Abstract

We study the interaction of focused radially-polarized light with metal nanospheres. By expanding the electromagnetic field in terms of multipoles, we gain insight on the excitation of localized surface plasmon-polariton resonances in the nanoparticle. We show that focused radially-polarized beams offer more opportunities than a focused plane wave or a Gaussian beam for tuning the near- and far-field system response. These results find applications in nano-optics, optical tweezers, and optical data storage.

© 2009 Optical Society of America

1. Introduction

Focused radially-polarized beams (FRBs) find application in many fields of optics. For instance, it has been demonstrated that they achieve a tighter focal spot compared to focused plane-wave (FPW) illumination [1, 2]. Moreover, Novotny et al. have used them to map the orientation of single molecules [3]. Van Enk has discussed FRBs in the context of quantum optics for efficient coupling of light to a single emitter [4]. Also in the area of optical tweezers Zhan has shown that metal nanoparticles (NPs) can be trapped in three dimensions under FRBs [5]. Meixner’s group has performed a set of experiments to explore the interaction of radially-, azimuthal- and linearly-polarized focused beams with metal NPs [6, 7, 8].

Recently, a number of theoretical works have studied the interaction of tightly-focused beams with spherical NPs. Şendur et al. have used diffraction integrals to analyze the near field produced by a silver NP under FPW and FRB illumination [9]. Likewise, van de Nes and Török have developed a generalized Mie theory for Gauss-Laguerre beams, including the case when the NP is displaced from the focus [10]. More recently, Moore and Alonso have chosen the multipole expansion approach to develop a generalized Mie theory where the incident beam is a complex focused field. In particular, they study the response of a dielectric NP under FRBs at a fixed wavelength [11]. Lastly, Lermé et al. have studied the extinction of light by silver NPs in the context of the spatial modulation spectroscopy technique [12].

In this paper we extend our earlier study on the localized surface plasmon-polariton (LSP) spectra of metal NPs [13] to the case of FRBs. Using the multipole expansion approach, we show that, contrary to a FPW [14] and a Gaussian beam [15], FRBs offer more opportunities for manipulating the excitation of higher-order LSP resonances in metal NPs. We discuss these features both in the far- and near-field regimes.

2. Generalized Mie theory

Radially-polarized doughnut beams are constructed by superposing the Gauss-Hermite modes with normal polarizations [16]. The effect of the lens is identified by two parameters, the focusing semi-angle α and the a factor defined as f/wo [4], where f is the focal length of the lens and wo is the beam waist (see inset in Fig.1). Using the Richards and Wolf formalism for an aplanatic system [17] and adapting it to radially-polarized illumination, the field right after passing the lens is given by E(a,θ)=Aexp(a2sin2θ)fsinθcosθθ̂, where cosθ represents the apodization function and θ̂ gives the orientation of the electric field in spherical coordinates (r,θ,ϕ) [1, 18]. Note that this field depends only on a and θ because f can be merged into the amplitude A, which only rescales the intensity of the beam.

The electric field in the image space can now be found by the multipole expansion, where the weight coefficient of each multipole is determined by matching the field at the lens boundary [13, 14, 19]. Since the electric field has only a θ component in the far field, the azimuthal number is m = 0, and the symmetry around the z axis implies that transverse electric (TE) multipoles have no contribution. The incident field and the coefficients are hence given by

Einc=l=1BlNe0l(1),
Bl=2l+12l(l+1)2kfilexp(ikf)0αE(a,θ)dPl(cosθ)dθsinθdθ,

where k is the wavevector and Pl(cosθ) are Legendre polynomials. Here the multipoles follow the notation of [20]. The expansion of Eq. (1) is a special case of tightly-focused spirally-polarized beams [19].

Now that the incident field is determined in terms of multipoles, the effect of placing a spherical NP at the origin can be solved by the generalized Mie theory [13]. The scattered and internal fields are respectively found to be

Es=l=1alBlNe0l(3),Ei=l=1dlBlNe0l(1),

where al and dl are the Mie coefficients as defined in [20]. Note that because the incident field does not contain TE multipoles, these are also absent in the scattered and internal fields. Having the fields expanded in all space, the total scattered (W s) and extinguished (W e) powers can also be calculated by integrating the corresponding Poynting vectors in the far field, giving

Ws=πZk2l=1Bl2al22l(l+1)2l+1,We=πZk2l=1Bl2Re{al}2l(l+1)2l+1,

where Z is the impedance of the background medium. According to Eq. (4), W s and W e not only depend on the NP size and material, but also vary with the multipole strength (the Bl term) [13]. This clearly shows that for a specific LSP resonance, one can in principle control the scattered and extinguished powers of the corresponding multipole by adjusting the illumination beam and the focusing system.

The cross sections associated with W s and W e are found by dividing these quantities by the average intensity incident on the xy section of the NP [13]. This averaging process is necessary because the field in the focal plane is strongly inhomogeneous in contrast to plane-wave illumination. However, the cross sections calculated by this approach lead to unrealistic values that are one or two orders of magnitude larger than those obtained by FPW illumination. The reason behind this delusive behavior is that although the electric field is strong near the focus, the Poynting vector is small and vanishes on axis (see inset in Fig. 2) [3]. We therefore use another quantity to study the LSP spectra, namely the scattering (𝓚s) and the extinction (𝓚e) efficiencies, defined by dividing W s and W e by the total incident power P inc [21]: 𝓚s = W s/P inc and 𝓚e = W e/P inc. The profiles obtained by this method slightly differ from the cross sections, but still exhibit the resonances of the system.

 

Fig. 1. Relative strength |Bl/B 1| of the multipoles in the incident field. The result for a FPW is also shown for comparison [13]. The inset sketches a radially-polarized beam focused onto a NP. The red curve is the beam intensity profile before the lens, f is the focal length, α the angular semi-aperture, wo the beam waist and a = f/wo.

