Spin-to-orbital angular momentum conversion in focusing, scattering, and imaging systems

doi:10.1364/OE.19.026132

Spin-to-orbital angular momentum conversion in focusing, scattering, and imaging systems

Konstantin Y. Bliokh, Elena A. Ostrovskaya, Miguel A. Alonso, Oscar G. Rodríguez-Herrera, David Lara, and Chris Dainty »View Author Affiliations

Optics Express, Vol. 19, Issue 27, pp. 26132-26149 (2011)
http://dx.doi.org/10.1364/OE.19.026132

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Abstract

We present a general theory of spin-to-orbital angular momentum (AM) conversion of light in focusing, scattering, and imaging optical systems. Our theory employs universal geometric transformations of non-paraxial optical fields in such systems and allows for direct calculation and comparison of the AM conversion efficiency in different physical settings. Observations of the AM conversions using local intensity distributions and far-field polarimetric measurements are discussed.

1. Introduction

Together with energy and momentum, angular momentum (AM) is one of the most important dynamical characteristics of light [1

L. Allen, S. M. Barnett, and M. J. Padgett, eds., Optical angular momentum (Taylor and Francis, 2003).

S. Franke-Arnold, L. Allen, and M. Padgett, “Advances in optical angular momentum,” Laser & Photon. Rev. 2(4), 299–313 (2008). [CrossRef]

]. For paraxial fields in free space, the eigenmodes of the AM operator are circularly-polarized vortex beams, where polarization helicity

σ = \pm 1

specifies the value of spin AM (SAM) per photon, whereas the vortex charge

ℓ = 0, \pm 1, \pm 2, ...

yields the orbital AM (OAM) per photon. Generation of the helicity-dependent vortices in optical systems signifies the spin-to-orbital AM conversion. This phenomenon is attracting noticeable attention in recent years and occurs in two basic situations: (A) upon interaction of paraxial light with anisotropic media possessing certain azimuthal symmetries [3

G. Biener, A. Niv, V. Kleiner, and E. Hasman, “Formation of helical beams by use of Pancharatnam-Berry phase optical elements,” Opt. Lett. 27(21), 1875–1877 (2002). [CrossRef] [PubMed]

–8

M. Y. Darsht, B. Y. Zel’dovich, I. V. Kataevskaya, and N. D. Kundikova, “Formation of an isolated wavefront dislocation,” Zh. Eksp. Theor. Fiz. 107, 1464 [JETP 80, 817 (1995)].

] and (B) in essentially non-paraxial optical fields in locally-isotropic media [8

M. Y. Darsht, B. Y. Zel’dovich, I. V. Kataevskaya, and N. D. Kundikova, “Formation of an isolated wavefront dislocation,” Zh. Eksp. Theor. Fiz. 107, 1464 [JETP 80, 817 (1995)].

–24

Y. Gorodetski, S. Nechayev, V. Kleiner, and E. Hasman, “Plasmonic Aharonov-Bohm effect: Optical spin as the magnetic flux parameter,” Phys. Rev. B 82(12), 125433 (2010). [CrossRef]

]. (Optical fibers [8

M. Y. Darsht, B. Y. Zel’dovich, I. V. Kataevskaya, and N. D. Kundikova, “Formation of an isolated wavefront dislocation,” Zh. Eksp. Theor. Fiz. 107, 1464 [JETP 80, 817 (1995)].

] can be regarded as an intermediate case.) In the case (B), nonparaxial AM states of light appear mostly upon (i) tight focusing [9

Z. Bomzon, M. Gu, and J. Shamir, “Angular momentum and geometric phases in tightly-focused circularly polarized plane waves,” Appl. Phys. Lett. 89(24), 241104 (2006). [CrossRef]

–17

M. R. Foreman and P. Török, “Spin-orbit coupling and conservation of angular momentum flux in non-paraxial imaging of forbidden radiation,” New J. Phys. 13(6), 063041 (2011). [CrossRef]

], (ii) scattering by small particles or apertures [16

O. G. Rodríguez-Herrera, D. Lara, K. Y. Bliokh, E. A. Ostrovskaya, and C. Dainty, “Optical nanoprobing via spin-orbit interaction of light,” Phys. Rev. Lett. 104(25), 253601 (2010). [CrossRef] [PubMed]

–24

Y. Gorodetski, S. Nechayev, V. Kleiner, and E. Hasman, “Plasmonic Aharonov-Bohm effect: Optical spin as the magnetic flux parameter,” Phys. Rev. B 82(12), 125433 (2010). [CrossRef]

], and (iii) in high numerical aperture (NA) imaging of small particles [16

,17

M. R. Foreman and P. Török, “Spin-orbit coupling and conservation of angular momentum flux in non-paraxial imaging of forbidden radiation,” New J. Phys. 13(6), 063041 (2011). [CrossRef]

]. Importantly, the case (iii), involving a combination of focusing and scattering processes, represents a fundamental mechanism for translating the fine spin-orbit effects at micro- and nano-scales to the far-field.

The spin-to-orbital AM conversion in anisotropic paraxial systems is an extrinsic phenomenon produced by the azimuthally-dependent phase difference between ordinary and extraordinary modes, which is well-studied and reviewed [25

E. Hasman, G. Biener, A. Niv, and V. Kleiner, “Space-variant polarization manipulation,” Prog. Opt. 47, 215–289 (2005). [CrossRef]

,26

L. Marrucci, E. Karimi, S. Slussarenko, B. Piccirillo, E. Santamato, E. Nagali, and F. Sciarrino, “Spin-to-orbital conversion of the angular momentum of light and its classical and quantum applications,” J. Opt. 13(6), 064001 (2011). [CrossRef]

]. On the other hand, the AM conversion in nonparaxial fields owes its origin to the intrinsic properties of light, geometric Berry phases, and fundamental separation of the SAM and OAM in the generic nonparaxial case [27

S. M. Barnett and L. Allen, “Orbital angular-momentum and nonparaxial light-beams,” Opt. Commun. 110(5-6), 670–678 (1994). [CrossRef]

–29

K. Y. Bliokh, M. A. Alonso, E. A. Ostrovskaya, and A. Aiello, “Angular momenta and spin-orbit interaction of nonparaxial light in free space,” Phys. Rev. A 82(6), 063825 (2010). [CrossRef]

]. The spin-to-orbital conversion in nonparaxial light has been considered for different systems using different ad hoc methods, such as Debye-Wolf theory for focusing or Mie theory for scattering. Furthermore, the spin-orbit interaction in a variety of similar imaging schemes is ascribed either to focusing [30

N. B. Baranova, A. Y. Savchenko, and B. Y. Zel'dovich, “Transverse shift of a focal spot due to switching of the sign of circular-polarization,” JETP Lett. 59, 232–234 (1994).

,31

B. Y. Zel’dovich, N. D. Kundikova, and L. F. Rogacheva, “Observed transverse shift of a focal spot upon a change in the sign of circular polarization,” JETP Lett. 59, 766–769 (1994).

], or to scattering [32

V. Garbin, G. Volpe, E. Ferrari, M. Versluis, D. Cojoc, and D. Petrov, “Mie scattering distinguishes the topological charge of an optical vortex: a homage to Gustav Mie,” New J. Phys. 11(1), 013046 (2009). [CrossRef]

], or to anisotropy [33

E. Brasselet, N. Murazawa, H. Misawa, and S. Juodkazis, “Optical vortices from liquid crystal droplets,” Phys. Rev. Lett. 103(10), 103903 (2009). [CrossRef] [PubMed]

,34

F. Manni, K. Lagoudakis, T. Paraïso, R. Cerna, Y. Léger, T. Liew, I. Shelykh, A. Kavokin, F. Morier-Genoud, and B. Deveaud-Plédran, “Spin-to-orbital angular momentum conversion in semiconductor microcavities,” Phys. Rev. B 83(24), 241307 (2011). [CrossRef]

]. Obviously, all these mechanisms co-exist and their unifying description and discrimination is necessary.

In the present paper we examine the spin-to-orbital AM conversion that appears in nonparaxial optical fields interacting with locally-isotropic media upon (i) focusing, (ii) scattering, and (iii) imaging. We develop a unifying theory of these effects based on universal, purely geometrical transformations of the fields and fundamental AM operators. Our approach highlights the common geometric origin of AM conversion due to different optical processes and allows us to compare the conversion efficiency in different physical settings depending on the aperture angles and properties of the incoming light.

2. Basic equations

We consider monochromatic wave electric fields

� (r, t)

characterized by their complex amplitudes

E (r)

� (r, t) = Re [E (r) e^{- i ω t}]

. Similar relations are implied when the wave magnetic field

H (r)

is involved. The Fourier or angular spectra of the fields are denoted by

\tilde{E} (k)

and

\tilde{H} (k)

. Throughout the paper we consider optical systems with axial symmetry about the

z

-axis, and, for the AM description, it is convenient to use the basis of circular polarizations with respect to the optical axis. Given that

(u_{x}, u_{y}, u_{z})

are the basic vectors of the laboratory Cartesian frame and

E_{L} = {(E_{x}, E_{y}, E_{z})}^{T}

is the vector of complex amplitudes of a monochromatic electric field in this basis, the basic vectors and electric field components in the circular basis are:

u^{\pm} = \frac{u_{x} \pm i u_{y}}{\sqrt{2}}, E^{\pm} = \frac{E_{x} \mp i E_{y}}{\sqrt{2}},

(1)

so that

E_{C} = {(E^{+}, E^{-}, E_{z})}^{T}

. Thus, transition from the Cartesian to circular basis is realized via the unitary transformation

E_{L} = \hat{V} E_{C}, \hat{V} = \frac{1}{\sqrt{2}} (\begin{matrix} 1 & 1 & 0 \\ i & - i & 0 \\ 0 & 0 & \sqrt{2} \end{matrix}) .

(2)

Throughout the paper we mostly analyze the fields in the circular basis, and the subscripts

C

are omitted.

The operators of the

z

-component of the OAM and SAM of light in the Cartesian basis are [1

L. Allen, S. M. Barnett, and M. J. Padgett, eds., Optical angular momentum (Taylor and Francis, 2003).

{\hat{L}}_{z} = - i \partial / \partial ϕ

and

{({\hat{S}}_{z})}_{i j} = - i ε_{z i j}

, where

ϕ

is the azimuthal angle (either in coordinate or momentum space, depending on representation) and

ε_{i j l}

is the Levi-Civita symbol. In the circular basis, these operator become, correspondingly

{\hat{L}}_{z} = - i \frac{\partial}{\partial ϕ}, {\hat{V}}^{†} {\hat{S}}_{z} \hat{V} = \hat{σ} = diag (1, - 1, 0) .

