Mie scattering as a cascade of Fano resonances Mikhail V. Rybin,1,2,∗ Kirill B. Samusev,1,2 Ivan S. Sinev,1,2 George Semouchkin,3 Elena Semouchkina,3 Yuri S. Kivshar,2,4 and Mikhail F. Limonov1,2 1 Ioffe Physical-Technical Institute, St. Petersburg 194021, Russia Research University for Information Technology, Mechanics and Optics (ITMO), St. Petersburg 197101, Russia, 3 Department of Electrical and Computer Engineering, Michigan Technological University, Houghton, Michigan 49931, USA, 4 Nonlinear Physics Center, Research School of Physics and Engineering, Australian National University, Canberra ACT 0200, Australia 2 National
∗ [email protected]
Abstract: We reveal that the resonant Mie scattering by high-index dielectric nanoparticles can be presented through cascades of Fano resonances. We employ the exact solution of Maxwell’s equations and demonstrate that the Lorenz-Mie coefficients of the Mie problem can be expressed generically as infinite series of Fano functions as they describe interference between the background radiation originated from an incident wave and narrow-spectrum Mie scattering modes that lead to Fano resonances. © 2013 Optical Society of America OCIS codes: (290.4020) Mie theory; (260.5740) Resonance.
References and links 1. C. M. Soukoulis, ed., Photonic Band Gap Materials, Vol. 315 of NATO ASI Series E (Springer, 1996). 2. V. Yannopapas and A. Moroz, “Negative refractive index metamaterials from inherently non-magnetic materials for deep infrared to terahertz frequency ranges,” J. Phys. Condens. Matt. 17, 3717–3734 (2005). 3. P. Garcia, M. Ibisate, R. Sapienza, D. Wiersma, and C. L´opez, “Mie resonances to tailor random lasers,” Phys. Rev. A 80, 013833 (2009). 4. M. V. Rybin, P. V. Kapitanova, D. S. Filonov, A. P. Slobozhanyuk, P. A. Belov, Y. S. Kivshar, and M. F. Limonov, “Fano resonances in antennas: General control over radiation patterns,” Phys. Rev. B 88, 205106 (2013). 5. S. Foteinopoulou, J. Vigneron, and C. Vandenbem, “Optical near-field excitations on plasmonic nanoparticlebased structures,” Opt. Express 15, 4253–4267 (2007). 6. I. Romero, J. Aizpurua, G. W. Bryant, and F. J. Garc´ıa de Abajo, “Plasmons in nearly touching metallic nanoparticles: singular response in the limit of touching dimers,” Opt. Express 14, 9988–9999 (2006). 7. S. A. Maier, P. G. Kik, and H. A. Atwater, “Optical pulse propagation in metal nanoparticle chain waveguides,” Phys. Rev. B 67, 205402 (2003). 8. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley-VCH, 1998). 9. M. F. Limonov and R. M. De La Rue, eds., Optical Properties of Photonic Structures: Interplay of Order and Disorder (CRC Press, 2012). 10. S. Gottardo, R. Sapienza, P. D. Garcia, A. Blanco, D. S. Wiersma, and C. L´opez, “Resonance-driven random lasing,” Nat. Photonics 2, 429–432 (2008). 11. U. Fano, “Effects of configuration interaction on intensities and phase shifts,” Phys. Rev. 124, 1866–1878 (1961). 12. V. Madhavan, W. Chen, T. Jamneala, M. F. Crommie, and N. S. Wingreen, “Tunneling into a single magnetic atom: Spectroscopic evidence of the Kondo resonance,” Science 280, 567–569 (1998). 13. M. F. Limonov, A. I. Rykov, S. Tajima, A. Yamanaka, “Raman scattering study on fully oxygenated YBa2 CuO7 single crystals: x-y anisotropy in the superconductivity-induced effects,” Phys. Rev. Lett. 80, 825–828 (1998). 14. M. Limonov, S. Lee, S. Tajima, and A. Yamanaka, “Superconductivity-induced resonant raman scattering in multilayer high-Tc superconductors,” Phys. Rev. B 66, 054509 (2002).
Received 14 Oct 2013; revised 7 Nov 2013; accepted 9 Nov 2013; published 27 Nov 2013 #199493 - $15.00 USD (C) 2013 OSA 2 December 2013 | Vol. 21, No. 24 | DOI:10.1364/OE.21.030107 | OPTICS EXPRESS 30107
15. M. I. Tribelsky, S. Flach, A. E. Miroshnichenko, A. V. Gorbach, and Y. S. Kivshar, “Light scattering by a finite obstacle and Fano resonances,” Phys. Rev. Lett. 100, 043903 (2008). 16. M. V. Rybin, A. B. Khanikaev, M. Inoue, K. B. Samusev, M. J. Steel, G. Yushin, and M. F. Limonov, “Fano resonance between Mie and Bragg scattering in photonic crystals,” Phys. Rev. Lett. 103, 023901 (2009). 17. M. V. Rybin, A. B. Khanikaev, M. Inoue, A. K. Samusev, M. J. Steel, G. Yushin, and M. F. Limonov, “Bragg scattering induces Fano resonance in photonic crystals,” Photonics Nanostruct. Fundam. Appl. 8, 86–93 (2010). 18. A. E. Miroshnichenko, S. Flach, and Y. S. Kivshar, “Fano resonances in nanoscale structures,” Rev. Mod. Phys. 82, 2257–2298 (2010). 19. B. Luk’yanchuk, N. I. Zheludev, S. A. Maier, N. J. Halas, P. Nordlander, H. Giessen, and C. T. Chong, “The Fano resonance in plasmonic nanostructures and metamaterials,” Nat. Mater. 9, 707–715 (2010). 20. J. A. Fan, C. Wu, K. Bao, J. Bao, R. Bardhan, N. J. Halas, V. N. Manoharan, P. Nordlander, G. Shvets, and F. Capasso, “Self-assembled plasmonic nanoparticle clusters,” Science 328, 1135–1138 (2010). 21. A. N. Poddubny, M. V. Rybin, M. F. Limonov, and Y. S. Kivshar, “Fano interference governs wave transport in disordered systems,” Nat. Commun. 3, 914 (2012). 22. A. E. Miroshnichenko, Y. S. Kivshar, “Fano resonances in all-dielectric oligomers,” Nano Lett. 12, 6459–6463 (2012). 23. Y. Francescato, V. Giannini, and S. A. Maier, “Plasmonic systems unveiled by Fano resonances,” ACS Nano 6, 1830–1838 (2012). 24. M. Tribelsky, A. Miroshnichenko, and Y. Kivshar, “Unconventional Fano resonances in light scattering by small particles,” Europhys. Lett. 97, 44005 (2012). 25. C. P. Burrows and W. L. Barnes, “Large spectral extinction due to overlap of dipolar and quadrupolar plasmonic modes of metallic nanoparticles in arrays,” Opt. Express 18, 3187–3198 (2010). 26. J.-P. Connerade, A. M. Lane, “Interacting resonances in atomic spectroscopy,” Rep. Prog. Phys. 51, 1439–1478 (1988).