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Many applications of plasmonic materials are based on their near-field properties [22], whereas cross sections or efficiencies are far-field quantities. For this reason, we use an alternative definition called the near-field average intensity enhancement K for characterizing the system response. This quantity is simply the total field intensity averaged over the NP surface, divided by the intensity at the origin in the absence of the NP [13]. By taking advantage of the orthogonality of the multipoles, K is separated into radial K r and tangential K t components,

Kr=916(kR)4lBlB12l2(l+1)22l+1[ψl2+al2χl2+2Re{al*χl*}ψl],
Kt=916(kR)2lBlB12l(l+1)22l+1[ψl2+al2χl2+2Re{al*χl*}ψl],

where R is the NPs radius. In these equations ψl and χl are respectively the Riccati-Bessel functions of the first and third kinds calculated at the NP surface, and the prime indicates the derivative with respect to kr [13, 20].

3. Results and discussion

In this section we discuss how FRBs can change the excitation of LSP resonances in metal NPs and its consequences on the far- and near-field optical response. As a case study we focus on the interaction of different types of FRBs with a 100 nm silver spherical NP [23] embedded in glass (background index n b = 1.5). This is a suitable NP size since it is neither too small to eliminate the excitation of higher-order modes, nor too large to suppress the near-field enhancement [24]. There is a freedom of changing f or wo to obtain different values of a = f/wo. We set the beam waist to wo = 5 mm and adjust f to select a. We use Mathematica for the calculations [25], taking into account up to 15 multipoles for the field expansions.

The relative strength |Bl/B 1| in Fig. 1 shows how much the l th multipole contributes to the fields in comparison to the dipole (l = 1). A tightly-focused beam (α = 90° and a = 0.5) mainly consists of a dipole and a quadrupole, the latter being 80% of the dipole strength. The next case has again the same focusing angle, but the beam waist is smaller. This is a special illumination because the quadrupole and the dipole coefficients exhibit nearly the same weight. Keeping the same value for a but focusing less tightly (α = 45°), the dipole strength becomes even less than the next three multipoles. In summary, the general trend in the incident-wave multipole content is as follows: by increasing the value of a, i.e. focusing more tightly, higher-order modes are suppressed and by increasing the value of a, i.e. longer focal length or smaller beam waist, higher-order modes get stronger. For the sake of comparison we also plot in Fig. 1 the multipole strength for a FPW. Note that for a FPW and a Gaussian beam the higher-order modes can only be suppressed with respect to an incident plane wave [13, 15].

 

Fig. 2. Extinction efficiency 𝓚e of a 100 nm silver NP in glass illuminated by three different FRBs and a FPW. D, Q and O respectively label the dipole, quadrupole and octupole resonances. Inset: electric field intensity (contours) and Poynting vector (arrows) in the focal region for α= 90°, a = 1.5, and λ = 520 nm.

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We next study the far-field LSP spectrum by looking at 𝓚e (see Fig. 2). The first property that catches one’s attention is that 𝓚e can exceed 2. This does not violate the conservation of energy, because one has to consider the total Poynting vector instead of only the scattered or extinguished components [20, 21]. Indeed the transmitted, reflected, and absorbed powers together are equal to the total input power. The effect of changing the multipoles strength can be observed here by studying the relative heights of the corresponding resonances. For example, the ratios of the quadrupole to dipole peaks are 1.63, 2.04, and 3.22 for the three cases presented in Fig. 1. As expected, the multipole strength determines the relative height of the LSP modes accordingly. Indeed for the FRB with a large content of higher-order multipoles, even the third resonance is clearly excited. To stress the difference with respect to conventional focused light, Fig. 2 shows that almost only the dipole resonance contributes to 𝓚e for a FPW when α= 90°.

Comparing these three cases reveals that the NP extinguishes more power when the beam is focused more tightly. This is simply because more light is concentrated onto the NP. Furthermore, the reason why 𝓚e is larger for a shorter focal length (smaller a) is that the electric field of the rays farther away from the z axis produce a stronger longitudinal field at the focus [1]. The interesting consequence of the multipole expansion is the correlation between the excited LSP resonances and the strength of the corresponding multipoles. This information can be used to control the excitation of each mode in the NP.

We next move our attention to the average near-field intensity enhancement and especially on the radial one (see Fig. 3). For a 100 nm silver NP, the higher-order excited modes yield better K r and K t with respect to the dipole one. Moreover, by focusing less tightly or making a larger, they exhibit a stronger enhancement. In comparison to 𝓚e of Fig. 2, the relative strength of the quadrupole peak with respect to the dipole is larger. This clearly shows the difference between the near- and the far-field regimes [13, 24]. As in the case of FPW illumination, which is shown in Fig. 3 for comparison, the enhancement near the dipole peak remains almost the same irrespective of the focusing parameters. For a FPW K r and K t are always less than for plane-wave illumination [13]. On the other hand, with FRBs one can control the enhancement at the quadrupole and higher-order resonances by changing the multipole content of the beam. More precisely, the ratio of quadrupole to dipole peak enhancements is 2.08, 3.13, and 5.71 for the three cases in Fig. 3. For instance, when α= 45° and a = 1.5 K r reaches a factor of 80, which is about 2 and 6 times the value for a plane wave and a FPW for α= 60°, respectively [13].

 

Fig. 3. Radial and tangential (inset) average intensity enhancement for a 100 nm silver NP in glass illuminated by three different FRBs and a FPW. D, Q and O respectively label the dipole, quadrupole and octupole resonances.

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4. Conclusion

We have studied the near- and far-field response of metal NPs illuminated by FRBs using a 100 nm silver spherical NP in glass as a representative system. The multipole expansion approach allows us to demonstrate the potential of FRBs for tailoring the excitation of LSP resonances and hence controlling the scattering and extinction efficiencies or the near-field enhancement. These findings show that FRBs offer more opportunities than conventional focused light. Our results might find application in the spectroscopy of metal NPs [8], optical tweezers [5] and optical data storage [26]. Furthermore, molecular fluorescence can also be improved by FRBs. Since the quadrupole mode exhibits a stronger electric field under appropriate illumination, one could use it to increase the excitation rate and the dipole resonance to amplify the emission rate without the need of designing complicated nanostructures [27].