(3)

Note that the vortex fields

E \propto \exp (i ℓ ϕ)

are eigenmodes of

{\hat{L}}_{z}

with the eigenvalues

ℓ

. At the same time, the circularly polarized paraxial field

E^{+} \propto (1, 0, 0) \equiv e^{+}

and

E^{-} \propto (0, 1, 0) \equiv e^{-}

are eigenmodes of

\hat{σ}

with the eigenvalues

σ = \pm 1

, whereas the third eigenvector of

\hat{σ}

corresponds to the

z

-polarized field,

E_{z} \propto (0, 0, 1)

, with the eigenvalue

σ = 0

. There are some fundamental mathematical difficulties in using canonical operators

\hat{L}

and

\hat{S}

, Eq. (3), for generic nonparaxial fields [29

K. Y. Bliokh, M. A. Alonso, E. A. Ostrovskaya, and A. Aiello, “Angular momenta and spin-orbit interaction of nonparaxial light in free space,” Phys. Rev. A 82(6), 063825 (2010). [CrossRef]

,35

S. J. van Enk and G. Nienhuis, “Spin and orbital angular momentum of photons,” Europhys. Lett. 25(7), 497–501 (1994). [CrossRef]

,36

S. J. van Enk and G. Nienhuis, “Commutation rules and eigenvalues of spin and orbital angular momentum of radiation fields,” J. Mod. Opt. 41(5), 963–977 (1994). [CrossRef]

], but they do not affect the OAM and SAM expectation values which will be calculated for different optical systems below.

3. High-NA focusing

Let us consider the Debye–Wolf theory of focusing with a spherical lens [37

E. Wolf, ““Electromagnetic diffraction in optical systems. I. An integral representation of the image field,” Proc. Roy. Soc. London,” Ser. A 253, 349–357 (1959).

,38

B. Richards and E. Wolf, ““Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system,” Proc. R. Soc. Lond. A Math. Phys. Sci. 253(1274), 358–379 (1959). [CrossRef]

], Fig. 1 . The incident field

E_{0} (r)

is paraxial, and one can neglect its

z

-component:

E_{0} ≃ {(E_{0}^{+}, E_{0}^{-}, 0)}^{T}

. Entering partial rays with the wave vector

k_{0} ≃ k u_{z}

are marked by coordinates at the entrance pupil, which can be expressed via spherical angles

(\tilde{θ}, \tilde{ϕ})

defined with respect to the origin in the focal point:

x = f \sin \tilde{θ} \cos \tilde{ϕ}

y = f \sin \tilde{θ} \sin \tilde{ϕ}

, and

z = f \cos \tilde{θ}

(

f

is the focal distance of the lens). After refraction, the partial rays converge at the focal point and have the nonparaxial k-vectors characterized by spherical coordinates

(θ, ϕ) = (π - \tilde{θ}, π + \tilde{ϕ})

in the momentum space:

k_{x} = k \sin θ \cos ϕ

k_{y} = k \sin θ \sin ϕ

, and

k_{z} = k \cos θ

(see Fig. 1) [38

]. The rays are distributed in the range

{θ \in (0, θ_{c}), ϕ \in (0, 2 π)}

, where

θ_{c}

is the aperture angle of the lens, and carry the electric fields

\tilde{E} (k)

. Thus,

k_{x} = - k \frac{x}{f}, k_{y} = - k \frac{y}{f}, k_{z} = - k \frac{z}{f},

(4)

and the lens performs a sort of Fourier transform translating the initial real-space distribution

E_{0} (r)

into the momentum distribution

\tilde{E} (k)

in the image space. In doing so, the angles

(θ, ϕ)

serve both as real-space coordinates for the incident field

E_{0}

and momentum-space coordinates for the refracted field

\tilde{E}

Fig. 1 Focusing of light by a spherical high-NA lens. The incident paraxial field

E_{0} (x, y)

is refracted in the meridional plane and transformed to a spectrum of plane waves (rays)

\tilde{E} (θ, ϕ)

with the

k

-vectors distributed on the sphere in k-space:

θ \in (0, θ_{c})

ϕ \in (0, 2 π)

. The electric field vectors are parallel-transported along each partial ray with the local helicity being conserved.

The Debye–Wolf theory assumes that partial waves do not change their polarization state in the local basis attached to the ray, and the electric fields experience pure meridional rotations by the angle

θ

together with their k-vectors. This is an adiabatic approximation which neglects the polarization dependence of the refraction coefficients, cf [39

M. V. Berry, “Interpreting the anholonomy of coiled light,” Nature 326(6110), 277–278 (1987). [CrossRef]

]. As a result, the focused field spectrum

\tilde{E} (k)

can be written using purely geometrical rotational transformation [16

,40

P. Török, P. D. Higdon, and T. Wilson, “On the general properties of polarized light conventional and confocal microscopes,” Opt. Commun. 148(4-6), 300–315 (1998). [CrossRef]

]

\hat{U} (θ, ϕ) = {\hat{V}}^{†} {\hat{R}}_{z} (- ϕ) {\hat{R}}_{y} (- θ) {\hat{R}}_{z} (ϕ) \hat{V}

, where

{\hat{R}}_{a} (α) = \exp (i α {\hat{S}}_{a})

a = x, y, z

, is the matrix of rotation about the a-axis by the angle

α

[

{({\hat{S}}_{a})}_{i j} = - i ε_{a i j}

are the SO(3) generators of rotations, i.e., spin-1 matrices]. Explicitly, the transformation of the field in the circular basis takes the form [16

,41

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

\tilde{E} \propto \sqrt{\cos θ} \hat{U} (θ, ϕ) E_{0}, \hat{U} (θ, ϕ) = (\begin{matrix} a & - b e^{- 2 i ϕ} & \sqrt{2 a b} e^{- i ϕ} \\ - b e^{2 i ϕ} & a & \sqrt{2 a b} e^{i ϕ} \\ - \sqrt{2 a b} e^{i ϕ} & - \sqrt{2 a b} e^{- i ϕ} & a - b \end{matrix}),

(5)

where

a = \cos^{2} (θ / 2)

b = \sin^{2} (θ / 2)

, and

\sqrt{\cos θ}

is the apodization factor that ensures the conservation of the energy flow [38

]. Remarkably, the same unitary transformation

\hat{U} (θ, ϕ)

describes transition from the global circular basis (1) to the local helicity basis attached to the partial wave vector for a generic nonparaxial field [29

K. Y. Bliokh, M. A. Alonso, E. A. Ostrovskaya, and A. Aiello, “Angular momenta and spin-orbit interaction of nonparaxial light in free space,” Phys. Rev. A 82(6), 063825 (2010). [CrossRef]

]. This reflects the fact that the refracted field does not change its polarization state in the helicity basis (labelled by subscript “H”):

{\tilde{E}}_{H} = {\hat{U}}^{†} \tilde{E} \propto E_{0} ≃ E_{0}_{H} .

(6)

It follows from here that the high-NA focusing of a paraxial circularly-polarized field generates a nonparaxial pure helicity state of light. Recently, some of us considered nonparaxial vector Bessel beams with well-defined helicity [29

K. Y. Bliokh, M. A. Alonso, E. A. Ostrovskaya, and A. Aiello, “Angular momenta and spin-orbit interaction of nonparaxial light in free space,” Phys. Rev. A 82(6), 063825 (2010). [CrossRef]

], which could be generated, e.g., via focusing by an axicon lens with fixed

θ = θ_{0}

. The focusing with a spherical lens differs only by a smooth

θ

-distribution of the produced plane-wave spectrum.

The three successive rotations,

{\hat{R}}_{z} (ϕ)

{\hat{R}}_{y} (- θ)

, and

{\hat{R}}_{z} (- ϕ)

, in the transformation

\hat{U}

indicate, respectively: (i) azimuthal rotation of the coordinate frame superimposing the

(x, z)

-plane with the local meridional plane, (ii) refraction of the field on the angle

θ

therein, and (iii) the reverse azimuthal rotation compensating the first one. Because of the noncommutativity of the rotations, this transformation is accompanied by a generation of geometrical phases which appear in the form of vortices in the off-diagonal elements of Eq. (5). These elements describe effective transitions between different AM modes. For instance, if the incident wave is

σ

-circularly polarized,

E_{0}^{σ} \propto e^{σ}

, i.e., has only the

E^{σ}

component, the refracted wave is

{\tilde{E}}^{σ} \propto \hat{U} E_{0}^{σ}

, i.e.,

{\tilde{E}}^{+} \propto {(a, - b e^{2 i ϕ}, - \sqrt{2 a b} e^{i ϕ})}^{T}

and

{\tilde{E}}^{-} \propto {(- b e^{- 2 i ϕ}, a, - \sqrt{2 a b} e^{- i ϕ})}^{T}

. This indicates the generation of the

{\tilde{E}}_{z}

component with the charge-

σ

vortex

e^{i σ ϕ}

and the oppositely-polarized component

{\tilde{E}}_{}^{- σ}

with the charge-

2 σ

vortex

e^{2 i σ ϕ}

. We emphasize once again that the actual helicities of partial waves remain unchanged, Eq. (6), and these components appear because of the observation of the redirected rays in the same laboratory frame. Nonetheless, the azimuthal phases signify real generation of the OAM in the laboratory frame, i.e., the spin-to-orbital AM conversion. Let the incident wave be a paraxial vortex beam with circular polarization

σ

and vortex charge

ℓ

, and let us denote the AM state of light with respect to the

z

-axis by the OAM and SAM quantum numbers:

| ℓ, σ 〉

. Then, the polarization transformation (5) can be symbolically written as

| ℓ, σ 〉 \to \sqrt{\cos θ} [a | ℓ, σ 〉 - b | ℓ + 2 σ, - σ 〉 - \sqrt{2 a b} | ℓ + σ, 0 〉] .

(7)

Equation (7) exhibits the conservation of the total AM quantum number in each term:

σ + ℓ = const

, where the last term

| ℓ + σ, 0 〉

corresponds to the longitudinal field

{\tilde{E}}_{z}

which carries no SAM. Both the transverse field with the opposite polarization (b-term) and the longitudinal field (

\sqrt{2 a b}

-term) contribute to the AM conversion. However, in the paraxial approximation,

θ ≪ 1

, one has

b ≃ θ^{2} / 4

and

\sqrt{2 a b} ≃ θ / \sqrt{2}

, so that the main contribution is due to the

z

-component of the field [11

Y. Zhao, J. S. Edgar, G. D. M. Jeffries, D. McGloin, and D. T. Chiu, “Spin-to-orbital angular momentum conversion in a strongly focused optical beam,” Phys. Rev. Lett. 99(7), 073901 (2007). [CrossRef] [PubMed]

First, we calculate the local (angle-resolved) OAM and SAM densities,

l_{z}

and

s_{z}

, without integration over the

(θ, ϕ)

-distribution of the field. Using operators (3) in the circular basis, one can write

l_{z} (θ, ϕ) = - i \frac{{\tilde{E}}^{*} \cdot \partial_{ϕ} \tilde{E}}{{\tilde{E}}^{*} \cdot \tilde{E}}, s_{z} (θ, ϕ) = \frac{{\tilde{E}}^{*} \cdot \hat{σ} \tilde{E}}{{\tilde{E}}^{*} \cdot \tilde{E}} .

(8)

The electric field of the incident circularly-polarized vortex beam can be written as

E_{0 ℓ}^{σ} ≃ e^{σ} E_{ℓ} (θ, ϕ), E_{ℓ} = F_{| ℓ |} (θ) e^{i ℓ ϕ},

(9)

and

F_{| ℓ |} (θ) \propto \sin^{| ℓ |} ​ θ

for the vortex beams with the waist much larger than the entrance pupil.From Eq. (5) the refracted field is

{\tilde{E}}_{ℓ}^{σ} \propto \sqrt{\cos θ} \hat{U} (θ, ϕ) e^{σ} E_{ℓ} (θ, ϕ)

, and Eqs. (8) brings about

l_{z} (θ) = ℓ + σ Φ_{B} (θ), s_{z} (θ) = σ [1 - Φ_{B} (θ)], Φ_{B} (θ) = (1 - \cos θ) = 2 b .

