Controlling near-field polarization distribution of a plasmonic prolate nanospheroid by its aspect ratio and polarization of the incident electromagnetic field Evgene D. Chubchev, Yulia V. Vladimirova∗ and Victor N. Zadkov International Laser Center and Faculty of Physics, M. V. Lomonosov Moscow State University, Moscow 119991, Russia ∗ [email protected]

Abstract: Near-field polarization distribution of a plasmonic prolate nanospheroid in an incident electromagnetic field versus its polarization and the spheroid’s aspect ratio is studied in detail. Polarization of the near-field is described with the help of the 3D generalized Stokes parameters, allowing simple visualization. It is shown that this distribution has a complex structure, which drastically depends on the incident field polarization and parameters of the plasmon resonance of the nanoparticle. Received analytical solutions cover the whole set of particles with shape varying from spherical to the nanoneedles and nanorods by changing the aspect ratio of the spheroid. An experiment for visualization of the vectorial near-field around a plasmonic nanoparticle is proposed. © 2014 Optical Society of America OCIS codes: (260.5430) Polarization; (160.4236) Nanomaterials; (180.4243) Near-field microscopy.

References and links 1. B. Hecht and L. Novotny, Principles of Nano-Optics (Cambridge University, 2006). 2. Alternatively, one can study the dependence of the vectorial near-field of the prolate nanospheroid at fixed value of its aspect ratio versus the polarization and frequency of the incident field. 3. L. Novotny and N. van Hulst, “Antennas for light,” Nat. Photonics 5, 83–90 (2011). 4. A. E. Krasnok, I. S. Maksymov, and A. I. Denisyuk, P. A. Belov, A. E. Miroshnichenko, C. R. Simovskii, and Yu. S. Kivshar, “Optical nanoantennas,” Phys. Usp. 56, 539–564 (2013). 5. V. Klimov, Nanoplasmonics (Pan Stanford, 2014). 6. O. L. Muskens, V. Giannini, J. A. S´anchez-Gil, and J. G´omez Rivas, “Strong enhancement of the radiative decay rate of emitters by single plasmonic nanoantennas,” Nano Lett. 7, 2871–2875 (2007). 7. J. N. Farahani, D. W. Pohl, H. J. Eisler, and B. Hecht, “Single quantum dot coupled to a scanning optical antenna: A tunable superemitter,” Phys. Rev. Lett. 95, 017402 (2005). 8. P. J. Schuck, D. P. Fromm, A. Sundaramurthy, G. S. Kino, and W. E. Moerner, “Improving the mismatch between light and nanoscale objects with gold bowtie nanoantennas,” Phys. Rev. Lett. 94, 017402 (2005). 9. T. H. Taminiau, R. J. Moerland, F. B. Segerink, L. Kuipers, and N. F. van Hulst, “λ /4 resonance of an optical monopole antenna probed by single molecule fluorescence,” Nano Lett. 7, 28–33 (2007). 10. S. Nie and S. R. Emory, “Probing single molecules and single nanoparticles by surface-enhanced raman scattering,” Science 275, 1102–1106 (1977). 11. K. Kneipp, M. Moskovits, and H. Kneipp, Surface-Enhanced Raman Scattering: Physics and Applications (Springer-Verlag, 2006).

#214368 - $15.00 USD Received 18 Jun 2014; revised 29 Jul 2014; accepted 1 Aug 2014; published 15 Aug 2014 (C) 2014 OSA 25 August 2014 | Vol. 22, No. 17 | DOI:10.1364/OE.22.020432 | OPTICS EXPRESS 20432

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Sonnefraud, Y. Kivshar, M. Hong, Ch. Phillips, S. Maier, and A. E. Miroshnichenko, “Plasmonic nanoclusters with rotational symmetry: polarization-invariant far-field response vs changing near-field distribution,” ACS Nano 7, 11138–11146 (2013). 31. V. Mizeikis, E. Kowalska, B. Ohtani, S. Juodkazis, “Frequency- and polarization-dependent optical response of asymmetric spheroidal silver nanoparticles on dielectric substrate,” Phys. Status Solidi RRL 4, 268–270 (2010). 32. Y. Y. Yu, S. S. Chang, C. L. Lee, and C. R. C. Wang, “Gold nanorods: electrochemical synthesis and optical properties,” J. Phys. Chem. B 101, 6661–6664 (1997). 33. J. Perez-Juste, L. M. Liz-Marzan, S. Carnie, D. Y. C. Chan, and P. Mulvaney, “Electric-field-directed growth for gold nanorods in aqueous surfactant solutions,” Adv. Funct. Mater. 14, 571–579 (2004). 34. T. Set¨al¨a, A. Shevchenko, M. Kaivola, and A. T. Friberg, “Degree of polarization for optical near fields,” Phys. Rev. E 66, 016615 (2002). 35. G. Hass and L. Hadley, Optical Properties of Metals (American Institute of Physics Handbook) ed. by D. E. Gray (McGraw-Hill, 1963). 36. M. S. Agranovich, B. Z. Katsenelenbaum, A. N. Sivov, and N. N.Voitovich, Generalized Method of Eigenoscillation in Diffraction Theory (Wiley, 1999). 37. V. V. Klimov, M. Ducloy and V. S. Letokhov, “Spontaneous emission of an atom placed near a prolate nanospheroid,” Eur. Phys. J. D 20, 133–148 (2002). 38. A. F. Stevenson, “Electromagnetic scattering by an ellipsoid in the third approximation,” Appl. Phys. 24, 1143– 1151 (1953). 39. It is worth to note here that the degree of polarization we introduced in the paper does not reflect the fluctuations of the incident field vector. In order to take these fluctuations into account one has to use different definition of the degree of polarization via the Stokes parameters, which for the plane wave has the form: P∗ = (S12 + S22 + S32 )0.5 /S0 . Without fluctuations, this polarization degree is always equal to 1. 40. P. J. S. Smith, I. Davis, C. G. Galbraith, and A. Stemmer, “Special issue on high-resolution optical imaging,” J. Opt. 15(9), 090201 (2013).