Acknowledgments

We thank V. Sandoghdar for continuous support and encouragement. We are also indebted to G. Zumofen and M. H. Eghlidi for fruitful discussions. This work was supported by ETH Zurich.

References and links

1. S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “The focus of light - theoretical calculation and experimental tomographic reconstruction,” Appl. Phys. B 72, 109–113 (2001).

2. R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91, 233901(4) (2003). [CrossRef]   [PubMed]  

3. L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, “Longitudinal field modes probed by single molecules,” Phys. Rev. Lett. 86, 5251–5254 (2001). [CrossRef]   [PubMed]  

4. S. J. van Enk, “Atoms, dipole waves, and strongly focused light beams,” Phys. Rev. A 69, 043813(8) (2004). [CrossRef]  

5. Q. Zhan, “Trapping metallic Rayleigh particles with radial polarization,” Opt. Express 12, 3377–3382 (2004). [CrossRef]   [PubMed]  

6. A. V. Failla, H. Qian, H. Qian, A. Hartschuh, and A. J. Meixner, “Orientational imaging of subwavelength Au particles with higher order laser modes,” Nano Lett. 61374–1378 (2006). [CrossRef]   [PubMed]  

7. A. V. Failla, S. Jäger, T. Züchner, M. Steiner, and A. J. Meixner, “Topology measurements of metal nanoparticles with 1 nm accuracy by Confocal Interference Scattering Microscopy,” Opt. Express 15, 8532–8542 (2007). [CrossRef]   [PubMed]  

8. T. Züchner, A. V. Failla, A. Hartschuh, and A. J. Meixner, “A novel approach to detect and characterize the scattering patterns of single Au nanoparticles using confocal microscopy,” J. Microsc. 229, 337–343 (2007). [CrossRef]  

9. K. Şendur, W. Challener, and O. Mryasov, “Interaction of spherical nanoparticles with highly focused beam of light,” Opt. Express 16, 2874–2886 (2008). [CrossRef]   [PubMed]  

10. A. S. van de Nes and P. Török, “Rigorous analysis of spheres in Gauss-Laguerre beams,” Opt. Express 15, 13360–13374 (2007). [CrossRef]   [PubMed]  

11. N. J. Moore and M. A. Alonso, “Closed form formula for Mie scattering of nonparaxial analogues of Gaussian beams,” Opt. Express 16, 5926–5933 (2008). [CrossRef]   [PubMed]  

12. J. Lermé, C. Bonnet, M. Broyer, E. Cottancin, S. Marhaba, and M. Pellarin, “Optical response of metal or dielectric nano-objects in strongly convergent light beams,” Phys. Rev. B 77, 245406-13 (2008). [CrossRef]  

13. N. M. Mojarad, V. Sandoghdar, and M. Agio, “Plasmon spectra of nanospheres under a tightly focused beam,” J. Opt. Soc. Am. B 25, 651–658 (2008). [CrossRef]  

14. C. J. R. Sheppard and P. Török, “Efficient calculation of electromagnetic diffraction in optical systems using a multipole expansion,” J. Mod. Opt. 44, 803–818 (1997). [CrossRef]  

15. J. A. Lock, J. T. Hodges, and G. Gouesbet, “Failure of the optical theorem for Gaussian-beam scattering by a spherical particle,” J. Opt. Soc. Am. A 12, 2708–2715 (1995). [CrossRef]  

16. R. Oron, S. Blit, N. Davidson, A. A. Friesem, Z. Bomzon, and E. Hasman, “The formation of laser beams with pure azimuthal or radial polarization,” Appl. Phys. Lett. 77, 3322–3324 (2000). [CrossRef]  

17. B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system,” Proc. R. Soc. A 253, 358–379 (1959). [CrossRef]  

18. K. S. Youngworth and T. G. Brown “Focusing of high numerical aperture cylindrical-vector beams,” Opt. Express 7, 77–87 (2000). [CrossRef]   [PubMed]  

19. R. Borghi, M. Santarsiero, and M. A. Alonso, “Highly focused spirally polarized beams,” J. Opt. Soc. Am. A 22, 1420–1431 (2005). [CrossRef]  

20. C. F. Bohren and D. R. Huffman, Absorption and scattering of light by small particles (Wiley, New York, 1983).

21. G. Zumofen, N. M. Mojarad, V. Sandoghdar, and M. Agio, “Perfect reflection of light by an oscillating dipole,” Phys. Rev. Lett. 101, 180404(4) (2008) and supplementary material. [CrossRef]   [PubMed]  

22. S. A. Maier and H. A. Atwater, “Plasmonic: Localization and guiding of electromagnetic energy in metal/dielectric structures,” J. Appl. Phys. 98, 011101(10) (2005). [CrossRef]  

23. P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370–4379 (1972). [CrossRef]  

24. B. J. Messinger, K. U. von Raben, R. K. Chang, and P. W. Barber, “Local fields at the surface of noble-metal microspheres,” Phys. Rev. B 24, 649–657 (1981). [CrossRef]  

25. Wolfram Research, Inc., MATHEMATICA, Version 5.1, Champaign, IL (2004).

26. M. Sugiyama, S. Inasawa, S. Koda, T. Hirose, T. Yonekawa, T. Omatsu, and A. Takami, “Optical recording media using laser-induced size reduction of Au nanoparticles,” Appl. Phys. Lett. 79, 1528–1530 (2001). [CrossRef]  

27. V. Giannini and J. A. Sánchez-Gil, “Excitation and emission enhancement of single molecule fluorescence through multiple surface-plasmon resonances on metal trimer nanoantennas,” Opt. Lett. 33, 899–901 (2008). [CrossRef]   [PubMed]  

