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Virtual Journal for Biomedical Optics

| EXPLORING THE INTERFACE OF LIGHT AND BIOMEDICINE

  • Editors: Andrew Dunn and Anthony Durkin
  • Vol. 8, Iss. 4 — May. 22, 2013
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Mie scattering and optical forces from evanescent fields: A complex-angle approach

Aleksandr Y. Bekshaev, Konstantin Y. Bliokh, and Franco Nori  »View Author Affiliations


Optics Express, Vol. 21, Issue 6, pp. 7082-7095 (2013)
http://dx.doi.org/10.1364/OE.21.007082


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Abstract

Mie theory is one of the main tools describing scattering of propagating electromagnetic waves by spherical particles. Evanescent optical fields are also scattered by particles and exert radiation forces which can be used for optical near-field manipulations. We show that the Mie theory can be naturally adopted for the scattering of evanescent waves via rotation of its standard solutions by a complex angle. This offers a simple and powerful tool for calculations of the scattered fields and radiation forces. Comparison with other, more cumbersome, approaches shows perfect agreement, thereby validating our theory. As examples of its application, we calculate angular distributions of the scattered far-field irradiance and radiation forces acting on dielectric and conducting particles immersed in an evanescent field.

© 2013 OSA

1. Introduction

The scattering of light by various particles appears in a variety of optical processes, with applications ranging from microscopy to astrophysics. A fundamental solution describing electromagnetic wave scattering by a spherical particle was found in 1908 by Gustav Mie [1

1. G. Mie, “Beiträge zur Optik trüber Medien, speziell kolloidaler Metallösungen,” Ann. Phys. (Leipzig) 25, 377–445 (1908).

]. Since then, the Mie theory has become the main tool for characterization of particle-induced light scattering [2

2. M. Born and E. Wolf, Principles of Optics, 7th ed. (London: Pergamon, 2005).

4

4. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (New York: Wiley, 1983).

]. In addition to the properties of scattered light, this theory allows calculation of radiation forces exerted on particles. Such forces are of great importance for optical manipulations [5

5. M. Dienerowitz, M. Mazilu, and K. Dholakia, “Optical manipulation of nanoparticles: a review,” J. Nanophotonics 2(1), 021875 (2008). [CrossRef]

] and for investigations of the fundamental physical properties of electromagnetic fields [6

6. I. Brevik, “Experiments in phenomenological electrodynamics and the electromagnetic energy-momentum tensor,” Phys. Rep. 52(3), 133–201 (1979). [CrossRef]

].

With the development of near-field optics and plasmonics [7

7. M. A. Paesler and P. J. Moyer, Near–Field Optics (New York: John Wiley & Sons, 1996).

9

9. S. A. Maier, Plasmonics: Fundamentals and Applications (New York: Springer, 2007).

], evanescent electromagnetic waves have attracted enormous interest, both for theory and applications. In particular, the evanescent-wave scattering by small particles and accompanying radiation forces are important in modern optics. Analogues of the Mie theory for evanescent waves were elaborated [10

10. H. Chew, D.-S. Wang, and M. Kerker, “Elastic scattering of evanescent electromagnetic waves,” Appl. Opt. 18(15), 2679–2687 (1979). [CrossRef] [PubMed]

13

13. E. Almaas and I. Brevik, “Radiation forces on a micrometer-sized sphere in an evanescent field,” J. Opt. Soc. Am. B 12(12), 2429–2438 (1995). [CrossRef]

], while the radiation forces from evanescent fields were extensively examined theoretically [13

13. E. Almaas and I. Brevik, “Radiation forces on a micrometer-sized sphere in an evanescent field,” J. Opt. Soc. Am. B 12(12), 2429–2438 (1995). [CrossRef]

23

23. M. Nieto-Vesperinas and J. R. Arias-González, “Theory of forces induced by evanescent fields,” arXiv: 1102.1613 (2011).

] and experimentally [24

24. S. Kawata and T. Sugiura, “Movement of micrometer-sized particles in the evanescent field of a laser beam,” Opt. Lett. 17(11), 772–774 (1992). [CrossRef] [PubMed]

31

31. S. Gaugiran, S. Gétin, J. M. Fedeli, and J. Derouard, “Polarization and particle size dependence of radiative forces on small metallic particles in evanescent optical fields. Evidences for either repulsive or attractive gradient forces,” Opt. Express 15(13), 8146–8156 (2007). [CrossRef] [PubMed]

]. Most of these Mie-type approaches are based on a straightforward expansion of the incident evanescent wave in a series of vector spherical harmonics and the subsequent reproduction of the Mie procedure, which results in rather cumbersome calculations [10

10. H. Chew, D.-S. Wang, and M. Kerker, “Elastic scattering of evanescent electromagnetic waves,” Appl. Opt. 18(15), 2679–2687 (1979). [CrossRef] [PubMed]

17

17. Y. G. Song, B. M. Han, and S. Chang, “Force of surface plasmon-coupled evanescent fields on Mie particles,” Opt. Commun. 198(1-3), 7–19 (2001). [CrossRef]

]. Alternatively, one can treat analytically the simplest approximation of the dipole Rayleigh scattering of evanescent waves by small (much less than the wavelength) particles [19

19. P. C. Chaumet and M. Nieto-Vesperinas, “Electromagnetic force on a metallic particle in the presence of a dielectric surface,” Phys. Rev. B 62(16), 11185–11191 (2000). [CrossRef]

23

23. M. Nieto-Vesperinas and J. R. Arias-González, “Theory of forces induced by evanescent fields,” arXiv: 1102.1613 (2011).

].

To demonstrate several applications of our method, we first calculate the far-field scattering diagrams for different evanescent-wave polarizations and particle sizes, and then compare them with the usual Mie-theory diagrams for the incident plane wave case. Second, we compute optical forces exerted on dielectric particles in an evanescent field from a totally-reflecting interface, and show that our results coincide with those previously reported in [13

13. E. Almaas and I. Brevik, “Radiation forces on a micrometer-sized sphere in an evanescent field,” J. Opt. Soc. Am. B 12(12), 2429–2438 (1995). [CrossRef]

,22

22. M. Nieto-Vesperinas and J. J. Saenz, “Optical forces from an evanescent wave on a magnetodielectric small particle,” Opt. Lett. 35(23), 4078–4080 (2010). [CrossRef] [PubMed]

,23

23. M. Nieto-Vesperinas and J. R. Arias-González, “Theory of forces induced by evanescent fields,” arXiv: 1102.1613 (2011).

