## Optical control of scattering, absorption and lineshape in nanoparticles |

Optics Express, Vol. 21, Issue 17, pp. 20322-20333 (2013)

http://dx.doi.org/10.1364/OE.21.020322

Acrobat PDF (1422 KB)

### Abstract

We find exact conditions for the enhancement or suppression of internal and/or scattered fields in any smooth particle and the determination of their spatial distribution or angular momentum through the combination of simple fields. The incident fields can be generated by a single monochromatic or broad band light source, or by several sources, which may also be impurities embedded in the nanoparticle. We can design the lineshape of a particle introducing very narrow features in its spectral response.

© 2013 OSA

## Introduction

1. M. Abb, P. Albella, J. Aizpurua, and O. Muskens, “All-optical control of a single plasmonic nanoantenna-ITO hybrid,” Nano Lett. **11**, 2457–2463 (2011). [CrossRef] [PubMed]

2. A. Kubo, K. Onda, H. Petek, Z. Sun, Y. Jung, and H. Kim, “Femtosecond imaging of surface plasmon dynamics in a nanostructured silver film,” Nano Lett. **5**, 1123–1127 (2005). [CrossRef] [PubMed]

3. M. Durach, A. Rusina, and M. Stockman, “Full spatiotemporal control on the nanoscale,” Nano Lett. **7**, 3145–3149 (2007). [CrossRef] [PubMed]

4. M. Martin Aeschlimann, M. Bauer, D. Bayer, T. Tobias Brixner, F. Garcia de Abajo, W. Pfeiffer, M. Rohmer, C. Spindler, and F. Felix Steeb, “Adaptive subwavelength control of nano-optical fields,” Nature **446**, 301–304 (2007). [CrossRef] [PubMed]

5. H. Noh, Y. Chomg, A. Stone, and H. Cao, “Perfect coupling of light to surface plasmons by coherent absorption,” Phys. Rev. Lett. **108**, 186805 (2012). [CrossRef] [PubMed]

6. R. Pierrat, C. Vandenbem, M. Fink, and R. Carminati, “Subwavelength focusing inside an open disordered medium by time reversal at a single point antenna,” Phys. Rev. A **87**, 041801 (2013). [CrossRef]

8. J. Zhang, K. MacDonald, and N. Zheludev, “Controlling light-with-light without nonlinearity,” Light: Science & Appl. **1**, e18 (2012). [CrossRef]

9. M. Mazilu, J. Baumgartl, S. Kosmeier, and K. Dholakia, “Optical eigenmodes; exploiting the quadratic nature of the energy flux and of scattering interactions,” Opt. Express **19**, 933–945 (2011). [CrossRef] [PubMed]

*N*+ 1 incident fields in order to control

*N*channels. Modes of the internal and scattered fields of nanoparticles are coupled pairwise, each pair forming an interaction channel for the incident light [10

10. F. Papoff and B. Hourahine, “Geometrical mie theory for resonances in nanoparticles of any shape,” Opt.Express **19**, 21432–21444 (2011). [CrossRef] [PubMed]

11. M. Doherty, A. Murphy, R. Pollard, and P. Dawson, “Surface-enhanced raman scattering from metallic nanostructures: Bridging the gap between the near-field and far-field responses,” Phys. Rev. X **3**, 011001 (2013). [CrossRef]

## Theory

17. B. F. Farrell and P. J. Ioannou, “Generalized stability theory. part i: Autonomous operators,” Journ. of Atm. Sc. **53**, 2025–2040 (1996). [CrossRef]

*s*but not

_{n}*i*(or

_{n}*i*and not

_{n}*s*), or that produce the largest amplitude for modes

_{n}*s*or

_{n}*i*(see Fig. 2). We recall that any incident field can be decomposed as

_{n}*f*=

*f*+

_{n}*f*

_{n⊥}, with

*f*the part of the incident field that couples only with the

_{n}*n*

^{th}modes and

*f*

_{n⊥}being the part that does not. We find that Eq. (1) give the requirement for incident fields that, irrespectively of

*f*

_{n⊥}, produce excitation of only the scattering mode, i.e., null amplitude for the corresponding internal mode. Alternatively the largest amplitude of the scattering mode is obtained for fields with the form of Eq. (2). Note that we are considering only incident fields with

*f*·

_{n}*f*= 1 in Eqs. (1,2) to avoid trivial effects due to the overall amplitude of the incident fields. Figure 2 depicts the corresponding incident fields and the associated amplitudes of the scattered light in mode

_{n}*s*for both types of incident field. The analogous conditions for

_{n}*i*are found by exchanging

_{n}*s*with

*i*in Eqs. (1, 2).