1.
Introduction
Mie scattering by small particles is one of the fundamental optical phenomena, which can govern the functionalities of forefront optical devices and innovative photonic structures such as photonic crystals [1], metamaterials [2], random lasers [3], and nanoantennas [4–6], nanoparticle-based waveguides [7], as well as SNOM tips [5]. The solution of the Mie scattering problem is represented by a sum √ of infinite series, and it is valid for all possible optical diameter-to-wavelength ratios (2r ε /λ ) and dielectric sphere-to-environment contrasts [8]. Important consequences of the Mie resonances are, for example, a strong reduction of the light group velocity in photonic glasses and monodisperse spheres [9], and modification of random lasing [10]. The Mie scattering theory can be applied not only to spherical particles, but also to other “bodies of revolution”, in particular, cylindrical nanorods. Resonant Mie scattering involves the excitation of localized modes in the scatterer when the frequency ω of the incident wave approaches one of the particle’s eigenmode frequencies. This leads to the emission of electromagnetic waves by the particle with the same frequency ω resulting in interference between the incident and scattered waves. If radiation with a narrow band of virtually any origin interacts with a continuum-spectrum radiation through interference effects constructively or destructively, we should expect Fano-type resonances [11]. Fano resonance is a general physical phenomenon well-known across many disciplines dealing with oscillations and waves [12–14]. In photonics, the Fano resonance emerges in many areas including the physics of plasmonic nanostructures and photonic crystals [15–23]. For the light scattering by small nanoparticles, the Fano resonance has also been investigated, in particular for the interference between different electromagnetic modes excited in the particle with the same or different multipole moments [15, 24, 25], as well as interference between resonant and non-resonant scattering from spheres. Surprisingly, the most general case of the Mie scattering that results in the interference between incident and scattered waves has never been investigated before, while such an interference constitutes the basic process in the structure of electromagnetic fields around the scatterer. In this work, we arrive at a new and unexpected conclusion that the resonant Mie scattering
Received 14 Oct 2013; revised 7 Nov 2013; accepted 9 Nov 2013; published 27 Nov 2013 #199493 - $15.00 USD (C) 2013 OSA 2 December 2013 | Vol. 21, No. 24 | DOI:10.1364/OE.21.030107 | OPTICS EXPRESS 30108
results in nothing but an infinite series of Fano resonances. We demonstrate analytically that the Lorenz-Mie coefficients describing Mie scattering and cascades of Fano profiles can be expressed by identically analytical formulas. For the sake of simplicity, we study the twodimensional case of the Fano resonance formation at the Mie scattering by an infinitely long dielectric cylindrical rod. However, the proposed approach is rather general, and it can be generalized to other systems. 2.
Lorenz-Mie coefficients
We consider the scattering of electromagnetic waves by a single homogeneous infinite dielectric rod with the radius r and with the purely real dielectric permittivity ε1 without dissipation. The surrounding medium is supposed to be transparent and homogeneous with dielectric permittivity ε2 . As is known, the far-field scattered by a circular rod can be expanded into orthogonal electromagnetic dipolar and multipolar terms, with cylindrical Lorenz-Mie coefficients an and bn corresponding to the electric and magnetic moments, respectively [8]. If only the TE polarization is considered, the magnetic field of incident waves is directed along the rod axis, while the electric field is normal to this axis. Then the scattered fields are defined by coefficients of only one type an , while bn vanish. The resonances excited in the rod are denoted as TEnk (n is integer and k is positive integer), where n is multipole order and k is resonance number. For the TE polarization, the Maxwell’s boundary conditions for the tangential components of both H and E fields at the rod surface can be written as: √ √ (1) √ En Jn (x ε2 ) + An Hn (x ε2 ) = Dn Jn (x ε1 ), (1) √ √ (1) √ ε1 En ∂∂r Jn (x ε2 ) + ε1 An ∂∂r Hn (x ε2 ) = ε2 Dn ∂∂r Jn (x ε1 ), where x = rω /c = 2π r/λ is the dimensionless size parameter, En , An and Dn are cylindrical harmonic amplitudes of the incident, scattered and internal magnetic fields, respectively, (1) expressed in terms of the Bessel Jn (ζ ) and Hankel Hn (ζ ) functions. The Lorenz-Mie coefficients can be expressed as an = An /En . In addition, we introduce new coefficients dn = Dn /En to characterize the fields inside the cylinder. The system (1) determines both coefficients an and dn for the scattered and internal magnetic fields as: an =
√ √ √ √ ε2 Jn (x ε2 ) ∂∂r Jn (x ε1 ) − ε1 ∂∂r Jn (x ε2 )Jn (x ε1 ) √ √ , (1) √ (1) √ ε1 ∂∂r Hn (x ε2 )Jn (x ε1 ) − ε2 Hn (x ε2 ) ∂∂r Jn (x ε1 )
√ √ (1) √ (1) √ Jn (x ε2 ) ∂∂r Hn (x ε2 ) − Hn (x ε2 ) ∂∂r Jn (x ε2 ) dn = √ √ . (1) √ (1) √ Jn (x ε1 ) ∂∂r Hn (x ε2 ) − εε21 Hn (x ε2 ) ∂∂r Jn (x ε1 )
(2)
(3)
Figure 1 presents the spectra of the coefficients a0 and d0 calculated by using Eqs. (2) and (3). As seen from the figure, these two coefficients are essentially different. In particular, the spectra of |d0 |2 demonstrate classical resonance responses with perfect Lorentzian shapes line at all values of ε1 and, in correspondence with the quasi-periodic character of Bessel and Hankel functions, are represented by quasi-equidistant bands on the frequency scale. In contrast to the spectra of |d0 |2 , the spectra of |a0 |2 demonstrate asymmetric profiles with either sharp increase or drop of |a0 |2 values at the resonance frequencies of the rod eigenmodes. The specific signs of the derivatives characterizing these changes depend on the position of the resonance on the frequency scale with respect to the maximum of the background (Fig. 1).