(10)

Here

Φ_{B} (θ) = 2 π Φ_{B} (θ)

stands for the spin-redirection Berry phase (see, e.g., [39

M. V. Berry, “Interpreting the anholonomy of coiled light,” Nature 326(6110), 277–278 (1987). [CrossRef]

,42

R. Bhandari, “Polarization of light and topological phases,” Phys. Rep. 281(1), 2–64 (1997). [CrossRef]

]) associated with the azimuthal distribution of partial rays with a fixed polar angle

θ

. Such distribution corresponds to the circular contour

ℂ = {θ = const, ϕ \in (0, 2 π)}

on the sphere of directions

S^{2} = {k / k}

and the Berry phase gained after traversing this contour is

Φ_{B} = \oint_{ℂ} (1 - \cos θ) d ϕ = 2 π (1 - \cos θ)

[29

K. Y. Bliokh, M. A. Alonso, E. A. Ostrovskaya, and A. Aiello, “Angular momenta and spin-orbit interaction of nonparaxial light in free space,” Phys. Rev. A 82(6), 063825 (2010). [CrossRef]

]. Equations (10) characterize a

θ

-dependent spin-to-orbital AM conversion which vanishes as

Φ_{B} \approx θ^{2} / 2

in the paraxial limit

θ ≪ 1

. This conversion is described by the Berry-phase term, whose appearance is explained in [29

K. Y. Bliokh, M. A. Alonso, E. A. Ostrovskaya, and A. Aiello, “Angular momenta and spin-orbit interaction of nonparaxial light in free space,” Phys. Rev. A 82(6), 063825 (2010). [CrossRef]

] in terms of phase matching of the geometrical-optics rays and quantization of caustics.

To calculate the OAM and SAM expectation values,

〈 L_{z} 〉

and

〈 S_{z} 〉

(throughout the paper we imply values per photon), one has to integrate both numerators and denominators of Eqs. (8) over the spherical angles

(θ, ϕ)

. Implying integration

\int d Ω \equiv \int_{0}^{2 π} d ϕ \int_{0}^{θ_{c}} d θ \sin θ

, the OAM and SAM expectation values are

〈 L_{z} 〉 = - i \frac{\int d Ω {\tilde{E}}^{*} \cdot \partial_{ϕ} \tilde{E}}{\int d Ω {\tilde{E}}^{*} \cdot \tilde{E}}, 〈 S_{z} 〉 = \frac{\int d Ω {\tilde{E}}^{*} \cdot \hat{σ} \tilde{E}}{\int d Ω {\tilde{E}}^{*} \cdot \tilde{E}} .

(11)

For the field

{\tilde{E}}_{ℓ}^{σ}

, this results in (cf. [15

P. B. Monteiro, P. A. M. Neto, and H. M. Nussenzveig, “Angular momentum of focused beams: Beyond the paraxial approximation,” Phys. Rev. A 79(3), 033830 (2009). [CrossRef]

,29

K. Y. Bliokh, M. A. Alonso, E. A. Ostrovskaya, and A. Aiello, “Angular momenta and spin-orbit interaction of nonparaxial light in free space,” Phys. Rev. A 82(6), 063825 (2010). [CrossRef]

])

〈 L_{z} 〉 = ℓ + σ {\bar{Φ}}_{B}, 〈 S_{z} 〉 = σ (1 - {\bar{Φ}}_{B}), {\bar{Φ}}_{B} = (1 - \cos \bar{θ}),

(12)

where the averaged polar angle represents a measure of the directional spread of the field and is defined as [43

M. A. Alonso and G. W. Forbes, “Uncertainty products for nonparaxial wave fields,” J. Opt. Soc. Am. A 17(12), 2391–2402 (2000). [CrossRef] [PubMed]

,44

M. A. Alonso, “The effect of orbital angular momentum and helicity in the uncertainty-type relations between focal spot size and angular spread,” J. Opt. 13(6), 064016 (2011). [CrossRef]

\cos \bar{θ} = \frac{c 〈 P_{z} 〉}{〈 W 〉} = \frac{\int_{0}^{θ_{c}} {| F_{| ℓ |} (θ) |}^{2} \cos^{2} ​ θ \sin θ d θ}{\int_{0}^{θ_{c}} {| F_{| ℓ |} (θ) |}^{2} \cos θ \sin θ d θ} .

(13)

Here

〈 P_{z} 〉

and

〈 W 〉

are the expectation values of the longitudinal momentum and energy of the focused field, which are based on the operators

{\hat{p}}_{z} = k \cos θ

and

\hat{w} = ω

. Evidently, both angle-resolved densities and integral values of the OAM and SAM satisfy the conservation of the total AM per one photon:

l_{z} + s_{z} = 〈 L_{z} 〉 + 〈 S_{z} 〉 = ℓ + σ .

(14)

Evaluation of Eq. (13) for

F_{| ℓ |} (θ) \propto {(\sin θ)}^{^{| ℓ |}}

(i.e., the vortex-core distribution) brings about

\cos \bar{θ} = \frac{2 (1 + | ℓ |)}{\sin^{(2 + 2 | ℓ |)} ​ θ_{c}} [\frac{2^{| ℓ |} | ℓ |!}{(2 | ℓ | + 3)!!} - \frac{1}{3} \cos^{3} ​ θ_{c} {}_{2}F_{1} (\frac{3}{2}, - | ℓ |, \frac{5}{2}; \cos^{2} ​ θ_{c})] .

(15)

where

{}_{2}F_{1} (α, β, γ; z)

is the Gauss hypergeometric function. In the simplest case of the incident plane wave with

ℓ = 0

and

F_{0} (θ) = 1

, Eqs. (15) yield

\cos \bar{θ} = 2 (1 - \cos^{3} ​ θ_{c}) / (3 \sin^{2} ​ θ_{c})

. Then, for small aperture angle,

θ_{c} ≪ 1

\cos \bar{θ} \approx 1 - θ_{c}^{2} / 4

and

〈 L_{z} 〉 \approx σ θ_{c}^{2} / 4

. For maximal aperture

θ_{c} = π / 2

, one has

\cos \bar{θ} = 2 / 3

〈 L_{z} 〉 = σ / 3

, and the efficiency of the spin-to-orbital AM conversion reaches the value of

1 / 3

. For higher values of

| ℓ |

, the efficiency of the conversion increases as the field becomes concentrated at higher angles

θ

(see Fig. 8 below).

Fig. 8 Left: OAM angle-resolved density

l_{z} (θ)

converted from SAM for the cases of focusing (red) [Eq. (10)], scattering (blue) [Eq. (24)], and imaging (black) [Eq. (30)], with circularly polarized incident light,

σ = + 1

ℓ = 0

. Right: The integral OAM,

〈 L_{z} 〉 - ℓ

, converted from SAM, vs. the aperture angle

θ_{c}

for focusing [Eqs. (12) and (13)], scattering [Eq. (25)], and imaging systems [Eq. (31)]. The solid, dashed, and dotted curves for focusing represent incident paraxial beams with

ℓ = 0

ℓ = 1

, and

ℓ = 2

(

σ = + 1

everywhere). For scattering, the dependence is obtained by formally replacing the upper limit of integration in Eqs. (25):

π \to θ_{c}

; analytical results above are recovered at

θ_{c} = π

π / 2

. The semitransparent curves indicate the corresponding quantities for the SAM, i.e.,

s_{z} (θ)

and

〈 S_{z} 〉

, satisfying the conservation law (14). For comparison, the right-hand panel also displays the OAM and SAM values for nonparaxial vector Bessel beams (yellow) [29

K. Y. Bliokh, M. A. Alonso, E. A. Ostrovskaya, and A. Aiello, “Angular momenta and spin-orbit interaction of nonparaxial light in free space,” Phys. Rev. A 82(6), 063825 (2010). [CrossRef]

], which achieve the total spin-to-orbital conversion at the aperture angle

θ_{c} = π / 2

, see [14

Y. Gorodetski, A. Niv, V. Kleiner, and E. Hasman, “Observation of the spin-based plasmonic effect in nanoscale structures,” Phys. Rev. Lett. 101(4), 043903 (2008). [CrossRef] [PubMed]

Let us compare Eqs. (12) and (13) with other calculations of the OAM and SAM of the tightly focused circularly-polarized light [9

Z. Bomzon, M. Gu, and J. Shamir, “Angular momentum and geometric phases in tightly-focused circularly polarized plane waves,” Appl. Phys. Lett. 89(24), 241104 (2006). [CrossRef]

,13

T. A. Nieminen, A. B. Stilgoe, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Angular momentum of a strongly focused Gaussian beam,” J. Opt. A, Pure Appl. Opt. 10(11), 115005 (2008). [CrossRef]

,15

P. B. Monteiro, P. A. M. Neto, and H. M. Nussenzveig, “Angular momentum of focused beams: Beyond the paraxial approximation,” Phys. Rev. A 79(3), 033830 (2009). [CrossRef]

,28

C.-F. Li, “Spin and orbital angular momentum of a class of nonparaxial light beams having a globally defined polarization,” Phys. Rev. A 80(6), 063814 (2009). [CrossRef]

,29

K. Y. Bliokh, M. A. Alonso, E. A. Ostrovskaya, and A. Aiello, “Angular momenta and spin-orbit interaction of nonparaxial light in free space,” Phys. Rev. A 82(6), 063825 (2010). [CrossRef]

]. First, our results differ from calculations [9

Z. Bomzon, M. Gu, and J. Shamir, “Angular momentum and geometric phases in tightly-focused circularly polarized plane waves,” Appl. Phys. Lett. 89(24), 241104 (2006). [CrossRef]

] based on approach of [27

S. M. Barnett and L. Allen, “Orbital angular-momentum and nonparaxial light-beams,” Opt. Commun. 110(5-6), 670–678 (1994). [CrossRef]

] with a nonconserved total AM:

〈 L_{z} 〉 + 〈 S_{z} 〉 \neq ℓ + σ

. At the same time, the post-paraxial estimation

〈 L_{z} 〉 \approx σ θ_{c}^{2} / 4

is analogous to the multipole expansion of a strongly focused Gaussian beam [13

T. A. Nieminen, A. B. Stilgoe, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Angular momentum of a strongly focused Gaussian beam,” J. Opt. A, Pure Appl. Opt. 10(11), 115005 (2008). [CrossRef]

]. Equations (12) and (13) are similar to calculations in [15

P. B. Monteiro, P. A. M. Neto, and H. M. Nussenzveig, “Angular momentum of focused beams: Beyond the paraxial approximation,” Phys. Rev. A 79(3), 033830 (2009). [CrossRef]

] and [28

C.-F. Li, “Spin and orbital angular momentum of a class of nonparaxial light beams having a globally defined polarization,” Phys. Rev. A 80(6), 063814 (2009). [CrossRef]

], but differ from the final result in [15

P. B. Monteiro, P. A. M. Neto, and H. M. Nussenzveig, “Angular momentum of focused beams: Beyond the paraxial approximation,” Phys. Rev. A 79(3), 033830 (2009). [CrossRef]

], apparently due to an arithmetic inaccuracy therein. Finally, Eq. (12) is entirely analogous to the OAM and SAM of nonparaxial Bessel beams with well-defined helicity [29

K. Y. Bliokh, M. A. Alonso, E. A. Ostrovskaya, and A. Aiello, “Angular momenta and spin-orbit interaction of nonparaxial light in free space,” Phys. Rev. A 82(6), 063825 (2010). [CrossRef]

] modified by averaging over the polar angles

θ

So far, we used only the plane-wave spectrum of the focused field,

\tilde{E} (k)

. The actual real-space electric field near the focal point is determined by the interference of the partial plane waves and is given by the Debye integral similar to Fourier transform [37

E. Wolf, ““Electromagnetic diffraction in optical systems. I. An integral representation of the image field,” Proc. Roy. Soc. London,” Ser. A 253, 349–357 (1959).