References

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  1. S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “The focus of light - theoretical calculation and experimental tomographic reconstruction,” Appl. Phys. B 72, 109–113 (2001).
  2. R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91, 233901(4) (2003).
    [CrossRef] [PubMed]
  3. L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, “Longitudinal field modes probed by single molecules,” Phys. Rev. Lett. 86, 5251–5254 (2001).
    [CrossRef] [PubMed]
  4. S. J. van Enk, “Atoms, dipole waves, and strongly focused light beams,” Phys. Rev. A 69, 043813(8) (2004).
    [CrossRef]
  5. Q. Zhan, “Trapping metallic Rayleigh particles with radial polarization,” Opt. Express 12, 3377–3382 (2004).
    [CrossRef] [PubMed]
  6. A. V. Failla, H. Qian, H. Qian, A. Hartschuh, and A. J. Meixner, “Orientational imaging of subwavelength Au particles with higher order laser modes,” Nano Lett. 61374–1378 (2006).
    [CrossRef] [PubMed]
  7. A. V. Failla, S. Jäger, T. Züchner, M. Steiner, and A. J. Meixner, “Topology measurements of metal nanoparticles with 1 nm accuracy by Confocal Interference Scattering Microscopy,” Opt. Express 15, 8532–8542 (2007).
    [CrossRef] [PubMed]
  8. T. Züchner, A. V. Failla, A. Hartschuh, and A. J. Meixner, “A novel approach to detect and characterize the scattering patterns of single Au nanoparticles using confocal microscopy,” J. Microsc. 229, 337–343 (2007).
    [CrossRef]
  9. K. Şendur, W. Challener, and O. Mryasov, “Interaction of spherical nanoparticles with highly focused beam of light,” Opt. Express 16, 2874–2886 (2008).
    [CrossRef] [PubMed]
  10. A. S. van de Nes and P. Török, “Rigorous analysis of spheres in Gauss-Laguerre beams,” Opt. Express 15, 13360–13374 (2007).
    [CrossRef] [PubMed]
  11. N. J. Moore and M. A. Alonso, “Closed form formula for Mie scattering of nonparaxial analogues of Gaussian beams,” Opt. Express 16, 5926–5933 (2008).
    [CrossRef] [PubMed]
  12. J. Lermé, C. Bonnet, M. Broyer, E. Cottancin, S. Marhaba, and M. Pellarin, “Optical response of metal or dielectric nano-objects in strongly convergent light beams,” Phys. Rev. B 77, 245406-13 (2008).
    [CrossRef]
  13. N. M. Mojarad, V. Sandoghdar, and M. Agio, “Plasmon spectra of nanospheres under a tightly focused beam,” J. Opt. Soc. Am. B 25, 651–658 (2008).
    [CrossRef]
  14. C. J. R. Sheppard and P. Török, “Efficient calculation of electromagnetic diffraction in optical systems using a multipole expansion,” J. Mod. Opt. 44, 803–818 (1997).
    [CrossRef]
  15. J. A. Lock, J. T. Hodges, and G. Gouesbet, “Failure of the optical theorem for Gaussian-beam scattering by a spherical particle,” J. Opt. Soc. Am. A 12, 2708–2715 (1995).
    [CrossRef]
  16. R. Oron, S. Blit, N. Davidson, A. A. Friesem, Z. Bomzon, and E. Hasman, “The formation of laser beams with pure azimuthal or radial polarization,” Appl. Phys. Lett. 77, 3322–3324 (2000).
    [CrossRef]
  17. B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system,” Proc. R. Soc. A 253, 358–379 (1959).
    [CrossRef]
  18. K. S. Youngworth and T. G. Brown “Focusing of high numerical aperture cylindrical-vector beams,” Opt. Express 7, 77–87 (2000).
    [CrossRef] [PubMed]
  19. R. Borghi, M. Santarsiero, and M. A. Alonso, “Highly focused spirally polarized beams,” J. Opt. Soc. Am. A 22, 1420–1431 (2005).
    [CrossRef]
  20. C. F. Bohren and D. R. Huffman, Absorption and scattering of light by small particles (Wiley, New York, 1983).
  21. G. Zumofen, N. M. Mojarad, V. Sandoghdar, and M. Agio, “Perfect reflection of light by an oscillating dipole,” Phys. Rev. Lett. 101, 180404(4) (2008) and supplementary material.
    [CrossRef] [PubMed]
  22. S. A. Maier and H. A. Atwater, “Plasmonic: Localization and guiding of electromagnetic energy in metal/dielectric structures,” J. Appl. Phys. 98, 011101(10) (2005).
    [CrossRef]
  23. P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370–4379 (1972).
    [CrossRef]
  24. B. J. Messinger, K. U. von Raben, R. K. Chang, and P. W. Barber, “Local fields at the surface of noble-metal microspheres,” Phys. Rev. B 24, 649–657 (1981).
    [CrossRef]
  25. Wolfram Research, Inc., MATHEMATICA, Version 5.1, Champaign, IL (2004).
  26. M. Sugiyama, S. Inasawa, S. Koda, T. Hirose, T. Yonekawa, T. Omatsu, and A. Takami, “Optical recording media using laser-induced size reduction of Au nanoparticles,” Appl. Phys. Lett. 79, 1528–1530 (2001).
    [CrossRef]
  27. V. Giannini and J. A. Sánchez-Gil, “Excitation and emission enhancement of single molecule fluorescence through multiple surface-plasmon resonances on metal trimer nanoantennas,” Opt. Lett. 33, 899–901 (2008).
    [CrossRef] [PubMed]

2008 (6)

2007 (3)

2006 (1)

A. V. Failla, H. Qian, H. Qian, A. Hartschuh, and A. J. Meixner, “Orientational imaging of subwavelength Au particles with higher order laser modes,” Nano Lett. 61374–1378 (2006).
[CrossRef] [PubMed]

2005 (2)

S. A. Maier and H. A. Atwater, “Plasmonic: Localization and guiding of electromagnetic energy in metal/dielectric structures,” J. Appl. Phys. 98, 011101(10) (2005).
[CrossRef]