*f*=

*s*can be in principle realized through time reversal of the lasing mode of an amplifier with the same shape as the particle and gain opposite to the loss [5

_{n}5. H. Noh, Y. Chomg, A. Stone, and H. Cao, “Perfect coupling of light to surface plasmons by coherent absorption,” Phys. Rev. Lett. **108**, 186805 (2012). [CrossRef] [PubMed]

*F*(

**r**) with tangent components

*f*=

*i*or

_{n}*f*=

*s*cannot be realized using common sources of radiation external to the particle, such as laser beams or SNOM tips, or even internal sources such as fluorescent or active hosts. This is because these sources emit waves that are neither outgoing radiating waves in the external medium, as is the scattering mode,

_{n}*s*, nor standing waves in the internal medium, such as the internal mode,

_{n}*i*. However, by combining two or more of these sources with appropriate phases and amplitudes, it is possible to control in a simple and effective way the few dominant interaction channels of any nanoparticle. To construct fields to realize the conditions of Eqs. (1) or (2) requires two linearly independent incident fields,

_{n}*Af*, and

*A*

_{1}

*f*

^{1}, both coupled to the channel

*n*, such that The condition for the suppression of

*i*is while condition for maximal excitation of

_{n}*s*for incident light such that |

_{n}*A*

_{1}

*f*

_{1}+

*Af*| is constant (the radius of the circle in Fig. 2) becomes with

*A*,

*A*

_{1}being complex amplitudes that can be experimentally adjusted through phase plates and dicroic elements (Fig. 1 suggests a geometry for constructing these fields, while a numerical example of scanning the relative complex amplitude to produce specific scattered light is shown later). Analogous conditions for optimization of

*i*and suppression of

_{n}*s*can be found by swapping

_{n}*i*,

_{n}*i′*with

_{n}*s*,

_{n}*s′*in Eqs. (4, 5): the generalization of Eq. (4) to a larger number of modes is presented in the Appendix.

_{n}*f*≠ 0,

_{n}*f*

_{n⊥}= 0 exist, the conservation of energy applies to the

*n*channel. From the Stratton-Chu representations [18

18. J. A. Stratton and L. J. Chu, “Scattering of light from a two-dimensional array of spherical particles on a substrate,” Phys. Rev. **56**, 99–107 (1939). [CrossRef]

*i*,

_{n}*s*,

_{n}*f*on the surface coordinates is the same, such as for sphere and infinite cylinders, the ratios in Eq. (3) depends on the flux of energy of the incident field into the particle, and Eq. (3) is

_{n}*always*violated when the sources are both either outside or inside the particle. For instance, in the case of a sphere any external source can be expanded by the set of regular multipoles with angular indexes

*l*,

*m*, while any internal source is expanded by the set of radiating spherical multipoles, as shown in the Appendix. In this case the amplitudes of internal and scattering modes cannot be controlled by independent external – or internal –sources: Eqs. (4, 5) become equivalent and lead to the simultaneous suppression of both

*i*and

_{n}*s*, while the maximization of the amplitudes of both scattering and internal modes is provided by ensuring that the contributions of the incident fields to the amplitudes add in phase. However, the condition for maximal excitation, Eq. (4), or suppression of a mode, Eq. (5), can be fulfilled also for spheres at any frequency provided one incident field is generated by an external source and the other by an internal source, or that one incident field is a regular wave with with a power flow of

_{n}*W*= 0 and the other an incoming wave with

_{n}*W*≠ 0. Examples of internal sources important for applications are impurities scattering light inelastically at the same frequency as the external control beam or active centers excited non-radiatively. Incoming waves are more difficult to realize, but could be in principle obtained through time reversal techniques [6