]. Finally, we calculate the optical force on metallic particles and address the problem of the positive vertical force repelling the particle from the surface [14

14. I. Brevik, T. A. Sivertsen, and E. Almaas, “Radiation forces on an absorbing micrometer-sized sphere in an evanescent field,” J. Opt. Soc. Am. B 20(8), 1739–1749 (2003). [CrossRef]

,15

15. H. Y. Jaising and O. G. Hellesø, “Radiation forces on a Mie particle in the evanescent field of an optical waveguide,” Opt. Commun. 246(4-6), 373–383 (2005). [CrossRef]

,19

19. P. C. Chaumet and M. Nieto-Vesperinas, “Electromagnetic force on a metallic particle in the presence of a dielectric surface,” Phys. Rev. B 62(16), 11185–11191 (2000). [CrossRef]

,20

20. J. R. Arias-González and M. Nieto-Vesperinas, “Optical forces on small particles: attractive and repulsive nature and plasmon-resonance conditions,” J. Opt. Soc. Am. A 20(7), 1201–1209 (2003). [CrossRef] [PubMed]

,23

23. M. Nieto-Vesperinas and J. R. Arias-González, “Theory of forces induced by evanescent fields,” arXiv: 1102.1613 (2011).

,24

24. S. Kawata and T. Sugiura, “Movement of micrometer-sized particles in the evanescent field of a laser beam,” Opt. Lett. 17(11), 772–774 (1992). [CrossRef] [PubMed]

,31

31. S. Gaugiran, S. Gétin, J. M. Fedeli, and J. Derouard, “Polarization and particle size dependence of radiative forces on small metallic particles in evanescent optical fields. Evidences for either repulsive or attractive gradient forces,” Opt. Express 15(13), 8146–8156 (2007). [CrossRef] [PubMed]

]. Note that, as in most other works, we neglect multiple reflections from the surface limiting the evanescent field. More accurate treatments [16

16. S. Chang, J. T. Kim, J. H. Jo, and S. S. Lee, “Optical force on a sphere caused by the evanescent field of a Gaussian beam; effects of multiple scattering,” Opt. Commun. 139(4-6), 252–261 (1997). [CrossRef]

,17

17. Y. G. Song, B. M. Han, and S. Chang, “Force of surface plasmon-coupled evanescent fields on Mie particles,” Opt. Commun. 198(1-3), 7–19 (2001). [CrossRef]

,19

19. P. C. Chaumet and M. Nieto-Vesperinas, “Electromagnetic force on a metallic particle in the presence of a dielectric surface,” Phys. Rev. B 62(16), 11185–11191 (2000). [CrossRef]

21

21. M. Nieto-Vesperinas, P. C. Chaumet, and A. Rahmani, “Near-field photonic forces,” Philos. Transact. A Math. Phys. Eng. Sci. 362(1817), 719–737 (2004). [CrossRef] [PubMed]

,23

23. M. Nieto-Vesperinas and J. R. Arias-González, “Theory of forces induced by evanescent fields,” arXiv: 1102.1613 (2011).

,32

32. D. C. Prieve and J. Y. Walz, “Scattering of an evanescent surface wave by a microscopic dielectric sphere,” Appl. Opt. 32(9), 1629–1641 (1993). [CrossRef] [PubMed]

] show that the influence of these reflections can be neglected in a wide range of parameters: e.g., in calculations of the parallel components of the force, and for particle sizes of the order of the wavelength and not exhibiting resonances. Even in cases where multiple-scattering effects must be taken into account, the single scattering of a pure evanescent wave is the first step, and it can be facilitated by the approach proposed here.

2. Incident field configuration

The standard formulation of the Mie scattering theory starts with an incident monochromatic plane wave propagating along the z axis, with wavevector k=(0,0,k), whereas the center of the particle is located at the origin (x,y,z)=0. Complex electric and magnetic field amplitudes of the incident wave are written as
E=(EE0)exp(ikz),H=(HH0)exp(ikz)=εμ(EE0)exp(ikz).
(1)
Here the subscripts and denote the p- and s- polarizations with respect to the (x,z) plane, k=ωn/cnk0 is the wave number [ω is the frequency, c is the velocity of light in vacuum, and throughout the paper we omit the common factor exp(iωt)], and we assume a lossless medium characterized by permittivity ε, permeability μ, and refractive index n=εμ. Note that in this paper we use the Gaussian system of units. For transition to the SI units, one should modify the field amplitudes as Eε0E, Hμ0H (ε0 and μ0 are the vacuum permittivity and permeability, ε0μ0=c2), and use the corresponding constant g=1/2 [see Eq. (8) below].

Let us consider a more generic situation, where an incident wave propagates at some angle γ with respect to the z axis in the (x,z) plane. The wave field is obtained using the corresponding rotation operator R^y(γ)=exp(iγJ^y/), where J^ is the total (spin plus orbital) angular momentum operator. Application of the operator R^y(γ) (which rotates both vector directions and function distributions) to the wave fields E(r) and H(r) results in the transformation:
E(r)R^y(γ)E[R^y(γ)r],H(r)R^y(γ)H[R^y(γ)r],
(2)
where
R^y(γ)=(cosγ0sinγ010sinγ0cosγ)
(3)
is the rotation matrix which acts on the Cartesian components of the vectors. Explicitly, Eqs. (2) and (3) yield
E=(EcosγEEsinγ)exp[ik(zcosγ+xsinγ)],H=εμ(EcosγEEsinγ)exp[ik(zcosγ+xsinγ)].
(4)
Here the transformation of coordinates rR^y(γ)r is equivalent to the wavevector rotation kR^y(γ)k.

Importantly, the simple expressions Eqs. (2)(4), which describe an obliquely-propagating plane wave, can also describe evanescent plane waves decaying away from the z=0 plane (Fig. 1
Fig. 1 Schematic of the Mie scattering problem. Incident wave (blue), scattered field (green), and radiation force exerted on the particle (yellow) are shown. (a) standard Mie theory with the incident plane wave propagating along the z-axis. (b) Rotation of the field, Eqs. (2) and (5), by the complex angle γ=π/2iα results in the modified Mie problem with evanescent incident wave Eq. (6). The parameter h indicates the distance to the surface where the evanescent wave is generated.
). Indeed, consider now the complex propagation angle γ given by:
γ=π2iα,α>0.
(5)
In this case, the rotation matrix (3) R^y(γ) takes the form
R^y(π2+iα)=(isinhα0coshα010coshα0isinhα),
(6)
and the incident field is obtained by the corresponding modifications of Eq. (4):
E=(iEsinhαEEcoshα)exp(ikxcoshαkzsinhα),H=εμ(iEsinhαEEcoshα)exp(ikxcoshαkzsinhα),
(7)
with wavevector k=(kcoshα,0,iksinhα). Equations (7) describes an evanescent plane wave propagating in the x-direction and decaying in the positive z-direction, see Fig. 1.