#214368 - $15.00 USD Received 18 Jun 2014; revised 29 Jul 2014; accepted 1 Aug 2014; published 15 Aug 2014 (C) 2014 OSA 25 August 2014 | Vol. 22, No. 17 | DOI:10.1364/OE.22.020432 | OPTICS EXPRESS 20433

1.

Introduction

Control of near-field in the proximity of plasmonic nanostructures is a key to shaping the spatial intensity of light and its polarization distribution at the nanoscale [1]. Near-field being formed by the interference of the incident electromagnetic field with the local field of the nanoparticle strongly depends both on the polarization of the incident field and aspect ratio of the prolate nanospheroid [2]. Dependence of the near-field intensity distribution on these parameters has been studied in detail [3–5] and it was shown that plasmon oscillations in the nanoparticle excited by the incident electromagnetic field could lead to a significant enhancement of the near-field. In the case of a plasmon resonance, the near-field can be enhanced in the tens or even hundreds of times [6–9]. Such plasmonic nanoparticles are able not only to efficiently convert incident optical radiation into a highly localized near-field, but also vice versa. That is why they are also called plasmonic nanoantennas. In experiment, the near-field enhancement was studied in detail near the dipole nanoantennas [6], bow-tie nanoantennas [7,8], monopole nanoantennas [9], etc. This near-field enhancement is widely used in various applications, such as enhanced Raman scattering [10, 11], spasers [12] and biosensors [13]. Plasmon oscillations in the metal nanoparticle excited by the incident electromagnetic field affect also the spontaneous relaxation rate of a quantum emitter (atom, molecule, quantum dot) located in close proximity of the nanoparticle [6–9]. Most attention, both experimentally and theoretically, has been paid to controling the properties of the spontaneous fluorescence of quantum emitters in the vicinity of a metal nanoparticle [7,14,15] and, recently, to the spectrum of resonance fluorescence [16]. The properties of a quantum emitter near a nanoantenna are determined by the position of the emitter relative to the nanoantenna and both, the amplitude and polarization of the near-field at the point where the emitter is located. Taking into account the actual polarization at the emitter’s location is crucial for the interaction of the nanoparticle with the quantum emitter [5, 17]. Moreover, controling the near-field polarization, one can selectively excite required quantum transition(s) of the emitter and control its fluorescence intensity in a specific experiment. Precision polarization control is also required for the tunneling time measurements [18], highresolution microscopy and spectroscopy up to the level of single atoms and molecules [19], ellipsometry [20] and magneto-optical effect [21], atom optics [22, 23], in quantum optics of atoms near plasmonic nanoparticles [16], near-field electromagnetic trapping [24], studying Fano resonances and vortexes [25], and nonlinear optical effects [26]. Despite this demanding request, how the near-field polarization distribution depends on the frequency and polarization of the incident electromagnetic field still remains largely a challenging question and related works are just at start. It was shown, for instance, that a symmetrical resonance cross nanoantenna with the gap in its center focuses the incident electromagnetic radiation in this gap in a way that the near-field in the gap retains the polarization of the incident field with a high degree of accuracy [27]. Moreover, it was shown that a similar, but non-symmetric nanoantenna allows to control the near-field polarization in the gap simply varying the geometrical parameters of the nanoantenna [28]. Recently, a detailed analysis of the near-field polarization of a plasmonic nanoparticle studied with the help of scattering near-field microscopy (SNOM) was made [29], formation of polarization-independent far-field from a pentamer-like cluster of plasmonic nanoparticles was demonstrated [30], and frequency- and polarization-dependent optical response of asymmetric spheroidal silver nanoparticles on dielectric substrate was studied [31]. In this paper, we study theoretically in detail how the near-field polarization distribution of metal prolate nanospheroid interacting with a plane electromagnetic wave depends on the polarization and frequency of the incident field. The arrangement of the problem is shown in

#214368 - $15.00 USD Received 18 Jun 2014; revised 29 Jul 2014; accepted 1 Aug 2014; published 15 Aug 2014 (C) 2014 OSA 25 August 2014 | Vol. 22, No. 17 | DOI:10.1364/OE.22.020432 | OPTICS EXPRESS 20434

(b)

(a) c a

k

Fig. 1. (a) Arrangement of the problem. An incident plane electromagnetic wave interacts with a prolate nanospheroid with c and a being the major and minor semiaxes of the ellipse, respectively, ε is the permittivity of the nanoparticle and εH is the permittivity of the media the nanoparticle is located in. (b) Prolate nanospheroid in an incident linearly polarized plane electromagnetic wave directed at angle θ towards x-axis.

Fig. 1(a). For simplicity, we will consider here only the nanoparticles, which are made of noble metals and allow to limit our treatment only by the dipole plasmonic resonance in such particles. Also, to avoid complications related to the retardation effect we will consider the nanoparticles in the range of the sizes of some 5 to 50 nm [5], which covers lots of practical applications. We chose a spheroidal shape for the plasmonic nanoparticle for the following reasons: i) it allows analytical solution for the calculation of 3D near-field distribution and ii) the nanoparticles of a spheroidal shape could well describe the whole set of particles whose shape varies from the spherical one to needle-shaped simply by changing the aspect ratio α = a/c of small to the large axis of the spheroid. At α → 0, the spherical nanoparticle mimics well the so called nanoneedles and nanorods. The plasmonic nanoparticles of all these geometries and required sizes can easily be nowadays synthesized experimentally [5, 32, 33]. The paper is organized as follows. In Sec. 2 we describe the method for calculation of the near-field spatial intensity and polarization distribution of the prolate nanospheroid with the help of the 3D generalized Stokes parameters [34]. Section 3 gives the results of the calculations of the near-field polarization degree distributions for the cases when the incident electromagnetic field is a plane wave with linear, elliptical and circular polarization. In conclusion, we summarize the received results, discuss limitations of our calculations and suggest an experiment for visualizing the vectorial near-field around a plasmonic nanoparticle in an incident electromagnetic field. 2.