_{n}6. R. Pierrat, C. Vandenbem, M. Fink, and R. Carminati, “Subwavelength focusing inside an open disordered medium by time reversal at a single point antenna,” Phys. Rev. A **87**, 041801 (2013). [CrossRef]

## Applications

10. F. Papoff and B. Hourahine, “Geometrical mie theory for resonances in nanoparticles of any shape,” Opt.Express **19**, 21432–21444 (2011). [CrossRef] [PubMed]

*m*= ±1 channels, where

*m*is the eigenvalue of the angular momentum around the axis of the disc. Hence for simple plane polarized axial light, an equal linear mixture of both

*m*channels results. However, a second plane-wave field which obeys the analogue of Eq. (3) for the scattering parts of the two different

*m*-channel resonances, can be added to selectively cancel one or the other

*m*-channel of the resonance. This leads to the middle sub-figure, where the square of the electric field in the vicinity of the particle and the scattering into the far field become close to cylindrically symmetric, consisting only of

*m*= +1 light. Finally by introducing a third field, both

*m*-channels of the resonance are no longer coupling to the incident light, leading to the loss of nearly all of the surface field and a weak scattering of light into the far field. The present

*m*= −1 canceling case could be experimentally realized by, for example, the situation depicted in Figure 1(a) – this is a direct analogue to the theoretical configuration of the fields used in Fig. 4(a). In the absence of knowledge of the principal mode amplitudes that are caused by the incident fields, scanning the relative complex amplitude of the two light sources can be used to identify the conditions for removal of one mode. The left figure of 4(b) shows the scattering cross section from the disc as both the relative phase and amplitude are varied for the two fields. The minimum scattering occurs close to the point at which only one of the

*m*channel resonances is active (the second sub-figure showing purely the light in the

*m*= +1 channel), but to actually reach the exact condition requires monitoring the amount of the light coming from the desired principal mode (for example using the approach of [20

20. H. Wei, X. Xue, J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, and a. J. C. E. Yao, “Simplified measurement of the orbital angular momentum of photons,” Opt. Commun. **223**, 117–122 (2003). [CrossRef]

*m*= ±1 channels, however, this approach also applies for other

*m*channels and does not require a resonance to be present at the wavelength, only that relatively few modes dominate the interaction with incident light.

*m*= ±1 channels. These consist of axially incident plane wave light, light incident equatorially and light incident at 45 degrees to the particle axis. A combination of the three fields which do not produce amplitude in the principal modes of the the resonance were then constructed, chosen such that the equatorial light source amplitude was 1, and effectively removing interaction of these three light fields with the particle. Then, on scanning the wavelength of the light over the range, the phase (but not the amplitude) of the axial and 45 degree incident light was modulated according to a Gaussian profile, shifting their phase from being at the canceling combination to

*π*out of phase with this over a narrow range of wavelengths. Figure 5 shows the results of choosing to remove the cancellation of the amplitudes at three different wavelengths. Equivalently different gaussian widths, or a different line profile could be chosen. Here, modulation of phase is being transferred into modulation of amplitude. This leads to a change between a situation similar to the final sub-figure of 4(a), when cancellation occurs, and a strong surface field and radiated light when the cancellation is disrupted. Experimentally, an approach similar to Fig. 1(b) may be able to produce the required change in phase of light over a narrow frequency range.

## Conclusions

## Appendix

*r*is the radius of the sphere, Ω the solid angle,

*k*,

_{i}*k*are the wavenumbers for internal and external medium, respectively,

*m*,

_{lm}*n*vector spherical harmonics.

_{lm}*j*,

_{l}*h*the spherical Bessel and Hankel functions of order

_{l}*l*.

*k*with

_{i}r*kr*in

*a/a*

_{1}, i.e. the inequality is violated. On the contrary, the equality can be satisfied when the two sources are such that

*kr*with

*k*in

_{i}r*i*and maximal excitation of

_{lm}*s*, can be applied and are The same analysis holds true also for infinite cylinders.

_{lm}*f*≠ 0,

_{n}*f*

_{n⊥}= 0.