Received 14 Oct 2013; revised 7 Nov 2013; accepted 9 Nov 2013; published 27 Nov 2013 #199493 - $15.00 USD (C) 2013 OSA 2 December 2013 | Vol. 21, No. 24 | DOI:10.1364/OE.21.030107 | OPTICS EXPRESS 30109
10
|d0|2 |a0|2
800
(a)
8
(b)
600
6 400
4 2
200
0 1.0
0 1.0
0.8
0.8
0.6
0.6
0.4
0.4
0.2 0
0
0.2
(c) 1
2
3
4
5
0 0
(d) 1
2
3
4
5
size parameter x Fig. 1. Spectra of the squared modules of the coefficients (a,b) |d0 |2 and (c,d) |a0 |2 for a single infinite dielectric rod at various values of the rod dielectric permittivity: (a, c) ε1 = 4 and (b, d) 50. of the surrounding Permittivity medium ε2 = 1. Blue curves represent the √ (1) √ background ∂∂r Jn (x ε2 ) / ∂∂r Hn (x ε2 ) . The size parameter x = 2π r/λ .
3.
Mie scattering and Fano resonances
If the interaction of a resonance mode with a continuum state involves an entire continuum, the following relation (named after Fano) can be derived [11]: I(ω ) =
(q + Ω)2 2 sin Δ, 1 + Ω2
(4)
where I(ω ) is the intensity, Δ(ω ) is the phase difference between the resonant and continuum states, q = cot Δ is the Fano asymmetry parameter, sin2 Δ represents a background produced by a plane wave. Below, we demonstrate that the spectrum of Mie scattering can be presented in the form of an infinite series of the Fano profiles. To demonstrate this result, we write simultaneously three expressions: (i) the Lorenz-Mie coefficient an which defines the scattered field in accord to Eq. (1), (ii) general condition for the Fano resonance; and (iii) analytical expression describing the Fano resonance at the interference of a narrow resonance (symmetric Lorentzian) with a slow varying background (plane wave): √ √ ∂ ∂ (1) √ Jn (x ε1 ) + −ε1 En ∂∂r Jn (x ε2 ) = ε1 an Hn (x ε2 ), ∂r ∂r Narrow resonance + Slow varying background = Fano profile, 1 (q + Ω)2 2 + B(ω ) exp [iϕB (ω )] = sin Δ, A(ω ) exp [iϕA (ω )] Ω+i 1 + Ω2
ε2 dn
(5a) (5b) (5c)
Analyzing the left-hand side of Eq. (5a), we identify two terms of different linewidth with √ the first term ε2 dn ∂∂r Jn (x ε1 ) as symmetric Lorentzian (see Fig. 1) and the second term √ −ε1 En ∂∂r Jn (x ε2 ) as a slowly changed background, both oscillating with the resonance frequency ω . Comparison with Eqs. (5b) and (5c) provides Eq. (5a) with the clear physical interpretation as a Fano resonance between an incident wave and the wave scattered from the Received 14 Oct 2013; revised 7 Nov 2013; accepted 9 Nov 2013; published 27 Nov 2013 #199493 - $15.00 USD (C) 2013 OSA 2 December 2013 | Vol. 21, No. 24 | DOI:10.1364/OE.21.030107 | OPTICS EXPRESS 30110
|q|→∞
q >0
q=0
10
Q-factor
8
Qsca,0
2
TE09
200 100
TE01
6 4
q 0
300
20
40
60
ε1
+4
80 100
ε1 = 50
+3
40
+2
30
+1
20 10
0 0
1
2
3
4
5
size parameter x Fig. 2. Spectrum of the Mie scattering efficiency Qsca,0 for the dipole mode TE0k for a single dielectric circular rod and different values of real dielectric permittivity ε1 . The rod is embedded in air, ε2 = 1. The corresponding values of the Fano parameter q are shown above the plot. Curves are shifted vertically by the values shown. Insert: Dependence of the Q factor for the TE0k resonance Mie modes (k = 1 to 9) on ε1 .