R. Borghi, M. Santarsiero, and M. A. Alonso, “Highly focused spirally polarized beams,” J. Opt. Soc. Am. A 22, 1420–1431 (2005).
[CrossRef]

2004 (2)

Q. Zhan, “Trapping metallic Rayleigh particles with radial polarization,” Opt. Express 12, 3377–3382 (2004).
[CrossRef] [PubMed]

S. J. van Enk, “Atoms, dipole waves, and strongly focused light beams,” Phys. Rev. A 69, 043813(8) (2004).
[CrossRef]

2003 (1)

R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91, 233901(4) (2003).
[CrossRef] [PubMed]

2001 (3)

L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, “Longitudinal field modes probed by single molecules,” Phys. Rev. Lett. 86, 5251–5254 (2001).
[CrossRef] [PubMed]

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “The focus of light - theoretical calculation and experimental tomographic reconstruction,” Appl. Phys. B 72, 109–113 (2001).

M. Sugiyama, S. Inasawa, S. Koda, T. Hirose, T. Yonekawa, T. Omatsu, and A. Takami, “Optical recording media using laser-induced size reduction of Au nanoparticles,” Appl. Phys. Lett. 79, 1528–1530 (2001).
[CrossRef]

2000 (2)

K. S. Youngworth and T. G. Brown “Focusing of high numerical aperture cylindrical-vector beams,” Opt. Express 7, 77–87 (2000).
[CrossRef] [PubMed]

R. Oron, S. Blit, N. Davidson, A. A. Friesem, Z. Bomzon, and E. Hasman, “The formation of laser beams with pure azimuthal or radial polarization,” Appl. Phys. Lett. 77, 3322–3324 (2000).
[CrossRef]

1997 (1)

C. J. R. Sheppard and P. Török, “Efficient calculation of electromagnetic diffraction in optical systems using a multipole expansion,” J. Mod. Opt. 44, 803–818 (1997).
[CrossRef]

1995 (1)

1981 (1)

B. J. Messinger, K. U. von Raben, R. K. Chang, and P. W. Barber, “Local fields at the surface of noble-metal microspheres,” Phys. Rev. B 24, 649–657 (1981).
[CrossRef]

1972 (1)

P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370–4379 (1972).
[CrossRef]

1959 (1)

B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system,” Proc. R. Soc. A 253, 358–379 (1959).
[CrossRef]

Agio, M.

G. Zumofen, N. M. Mojarad, V. Sandoghdar, and M. Agio, “Perfect reflection of light by an oscillating dipole,” Phys. Rev. Lett. 101, 180404(4) (2008) and supplementary material.
[CrossRef] [PubMed]

N. M. Mojarad, V. Sandoghdar, and M. Agio, “Plasmon spectra of nanospheres under a tightly focused beam,” J. Opt. Soc. Am. B 25, 651–658 (2008).
[CrossRef]

Alonso, M. A.

Atwater, H. A.

S. A. Maier and H. A. Atwater, “Plasmonic: Localization and guiding of electromagnetic energy in metal/dielectric structures,” J. Appl. Phys. 98, 011101(10) (2005).
[CrossRef]

Barber, P. W.

B. J. Messinger, K. U. von Raben, R. K. Chang, and P. W. Barber, “Local fields at the surface of noble-metal microspheres,” Phys. Rev. B 24, 649–657 (1981).
[CrossRef]

Beversluis, M. R.

L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, “Longitudinal field modes probed by single molecules,” Phys. Rev. Lett. 86, 5251–5254 (2001).
[CrossRef] [PubMed]

Blit, S.

R. Oron, S. Blit, N. Davidson, A. A. Friesem, Z. Bomzon, and E. Hasman, “The formation of laser beams with pure azimuthal or radial polarization,” Appl. Phys. Lett. 77, 3322–3324 (2000).
[CrossRef]

Bohren, C. F.

C. F. Bohren and D. R. Huffman, Absorption and scattering of light by small particles (Wiley, New York, 1983).

Bomzon, Z.

R. Oron, S. Blit, N. Davidson, A. A. Friesem, Z. Bomzon, and E. Hasman, “The formation of laser beams with pure azimuthal or radial polarization,” Appl. Phys. Lett. 77, 3322–3324 (2000).
[CrossRef]

Bonnet, C.

J. Lermé, C. Bonnet, M. Broyer, E. Cottancin, S. Marhaba, and M. Pellarin, “Optical response of metal or dielectric nano-objects in strongly convergent light beams,” Phys. Rev. B 77, 245406-13 (2008).
[CrossRef]

Borghi, R.

Brown, T. G.

L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, “Longitudinal field modes probed by single molecules,” Phys. Rev. Lett. 86, 5251–5254 (2001).
[CrossRef] [PubMed]

K. S. Youngworth and T. G. Brown “Focusing of high numerical aperture cylindrical-vector beams,” Opt. Express 7, 77–87 (2000).
[CrossRef] [PubMed]

Broyer, M.

J. Lermé, C. Bonnet, M. Broyer, E. Cottancin, S. Marhaba, and M. Pellarin, “Optical response of metal or dielectric nano-objects in strongly convergent light beams,” Phys. Rev. B 77, 245406-13 (2008).
[CrossRef]

Challener, W.

Chang, R. K.

B. J. Messinger, K. U. von Raben, R. K. Chang, and P. W. Barber, “Local fields at the surface of noble-metal microspheres,” Phys. Rev. B 24, 649–657 (1981).
[CrossRef]

Christy, R. W.

P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370–4379 (1972).
[CrossRef]

Cottancin, E.

J. Lermé, C. Bonnet, M. Broyer, E. Cottancin, S. Marhaba, and M. Pellarin, “Optical response of metal or dielectric nano-objects in strongly convergent light beams,” Phys. Rev. B 77, 245406-13 (2008).
[CrossRef]

Davidson, N.

