## Field enhancement in a chain of optically bound dipoles

Optics Express, Vol. 14, Issue 7, pp. 3045-3055 (2006)

http://dx.doi.org/10.1364/OE.14.003045

Acrobat PDF (211 KB)

### 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

## 1. Introduction

1. V.A. Markel, “Divergence of dipole sums and the nature of non-Lorentzian exponentially narrow resonances in one-dimensional periodic arrays of nanospheres,” Let. Ed., J. Phys. B: At. Mol. Phys. **38**, 7, L115–L121 (2005). [CrossRef]

2. M. Burns, J.-M. Fournier, and J.A. Golovchenko, “Optical matter: crystallization and binding in intense optical fields,” Science **249**, 749–754 (Aug. 1990). [CrossRef] [PubMed]

3. S.A. Tatarkova, A.E. Carruthers, and K. Dholakia, “One-dimensional optically bound arrays of microscopic particles,” Phys. Rev. Lett. **89**, 283901, 1–4 (2002). [CrossRef]

4. V. Garcés-Chávez, K. Dholakia, and G.C. Spalding, “Extended-area optically induced organization of microparticles on a surface,” Appl. Phys. Lett. **86**, 031106 (2005). [CrossRef]

5. V. Garcés-Chávez, R. Quidant, P.J. Reece, G. Badenes, L. Torner, and K. Dholakia, “Extended organization of colloidal microparticles by surface plasmon polariton excitation,” Phys. Rev. B **73**, 085417 (2006). [CrossRef]

6. D. Rogovin, R. McGraw, and P. Yeh, “Harmonic phase conjugation in liquid suspensions of microparticles via higher-order gratings,” Phys. Rev. Lett. **55**, 2864–2867 (1985). [CrossRef] [PubMed]

7. P.W. Smith, A. Ashkin, J.E. Bjorkholm, and D.J. Eilenberger, “Studies of self-focusing bistable devices using liquid suspensions of dielectric particles,” Opt. Lett. **10**, 131–133 (1984). [CrossRef]

8. C.D. Mellor and C.D. Bain, “Array formation in evanescent wave,” ChemPhysChem **7**, 329–332 (2005). [CrossRef] [PubMed]

*r*long range interaction dependency of the field scattered by a particle in the Rayleigh range [9

9. M. M. Burns, J.-M. Fournier, and J. A. Golovchenko, “Optical binding,” Phys. Rev. Lett. **63**, 12, 1233–1296 (1989). [CrossRef] [PubMed]

10. W. Singer, M. Frick, S. Bernet, and M. Ritsch-Marte, “Self-organized array of regularly spaced microbeads in a fiber-optical trap,” J. Opt. Soc. Am. B **20**, 7, 1568–1574 (2003). [CrossRef]

9. M. M. Burns, J.-M. Fournier, and J. A. Golovchenko, “Optical binding,” Phys. Rev. Lett. **63**, 12, 1233–1296 (1989). [CrossRef] [PubMed]

12. F. Depasse and J.-M. Vigoureux, “Optical binding force between two Rayleigh particles,” J. Phys. D: Appl. Phys. **27**, 914–919 (1994). [CrossRef]

13. P. C. Chaumet and M. Nieto-Vesperinas, “Optical binding of particles with or without the presence of a flat dielectric surface,” Phys. Rev. B **64**, 035422, 1–7 (2001). [CrossRef]

*ka*< 1 (

*k*being the wave vector and

*a*the radius of the sphere), one can neglect radiation pressure, proportional to the squared volume of the sphere while gradient force is proportional to its volume. In this approximation, it can be checked numerically that potential minima are distributed every lambda with a good approximation even when the chain is finite (Fig. 2). This calculation has been performed neglecting scattering force and looking for the energy minimum of each dipole, starting from a

*λ*periodicity. Then, we apply infinitesimal variations in a multi-iterative process.

*λ*: dipoles in the middle of chain see an infinite-like chain. On the edges, the separation distance increases: edge effects are similar to periodicity defects in solid state physics, for layers close to the surface (the bulk appears after five or six layers). This result also looks like periodicity defects experimentally observed for an optically levitated chain of microspheres in the Mie regime[14

14. V. Garcés-Chávez, D. Roskey, M.D. Summers, H. Melville, D. McGloin, E.M. Wright, and K. Dholakia, “Optical levitation in a Bessel light beam,” Appl. Phys. Lett. **85**, 4001–4003 (2004). [CrossRef]

## 2. Finite number of interacting induced coherent dipoles

*E*

_{j}is the field seen by the

*j*

^{th}dipole,

*r*the distance to this dipole, and

*ka*< 1),

*m*being the refractive index of the dielectric sphere and

*a*its radius. Assuming these spheres self-organize into a “d-periodic” linear chain, the self-consistent equations to solve are then scalar:

*n*∈〚1,

*N*〛,

*N*dipoles of the chain except the

*n*

^{th}for which we calculate the field. Working with identical spheres, and the incident fields

*E*

_{0,n}being the same for every dipole we assume, when the chain is big, that the field seen by every dipole is the same (translation invariance). In this case, the system is easily solved and gives:

*kd*= 2

*π*), the real part of the summation at the denominator covers the range from zero to infinity when increasing the number

*N*of interacting dipoles. The field

*E*

_{n}is then maximized when the real part of the denominator is zero. This enhancement also depends on the polarizability

*k*

^{3}

*α*as already numerically observed [16

16. Mufei Xiao and Sergey I. Bozhevolnyi, “Resonant field enhancement by a finite-size periodic array of surface scatterers,” J.Phys.: Condens. Matter **13**, 3001–3010 (2001). [CrossRef]

*kd*= 2

*π*and the summation

*n*×

*n*matrix must be inversed. As we are interested in translation-invarianced long structures, we set particles every lambda. A more realistic model should take into account periodicity defects, especially close to boundaries. However, looking for potential minima of each dipole requires much longer computational processes. Moreover, there may be not a single stable solution.

*n*

^{th}dipole feels mostly the incident field, slightly enhanced by other dipoles. In this regime, the field increases as the logarithm of the number of interacting dipoles. As the scattered field of a dipole has a 1/

*r*dependency, for a given number of dipoles, the scattered field becomes stronger than the incident field. When the number of dipoles is too big, the impedance of the resonator is too high to be efficiently coupled with the incident field. The field

*E*

_{n}is then phase-opposed to the incident field

*E*

_{0,n}. For a finite chain, all the dipoles might not be in the same regime. Indeed, spheres at the edges interact with roughly half less dipoles than the ones in the middle. The impedance of the chain can be defined as:

*α*|

*E*

_{n,res}|

^{2}) increases when the polarizability decreases. Finally, in this case, the rate between excitation and response is purely imaginary.

## 3. Coherence length of the trapping light

### 3.1. Two dipoles case

*Ẽ*

_{1}and

*Ẽ*

_{2}:

*W*>

_{t}= -4

*παε*<|

*E*(

*t*)|

^{2}>

_{t}. According to the Wiener-Khinchin theorem, <

*W*>

_{t}can be expressed in reciprocal domain, as:

*Ẽ*

_{1}, we get:

*d*is very small compared to the coherence length of the incident light, the pair of dipole sees a perfectly monochromatic incident light. In the former calculation, the characteristic widths of

*Ẽ*

_{10}and

*Ẽ*

_{20}are much smaller than variations of the impedance and than

*e*

^{ikd}

*f*(

*kd*). These terms can then be set as constants in the integral, which gives the results for the monochromatic case.

*Ẽ*

_{0j}, centered on

*ω*

_{0}. For large distances,

*f*(

*kd*) can be approximated by

*Ẽ*

_{10}=

*Ẽ*

_{20}=

*Ẽ*

_{0}. That is to say that the two dipoles are coherently excited. In this approximation, the calculation of the three parts of <

*W*>

_{t}gives:

*Ẽ*

_{0}|

^{2}>

_{ω}, is a constant, and is the energy well where dipole 1 sits, due to the single incident field. The second one of the same equation is the auto-coherence of the twice scattered field

*Ẽ*

_{01}. It is the product between the incident field and the field scattered by dipole 1, which has traveled the distance

*d*twice, after a reflection on the second dipole. This double scattering explains the

*d*is smaller than the coherence length of the source. The relevant parameter is then the first harmonic of the autocorrelation function of the incident field. When

*d*is bigger than the coherence length of the binding light, the interaction between the two dipoles is weaker and more precisely, averaged interaction over time has a

### 3.2. Infinite chain of dipoles

*λ*(see Fig. 7). For a self-organized chain of dipoles, the step will self-tune so as to keep field enhancement. There are here two possibilities, two stable crystalline states, similar to allotropic phases in solid state physics. The difference between these two phases will be a slightly different step. On the curve presented on Fig. 7, the relative step difference is typically 0.2%. According to the first paragraph, the binding energy is maximum for a finite number of interacting dipoles. A solution is to widen the spectrum of the source in order to reduce the number of coherently interacting dipoles. Multiplying the presented curve (Fig. 7) by a Gaussian spectrum, the spectrum of the propagating field in the chain has two peaks. In time domain, the field beats. In this configuration as in the one presented in the first part, the enhancement of the field increases when

*k*

^{3}

*α*decreases. Meanwhile, the infinite chain approximation is correct for a larger number of dipoles when

*k*

^{3}

*α*is small.