cylinder. In this case the total spectral dependence of the scattering amplitudes TEn as a function of the incident light frequency ω exhibits an infinite cascade of the Fano resonances, with each individual resonance described by the conventional Fano formula. Scattering efficiency of a circular cylinder for the TE polarized waves and normal incidence 2 is defined by the squared modules of the scattering amplitude, Qsca = (2/x) ∑∞ n=−∞ |an | [8]. Figure 2 presents the spectra of the Mie scattering efficiency Qsca,0 = (2/x) |a0 |2 for the most intense dipole mode TE0 . We observe the spectrum transformation with a change of the dielectric permittivity ε1 , manifested in a shift of the resonances ω0k to the low-frequency region and also in a substantial narrowing of the resonance peaks. These two features define the dependence of the Q factor Qk = ω0k /Γk on ε1 for TE0k resonances, shown in the insert of Fig. 2. Independence of the Mie band asymmetry on ε1 is explained by the fact that the Fano parameter q = cot Δ depends only on the phase difference [26] between the resonant band and the background Δ(ω ) = ϕA (ω ) − ϕB (ω ). Thus, for the resonant frequency ω0 the phase of the narrow band is always ϕA (ω0 ) = π /2 = const. In our case, the role of a slowly varying background is played by the incoming waves which propagate in a surrounding medium with the permittivity ε2 . Correspondingly, the phase of the background defined by the function dielectric √ ∂ (1) √ ∂ ∂ r Jn (x ε2 ) / ∂ r Hn (x ε2 ) does not depend of the dielectric permittivity ε1 . 4.
Fano parameter
To provide the justification for our approach and to extract the Fano asymmetry parameter q, we fit the calculated spectra with the Fano formula (4). In the vicinity of the Mie resonances, the intensity of waves scattered by the rod is comparable to the sin-type background. As a result, interference of the resonant scattered wave with the incident wave produces strong corrections to the resonant mode. Figure 3(a) shows the TE0n spectrum |a0 |2 for the dipole mode TE0k and the sin-type background. Additionally, we present three fitting Fano profiles for TE02 , TE05 and TE07 modes as examples to demonstrate the accuracy of our approach. All these spectra allow us to analyze the shape of the Mie resonances with particular thoroughness. The results clearly
Received 14 Oct 2013; revised 7 Nov 2013; accepted 9 Nov 2013; published 27 Nov 2013 #199493 - $15.00 USD (C) 2013 OSA 2 December 2013 | Vol. 21, No. 24 | DOI:10.1364/OE.21.030107 | OPTICS EXPRESS 30111
q02= 2.82 q05= 0.08 q07= -0.95
Jn(x)
1.0
(a)
0.8
1 0
Fano parameter q
-1 8
0.6 0.4 0.2
(b)
J0(x)
J1(x)
J2(x)
4 0
-4
0 0
-8 (c)
1
2
3
4
5
0
2
4
6
8
10
size parameter x Fig. 3. (a) Red: squared Lorenz-Mie coefficient |a0 |2 for a circular rod (ε1 = 50) embedded in air (ε2 = 1). Blue, black and green curves present the results for the Fano fitting of the TE02 , TE05 and TE07 modes. The corresponding values of the Fano parameter q are shown on the top. (b) Bessel functions Jn (x) for n = 0, 1, 2. (c) Dependence of the Fano parameter q on the size parameter x = 2π r/λ for the dipole mode TE0k (red) and multipole modes TE1k (green) and TE2k (blue) for a circular rod (ε1 = 50) embedded in air (ε2 = 1). The Fano line shapes for selected values of q are shown on the right.
indicate a remarkable transformation of the resonant TE0n profile when the size parameter x = 2π r/λ is changing. The results of our calculations of the spectral dependence of the Fano parameter q for the dipole mode TE0k , quadrupole mode TE1k and octuple mode TE2k of a high-dielectric circular rod are presented in Fig. 3(c). To obtain these results, we use the profiles of large number of resonances (1 k 9) in a wide range of the value of the dielectric permittivity ε1 = 1 to 100. This allow obtaining sets of calculated values of the Fano parameter q describing the evolution of the Mie resonances for changing refractive index of the rod material. The characteristic dependence q(x) ∼ − cot x is revealed for the dipole mode TE0k and multipole modes TE1k , TE2k resembles the familiar Fano-type dependence q(Δ) = − cot Δ [26], demonstrating directly a link between the shape of a narrow resonance with the phase difference between resonant and continuum states for given ω . Similar dependence of the Fano parameter q is observed for all other types of Fano resonances, in particular for the case of Fabry-P´erot resonances in photonic crystal created by an array of two types of slabs with disordered dielectric permittivity ε [21]. The most intriguing result here is the difference in the line shape and q(x) – dependence between the dipole mode TE0k and multipole modes TEnk at small x. We notice that with a growth of x, the multipole modes retrace the initial q(x) dependence of the dipole mode TE0k . For understanding of the behavior of the Fano parameter q, in Fig. 3(b) we show Bessel functions Jn (x) for n = 0, 1, 2. One can observe a perfect analogy between Figs. 3(b) and 3(c) namely the difference between J0 (x) and J1 (x), J2 (x) in the half of the first period and a perfect analogy at higher x. We notice that the Fano dependence q(Δ) = − cot Δ is satisfied for all multipole modes. 5.
Conclusions
We have demonstrated analytically that the results of the conventional Mie scattering of electromagnetic waves by high-index dielectric circular rods can be represented as cascades of Fano
Received 14 Oct 2013; revised 7 Nov 2013; accepted 9 Nov 2013; published 27 Nov 2013 #199493 - $15.00 USD (C) 2013 OSA 2 December 2013 | Vol. 21, No. 24 | DOI:10.1364/OE.21.030107 | OPTICS EXPRESS 30112
resonances. This result has a general nature, and it is applicable to any wave scattering by a body of revolution exhibiting Mie resonances. Our conclusions are based on Eqs. (5a)–(5c) that demonstrate a complete equivalence between the conditions which define the Fano resonance and Mie scattering. We believe that our study may change substantially the perception of the seemingly well-known Mie scattering of light, and it may open novel opportunities for the manipulation and control of electromagnetic waves by high-index dielectric particles. Acknowledgments We thank A.E. Mrioshnichenko for fruitful discussions. This work was supported by the Ministry of Education and Science of the Russian Federation (Projects No. 11.G34.31.0020, 14.B37.21.1964), the Russian Foundation of Basic Research (grants 11-02-00865 and 13-0200186), and the Australian National University.