R. Oron, S. Blit, N. Davidson, A. A. Friesem, Z. Bomzon, and E. Hasman, “The formation of laser beams with pure azimuthal or radial polarization,” Appl. Phys. Lett. 77, 3322–3324 (2000).
[CrossRef]

Dorn, R.

R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91, 233901(4) (2003).
[CrossRef] [PubMed]

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “The focus of light - theoretical calculation and experimental tomographic reconstruction,” Appl. Phys. B 72, 109–113 (2001).

Eberler, M.

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “The focus of light - theoretical calculation and experimental tomographic reconstruction,” Appl. Phys. B 72, 109–113 (2001).

Failla, A. V.

T. Züchner, A. V. Failla, A. Hartschuh, and A. J. Meixner, “A novel approach to detect and characterize the scattering patterns of single Au nanoparticles using confocal microscopy,” J. Microsc. 229, 337–343 (2007).
[CrossRef]

A. V. Failla, S. Jäger, T. Züchner, M. Steiner, and A. J. Meixner, “Topology measurements of metal nanoparticles with 1 nm accuracy by Confocal Interference Scattering Microscopy,” Opt. Express 15, 8532–8542 (2007).
[CrossRef] [PubMed]

A. V. Failla, H. Qian, H. Qian, A. Hartschuh, and A. J. Meixner, “Orientational imaging of subwavelength Au particles with higher order laser modes,” Nano Lett. 61374–1378 (2006).
[CrossRef] [PubMed]

Friesem, A. A.

R. Oron, S. Blit, N. Davidson, A. A. Friesem, Z. Bomzon, and E. Hasman, “The formation of laser beams with pure azimuthal or radial polarization,” Appl. Phys. Lett. 77, 3322–3324 (2000).
[CrossRef]

Giannini, V.

Glöckl, O.

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “The focus of light - theoretical calculation and experimental tomographic reconstruction,” Appl. Phys. B 72, 109–113 (2001).

Gouesbet, G.

Hartschuh, A.

T. Züchner, A. V. Failla, A. Hartschuh, and A. J. Meixner, “A novel approach to detect and characterize the scattering patterns of single Au nanoparticles using confocal microscopy,” J. Microsc. 229, 337–343 (2007).
[CrossRef]

A. V. Failla, H. Qian, H. Qian, A. Hartschuh, and A. J. Meixner, “Orientational imaging of subwavelength Au particles with higher order laser modes,” Nano Lett. 61374–1378 (2006).
[CrossRef] [PubMed]

Hasman, E.

R. Oron, S. Blit, N. Davidson, A. A. Friesem, Z. Bomzon, and E. Hasman, “The formation of laser beams with pure azimuthal or radial polarization,” Appl. Phys. Lett. 77, 3322–3324 (2000).
[CrossRef]

Hirose, T.

M. Sugiyama, S. Inasawa, S. Koda, T. Hirose, T. Yonekawa, T. Omatsu, and A. Takami, “Optical recording media using laser-induced size reduction of Au nanoparticles,” Appl. Phys. Lett. 79, 1528–1530 (2001).
[CrossRef]

Hodges, J. T.

Huffman, D. R.

C. F. Bohren and D. R. Huffman, Absorption and scattering of light by small particles (Wiley, New York, 1983).

Inasawa, S.

M. Sugiyama, S. Inasawa, S. Koda, T. Hirose, T. Yonekawa, T. Omatsu, and A. Takami, “Optical recording media using laser-induced size reduction of Au nanoparticles,” Appl. Phys. Lett. 79, 1528–1530 (2001).
[CrossRef]

Jäger, S.

Johnson, P. B.

P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370–4379 (1972).
[CrossRef]

Koda, S.

M. Sugiyama, S. Inasawa, S. Koda, T. Hirose, T. Yonekawa, T. Omatsu, and A. Takami, “Optical recording media using laser-induced size reduction of Au nanoparticles,” Appl. Phys. Lett. 79, 1528–1530 (2001).
[CrossRef]

Lermé, J.

J. Lermé, C. Bonnet, M. Broyer, E. Cottancin, S. Marhaba, and M. Pellarin, “Optical response of metal or dielectric nano-objects in strongly convergent light beams,” Phys. Rev. B 77, 245406-13 (2008).
[CrossRef]

Leuchs, G.

R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91, 233901(4) (2003).
[CrossRef] [PubMed]

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “The focus of light - theoretical calculation and experimental tomographic reconstruction,” Appl. Phys. B 72, 109–113 (2001).

Lock, J. A.

Maier, S. A.

S. A. Maier and H. A. Atwater, “Plasmonic: Localization and guiding of electromagnetic energy in metal/dielectric structures,” J. Appl. Phys. 98, 011101(10) (2005).
[CrossRef]

Marhaba, S.

J. Lermé, C. Bonnet, M. Broyer, E. Cottancin, S. Marhaba, and M. Pellarin, “Optical response of metal or dielectric nano-objects in strongly convergent light beams,” Phys. Rev. B 77, 245406-13 (2008).
[CrossRef]

Meixner, A. J.

T. Züchner, A. V. Failla, A. Hartschuh, and A. J. Meixner, “A novel approach to detect and characterize the scattering patterns of single Au nanoparticles using confocal microscopy,” J. Microsc. 229, 337–343 (2007).
[CrossRef]

A. V. Failla, S. Jäger, T. Züchner, M. Steiner, and A. J. Meixner, “Topology measurements of metal nanoparticles with 1 nm accuracy by Confocal Interference Scattering Microscopy,” Opt. Express 15, 8532–8542 (2007).
[CrossRef] [PubMed]

A. V. Failla, H. Qian, H. Qian, A. Hartschuh, and A. J. Meixner, “Orientational imaging of subwavelength Au particles with higher order laser modes,” Nano Lett. 61374–1378 (2006).
[CrossRef] [PubMed]

Messinger, B. J.