*i*,

_{n}*s*,

_{n}*f*are exact solutions of the Maxwell’s equations that, for particles with continuously varying surface tangents (particles of class

_{n}*C*

^{1}), have Stratton-Chu representation [18

18. J. A. Stratton and L. J. Chu, “Scattering of light from a two-dimensional array of spherical particles on a substrate,” Phys. Rev. **56**, 99–107 (1939). [CrossRef]

*μ*the magnetic permitivity of the external medium,

*ω*the frequency of light, the outgoing normal,

*n*, the fields

*F*,

*S*and the infinitesimal area

*dσ*depend on a point

*y*on the surface of the particle,

*x*is a point inside the particle, ∇ acts on

*x*and

*g*(

*x,y,k*) is the Green function of the scalar Helmholtz equation for the external medium [28]. For

*x*infinitesimally close to the surface, using Eq. (13) and the boundary conditions in Eq. (14) and the conservation of energy we get where

*ε*,

_{i}*ε*are the dielectric constant of the internal and external media and

*W*,

_{n}*f*,

_{n}*i*,

_{n}*s*[28]. Eq. (15) and its magnetic equivalent are independent complex equations that must be solved up to a factor by

_{n}*x*, these equations can be solved only if the dependence on

*x*in all the terms in Eq. (14) can be factored out. When this happens, as for spheres and cylinders, Eq. (16) shows that the condition in Eq. (3) cannot be satisfied by incident fields generated by sources outside the particle.

*m*internal modes and

*l*scattering modes using

*m*+

*l*+ 2 incident fields that satisfy the following equation

*f*

^{1},...,

*f*

^{m+l+1}are arbitrary. The solution for the amplitudes

*A*

^{1},...,

*A*

^{m+l+1}is unique if the determinant of the matrix is not null.

## References and links

1. | M. Abb, P. Albella, J. Aizpurua, and O. Muskens, “All-optical control of a single plasmonic nanoantenna-ITO hybrid,” Nano Lett. |

2. | A. Kubo, K. Onda, H. Petek, Z. Sun, Y. Jung, and H. Kim, “Femtosecond imaging of surface plasmon dynamics in a nanostructured silver film,” Nano Lett. |

3. | M. Durach, A. Rusina, and M. Stockman, “Full spatiotemporal control on the nanoscale,” Nano Lett. |

4. | M. Martin Aeschlimann, M. Bauer, D. Bayer, T. Tobias Brixner, F. Garcia de Abajo, W. Pfeiffer, M. Rohmer, C. Spindler, and F. Felix Steeb, “Adaptive subwavelength control of nano-optical fields,” Nature |

5. | H. Noh, Y. Chomg, A. Stone, and H. Cao, “Perfect coupling of light to surface plasmons by coherent absorption,” Phys. Rev. Lett. |

6. | R. Pierrat, C. Vandenbem, M. Fink, and R. Carminati, “Subwavelength focusing inside an open disordered medium by time reversal at a single point antenna,” Phys. Rev. A |

7. | J. Jeffers, “Interference and the lossless lossy beam splitter,” Journ. Mod. Opt. |

8. | J. Zhang, K. MacDonald, and N. Zheludev, “Controlling light-with-light without nonlinearity,” Light: Science & Appl. |

9. | M. Mazilu, J. Baumgartl, S. Kosmeier, and K. Dholakia, “Optical eigenmodes; exploiting the quadratic nature of the energy flux and of scattering interactions,” Opt. Express |

10. | F. Papoff and B. Hourahine, “Geometrical mie theory for resonances in nanoparticles of any shape,” Opt.Express |

11. | M. Doherty, A. Murphy, R. Pollard, and P. Dawson, “Surface-enhanced raman scattering from metallic nanostructures: Bridging the gap between the near-field and far-field responses,” Phys. Rev. X |

12. | P. C. Waterman, “The T-matrix revisited,” JOSA A |

13. | P. Barber and C. Yeh, “Scattering of electromagnetic waves by arbitrarily shaped dielectric bodies,” Appl. Opt. |

14. | M.I. Mishchenko, J. H. Hovernier, and L. D. Travis, eds., |

15. | A. Taflove, |

16. | R. Rodríguez-Oliveros and J. A. Sanchez-Gil, “Localized surface-plasmon resonances on single and coupled ′ nanoparticles through surface integral equations for flexible surfaces,” Opt. Express |

