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Field enhancement in a chain of optically bound dipoles

M. Guillon

Open Access Open Access

Abstract

A one-dimensional array of dipoles, optically trapped and bound in a fringe, is considered. The coupling with the incident field is studied as a function of the number of interacting dipoles. This coupling exhibits an enhancement which collapses when the chain is too long. Two possibilities are explored to keep enhancement: shrinking the coherence and spatially phase modulating the trapping light.

©2006 Optical Society of America

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Figures (9)
Fig. 1.
Fig. 1. Configuration of the trapped chain made of induced scattering dipoles.

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Fig. 2.
Fig. 2. Deviation of position from lambda periodicity for a finite chain. The longer the chain, the weaker the edge effects.

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Fig. 3.
Fig. 3. Numerical calculation of field enhancement (blue curve) for the dipole in the middle of the chain, is compared to the presented simple approximation (red curve) of uniform field. The maximum enhancement does not happen exactly for the same number of dipoles. The difference between these two curves is due to edge effects. For small Ndip , the field increases logarithmically. Curves plotted for k 3 α = 0.86.

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Fig. 4.
Fig. 4. Field seen by dipoles as a function of their position in the chain (starting from the middle). The red line represents the numerical value obtained by the infinite chain model.

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Fig. 5.
Fig. 5. Inverse of squared impedance ( 1 ∣ 1 − e 2 ikd f 2 ( k d ) ∣ 2 ) of a pair of interacting dipoles for different values of distance: d=λ 0 (blue), 2λ 0 (green), 3λ 0 (red), 4λ 0 (yellow), 5λ 0 (cyan). Curves plotted as a function of λ 0/λ.

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Fig. 6.
Fig. 6. Interaction between two dipoles.

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Fig. 7.
Fig. 7. ( E n E 0 ) 2 for an infinite chain of dipoles as a function of λ 0/λ. Curve plotted for k03 α= 0.4.

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Fig. 8.
Fig. 8. Color coded intensity as a function of the grating step k d 2 π and the phase modulation period N. Curve plotted for k 3 α = 0.86. One of the tails is enlarged in the upper right corner. Field enhancement is the strongest in this part of the curve.

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Fig. 9.
Fig. 9. Cross section of the curve on Fig. 8 for N=100 (blue curve), N=150 (green curve) and N=200 (red curve).

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Equations (16)

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E s , j = k 3 α j E j e ikr ( 1 k r − 1 ( k r ) 3 + i ( k r ) 2 )
E n = E 0 , n + ∑ j ≠ n ; j = 1 N k 3 α j E j e ikd ∣ n − j ∣ ( 1 k d ∣ n − j ∣ − 1 ( k d ∣ n − j ∣ ) 3 + i ( k d ∣ n − j ∣ ) 2 )
∀ n ∈ 〚 1 , N 〛 , E n = E 0 1 − k 3 α ∑ j ≠ 0 ; j = − N / 2 j = N / 2 e ikd ∣ j ∣ ( 1 k d ∣ j ∣ − 1 ( k d ∣ j ∣ ) 3 + i ( k d ∣ j ∣ ) 2 )
E n , res E 0 = i 3 π 2 ( 2 π ) 2 k 0 3 α = i 12 k 0 3 α
E ˜ 1 ( ω ) = E ˜ 10 ( ω ) + e ikd f ( k d ) E ˜ 2 ( ω )
E ˜ 2 ( ω ) = E ˜ 20 ( ω ) + e ikd f ( k d ) E ˜ 1 ( ω )
E ˜ 1 ( ω ) = E ˜ 10 ( ω ) + e ikd f ( k d ) E ˜ 20 ( ω ) 1 − e 2 ikd f 2 ( k d )
E ˜ 2 ( ω ) = E ˜ 20 ( ω ) + e ikd f ( k d ) E ˜ 10 ( ω ) 1 − e 2 ikd f 2 ( k d )
< W 1 > t = − 4 παε ∫ − ∞ ∞ ∣ E ˜ 1 ( ω ) ∣ 2 d ω
< W 1 > t = − 4 παε ( ∫ − ∞ ∞ ∣ E ˜ 10 ( ω ) ∣ 2 + ∣ f ( k d ) E ˜ 20 ( ω ) ∣ 2 ∣ 1 − e 2 ikd f 2 ( k d ) ∣ 2 d ω + ∫ − ∞ ∞ 2 R ( E ˜ 10 ( ω ) e − ikd f ( k d ) E ˜ 20 * ( ω ) ) ∣ 1 − e 2 ikd f 2 ( k d ) ∣ 2 d ω )
∫ − ∞ ∞ ∣ E ˜ 10 ( ω ) ∣ 2 ∣ 1 − e 2 ikd f 2 ( k d ) ∣ 2 d ω = < ∣ E ˜ 0 ( ω ) ∣ 2 > ω
+ 1 ( 2 k 0 d ) 2 ∫ − ∞ ∞ ∣ E ˜ 10 ( ω ) ∣ 2 cos ( 2 k d ) d ω + o ( 1 ( k 0 d ) 2 )
∫ − ∞ ∞ ∣ f ( k d ) E ˜ 20 ( ω ) ∣ 2 ∣ 1 − e 2 ikd f 2 ( k d ) ∣ 2 d ω = < ∣ E ˜ 0 ∣ 2 > ω ( k 0 d ) 2 + o ( 1 ( k 0 d ) 2 )
∫ − ∞ ∞ 2 R ( E ˜ 10 ( ω ) e − ikd f ( k d ) E ˜ 20 * ( ω ) ) ∣ 1 − e 2 ikd f 2 ( k d ) ∣ 2 d ω = 2 k 0 d ∫ − ∞ ∞ ∣ E ˜ 0 ( ω ) ∣ 2 cos ( k d ) d ω + o ( 1 ( k 0 d ) 2 )
E ˜ n ( ω ) = E ˜ 0 ( ω ) 1 − 2 ∑ j > 0 e i jkd f ( jkd )
E n = E 0 1 − 2 ∑ j > 0 cos ( 2 iπj / N ) e i jkd f ( jkd )
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