Received 14 Oct 2013; revised 7 Nov 2013; accepted 9 Nov 2013; published 27 Nov 2013 #199493 - $15.00 USD (C) 2013 OSA 2 December 2013 | Vol. 21, No. 24 | DOI:10.1364/OE.21.030107 | OPTICS EXPRESS 30113
∗ [email protected]
Abstract: We reveal that the resonant Mie scattering by high-index dielectric nanoparticles can be presented through cascades of Fano resonances. We employ the exact solution of Maxwell’s equations and demonstrate that the Lorenz-Mie coefficients of the Mie problem can be expressed generically as infinite series of Fano functions as they describe interference between the background radiation originated from an incident wave and narrow-spectrum Mie scattering modes that lead to Fano resonances. © 2013 Optical Society of America OCIS codes: (290.4020) Mie theory; (260.5740) Resonance.
References and links 1. C. M. Soukoulis, ed., Photonic Band Gap Materials, Vol. 315 of NATO ASI Series E (Springer, 1996). 2. V. Yannopapas and A. Moroz, “Negative refractive index metamaterials from inherently non-magnetic materials for deep infrared to terahertz frequency ranges,” J. Phys. Condens. Matt. 17, 3717–3734 (2005). 3. P. Garcia, M. Ibisate, R. Sapienza, D. Wiersma, and C. L´opez, “Mie resonances to tailor random lasers,” Phys. Rev. A 80, 013833 (2009). 4. M. V. Rybin, P. V. Kapitanova, D. S. Filonov, A. P. Slobozhanyuk, P. A. Belov, Y. S. Kivshar, and M. F. Limonov, “Fano resonances in antennas: General control over radiation patterns,” Phys. Rev. B 88, 205106 (2013). 5. S. Foteinopoulou, J. Vigneron, and C. Vandenbem, “Optical near-field excitations on plasmonic nanoparticlebased structures,” Opt. Express 15, 4253–4267 (2007). 6. I. Romero, J. Aizpurua, G. W. Bryant, and F. J. Garc´ıa de Abajo, “Plasmons in nearly touching metallic nanoparticles: singular response in the limit of touching dimers,” Opt. Express 14, 9988–9999 (2006). 7. S. A. Maier, P. G. Kik, and H. A. Atwater, “Optical pulse propagation in metal nanoparticle chain waveguides,” Phys. Rev. B 67, 205402 (2003). 8. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley-VCH, 1998). 9. M. F. Limonov and R. M. De La Rue, eds., Optical Properties of Photonic Structures: Interplay of Order and Disorder (CRC Press, 2012). 10. S. Gottardo, R. Sapienza, P. D. Garcia, A. Blanco, D. S. Wiersma, and C. L´opez, “Resonance-driven random lasing,” Nat. Photonics 2, 429–432 (2008). 11. U. Fano, “Effects of configuration interaction on intensities and phase shifts,” Phys. Rev. 124, 1866–1878 (1961). 12. V. Madhavan, W. Chen, T. Jamneala, M. F. Crommie, and N. S. Wingreen, “Tunneling into a single magnetic atom: Spectroscopic evidence of the Kondo resonance,” Science 280, 567–569 (1998). 13. M. F. Limonov, A. I. Rykov, S. Tajima, A. Yamanaka, “Raman scattering study on fully oxygenated YBa2 CuO7 single crystals: x-y anisotropy in the superconductivity-induced effects,” Phys. Rev. Lett. 80, 825–828 (1998). 14. M. Limonov, S. Lee, S. Tajima, and A. Yamanaka, “Superconductivity-induced resonant raman scattering in multilayer high-Tc superconductors,” Phys. Rev. B 66, 054509 (2002).
Received 14 Oct 2013; revised 7 Nov 2013; accepted 9 Nov 2013; published 27 Nov 2013 #199493 - $15.00 USD (C) 2013 OSA 2 December 2013 | Vol. 21, No. 24 | DOI:10.1364/OE.21.030107 | OPTICS EXPRESS 30107
15. M. I. Tribelsky, S. Flach, A. E. Miroshnichenko, A. V. Gorbach, and Y. S. Kivshar, “Light scattering by a finite obstacle and Fano resonances,” Phys. Rev. Lett. 100, 043903 (2008). 16. M. V. Rybin, A. B. Khanikaev, M. Inoue, K. B. Samusev, M. J. Steel, G. Yushin, and M. F. Limonov, “Fano resonance between Mie and Bragg scattering in photonic crystals,” Phys. Rev. Lett. 103, 023901 (2009). 17. M. V. Rybin, A. B. Khanikaev, M. Inoue, A. K. Samusev, M. J. Steel, G. Yushin, and M. F. Limonov, “Bragg scattering induces Fano resonance in photonic crystals,” Photonics Nanostruct. Fundam. Appl. 8, 86–93 (2010). 18. A. E. Miroshnichenko, S. Flach, and Y. S. Kivshar, “Fano resonances in nanoscale structures,” Rev. Mod. Phys. 82, 2257–2298 (2010). 19. B. Luk’yanchuk, N. I. Zheludev, S. A. Maier, N. J. Halas, P. Nordlander, H. Giessen, and C. T. Chong, “The Fano resonance in plasmonic nanostructures and metamaterials,” Nat. Mater. 9, 707–715 (2010). 20. J. A. Fan, C. Wu, K. Bao, J. Bao, R. Bardhan, N. J. Halas, V. N. Manoharan, P. Nordlander, G. Shvets, and F. Capasso, “Self-assembled plasmonic nanoparticle clusters,” Science 328, 1135–1138 (2010). 21. A. N. Poddubny, M. V. Rybin, M. F. Limonov, and Y. S. Kivshar, “Fano interference governs wave transport in disordered systems,” Nat. Commun. 3, 914 (2012). 22. A. E. Miroshnichenko, Y. S. Kivshar, “Fano resonances in all-dielectric oligomers,” Nano Lett. 12, 6459–6463 (2012). 23. Y. Francescato, V. Giannini, and S. A. Maier, “Plasmonic systems unveiled by Fano resonances,” ACS Nano 6, 1830–1838 (2012). 24. M. Tribelsky, A. Miroshnichenko, and Y. Kivshar, “Unconventional Fano resonances in light scattering by small particles,” Europhys. Lett. 97, 44005 (2012). 25. C. P. Burrows and W. L. Barnes, “Large spectral extinction due to overlap of dipolar and quadrupolar plasmonic modes of metallic nanoparticles in arrays,” Opt. Express 18, 3187–3198 (2010). 26. J.-P. Connerade, A. M. Lane, “Interacting resonances in atomic spectroscopy,” Rep. Prog. Phys. 51, 1439–1478 (1988).