Calculation method for the near-field of a prolate nanospheroid

Consider a plasmonic nanoparticle having a shape of a prolate nanospheroid with linear dimensions much smaller than the wavelength of the incident electromagnetic field, large axis of which lies along z-axes (Fig. 1(a)). In all our numerical calculations throughout the paper we will assume that the nanoparticle is made of gold whose optical and plasmonic properties are well known [35]. In order to calculate the near-field of the nanoparticle we will use the so-called ε-method of solving the Maxwell equations [5, 36], which allows finding the near-field of a nanoparticle from its plasmonic spectrum, i.e., from eigenfunctions en and hn that satisfy the homogeneous Maxwell equations: rothn + ikεn en = 0, roten − ikhn = 0. (1) As a result, the electric field in the proximity of a nanoparticle of an arbitrary shape is defined as R ε(ω) − εH V (en , E0 )dV R E(r,t) = E0 (r) + ∑ en , (2) 2 εn − ε(ω) n V en dV #214368 - $15.00 USD Received 18 Jun 2014; revised 29 Jul 2014; accepted 1 Aug 2014; published 15 Aug 2014 (C) 2014 OSA 25 August 2014 | Vol. 22, No. 17 | DOI:10.1364/OE.22.020432 | OPTICS EXPRESS 20435

where ε(ω) is the permittivity of the material the nanoparticle is made of, which depends on the frequency ω of the incident electromagnetic field, en and εn are the unit-vector of the plasmon oscillations mode n and the eigenvalue corresponding to this mode, respectively, E0 is the incoming field and V is the nanoparticle volume. A detailed example of using the ε-method for calculation of the near-field of a plasmonic nanospheroid can be found in Ref. [37]. In our case, we can use for calculation of the fields en a quasistatic approximation [38] because the wavelength of the incident field significantly exceeds the linear sizes of the nanoparticle, i.e., λ  l. Therefore, the near-field of the nanoparticle can be written in accordance with Eq. (2) as R ε(ω) − εH V (−∇ϕn , E0 )dV R (−∇ϕn ), (3) E = E0 + ∑ 2 n εn − ε(ω) V ∇ϕn dV where functions ϕn are the potentials of the respective plasmonic modes that can be conveniently described with the help of the prolate spheroidal coordinates (see Ref. [5], chapter 7). Using correspondence between the prolate spheroidal (ξ , η, ψ) and Cartesian coordinates the potentials ϕ in the prolate spheroidal coordinates can be written as   cos mψ (m) (m) (m) in (4) ϕnm = Qn (ξ0 )Pn (η)Pn (ξ ) sin mψ inside the nanospheroid and (m)

(m)

(m)

out ϕnm = Qn (ξ )Pn (η)Pn (ξ0 ) (m)



cos mψ sin mψ

 (5)

(m)

outside it. In Eqs. (4) and (5) functions Pn (ξ ), Qn (ξ ) are the associated Legendre functions of the first and second kind, respectively, which are defined at [1, ∞). The eigenfunctions εnm corresponding to these solutions have the form: (m)

εnm =

(m)

Pn (ξ0 ) dξd Qn (ξ0 ) 0

εH , (m) (m) d P (ξ )Q (ξ ) n n 0 0 dξ0

(6)

where the nanospheroid’s surface is described by the coordinate ξ = ξ0 , which depends on the semiaxes lengths as 1 c =√ (7) ξ0 = √ 2 2 c −a 1 − α2 and their ratio α = a/c is called the aspect ratio. Note that εnm at the given εH depends only on ξ0 , i.e., only on the aspect ratio. This means that the optical properties of the nanospheroid are determined only by the aspect ratio α and the permittivity of the media the nanoparticle is located in under condition when we can neglect all terms in the field expansion by the parameter (ka), except the first one, i.e., when the linear size of the nanoparticle is much smaller than the wavelength of the incident field. This is exactly the case when the quasistatic approximation works well. For the nanoparticles whose size is comparable or larger than the wavelength of the incident field this approximation gives only qualitative description. From now on we will consider in our calculations a prolate nanospheroid of 30 nm in length, so that the quasistatic approximation is readily satisfied. It is also worth to note that a possibility of plasmon resonance excitation for the specific plasmon mode is defined by the value of εH . For the gold nanoparticle, for instance, the permittivity ε = −8.37 + 1.15i at the incident field wavelength λ = 600 nm [35] and the plasmon resonance is therefore excited at the aspect ratio αres = 0.32. The plasmon resonance width #214368 - $15.00 USD Received 18 Jun 2014; revised 29 Jul 2014; accepted 1 Aug 2014; published 15 Aug 2014 (C) 2014 OSA 25 August 2014 | Vol. 22, No. 17 | DOI:10.1364/OE.22.020432 | OPTICS EXPRESS 20436

is determined by the imaginary part of the nanoparticle’s permittivity and enhancement of the incident electromagnetic field would be expected for the nanospheroids with the aspect ratio in the range of 0.30 to 0.36. The maximum enhancement of the incident field is achieved at the very localized area around the nanospheroid’s tips, where it is enhanced by 102 times. However, even at the distance of about 10 nm from the tip the local field significantly weakens and the field enhancement drops by the order of magnitude. 2.1.