It is useful to consider the time-averaged densities of the electromagnetic energy and momentum in the evanescent field (7). They are determined by the well-known relations [2

2. M. Born and E. Wolf, Principles of Optics, 7th ed. (London: Pergamon, 2005).

]
w=g2(ε|E|2+μ|H|2),p=gcRe(E*×H),
(8)
where g=(8π)1 in the Gaussian system of units (g=1/2 in SI units). Substituting the evanescent fields (7) into Eq. (8), we obtain
w=εgcosh2α(|E|2+|E|2)exp(2kzsinhα),px=gncμcoshα(|E|2+|E|2)exp(2kzsinhα),pz=0,py=2gncμsinhαcoshαIm(E*E)exp(2kzsinhα).
(9)
As expected for an evanescent field, the z-directed momentum component vanishes and the energy flows parallel to the z=0 plane. It is worth noticing that the x component of the momentum transports the energy and can be written as px=vgw/c2. Here vg=ck0/kx (kx=kcoshα>k) is the wave group velocity, and px is essentially combined from orbital and spin contributions as described in [33

33. A. Bekshaev, K. Bliokh, and M. Soskin, “Internal flows and energy circulation in light beams,” J. Opt. 13(5), 053001 (2011). [CrossRef]

,34

34. K. Y. Bliokh and F. Nori, “Transverse spin of a surface polariton,” Phys. Rev. A 85(6), 061801 (2012). [CrossRef]

]. At the same time, the transverse momentum py was described by Fedorov and Imbert in the total internal reflection problem [35

35. F. I. Fedorov, “To the theory of total reflection,” Dokl. Akad. Nauk SSSR 105, 465–468 (1955) (reprinted in J. Opt. 15, 014002 (2013)).

,36

36. C. Imbert, “Calculation and experimental proof of the transverse shift induced by total internal reflection of a circularly polarized light beam,” Phys. Rev. D Part. Fields 5(4), 787–796 (1972). [CrossRef]

]. This transverse momentum py is proportional to the ellipticity of the wave polarization 2Im(E*E), it is of purely spin nature, and therefore does not transport energy [33

33. A. Bekshaev, K. Bliokh, and M. Soskin, “Internal flows and energy circulation in light beams,” J. Opt. 13(5), 053001 (2011). [CrossRef]

,34

34. K. Y. Bliokh and F. Nori, “Transverse spin of a surface polariton,” Phys. Rev. A 85(6), 061801 (2012). [CrossRef]

].

In practice, one of the standard ways to generate the incident evanescent wave (7) is to use the total internal reflection. For instance, let the plane z=h be the dielectric interface separating two media, with parameters ε1, μ1, n1=ε1μ1 (z<h) and ε, μ, n=εμ<n1 (z>h). A propagating plane wave E0 with the p- and s-polarized field components E0 and E0 impinges on the interface from the z<h half-space at an angle of incidence θ1, such that the condition for total internal reflection n1sinθ1>n is realized. Then, the transmitted field at z>h calculated via the corresponding Snell-Fresnel formulae [2

2. M. Born and E. Wolf, Principles of Optics, 7th ed. (London: Pergamon, 2005).

] will be the evanescent wave (7) with parameters
coshα=n1nsinθ1,sinhα=(n1n)2sin2θ11,
(10)
E=2nn1cosθ1εε1cosθ1+inn1sinhαekhsinhαE0,E=2μμ1cosθ1μμ1cosθ1+inn1sinhαekhsinhαE0.
(11)
One can notice that the complex angle of rotation γ, determined by Eqs. (5) and (10), is just the complex angle of refraction which formally follows from Snell’s law under the total-reflection conditions [2

2. M. Born and E. Wolf, Principles of Optics, 7th ed. (London: Pergamon, 2005).

]. Note also that the evanescent wave generated at the distance h from the z=0 plane acquires the amplitude attenuation factor exp(khsinhα).

3. Complex-angle Mie theory: Scattered field and radiation forces

Considering light scattering by a spherical particle of radius a with electromagnetic parameters εp, μp, and np=εpμp, the standard Mie equations establish linear relation between the amplitudes of the incident plane wave (1), E, H and scattered fields Es, Hs. These known equations [4

4. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (New York: Wiley, 1983).

] are collected in Appendix A in a complete form, keeping the scattered near field and radial components (which are typically omitted). One can write these equations in a symbolic operator form as
Es(r)=S^E(r)E,Hs(r)=S^H(r)E,
(12)
where the Mie scattering operators S^E,H(r) are well defined in the complex domain.

One of the important applications of the Mie theory is the calculation of the optical force exerted on a particle by the total electromagnetic field (14). This force is determined by the Maxwell stress tensor T^={Tij}, i,j=x,y,z:
Tij=gRe[εEitot*Ejtot+μHitot*Hjtot12δij(ε|Etot|2+μ|Htot|2)].
(15)
Integrating the stress tensor components over any surface A enclosing the particle (e.g., a sphere S={r=R}, R>a), we obtain the optical force:
F=AT^ndA=R2ST^ndΩ,
(16)
where dΩ=sinθdθdϕ is the elementary solid angle, n=(sinθcosϕ,sinθsinϕ,cosθ)T is the unit vector of the outer normal to the sphere surface. Below we calculate the optical force from the evanescent incident field using the complex-angle Mie theory, and compare these results with the results of previous, more cumbersome, approaches.

4. Radiation forces: Comparison with other approaches and new applications

To verify the validity of our method, we apply the complex-angle Mie theory to problems involving optical forces from evanescent fields. We assume that a spherical particle of radius a lies on the totally-reflecting surface z=a [13

13. E. Almaas and I. Brevik, “Radiation forces on a micrometer-sized sphere in an evanescent field,” J. Opt. Soc. Am. B 12(12), 2429–2438 (1995). [CrossRef]

,18

18. J. Y. Walz, “Ray optics calculation of the radiation forces exerted on a dielectric sphere in an evanescent field,” Appl. Opt. 38(25), 5319–5330 (1999). [CrossRef] [PubMed]

], so that its center is positioned at z=0, and the incident evanescent wave is described by Eqs. (7), (10), and (11) with h=a.