1.
Introduction
Mie scattering by small particles is one of the fundamental optical phenomena, which can govern the functionalities of forefront optical devices and innovative photonic structures such as photonic crystals [1], metamaterials [2], random lasers [3], and nanoantennas [4–6], nanoparticle-based waveguides [7], as well as SNOM tips [5]. The solution of the Mie scattering problem is represented by a sum √ of infinite series, and it is valid for all possible optical diameter-to-wavelength ratios (2r ε /λ ) and dielectric sphere-to-environment contrasts [8]. Important consequences of the Mie resonances are, for example, a strong reduction of the light group velocity in photonic glasses and monodisperse spheres [9], and modification of random lasing [10]. The Mie scattering theory can be applied not only to spherical particles, but also to other “bodies of revolution”, in particular, cylindrical nanorods. Resonant Mie scattering involves the excitation of localized modes in the scatterer when the frequency ω of the incident wave approaches one of the particle’s eigenmode frequencies. This leads to the emission of electromagnetic waves by the particle with the same frequency ω resulting in interference between the incident and scattered waves. If radiation with a narrow band of virtually any origin interacts with a continuum-spectrum radiation through interference effects constructively or destructively, we should expect Fano-type resonances [11]. Fano resonance is a general physical phenomenon well-known across many disciplines dealing with oscillations and waves [12–14]. In photonics, the Fano resonance emerges in many areas including the physics of plasmonic nanostructures and photonic crystals [15–23]. For the light scattering by small nanoparticles, the Fano resonance has also been investigated, in particular for the interference between different electromagnetic modes excited in the particle with the same or different multipole moments [15, 24, 25], as well as interference between resonant and non-resonant scattering from spheres. Surprisingly, the most general case of the Mie scattering that results in the interference between incident and scattered waves has never been investigated before, while such an interference constitutes the basic process in the structure of electromagnetic fields around the scatterer. In this work, we arrive at a new and unexpected conclusion that the resonant Mie scattering
Received 14 Oct 2013; revised 7 Nov 2013; accepted 9 Nov 2013; published 27 Nov 2013 #199493 - $15.00 USD (C) 2013 OSA 2 December 2013 | Vol. 21, No. 24 | DOI:10.1364/OE.21.030107 | OPTICS EXPRESS 30108
results in nothing but an infinite series of Fano resonances. We demonstrate analytically that the Lorenz-Mie coefficients describing Mie scattering and cascades of Fano profiles can be expressed by identically analytical formulas. For the sake of simplicity, we study the twodimensional case of the Fano resonance formation at the Mie scattering by an infinitely long dielectric cylindrical rod. However, the proposed approach is rather general, and it can be generalized to other systems. 2.
Lorenz-Mie coefficients
We consider the scattering of electromagnetic waves by a single homogeneous infinite dielectric rod with the radius r and with the purely real dielectric permittivity ε1 without dissipation. The surrounding medium is supposed to be transparent and homogeneous with dielectric permittivity ε2 . As is known, the far-field scattered by a circular rod can be expanded into orthogonal electromagnetic dipolar and multipolar terms, with cylindrical Lorenz-Mie coefficients an and bn corresponding to the electric and magnetic moments, respectively [8]. If only the TE polarization is considered, the magnetic field of incident waves is directed along the rod axis, while the electric field is normal to this axis. Then the scattered fields are defined by coefficients of only one type an , while bn vanish. The resonances excited in the rod are denoted as TEnk (n is integer and k is positive integer), where n is multipole order and k is resonance number. For the TE polarization, the Maxwell’s boundary conditions for the tangential components of both H and E fields at the rod surface can be written as: √ √ (1) √ En Jn (x ε2 ) + An Hn (x ε2 ) = Dn Jn (x ε1 ), (1) √ √ (1) √ ε1 En ∂∂r Jn (x ε2 ) + ε1 An ∂∂r Hn (x ε2 ) = ε2 Dn ∂∂r Jn (x ε1 ), where x = rω /c = 2π r/λ is the dimensionless size parameter, En , An and Dn are cylindrical harmonic amplitudes of the incident, scattered and internal magnetic fields, respectively, (1) expressed in terms of the Bessel Jn (ζ ) and Hankel Hn (ζ ) functions. The Lorenz-Mie coefficients can be expressed as an = An /En . In addition, we introduce new coefficients dn = Dn /En to characterize the fields inside the cylinder. The system (1) determines both coefficients an and dn for the scattered and internal magnetic fields as: an =
√ √ √ √ ε2 Jn (x ε2 ) ∂∂r Jn (x ε1 ) − ε1 ∂∂r Jn (x ε2 )Jn (x ε1 ) √ √ , (1) √ (1) √ ε1 ∂∂r Hn (x ε2 )Jn (x ε1 ) − ε2 Hn (x ε2 ) ∂∂r Jn (x ε1 )
√ √ (1) √ (1) √ Jn (x ε2 ) ∂∂r Hn (x ε2 ) − Hn (x ε2 ) ∂∂r Jn (x ε2 ) dn = √ √ . (1) √ (1) √ Jn (x ε1 ) ∂∂r Hn (x ε2 ) − εε21 Hn (x ε2 ) ∂∂r Jn (x ε1 )
(2)
(3)
Figure 1 presents the spectra of the coefficients a0 and d0 calculated by using Eqs. (2) and (3). As seen from the figure, these two coefficients are essentially different. In particular, the spectra of |d0 |2 demonstrate classical resonance responses with perfect Lorentzian shapes line at all values of ε1 and, in correspondence with the quasi-periodic character of Bessel and Hankel functions, are represented by quasi-equidistant bands on the frequency scale. In contrast to the spectra of |d0 |2 , the spectra of |a0 |2 demonstrate asymmetric profiles with either sharp increase or drop of |a0 |2 values at the resonance frequencies of the rod eigenmodes. The specific signs of the derivatives characterizing these changes depend on the position of the resonance on the frequency scale with respect to the maximum of the background (Fig. 1).
