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Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 22, Iss. 6 — Mar. 24, 2014
  • pp: 6586–6596
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Surface angular momentum of light beams

Marco Ornigotti and Andrea Aiello  »View Author Affiliations


Optics Express, Vol. 22, Issue 6, pp. 6586-6596 (2014)
http://dx.doi.org/10.1364/OE.22.006586


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Abstract

Traditionally, the angular momentum of light is calculated for “bullet-like” electromagnetic wave packets, although in actual optical experiments “pencil-like” beams of light are more commonly used. The fact that a wave packet is bounded transversely and longitudinally while a beam has, in principle, an infinite extent along the direction of propagation, renders incomplete the textbook calculation of the spin/orbital separation of the angular momentum of a light beam. In this work we demonstrate that a novel, extra surface part must be added in order to preserve the gauge invariance of the optical angular momentum per unit length. The impact of this extra term is quantified by means of two examples: a Laguerre-Gaussian and a Bessel beam, both circularly polarized.

© 2014 Optical Society of America

1. Introduction

In his Treatise of Electricity and Magnetism [1

1. J. C. Maxwell, A Treatise on Electricity and Magnetism(Dover, 1954).

], Maxwell pointed out that the electromagnetic field carries both energy and momentum, and that the momentum can have both linear and angular contributions. Today it is common knowledge that the angular momentum (AM) of light can be thought of having two contributions: a spin part associated to polarization (firstly theoretically investigated by Poynting [2

2. J. H. Poynting, “The wave motion of a revolving shaft, and a suggestion as to the angular momentum in a beam of circularly polarised light,” Proc. R. Soc. London Ser. A 82, 560–567 (1909). [CrossRef]

] and then experimentally demonstrated by Beth [3

3. R. A. Beth, “Mechanical detection and measurement of the angular momentum of light,” Phys. Rev. 50, 115–125 (1936). [CrossRef]

]), and an orbital part associated with the spatial distribution of the field, as first recognised by Darwin [4

4. C. G. Darwin, “Notes on the theory of radiation,” Proc. R. Soc. London A 136, 36–52 (1932). [CrossRef]

]. About sixty years later, thanks to the seminal work of Allen and Woerdman [5

5. L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992). [CrossRef] [PubMed]

], the topic of AM of light experienced a new renaissance, producing a considerable amount of literature on the subject, including theoretical discussions about the separation of AM in spin and orbital parts for optical beams [6

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

14

14. I. Bialynicki-Birula and Z. Bialynicki-Birula, “Canonical separation of angular momentum of light into its orbital and spin parts,” J. Opt. 13, 064014 (2011). [CrossRef]

], the spin-Hall effect of light [15

15. O. Hosten and P. Kwiat, “Observation of the spin Hall effect of light via weak measurements,” Science 319, 787–790 (2008). [CrossRef] [PubMed]

18

18. K. Y. Bliokh and Y. P. Bliokh, “Conservation of angular momentum, transverse shift, and spin Hall effect in reflection and refraction of an electromagnetic wave packet,” Phys. Rev. Lett. 96, 073903 (2006). [CrossRef] [PubMed]

], the geometric spin-Hall effect of light [19

19. A. Y. Bekshaev, “Oblique section of a paraxial light beam: criteria for azimuthal energy flow and orbital angular momentum,” J. Opt. A Pure Appl. Opt. 11, 094003 (2009). [CrossRef]

,20

20. A. Aiello, N. Lindlein, C. Marquardt, and G. Leuchs, “Transverse angular momentum and geometric spin Hall effect of light,” Phys. Rev. Lett. 103, 100401 (2009). [CrossRef] [PubMed]

], and a huge variety of experimental applications such as optical tweezers, spatial light modulators and vortex beams. A satisfactory review of the development of the field of optical AM of light can be found in Refs. [21

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

23

23. K. Y. Bliokh, A. Aiello, and M. Alonso, “Spin-orbit interactions of light in isotropic media,” in The Angular Momentum of Light, D. L. Andrews and M. Babiker, eds. (Cambridge University, 2012), 174–245.

].

The AM of light has been frequently separated, by many authors, into a spin part and an orbital part (see, e.g., [24

24. H. A. Haus and J. L. Pan, “Photon spin and the paraxial wave equation,” Am. J. Phys. 61, 818–821 (1993). [CrossRef]

] for a didactic presentation of the subject). As emphasized by Birula&Birula [14

14. I. Bialynicki-Birula and Z. Bialynicki-Birula, “Canonical separation of angular momentum of light into its orbital and spin parts,” J. Opt. 13, 064014 (2011). [CrossRef]

] such separation can be either gauge dependent or gauge invariant. The latter result is achieved, according to Darwin [4

4. C. G. Darwin, “Notes on the theory of radiation,” Proc. R. Soc. London A 136, 36–52 (1932). [CrossRef]

], by expressing the optical AM in terms of the Fourier transform of the electromagnetic fields. On the contrary, the gauge dependent separation is obtained, traditionally, when in the expression of the optical AM
𝒥=ε0r×(E×B)d3r,
(1)
the magnetic field B is replaced by the curl of the vector potential A, namely B = × A. After partial integration Eq. (1) takes the split form [25

25. C. Cohen-Tannoudji, J. Dupont-Roc, and G. Grynberg, Photons and Atoms (Wiley-VCH, 2004), Chap. I.

, 26

26. J. B. Götte and S. M. Barnett, “Light beams carrying orbital angular momentum,” in The Angular Momentum of Light, D. L. Andrews and M. Babiker, eds. (Cambridge University, 2012), 1–30.

]
𝒥=ε0E×Ad3r+ε0ξEξ(r×)Aξd3rε0ξξ(Eξr×A)d3r,
(2)
where ξ ∈ {x, y, z} and the first and the second integral represent the spin part and the orbital part, respectively, of the optical AM. The third integrals is usually omitted in the textbook expressions of 𝒥 because, as consequence of the Gauss’ theorem, it will be identically zero if the fields vanish sufficiently quickly as |r| → ∞. This requirement, as pointed out by Crichton&Marston [27

27. J. H. Crichton and P. L. Marston, “The measurable distinction between the spin and orbital angular momenta of electromagnetic radiation,” Electron. J. Diff. Eq. Conf. 04, 37–50 (2000).