*λ*over an infinite distance, the chain should be a melting of the two possible allotropic crystalline states.

## 4. Space modulation of the phase of the trapping light

*E*

_{0}in the former equations by

*E*

_{0}

*e*

^{2iπn/N}. Now, the “

*N*” is the phase modulation period of the incident field and not the number of dipoles (like in the first paragraph) as the chain we consider is infinite. We assume the same periodic dependency for the amplitude of the field seen by the dipoles. In Eq. (1),

*E*

_{n}is then replaced by

*E*

_{n}

*e*

^{2iπn/N}. With this hypothesis, the field seen by the

*n*

^{th}dipole is:

*kd*. In this summation, only a few terms have a simple mathematical expression [17]. However, some numerical results are presented on Figs. 8 and 9.

*kd*= 2

*π*and

*N*tends to infinity, the field falls down to zero as already observed in the previous paragraphs. When the step of the phase modulation is short enough, a second maximum appears, two crystalline states are then stable. In this experimental configuration, the chain will have very likely two allotropic crystalline phases. We assume the chain could be formed with a single phase or a binary melting phase. The final separation distance between particles will depend on the building process of the chain, on the relative stability of the two allotropic states and on external fluctuations as already discussed in the previous paragraph. We can also notice that maximum field enhancement is only one half than the one for a finite chain. By using a more complicated structure for phase modulation, it should be possible to improve this ratio and to get closer to the finite chain case.

## 5. Conclusion

18. O. Moine and B. Stout, “Optical force calculations in arbitrary beams using the vector addition theorem,” J. Opt. Soc. Am. B **22**, 1620–1631 (2005). [CrossRef]

## Acknowledgments

## References and links

1. | V.A. Markel, “Divergence of dipole sums and the nature of non-Lorentzian exponentially narrow resonances in one-dimensional periodic arrays of nanospheres,” Let. Ed., J. Phys. B: At. Mol. Phys. |

2. | M. Burns, J.-M. Fournier, and J.A. Golovchenko, “Optical matter: crystallization and binding in intense optical fields,” Science |

3. | S.A. Tatarkova, A.E. Carruthers, and K. Dholakia, “One-dimensional optically bound arrays of microscopic particles,” Phys. Rev. Lett. |

4. | V. Garcés-Chávez, K. Dholakia, and G.C. Spalding, “Extended-area optically induced organization of microparticles on a surface,” Appl. Phys. Lett. |

5. | V. Garcés-Chávez, R. Quidant, P.J. Reece, G. Badenes, L. Torner, and K. Dholakia, “Extended organization of colloidal microparticles by surface plasmon polariton excitation,” Phys. Rev. B |

6. | D. Rogovin, R. McGraw, and P. Yeh, “Harmonic phase conjugation in liquid suspensions of microparticles via higher-order gratings,” Phys. Rev. Lett. |

7. | P.W. Smith, A. Ashkin, J.E. Bjorkholm, and D.J. Eilenberger, “Studies of self-focusing bistable devices using liquid suspensions of dielectric particles,” Opt. Lett. |

8. | C.D. Mellor and C.D. Bain, “Array formation in evanescent wave,” ChemPhysChem |

9. | M. M. Burns, J.-M. Fournier, and J. A. Golovchenko, “Optical binding,” Phys. Rev. Lett. |

10. | W. Singer, M. Frick, S. Bernet, and M. Ritsch-Marte, “Self-organized array of regularly spaced microbeads in a fiber-optical trap,” J. Opt. Soc. Am. B |

11. | M. Guillon, “Optical trapping in rarefied media: towards laser-trapped space telescopes,” in |

12. | F. Depasse and J.-M. Vigoureux, “Optical binding force between two Rayleigh particles,” J. Phys. D: Appl. Phys. |

13. | P. C. Chaumet and M. Nieto-Vesperinas, “Optical binding of particles with or without the presence of a flat dielectric surface,” Phys. Rev. B |

14. | V. Garcés-Chávez, D. Roskey, M.D. Summers, H. Melville, D. McGloin, E.M. Wright, and K. Dholakia, “Optical levitation in a Bessel light beam,” Appl. Phys. Lett. |

15. | J.D. Jackson, “Classical electrodynamics,” second edition, New York, John Wiley & Son, Chpt 9. |

16. | Mufei Xiao and Sergey I. Bozhevolnyi, “Resonant field enhancement by a finite-size periodic array of surface scatterers,” J.Phys.: Condens. Matter |

17. | I.S. Gradshteyn and I.M. Ryzhik, “Table of integrals series and products,” (sixth edition, Alan Jeffrey, Academic Press, 2000). |

