξ ( t ) ,
ξ ( t ) ξ ( s ) = 2 k B T γ δ ( t s ) .
x ( t ) = x 0 exp ( t τ ) + 0 t ξ ( s ) exp ( t s τ ) ds .
x j + 1 = ϕ x j + a j + 1 ,
ϕ = exp ( Δ t τ ) ,
σ a 2 = k B T κ [ 1 exp ( 2 Δ t τ ) ] .
x j = ϕ x j 1 + a j and y j = x j + b j ,
p ( { x i } , { y i } ϕ , σ a 2 , σ b 2 ) = j = 2 N [ exp ( a j 2 2 σ a 2 ) 2 π σ a 2 ] j = 1 N [ exp ( b j 2 2 σ b 2 ) 2 π σ b 2 ] .
p ( { y j } ϕ , σ a 2 , σ b 2 ) = p ( { x j } , { y j } ϕ , σ a 2 , σ b 2 ) d x 1 d x N
= ( 2 π σ a 2 σ b 2 ) N 1 2 σ b 2 det ( A ϕ ) exp ( 1 2 σ b 2 ( y ) T [ I A σ 1 σ b 2 ] y ) ,
A ϕ = I σ b 2 + M ϕ σ a 2 ,
M ϕ = ( ϕ 2 ϕ 0 0 0 ϕ 1 + ϕ 2 ϕ 0 0 ϕ 1 + ϕ 2 ϕ 0 0 ϕ ϕ 1 + ϕ 2 ϕ 0 0 0 ϕ 1 ) .
det ( A ϕ ) = n = 1 N { 1 σ b 2 + 1 σ a 2 [ 1 + ϕ 2 2 ϕ cos ( 2 π n N ) ] }
( A ϕ 1 ) α β = 1 N n = 1 N σ a 2 σ b 2 exp ( i 2 π N n ( α β ) ) σ a 2 + σ b 2 [ 1 + ϕ 2 2 ϕ cos ( 2 π n N ) ] ,
p ( { y j } ϕ , σ a 2 , σ b 2 ) = ( 2 π ) N 2 n = 1 N { σ a 2 + σ b 2 [ 1 + ϕ 2 2 ϕ cos ( 2 π n N ) ] } 1 2
× exp ( 1 2 σ b 2 n = 1 N y n 2 ) exp ( 1 2 σ b 2 1 N m = 1 N y ˜ m 2 σ a 2 σ a 2 + σ b 2 [ 1 + ϕ 2 2 ϕ cos ( 2 π m N ) ] ) ,
L ( ϕ , σ a 2 , σ b 2 { y i } ) = N 2 ln 2 π 1 2 σ b 2 n = 1 N y n 2 + σ a 2 2 σ b 2 1 N n = 1 N y ˜ n 2 σ a 2 σ a 2 + σ b 2 [ 1 + ϕ 2 2 ϕ cos ( 2 π n N ) ]
1 2 n = 1 N ln ( σ a 2 + σ b 2 [ 1 + ϕ 2 2 ϕ cos ( 2 π n N ) ] ) .
L ϕ = L σ a 2 = L σ b 2 = 0 .
ϕ ̂ 0 = c 1 c 0 and σ ̂ a 0 2 = c 0 [ 1 ( c 1 c 0 ) 2 ] ,
c m = 1 N j = 1 N y j y ( j + m ) mod N
Δ ϕ ̂ 0 = σ ̂ a 0 2 N c 0 and Δ σ ̂ a 0 2 = σ ̂ a 0 2 2 N .
ϕ ̂ ϕ ̂ 0 { 1 + σ b 2 σ ̂ a 0 2 [ 1 ϕ ̂ 0 2 + c 2 c 0 ] } and σ ̂ a 2 σ ̂ a 0 2 σ b 2 σ ̂ a 0 2 c 0 [ 1 5 ϕ ̂ 0 4 + 4 ϕ ̂ 0 2 c 2 c 0 ] ,
κ k B T = 1 ϕ ̂ 2 σ ̂ a 2 and γ k B T Δ t = 1 ϕ ̂ 2 σ ̂ a 2 ln ϕ ̂ ,
( Δ κ κ ) 2 = ( Δ σ ̂ a 2 σ ̂ a 2 ) 2 + ( 2 ϕ ̂ 2 1 ϕ ̂ 2 ) 2 ( Δ ϕ ̂ ϕ ̂ ) 2 and
( Δγ γ ) 2 = ( Δ σ ̂ a 2 σ ̂ a 2 ) 2 + ( 2 ϕ ̂ 2 1 ϕ ̂ 2 + 1 ln ϕ ̂ ) 2 ( Δ ϕ ̂ ϕ ̂ ) 2 .
κ 0 k B T = 1 c 0 [ 1 ± 2 N ( 1 + 2 c 1 2 c 0 2 c 1 2 ) ] and γ 0 k B T Δ t = 1 c 0 ln ( c 0 c 1 ) ( 1 ± Δ γ 0 γ 0 )
N ( Δ γ 0 γ 0 ) 2 = 2 + 1 c 0 2 c 1 2 [ c 0 2 + 2 c 1 2 ln ( c 1 c 0 ) c 1 2 c 1 ln ( c 1 c 0 ) ] 2 .
α m α m i = 1 N κ i κ m .

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