] and Nieminen et al. [28

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

], is often understood while its fulfillment should be carefully checked case by case. For example, the electric and magnetic fields of an optical “bullet-like” wave packet with a finite transverse and longitudinal (with respect to the direction of propagation) extent, by definition vanish as |r| → ∞ and the third term in Eq. (2) can be safely neglected. However, this is no longer true for a “pencil-like” beam of light whose span along the direction of propagation is virtually infinite. This simple but important fact was already noticed by Barnett&Allen [6

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

] (see Eqs. (3.15–18) in their paper) who, nevertheless, simply classified the third integral in Eq. (2) as a nonparaxial contribution to the AM per unit length of the beam and did not investigate its physical content.

In this paper we thoroughly investigate the physical nature of the latter term in Eq. (2) that we call the “Surface Angular Momentum” (SuAM) of light. At the fundamental level, we demonstrate that even for an optical beam with a small angular spread around the direction of propagation (paraxial beams), the SuAM term must be retained in order to guarantee the gauge invariance of the theory. From an experimental point of view, we show that SuAM arises whenever the intensity of an optical beam is recorded by a planar detector which, in practice, performs a two dimensional integration over a cross section of the beam perpendicular to its direction of propagation. As this is the case occurring in the majority of experimental optical setups, we believe that the impact of our new results may be significant. All our conclusions are valid for light beams propagating in free space.

The structure of this Article is as follows. In Sec. II we first calculate the total AM density j(r) of a monochromatic beam of light described in terms of the vector potential A. Then, by following the standard procedure, we decompose j(r) into the sum of three terms: the well-known orbital and spin AM parts and the third term, leading to the SuAM. Next, we integrate j(r) upon a two-dimensional surface at z = const., mimicking the detection process, thus obtaining the total AM per unit length J. This unravels the nature of the SuAM term and how it manifests in the actual detection of the AM of a light beam. For the sake of completeness, we conclude this section by calculating the explicit form of the SuAM for the cases of Laguerre-Gaussian and Bessel beams. In Sec. III we calculate again J but, this time, in a gauge invariant manner by using the angular spectrum representation of the electric and magnetic fields. Then, by comparing the expressions for J obtained in Secs. II and III we show that the SuAM term cannot be neglected even for paraxial optical beams. Finally, we draw our conclusions.

2. Theory of SuAM

To begin with, consider a monochromatic beam of light with angular frequency ω whose real electric and magnetic fields (r, t) and (r, t) are expressed in terms of the complex field amplitudes E(r) and B(r) as (r, t) = Re[E(r) exp(−iωt)] and (r, t) = Re[B(r) exp(−iωt)]. Following van Enk and Nienhuis [8

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

] we write the cycle-averaged linear momentum, angular momentum and energy densities, respectively,
p(r)=ε02Re(E*×B),
(3a)
j(r)=ε02r×Re(E*×B),
(3b)
u(r)=ε04(EE*+c2BB*),
(3c)
where E and B are written in terms of the transverse vector potential A(r) as [25

25. C. Cohen-Tannoudji, J. Dupont-Roc, and G. Grynberg, Photons and Atoms (Wiley-VCH, 2004), Chap. I.

]
E=iωAand B=×Awith A=0.
(4)
By using Eqs. (3) and (4) we can express the AM densities j(r) as [27

27. J. H. Crichton and P. L. Marston, “The measurable distinction between the spin and orbital angular momenta of electromagnetic radiation,” Electron. J. Diff. Eq. Conf. 04, 37–50 (2000).

, 29

29. J. Humblet, “Sur le moment d’impulsion d’une onde électromagnètique,” Physica 10, 585–603 (1943). [CrossRef]

]:
j(r)=ωε02Re{A*(ir×)AiA*×A+iξ{x,y,z}ξ[Aξ*(r×A)]}.
(5)
According to [27

27. J. H. Crichton and P. L. Marston, “The measurable distinction between the spin and orbital angular momenta of electromagnetic radiation,” Electron. J. Diff. Eq. Conf. 04, 37–50 (2000).

], the first term in Eq. (5) represents the orbital part of the AM and the second term gives the spin AM of light. The third term is a three-divergence that vanishes when integrated over a volume V whose surface boundary S is far enough from the field sources to give A(r)|S0. However, as noticed in [27

27. J. H. Crichton and P. L. Marston, “The measurable distinction between the spin and orbital angular momenta of electromagnetic radiation,” Electron. J. Diff. Eq. Conf. 04, 37–50 (2000).

], the latter relation should be checked whenever the decomposition (5) is used. In the remainder of this Article, we will show some consequences of the failure of the relation A(r)|S0.

Haus and Pan [24

24. H. A. Haus and J. L. Pan, “Photon spin and the paraxial wave equation,” Am. J. Phys. 61, 818–821 (1993). [CrossRef]

] pointed out that for a monochromatic beam of light of angular frequency ω propagating in the z direction, the classical optics analogous of the helicity ± of a photon, namely the projection of the spin AM along the direction of the linear momentum, is given by the ratio êz · J/U = ±1/ω, where J and U denote the cycle-averaged AM and energy per unit length:
J=j(r)d2r,U=u(r)d2r,
(6)
where d2r = dx dy and the integrations extend over all the plane xy at z = const. It should be noticed that for a monochromatic optical beam that can be represented by a superposition of homogeneous plane waves solely (namely, without contributions from evanescent waves) both J and U do not depend upon the longitudinal coordinate z and, therefore, are conserved during free propagation. Henceforth, we shall use over line symbol to denote normalization according to the rule
J¯=JU/ω.
(7)
This definition of helicity is consistent with the physical picture of a beam of light as observed on a laboratory bench, where it travels across plates, lenses and other optical devices before eventually shining the surface of a planar detector. Last but not least, it should be noticed that the calculations by Haus and Pan [24

24. H. A. Haus and J. L. Pan, “Photon spin and the paraxial wave equation,” Am. J. Phys. 61, 818–821 (1993). [CrossRef]

] are valid not only in the strict paraxial limit where the fields are purely transverse with respect to the propagation axis z, but also when first-order corrections (in the sense of Lax et al., [30

30. M. Lax, W. H. Louisell, and W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975). [CrossRef]

]), leading to small longitudinal fields components, are accounted for.