Polarization of the near-field

In this subsection, we will describe polarization of the near-field and will start with an assumption that the temporal dependence of the near-field in a point r in space around the nanoparticle can be written as Ex (r,t) = Ax (r) cos(ωt + φx (r)), Ey (r,t) = Ay (r) cos(ωt + φy (r)), Ez (r,t) = Az (r) cos(ωt + φz (r)),

(8)

where Ai (r) (i = x, y, z) is the actual amplitude of the i-th component of the near-field E, φx (r), φy (r) and φz (r) are the phases of the near-field projections onto the respective coordinate axes. All these values can be calculated from Eq. (3). Eliminating the time dependence, one can show that the curve, which vector E draws in space for the period of oscillation lies in plane orientation of which is determined by the amplitudes Ai and phases shifts φx , φy and φz . Keeping in mind that all the components of the near-field E vary with the same frequency, the vector E draws in the plane of oscillations an ellipse semiaxes of which are equal to the minimum Emin and maximum Emax values of the near-field amplitude in the given point in space. If the minimum value of the near-field amplitude is equal to zero, then the near-field has linear polarization in this point; in the case of Emin = Emax , it has a circular polarization. In order to determine the near-field polarization in a point of space around the nanospheroid let us calculate the polarization degree, which is defined as P=

Imax − Imin , Imax + Imin

(9)

where Imax and Imin are the maximum and minimum values of the module of the Umov–Pointing vector in a specific point of space at the oscillation period of the electromagnetic wave. This value shows how the polarization field is close to linear. For the linear polarization, P = 1, for the circular — P = 0, and for the elliptical one — takes values from 0 to 1 [39]. Let us now define Imax and Imin in the given point r in space via the field components (8). We will start with the intensity of the near-field in this given point and substituting Eq. (8) in this formula receive I(t) = (A2x + A2y + A2z ) + (A2x + A2y cos 2φy + A2z cos 2φz ) cos 2ωt −(A2y sin 2φy + A2z sin 2φz ) sin 2ωt. Calculating Imax and Imin and substituting them into Eq. (9), we receive for the polarization degree  −1/2 (|A˜ x |2 + |A˜ y |2 + |A˜ z |2 )2 − 4Im2 (A˜ ∗x A˜ y ) − 4Im2 (A˜ ∗x A˜ z ) − 4Im2 (A˜ ∗y A˜ z ) P= , (10) |A˜ x |2 + |A˜ y |2 + |A˜ z |2 where A˜ k = Ak exp(iφk ) are the complex amplitudes. #214368 - $15.00 USD Received 18 Jun 2014; revised 29 Jul 2014; accepted 1 Aug 2014; published 15 Aug 2014 (C) 2014 OSA 25 August 2014 | Vol. 22, No. 17 | DOI:10.1364/OE.22.020432 | OPTICS EXPRESS 20437

Note here that the polarization degree P does not provide full information on the polarization of the near-field, i.e., does not characterize the rotation direction of the vector E. To determine this rotation direction it is convenient to use the Stokes parameters and because the near-field in general case has three field components, it is convenient to describe its polarization with the so called 3D generalized Stokes parameters [34]. In 3D case, the number of Stokes parameters is equal to nine of which three play role similar to the parameter S3 for each of the three planes xy, yz and xz and describe the excess of the right-hand circularly polarized field component over the left-hand circularly polarized one. Expressions for these parameters have the form:



S3xy = i( Ex∗ (r, ω)Ey (r, ω) − Ex (r, ω)Ey∗ (r, ω) ),

S3xz = i(hEx∗ (r, ω)Ez (r, ω)i − Ex (r, ω)Ez∗ (r, ω) ), (11)



∗ S3yz = i( Ey (r, ω)Ez (r, ω) − Ey (r, ω)Ez (r, ω) ), where Ei = A˜ i exp(iωt) and the angle brackets denote the time averaging. This averaging can be omitted because we have a stationary electromagnetic field. The parameter S3 takes negative values when left circular polarization dominates, and takes positive values in the case when dominates the right circular polarization. Using Eq. (11) the polarization degree can be rewritten in the following form: s       S3xy 2 S3yz 2 S3xz 2 − − , (12) P = 1− I I I where the Stokes parameters are normalized to the intensity of the near-field at the observation point for the convenience. 3.

Near-field polarization distribution calculation results

In this section, we will analyze in detail the distribution of the near-field polarization of the plasmonic prolate nanospheroid interacting with an incident plane electromagnetic wave of linear, elliptical and circular polarization. We will assume for simplicity that εH = 1. The obtained results are also valid for a specific case of a nanosphere. 3.1.

The case of a plane incident electromagnetic wave linearly polarized along z-axis

In the case of interaction of a plasmonic nanoparticle with a plane electromagnetic wave, in a spheroidal nanoparticle only plasmon modes with n = 1 (dipole modes) are excited, since the corresponding potentials ϕ1,0 and ϕ1,1 show linear dependence versus the Cartesian coordinates of the system. For the linearly polarized along the z-axis incident field, the only one plasmon mode with n = 1, m = 0 is excited. Let us consider the case of a prolate plasmonic nanospheroid in the linearly-polarized along z-axis incident field E0 = {0, 0, E0z } with the wave vector k directed along x-axis. Substituting E0 in Eq. (3) gives us the distribution of the near-field E = E0,z ez +

ε(ω) − 1 P1 (ξ0 ) f E0,z ∇P1 (η)Q1 (ξ ). ε1,0 − ε(ω) Q1 (ξ0 )

(13)

and expressions for the amplitudes of the projections of the near-field have the form: ε(ω) − 1 P1 (ξ0 ) ∂ A˜ x (r) = f E0,z P1 (η)Q1 (ξ ), ε1,0 − ε(ω) Q1 (ξ0 ) ∂x

(14)

#214368 - $15.00 USD Received 18 Jun 2014; revised 29 Jul 2014; accepted 1 Aug 2014; published 15 Aug 2014 (C) 2014 OSA 25 August 2014 | Vol. 22, No. 17 | DOI:10.1364/OE.22.020432 | OPTICS EXPRESS 20438

ε(ω) − 1 P1 (ξ0 ) ∂ A˜ y (r) = f E0,z P1 (η)Q1 (ξ ), ε1,0 − ε(ω) Q1 (ξ0 ) ∂y

(15)