First, following the well-established approach of Refs. [13

13. E. Almaas and I. Brevik, “Radiation forces on a micrometer-sized sphere in an evanescent field,” J. Opt. Soc. Am. B 12(12), 2429–2438 (1995). [CrossRef]

,14

14. I. Brevik, T. A. Sivertsen, and E. Almaas, “Radiation forces on an absorbing micrometer-sized sphere in an evanescent field,” J. Opt. Soc. Am. B 20(8), 1739–1749 (2003). [CrossRef]

], we consider a dielectric particle and either p-polarized (E0=0) or s-polarized (E0=0) incident wave. The input parameters possess the following numerical values [13

13. E. Almaas and I. Brevik, “Radiation forces on a micrometer-sized sphere in an evanescent field,” J. Opt. Soc. Am. B 12(12), 2429–2438 (1995). [CrossRef]

]:
μ1=μ=μp=1,n1=1.75,n=1,np=1.5,θ1=51°.
(17)
(Recall that parameters with the subscript “1”, without subscript, and with subscript “p” correspond to the high-index medium, low-index medium, and particle, respectively.). Using Eq. (10), this yields sinhα0.92. The calculated force (16) will be normalized by
P0=a24π(|E0|2+|E0|2).
(18)
This quantity is proportional to the time-average momentum flux of the incident plane wave E0 through the area a2 and represents the Gaussian-unit counterpart to the SI-unit normalization divider ε0a2E02 used in [13

13. E. Almaas and I. Brevik, “Radiation forces on a micrometer-sized sphere in an evanescent field,” J. Opt. Soc. Am. B 12(12), 2429–2438 (1995). [CrossRef]

,14

14. I. Brevik, T. A. Sivertsen, and E. Almaas, “Radiation forces on an absorbing micrometer-sized sphere in an evanescent field,” J. Opt. Soc. Am. B 20(8), 1739–1749 (2003). [CrossRef]

,18

18. J. Y. Walz, “Ray optics calculation of the radiation forces exerted on a dielectric sphere in an evanescent field,” Appl. Opt. 38(25), 5319–5330 (1999). [CrossRef] [PubMed]

]. Figure 4
Fig. 4 Dimensionless radiation force components Fx,z/P0 versus the particle-size parameter ka for a dielectric particle lying on a total-reflecting surface. The parameters of the particle and incident field are given by Eqs. (7), (10), (11), and (17). The cases of the s-polarization (solid lines) and p-polarization (dashed lines) are shown. The results of our calculations based on the complex-angle Mie theory Eqs. (12)(16) (red curves) are superimposed over the data taken from Figs. 8 and 9 of Ref [13]. (black curves).
shows the optical force components Fx and Fz versus the particle size parameter ka, calculated using the complex-angle Mie theory (6), (7), (10)–(16), i.e., via numerical evaluation of the scattered field (12), (13) and the force integral (16). These data are superimposed over the data obtained in [13

13. E. Almaas and I. Brevik, “Radiation forces on a micrometer-sized sphere in an evanescent field,” J. Opt. Soc. Am. B 12(12), 2429–2438 (1995). [CrossRef]

] using a considerably more complex theory. Evidently, there is an excellent agreement between the two approaches. (Small deviations can be attributed to the accuracy of numerical calculations and to graphic distortions in the printed copy of [13

13. E. Almaas and I. Brevik, “Radiation forces on a micrometer-sized sphere in an evanescent field,” J. Opt. Soc. Am. B 12(12), 2429–2438 (1995). [CrossRef]

].)

Second, we compare the results of our calculations of radiation forces for small particles (ka<˜0.1) with what follows from the known Rayleigh-scattering formulae for evanescent fields. From the equations derived in [21

21. M. Nieto-Vesperinas, P. C. Chaumet, and A. Rahmani, “Near-field photonic forces,” Philos. Transact. A Math. Phys. Eng. Sci. 362(1817), 719–737 (2004). [CrossRef] [PubMed]

23

23. M. Nieto-Vesperinas and J. R. Arias-González, “Theory of forces induced by evanescent fields,” arXiv: 1102.1613 (2011).

], it follows that for non-magnetic media and particles (μ1=μ=μp=1) the optical force components are given by
Fx=12[|E|2+|E|2(2cosh2α1)]ka3coshα×[Im(εpεεp+2ε)+23(ka)3|εpεεp+2ε|2]e2khsinhα,
(19)
Fz=12[|E|2+|E|2(2cosh2α1)]ka3sinhαRe(εpεεp+2ε)e2khsinhα.
(20)
The same expressions can be derived in the first electric-dipole-scattering approximation [i.e., keeping only the terms with the coefficient a1 of Eq. (A4)] of our complex-angle Mie theory Eqs. (12)(16). Taking the numerical values of the parameters from [23

23. M. Nieto-Vesperinas and J. R. Arias-González, “Theory of forces induced by evanescent fields,” arXiv: 1102.1613 (2011).

]:
n1=1.5,n=1,θ1=42°,λ=632.8nm,a=10nm(ka0.1),
(21)
(which yields sinhα0.086), we calculate the optical force components for various complex values of the particle permittivity εp. In Table 1

Table 1. Comparison of radiation forces for a particle with the parameters (21) and different permittivities εp, s-polarized incident wave, calculated using: (i) the dipole approximation [23], (19) and (20), and (ii) the exact complex-angle Mie theory (10)–(16).

table-icon
View This Table
we show the comparison of the forces obtained from the exact complex-angle Mie-scattering calculations and from the dipole-approximation Eqs. (19) and (20), both for the s-polarized incident wave. Evidently, the agreement is very good, with deviations within a few percent caused by the accuracy of the dipole approximation.

