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Dynamically reconfigurable optical lattices

Peter John Rodrigo, Vincent Ricardo Daria, and Jesper Glückstad

Open Access Open Access

Abstract

We present a theoretical framework for two approaches of generating light-efficient optical lattices via a generic Fourier-filtering operation. We demonstrate how lattice geometry can be dynamically changed from fully discrete to interconnected optical arrays conveniently achieved with virtually zero computational resources. Both approaches apply a real-time reconfigurable phase-only spatial light modulator to set up dynamic input phase patterns for a 4-f spatial filtering system that synthesizes the optical lattices. The first method is based on lossless phase-only Fourier-filtering; the second, on amplitude-only Fourier-filtering. We show numerically generated optical lattices rendered by both schemes and quantify the strength of the light throughput that can be achieved by each filtering alternative.

©2005 Optical Society of America

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Figures (9)
Fig. 1.
Fig. 1. 4-f Fourier-filtering setup with phase-only SLM.

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Fig. 2.
Fig. 2. Plot of the SRW for η=2.0 (red), η=4.0 (green), and η=20.0 (blue). As η is increased, the SRW approaches a flattop profile (black).

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Fig. 3.
Fig. 3. Optical lattices generated using a phase-only filter{{A=1,B=1,θ=π,η=4} with (a) binary (75% zero and 25% π levels) and (b) ternary (50% ϕ0=π/2, 25% ϕ1=0, and 25% ϕ2=π) phase input, respectively. (c) Intensity plot of incident beam to the phase input. Line scan along the horizontal direction in (a) and in (b) shows maximum intensity four times and twice, respectively, the maximum incident intensity (c) indicating light efficiency of close to 100%.

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Fig. 4.
Fig. 4. (a) Input ternary phase pattern and (b) corresponding high contrast intensity lattice at the observation plane. Setting ϕ0=π/2, ϕ1=0, and ϕ2=π, results in I 0=0 and I 1≈I 2>0.

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Fig. 5.
Fig. 5. (a) Binary rectangular-array phase pattern with fill factor (Δxc Δyc )/(ΔxΔy) and (b) its possible spatial Fourier components having diffraction orders indicated by indices m, n. The applied filter transmits only the zero-order (open circle) and the four first-orders (black filled circles).

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Fig. 6.
Fig. 6. Light throughput at various fill factor F (solid curve) measured by taking the sum of the intensities of the allowed diffraction orders. The intensity axis is normalised with respect to the incident intensity.

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Fig. 7.
Fig. 7. Intensity lattices taken at the observation plane formed by amplitude-only filtering at the Fourier plane. The structure of the output intensity lattices varies with different input phase fill factor, F, (a) F=0.25, (b) F=0.5 and (c) F=0.

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Fig. 8.
Fig. 8. Dependence of the coefficients a 2, -2ab, and b 2 from Eq. (13) on fill factor. The intensity axis is normalized with the intensity of the input field of unity-amplitude.

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Fig. 9.
Fig. 9. The optical lattice at F=0.12. Unlike in Fig. 7(a), secondary intensity maxima are no longer present. The intensity islands are interconnected along the horizontal and vertical directions.

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Equations (18)

