## Efficient and intuitive method for the analysis of light scattering by a resonant nanostructure |

Optics Express, Vol. 21, Issue 22, pp. 27371-27382 (2013)

http://dx.doi.org/10.1364/OE.21.027371

Acrobat PDF (1518 KB)

### Abstract

We present a semi-analytical formalism capable of handling the coupling of electromagnetic sources, such as point dipoles or free-propagating fields, with various kinds of dissipative resonances with radiation leakage, Ohmic losses or both. Due to its analyticity, the approach is very intuitive and physically-sound. It is also very economic in computational resources, since once the resonances of a plasmonic or photonic resonator are known, their excitation coefficients are obtained analytically, independently of the polarization, frequency or location of the excitation source. To evidence that the present formalism is very general and versatile, we implement it with the commercial software COMSOL, rather than with our in-house numerical tools.

© 2013 Optical Society of America

## 1. Introduction

3. P. T. Leung, S. Y. Liu, and K. Young, “Completeness and orthogonality of quasinormal modes in leaky optical cavities,” Phys. Rev. A **49**(4), 3057–3067 (1994). [CrossRef] [PubMed]

*t*) notation is adopted for the time-harmonic fields. The eigenfrequency is a pole of the Green tensor of the system and it admits a classical interpretation in terms of the quality factor

4. P. Lalanne, C. Sauvan, and J. P. Hugonin, “Photon confinement in photonic crystal nanocavities,” Laser & Photon. Rev. **2**(6), 514–526 (2008). [CrossRef]

3. P. T. Leung, S. Y. Liu, and K. Young, “Completeness and orthogonality of quasinormal modes in leaky optical cavities,” Phys. Rev. A **49**(4), 3057–3067 (1994). [CrossRef] [PubMed]

5. P. T. Leung, K. M. Pang, and K. Young, “Two-component wave formalism in spherical open systems,” J. Phys. Math. Gen. **39**(1), 247–267 (2006). [CrossRef]

10. C. Sauvan, J. P. Hugonin, I. S. Maksymov, and P. Lalanne, “Theory of the spontaneous optical emission of nanosize photonic and plasmon resonators,” Phys. Rev. Lett. **110**(23), 237401 (2013). [CrossRef]

7. P. T. Kristensen, C. Van Vlack, and S. Hughes, “Generalized effective mode volume for leaky optical cavities,” Opt. Lett. **37**(10), 1649–1651 (2012). [CrossRef] [PubMed]

8. A. F. Koenderink, “On the use of Purcell factors for plasmon antennas,” Opt. Lett. **35**(24), 4208–4210 (2010). [CrossRef] [PubMed]

9. S. Derom, R. Vincent, A. Bouhelier, and G. Colas des Francs, “Resonance quality, radiative/ohmic losses and modal volume of Mie plasmons,” Europhys. Lett. **98**(4), 47008 (2012). [CrossRef]

10. C. Sauvan, J. P. Hugonin, I. S. Maksymov, and P. Lalanne, “Theory of the spontaneous optical emission of nanosize photonic and plasmon resonators,” Phys. Rev. Lett. **110**(23), 237401 (2013). [CrossRef]

- • we generalize the approach to encompass more general interconversion processes between free-propagating and localized fields. In particular we derive new formulas for the coupling of QNMs with plane waves (or free-propagative beams in general), and obtain analytical formulas for the extinction and absorption cross sections, two fundamental quantities of scattering processes.
- • we also propose a method to normalize the QNMs that is more general than the one in [10]. The net benefit is the derivation of a versatile and practical method for normalizing the QNM, which is suitable for any software and in particular commercial electromagnetic solvers. We illustrate how to calculate normalized QNMs with COMSOL Multiphysics (http://en.wikipedia.org/wiki/COMSOL_Multiphysics).
**110**(23), 237401 (2013). [CrossRef]

**110**(23), 237401 (2013). [CrossRef]

**110**(23), 237401 (2013). [CrossRef]

11. F. L. Teixeira and W. C. Chew, “General closed-form PML constitutive tensors to match arbitrary bianisotropic and dispersive linear media,” IEEE Microw. Guided Wave Lett. **8**(6), 223–225 (1998). [CrossRef]