Thus, we have shown that the complex-angle Mie theory results for the radiation forces are fully consistent with other approaches and approximations. Now we demonstrate an application of the proposed theory to the study of optical forces in evanescent fields. The most interesting situations occur in the case of conducting particles. For dielectric particles, the force Fz is usually negative in the range of parameters considered [13

13. E. Almaas and I. Brevik, “Radiation forces on a micrometer-sized sphere in an evanescent field,” J. Opt. Soc. Am. B 12(12), 2429–2438 (1995). [CrossRef]

,14

14. I. Brevik, T. A. Sivertsen, and E. Almaas, “Radiation forces on an absorbing micrometer-sized sphere in an evanescent field,” J. Opt. Soc. Am. B 20(8), 1739–1749 (2003). [CrossRef]

,18

18. J. Y. Walz, “Ray optics calculation of the radiation forces exerted on a dielectric sphere in an evanescent field,” Appl. Opt. 38(25), 5319–5330 (1999). [CrossRef] [PubMed]

,20

20. J. R. Arias-González and M. Nieto-Vesperinas, “Optical forces on small particles: attractive and repulsive nature and plasmon-resonance conditions,” J. Opt. Soc. Am. A 20(7), 1201–1209 (2003). [CrossRef] [PubMed]

,23

23. M. Nieto-Vesperinas and J. R. Arias-González, “Theory of forces induced by evanescent fields,” arXiv: 1102.1613 (2011).

] (see Fig. 4), i.e. attracts a particle towards the surface. It was suggested in [14

14. I. Brevik, T. A. Sivertsen, and E. Almaas, “Radiation forces on an absorbing micrometer-sized sphere in an evanescent field,” J. Opt. Soc. Am. B 20(8), 1739–1749 (2003). [CrossRef]

] that the particle’s conductivity can be a source of the positive Fz; here we consider how this effect can be evaluated using our complex-angle Mie theory.

We first consider a gold particle in water, Fig. 5(a)
Fig. 5 Dimensionless radiation force components Fx (black curves) and Fz (red curves) versus the particle size parameter ka for the s-polarized (solid curves) and p-polarized (dashed curves) incident wave. The parameters are the same as in Eq. (17) but with (a) n=1.33, np=0.43+3.52i (gold particle in water at λ=650nm [3]) and (b) n=1, np=i (“perfect metal” particle).
. This case is characterized by a significant imaginary part of the permittivity εp=np2 [37

37. S. H. Simpson and S. Hanna, “Orbital motion of optically trapped particles in Laguerre-Gaussian beams,” J. Opt. Soc. Am. A 27(9), 2061–2071 (2010). [CrossRef] [PubMed]

,38

38. A. Y. Bekshaev, O. V. Angelsky, S. G. Hanson, and C. Y. Zenkova, “Scattering of inhomogeneous circularly polarized optical field and mechanical manifestation of the internal energy flows,” Phys. Rev. A 86(2), 023847 (2012). [CrossRef]

], which likely promotes high absolute values of the optical forces. At the same time, the considerable difference in refraction indices np and n contributes to the oscillatory behavior of the curves. Note that |Fx|>|Fz| for gold particles for almost the whole range of particle sizes. Furthermore, Fig. 5(a) shows the possibility of positive Fz for large enough particles ka>2 and s-polarized illumination. This tendency becomes dominant for a “perfect metal” particle with εp=1, Fig. 5(b), where the vertical force Fz is always positive. The perfect-metal model qualitatively represents optical properties of some well-conducting metals at frequencies below the plasmon resonance [2

2. M. Born and E. Wolf, Principles of Optics, 7th ed. (London: Pergamon, 2005).

]. All components of the normalized optical force exerted on conducting particles show a rather fast attenuation when the particle size increases, which can be attributed to the influence of absorption. The suppressed penetration of the radiation inside the particle and thus absence of the in-particle resonances is likely responsible for the fact that in Fig. 5(b) there are no oscillations, in contrast to the case of a dielectric particle (Fig. 4).

Finally, we characterize the physical origin of the optical forces considered above. In all cases, Fx>0, i.e., the force Fx is directed along the field momentum px (9). This enables us to associate the horizontal force with the surface energy flow of the evanescent field [the momentum component py (9) vanishes in the case of s- or p-polarizations]. At ka1, the horizontal force grows as Fxa6 in Fig. 4 and in Fig. 5(b), which is typical for the electromagnetic momentum action on particles with real polarizability in the dipole approximation [37

37. S. H. Simpson and S. Hanna, “Orbital motion of optically trapped particles in Laguerre-Gaussian beams,” J. Opt. Soc. Am. A 27(9), 2061–2071 (2010). [CrossRef] [PubMed]

,38

38. A. Y. Bekshaev, O. V. Angelsky, S. G. Hanson, and C. Y. Zenkova, “Scattering of inhomogeneous circularly polarized optical field and mechanical manifestation of the internal energy flows,” Phys. Rev. A 86(2), 023847 (2012). [CrossRef]

]. In contrast, Fxa3 in Fig. 5(a) due to the non-zero imaginary part of the complex polarizability εa3εpεεp+2ε [22

22. M. Nieto-Vesperinas and J. J. Saenz, “Optical forces from an evanescent wave on a magnetodielectric small particle,” Opt. Lett. 35(23), 4078–4080 (2010). [CrossRef] [PubMed]

,23

23. M. Nieto-Vesperinas and J. R. Arias-González, “Theory of forces induced by evanescent fields,” arXiv: 1102.1613 (2011).

,37

37. S. H. Simpson and S. Hanna, “Orbital motion of optically trapped particles in Laguerre-Gaussian beams,” J. Opt. Soc. Am. A 27(9), 2061–2071 (2010). [CrossRef] [PubMed]

] [see Eq. (19)]. At the same time, the vertical force shows a characteristic gradient-force behavior: Fza3 at ka1 [see Eq. (20)], which is not surprising since pz=0, and the force appears due to the inhomogeneous distribution of the energy density w (9).

5. Conclusion

Appendix A: Mie scattering formulas

Here we collect formulas of the standard Mie theory. We mostly follow [4

4. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (New York: Wiley, 1983).