ε(ω) − 1 Q1 (ξ0 ) ∂ A˜ z (r) = E0,z + P1 (ξ0 ) E0,z P1 (η)Q1 (ξ ), ε1,0 − ε(ω) f ∂z

(16)

where f is the distance between the foci of the prolate spheroid. As seen from Eq. (16), at the plasmon resonance the phase shift of the near-field with respect to the incident field occurs and, therefore, if the local field and the incident fields have the amplitudes of the same order in magnitude, then the projection of the total near-field at the z-axis has a phase shift relative to the projections on the other axes. As a result, the total field becomes elliptically polarized. Substituting Eqs. (14)–(16) in Eqs. (10) and (11), we obtain expressions describing the distribution of the degree of polarization of the near-field P and the Stokes parameter S3xz . Figure 2 (right plot) shows the distribution of the degree of polarization of the near-field under the linearly polarized incident field in the case of the plasmon resonance. As can be seen from the figure, this distribution is symmetrical about the z-axis and has the form of two toroidal areas with the radius of about 20 nm arranged symmetrically relative to the plane xy. The torus generating line has a shape close to a circle with a radius of about 20 nm. In the center of the torus the polarization degree P is close to zero (purple color). As the distance from the center increases, the polarization degree P → 1 (red color), which corresponds to the linear polarization of the external field. Also, Fig. 2 shows the distribution of the degree of polarization P in three planes: xz, yz and z = 15 nm (horizontal section of the toroidal area). 1,0

1.0 40

40

0.8

0.5

20

20

0

z, nm 0 -20

-0.5

S3xz I

0.6

z, nm 0

P

-20 0.4

-40 -40 40

-20

-40

-1.0 -40

-20

0

x, nm

20

40

x, nm

0

0

20

20

0.2

y, nm

-20 40

-40

0

Fig. 2. Polarization of the near-field of the plasmonic prolate nanospheroid under linearly polarized along the z-axis incident plane electromagnetic wave in the case of the plasmon resonance at α = αres ; E0,y = 0 V/m and E0,z = 2000 V/m (about 7 mW/mm2 ). Left figure shows the distribution of the normalized Stokes parameter S3xz /I. In areas where S3xz takes positive values, the field vector rotates clockwise, whereas in areas with negative values of the Stokes parameter—it rotates anti-clockwise. The right figure shows the distribution of the degree of polarization of the near-field. 3D surface corresponds to the surface at the constant degree of polarization P = 0.8. 2D distributions show respective sections at three planes: xz, yz, and z = 15 nm.

The Stokes parameter S3xz /I characterizes the rotation direction of the vector E, which is indicated by color in the left plot of Fig. 2 and by arrows in the right plot. From Fig. 2 (left plot) one can see that the field vector curled around the torus and directed toward the center of the torus. As we tune the system away from the plasmon resonance, the areas of polarization changes are significantly reduced in size and shifted towards the tips of the nanospheroid at α < αres (Fig. 3(a)) or towards the equator at α > αres (Fig. 3(b)). The electric field vectors, as in the case of a plasmon resonance, rotate in the direction towards the center of the torus. #214368 - $15.00 USD Received 18 Jun 2014; revised 29 Jul 2014; accepted 1 Aug 2014; published 15 Aug 2014 (C) 2014 OSA 25 August 2014 | Vol. 22, No. 17 | DOI:10.1364/OE.22.020432 | OPTICS EXPRESS 20439

1,0

b.

a. 20

20

10

0.8

10

z, nm 0

z, nm 0

-10

-10

-20

0.6

P 0.4

-20

-20

20

-10 x, nm

10

0

0 10

-10

-20

x, nm

y, nm

20

-10

10

0

0 10

20 -20

-10

0.2

y, nm

20 -20

0

Fig. 3. The distribution of the degree of polarization of the near-field under the linearly polarized along the z-axis incident field in the case of far away from the plasmon resonance at α = 0.25 < αres (a) and α = 0.41 > αres (b).

3.2.

The case of plane, linearly polarized incident electromagnetic wave directed at an angle to the axis y

In this subsection, we will consider distribution of the degree of polarization of the near-field when the vector k of the linearly polarized incident electromagnetic wave directed at an angle to the axis y (Fig. 1(b)). In this case, the external field E0 = {E0x , 0, E0z } excites in the plasmonic nanospheroid a mode with n = 1, m = 1 due to the presence of nonzero component E0,x , which leads to the presence a nanoparticle’s dipole moment along the x-axis. In view of Eq. (3) the expression for the near-field can be written as E = E0,x ex + E0,z ez +

ε(ω) − 1 P1 (ξ0 ) f E0,z ∇P1 (η)Q1 (ξ ) ε1,0 − ε(ω) Q1 (ξ0 ) (1)

+

ε(ω) − 1 P1 (ξ0 ) (1) (1) f E0,x ∇P1 (η)Q1 (ξ ) cos ψ ε1,1 − ε(ω) Q(1) (ξ0 )

(17)

1

and the projections for the field amplitudes are given by ∂ ε(ω) − 1 P1 (ξ0 ) f E0,z [P1 (η)Q1 (ξ )] + A˜ x (r) = E0,x + ε1,0 − ε(ω) Q1 (ξ0 ) ∂x (1) h i ε(ω) − 1 P1 (ξ0 ) ∂ (1) (1) f E P (η)Q (ξ ) cos ψ , 0,x 1 ε1,1 − ε(ω) Q(1) (ξ0 ) ∂x 1 1

ε(ω) − 1 P1 (ξ0 ) ∂ A˜ y (r) = f E0,z [P1 (η)Q1 (ξ )] + ε1,0 − ε(ω) Q1 (ξ0 ) ∂y (1) i ε(ω) − 1 P1 (ξ0 ) ∂ h (1) (1) f E P (η)Q (ξ ) cos ψ , 0,x 1 1 ε1,1 − ε(ω) Q(1) (ξ0 ) ∂y 1

(18)