Received 14 Oct 2013; revised 7 Nov 2013; accepted 9 Nov 2013; published 27 Nov 2013 #199493 - $15.00 USD (C) 2013 OSA 2 December 2013 | Vol. 21, No. 24 | DOI:10.1364/OE.21.030107 | OPTICS EXPRESS 30109
10
|d0|2 |a0|2
800
(a)
8
(b)
600
6 400
4 2
200
0 1.0
0 1.0
0.8
0.8
0.6
0.6
0.4
0.4
0.2 0
0
0.2
(c) 1
2
3
4
5
0 0
(d) 1
2
3
4
5
size parameter x Fig. 1. Spectra of the squared modules of the coefficients (a,b) |d0 |2 and (c,d) |a0 |2 for a single infinite dielectric rod at various values of the rod dielectric permittivity: (a, c) ε1 = 4 and (b, d) 50. of the surrounding Permittivity medium ε2 = 1. Blue curves represent the √ (1) √ background ∂∂r Jn (x ε2 ) / ∂∂r Hn (x ε2 ) . The size parameter x = 2π r/λ .
3.
Mie scattering and Fano resonances
If the interaction of a resonance mode with a continuum state involves an entire continuum, the following relation (named after Fano) can be derived [11]: I(ω ) =
(q + Ω)2 2 sin Δ, 1 + Ω2
(4)
where I(ω ) is the intensity, Δ(ω ) is the phase difference between the resonant and continuum states, q = cot Δ is the Fano asymmetry parameter, sin2 Δ represents a background produced by a plane wave. Below, we demonstrate that the spectrum of Mie scattering can be presented in the form of an infinite series of the Fano profiles. To demonstrate this result, we write simultaneously three expressions: (i) the Lorenz-Mie coefficient an which defines the scattered field in accord to Eq. (1), (ii) general condition for the Fano resonance; and (iii) analytical expression describing the Fano resonance at the interference of a narrow resonance (symmetric Lorentzian) with a slow varying background (plane wave): √ √ ∂ ∂ (1) √ Jn (x ε1 ) + −ε1 En ∂∂r Jn (x ε2 ) = ε1 an Hn (x ε2 ), ∂r ∂r Narrow resonance + Slow varying background = Fano profile, 1 (q + Ω)2 2 + B(ω ) exp [iϕB (ω )] = sin Δ, A(ω ) exp [iϕA (ω )] Ω+i 1 + Ω2
ε2 dn
(5a) (5b) (5c)
Analyzing the left-hand side of Eq. (5a), we identify two terms of different linewidth with √ the first term ε2 dn ∂∂r Jn (x ε1 ) as symmetric Lorentzian (see Fig. 1) and the second term √ −ε1 En ∂∂r Jn (x ε2 ) as a slowly changed background, both oscillating with the resonance frequency ω . Comparison with Eqs. (5b) and (5c) provides Eq. (5a) with the clear physical interpretation as a Fano resonance between an incident wave and the wave scattered from the Received 14 Oct 2013; revised 7 Nov 2013; accepted 9 Nov 2013; published 27 Nov 2013 #199493 - $15.00 USD (C) 2013 OSA 2 December 2013 | Vol. 21, No. 24 | DOI:10.1364/OE.21.030107 | OPTICS EXPRESS 30110
|q|→∞
q >0
q=0
10
Q-factor
8
Qsca,0
2
TE09
200 100
TE01
6 4
q 0
300
20
40
60
ε1
+4
80 100
ε1 = 50
+3
40
+2
30
+1
20 10
0 0
1
2
3
4
5
size parameter x Fig. 2. Spectrum of the Mie scattering efficiency Qsca,0 for the dipole mode TE0k for a single dielectric circular rod and different values of real dielectric permittivity ε1 . The rod is embedded in air, ε2 = 1. The corresponding values of the Fano parameter q are shown above the plot. Curves are shifted vertically by the values shown. Insert: Dependence of the Q factor for the TE0k resonance Mie modes (k = 1 to 9) on ε1 .
cylinder. In this case the total spectral dependence of the scattering amplitudes TEn as a function of the incident light frequency ω exhibits an infinite cascade of the Fano resonances, with each individual resonance described by the conventional Fano formula. Scattering efficiency of a circular cylinder for the TE polarized waves and normal incidence 2 is defined by the squared modules of the scattering amplitude, Qsca = (2/x) ∑∞ n=−∞ |an | [8]. Figure 2 presents the spectra of the Mie scattering efficiency Qsca,0 = (2/x) |a0 |2 for the most intense dipole mode TE0 . We observe the spectrum transformation with a change of the dielectric permittivity ε1 , manifested in a shift of the resonances ω0k to the low-frequency region and also in a substantial narrowing of the resonance peaks. These two features define the dependence of the Q factor Qk = ω0k /Γk on ε1 for TE0k resonances, shown in the insert of Fig. 2. Independence of the Mie band asymmetry on ε1 is explained by the fact that the Fano parameter q = cot Δ depends only on the phase difference [26] between the resonant band and the background Δ(ω ) = ϕA (ω ) − ϕB (ω ). Thus, for the resonant frequency ω0 the phase of the narrow band is always ϕA (ω0 ) = π /2 = const. In our case, the role of a slowly varying background is played by the incoming waves which propagate in a surrounding medium with the permittivity ε2 . Correspondingly, the phase of the background defined by the function dielectric √ ∂ (1) √ ∂ ∂ r Jn (x ε2 ) / ∂ r Hn (x ε2 ) does not depend of the dielectric permittivity ε1 . 4.