*simpler and more general*method that is independent of the numerical technique used for satisfying the outgoing wave conditions (transparent boundary conditions, PMLs with linear or even non-linear coordinate transforms or wave absorbers). The new method is expected to remain valid even for the split-field PML often used with FDTD [12

12. J. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. **114**(2), 185–200 (1994). [CrossRef]

## 2. Theoretical background

**110**(23), 237401 (2013). [CrossRef]

**J**(

**r**) located in the vicinity of the resonant system. The field

**110**(23), 237401 (2013). [CrossRef]

*where*

_{m}*normalization*conditionEquation (4) is not as straightforward as it seems, simply because the QNM diverges exponentially for |

**r**| → ∞. In [10

**110**(23), 237401 (2013). [CrossRef]

**110**(23), 237401 (2013). [CrossRef]

13. A COMSOL model file associated to the article can be found at http://www.lp2n.institutoptique.fr/Membres-Services/Responsables-d-equipe/LALANNE-Philippe.

*small set*of resonances. Such an expansion is extremely useful to get a deep insight into the problem and, since it can be handled analytically, it enables also ultra-efficient electromagnetic simulations specifically tailored for design and fast optimizations. Hereafter, a single QNM will be considered for simplicity. Thus we will drop the subscript

*m*and denote by

**110**(23), 237401 (2013). [CrossRef]

## 3. Calculation and normalization of the QNMs

**r**| → ∞. For the sake of illustration, we use the COMSOL Multiphysics software and its RF toolbox for which we provide a model sheet for the nanorod geometry of Fig. 1 [14

14. P. Nordlander and J. C. Tully, “Energy shifts and broadening of atomic levels near metal surfaces,” Phys. Rev. B Condens. Matter **42**(9), 5564–5578 (1990). [CrossRef] [PubMed]

### 3.1 Scattering formulation

**E**,

**H**), satisfieswhere

**J**(

**r**) is the source term, potentially located at infinity. The field

_{z}(

**r**

_{1}) of the total field calculated at some specific location

**r**=

**r**

_{1}.

### 3.2 Pole calculation

_{p}= 1.26 × 10

^{16}s

^{−1}and γ = 1.41 × 10

^{14}s

^{−1}, to define an analytical continuation of the gold permittivity for complex frequencies. The values of ω

_{p}and γ are fitted from tabulated data in [16], and

*c*/

_{1}≈921 −48.6i nm (

*Q*= 9.5) estimated from the data obtained with the fine mesh in Fig. 1b. We then use an iterative method to calculate the pole. The method relies on the fact that, as the frequency approaches the pole frequency

*a*

_{0},

*a*

_{1}and

^{15}+ i0.1046416051442 × 10

^{15}s

^{−1}. This is intentional, since our purpose is to show that the present semi-analytical method is very fast. Actually as shown in Figs. 1(b), 1(c), 2, 3(b) and 3(c) the main physical quantities attached to the scattering by resonance are accurately predicted even if the resonance frequency is only approximately calculated with the coarse discretization. This evidences the soundness of the approach.

### 3.3 Normalization of the QNMs

**110**(23), 237401 (2013). [CrossRef]

**J**and using

**r**=

**r**

_{0}, one gets

*normalized*modeEquation (8) is very useful as it allows us to calculate a

*normalized*QNM from the sole knowledge of the field scattered by a resonant structure at any frequency ω close to the QNM frequency

*Q*-factor), the frequency ω has to be taken in the complex plane.

_{z}-component. The distributions are invariant by rotation along the z-axis, and only one half of a cross-section plane is shown. They are obtained for the pole estimates using Eq. (8), with

*P*(ω) of a z-polarized dipole

**110**(23), 237401 (2013). [CrossRef]

*n*= 1.5 is the refractive index of the background,

*c*is the speed of light in vacuum, α is given by Eq. (3) and the QNM field

**110**(23), 237401 (2013). [CrossRef]

## 4. Absorption and scattering cross sections

**110**(23), 237401 (2013). [CrossRef]

### 4.1 Problem formulation

*in the presence of the structure*by a source distribution

**J**used to derive the coupling coefficient in Eq. (3) with the new distribution source

*is easily calculated as an overlap integral between the normalized QNM and the incident field. Note that the integral has a simpler expression for a homogeneous scatterer where*

_{m}**r**and can be put outside the integral.