,38

E (r) \propto \int d Ω \tilde{E} (θ, ϕ) e^{i Φ (θ, ϕ, r)}, Φ = k (x \sin θ \cos ϕ + y \sin θ \sin ϕ + z \cos θ) .

(16)

Here

Φ = k | r - R | - k f ≃ - k r \cdot R / f

is the optical phase gained along the path from the refraction point

R = - f (\sin θ \cos ϕ, \sin θ \sin ϕ, \cos θ)

to the observation point

r = (x, y, z)

(

| r | ≪ \sqrt{f / k}

and the common phase

k f

is subtracted). Calculating the Debye integral (16) and then the intensity,

I_{ℓ}^{σ} = E_{ℓ}^{σ *} \cdot E_{ℓ}^{σ}

, for the incident paraxial beam (9) with the focusing transformation (5), one can derive (cf [15

P. B. Monteiro, P. A. M. Neto, and H. M. Nussenzveig, “Angular momentum of focused beams: Beyond the paraxial approximation,” Phys. Rev. A 79(3), 033830 (2009). [CrossRef]

,29

K. Y. Bliokh, M. A. Alonso, E. A. Ostrovskaya, and A. Aiello, “Angular momenta and spin-orbit interaction of nonparaxial light in free space,” Phys. Rev. A 82(6), 063825 (2010). [CrossRef]

].):

I_{ℓ}^{σ} (ρ, z) \propto {〈 a J_{ℓ} (ξ) 〉}^{2} + {〈 b J_{ℓ + 2 σ} (ξ) 〉}^{2} + 2 {〈 \sqrt{a b} J_{ℓ + σ} (ξ) 〉}^{2} .

(17)

Here we denoted

ξ = k ρ \sin θ

(

ρ

is the radial cylindrical coordinate) and

〈 ... 〉 \equiv \int_{0}^{θ_{c}} d θ \sin θ \sqrt{\cos θ} F_{| ℓ |} (θ) e^{i k z \cos θ}

. Note that for pure helicity states under consideration the magnetic-field intensity in real space coincides with the electric-field intensity, so that Eq. (17) can be regarded as the total intensity of the electromagnetic field.

It is seen that the intensity distribution of the focused field depends on both the OAM (vortex) and SAM (helicity) quantum numbers,

ℓ

and

σ

. It is invariant with respect to the transformation

(ℓ, σ) \to (- ℓ, - σ)

but neither with respect to

(ℓ, σ) \to (ℓ, - σ)

nor

(ℓ, σ) \to (- ℓ, σ)

, which reflects

ℓ \cdot σ

symmetry typical of spin-orbit interaction. For a scalar wave, the intensity (17) would be

I_{ℓ}^{} (ρ, z) \propto {〈 J_{ℓ} (ξ) 〉}^{2}

. Nonparaxial vector terms proportional to

b = Φ_{B} / 2

describe spin-orbit coupling between the main mode of the order

ℓ

to the modes of the order

(ℓ + σ)

and

(ℓ + 2 σ)

which appear owing to the azimuthal geometric phases in the matrix

\hat{U}

, Eq. (5). Dependence of the intensity distribution (17) on

ℓ

and

σ

is closely related to the spin-to-orbital AM conversion and the OAM expectation value

〈 L_{z} 〉

, Eq. (12). Namely, generalizing the Bessel-beam results of [29

K. Y. Bliokh, M. A. Alonso, E. A. Ostrovskaya, and A. Aiello, “Angular momenta and spin-orbit interaction of nonparaxial light in free space,” Phys. Rev. A 82(6), 063825 (2010). [CrossRef]

] for the averaged polar angle

\bar{θ}

, Eq. (13), the “mechanical” radius

R

corresponding to the orbital angular momentum

〈 L_{z} 〉

and underlying the focal intensity (17) can be estimated as

R_{ℓ}^{σ} ≃ \frac{〈 L_{z} 〉}{k \sin \bar{θ}} = \frac{ℓ + σ (1 - \cos \bar{θ})}{k \sin \bar{θ}} .

(18)

Indeed, the transverse momentum

〈 P_{⊥} 〉 = k \sin \bar{θ}

represents the azimuthal momentum in the focal spot (the mean radial momentum vanishes there), and using mechanical definition

L = P \times R

one can write

〈 L_{z} 〉 = 〈 P_{⊥} 〉 R

. Comparison of the

σ

-dependent fine structure of radii (18) with the wave intensities (17) is given in Fig. 2 . Similar polarization-dependent properties of the focal intensities have appeared in [10

Z. Bomzon and M. Gu, “Space-variant geometrical phases in focused cylindrical light beams,” Opt. Lett. 32(20), 3017–3019 (2007). [CrossRef] [PubMed]

,11

,14

Y. Gorodetski, A. Niv, V. Kleiner, and E. Hasman, “Observation of the spin-based plasmonic effect in nanoscale structures,” Phys. Rev. Lett. 101(4), 043903 (2008). [CrossRef] [PubMed]

,15

P. B. Monteiro, P. A. M. Neto, and H. M. Nussenzveig, “Angular momentum of focused beams: Beyond the paraxial approximation,” Phys. Rev. A 79(3), 033830 (2009). [CrossRef]

]. In particular, it follows from Eq. (17) that the intensity of the beams with antiparallel OAM and SAM,

ℓ σ < 0

, does not vanish on the axis

ρ = 0

for

ℓ = 1, 2

. This was experimentally verified in [45

N. Bokor, Y. Iketaki, T. Watanabe, and M. Fujii, “Investigation of polarization effects for high-numerical-aperture first-order Laguerre-Gaussian beams by 2D scanning with a single fluorescent microbead,” Opt. Express 13(26), 10440–10447 (2005). [CrossRef] [PubMed]

,46

Y. Iketaki, T. Watanabe, N. Bokor, and M. Fujii, “Investigation of the center intensity of first- and second-order Laguerre-Gaussian beams with linear and circular polarization,” Opt. Lett. 32(16), 2357–2359 (2007). [CrossRef] [PubMed]

]. In general, the size of the focal spot for a strongly focused beam has a fundamental lower bound that depends not only on the beam’s directional spread but also on its OAM and SAM, such that beams with antiparallel OAM and SAM can achieve tighter focal spots than those for which these AM are parallel [44

M. A. Alonso, “The effect of orbital angular momentum and helicity in the uncertainty-type relations between focal spot size and angular spread,” J. Opt. 13(6), 064016 (2011). [CrossRef]

]. Note that asymmetric focusing with a truncated lens or distribution of the incident light brings about the

ℓ

- and

σ

-dependent transverse shifts of the focal-spot intensity centroid, which is proportional to

〈 L_{z} 〉

, i.e., orbital and spin Hall effects [29

K. Y. Bliokh, M. A. Alonso, E. A. Ostrovskaya, and A. Aiello, “Angular momenta and spin-orbit interaction of nonparaxial light in free space,” Phys. Rev. A 82(6), 063825 (2010). [CrossRef]

,30

N. B. Baranova, A. Y. Savchenko, and B. Y. Zel'dovich, “Transverse shift of a focal spot due to switching of the sign of circular-polarization,” JETP Lett. 59, 232–234 (1994).

,47

K. Y. Bliokh, Y. Gorodetski, V. Kleiner, and E. Hasman, “Coriolis effect in optics: unified geometric phase and spin-Hall effect,” Phys. Rev. Lett. 101(3), 030404 (2008). [CrossRef] [PubMed]

Fig. 2 Focal intensity distributions

I_{ℓ}^{σ} (ρ, 0)

, Eq. (17), in comparison with the radii

R_{ℓ}^{σ}

, Eq. (18), (vertical lines) for

ℓ = 1, 2, 3

σ = \pm 1

, and different values of the aperture angles

θ_{c}

. It is seen that the structure of radii (18) underlies positions of the first maxima of intensity (17), cf [29

K. Y. Bliokh, M. A. Alonso, E. A. Ostrovskaya, and A. Aiello, “Angular momenta and spin-orbit interaction of nonparaxial light in free space,” Phys. Rev. A 82(6), 063825 (2010). [CrossRef]

4. Dipole scattering

Scattering of paraxial light by a nano-particle located at the origin essentially represents the spherical redirection of partial plane waves, Fig. 3 . We examine the simplest dipole approximation when the scattered spherical wave is generated by the dipole moment

� \propto E_{0} (0)

proportional to the incident field

E_{0}

at the origin. Since higher-order paraxial beams (9) with

ℓ \neq 0

have zero field in the center,

E_{0 ℓ}^{σ} (0) ≃ 0

, below we consider an incident plane wave with circular polarization,

E_{0}^{σ} (0) \propto e^{σ}

. The electric far field of the scattered wave is [48

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

]

\tilde{E} (θ, ϕ, r) \propto - \frac{\overset{⌢}{r} \times [\overset{⌢}{r} \times E_{0} (0)]}{r},

(19)

where

\overset{⌢}{r} = r / r

is the unit radial vector with spherical coordinates

(θ, ϕ)

of the observation point. Akin to focusing, the real-space spherical angles

(θ, ϕ)

serve as the coordinates in momentum space for the scattered far field. In this manner, the

\overset{⌢}{r}

-vector plays the role of the

k / k

-vector for scattered waves and transformation (19) is reminiscent of the vector transformation

k \times (k \times \tilde{E})

in free-space Maxwell equations

k \times (k \times \tilde{E}) + k^{2} \tilde{E} = 0

. The double vector product

- \overset{⌢}{r} \times (\overset{⌢}{r} \times E_{0}) = E_{0} - \overset{⌢}{r} (\overset{⌢}{r} \cdot E_{0})

represents a spherical projection of

E_{0} (0)

onto the

\overset{⌢}{r}

-sphere and can be written, in the laboratory circular basis (1), as the action of a nondiagonal operator

\hat{Π} (θ, ϕ) = \hat{U} (θ, ϕ) {\hat{p}}_{z} {\hat{U}}^{†} (θ, ϕ)

[16

\tilde{E} (θ, ϕ) \propto \frac{1}{r} \hat{Π} (θ, ϕ) E_{0} (0), \hat{Π} = \frac{1}{2} (\begin{matrix} 1 + a_{1} & - b_{1} e^{- 2 i ϕ} & - \sqrt{2 a_{1} b_{1}} e^{- i ϕ} \\ - b_{1} e^{2 i ϕ} & 1 + a_{1} & - \sqrt{2 a_{1} b_{1}} e^{i ϕ} \\ - \sqrt{2 a_{1} b_{1}} e^{i ϕ} & - \sqrt{2 a_{1} b_{1}} e^{- i ϕ} & 2 b_{1}^{2} \end{matrix}) .