Fano parameter
To provide the justification for our approach and to extract the Fano asymmetry parameter q, we fit the calculated spectra with the Fano formula (4). In the vicinity of the Mie resonances, the intensity of waves scattered by the rod is comparable to the sin-type background. As a result, interference of the resonant scattered wave with the incident wave produces strong corrections to the resonant mode. Figure 3(a) shows the TE0n spectrum |a0 |2 for the dipole mode TE0k and the sin-type background. Additionally, we present three fitting Fano profiles for TE02 , TE05 and TE07 modes as examples to demonstrate the accuracy of our approach. All these spectra allow us to analyze the shape of the Mie resonances with particular thoroughness. The results clearly
Received 14 Oct 2013; revised 7 Nov 2013; accepted 9 Nov 2013; published 27 Nov 2013 #199493 - $15.00 USD (C) 2013 OSA 2 December 2013 | Vol. 21, No. 24 | DOI:10.1364/OE.21.030107 | OPTICS EXPRESS 30111
q02= 2.82 q05= 0.08 q07= -0.95
Jn(x)
1.0
(a)
0.8
1 0
Fano parameter q
-1 8
0.6 0.4 0.2
(b)
J0(x)
J1(x)
J2(x)
4 0
-4
0 0
-8 (c)
1
2
3
4
5
0
2
4
6
8
10
size parameter x Fig. 3. (a) Red: squared Lorenz-Mie coefficient |a0 |2 for a circular rod (ε1 = 50) embedded in air (ε2 = 1). Blue, black and green curves present the results for the Fano fitting of the TE02 , TE05 and TE07 modes. The corresponding values of the Fano parameter q are shown on the top. (b) Bessel functions Jn (x) for n = 0, 1, 2. (c) Dependence of the Fano parameter q on the size parameter x = 2π r/λ for the dipole mode TE0k (red) and multipole modes TE1k (green) and TE2k (blue) for a circular rod (ε1 = 50) embedded in air (ε2 = 1). The Fano line shapes for selected values of q are shown on the right.
indicate a remarkable transformation of the resonant TE0n profile when the size parameter x = 2π r/λ is changing. The results of our calculations of the spectral dependence of the Fano parameter q for the dipole mode TE0k , quadrupole mode TE1k and octuple mode TE2k of a high-dielectric circular rod are presented in Fig. 3(c). To obtain these results, we use the profiles of large number of resonances (1 k 9) in a wide range of the value of the dielectric permittivity ε1 = 1 to 100. This allow obtaining sets of calculated values of the Fano parameter q describing the evolution of the Mie resonances for changing refractive index of the rod material. The characteristic dependence q(x) ∼ − cot x is revealed for the dipole mode TE0k and multipole modes TE1k , TE2k resembles the familiar Fano-type dependence q(Δ) = − cot Δ [26], demonstrating directly a link between the shape of a narrow resonance with the phase difference between resonant and continuum states for given ω . Similar dependence of the Fano parameter q is observed for all other types of Fano resonances, in particular for the case of Fabry-P´erot resonances in photonic crystal created by an array of two types of slabs with disordered dielectric permittivity ε [21]. The most intriguing result here is the difference in the line shape and q(x) – dependence between the dipole mode TE0k and multipole modes TEnk at small x. We notice that with a growth of x, the multipole modes retrace the initial q(x) dependence of the dipole mode TE0k . For understanding of the behavior of the Fano parameter q, in Fig. 3(b) we show Bessel functions Jn (x) for n = 0, 1, 2. One can observe a perfect analogy between Figs. 3(b) and 3(c) namely the difference between J0 (x) and J1 (x), J2 (x) in the half of the first period and a perfect analogy at higher x. We notice that the Fano dependence q(Δ) = − cot Δ is satisfied for all multipole modes. 5.
Conclusions
We have demonstrated analytically that the results of the conventional Mie scattering of electromagnetic waves by high-index dielectric circular rods can be represented as cascades of Fano
Received 14 Oct 2013; revised 7 Nov 2013; accepted 9 Nov 2013; published 27 Nov 2013 #199493 - $15.00 USD (C) 2013 OSA 2 December 2013 | Vol. 21, No. 24 | DOI:10.1364/OE.21.030107 | OPTICS EXPRESS 30112
resonances. This result has a general nature, and it is applicable to any wave scattering by a body of revolution exhibiting Mie resonances. Our conclusions are based on Eqs. (5a)–(5c) that demonstrate a complete equivalence between the conditions which define the Fano resonance and Mie scattering. We believe that our study may change substantially the perception of the seemingly well-known Mie scattering of light, and it may open novel opportunities for the manipulation and control of electromagnetic waves by high-index dielectric particles. Acknowledgments We thank A.E. Mrioshnichenko for fruitful discussions. This work was supported by the Ministry of Education and Science of the Russian Federation (Projects No. 11.G34.31.0020, 14.B37.21.1964), the Russian Foundation of Basic Research (grants 11-02-00865 and 13-0200186), and the Australian National University.
Received 14 Oct 2013; revised 7 Nov 2013; accepted 9 Nov 2013; published 27 Nov 2013 #199493 - $15.00 USD (C) 2013 OSA 2 December 2013 | Vol. 21, No. 24 | DOI:10.1364/OE.21.030107 | OPTICS EXPRESS 30113