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e ( x , y ) = circ ( r ⁄ Δ r ) exp ( i ϕ ( x , y ) )
H ( f x , f y ) = A ⌈ 1 + ( BA − 1 exp ( i θ ) − 1 ) circ ( f r ⁄ Δ f r ) ⌉ .
I ( x ′ , y ′ ) = A 2 ∣ exp ( i ϕ ˜ ( x ′ , y ′ ) ) circ ( r ′ ⁄ Δ r ) + ∣ α ̅ ∣ ( BA − 1 exp ( i θ ) − 1 ) g ( r ′ ) ∣ 2 ,
g ( r ′ ) = 2 π Δ r ∫ 0 Δ f r J 1 ( 2 π Δ r f r ) J 0 ( 2 π r ′ f r ) d f r
{ α ̅ = ( π ( Δ r ) 2 ) − 1 ∬ x 2 + y 2 ≤ Δ r exp ( i ϕ ( x , y ) ) d x d y = ∣ α ̅ ∣ exp ( i ϕ α ̅ ) ϕ ˜ ( x ′ , y ′ ) = ϕ ( x ′ , y ′ ) − ϕ α ̅
I ( x ′ , y ′ ) = ∣ exp ( i ϕ ˜ ( x ′ , y ′ ) ) circ ( r ′ ⁄ Δ r ) − 2 ∣ α ̅ ∣ g ( r ′ ) ∣ 2 .
g ( r ′ Δ r ) = 2 π ∫ 0 0.61 η J 1 ( 2 π f r ) J 0 ( 2 π r ′ Δ r f r ) d f r .
e ( x , y ) = exp ( i φ ⁄ 2 )
+ ( exp ( − i φ ⁄ 2 ) − exp ( i φ ⁄ 2 ) ) · rect ( x Δ x c , y Δ y c ) ⊗ ∑ m , n δ ( x − m Δ x , y − Δ y )
E ( f x , f y ) = exp ( i φ ⁄ 2 ) δ ( f x , f y )
− 2 i sin ( φ ⁄ 2 ) Δ x c Δ x Δ y c Δ y · ∑ m , n sinc ( m Δ x c Δ x , n Δ y c Δ y ) δ ( f x − m Δ x , f y − n Δ y ) .
E filtered ( f x , f y ) = exp ( i φ ⁄ 2 ) δ ( f x , f y )
− 2 i sin ( φ ⁄ 2 ) Δ x c Δ x Δ y c Δ y . ∑ ( m , n ) = ( 0 , 0 ) , ( 1 , 0 ) ( 1 , 0 ) , ( − 1 , 0 ) , ( 0 , − 1 ) sinc ( m Δ x c Δ x , n Δ y c Δ y ) δ ( f x − m Δ x , f y − n Δ y ) .
E filtered , π ( f x , f y ) = i [ δ ( f x , f y ) − 2 Δ x c Δ x Δ y c Δ y ∑ ( m , n ) = ( 0 , 0 ) , ( 1 , 0 ) ( 1 , 0 ) , ( − 1 , 0 ) , ( 0 , − 1 ) sinc ( m Δ x c Δ x , n Δ y c Δ y ) δ ( f x − m Δ x , f y − n Δ y ) ] .
{ E 0 , 0 ( f x , f y ) = i [ 1 − 2 Δ x c Δ x Δ y c Δ y ] δ ( f x , f y ) E 1 , 0 ( f x , f y ) = − i [ 2 Δ x c Δ x Δ y c Δ y . sinc ( Δ x c Δ x ) δ ( f x − 1 Δ x , f y ) ] E 0 , 1 ( f x , f y ) = − i [ 2 Δ x c Δ x Δ y c Δ y . sinc ( Δ y c Δ x ) δ ( f x , f y − 1 Δ y ) ] E − 1 , 0 ( f x , f y ) = − i [ 2 Δ x c Δ x Δ y c Δ y . sinc ( Δ x c Δ x ) δ ( f x − 1 Δ x , f y ) ] E 0 , − 1 ( f x , f y ) = − i [ 2 Δ x c Δ x Δ y c Δ y . sinc ( Δ y c Δ y ) δ ( f x , f y + 1 Δ y ) ]
I ( x ′ , y ′ ) = a 2 − 2 ab cos [ π Δ x ( x ′ + y ′ ) ] cos [ π Δ x ( x ′ − y ′ ) ]
+ b 2 cos 2 [ π Δ x ( x ′ + y ′ ) ] cos 2 [ π Δ x ( x ′ − y ′ ) ]
{ a = 1 − 2 F b = 8 F sinc ( F 1 ⁄ 2 )
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