] but modify and adapt the Mie equations with special attention to the scattering near field and to the radial components which are typically omitted. As usual, it is assumed that a spherical particle of radius a (placed at the origin) with electromagnetic parameters εp, μp, np=εpμp scatters the incident z-propagating plane wave (1) in a medium with parameters ε, μ, and n=εμ. Using spherical coordinates (θ,ϕ,r) introduced with respect to the (x,y,z) coordinates (see Fig. 1), the scattering is described with respect to planes determined by the azimuthal cross-sections ϕ=const. In this manner, the in-plane and out-of-plane components of the amplitudes of the incident field (1) are re-defined as

(E˜E˜)=(cosϕsinϕsinϕcosϕ)(EE),(H˜H˜)=εμ(cosϕsinϕsinϕcosϕ)(EE)
(A1)

In spherical coordinates, the components of the scattered fields, Es and Hs, are calculated via the following expansions:
Eθs=E˜1r=1A(iaξτbξπ),Hθs=E˜1rεμ=1A(ibξτaξπ),Eϕs=E˜1r=1A(bξτiaξπ),Hϕs=E˜1rεμ=1A(ibξπaξτ),Ers=E˜sinθ1r2=1A(+1)iaξπ,Hrs=E˜sinθ1r2εμ=1A(+1)ibξπ.
(A2)
Here A=i2+1(+1), and each term in the sums describes a certain order of multipole radiation, namely, the a- and b-terms represent the electric and magnetic 2-poles located at the origin. The radial and polar dependences of the solutions Eq. (A2) are contained in the functions
ξ(kr)=krh(1)(kr),ξ(kr)=d[krh(1)(kr)]d(kr),π(cosθ)=P1(cosθ)sinθ,τ(cosθ)=dP1(cosθ)dθ.
(A3)
where h(1)(u) are the spherical Hankel functions and P1(u) are the adjoint Legendre polynomials (well-defined in the complex domain). The scattering coefficients a(χ) and b(χ) depend on the dimensionless particle radius χ=ka and are given by
a=mεψ(mχ)ψ(χ)mμψ(χ)ψ(mχ)mεψ(mχ)ξ(χ)mμξ(χ)ψ(mχ),b=mμψ(mχ)ψ(χ)mεψ(χ)ψ(mχ)mμψ(mχ)ξ(χ)mεξ(χ)ψ(mχ).
(A4)
Here the following parameters and functions are used:
mε=εpε,mμ=μpμ,m=npn,ψ(u)=uj(u),ψ(u)=d[uj(u)]du,
(A5)
where j(u) are the spherical Bessel functions.

Finally, the Cartesian components of the scattered field are obtained via the standard rotational transformation connecting Cartesian and spherical coordinates:
Es=(ExsEysEzs)=R^(θ,ϕ)(EθsEϕsErs),Hs=(HxsHysHzs)=R^(θ,ϕ)(HθsHϕsHrs),R^(θ,ϕ)=R^z(ϕ)R^y(θ)=(cosθcosϕsinϕsinθcosϕcosθsinϕcosϕsinθsinϕsinθ0cosθ).
(A6)
Equations (1), (A1)(A6) represent the standard Mie scattering solution establishing the linear relation (12) between the scattered and incident (z-propagating plane wave) fields.

Acknowledgments

This work was supported by the European Commission (Marie Curie Action), ARO, JSPS-RFBR contract No. 12-02-92100, Grant-in-Aid for Scientific Research (S), MEXT Kakenhi on Quantum Cybernetics, and the JSPS via its FIRST program.

References and links

1.

G. Mie, “Beiträge zur Optik trüber Medien, speziell kolloidaler Metallösungen,” Ann. Phys. (Leipzig) 25, 377–445 (1908).

2.

M. Born and E. Wolf, Principles of Optics, 7th ed. (London: Pergamon, 2005).

3.

H. C. Van de Hulst, Light Scattering by Small Particles (New York: Chapman & Hall, 1957).

4.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (New York: Wiley, 1983).

5.

M. Dienerowitz, M. Mazilu, and K. Dholakia, “Optical manipulation of nanoparticles: a review,” J. Nanophotonics 2(1), 021875 (2008). [CrossRef]

6.

I. Brevik, “Experiments in phenomenological electrodynamics and the electromagnetic energy-momentum tensor,” Phys. Rep. 52(3), 133–201 (1979). [CrossRef]

7.

M. A. Paesler and P. J. Moyer, Near–Field Optics (New York: John Wiley & Sons, 1996).

8.

M. Nieto-Vesperinas and N. Garcia, eds., Optics at the nanometer scale, NATO ASI Series (Dordrecht: Kluwer Academic Publishing, 1996).

9.

S. A. Maier, Plasmonics: Fundamentals and Applications (New York: Springer, 2007).

10.

H. Chew, D.-S. Wang, and M. Kerker, “Elastic scattering of evanescent electromagnetic waves,” Appl. Opt. 18(15), 2679–2687 (1979). [CrossRef] [PubMed]

11.

C. Liu, T. Kaiser, S. Lange, and G. Schweiger, “Structural resonances in a dielectric sphere illuminated by an evanescent wave,” Opt. Commun. 117(5-6), 521–531 (1995). [CrossRef]

12.

M. Quinten, A. Pack, and R. Wannemacher, “Scattering and extinction of evanescent waves by small particles,” Appl. Phys. B 68(1), 87–92 (1999). [CrossRef]

13.

E. Almaas and I. Brevik, “Radiation forces on a micrometer-sized sphere in an evanescent field,” J. Opt. Soc. Am. B 12(12), 2429–2438 (1995). [CrossRef]

14.

I. Brevik, T. A. Sivertsen, and E. Almaas, “Radiation forces on an absorbing micrometer-sized sphere in an evanescent field,” J. Opt. Soc. Am. B 20(8), 1739–1749 (2003). [CrossRef]

15.

H. Y. Jaising and O. G. Hellesø, “Radiation forces on a Mie particle in the evanescent field of an optical waveguide,” Opt. Commun. 246(4-6), 373–383 (2005). [CrossRef]

16.

S. Chang, J. T. Kim, J. H. Jo, and S. S. Lee, “Optical force on a sphere caused by the evanescent field of a Gaussian beam; effects of multiple scattering,” Opt. Commun. 139(4-6), 252–261 (1997). [CrossRef]

17.

Y. G. Song, B. M. Han, and S. Chang, “Force of surface plasmon-coupled evanescent fields on Mie particles,” Opt. Commun. 198(1-3), 7–19 (2001). [CrossRef]

18.

J. Y. Walz, “Ray optics calculation of the radiation forces exerted on a dielectric sphere in an evanescent field,” Appl. Opt. 38(25), 5319–5330 (1999). [CrossRef] [PubMed]

19.

P. C. Chaumet and M. Nieto-Vesperinas, “Electromagnetic force on a metallic particle in the presence of a dielectric surface,” Phys. Rev. B 62(16), 11185–11191 (2000). [CrossRef]

20.

J. R. Arias-González and M. Nieto-Vesperinas, “Optical forces on small particles: attractive and repulsive nature and plasmon-resonance conditions,” J. Opt. Soc. Am. A 20(7), 1201–1209 (2003). [CrossRef] [PubMed]

21.