As a consequence of the surface integration, a beam-like field “feels” the contribution given by the z-part of the latter term in Eq. (5), namely z[Az*(r×A)], because such a term is not integrated when one calculates J. In practice, since ξ[Aξ*(r×A)]d2r=zAz*(r×A)d2r, we can write the SuAM contribution in the following form:
Jsurf=zRe{iAz*(r×A)}d2r,
(8)
where Re{iAz*(r×A)}=i[z(A*×A)r(A*×A)e^z]/2. From Eq. (8) it appears that the Jsurf depends both on the potential vector A and the position vector r, thus revealing a seemingly hybrid spin/orbital nature. Now, we can calculate Jsurf from Eq. (8) by writing the vector potential as a superposition of homogeneous plane waves
A(r)=12πA˜(k)exp(irk+izkz)d2k,
(9)
where Ã(k) : k · Ã(k) = 0 is the so-called angular spectrum of the field evaluated at z = 0 [31

31. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995). [CrossRef]

] and k = (kx, ky, kz) is the wave vector with k = |k| and ω = kc. For the sake of consistency, henceforth we shall use the following notation: r = (x, y, 0), k = (kx, ky, 0), r · k = xkx + yky, k = |k| and d2k = dkx dky. By definition, kz is not an independent variable and can be fixed to the non-negative value kz=+(k2k2)1/20, which is appropriate for the forward homogeneous fields considered here. For a circularly polarized optical beam, the angular spectrum can be chosen of the form
A˜(k)=A˜(k)ε(σ,k),=|A˜(k)|exp[iα˜(k)]ε(σ,k),
(10)
where the real-valued amplitude |Ã(k)| and phase α̃(k) determines the spatial profile of the beam and ε (σ, k) is the complex-valued polarization vector defined as:
ε(σ,k)=nσk(knσ)/k2=k×(k×nσ)/k2.
(11)
In the equation above nσ=(e^x+iσe^y)/2 with σ = ±1, represents the wave vector-independent configuration-space helicity assigned in the (global) laboratory reference frame {êx, êy, êz}. By definition ε (σ, k) is not normalized; however, since its norm
ε*(σ,k)ε(σ,k)=1|knσ|2/k2=1k22k21ϑ2,
(12)
is polarization-independent, we can assume that the normalization factor is tacitly contained in the expression of Ã(k). Here, the dimensionless parameter 0ϑ<1/2:
ϑ2=kx2+ky22k2,
(13)
defines the angular spread of the beam around the direction of propagation z, and the paraxial regime is characterized by angular spectra such that the condition ϑ ≪ 1 is satisfied. At this point, by substituting Eq. (9) into Eq. (5) and using Eqs. (10) and (11) one obtains after a straightforward calculation the orbital part
Jxorb=ε0ω2|A˜|2[σkxkzk2+kz2+kzα˜ky]d2k,
(14a)
Jyorb=ε0ω2|A˜|2[σkykzk2+kz2kzα˜ky]d2k,
(14b)
Jzorb=ε0ω2|A˜|2[σϑ21ϑ2+(kxα˜kykyα˜ky)]d2k,
(14c)
the spin part
Jxspin=ε0ω22σ|A˜|2kxkzk2+kz2d2k,
(15a)
Jyspin=ε0ω22σ|A˜|2kykzk2+kz2d2k,
(15b)
Jzspin=ε0ω2σ|A˜|212ϑ21ϑ2d2k,
(15c)
and the surface part of the optical AM per unit of length:
Jxsurf=ε0ω2σ|A˜|2kxkzk2+kz2d2k,
(16a)
Jxsurf=ε0ω2σ|A˜|2kykzk2+kz2d2k,
(16b)
Jzsurf=ε0ω2σ|A˜|2ϑ21ϑ2d2k.
(16c)
In addition, the energy per unit length is given by
Uω=ε0ω2|A|2d2k.
(17)
Note that all the quantities (14–17) are z-independent, so they cannot vanish if one would perform an additional integration along the axis z to obtain the AMs in a given, finite, volume.

The explicit form for Jsurf is the first main result of this Article. What does it represent? First of all, it should be noticed that at this stage the angular spectrum Ã(k) is perfectly general and may represent, for example, an electromagnetic beam with arbitrary orbital AM. The z-component of the integrand in Eq. (16c) is proportional to ϑ2/(1 − ϑ2) as noticed already in [6

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

]. This contribution is O(ϑ2) and therefore disappears in the strict paraxial limit where such terms are not retained [24

24. H. A. Haus and J. L. Pan, “Photon spin and the paraxial wave equation,” Am. J. Phys. 61, 818–821 (1993). [CrossRef]

]. In order to understand better how Jsurf affects the total AM per unit length we sum the three contribution and obtain
Jzorb+Jzspin+Jzsurf=ε0ω2|A˜|2[σ1ϑ2+(kxα˜kykyα˜kx)]d2k.
(18)
It is instructive to compare Eq. (18) with Eq. (38) in [9