(20)

Here

{\hat{p}}_{z} = diag (1, 1, 0)

is the projector onto the orthogonal plane,

a_{1} = \cos^{2} θ

, and

b_{1} = \sin^{2} θ

Fig. 3 Akin to focusing, Fig. 1, dipole scattering of the incident paraxial field

E_{0} (x, y)

by a small spherical particle transforms it into a spectrum of plane waves

\tilde{E} (θ, ϕ)

with the spherically-distributed

k

-vectors.

Projection (20) is a nonunitary transformation, and it does not preserve the local helicity of the scattered field. Indeed, in the helicity basis attached to the sphere of partial

k

-vectors, the scattered field is [cf. Equation (6)]:

{\tilde{E}}_{H} = {\hat{U}}^{†} \tilde{E} \propto {\hat{p}}_{z} {\hat{U}}^{†} E_{0} ≃ \hat{p} {\hat{U}}^{†} E_{0 H}

(21)

Other than that, Eq. (20) demonstrates features that are quite similar to those of the focusing transformation (5). In particular, the off-diagonal geometric-phase elements of the matrix

\hat{Π}

produces spin-to-orbital AM conversion which can be symbolically written similarly to Eq. (7):

| 0, σ 〉 \to \frac{1}{2} [(1 + a_{1}) | 0, σ 〉 - b_{1} | 2 σ, - σ 〉 - \sqrt{2 a_{1} b_{1}} | σ, 0 〉] .

(22)

The AM conversion upon scattering of light on various objects was analyzed in several papers [16

–24

Y. Gorodetski, S. Nechayev, V. Kleiner, and E. Hasman, “Plasmonic Aharonov-Bohm effect: Optical spin as the magnetic flux parameter,” Phys. Rev. B 82(12), 125433 (2010). [CrossRef]

]. While in the case of focusing the polarization is not changed in the helicity basis, the transformation (21) of the scattered field can be written as the following helicity transition:

{| σ 〉}_{H} \to a {| σ 〉}_{H} - b e^{2 i σ ϕ} {| - σ 〉}_{H} .

(23)

In the linear approximation in

θ

(for small scattering angles

θ ≪ 1

), the helicity (23) is conserved [19

C. Schwartz and A. Dogariu, “Backscattered polarization patterns, optical vortices, and the angular momentum of light,” Opt. Lett. 31(8), 1121–1123 (2006). [CrossRef] [PubMed]

,20

C. Schwartz and A. Dogariu, “Backscattered polarization patterns determined by conservation of angular momentum,” J. Opt. Soc. Am. A 25(2), 431–436 (2008). [CrossRef] [PubMed]

], and the scattering transformation (20) becomes similar to the focusing one, Eq. (5).

Assuming an incident plane wave with circular polarization,

E_{0}^{σ} (0) \propto e^{σ}

, we determine the scattered field (20) and calculate the OAM and SAM angle-resolved densities using Eqs. (8):

l_{z} = σ \frac{1 - \cos^{2} ​ θ}{1 + \cos^{2} ​ θ}, s_{z} = σ \frac{2 \cos^{2} ​ θ}{1 + \cos^{2} ​ θ} .

(24)

Identical results were obtained for the field radiated by a rotating dipole [49

G. Moe and W. Happer, “Conservation of angular momentum for light propagating in a transparent anisotropic medium,” J. Phys. B 10(7), 1191–1208 (1977). [CrossRef]

], and similar results [but without

{(1 + \cos^{2} θ)}^{- 1}

factor] have appeared for the Rayleigh scattering [18

A. Dogariu and C. Schwartz, “Conservation of angular momentum of light in single scattering,” Opt. Express 14(18), 8425–8433 (2006). [CrossRef] [PubMed]

]. Equations (24) yield

l_{z} \approx θ^{2} σ

for the paraxial angles

θ ≪ 1

and the total conversion

l_{z} = σ

θ = π / 2

(see Fig. 8 below). The integral OAM and SAM values are determined via Eqs. (11) with the spherical integration

\int d Ω \equiv \int_{0}^{2 π} d ϕ \int_{0}^{π} d θ \sin θ

, which results in

〈 L_{z} 〉 = σ \frac{\int_{0}^{π} (1 - \cos^{2} ​ θ) \sin θ d θ}{\int_{0}^{π} (1 + \cos^{2} ​ θ) \sin θ d θ} = \frac{1}{2} σ, 〈 S_{z} 〉 = σ \frac{2 \int_{0}^{π} \cos^{2} ​ θ \sin θ d θ}{\int_{0}^{π} (1 + \cos^{2} ​ θ) \sin θ d θ} = \frac{1}{2} σ .

(25)

Thus, the spin-to-orbital AM conversion has the efficiency

1 / 2

[18

A. Dogariu and C. Schwartz, “Conservation of angular momentum of light in single scattering,” Opt. Express 14(18), 8425–8433 (2006). [CrossRef] [PubMed]

,49

G. Moe and W. Happer, “Conservation of angular momentum for light propagating in a transparent anisotropic medium,” J. Phys. B 10(7), 1191–1208 (1977). [CrossRef]

] (see Fig. 8 below). Equations (24) and (25) demonstrate conservation of the local and integral total AM per photon, Eq. (14).

It is worth remarking that the converted part of the AM in Eq. (24) can be expressed as

l_{z} = σ P (θ), s_{z} = σ [1 - P (θ)], P (θ) = \frac{2 �^{- σ}}{�^{σ} + �^{- σ}} = \frac{\sin^{2} ​ θ}{1 + \cos^{2} ​ θ},

(24')

where

�^{σ} = a^{2}

and

�^{- σ} = b^{2}

stand for local intensities of the helicity components, Eq. (23). Here the quantity

P (θ)

coincides with the degree of polarization of the scattered unpolarized light [48

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

]. Thus, the changes in the degree of polarization upon scattering are also connected with the above-considered geometric transformations of the wave field. In particular, the depolarization of multiply scattered polarized light and typical four-fold polarization patterns of the backscattered light are intimately related to the spin-to-orbital AM conversion [19

C. Schwartz and A. Dogariu, “Backscattered polarization patterns, optical vortices, and the angular momentum of light,” Opt. Lett. 31(8), 1121–1123 (2006). [CrossRef] [PubMed]

,20

C. Schwartz and A. Dogariu, “Backscattered polarization patterns determined by conservation of angular momentum,” J. Opt. Soc. Am. A 25(2), 431–436 (2008). [CrossRef] [PubMed]

]. In the weak-scattering approximation of small scattering angles (

θ ≪ 1

in a single scattering), these depolarization effects can be explained via Berry-phase accumulation along the partial scattering paths [19

C. Schwartz and A. Dogariu, “Backscattered polarization patterns, optical vortices, and the angular momentum of light,” Opt. Lett. 31(8), 1121–1123 (2006). [CrossRef] [PubMed]

,20

C. Schwartz and A. Dogariu, “Backscattered polarization patterns determined by conservation of angular momentum,” J. Opt. Soc. Am. A 25(2), 431–436 (2008). [CrossRef] [PubMed]

,50

V. Rossetto and A. C. Maggs, “Writhing geometry of stiff polymers and scattered light,” Eur. Phys. J. B 29(2), 323–326 (2002). [CrossRef]

,51

D. Lacoste, V. Rossetto, F. Jaillon, and H. Saint-Jalmes, “Geometric depolarization in patterns formed by backscattered light,” Opt. Lett. 29(17), 2040–2042 (2004). [CrossRef] [PubMed]

]. This establishes a geometric-phase link between the AM conversions in focusing and scattering processes. For strong single scattering event (

θ ~ 1

), the Berry-phase (adiabatic) approximation is not applicable because the geometric transformation (20) represents a projection rather than parallel-transport rotation (5) of the field.

5. Imaging of nanoparticles

Strongly focused or scattered fields are essentially nonparaxial, which is the main source of the spin-orbit phenomena. Accordingly, the AM conversion can be detected via various near-field methods: e.g., tracing motion of testing particles [11

,12

Y. Zhao, D. Shapiro, D. McGloin, D. T. Chiu, and S. Marchesini, “Direct observation of the transfer of orbital angular momentum to metal particles from a focused circularly polarized Gaussian beam,” Opt. Express 17(25), 23316–23322 (2009). [CrossRef] [PubMed]

] or using near-field probes [14

Y. Gorodetski, A. Niv, V. Kleiner, and E. Hasman, “Observation of the spin-based plasmonic effect in nanoscale structures,” Phys. Rev. Lett. 101(4), 043903 (2008). [CrossRef] [PubMed]

,21

D. Haefner, S. Sukhov, and A. Dogariu, “Spin hall effect of light in spherical geometry,” Phys. Rev. Lett. 102(12), 123903 (2009). [CrossRef] [PubMed]

,24

Y. Gorodetski, S. Nechayev, V. Kleiner, and E. Hasman, “Plasmonic Aharonov-Bohm effect: Optical spin as the magnetic flux parameter,” Phys. Rev. B 82(12), 125433 (2010). [CrossRef]

,47

K. Y. Bliokh, Y. Gorodetski, V. Kleiner, and E. Hasman, “Coriolis effect in optics: unified geometric phase and spin-Hall effect,” Phys. Rev. Lett. 101(3), 030404 (2008). [CrossRef] [PubMed]

]. It should be noted that the use of testing particles is somewhat ambiguous as they can undergo orbital motion due to both orbital and spin energy flows, i.e., OAM and SAM [52

M. V. Berry, “Optical currents,” J. Opt. A, Pure Appl. Opt. 11(9), 094001 (2009). [CrossRef]

,53

A. Bekshaev and S. Sviridova, “Mechanical action of inhomogeneously polarized optical fields and detection of the internal energy flows,” arXiv:1102.3514.

]. At the same time, traditional paraxial-optics detectors are unable to measure adequately the spin-orbit phenomena in nonparaxial fields with strong longitudinal field component. It turns out, however, that a standard imaging system successfully resolves this dilemma by transfering the spin-orbit coupling into a paraxial far-field, where it can be easily detected. There were several experimental observations [16

,31

B. Y. Zel’dovich, N. D. Kundikova, and L. F. Rogacheva, “Observed transverse shift of a focal spot upon a change in the sign of circular polarization,” JETP Lett. 59, 766–769 (1994).