M. Nieto-Vesperinas, P. C. Chaumet, and A. Rahmani, “Near-field photonic forces,” Philos. Transact. A Math. Phys. Eng. Sci. 362(1817), 719–737 (2004). [CrossRef] [PubMed]

22.

M. Nieto-Vesperinas and J. J. Saenz, “Optical forces from an evanescent wave on a magnetodielectric small particle,” Opt. Lett. 35(23), 4078–4080 (2010). [CrossRef] [PubMed]

23.

M. Nieto-Vesperinas and J. R. Arias-González, “Theory of forces induced by evanescent fields,” arXiv: 1102.1613 (2011).

24.

S. Kawata and T. Sugiura, “Movement of micrometer-sized particles in the evanescent field of a laser beam,” Opt. Lett. 17(11), 772–774 (1992). [CrossRef] [PubMed]

25.

S. Kawata and T. Tani, “Optically driven Mie particles in an evanescent field along a channeled waveguide,” Opt. Lett. 21(21), 1768–1770 (1996). [CrossRef] [PubMed]

26.

M. Vilfan, I. Muševič, and M. Čopič, “AFM observation of force on a dielectric sphere in the evanescent field of totally reflected light,” Europhys. Lett. 43(1), 41–46 (1998). [CrossRef]

27.

K. Sasaki, J.-I. Hotta, K.-I. Wada, and H. Masuhara, “Analysis of radiation pressure exerted on a metallic particle within an evanescent field,” Opt. Lett. 25(18), 1385–1387 (2000). [CrossRef] [PubMed]

28.

L. N. Ng, M. N. Zervas, J. S. Wilkinson, and B. J. Luff, “Manipulation of colloidal gold nanoparticles in the evanescent field of a channel waveguide,” Appl. Phys. Lett. 76(15), 1993–1995 (2000). [CrossRef]

29.

G. Volpe, R. Quidant, G. Badenes, and D. Petrov, “Surface plasmon radiation forces,” Phys. Rev. Lett. 96(23), 238101 (2006). [CrossRef] [PubMed]

30.

M. Šiler, T. Čižmár, M. Šerý, and P. Zemánek, “Optical forces generated by evanescent standing waves and their usage for sub-micron particle delivery,” Appl. Phys. B 84(1-2), 157–165 (2006). [CrossRef]

31.

S. Gaugiran, S. Gétin, J. M. Fedeli, and J. Derouard, “Polarization and particle size dependence of radiative forces on small metallic particles in evanescent optical fields. Evidences for either repulsive or attractive gradient forces,” Opt. Express 15(13), 8146–8156 (2007). [CrossRef] [PubMed]

32.

D. C. Prieve and J. Y. Walz, “Scattering of an evanescent surface wave by a microscopic dielectric sphere,” Appl. Opt. 32(9), 1629–1641 (1993). [CrossRef] [PubMed]

33.

A. Bekshaev, K. Bliokh, and M. Soskin, “Internal flows and energy circulation in light beams,” J. Opt. 13(5), 053001 (2011). [CrossRef]

34.

K. Y. Bliokh and F. Nori, “Transverse spin of a surface polariton,” Phys. Rev. A 85(6), 061801 (2012). [CrossRef]

35.

F. I. Fedorov, “To the theory of total reflection,” Dokl. Akad. Nauk SSSR 105, 465–468 (1955) (reprinted in J. Opt. 15, 014002 (2013)).

36.

C. Imbert, “Calculation and experimental proof of the transverse shift induced by total internal reflection of a circularly polarized light beam,” Phys. Rev. D Part. Fields 5(4), 787–796 (1972). [CrossRef]

37.

S. H. Simpson and S. Hanna, “Orbital motion of optically trapped particles in Laguerre-Gaussian beams,” J. Opt. Soc. Am. A 27(9), 2061–2071 (2010). [CrossRef] [PubMed]

38.

A. Y. Bekshaev, O. V. Angelsky, S. G. Hanson, and C. Y. Zenkova, “Scattering of inhomogeneous circularly polarized optical field and mechanical manifestation of the internal energy flows,” Phys. Rev. A 86(2), 023847 (2012). [CrossRef]

OCIS Codes
(240.6690) Optics at surfaces : Surface waves
(290.4020) Scattering : Mie theory
(350.4855) Other areas of optics : Optical tweezers or optical manipulation

ToC Category:
Optical Trapping and Manipulation

History
Original Manuscript: November 27, 2012
Revised Manuscript: February 6, 2013
Manuscript Accepted: February 6, 2013
Published: March 13, 2013

Virtual Issues
Vol. 8, Iss. 4 Virtual Journal for Biomedical Optics

Citation
Aleksandr Y. Bekshaev, Konstantin Y. Bliokh, and Franco Nori, "Mie scattering and optical forces from evanescent fields: A complex-angle approach," Opt. Express 21, 7082-7095 (2013)
http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=oe-21-6-7082