9. S. M. Barnett, “Optical angular-momentum flux,” J. Opt. B Quantum Semiclass. Opt. 4, S7–S16 (2002). [CrossRef]

] where Barnett calculates the ratio between the cycle-averaged angular momentum density flux zz and the energy density flux through the xy plane, namely zz/ℱ. In our case, for a circularly polarized beam with helicity σ and by using Eq. (9), it follows that
zz=ε0c22|A˜|2kz[σ+(kxα˜kykyα˜kx)]d2k,
(19)
and
=c2ε0ω2|A˜|2kzd2k.
(20)
The main difference between Eq. (18) and Eq. (19) resides, apart from the trivial factor c2/ω, in the extra multiplicative term kz = k(1 − 2ϑ2)1/2k(1 − ϑ2) in the integrand. Of course, in the paraxial limit kzk and the two expressions coincides. There is, however, a profound conceptual and practical difference between the energy density U and the energy density flux (Poynting vector flux). The physical quantity that is actually measured by common detection devices as CCD detectors, photographic plates, photoresists, etc., is the time-averaged value of the scalar energy density (namely, loosely speaking, the number of photons in the unit volume) integrated over the detector surface, rather than the Poynting vector flux [32

32. R. Loudon, The Quantum Theory of Light, 3rd ed. (Oxford University, 2000).

34

34. J. J. M. Braat, S. van Haver, A. J. E. M. Janssen, and P. Dirksen, “Energy and momentum flux in a high-numerical-aperture beam using the extended Nijboer-Zernike diffraction formalism,” J. Eur. Opt. Soc. Rapid 2, 07032 (2007). [CrossRef]

]. This distinction between scalar and vector quantities becomes crucial when measuring spin-dependent non-paraxial optical phenomena as, e.g., the geometric spin Hall effect of light [20

20. A. Aiello, N. Lindlein, C. Marquardt, and G. Leuchs, “Transverse angular momentum and geometric spin Hall effect of light,” Phys. Rev. Lett. 103, 100401 (2009). [CrossRef] [PubMed]

]. In fact, the standard theory of photo-detection (see, e.g., sec. 4.11 of [32

32. R. Loudon, The Quantum Theory of Light, 3rd ed. (Oxford University, 2000).

] and chap. III of [25

25. C. Cohen-Tannoudji, J. Dupont-Roc, and G. Grynberg, Photons and Atoms (Wiley-VCH, 2004), Chap. I.

]) shows that the observable intensity of the electromagnetic field is proportional to the probability of observing a photoionization in a phototube detector which is given by the expectation value of the electric-field energy density operator Ê(−)(r, t) · Ê(+)(r, t). As shown in [32

32. R. Loudon, The Quantum Theory of Light, 3rd ed. (Oxford University, 2000).

], only when a light beam is at least approximately paraxial the observable intensity coincides with the flux across the detector surface of the Poynting vector operator Î(r, t) = ε0c2 {Ê(−)(r, t) × (+)(r, t) − (−)(r, t) × Ê(+)(r, t)}. Therefore, in general, is the electromagnetic energy density and not the Poynting vector flux that matters in ordinary photodetection.

It should be noticed that both Jsurf and Jspin are proportional to the helicity σ of the incident beam. However, this should not be interpreted as a signature of the spin nature of the SuAM, since such a dependence also appears in spin-to-orbit conversion phenomena [35

35. D. B. Ruffner and D. G. Grier, “Optical forces and torques in nonuniform beams of light,” Phys. Rev. Lett. 108, 173602 (2012). [CrossRef] [PubMed]

]. From Eqs. (15) and (16) it follows that
Jzspin+Jzsurf=ε0ω2σ|A˜|2d2k,
(21)
from which, by using Eqs. (7) and (17), we obtain
Jzspin¯+Jzsurf¯=σ.
(22)
Equation (22) shows that adding the z-component of the SuAM term to the z-component of the spin AM per unit length ensures that the total “spin” AM per unit length along the propagation axis z is exactly equal to σ times the energy per unit length of the beam, the latter being completely arbitrary. This our second main result.

2.1. Examples

In the remainder of this section, we calculate explicitly the SuAM for Laguerre-Gaussian and Bessel beams to investigate how orbital AM affects Jsurf¯. Consider first the case of a Laguerre-Gaussian beam of waist w0, angular aperture θ0 = 2/(kw0) and with azimuthal and radial mode indexes and p, respectively [36

36. A. E. Siegman, Lasers (University Science Books, 1986).

]. In this case, from Eqs. (15) and (16) we obtain
Jsurf¯=σβ1βe^z,Jspin¯=σ12β1βe^z,
(23)
where β = (θ0/2)2(2p + || + 1) [37

37. The same “correction” factor β was found in [6]. However, the expression (4.4) in [6], when evaluated for l= 0 = p in the paraxial limit 1/(2kzR)=θ02/41 gives 𝒥z/(σz/ω)(1+θ02/4) instead of 𝒥z/(σz/ω)(1θ02/4), as given by Eq. (8) in [28]. The difference between the two results resides in the fact that different types of beams and geometries (planar versus spherical) are considered.

]. Although β depends on || such a dependence is not a consequence of the orbital AM of the beam (this would lead to a dependence from the signed quantity ), but it comes from the radial dependence of the beam profile. For a fundamental Gaussian beam = 0 = p and βθ02/4. In this case, for θ0 ≪ 1, the small angle approximation furnishes Jzspin¯/σ1θ02/4, in agreement with [28

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

].

Next, we consider the case of a th-order Bessel beam that, in the cylindrical polar coordinates (r, ϕ, z), takes the form E(r, ϕ, z) = J(rksinϑ0) exp (iℓϕ)exp(izkcosϑ0), where ϑ0 is the angular aperture of the characteristic Bessel cone. A straightforward calculation shows that Eq. (23) are still valid providing that β = sin2ϑ0/2. Note that for both Laguerre-Gaussian and Bessel paraxial beams with θ0 = ϑ0 ≪ 1, one has Jsurf¯~O(θ02), namely Jsurf¯ becomes negligible. The absolute value of the “surface helicity” σsurfJsurf¯e^z, is depicted in Fig. 1 for both the case of a fundamental Gaussian beam (green solid line) and of a Bessel beam (blue solid line) for ϑ0 = θ0 ∈ [0, π/2]. In the paraxial domain σsurf approaches zero and its contribution can be neglected for θ0 ≪ 1. However, for non paraxial values of θ0, |σsurf| becomes comparable with the standard helicity value |σ| = 1 [24

24. H. A. Haus and J. L. Pan, “Photon spin and the paraxial wave equation,” Am. J. Phys. 61, 818–821 (1993). [CrossRef]

].