∂ ε(ω) − 1 P1 (ξ0 ) A˜ z (r) = E0,z + f E0,z [P1 (η)Q1 (ξ )] + ε1,0 − ε(ω) Q1 (ξ0 ) ∂z #214368 - $15.00 USD Received 18 Jun 2014; revised 29 Jul 2014; accepted 1 Aug 2014; published 15 Aug 2014 (C) 2014 OSA 25 August 2014 | Vol. 22, No. 17 | DOI:10.1364/OE.22.020432 | OPTICS EXPRESS 20440

(1)

i ε(ω) − 1 P1 (ξ0 ) ∂ h (1) (1) P1 (η)Q1 (ξ ) cos ψ . f E0,x (1) ε1,1 − ε(ω) Q (ξ0 ) ∂z 1

Case of θ = 0 corresponds to the example considered in the previous subsection. The distribution of the degree of polarization of the near-field changes as changing the direction of propagation of the field. Figure 4 shows the distribution of the near-field polarization for two different incident field propagation directions. Plane of the toroidal area is always parallel to the vector k, so when you turn the vector k at an angle θ the plane of the toroidal area is also rotated by this angle. When θ & π/6, the toroidal areas corresponding to the same values of P begin to overlap (Fig. 4). Since the axial symmetry of the system in this case is absent, to characterize the direction of rotation of the vector E the distributions of the Stokes parameters S3xz /I and S3yz /I are used. Vectors of the field turn around the areas, where polarization changes, and directed to the centers of these areas. The direction of rotation is indicated by arrows. a.

1.0 1,0

b. b.

a.

40 40

40 40 20 20

20 20

z, z, nm nm 00

nm 00 z,z,nm

0.8 0.8

0.6 0.6

-40 -40 -20 -20 nm x,x, nm

0.4 0.4

-40 -40

-40

40 40

00

00

20 20 40 40

-40 -40

-20 -20

20 20

-40 -40 -20 -20 0 x, nm 0

x, nm

y,y,nm nm

P

P

-20 -20

-20 -20

00

20 20

-20 -20 40 40

20 20

40 40

0.2 0.2

y, y, nm nm

-40 -40

0 0

Fig. 4. The distribution of the degree of polarization of the near-field under the linearly polarized along the angle θ = 40◦ (a) and θ = 70◦ (b) towards z-axis in the case of the plasmon resonance at α = αres , E0,y = 0 V/m and E0,z = 2000 V/m (about 7 mW/mm2 ). 3D surface corresponds to the surface at the constant degree of polarization P = 0.8. 2D distributions show respective sections at three planes: xz, yz, and z = 0.

3.3.

The case of plane, elliptically polarized incident electromagnetic wave

Here we will consider a plasmonic nanospheroid in the field of elliptically polarized plane wave, when k||x, E0 = {0, iE0,y , E0,z } and the phase shift between the components E0,y and E0,z is equal to π/2. Then, in accordance with Eq. (3) the near-field takes the form E = iE0,y ey + E0,z ez +

ε(ω) − 1 P1 (ξ0 ) f E0,z ∇P1 (η)Q1 (ξ ) ε1,0 − ε(ω) Q1 (ξ0 ) (1)

+

ε(ω) − 1 P1 (ξ0 ) (1) (1) f iE0,y ∇P1 (η)Q1 (ξ ) sin ψ, ε1,1 − ε(ω) Q(1) (ξ0 )

(19)

1

and the respective expressions for the projections of the field amplitudes ε(ω) − 1 P1 (ξ0 ) ∂ A˜ x (r) = f E0,z [P1 (η)Q1 (ξ )] + ε(ω) − ε1,0 Q1 (ξ0 ) ∂x (1) i ε(ω) − 1 P1 (ξ0 ) ∂ h (1) (1) P (η)Q (ξ ) sin ψ , f iE 0,y 1 1 ε1,1 − ε(ω) Q(1) (ξ0 ) ∂x 1

(20)

#214368 - $15.00 USD Received 18 Jun 2014; revised 29 Jul 2014; accepted 1 Aug 2014; published 15 Aug 2014 (C) 2014 OSA 25 August 2014 | Vol. 22, No. 17 | DOI:10.1364/OE.22.020432 | OPTICS EXPRESS 20441

ε(ω) − 1 P1 (ξ0 ) ∂ A˜ y (r) = iEy,0 + f E0,z [P1 (η)Q1 (ξ )] + ε(ω) − ε1,0 Q1 (ξ0 ) ∂y (1) i ε(ω) − 1 P1 (ξ0 ) ∂ h (1) (1) P (η)Q (ξ ) sin ψ , f iE 0,y 1 1 ε1,1 − ε(ω) Q(1) (ξ0 ) ∂y 1 ε(ω) − 1 P1 (ξ0 ) ∂ A˜ z (r) = E0,z + f E0,z [P1 (η)Q1 (ξ )] + ε(ω) − ε1,0 Q1 (ξ0 ) ∂z (1) i ε(ω) − 1 P1 (ξ0 ) ∂ h (1) (1) P f iE (η)Q (ξ ) sin ψ . 0,y 1 ε1,1 − ε(ω) Q(1) (ξ0 ) ∂z 1 1

(21)

(22)

Fig. 5. The distribution of the degree of polarization of the near-field under the incident field with the left elliptically polarization in the case of the plasmon resonance at α = αres , E0,x = 0 V/m, E0,y = 400i V/m and E0,z = 2000 V/m (about 7 mW/mm2 ). 3D surface corresponds to the surface at the constant degree of polarization P = 0.8. 2D distributions show respective sections at three planes: xz, yz, and z = 15 nm.

The near-field polarization distribution calculated with the help of the above expressions for the case of an elliptically polarized incident plane wave interacting with the plasmonic prolate nanospheroid is shown in Fig. 5 for the resonant case at α = αres . From this figure one can readily see that the toroidal areas change their shape: one part of the ring thickens, and the other, on the contrary, narrows, whereas the picture remains symmetric with respect to the origin of the frame. In the center of the torus the polarization remains close to linear, however, the transition from linear to elliptical polarization, as the distance from the nanoparticle surface increases, is no longer centrosymmetrical (see the cross-sections in Fig. 5). This means that transition from the linear polarization (in the center of the torus) to the elliptic one (on the surface P = const) is not uniform in all directions. 3.4.

