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Optimized holographic optical traps

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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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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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