### 4.2 Absorption cross-section

*of the plasmonic nanorod can be calculated with the following formula [15]considering the approximate expressions of Eqs. (11) and (12) for the scattered field. In Eq. (13),*

_{A}*S*

_{0}is the time-averaged Poynting vector of the incident plane-wave and the integral is performed over the volume

*V*of the resonant structure, where the QNM expansion is a very good approximation of the scattered field.

### 4.3 Scattering cross-section

*or the extinction cross-section σ*

_{S}*, and because Eq. (11) is approximate they do not have all the same accuracy. For instance as |*

_{E}**r**| → ∞, the scattered field vanishes as 1/|

**r**|

^{2}far away from the scatterer, whereas the finite sum in Eq. (11) exponentially diverges. As we checked, calculating the scattering cross-section with the classical formula [15]

*A*is a closed surface surrounding the scatterer, is inaccurate when

*A*is far away from the scatterer and it slightly depends on of the specific choice of

*A*when

*A*is close to the scatterer. From our tests, we find that best accuracy and consistency are achieved by first calculating the extinction cross-section with the following expressionwhich is valid if the resonant structure is immersed in a transparent medium,

*is obtained by [15]*

_{S}*calculated with Eqs. (14) and (15). Again quantitative agreement is achieved with fully-vectorial data obtained with the fine mesh, especially for the peak values for*

_{S}_{S}is expressed as a difference of two terms (see Eq. (15)) that are each approximately predicted. As we checked, the inaccuracy is not due to the single mode approximation (including other higher-order QNMs in the expansion has a negligible impact on the prediction) but to a non-resonant background. Far away from any resonance, the scattering is dominantly due to a continuum of states (of plane waves in a uniform material). This background, which is for instance responsible for the well-known 1/λ

^{4}Rayleigh scattering of small particles, is difficult to represent as a sum of QNMs. Such a difficulty also holds for Purcell-factor calculations. When the dipole source is located at very small distances from a metal particle and experiences a direct electrostatic coupling with the metal (quenching), the QNM expansion becomes inaccurate, see Section 3.3. of the Suppl. Inf. in [10

**110**(23), 237401 (2013). [CrossRef]

**110**(23), 237401 (2013). [CrossRef]

**110**(23), 237401 (2013). [CrossRef]

### 4.4 Purcell factor and Green tensor

## 5. Conclusion

**110**(23), 237401 (2013). [CrossRef]

**110**(23), 237401 (2013). [CrossRef]

## Appendix 1: Implementation of complex frequencies with COMSOL

## Appendix 2: Iterative approach for the pole calculation

_{0}, a

_{1}and

*c*/

_{1}, Z

_{2}, Z

_{3}for three frequencies ω

_{1}=

^{−5}), ω

_{2}=

_{3}=

^{−5}). By solving a linear system of three equations with three unknowns, we get a new frequency

## Acknowledgments

## References and links

1. | P. M. Morse and H. Feshbach, |

2. | R. K. Chang and A. J. Campillo, |

3. | P. T. Leung, S. Y. Liu, and K. Young, “Completeness and orthogonality of quasinormal modes in leaky optical cavities,” Phys. Rev. A |

4. | P. Lalanne, C. Sauvan, and J. P. Hugonin, “Photon confinement in photonic crystal nanocavities,” Laser & Photon. Rev. |

5. | P. T. Leung, K. M. Pang, and K. Young, “Two-component wave formalism in spherical open systems,” J. Phys. Math. Gen. |

6. | E. M. Purcell, “Spontaneous emission probabilities at radio frequencies,” Phys. Rev. |

7. | P. T. Kristensen, C. Van Vlack, and S. Hughes, “Generalized effective mode volume for leaky optical cavities,” Opt. Lett. |

8. | A. F. Koenderink, “On the use of Purcell factors for plasmon antennas,” Opt. Lett. |

9. | S. Derom, R. Vincent, A. Bouhelier, and G. Colas des Francs, “Resonance quality, radiative/ohmic losses and modal volume of Mie plasmons,” Europhys. Lett. |

10. | C. Sauvan, J. P. Hugonin, I. S. Maksymov, and P. Lalanne, “Theory of the spontaneous optical emission of nanosize photonic and plasmon resonators,” Phys. Rev. Lett. |