Fig. 1 SuAM for a circularly polarized fundamental Gaussian beam (green line) and Bessel beam (blue line). At θ0=θc2rad81° the tails of the Gaussian angular spectrum distribution |A˜(k)2|exp[2(kx2+ky2)/θ02] becomes no longer negligible for kx2+ky2k2 where evanescent waves occur [31]. The angular spectrum representation Eq. (9) with kz=+(k2kx2ky2)1/2 does not account for evanescent waves, therefore it breaks down for θ0 > θc. This critical value is marked by a dashed vertical line. Such a problem does not occur for a Bessel beam because in this case the angular spectrum does not possess tails but is sharply peaked about ϑ0.

3. Gauge invariant calculation

In this section we calculate again the AM per unit length for a monochromatic beam of light. However, this time we will not use the vector potential and, thus, our calculation will be manifestly gauge invariant. Having this goal in mind, we write the electromagnetic fields in their (homogeneous) angular spectrum form as
E(r)=12πE˜(k)exp(irk+izkz)d2k,
(24a)
B(r)=12πB˜(k)exp(irk+izkz)d2k,
(24b)
where and fulfill the following relations stemming from Maxwell equations:
kE˜=0,
(25a)
kB˜=0,
(25b)
cB˜=kk×E˜.
(25c)
As before, for a circularly polarized beam of light we can choose
E˜(k)=E˜(k)ε(σ,k)=|E˜(k)|exp[iθ˜(k)]ε(σ,k)
(26)
and from Eq. (25c) it follows that c(k) = (k)β(σ, k), where
β(σ,k)=k×ε(σ,k)/k=(k×nσ)/k.
(27)
Once again, we assume that the normalization factor for the polarization vector ε(σ, k) is contained in the complex-valued amplitude (k).

By substituting Eqs. (24)(27) into Eq. (3) we obtain, after a straightforward calculation, the following expressions for the energy per unit length
Uω=ε02ω|E˜|2d2k,
(28)
for the Poynting vector
S=c2ε02ω|E|2kd2k,
(29)
and for the AM per unit length:
Jx=ε02ω|E˜|2kzθ˜kyd2k,
(30a)
Jy=ε02ω|E˜|2kzθ˜kxd2k,
(30b)
Jz=ε02ω|E˜|2[σ1ϑ2+(kxθ˜kykyθ˜kx)]d2k.
(30c)
By comparing Eq. (30c) with the expression of the AM flux below
zz=c2ε02ω2|E˜|2kz[σ+(kxθ˜kykyθ˜kx)]d2k,
(31)
one may be tempted to identify the term proportional to σ in Eq. (30c) with the spin part of the AM per unit length. However, such a term differs from the expression of Jzspin given by Eq. (15). Moreover, from Eqs. (14) and (15) it follows that
Jzorb+Jzspin=ε0ω2|A˜|2[σ+(kxα˜kykyα˜kx)]d2k,
(32)
which does not coincides with Eq. (30c). However, if we add Jzsurf to Eq. (32) we obtain the result given by Eq. (18) that we rewrite below:
Jzorb+Jzspin+Jzsurf=ε0ω2|A˜|2[σ1ϑ2+(kxα˜kykyα˜kx)]d2k.
(33)
Finally, the two expressions given by Eq. (30c) and Eq. (33) coincide (apart from the trivial factor ω linking the electric field and the vector potential amplitudes and Ã, respectively). An analogous result is found for the other two components Jx and Jy. It should be noted that Eq. (33) is independent from z, thus indicating that the total (spin plus orbital plus surface) AM per unit of length is conserved during the free propagation of the beam.

In summary, we can recollect the results obtained in this section as follows: By using the angular spectrum representation for the electromagnetic fields, we have calculated the AM per unit length J for an arbitrary beam of light, either paraxial or non paraxial. Then, we compared this result given by Eq. (30) with the expressions (14–16) of Jorb, Jspin and Jsurf and we found that
J=Jorb+Jspin+Jsurf.
(34)
The equation above unambiguously shows that the gauge invariant expression of J on the left side can be obtained via the gauge dependent calculation on the right side, only when the SuAM is accounted for, irrespective of the paraxial or non paraxial nature of the beam of light.

4. Conclusions

One of the key aspects of this work, is the illustration of the interplay between paraxial approximation, AM of light and gauge invariance. The actual situation can be schematically illustrated as follows. In the simplest electromagnetic field, the plane wave, the electric and magnetic fields are strictly perpendicular to the direction of propagation of the wave. Differently, for a beam of finite waist, a nonzero component of the electric field Ez parallel to the direction of propagation z of the beam, is unavoidable [38

38. W. L. Erikson and S. Singh, “Polarization properties of Maxwell-Gaussian laser beams,” Phys. Rev. A 49, 5778–5786 (1994).

]. This component, although small with respect to Ex and Ey in collimated beams, is responsible for the difference between A) linear and angular momentum fluxes from one side, and B) energy density and AM per unit of length from the other side. In fact, as shown in [9

9. S. M. Barnett, “Optical angular-momentum flux,” J. Opt. B Quantum Semiclass. Opt. 4, S7–S16 (2002). [CrossRef]

], both fluxes depends only upon the transverse fields components Ex, Ey and Bx, By. On the opposite, it can be shown that the energy density and the AM per unit of length, receive a contribution from Ez and Bz, as well [20

20. A. Aiello, N. Lindlein, C. Marquardt, and G. Leuchs, “Transverse angular momentum and geometric spin Hall effect of light,” Phys. Rev. Lett. 103, 100401 (2009). [CrossRef] [PubMed]

]. What quantities are more physically meaningful, A) or B)? There is not a definite answer to this question. In principle, the fluxes obey more general conservation laws and therefore, from a theoretical point of view, are more appealing. On the other hand, the quantities more commonly measured in standard laboratories, are the energy density and the AM per unit of length. In practice, it will be the experimental configuration at hand to set the choice between A) and B), because spin, orbital and total AM of light may be measured by means of different techniques [23

23. K. Y. Bliokh, A. Aiello, and M. Alonso, “Spin-orbit interactions of light in isotropic media,” in The Angular Momentum of Light, D. L. Andrews and M. Babiker, eds. (Cambridge University, 2012), 174–245.