17. | B. F. Farrell and P. J. Ioannou, “Generalized stability theory. part i: Autonomous operators,” Journ. of Atm. Sc. |

18. | J. A. Stratton and L. J. Chu, “Scattering of light from a two-dimensional array of spherical particles on a substrate,” Phys. Rev. |

19. | I. Malitson, “Interspecimen comparison of the refractive index of fused silica,” Journal of the Optical Society of America |

20. | H. Wei, X. Xue, J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, and a. J. C. E. Yao, “Simplified measurement of the orbital angular momentum of photons,” Opt. Commun. |

21. | K. Holms, B. Hourahine, and F. Papoff, “Calculation of internal and scattered fields of axisymmetric nanoparticles at any point in space,” J. Opt. A: Pure Appl. Opt. |

22. | T. Rother, M. Kahnert, A. Doicu, and J. Wauer, “Surface Green’s Function of the Helmholtz Equation in Spherical Coordinates,” Prog. Electromag. Res. |

23. | E. Hannan, “The general theory of canonical correlation and its relation to functional analysis,” J. Aust. Math. Soc. |

24. | A. Knyazev, A. Jujusnashvili, and M. Argentati, “Angles between Infinite Dimensional Subspaces with Applications to the Rayleigh-Ritz and Alternating Projectors Methods,” Journ. of Func. Analys. |

25. | W. J. Firth and A. Yao, “Giant excess noise and transient gain in misaligned laser cavities,” Phys. Rev. Lett. |

26. | F. Papoff, G. D’Alessandro, and G.-L. Oppo, “State dependent pseudoresonances and excess noise,” Phys. Rev. Lett. |

27. | F. Papoff and G. Robb, “Rapid coherent optical modulation of atomic momenta via pseudoresonances,” Phys. Rev. Lett. |

28. | A. Doicu, Y. Eremin, and T. Wreidt, |

**OCIS Codes**

(020.3690) Atomic and molecular physics : Line shapes and shifts

(290.0290) Scattering : Scattering

(160.4236) Materials : Nanomaterials

(290.5825) Scattering : Scattering theory

**ToC Category:**

Scattering

**History**

Original Manuscript: June 12, 2013

Revised Manuscript: August 9, 2013

Manuscript Accepted: August 13, 2013

Published: August 22, 2013

**Citation**

Benjamin Hourahine and Francesco Papoff, "Optical control of scattering, absorption and lineshape in nanoparticles," Opt. Express **21**, 20322-20333 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-17-20322