B. J. Messinger, K. U. von Raben, R. K. Chang, and P. W. Barber, “Local fields at the surface of noble-metal microspheres,” Phys. Rev. B 24, 649–657 (1981).
[CrossRef]

Mojarad, N. M.

G. Zumofen, N. M. Mojarad, V. Sandoghdar, and M. Agio, “Perfect reflection of light by an oscillating dipole,” Phys. Rev. Lett. 101, 180404(4) (2008) and supplementary material.
[CrossRef] [PubMed]

N. M. Mojarad, V. Sandoghdar, and M. Agio, “Plasmon spectra of nanospheres under a tightly focused beam,” J. Opt. Soc. Am. B 25, 651–658 (2008).
[CrossRef]

Moore, N. J.

Mryasov, O.

Novotny, L.

L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, “Longitudinal field modes probed by single molecules,” Phys. Rev. Lett. 86, 5251–5254 (2001).
[CrossRef] [PubMed]

Omatsu, T.

M. Sugiyama, S. Inasawa, S. Koda, T. Hirose, T. Yonekawa, T. Omatsu, and A. Takami, “Optical recording media using laser-induced size reduction of Au nanoparticles,” Appl. Phys. Lett. 79, 1528–1530 (2001).
[CrossRef]

Oron, R.

R. Oron, S. Blit, N. Davidson, A. A. Friesem, Z. Bomzon, and E. Hasman, “The formation of laser beams with pure azimuthal or radial polarization,” Appl. Phys. Lett. 77, 3322–3324 (2000).
[CrossRef]

Pellarin, M.

J. Lermé, C. Bonnet, M. Broyer, E. Cottancin, S. Marhaba, and M. Pellarin, “Optical response of metal or dielectric nano-objects in strongly convergent light beams,” Phys. Rev. B 77, 245406-13 (2008).
[CrossRef]

Qian, H.

A. V. Failla, H. Qian, H. Qian, A. Hartschuh, and A. J. Meixner, “Orientational imaging of subwavelength Au particles with higher order laser modes,” Nano Lett. 61374–1378 (2006).
[CrossRef] [PubMed]

A. V. Failla, H. Qian, H. Qian, A. Hartschuh, and A. J. Meixner, “Orientational imaging of subwavelength Au particles with higher order laser modes,” Nano Lett. 61374–1378 (2006).
[CrossRef] [PubMed]

Quabis, S.

R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91, 233901(4) (2003).
[CrossRef] [PubMed]

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “The focus of light - theoretical calculation and experimental tomographic reconstruction,” Appl. Phys. B 72, 109–113 (2001).

Richards, B.

B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system,” Proc. R. Soc. A 253, 358–379 (1959).
[CrossRef]

Sánchez-Gil, J. A.

Sandoghdar, V.

G. Zumofen, N. M. Mojarad, V. Sandoghdar, and M. Agio, “Perfect reflection of light by an oscillating dipole,” Phys. Rev. Lett. 101, 180404(4) (2008) and supplementary material.
[CrossRef] [PubMed]

N. M. Mojarad, V. Sandoghdar, and M. Agio, “Plasmon spectra of nanospheres under a tightly focused beam,” J. Opt. Soc. Am. B 25, 651–658 (2008).
[CrossRef]

Santarsiero, M.

Sendur, K.

Sheppard, C. J. R.

C. J. R. Sheppard and P. Török, “Efficient calculation of electromagnetic diffraction in optical systems using a multipole expansion,” J. Mod. Opt. 44, 803–818 (1997).
[CrossRef]

Steiner, M.

Sugiyama, M.

M. Sugiyama, S. Inasawa, S. Koda, T. Hirose, T. Yonekawa, T. Omatsu, and A. Takami, “Optical recording media using laser-induced size reduction of Au nanoparticles,” Appl. Phys. Lett. 79, 1528–1530 (2001).
[CrossRef]

Takami, A.

M. Sugiyama, S. Inasawa, S. Koda, T. Hirose, T. Yonekawa, T. Omatsu, and A. Takami, “Optical recording media using laser-induced size reduction of Au nanoparticles,” Appl. Phys. Lett. 79, 1528–1530 (2001).
[CrossRef]

Török, P.

A. S. van de Nes and P. Török, “Rigorous analysis of spheres in Gauss-Laguerre beams,” Opt. Express 15, 13360–13374 (2007).
[CrossRef] [PubMed]

C. J. R. Sheppard and P. Török, “Efficient calculation of electromagnetic diffraction in optical systems using a multipole expansion,” J. Mod. Opt. 44, 803–818 (1997).
[CrossRef]

van de Nes, A. S.

van Enk, S. J.

S. J. van Enk, “Atoms, dipole waves, and strongly focused light beams,” Phys. Rev. A 69, 043813(8) (2004).
[CrossRef]

von Raben, K. U.

B. J. Messinger, K. U. von Raben, R. K. Chang, and P. W. Barber, “Local fields at the surface of noble-metal microspheres,” Phys. Rev. B 24, 649–657 (1981).
[CrossRef]

Wolf, E.

B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system,” Proc. R. Soc. A 253, 358–379 (1959).
[CrossRef]

Yonekawa, T.

M. Sugiyama, S. Inasawa, S. Koda, T. Hirose, T. Yonekawa, T. Omatsu, and A. Takami, “Optical recording media using laser-induced size reduction of Au nanoparticles,” Appl. Phys. Lett. 79, 1528–1530 (2001).
[CrossRef]

Youngworth, K. S.

L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, “Longitudinal field modes probed by single molecules,” Phys. Rev. Lett. 86, 5251–5254 (2001).
[CrossRef] [PubMed]

K. S. Youngworth and T. G. Brown “Focusing of high numerical aperture cylindrical-vector beams,” Opt. Express 7, 77–87 (2000).
[CrossRef] [PubMed]

Zhan, Q.

Züchner, T.