]. Many detection apparatuses yield the ratio between the angular momentum per unit length and the energy density per unit length. Typical examples thereof are Stokes parameter measurements in paraxial polarization optics (the third Stokes parameter, usually denotes s3, gives a measure of σ; see, e.g., [27

27. J. H. Crichton and P. L. Marston, “The measurable distinction between the spin and orbital angular momenta of electromagnetic radiation,” Electron. J. Diff. Eq. Conf. 04, 37–50 (2000).

] and Sec. 6.2 of [31

31. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995). [CrossRef]

]) and interferometric methods in single photon AM detection [39

39. J. Leach, J. Courtial, K. Skeldon, S. M. Barnett, S. Franke-Arnold, and M. J. Padgett, “Interferometric methods to measure orbital and spin, or the total angular momentum of a single photon,” Phys. Rev. Lett. 92, 013601 (2004). [CrossRef] [PubMed]

]. In all these cases, the energy density of light per unit length is measured by means of commonplace photo-detectors as CCD cameras, photographic plates, photoresists, etc. and the quantities calculated in this work are the relevant ones.

How gauge invariance enters this discussion? In the paraxial approximations the quantities A) and B) coincide and the spin/orbital separation of the AM is manifestly gauge-invariant in both cases. However, beyond paraxial approximation only the fluxes A) remain explicitly gauge-invariant, as shown by Barnett [9

9. S. M. Barnett, “Optical angular-momentum flux,” J. Opt. B Quantum Semiclass. Opt. 4, S7–S16 (2002). [CrossRef]

]. Oppositely, gauge-invariance is no longer guaranteed for the AM per unit of length outside paraxial approximation. In fact, in this paper we have demonstrated that the textbook expression for the angular momentum per unit length of a beam of light, containing a spin and an orbital part solely, is not complete and that a third part must be included in order to preserve the gauge invariance of the theory. We call this new term the surface AM (SuAM). This quantity is derived considering the virtually infinite extent of the beam along the direction of propagation. We believe that our results may have a relevant conceptual impact upon the very lively and timely research field about AM of light [40

40. 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, 253601 (2010). [CrossRef] [PubMed]

, 41

41. P. Banzer, U. Peschel, S. Quabis, and G. Leuchs, “On the experimental investigation of the electric and magnetic response of a single nano-structure,” Opt. Express 18, 10905–10923 (2010). [CrossRef] [PubMed]

].

Acknowledgments

We are grateful to Konstantin Bliokh for fruitful and stimulating discussions.

References and links

1.

J. C. Maxwell, A Treatise on Electricity and Magnetism(Dover, 1954).

2.

J. H. Poynting, “The wave motion of a revolving shaft, and a suggestion as to the angular momentum in a beam of circularly polarised light,” Proc. R. Soc. London Ser. A 82, 560–567 (1909). [CrossRef]

3.

R. A. Beth, “Mechanical detection and measurement of the angular momentum of light,” Phys. Rev. 50, 115–125 (1936). [CrossRef]

4.

C. G. Darwin, “Notes on the theory of radiation,” Proc. R. Soc. London A 136, 36–52 (1932). [CrossRef]

5.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992). [CrossRef] [PubMed]

6.

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

7.

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

8.

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

9.

S. M. Barnett, “Optical angular-momentum flux,” J. Opt. B Quantum Semiclass. Opt. 4, S7–S16 (2002). [CrossRef]

10.

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

11.

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

12.

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, 063825 (2010). [CrossRef]

13.

S. M. Barnett, “Rotation of electromagnetic fields and the nature of optical angular momentum,” J. Mod. Opt. 57, 1339–1343 (2010). [CrossRef]

14.

I. Bialynicki-Birula and Z. Bialynicki-Birula, “Canonical separation of angular momentum of light into its orbital and spin parts,” J. Opt. 13, 064014 (2011). [CrossRef]

15.

O. Hosten and P. Kwiat, “Observation of the spin Hall effect of light via weak measurements,” Science 319, 787–790 (2008). [CrossRef] [PubMed]

16.

A. Aiello and J. P. Woerdman, “Role of beam propagation in Goos-Hänchen and Imbert-Fedorov shifts,” Opt. Lett. 33, 1437–1439 (2008). [CrossRef] [PubMed]

17.

M. Onoda, S. Murakami, and N. Nagaosa, “Hall effect of light,” Phys. Rev. Lett. 93, 083901 (2004). [CrossRef] [PubMed]

18.

K. Y. Bliokh and Y. P. Bliokh, “Conservation of angular momentum, transverse shift, and spin Hall effect in reflection and refraction of an electromagnetic wave packet,” Phys. Rev. Lett. 96, 073903 (2006). [CrossRef] [PubMed]

19.

A. Y. Bekshaev, “Oblique section of a paraxial light beam: criteria for azimuthal energy flow and orbital angular momentum,” J. Opt. A Pure Appl. Opt. 11, 094003 (2009). [CrossRef]

20.

A. Aiello, N. Lindlein, C. Marquardt, and G. Leuchs, “Transverse angular momentum and geometric spin Hall effect of light,” Phys. Rev. Lett. 103, 100401 (2009). [CrossRef] [PubMed]

21.

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

22.

Optical Angular Momentum, L. Allen, S. M. Barnett, and M. J. Padgett, eds. (Institute of Physics, 2003). [CrossRef]

23.