–34

,54

O. G. Rodríguez-Herrera, D. Lara, and C. Dainty, “Far-field polarization-based sensitivity to sub-resolution displacements of a sub-resolution scatterer in tightly focused fields,” Opt. Express 18(6), 5609–5628 (2010). [CrossRef] [PubMed]

] of the spin-orbit interactions of light using imaging scheme: (i) focusing of the incident paraxial light with a high-NA lens; (ii) scattering by a small specimen; (iii) collection of the scattered light by another high-NA lens transforming it to the outgoing paraxial light, see Fig. 4 . In most cases the observed effects were ascribed either to focusing or to scattering process, although all the three elements of the imaging system contributed to the effect. The ‘lens-scatterer-lens’ system (Fig. 4) represents the basis for optical microscopy and it is important to describe the spin-orbit effects in such imaging system taking into account all its elements [16

,17

M. R. Foreman and P. Török, “Spin-orbit coupling and conservation of angular momentum flux in non-paraxial imaging of forbidden radiation,” New J. Phys. 13(6), 063041 (2011). [CrossRef]

Fig. 4 Scheme of the ‘lens-scatterer-lens’ imaging system. First, the incident paraxial light

E_{0}

is focused by a high-NA lens, then a small specimen in the focus scatters the nonparaxial focused field, and finally the scattered light is collected by the second high-NA lens. The output paraxial field

E (θ, ϕ)

has a space-variant polarization distribution and bears information about the spin-orbit coupling inside the system. This offers an efficient tool to retrieve fine subwavelength information about the specimen.

Since both the input and output fields in the imaging system are paraxial, the transformation of the field by the system can be described by an effective Jones matrix which gives the angle-resolved polarization state of the output field. Using successive applications of the geometric transformations of the first lens, Eq. (5), together with the Debye integral (16), dipole-scattering transformation (20), and the inverse transformation (5) of the collector lens, one can obtain the 3D transformation operator of the system [16

E (θ, ϕ) \propto \hat{T} (θ, ϕ) E_{0} (θ^{'}, ϕ^{'})

\hat{T} = \frac{1}{\sqrt{\cos θ}} {\hat{p}}_{z} {\hat{U}}^{†} (θ, ϕ) e^{i Φ (θ, ϕ, r_{s})} \int d Ω^{'} \sqrt{\cos θ^{'}} \hat{U} (θ^{'}, ϕ^{'}) e^{i Φ^{'} (θ^{'}, ϕ^{'}, r_{s})}

. (26)Here

\int d Ω^{'} \equiv \int_{0}^{2 π} d ϕ^{'} \int_{0}^{θ_{c}} d θ^{'} \sin θ^{'}

, the angles

(θ^{'}, ϕ^{'})

and

(θ, ϕ)

mark the partial rays at the entrance and exit pupils, respectively (Fig. 4),

r_{s} = (x_{s}, y_{s}, z_{s})

is the position of the scattering particle near the common focus of the two lenses, and the phases are

\begin{array}{l} Φ^{'} ≃ - k r_{s} \cdot R^{'} / f = k (x_{s} \sin θ^{'} \cos ϕ^{'} + y_{s} \sin θ^{'} \sin ϕ^{'} + z_{s} \cos θ^{'}), \\ Φ ≃ - k r_{s} \cdot R / f = - k (x_{s} \sin θ \cos ϕ + y_{s} \sin θ \sin ϕ + z_{s} \cos θ) . \end{array}

(27)

Here

R^{'} = - f (\sin θ^{'} \cos ϕ^{'}, \sin θ^{'} \sin ϕ^{'}, \cos θ^{'})

and

R = f (\sin θ \cos ϕ, \sin θ \sin ϕ, \cos θ)

are the refraction points at the focusing and collecting lenses, respectively. The projector

{\hat{p}}_{z}

in Eq. (26) ensures transversality of the output field, and the upper left

2 \times 2

sector of the operator (26),

{\hat{T}}_{⊥} (θ, ϕ)

, provides the effective space-variant Jones matrix connecting the transverse components of the input and output fields in the basis of circular polarizations. In what follows we denote the transverse components of the vectors and matrices by the subscript “

⊥

”:

E_{⊥} = {(E^{+}, E^{-})}^{T}

, etc. Note that even if the incident light represents a homogeneous plane wave

E_{0}

, the output field will have nonuniform polarization

E (θ, ϕ)

bearing information about the spin-orbit coupling in the system and specimen properties.

First, let us consider the symmetric case when the scatterer is located precisely in the focus:

r_{s} = 0

. If the incident field is a homogeneous plane wave

E_{0}

(which implies

ℓ = 0

), one can evaluate the integral (26) analytically. The resulting Jones operator is [16

{\hat{T}}_{⊥}^{(0)} = \frac{A (θ_{c})}{\sqrt{\cos θ}} (\begin{matrix} a & - b e^{- 2 i ϕ} \\ - b e^{2 i ϕ} & a \end{matrix}),

(28)

where the aperture-dependent coefficient is

A (θ_{c}) = \frac{2 π}{15} [8 - {(\cos θ_{c})}^{3 / 2} (5 + 3 \cos θ_{c})]

and

A \approx π θ_{c}^{2}

θ_{c} ≪ 1

. The Jones matrix (28) is proportional to the transverse part of

\hat{U}

, Eq. (5), and it describes the spin-to-orbital AM conversion between two paraxial states of light:

| 0, σ 〉 \to a | 0, σ 〉 - b | 2 σ, - σ 〉 .

(29)

Comparing this equation with Eq. (23), one can see that, seemingly, the imaging scheme transfers the scattering-induced AM conversion which appears in the local helicity basis to the paraxial field and laboratory circular basis. AM conversion (29) strongly resembles paraxial spin-to-orbital convertors based on locally-anisotropic axially symmetric structures [3

G. Biener, A. Niv, V. Kleiner, and E. Hasman, “Formation of helical beams by use of Pancharatnam-Berry phase optical elements,” Opt. Lett. 27(21), 1875–1877 (2002). [CrossRef] [PubMed]

–7

E. Brasselet, Y. Izdebskaya, V. Shvedov, A. S. Desyatnikov, W. Krolikowski, and Y. S. Kivshar, “Dynamics of optical spin-orbit coupling in uniaxial crystals,” Opt. Lett. 34(7), 1021–1023 (2009). [CrossRef] [PubMed]

,25

E. Hasman, G. Biener, A. Niv, and V. Kleiner, “Space-variant polarization manipulation,” Prog. Opt. 47, 215–289 (2005). [CrossRef]

,26

]. In our case, the

ϕ

-dependent helical phases arise from purely geometrical 3D transformations of the field inside the imaging system, which demonstrate effective birefringence of the radially (TM) and azimuthally (TE) polarized modes. Assuming

σ

-circularly polarized incident wave,

E_{0}^{σ} \propto e^{σ}

, the output field is

E_{⊥}^{σ} \propto {\hat{T}}_{⊥}^{(0)} e_{⊥}^{σ}

, i.e.,

E_{⊥}^{+} \propto {(a, - b e^{2 i ϕ})}^{T} / \sqrt{\cos θ}

and

E_{⊥}^{-} \propto {(- b e^{- 2 i ϕ}, a)}^{T} / \sqrt{\cos θ}

. Using Eqs. (8), we determine the local OAM and SAM densities at the exit pupil:

l_{z} = σ (1 - \frac{2 \cos θ}{1 + \cos^{2} ​ θ}), s_{z} = σ \frac{2 \cos θ}{1 + \cos^{2} ​ θ} .

(30)

From Eqs. (11), implying integration over the exit pupil,

\int d x d y \propto \int d Ω_{⊥} \equiv \int_{0}^{2 π} d ϕ \int_{0}^{θ_{c}} d θ \sin θ \cos θ

, one can obtain the integral OAM and SAM values:

〈 L_{z} 〉 = σ \frac{1 - 3 \cos θ_{c} + 3 \cos^{2} ​ θ_{c} - \cos^{3} ​ θ_{c}}{4 - 3 \cos θ_{c} - \cos^{3} ​ θ_{c}}, 〈 S_{z} 〉 = σ \frac{3 \sin^{2} ​ θ_{c}}{4 - 3 \cos θ_{c} - \cos^{3} ​ θ_{c}},

(31)

which satisfies the total AM conservation (14). The efficiency of the spin-to-orbital AM conversion is tiny in the imaging system with small aperture:

〈 L_{z} 〉 \approx σ θ_{c}^{4} / 24

θ_{c} ≪ 1

, and it reaches the value

〈 L_{z} 〉 = σ / 4

θ_{c} = π / 2

Since the output field

E (θ, ϕ)

is paraxial, its polarization properties can be characterized by space-variant Stokes parameters. The normalized Stokes vector

\vec{S} = (S_{1}, S_{2}, S_{3})

can be calculated using the Pauli matrices

\hat{\vec{σ}} = ({\hat{σ}}_{1}, {\hat{σ}}_{2}, {\hat{σ}}_{3})

[55

Y. A. Kravtsov, B. Bieg, and K. Y. Bliokh, “Stokes-vector evolution in a weakly anisotropic inhomogeneous medium,” J. Opt. Soc. Am. A 24(10), 3388–3396 (2007). [CrossRef] [PubMed]

\vec{S} (θ, ϕ) = \frac{{\tilde{E}}^{*} \cdot \hat{\vec{σ}} \tilde{E}}{{\tilde{E}}^{*} \cdot \tilde{E}},

(32)

Note that the spin density

s_{z}

, Eq. (30), coincides with the local degree of circular polarization, i.e., the third component of the Stokes vector:

S_{3} = s_{z}

. Indeed, for paraxial fields it is determined by the transverse part of the operator

\hat{σ}

{\hat{σ}}_{⊥} = diag (1, - 1) \equiv {\hat{σ}}_{3}

. The complete description of the polarization contains more information than the spin AM of light [56

M. V. Berry, “Paraxial beams of spinning light,” Proc. SPIE 3487, 6–11 (1998). [CrossRef]

]. In particular, the first two components of the Stokes vector (32) can evidence conversion to the OAM, Eq. (29). Indeed, calculating Eq. (32) for the incident circularly-polarized plane wave, we obtain

\vec{S} (θ, ϕ) = \frac{1}{a^{2} + b^{2}} [- 2 a b \cos 2 ϕ, - 2 a b \sin 2 ϕ, σ (a^{2} - b^{2})] .