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### References

- M. Abb, P. Albella, J. Aizpurua, and O. Muskens, “All-optical control of a single plasmonic nanoantenna-ITO hybrid,” Nano Lett.11, 2457–2463 (2011). [CrossRef] [PubMed]
- A. Kubo, K. Onda, H. Petek, Z. Sun, Y. Jung, and H. Kim, “Femtosecond imaging of surface plasmon dynamics in a nanostructured silver film,” Nano Lett.5, 1123–1127 (2005). [CrossRef] [PubMed]
- M. Durach, A. Rusina, and M. Stockman, “Full spatiotemporal control on the nanoscale,” Nano Lett.7, 3145–3149 (2007). [CrossRef] [PubMed]
- M. Martin Aeschlimann, M. Bauer, D. Bayer, T. Tobias Brixner, F. Garcia de Abajo, W. Pfeiffer, M. Rohmer, C. Spindler, and F. Felix Steeb, “Adaptive subwavelength control of nano-optical fields,” Nature446, 301–304 (2007). [CrossRef] [PubMed]
- H. Noh, Y. Chomg, A. Stone, and H. Cao, “Perfect coupling of light to surface plasmons by coherent absorption,” Phys. Rev. Lett.108, 186805 (2012). [CrossRef] [PubMed]
- R. Pierrat, C. Vandenbem, M. Fink, and R. Carminati, “Subwavelength focusing inside an open disordered medium by time reversal at a single point antenna,” Phys. Rev. A87, 041801 (2013). [CrossRef]
- J. Jeffers, “Interference and the lossless lossy beam splitter,” Journ. Mod. Opt.47, 1819–1824 (2000).
- J. Zhang, K. MacDonald, and N. Zheludev, “Controlling light-with-light without nonlinearity,” Light: Science & Appl.1, e18 (2012). [CrossRef]
- M. Mazilu, J. Baumgartl, S. Kosmeier, and K. Dholakia, “Optical eigenmodes; exploiting the quadratic nature of the energy flux and of scattering interactions,” Opt. Express19, 933–945 (2011). [CrossRef] [PubMed]
- F. Papoff and B. Hourahine, “Geometrical mie theory for resonances in nanoparticles of any shape,” Opt.Express19, 21432–21444 (2011). [CrossRef] [PubMed]
- M. Doherty, A. Murphy, R. Pollard, and P. Dawson, “Surface-enhanced raman scattering from metallic nanostructures: Bridging the gap between the near-field and far-field responses,” Phys. Rev. X3, 011001 (2013). [CrossRef]
- P. C. Waterman, “The T-matrix revisited,” JOSA A24, 2257–2267 (2007). [CrossRef] [PubMed]
- P. Barber and C. Yeh, “Scattering of electromagnetic waves by arbitrarily shaped dielectric bodies,” Appl. Opt.14, 2864–2872 (1975). [CrossRef] [PubMed]
- M.I. Mishchenko, J. H. Hovernier, and L. D. Travis, eds., Light Scattering by Nonspherical Particles: Theory, Measurements and Applications (Academic Press, 2000).
- A. Taflove, Computational Electrodynamics: The Finite Difference Time-Domain Method (Artech House Publishers , Boston, MA, 1995).
- R. Rodríguez-Oliveros and J. A. Sanchez-Gil, “Localized surface-plasmon resonances on single and coupled ′ nanoparticles through surface integral equations for flexible surfaces,” Opt. Express19, 12208–12219 (2011). [CrossRef]
- B. F. Farrell and P. J. Ioannou, “Generalized stability theory. part i: Autonomous operators,” Journ. of Atm. Sc.53, 2025–2040 (1996). [CrossRef]
- J. A. Stratton and L. J. Chu, “Scattering of light from a two-dimensional array of spherical particles on a substrate,” Phys. Rev.56, 99–107 (1939). [CrossRef]
- I. Malitson, “Interspecimen comparison of the refractive index of fused silica,” Journal of the Optical Society of America55, 1205–1209 (1965). [CrossRef]
- H. Wei, X. Xue, J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, and a. J. C. E. Yao, “Simplified measurement of the orbital angular momentum of photons,” Opt. Commun.223, 117–122 (2003). [CrossRef]
- K. Holms, B. Hourahine, and F. Papoff, “Calculation of internal and scattered fields of axisymmetric nanoparticles at any point in space,” J. Opt. A: Pure Appl. Opt.11, 054009 (2009). [CrossRef]
- T. Rother, M. Kahnert, A. Doicu, and J. Wauer, “Surface Green’s Function of the Helmholtz Equation in Spherical Coordinates,” Prog. Electromag. Res.38, 47–95 (2002). [CrossRef]
- E. Hannan, “The general theory of canonical correlation and its relation to functional analysis,” J. Aust. Math. Soc.2, 229–242 (1961/1962). [CrossRef]
- A. Knyazev, A. Jujusnashvili, and M. Argentati, “Angles between Infinite Dimensional Subspaces with Applications to the Rayleigh-Ritz and Alternating Projectors Methods,” Journ. of Func. Analys.259, 1323–1345 (2010). [CrossRef]
- W. J. Firth and A. Yao, “Giant excess noise and transient gain in misaligned laser cavities,” Phys. Rev. Lett.95, 073903 (2005). [CrossRef] [PubMed]
- F. Papoff, G. D’Alessandro, and G.-L. Oppo, “State dependent pseudoresonances and excess noise,” Phys. Rev. Lett.100, 123905 (2008). [CrossRef] [PubMed]
- F. Papoff and G. Robb, “Rapid coherent optical modulation of atomic momenta via pseudoresonances,” Phys. Rev. Lett.108, 113902 (2012). [CrossRef] [PubMed]
- A. Doicu, Y. Eremin, and T. Wreidt, Acoustic and Electromagnetic Scattering Analysis Using Discrete Sources(Accademic Press, 2000).

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