K. Y. Bliokh, A. Aiello, and M. Alonso, “Spin-orbit interactions of light in isotropic media,” in The Angular Momentum of Light, D. L. Andrews and M. Babiker, eds. (Cambridge University, 2012), 174–245.

24.

H. A. Haus and J. L. Pan, “Photon spin and the paraxial wave equation,” Am. J. Phys. 61, 818–821 (1993). [CrossRef]

25.

C. Cohen-Tannoudji, J. Dupont-Roc, and G. Grynberg, Photons and Atoms (Wiley-VCH, 2004), Chap. I.

26.

J. B. Götte and S. M. Barnett, “Light beams carrying orbital angular momentum,” in The Angular Momentum of Light, D. L. Andrews and M. Babiker, eds. (Cambridge University, 2012), 1–30.

27.

J. H. Crichton and P. L. Marston, “The measurable distinction between the spin and orbital angular momenta of electromagnetic radiation,” Electron. J. Diff. Eq. Conf. 04, 37–50 (2000).

28.

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

29.

J. Humblet, “Sur le moment d’impulsion d’une onde électromagnètique,” Physica 10, 585–603 (1943). [CrossRef]

30.

M. Lax, W. H. Louisell, and W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975). [CrossRef]

31.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995). [CrossRef]

32.

R. Loudon, The Quantum Theory of Light, 3rd ed. (Oxford University, 2000).

33.

J. J. M. Braat, P. Dirksen, A. J. E. M. Janssen, and A. S. van de Nes, “Extended Nijboer-Zernike representation of the vector field in the focal region of an aberrated high-aperture optical system,” J. Opt. Soc. Am. A 20, 2281–2292 (2003). [CrossRef]

34.

J. J. M. Braat, S. van Haver, A. J. E. M. Janssen, and P. Dirksen, “Energy and momentum flux in a high-numerical-aperture beam using the extended Nijboer-Zernike diffraction formalism,” J. Eur. Opt. Soc. Rapid 2, 07032 (2007). [CrossRef]

35.

D. B. Ruffner and D. G. Grier, “Optical forces and torques in nonuniform beams of light,” Phys. Rev. Lett. 108, 173602 (2012). [CrossRef] [PubMed]

36.

A. E. Siegman, Lasers (University Science Books, 1986).

37.

The same “correction” factor β was found in [6]. However, the expression (4.4) in [6], when evaluated for l= 0 = p in the paraxial limit 1/(2kzR)=θ02/41 gives 𝒥z/(σz/ω)(1+θ02/4) instead of 𝒥z/(σz/ω)(1θ02/4), as given by Eq. (8) in [28]. The difference between the two results resides in the fact that different types of beams and geometries (planar versus spherical) are considered.

38.

W. L. Erikson and S. Singh, “Polarization properties of Maxwell-Gaussian laser beams,” Phys. Rev. A 49, 5778–5786 (1994).

39.

J. Leach, J. Courtial, K. Skeldon, S. M. Barnett, S. Franke-Arnold, and M. J. Padgett, “Interferometric methods to measure orbital and spin, or the total angular momentum of a single photon,” Phys. Rev. Lett. 92, 013601 (2004). [CrossRef] [PubMed]

40.

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, 253601 (2010). [CrossRef] [PubMed]

41.

P. Banzer, U. Peschel, S. Quabis, and G. Leuchs, “On the experimental investigation of the electric and magnetic response of a single nano-structure,” Opt. Express 18, 10905–10923 (2010). [CrossRef] [PubMed]

OCIS Codes
(260.0260) Physical optics : Physical optics
(260.2110) Physical optics : Electromagnetic optics
(260.6042) Physical optics : Singular optics

ToC Category:
Physical Optics

History
Original Manuscript: January 22, 2014
Revised Manuscript: March 4, 2014
Manuscript Accepted: March 4, 2014
Published: March 13, 2014

Citation
Marco Ornigotti and Andrea Aiello, "Surface angular momentum of light beams," Opt. Express 22, 6586-6596 (2014)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-22-6-6586