A. V. Failla, S. Jäger, T. Züchner, M. Steiner, and A. J. Meixner, “Topology measurements of metal nanoparticles with 1 nm accuracy by Confocal Interference Scattering Microscopy,” Opt. Express 15, 8532–8542 (2007).
[CrossRef] [PubMed]

T. Züchner, A. V. Failla, A. Hartschuh, and A. J. Meixner, “A novel approach to detect and characterize the scattering patterns of single Au nanoparticles using confocal microscopy,” J. Microsc. 229, 337–343 (2007).
[CrossRef]

Zumofen, G.

G. Zumofen, N. M. Mojarad, V. Sandoghdar, and M. Agio, “Perfect reflection of light by an oscillating dipole,” Phys. Rev. Lett. 101, 180404(4) (2008) and supplementary material.
[CrossRef] [PubMed]

Appl. Phys. B (1)

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “The focus of light - theoretical calculation and experimental tomographic reconstruction,” Appl. Phys. B 72, 109–113 (2001).

Appl. Phys. Lett. (2)

R. Oron, S. Blit, N. Davidson, A. A. Friesem, Z. Bomzon, and E. Hasman, “The formation of laser beams with pure azimuthal or radial polarization,” Appl. Phys. Lett. 77, 3322–3324 (2000).
[CrossRef]

M. Sugiyama, S. Inasawa, S. Koda, T. Hirose, T. Yonekawa, T. Omatsu, and A. Takami, “Optical recording media using laser-induced size reduction of Au nanoparticles,” Appl. Phys. Lett. 79, 1528–1530 (2001).
[CrossRef]

J. Appl. Phys. (1)

S. A. Maier and H. A. Atwater, “Plasmonic: Localization and guiding of electromagnetic energy in metal/dielectric structures,” J. Appl. Phys. 98, 011101(10) (2005).
[CrossRef]

J. Microsc. (1)

T. Züchner, A. V. Failla, A. Hartschuh, and A. J. Meixner, “A novel approach to detect and characterize the scattering patterns of single Au nanoparticles using confocal microscopy,” J. Microsc. 229, 337–343 (2007).
[CrossRef]

J. Mod. Opt. (1)

C. J. R. Sheppard and P. Török, “Efficient calculation of electromagnetic diffraction in optical systems using a multipole expansion,” J. Mod. Opt. 44, 803–818 (1997).
[CrossRef]

J. Opt. Soc. Am. A (2)

J. Opt. Soc. Am. B (1)

Nano Lett. (1)

A. V. Failla, H. Qian, H. Qian, A. Hartschuh, and A. J. Meixner, “Orientational imaging of subwavelength Au particles with higher order laser modes,” Nano Lett. 61374–1378 (2006).
[CrossRef] [PubMed]

Opt. Express (6)

Opt. Lett. (1)

Phys. Rev. A (1)

S. J. van Enk, “Atoms, dipole waves, and strongly focused light beams,” Phys. Rev. A 69, 043813(8) (2004).
[CrossRef]

Phys. Rev. B (3)

J. Lermé, C. Bonnet, M. Broyer, E. Cottancin, S. Marhaba, and M. Pellarin, “Optical response of metal or dielectric nano-objects in strongly convergent light beams,” Phys. Rev. B 77, 245406-13 (2008).
[CrossRef]

P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370–4379 (1972).
[CrossRef]

B. J. Messinger, K. U. von Raben, R. K. Chang, and P. W. Barber, “Local fields at the surface of noble-metal microspheres,” Phys. Rev. B 24, 649–657 (1981).
[CrossRef]

Phys. Rev. Lett. (3)

R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91, 233901(4) (2003).
[CrossRef] [PubMed]

L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, “Longitudinal field modes probed by single molecules,” Phys. Rev. Lett. 86, 5251–5254 (2001).
[CrossRef] [PubMed]

G. Zumofen, N. M. Mojarad, V. Sandoghdar, and M. Agio, “Perfect reflection of light by an oscillating dipole,” Phys. Rev. Lett. 101, 180404(4) (2008) and supplementary material.
[CrossRef] [PubMed]

Proc. R. Soc. A (1)

B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system,” Proc. R. Soc. A 253, 358–379 (1959).
[CrossRef]

Other (2)

C. F. Bohren and D. R. Huffman, Absorption and scattering of light by small particles (Wiley, New York, 1983).

Wolfram Research, Inc., MATHEMATICA, Version 5.1, Champaign, IL (2004).

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Figures (3)

Fig. 1.
Fig. 1.

Relative strength |Bl /B 1| of the multipoles in the incident field. The result for a FPW is also shown for comparison [13]. The inset sketches a radially-polarized beam focused onto a NP. The red curve is the beam intensity profile before the lens, f is the focal length, α the angular semi-aperture, wo the beam waist and a = f/wo .

Fig. 2.
Fig. 2.

Extinction efficiency ��e of a 100 nm silver NP in glass illuminated by three different FRBs and a FPW. D, Q and O respectively label the dipole, quadrupole and octupole resonances. Inset: electric field intensity (contours) and Poynting vector (arrows) in the focal region for α= 90°, a = 1.5, and λ = 520 nm.

Fig. 3.
Fig. 3.

Radial and tangential (inset) average intensity enhancement for a 100 nm silver NP in glass illuminated by three different FRBs and a FPW. D, Q and O respectively label the dipole, quadrupole and octupole resonances.

Equations (6)

Equations on this page are rendered with MathJax. Learn more.

Einc=l=1BlNe0l(1),
Bl=2l+12l(l+1)2kfilexp(ikf)0αE(a,θ)dPl(cosθ)dθsinθdθ,
Es=l=1alBlNe0l(3),Ei=l=1dlBlNe0l(1),
Ws=πZk2l=1Bl2al22l(l+1)2l+1,We=πZk2l=1Bl2Re{al}2l(l+1)2l+1,
Kr=916(kR)4lBlB12l2(l+1)22l+1[ψl2+al2χl2+2Re{al*χl*}ψl],
Kt=916(kR)2lBlB12l(l+1)22l+1[ψl2+al2χl2+2Re{al*χl*}ψl],

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