(33)

This distribution of the Stokes parameters is shown in Fig. 5 . Parameters

S_{1}

and

S_{2}

demonstrate typical four-fold patters which signify generation of the oppositely polarized component

| - σ 〉

with the optical vortex

e^{2 i σ ϕ}

[16

,19

C. Schwartz and A. Dogariu, “Backscattered polarization patterns, optical vortices, and the angular momentum of light,” Opt. Lett. 31(8), 1121–1123 (2006). [CrossRef] [PubMed]

,20

C. Schwartz and A. Dogariu, “Backscattered polarization patterns determined by conservation of angular momentum,” J. Opt. Soc. Am. A 25(2), 431–436 (2008). [CrossRef] [PubMed]

,33

E. Brasselet, N. Murazawa, H. Misawa, and S. Juodkazis, “Optical vortices from liquid crystal droplets,” Phys. Rev. Lett. 103(10), 103903 (2009). [CrossRef] [PubMed]

,50

V. Rossetto and A. C. Maggs, “Writhing geometry of stiff polymers and scattered light,” Eur. Phys. J. B 29(2), 323–326 (2002). [CrossRef]

Fig. 5 Distributions of the Stokes parameters

\vec{S} = (S_{1}, S_{2}, S_{3})

, Eqs. (32) and (33), in the exit pupil of the optical microscope (Fig. 4) in the case of the on-axis location of the specimen,

r_{s} = 0

, and left-hand circular polarization of the incident light,

σ = - 1

. The normalized coordinates

\tilde{x} = x / (f \sin θ_{c})

and

\tilde{y} = y / (f \sin θ_{c})

are used. The four-fold patterns in the

S_{1}

and

S_{2}

distributions are the signature of the generation of the right-hand polarized component with optical vortex

e^{- 2 i ϕ}

, i.e., the spin-to-orbital AM conversion (29). The aperture angle is

θ_{c} = 3 π / 8

The cylindrical symmetry is broken in the imaging system if the specimen is transversely shifted in the focal plane. In this case, the system is not rotationally-invariant about the

z

-axis, and the total AM is no longer conserved. This results in the AM-dependent orthogonal shift of the center of gravity of the output field, i.e., the Hall effect of light. Considering a small subwavelength displacement of the specimen in the

(x, y)

plane,

r_{s} = (x_{s}, y_{s}, 0)

k | r_{s} | ≪ 1

, the effective Jones matrix can be evaluated analytically from Eq. (26) as a correction to Eq. (28) [16

{\hat{T}}_{⊥} ≃ {\hat{T}}_{⊥}^{(0)} + {\hat{T}}_{⊥}^{(1)}

{\hat{T}}_{⊥}^{(1)} = - i k \frac{B (θ_{c}) \sin θ}{2 \sqrt{\cos θ}} (\begin{matrix} ρ_{s} e^{- i ϕ} & ρ_{s}^{*} e^{- i ϕ} \\ ρ_{s} e^{i ϕ} & ρ_{s}^{*} e^{i ϕ} \end{matrix}) .

(34)

Here we introduced

ρ_{s} = x_{s} + i y_{s}

and the aperture-dependent coefficient is

B (θ_{c}) = \frac{π}{21} [8 - {(\cos θ_{c})}^{3 / 2} (11 - 3 \cos 2 θ_{c})]

, with

B \approx π θ_{c}^{4} / 4

θ_{c} ≪ 1

. The Jones matrix (34) reveals coupling between the position of the scatterer and polarization of light, which results in the spin Hall effect. Taking the incident circularly polarized plane wave

E_{0}^{σ} \propto e^{σ}

, we assume

ρ_{s} = x_{s}

, determine the output field

E_{⊥}^{σ} \propto {\hat{T}}_{⊥}^{} e_{⊥}^{σ}

, and calculate the transverse position of centre of gravity of the field. It is determined as

〈 R_{⊥} 〉 = \frac{\int d Ω_{⊥} R_{⊥} (E^{σ *} \cdot E^{σ})}{\int d Ω_{⊥} E^{σ *} \cdot E^{σ}},

(35)

where

R_{⊥} \equiv (X, Y) = f (\sin θ \cos ϕ, \sin θ \sin ϕ)

. Substituting the output field calculated through the Jones matrix (28) and (34) into Eq. (35), we obtain

〈 X 〉 = 0, 〈 Y 〉 = - σ f k x_{s} \frac{3 B \sin^{4} ​ θ_{c}}{2 A (4 - 3 \cos θ_{c} - \cos^{3} ​ θ_{c})} .

(36)

The

σ

-dependent shift, orthogonal to the displacement of the specimen, represents the spin Hall effect of light. It is also accompanied by a nonzero mean momentum

〈 P_{x} 〉 \neq 0

. The shift (36) behaves as

〈 Y 〉 \approx - σ f k x_{s} θ_{c}^{4} / 8

θ_{c} ≪ 1

and reaches value

〈 Y 〉 ≃ - 0.13 σ f k x_{s}

θ_{c} = π / 2

. Thus, for high-NA systems and subwavelength displacements of the particle,

k x_{s} ~ 1

, the shift of the centre of gravity is of the order of a fraction of the focal length

f ≫ λ

. In other words, high-NA optical microscope dramatically magnifies the spin Hall effect from the typical subwavelength scale to the scale of the exit pupil [16

]. Figure 6 displays the intensity distributions

I = E^{σ *} \cdot E^{σ}

and shifts of the field centroid at different positions

x_{s}

of the scatterer. Similar

ℓ

-dependent deformations of intensity in the imaging system with incident paraxial vortex beams, i.e., orbital Hall effect, were observed in [32

]. Another striking manifestation of the spin Hall effect is a strong separation of local SAM densities in the incident linearly polarized light [16

Fig. 6 Distributions of the output field intensity

I (θ, ϕ)

in the imaging system Fig. 4 for the incident right-hand circularly polarized light (

σ = 1

) at different subwavelength displacements of the scattering particle:

x_{s} = 0, - λ / 3, λ / 3

. The transverse shift of the center of gravity of the field,

〈 Y 〉 \propto σ x_{s}

signifies the spin Hall effect of light. Parameters are the same as in Fig. 5.

Alongside the Stokes polarimetry, which reveals conversion to different polarization and OAM modes, the state of the output light can be fully described via intensities and phases of the two polarization components following from the Jones matrices (28) and (34). Distributions of the intensities and phases of the right- and left-hand circularly polarized field components for the on-axis and off-axis scatterer are shown in Fig. 7 . It is seen that the nonzero intensity and the vortices in the oppositely polarized (with respect to the incident light) component indicates the spin-to-orbital AM conversion (29). At the same time, the intensity redistribution and phase gradient in the main polarization component are responsible for the shifts (36),

〈 Y 〉 \neq 0

〈 P_{x} 〉 \neq 0

for an off-axis scatterer.

Fig. 7 Intensities and phases of the right- and left-hand circularly polarized components in the output field for the on-axis (

r_{s} = 0

) and off-axis (

x_{s} = 0.28 λ

) positions of the scatterer (Fig. 4). The incident field is right-hand circularly polarized (

σ = 1

), the aperture of the system corresponds to

\sin θ_{c} = 0.922

. The charge-2 optical vortex and nonzero intensity in the left-hand polarized component is clearly seen for the on-axis particle. Displacement of the scatterer induces splitting of the charge-2 vortex into two charge-1 vortices (cf [34

].), strong deformation of the intensity of the right-hand component (responsible for the spin-Hall effect and

〈 Y 〉 \neq 0

), and smooth gradient of the phase in the right-hand component which yields

〈 P_{x} 〉 \neq 0

6. Conclusions

We have examined the spin-to-orbital AM conversion in basic optical systems involving nonparaxial fields. The AM conversion originates from geometric transformations of the wave field: parallel-transport rotations in the case of focusing and spherical projections in the case of scattering. Despite the fact that problems of spin-to-orbital conversion were considered in recent years both in focusing and scattering systems, a number of controversies and the use of dissimilar approaches did not allow one to obtain solid quantitative results and compare the efficiency of the conversion in different systems. Our approach unifies various treatments of the AM in different nonparaxial optical systems and is based on the geometric field transformations and canonical AM operators. As a result, one can compare the efficiency of the spin-to-orbital conversion in high-NA lens focusing, Rayleigh (dipole) scattering, and in far-field “lens-scatterer-lens” imaging system. Figure 8 shows both angle-resolved OAM and SAM densities and integral OAM and SAM values (per photon) as dependent on the aperture angle. For comparison we also plot there the OAM and SAM of nonparaxial Bessel beams calculated in [29

K. Y. Bliokh, M. A. Alonso, E. A. Ostrovskaya, and A. Aiello, “Angular momenta and spin-orbit interaction of nonparaxial light in free space,” Phys. Rev. A 82(6), 063825 (2010). [CrossRef]

]. One can see that for the plane incident wave (

ℓ = 0

) the conversion efficiency decreases in a sequence: “Bessel beam” – “scattering” – “focusing” – “imaging”. Curiously, the corresponding maximum efficiencies reached at the aperture angle

θ_{c} = π / 2

are equal to 1, 1/2, 1/3, and 1/4. The total spin-to-orbital conversion for Bessel modes with

θ_{c} = π / 2

was observed in a plasmonic experiment [14

Y. Gorodetski, A. Niv, V. Kleiner, and E. Hasman, “Observation of the spin-based plasmonic effect in nanoscale structures,” Phys. Rev. Lett. 101(4), 043903 (2008). [CrossRef] [PubMed]

Although the OAM and SAM are important dynamical characteristics of light, it is difficult to measure them directly in optical systems. In particular, orbital motion of small testing particles [11

,12

] cannot be used as a measure of the OAM because both the spin and orbital energy flows can be responsible for it [52

M. V. Berry, “Optical currents,” J. Opt. A, Pure Appl. Opt. 11(9), 094001 (2009). [CrossRef]

,53

A. Bekshaev and S. Sviridova, “Mechanical action of inhomogeneously polarized optical fields and detection of the internal energy flows,” arXiv:1102.3514.

]. However, the close connection between the intensity distributions (in particular the radius of the focal spot) and the OAM values, Eq. (18) and Fig. 2, enables one to quantify the AM conversion via spin-dependent intensity profiles. Also, as we have shown, the angle-resolved polarimetry in the far-field imaging systems enables one to reconstruct the field distribution and unambiguously characterize its AM features [16

] (Section 5).

Importantly, universal mechanisms of the spin-to-orbital AM conversion manifest themselves not only in optical but also in plasmonic [14

Y. Gorodetski, A. Niv, V. Kleiner, and E. Hasman, “Observation of the spin-based plasmonic effect in nanoscale structures,” Phys. Rev. Lett. 101(4), 043903 (2008). [CrossRef] [PubMed]

,22

Y. Gorodetski, N. Shitrit, I. Bretner, V. Kleiner, and E. Hasman, “Observation of optical spin symmetry breaking in nanoapertures,” Nano Lett. 9(8), 3016–3019 (2009). [CrossRef] [PubMed]

–24

Y. Gorodetski, S. Nechayev, V. Kleiner, and E. Hasman, “Plasmonic Aharonov-Bohm effect: Optical spin as the magnetic flux parameter,” Phys. Rev. B 82(12), 125433 (2010). [CrossRef]

,47

K. Y. Bliokh, Y. Gorodetski, V. Kleiner, and E. Hasman, “Coriolis effect in optics: unified geometric phase and spin-Hall effect,” Phys. Rev. Lett. 101(3), 030404 (2008). [CrossRef] [PubMed]

] and semiconductor [34

] systems. Therefore, the presented unified geometric and operator description of the AM conversions provides a useful link between spin-orbit phenomena in nonparaxial wave fields of various nature.

Acknowledgements

This work was supported by the European Commission (Marie Curie Action), Science Foundation Ireland (Grant No. 07/IN.1/I906), and the Australian Research Council.