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References

  1. J. C. Maxwell, A Treatise on Electricity and Magnetism(Dover, 1954).
  2. J. H. Poynting, “The wave motion of a revolving shaft, and a suggestion as to the angular momentum in a beam of circularly polarised light,” Proc. R. Soc. London Ser. A 82, 560–567 (1909). [CrossRef]
  3. R. A. Beth, “Mechanical detection and measurement of the angular momentum of light,” Phys. Rev. 50, 115–125 (1936). [CrossRef]
  4. C. G. Darwin, “Notes on the theory of radiation,” Proc. R. Soc. London A 136, 36–52 (1932). [CrossRef]
  5. L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992). [CrossRef] [PubMed]
  6. S. M. Barnett, L. Allen, “Orbital angular momentum and nonparaxial light beams,” Opt. Commun. 110, 670–678 (1994). [CrossRef]
  7. S. J. van Enk, G. Nienhuis, “Spin and orbital angular momentum of photons,” Europhys. Lett. 25, 497–501 (1994). [CrossRef]
  8. S. J. van Enk, G. Nienhuis, “Commutation rules and eigenvalues of spin and orbital angular momentum of radiation fields,” J. Mod. Opt. 41, 963–977 (1994). [CrossRef]
  9. S. M. Barnett, “Optical angular-momentum flux,” J. Opt. B Quantum Semiclass. Opt. 4, S7–S16 (2002). [CrossRef]
  10. M. V. Berry, “Optical currents,” J. Opt. A Pure Appl. Opt. 11, 094001 (2009). [CrossRef]
  11. C.-F. Li, “Spin and orbital angular momentum of a class of nonparaxial light beams having a globally defined polarization,” Phys. Rev. A 80, 063814 (2009). [CrossRef]
  12. K. Y. Bliokh, M. A. Alonso, E. A. Ostrovskaya, A. Aiello, “Angular momenta and spin-orbit interaction of nonparaxial light in free space,” Phys. Rev. A 82, 063825 (2010). [CrossRef]
  13. S. M. Barnett, “Rotation of electromagnetic fields and the nature of optical angular momentum,” J. Mod. Opt. 57, 1339–1343 (2010). [CrossRef]
  14. I. Bialynicki-Birula, Z. Bialynicki-Birula, “Canonical separation of angular momentum of light into its orbital and spin parts,” J. Opt. 13, 064014 (2011). [CrossRef]
  15. O. Hosten, P. Kwiat, “Observation of the spin Hall effect of light via weak measurements,” Science 319, 787–790 (2008). [CrossRef] [PubMed]
  16. A. Aiello, J. P. Woerdman, “Role of beam propagation in Goos-Hänchen and Imbert-Fedorov shifts,” Opt. Lett. 33, 1437–1439 (2008). [CrossRef] [PubMed]
  17. M. Onoda, S. Murakami, N. Nagaosa, “Hall effect of light,” Phys. Rev. Lett. 93, 083901 (2004). [CrossRef] [PubMed]
  18. K. Y. Bliokh, Y. P. Bliokh, “Conservation of angular momentum, transverse shift, and spin Hall effect in reflection and refraction of an electromagnetic wave packet,” Phys. Rev. Lett. 96, 073903 (2006). [CrossRef] [PubMed]
  19. A. Y. Bekshaev, “Oblique section of a paraxial light beam: criteria for azimuthal energy flow and orbital angular momentum,” J. Opt. A Pure Appl. Opt. 11, 094003 (2009). [CrossRef]
  20. A. Aiello, N. Lindlein, C. Marquardt, G. Leuchs, “Transverse angular momentum and geometric spin Hall effect of light,” Phys. Rev. Lett. 103, 100401 (2009). [CrossRef] [PubMed]
  21. S. Franke-Arnold, L. Allen, M. Padgett, “Advances in optical angular momentum,” Laser Photonics Rev. 2, 299–313 (2008). [CrossRef]
  22. Optical Angular Momentum, L. Allen, S. M. Barnett, M. J. Padgett, eds. (Institute of Physics, 2003). [CrossRef]
  23. K. Y. Bliokh, A. Aiello, M. Alonso, “Spin-orbit interactions of light in isotropic media,” in The Angular Momentum of Light, D. L. Andrews, M. Babiker, eds. (Cambridge University, 2012), 174–245.
  24. H. A. Haus, J. L. Pan, “Photon spin and the paraxial wave equation,” Am. J. Phys. 61, 818–821 (1993). [CrossRef]
  25. C. Cohen-Tannoudji, J. Dupont-Roc, G. Grynberg, Photons and Atoms (Wiley-VCH, 2004), Chap. I.
  26. J. B. Götte, S. M. Barnett, “Light beams carrying orbital angular momentum,” in The Angular Momentum of Light, D. L. Andrews, M. Babiker, eds. (Cambridge University, 2012), 1–30.
  27. J. H. Crichton, P. L. Marston, “The measurable distinction between the spin and orbital angular momenta of electromagnetic radiation,” Electron. J. Diff. Eq. Conf. 04, 37–50 (2000).
  28. T. A. Nieminen, A. B. Stilgoe, N. R. Heckenberg, A. Rubinsztein-Dunlop, “Angular momentum of a strongly focused Gaussian beam,” J. Opt. A Pure Appl. Opt. 10, 115005 (2008). [CrossRef]
  29. J. Humblet, “Sur le moment d’impulsion d’une onde électromagnètique,” Physica 10, 585–603 (1943). [CrossRef]
  30. M. Lax, W. H. Louisell, W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975). [CrossRef]
  31. L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995). [CrossRef]
  32. R. Loudon, The Quantum Theory of Light, 3rd ed. (Oxford University, 2000).
  33. J. J. M. Braat, P. Dirksen, A. J. E. M. Janssen, A. S. van de Nes, “Extended Nijboer-Zernike representation of the vector field in the focal region of an aberrated high-aperture optical system,” J. Opt. Soc. Am. A 20, 2281–2292 (2003). [CrossRef]
  34. J. J. M. Braat, S. van Haver, A. J. E. M. Janssen, P. Dirksen, “Energy and momentum flux in a high-numerical-aperture beam using the extended Nijboer-Zernike diffraction formalism,” J. Eur. Opt. Soc. Rapid 2, 07032 (2007). [CrossRef]
  35. D. B. Ruffner, D. G. Grier, “Optical forces and torques in nonuniform beams of light,” Phys. Rev. Lett. 108, 173602 (2012). [CrossRef] [PubMed]
  36. A. E. Siegman, Lasers (University Science Books, 1986).
  37. The same “correction” factor β was found in [6]. However, the expression (4.4) in [6], when evaluated for l= 0 = p in the paraxial limit 1/(2kzR)=θ02/4≪1 gives 𝒥z/ℰ≃(σz/ω)(1+θ02/4) instead of 𝒥z/ℰ≃(σz/ω)(1−θ02/4), as given by Eq. (8) in [28]. The difference between the two results resides in the fact that different types of beams and geometries (planar versus spherical) are considered.
  38. W. L. Erikson, S. Singh, “Polarization properties of Maxwell-Gaussian laser beams,” Phys. Rev. A 49, 5778–5786 (1994).
  39. J. Leach, J. Courtial, K. Skeldon, S. M. Barnett, S. Franke-Arnold, M. J. Padgett, “Interferometric methods to measure orbital and spin, or the total angular momentum of a single photon,” Phys. Rev. Lett. 92, 013601 (2004). [CrossRef] [PubMed]
  40. O. G. Rodríguez-Herrera, D. Lara, K. Y. Bliokh, E. A. Ostrovskaya, C. Dainty, “Optical nanoprobing via spin-orbit interaction of light,” Phys. Rev. Lett. 104, 253601 (2010). [CrossRef] [PubMed]
  41. P. Banzer, U. Peschel, S. Quabis, G. Leuchs, “On the experimental investigation of the electric and magnetic response of a single nano-structure,” Opt. Express 18, 10905–10923 (2010). [CrossRef] [PubMed]

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