The case of plane, circularly polarized incident electromagnetic wave

Qualitatively different near-field polarization distribution occurs when the plasmonic nanospheroid interacts with the circularly polarized external field. Projections field amplitudes in the case of an incident field with left circular polarization are described by Eqs. (20)–(22) with E0,y = iE0,z = iE0 . Dependence of the degree of polarization on the coordinates in the case of the plasmon resonance in an external field with the left circular polarization is shown in Fig. 6. The figure shows that in this case, a region of complex shape around the nanoparticle is formed, inside #214368 - $15.00 USD Received 18 Jun 2014; revised 29 Jul 2014; accepted 1 Aug 2014; published 15 Aug 2014 (C) 2014 OSA 25 August 2014 | Vol. 22, No. 17 | DOI:10.1364/OE.22.020432 | OPTICS EXPRESS 20442

1,0

1.0 40 40

0.8

0.5

20

20

0

z, nm 0 -20

S3yz I

0.6

z, nm 0

P

-20 0.4

-40

-0.5

-40 40

-20

-40

-1.0 -40

-20

0

20

x, nm

40

0 20

y, nm

20

0

0.2

y, nm

-20 40

0

-40

Fig. 6. Polarization of the near-field of the plasmonic prolate nanospheroid under the incident plane electromagnetic wave of left-handed circular polarization (E0,x = 0 V/m, E0,y = 2000i V/m, E0,z = 2000 V/m) (2000 V/m ≈ 7 mW/mm2 ) in the case of the plasmon resonance at α = 0.32. Left figure show the distribution of the normalized Stokes parameter S3yz /I versus y, z coordinates. In the light lilac color areas the electric field has left-handed circular polarization, whereas in the red areas—right-handed circular polarization, and in the green areas—the field projections on the axes y and z oscillate with the same phase. Right figure shows the distribution of the degree of polarization of the nearfield. 3D surface corresponds to the surface at the constant degree of polarization P = 0.8. 2D distributions show respective sections at three planes: xz, yz, and xy.

which the near-field polarization is close to the linear one. Within this region in the plane yz, occur two small field areas with the size of about 10 nm, in which the polarization remains circular, but the phase between the projections of the total field Ey , Ez changes the sign. It should be noted that from Eq. (12) at P = 0 and S3yz /I = ±1 it follows that S3xz = S3xy = 0, i.e., Ax = 0 and Ay = ±Az . Thus, in these regions the direction of rotation of the polarization plane of the near-field is opposite the direction of rotation of the plane of polarization of the incident field. For the incident field with right-handed polarization, the polarization distribution is inverted relative to the z-axis. In the nonresonant case, when α < αres , the polarization distribution changes only at a distance of 5 nm and further from the nanoparticle’s surface (Fig. 7). 1.0 1,0

b.

a. 40 40

40 40

20 20

20 20

z,nm nm 00 z,

z, nm nm 0 0 z,

-20 -20

0.8 0.8

0.6 0.6

-40 -40

0.4 0.4

-40 -40

-40 -40 -20 -20

x,x,nm nm

00 20 20

40 40

-40 -40

-20 -20

0 0

20 20

40 40

y, nm

y, nm

P

P

-20 -20

-40 -40 -20 -20 0 x, nm

x, nm

00

20 20 40 40

-20 -20 -40

20 20

40 40

0.2 0.2

y, nm

y, nm

0 0

Fig. 7. The distribution of the degree of polarization of the near-field under the incident plane electromagnetic wave of left circular polarization (E0,x = 0 V/m, E0,y = 2000i V/m, E0,z = 2000 V/m) (2000 V/m ≈ 7 mW/mm2 ) at α = 0.25 < αres (a) and α = 0.42 > αres (b). 3D surface corresponds to the surface at the constant degree of polarization P = 0.8. 2D distributions show respective sections at three planes: xz, yz, and xy.

#214368 - $15.00 USD Received 18 Jun 2014; revised 29 Jul 2014; accepted 1 Aug 2014; published 15 Aug 2014 (C) 2014 OSA 25 August 2014 | Vol. 22, No. 17 | DOI:10.1364/OE.22.020432 | OPTICS EXPRESS 20443

4.