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32.	V. Garbin, G. Volpe, E. Ferrari, M. Versluis, D. Cojoc, and D. Petrov, “Mie scattering distinguishes the topological charge of an optical vortex: a homage to Gustav Mie,” New J. Phys. 11(1), 013046 (2009). [CrossRef]
33.	E. Brasselet, N. Murazawa, H. Misawa, and S. Juodkazis, “Optical vortices from liquid crystal droplets,” Phys. Rev. Lett. 103(10), 103903 (2009). [CrossRef] [PubMed]
34.	F. Manni, K. Lagoudakis, T. Paraïso, R. Cerna, Y. Léger, T. Liew, I. Shelykh, A. Kavokin, F. Morier-Genoud, and B. Deveaud-Plédran, “Spin-to-orbital angular momentum conversion in semiconductor microcavities,” Phys. Rev. B 83(24), 241307 (2011). [CrossRef]
35.	S. J. van Enk and G. Nienhuis, “Spin and orbital angular momentum of photons,” Europhys. Lett. 25(7), 497–501 (1994). [CrossRef]
36.	S. J. van Enk and G. Nienhuis, “Commutation rules and eigenvalues of spin and orbital angular momentum of radiation fields,” J. Mod. Opt. 41(5), 963–977 (1994). [CrossRef]
37.	E. Wolf, ““Electromagnetic diffraction in optical systems. I. An integral representation of the image field,” Proc. Roy. Soc. London,” Ser. A 253, 349–357 (1959).
38.	B. Richards and E. Wolf, ““Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system,” Proc. R. Soc. Lond. A Math. Phys. Sci. 253(1274), 358–379 (1959). [CrossRef]
39.	M. V. Berry, “Interpreting the anholonomy of coiled light,” Nature 326(6110), 277–278 (1987). [CrossRef]
40.	P. Török, P. D. Higdon, and T. Wilson, “On the general properties of polarized light conventional and confocal microscopes,” Opt. Commun. 148(4-6), 300–315 (1998). [CrossRef]
41.	A. Bekshaev, K. Y. Bliokh, and M. Soskin, “Internal flows and energy circulation in light beams,” J. Opt. 13(5), 053001 (2011). [CrossRef]
42.	R. Bhandari, “Polarization of light and topological phases,” Phys. Rep. 281(1), 2–64 (1997). [CrossRef]
43.	M. A. Alonso and G. W. Forbes, “Uncertainty products for nonparaxial wave fields,” J. Opt. Soc. Am. A 17(12), 2391–2402 (2000). [CrossRef] [PubMed]
44.	M. A. Alonso, “The effect of orbital angular momentum and helicity in the uncertainty-type relations between focal spot size and angular spread,” J. Opt. 13(6), 064016 (2011). [CrossRef]
45.	N. Bokor, Y. Iketaki, T. Watanabe, and M. Fujii, “Investigation of polarization effects for high-numerical-aperture first-order Laguerre-Gaussian beams by 2D scanning with a single fluorescent microbead,” Opt. Express 13(26), 10440–10447 (2005). [CrossRef] [PubMed]
46.	Y. Iketaki, T. Watanabe, N. Bokor, and M. Fujii, “Investigation of the center intensity of first- and second-order Laguerre-Gaussian beams with linear and circular polarization,” Opt. Lett. 32(16), 2357–2359 (2007). [CrossRef] [PubMed]
47.	K. Y. Bliokh, Y. Gorodetski, V. Kleiner, and E. Hasman, “Coriolis effect in optics: unified geometric phase and spin-Hall effect,” Phys. Rev. Lett. 101(3), 030404 (2008). [CrossRef] [PubMed]
48.	M. Born and E. Wolf, Principles of Optics, 7th edn. (Pergamon, 2005).
49.	G. Moe and W. Happer, “Conservation of angular momentum for light propagating in a transparent anisotropic medium,” J. Phys. B 10(7), 1191–1208 (1977). [CrossRef]
50.	V. Rossetto and A. C. Maggs, “Writhing geometry of stiff polymers and scattered light,” Eur. Phys. J. B 29(2), 323–326 (2002). [CrossRef]
51.	D. Lacoste, V. Rossetto, F. Jaillon, and H. Saint-Jalmes, “Geometric depolarization in patterns formed by backscattered light,” Opt. Lett. 29(17), 2040–2042 (2004). [CrossRef] [PubMed]
52.	M. V. Berry, “Optical currents,” J. Opt. A, Pure Appl. Opt. 11(9), 094001 (2009). [CrossRef]
53.	A. Bekshaev and S. Sviridova, “Mechanical action of inhomogeneously polarized optical fields and detection of the internal energy flows,” arXiv:1102.3514.
54.	O. G. Rodríguez-Herrera, D. Lara, and C. Dainty, “Far-field polarization-based sensitivity to sub-resolution displacements of a sub-resolution scatterer in tightly focused fields,” Opt. Express 18(6), 5609–5628 (2010). [CrossRef] [PubMed]
55.	Y. A. Kravtsov, B. Bieg, and K. Y. Bliokh, “Stokes-vector evolution in a weakly anisotropic inhomogeneous medium,” J. Opt. Soc. Am. A 24(10), 3388–3396 (2007). [CrossRef] [PubMed]
56.	M. V. Berry, “Paraxial beams of spinning light,” Proc. SPIE 3487, 6–11 (1998). [CrossRef]

OCIS Codes
(260.5430) Physical optics : Polarization
(350.1370) Other areas of optics : Berry's phase
(260.6042) Physical optics : Singular optics

ToC Category:
Physical Optics

History
Original Manuscript: August 23, 2011
Manuscript Accepted: October 6, 2011
Published: December 7, 2011

Citation
Konstantin Y. Bliokh, Elena A. Ostrovskaya, Miguel A. Alonso, Oscar G. Rodríguez-Herrera, David Lara, and Chris Dainty, "Spin-to-orbital angular momentum conversion in focusing, scattering, and imaging systems," Opt. Express 19, 26132-26149 (2011)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-27-26132

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References

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C.-F. Li, “Spin and orbital angular momentum of a class of nonparaxial light beams having a globally defined polarization,” Phys. Rev. A80(6), 063814 (2009). [CrossRef]
K. Y. Bliokh, M. A. Alonso, E. A. Ostrovskaya, and A. Aiello, “Angular momenta and spin-orbit interaction of nonparaxial light in free space,” Phys. Rev. A82(6), 063825 (2010). [CrossRef]
N. B. Baranova, A. Y. Savchenko, and B. Y. Zel'dovich, “Transverse shift of a focal spot due to switching of the sign of circular-polarization,” JETP Lett.59, 232–234 (1994).
B. Y. Zel’dovich, N. D. Kundikova, and L. F. Rogacheva, “Observed transverse shift of a focal spot upon a change in the sign of circular polarization,” JETP Lett.59, 766–769 (1994).
V. Garbin, G. Volpe, E. Ferrari, M. Versluis, D. Cojoc, and D. Petrov, “Mie scattering distinguishes the topological charge of an optical vortex: a homage to Gustav Mie,” New J. Phys.11(1), 013046 (2009). [CrossRef]
E. Brasselet, N. Murazawa, H. Misawa, and S. Juodkazis, “Optical vortices from liquid crystal droplets,” Phys. Rev. Lett.103(10), 103903 (2009). [CrossRef] [PubMed]
F. Manni, K. Lagoudakis, T. Paraïso, R. Cerna, Y. Léger, T. Liew, I. Shelykh, A. Kavokin, F. Morier-Genoud, and B. Deveaud-Plédran, “Spin-to-orbital angular momentum conversion in semiconductor microcavities,” Phys. Rev. B83(24), 241307 (2011). [CrossRef]
S. J. van Enk and G. Nienhuis, “Spin and orbital angular momentum of photons,” Europhys. Lett.25(7), 497–501 (1994). [CrossRef]
S. J. van Enk and G. Nienhuis, “Commutation rules and eigenvalues of spin and orbital angular momentum of radiation fields,” J. Mod. Opt.41(5), 963–977 (1994). [CrossRef]
E. Wolf, ““Electromagnetic diffraction in optical systems. I. An integral representation of the image field,” Proc. Roy. Soc. London,” Ser. A253, 349–357 (1959).
B. Richards and E. Wolf, ““Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system,” Proc. R. Soc. Lond. A Math. Phys. Sci.253(1274), 358–379 (1959). [CrossRef]
M. V. Berry, “Interpreting the anholonomy of coiled light,” Nature326(6110), 277–278 (1987). [CrossRef]
P. Török, P. D. Higdon, and T. Wilson, “On the general properties of polarized light conventional and confocal microscopes,” Opt. Commun.148(4-6), 300–315 (1998). [CrossRef]
A. Bekshaev, K. Y. Bliokh, and M. Soskin, “Internal flows and energy circulation in light beams,” J. Opt.13(5), 053001 (2011). [CrossRef]
R. Bhandari, “Polarization of light and topological phases,” Phys. Rep.281(1), 2–64 (1997). [CrossRef]
M. A. Alonso and G. W. Forbes, “Uncertainty products for nonparaxial wave fields,” J. Opt. Soc. Am. A17(12), 2391–2402 (2000). [CrossRef] [PubMed]
M. A. Alonso, “The effect of orbital angular momentum and helicity in the uncertainty-type relations between focal spot size and angular spread,” J. Opt.13(6), 064016 (2011). [CrossRef]
N. Bokor, Y. Iketaki, T. Watanabe, and M. Fujii, “Investigation of polarization effects for high-numerical-aperture first-order Laguerre-Gaussian beams by 2D scanning with a single fluorescent microbead,” Opt. Express13(26), 10440–10447 (2005). [CrossRef] [PubMed]
Y. Iketaki, T. Watanabe, N. Bokor, and M. Fujii, “Investigation of the center intensity of first- and second-order Laguerre-Gaussian beams with linear and circular polarization,” Opt. Lett.32(16), 2357–2359 (2007). [CrossRef] [PubMed]
K. Y. Bliokh, Y. Gorodetski, V. Kleiner, and E. Hasman, “Coriolis effect in optics: unified geometric phase and spin-Hall effect,” Phys. Rev. Lett.101(3), 030404 (2008). [CrossRef] [PubMed]
M. Born and E. Wolf, Principles of Optics, 7th edn. (Pergamon, 2005).
G. Moe and W. Happer, “Conservation of angular momentum for light propagating in a transparent anisotropic medium,” J. Phys. B10(7), 1191–1208 (1977). [CrossRef]
V. Rossetto and A. C. Maggs, “Writhing geometry of stiff polymers and scattered light,” Eur. Phys. J. B29(2), 323–326 (2002). [CrossRef]
D. Lacoste, V. Rossetto, F. Jaillon, and H. Saint-Jalmes, “Geometric depolarization in patterns formed by backscattered light,” Opt. Lett.29(17), 2040–2042 (2004). [CrossRef] [PubMed]
M. V. Berry, “Optical currents,” J. Opt. A, Pure Appl. Opt.11(9), 094001 (2009). [CrossRef]
A. Bekshaev and S. Sviridova, “Mechanical action of inhomogeneously polarized optical fields and detection of the internal energy flows,” arXiv:1102.3514.
O. G. Rodríguez-Herrera, D. Lara, and C. Dainty, “Far-field polarization-based sensitivity to sub-resolution displacements of a sub-resolution scatterer in tightly focused fields,” Opt. Express18(6), 5609–5628 (2010). [CrossRef] [PubMed]
Y. A. Kravtsov, B. Bieg, and K. Y. Bliokh, “Stokes-vector evolution in a weakly anisotropic inhomogeneous medium,” J. Opt. Soc. Am. A24(10), 3388–3396 (2007). [CrossRef] [PubMed]
M. V. Berry, “Paraxial beams of spinning light,” Proc. SPIE3487, 6–11 (1998). [CrossRef]

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