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References

  1. G. Mie, “Beiträge zur Optik trüber Medien, speziell kolloidaler Metallösungen,” Ann. Phys. (Leipzig) 25, 377–445 (1908).
  2. M. Born and E. Wolf, Principles of Optics, 7th ed. (London: Pergamon, 2005).
  3. H. C. Van de Hulst, Light Scattering by Small Particles (New York: Chapman & Hall, 1957).
  4. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (New York: Wiley, 1983).
  5. M. Dienerowitz, M. Mazilu, and K. Dholakia, “Optical manipulation of nanoparticles: a review,” J. Nanophotonics 2(1), 021875 (2008). [CrossRef]
  6. I. Brevik, “Experiments in phenomenological electrodynamics and the electromagnetic energy-momentum tensor,” Phys. Rep. 52(3), 133–201 (1979). [CrossRef]
  7. M. A. Paesler and P. J. Moyer, Near–Field Optics (New York: John Wiley & Sons, 1996).
  8. M. Nieto-Vesperinas and N. Garcia, eds., Optics at the nanometer scale, NATO ASI Series (Dordrecht: Kluwer Academic Publishing, 1996).
  9. S. A. Maier, Plasmonics: Fundamentals and Applications (New York: Springer, 2007).
  10. H. Chew, D.-S. Wang, and M. Kerker, “Elastic scattering of evanescent electromagnetic waves,” Appl. Opt. 18(15), 2679–2687 (1979). [CrossRef] [PubMed]
  11. C. Liu, T. Kaiser, S. Lange, and G. Schweiger, “Structural resonances in a dielectric sphere illuminated by an evanescent wave,” Opt. Commun. 117(5-6), 521–531 (1995). [CrossRef]
  12. M. Quinten, A. Pack, and R. Wannemacher, “Scattering and extinction of evanescent waves by small particles,” Appl. Phys. B 68(1), 87–92 (1999). [CrossRef]
  13. E. Almaas and I. Brevik, “Radiation forces on a micrometer-sized sphere in an evanescent field,” J. Opt. Soc. Am. B 12(12), 2429–2438 (1995). [CrossRef]
  14. I. Brevik, T. A. Sivertsen, and E. Almaas, “Radiation forces on an absorbing micrometer-sized sphere in an evanescent field,” J. Opt. Soc. Am. B 20(8), 1739–1749 (2003). [CrossRef]
  15. H. Y. Jaising and O. G. Hellesø, “Radiation forces on a Mie particle in the evanescent field of an optical waveguide,” Opt. Commun. 246(4-6), 373–383 (2005). [CrossRef]
  16. S. Chang, J. T. Kim, J. H. Jo, and S. S. Lee, “Optical force on a sphere caused by the evanescent field of a Gaussian beam; effects of multiple scattering,” Opt. Commun. 139(4-6), 252–261 (1997). [CrossRef]
  17. Y. G. Song, B. M. Han, and S. Chang, “Force of surface plasmon-coupled evanescent fields on Mie particles,” Opt. Commun. 198(1-3), 7–19 (2001). [CrossRef]
  18. J. Y. Walz, “Ray optics calculation of the radiation forces exerted on a dielectric sphere in an evanescent field,” Appl. Opt. 38(25), 5319–5330 (1999). [CrossRef] [PubMed]
  19. P. C. Chaumet and M. Nieto-Vesperinas, “Electromagnetic force on a metallic particle in the presence of a dielectric surface,” Phys. Rev. B 62(16), 11185–11191 (2000). [CrossRef]
  20. J. R. Arias-González and M. Nieto-Vesperinas, “Optical forces on small particles: attractive and repulsive nature and plasmon-resonance conditions,” J. Opt. Soc. Am. A 20(7), 1201–1209 (2003). [CrossRef] [PubMed]
  21. M. Nieto-Vesperinas, P. C. Chaumet, and A. Rahmani, “Near-field photonic forces,” Philos. Transact. A Math. Phys. Eng. Sci. 362(1817), 719–737 (2004). [CrossRef] [PubMed]
  22. M. Nieto-Vesperinas and J. J. Saenz, “Optical forces from an evanescent wave on a magnetodielectric small particle,” Opt. Lett. 35(23), 4078–4080 (2010). [CrossRef] [PubMed]
  23. M. Nieto-Vesperinas and J. R. Arias-González, “Theory of forces induced by evanescent fields,” arXiv: 1102.1613 (2011).
  24. S. Kawata and T. Sugiura, “Movement of micrometer-sized particles in the evanescent field of a laser beam,” Opt. Lett. 17(11), 772–774 (1992). [CrossRef] [PubMed]
  25. S. Kawata and T. Tani, “Optically driven Mie particles in an evanescent field along a channeled waveguide,” Opt. Lett. 21(21), 1768–1770 (1996). [CrossRef] [PubMed]
  26. M. Vilfan, I. Muševi?, and M. ?opi?, “AFM observation of force on a dielectric sphere in the evanescent field of totally reflected light,” Europhys. Lett. 43(1), 41–46 (1998). [CrossRef]
  27. K. Sasaki, J.-I. Hotta, K.-I. Wada, and H. Masuhara, “Analysis of radiation pressure exerted on a metallic particle within an evanescent field,” Opt. Lett. 25(18), 1385–1387 (2000). [CrossRef] [PubMed]
  28. L. N. Ng, M. N. Zervas, J. S. Wilkinson, and B. J. Luff, “Manipulation of colloidal gold nanoparticles in the evanescent field of a channel waveguide,” Appl. Phys. Lett. 76(15), 1993–1995 (2000). [CrossRef]
  29. G. Volpe, R. Quidant, G. Badenes, and D. Petrov, “Surface plasmon radiation forces,” Phys. Rev. Lett. 96(23), 238101 (2006). [CrossRef] [PubMed]
  30. M. Šiler, T. ?ižmár, M. Šerý, and P. Zemánek, “Optical forces generated by evanescent standing waves and their usage for sub-micron particle delivery,” Appl. Phys. B 84(1-2), 157–165 (2006). [CrossRef]
  31. S. Gaugiran, S. Gétin, J. M. Fedeli, and J. Derouard, “Polarization and particle size dependence of radiative forces on small metallic particles in evanescent optical fields. Evidences for either repulsive or attractive gradient forces,” Opt. Express 15(13), 8146–8156 (2007). [CrossRef] [PubMed]
  32. D. C. Prieve and J. Y. Walz, “Scattering of an evanescent surface wave by a microscopic dielectric sphere,” Appl. Opt. 32(9), 1629–1641 (1993). [CrossRef] [PubMed]
  33. A. Bekshaev, K. Bliokh, and M. Soskin, “Internal flows and energy circulation in light beams,” J. Opt. 13(5), 053001 (2011). [CrossRef]
  34. K. Y. Bliokh and F. Nori, “Transverse spin of a surface polariton,” Phys. Rev. A 85(6), 061801 (2012). [CrossRef]
  35. F. I. Fedorov, “To the theory of total reflection,” Dokl. Akad. Nauk SSSR 105, 465–468 (1955) (reprinted in J. Opt. 15, 014002 (2013)).
  36. C. Imbert, “Calculation and experimental proof of the transverse shift induced by total internal reflection of a circularly polarized light beam,” Phys. Rev. D Part. Fields 5(4), 787–796 (1972). [CrossRef]
  37. S. H. Simpson and S. Hanna, “Orbital motion of optically trapped particles in Laguerre-Gaussian beams,” J. Opt. Soc. Am. A 27(9), 2061–2071 (2010). [CrossRef] [PubMed]
  38. A. Y. Bekshaev, O. V. Angelsky, S. G. Hanson, and C. Y. Zenkova, “Scattering of inhomogeneous circularly polarized optical field and mechanical manifestation of the internal energy flows,” Phys. Rev. A 86(2), 023847 (2012). [CrossRef]

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