Conclusion

In conclusion, we have studied in detail how the near-field polarization distribution of a plasmonic prolate nanospheroid in an incident plane electromagnetic wave depends on its polarization (linear, elliptical or circular) and aspect ration of the nanospheroid (at the plasmon resonance in the nanoparticle and away from it). Polarization properties of the near-field are described with the help of the 3D generalized Stokes parameters, which allow simple visualization. Our results show that the near-field polarization changes significantly from linear to the circular one in certain areas in the vicinity of the nanoparticle’s surface and that the shape of these areas drastically depends on the polarization of the incident field. It is shown that in all considered cases the near-field of the prolate nanospheroid has the areas with the polarization opposite to the incident field polarization. For instance, in the case of linearly polarized incident field the near-field polarization distribution has the form of two symmetric tori in the center of which the polarization of the near-field is circular and vice versa: when the polarization of the incident field is circular, the near-field polarization distribution has a complex structure in the center of which the polarization of the near-field is linear. It was also shown that for a linearly polarized incident field, the areas in which the near-field polarization changes decrease and shift to the tips of the prolate nanospheroid at α < αres and to the equator of the nanospheroid at α > αres . The location of these areas can be controlled by changing the direction of the incident field towards the nanospheroid. In the case of elliptically polarized incident field, the size of the near-field area in which its polarization is close to the circular one increases and in the resting areas around the nanospheroid the near-field polarization becomes close to linear. At the plasmon resonance, a circularly polarized incident field forms around the nanospheroid a region of a complex shape in which the near-field polarization changes in respect to the polarization of the incident field. The shape of this area is inverted when the polarization of the incident field changes from left to right helicity. Within this area there are two small subareas in which the near-field polarization is circular and its helicity is opposite to the helicity of the incident field. Away from the plasmon resonance, the near-field polarization is different from the circular one only near the particle’s surface. Our analysis is also valid for an oblate nanospheroid simply by replacing the coordinate ξ (ξ > 1) → iξ (ξ > 0). Moreover, it covers the whole set of particles whose shape varies from the spherical one to the nanoneedles and nanorods by changing the aspect ratio of the spheroid. For simplicity, we considered here only the nanoparticles, which are made of noble metals (like gold or silver) imaginary part of the permittivity of which at the frequency of the quadrupole plasmon resonance in the nanoparticle is rather high, so that we can limit our treatment only by the dipole plasmon resonance in such particles. However, this analysis can also be extended to the case of low-loss metals (like Na, Al, K, etc.) in which both the dipole and quadrupole plasmon resonances can be observed simultaneously in some region of parameters, so that both these plasmon resonances are to be taken into account in our calculations for such particles. Interference of the dipole and quadrupole resonances and with the incoming electromagnetic field could lead to an extremely interesting and complex behavior of the vectorial near-field around the nanoparticle featuring Fano resonances and vortexes [25] and studying the behavior of the polarization distribution in such systems in terms of the 3D generalized Stokes parameters deserves a separate study. Also, we considered the nanoparticles in the range of 5 to 50 nm in size that cover lots of important practical applications, but still are much smaller than the wavelength of the incoming electromagnetic field and make valid both the used quasistatic approximation and neglecting the retardation effects (for details, see [5], p. 185). It is worth also to note here that for the case of a spherical nanoparticle Mie scattering approach gives similar results and allows to take into

#214368 - $15.00 USD Received 18 Jun 2014; revised 29 Jul 2014; accepted 1 Aug 2014; published 15 Aug 2014 (C) 2014 OSA 25 August 2014 | Vol. 22, No. 17 | DOI:10.1364/OE.22.020432 | OPTICS EXPRESS 20444

account retardation effects and consider particles of bigger sizes. We however intentionally did not use it for our analysis as it offers nothing new by contrast with our analytical analysis in the limits we consider and if one apply Mie approach for the larger plasmonic nanoparticles, this will force us not only to take into account the retardation effects, but also consider possible whispering gallery modes in such nanoparticles, etc. [5]. At the same time, the ε-method we used in our calculations also allow to treat particles of any size (if we do not use quasistatic approximation), shape and material the nanoparticle is made of. In this approach, major role is played by the permittivities of the nanoparticle that correspond to the respective plasmon resonances frequencies, which makes it ideal for describing plasmonic properties of the nanoparticle and allows simultaneous description of the nanoparticles of the same shape, but made of different materials [5, 36]. In many experimental cases, the external electromagnetic field cannot be approximated by a plane wave and using a beam instead of the plane wave can drastically affect the results of the measurements. In the format of this short publication we omitted our results obtained for the case of a Gaussian laser beam with high-order transverse modes interacting with the metal prolate nanospheroid. These results will be published elsewhere, we will only note here that using the beam can lead to the excitation of plasmon oscillations of higher order if the condition of the plasmon resonance is fulfilled. This will lead to the very complicated distribution of the polarization degree of the near-field. Finally, let us discuss how the spatial near-field polarization distribution of a plasmonic nanoparticle interacting with the incoming electromagnetic field can potentially be measured in an experiment in the vein of visualizing the magnetic field with the help of sprinkled on the top of the paper iron fillings that line up along the field lines. Consider, for example, a plasmonic nanoparticle, which is embedded into the transparent to the incident light polymer layer with scattered in the layer fluorescent molecules. This system is then illuminated by the optical field polarization, wavelength and incident direction of which can be varied in the experiment. The fluorescent molecules randomly scattered around the nanoparticle will act as single quantum emitters emitting fluorescence, which can be measured with single-molecule resolution, especially at the low temperatures, with the help of modern spectroscopical techniques such as single-molecule spectromicroscopy (SMSM) that allow precise spatial detection (with shingle molecule spatial resolution) of the whole myriad of molecules located in the vicinity of the nanoparticle by detecting their fluorescence, providing spatial 3D picture of the fluorescence signals in the sample [40]. Fluorescence intensity of each such quantum emitter depends on the spatial location of the molecule around the nanoparticle, polarization of the incident field, its wavelength, geometrical parameters of the nanoparticle and the materials it is made of [5, 16]. If we know all these parameters, but the polarization of the near-field in the location of the molecule, we can readily reconstruct the generalized 3D Stokes parameters in this point or any other characteristic, including Poynting vector, etc. As a result, a spatial near-field polarization distribution can be reconstructed. In addition, the initial measured spatial distribution of the fluorescence intensities of the molecules allows simple reconstruction of the spatial near-field intensity distribution, which compliments the reconstructed generalized 3D Stokes parameters. Acknowledgments The authors would like to thank Victor Balykin, Vasily Klimov and Andrey Naumov for fruitful discussions and acknowledge funding from the Russian Foundation for Basic Research under the grant No. 13-02-00446. E.D.C. also acknowledges support from the Dynasty foundation.

#214368 - $15.00 USD Received 18 Jun 2014; revised 29 Jul 2014; accepted 1 Aug 2014; published 15 Aug 2014 (C) 2014 OSA 25 August 2014 | Vol. 22, No. 17 | DOI:10.1364/OE.22.020432 | OPTICS EXPRESS 20445

Controlling near-field polarization distribution of a plasmonic prolate nanospheroid by its aspect ratio and polarization of the incident electromagnetic field.

Near-field polarization distribution of a plasmonic prolate nanospheroid in an incident electromagnetic field versus its polarization and the spheroid...
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