18. | O. Moine and B. Stout, “Optical force calculations in arbitrary beams using the vector addition theorem,” J. Opt. Soc. Am. B |

**OCIS Codes**

(020.7010) Atomic and molecular physics : Laser trapping

(120.5060) Instrumentation, measurement, and metrology : Phase modulation

(170.4520) Medical optics and biotechnology : Optical confinement and manipulation

(290.5870) Scattering : Scattering, Rayleigh

**ToC Category:**

Trapping

**History**

Original Manuscript: February 21, 2006

Revised Manuscript: March 23, 2006

Manuscript Accepted: March 23, 2006

Published: April 3, 2006

**Citation**

M. Guillon, "Field enhancement in a chain of optically bound dipoles," Opt. Express **14**, 3045-3055 (2006)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-7-3045

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### References

- V.A. Markel, "Divergence of dipole sums and the nature of non-Lorentzian exponentially narrow resonances in one-dimensional periodic arrays of nanospheres," Let. Ed., J. Phys. B: At. Mol. Phys. 38,7, L115-L121 (2005). [CrossRef]
- M. Burns, J.-M. Fournier, J.A. Golovchenko, "Optical matter: crystallization and binding in intense optical fields," Science 249,749-754 (Aug. 1990). [CrossRef] [PubMed]
- S.A. Tatarkova, A.E. Carruthers, K. Dholakia, "One-dimensional optically bound arrays of microscopic particles," Phys. Rev. Lett. 89,283901, 1-4 (2002). [CrossRef]
- V. Garcés-Chávez, K. Dholakia, G.C. Spalding, "Extended-area optically induced organization of microparticles on a surface," Appl. Phys. Lett. 86,031106 (2005). [CrossRef]
- V. Garcés-Chávez, R. Quidant, P.J. Reece, G. Badenes, L. Torner, K. Dholakia, "Extended organization of colloidal microparticles by surface plasmon polariton excitation," Phys. Rev. B 73,085417 (2006). [CrossRef]
- D. Rogovin, R. McGraw, P. Yeh, "Harmonic phase conjugation in liquid suspensions of microparticles via higherorder gratings," Phys. Rev. Lett. 55,2864-2867 (1985). [CrossRef] [PubMed]
- P.W. Smith, A. Ashkin, J.E. Bjorkholm, D.J. Eilenberger, "Studies of self-focusing bistable devices using liquid suspensions of dielectric particles," Opt. Lett. 10,131-133 (1984). [CrossRef]
- C.D. Mellor, C.D. Bain, "Array formation in evanescent wave," ChemPhysChem 7,329-332 (2005). [CrossRef] [PubMed]
- M. M. Burns, J.-M. Fournier, and J. A. Golovchenko, "Optical binding," Phys. Rev. Lett. 63,12, 1233-1236 (1989). [CrossRef] [PubMed]
- W. Singer, M. Frick, S. Bernet, M Ritsch-Marte, "Self-organized array of regularly spaced microbeads in a fiberoptical trap," J. Opt. Soc. Am. B 20,7, 1568-1574 (2003). [CrossRef]
- M. Guillon, "Optical trapping in rarefied media: towards laser-trapped space telescopes," in Optical Trapping and Optical Micromanipulation II, K. Dholakia and G.C. Spalding, eds., Proc. SPIE, 59301T, 1-7 (2005).
- F. Depasse, J.-M. Vigoureux, "Optical binding force between two Rayleigh particles," J. Phys. D: Appl. Phys. 27,914-919 (1994). [CrossRef]
- P. C. Chaumet,M. Nieto-Vesperinas, "Optical binding of particles with or without the presence of a flat dielectric surface," Phys. Rev. B 64,035422, 1-7 (2001). [CrossRef]
- V. Garcés-Chávez, D. Roskey, M.D. Summers, H. Melville, D. McGloin, E.M. Wright, K. Dholakia, "Optical levitation in a Bessel light beam," Appl. Phys. Lett. 85,4001-4003 (2004). [CrossRef]
- J.D. Jackson, "Classical electrodynamics," second edition, New York, John Wiley & Son, Chpt 9.
- Mufei Xiao and Sergey I. Bozhevolnyi, "Resonant field enhancement by a finite-size periodic array of surface scatterers," J.Phys.: Condens. Matter 13,3001-3010 (2001). [CrossRef]
- I.S. Gradshteyn, I.M. Ryzhik, "Table of integrals series and products," (sixth edition, Alan Jeffrey, Academic Press, 2000).
- O. Moine, B. Stout, "Optical force calculations in arbitrary beams using the vector addition theorem," J. Opt. Soc. Am. B 22,1620-1631 (2005). [CrossRef]

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