11. | F. L. Teixeira and W. C. Chew, “General closed-form PML constitutive tensors to match arbitrary bianisotropic and dispersive linear media,” IEEE Microw. Guided Wave Lett. |

12. | J. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. |

13. | A COMSOL model file associated to the article can be found at http://www.lp2n.institutoptique.fr/Membres-Services/Responsables-d-equipe/LALANNE-Philippe. |

14. | P. Nordlander and J. C. Tully, “Energy shifts and broadening of atomic levels near metal surfaces,” Phys. Rev. B Condens. Matter |

15. | C. F. Bohren and D. R. Huffman, |

16. | E. D. Palik, |

17. | W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, |

**OCIS Codes**

(260.5740) Physical optics : Resonance

(290.2200) Scattering : Extinction

(290.5825) Scattering : Scattering theory

**ToC Category:**

Plasmonics

**History**

Original Manuscript: July 29, 2013

Revised Manuscript: September 23, 2013

Manuscript Accepted: September 23, 2013

Published: November 4, 2013

**Virtual Issues**

Vol. 9, Iss. 1 *Virtual Journal for Biomedical Optics*

Surface Plasmon Photonics (2013) *Optics Express*

**Citation**

Q. Bai, M. Perrin, C. Sauvan, J-P Hugonin, and P. Lalanne, "Efficient and intuitive method for the analysis of light scattering by a resonant nanostructure," Opt. Express **21**, 27371-27382 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-22-27371

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### References

- P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, 1953).
- R. K. Chang and A. J. Campillo, Optical Processes in Microcavities, Chap. 1 (World Scientific, 1996).
- P. T. Leung, S. Y. Liu, and K. Young, “Completeness and orthogonality of quasinormal modes in leaky optical cavities,” Phys. Rev. A49(4), 3057–3067 (1994). [CrossRef] [PubMed]
- P. Lalanne, C. Sauvan, and J. P. Hugonin, “Photon confinement in photonic crystal nanocavities,” Laser & Photon. Rev.2(6), 514–526 (2008). [CrossRef]
- P. T. Leung, K. M. Pang, and K. Young, “Two-component wave formalism in spherical open systems,” J. Phys. Math. Gen.39(1), 247–267 (2006). [CrossRef]
- E. M. Purcell, “Spontaneous emission probabilities at radio frequencies,” Phys. Rev.69, 681 (1946).
- P. T. Kristensen, C. Van Vlack, and S. Hughes, “Generalized effective mode volume for leaky optical cavities,” Opt. Lett.37(10), 1649–1651 (2012). [CrossRef] [PubMed]
- A. F. Koenderink, “On the use of Purcell factors for plasmon antennas,” Opt. Lett.35(24), 4208–4210 (2010). [CrossRef] [PubMed]
- S. Derom, R. Vincent, A. Bouhelier, and G. Colas des Francs, “Resonance quality, radiative/ohmic losses and modal volume of Mie plasmons,” Europhys. Lett.98(4), 47008 (2012). [CrossRef]
- C. Sauvan, J. P. Hugonin, I. S. Maksymov, and P. Lalanne, “Theory of the spontaneous optical emission of nanosize photonic and plasmon resonators,” Phys. Rev. Lett.110(23), 237401 (2013). [CrossRef]
- F. L. Teixeira and W. C. Chew, “General closed-form PML constitutive tensors to match arbitrary bianisotropic and dispersive linear media,” IEEE Microw. Guided Wave Lett.8(6), 223–225 (1998). [CrossRef]
- J. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys.114(2), 185–200 (1994). [CrossRef]
- A COMSOL model file associated to the article can be found at http://www.lp2n.institutoptique.fr/Membres-Services/Responsables-d-equipe/LALANNE-Philippe .
- P. Nordlander and J. C. Tully, “Energy shifts and broadening of atomic levels near metal surfaces,” Phys. Rev. B Condens. Matter42(9), 5564–5578 (1990). [CrossRef] [PubMed]
- C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Pparticles (Wiley 1983).
- E. D. Palik, Handbook of Optical Constants of Solids, Part II (Academic Press, 1985).
- W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes: The Art of Scientific Computing (Cambridge University Press, 1989).

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