⬇ Download Markdown · ⬇ Download LaTeX · ⬇ original · view original · CSV: table 1

ARTICLE

Received 25 Oct 2013 | Accepted 22 Jan 2014 | Published 6 Mar 2014 DOI: 10.1038/ncomms4300

Extraordinary momentum and spin in evanescent waves

Konstantin Y. Bliokh 1,2 , Aleksandr Y. Bekshaev 3,4 & Franco Nori 3,5,6 Momentum and spin represent fundamental dynamic properties of quantum particles and ﬁelds. In particular, propagating optical waves (photons) carry momentum and longitudinal spin determined by the wave vector and circular polarization, respectively. Here we show that exactly the opposite can be the case for evanescent optical waves. A single evanescent wave possesses a spin component, which is independent of the polarization and is orthogonal to the wave vector. Furthermore, such a wave carries a momentum component, which is determined by the circular polarization and is also orthogonal to the wave vector. We show that these extraordinary properties reveal a fundamental Belinfante’s spin momentum, known in ﬁeld theory and unobservable in propagating ﬁelds. We demonstrate that the transverse momentum and spin push and twist a probe Mie particle in an evanescent ﬁeld. This allows the observation of ‘impossible’ properties of light and of a fundamental ﬁeld-theory quantity, which was previously considered as ‘virtual’.

1 iTHES Research Group, RIKEN, Wako-shi, Saitama 351-0198, Japan. 2 A. Usikov Institute of Radiophysics and Electronics, NASU, Kharkov 61085, Ukraine. 3 Center for Emergent Matter Science, RIKEN, Wako-shi, Saitama 351-0198, Japan. 4 I. I. Mechnikov National University, Dvorianska 2, Odessa 65082, Ukraine. 5 Physics Department, University of Michigan, Ann Arbor, Michigan 48109-1040, USA. 6 Department of Physics, Korea University, Seoul 136-713, Korea. Correspondence and requests for materials should be addressed to K.B. (email: [email protected]) or to F.N. (email: [email protected]).

NATURE COMMUNICATIONS | 5:3300 | DOI: 10.1038/ncomms4300 | www.nature.com/naturecommunications

1 ARTICLE NATURE COMMUNICATIONS | DOI: 10.1038/ncomms4300 I t has been known for more than a century, since the seminal works by J.H. Poynting 1 , that light carries momentum and angular momentum (AM) 2,3 . These are the main dynamical properties of electromagnetic waves, which are also preserved in the quantum-mechanical picture of photons 4 . Optical momentum and AM play a crucial role in various light-matter interactions 5–7 , including laser cooling 8–10 , optical manipulation of atoms or small particles 8–13 , and in optomechanical systems 14 .

The simplest example of an optical ﬁeld carrying momentum and spin AM is an elliptically polarized plane wave. Assuming free space propagation along the z axis, the complex electric ﬁeld of such wave can be written as

ð x þ m y Þ E ð r Þ ¼ A q ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ exp ð ikz Þ:

1 þ j m j 2 ð1Þ Here A is the wave amplitude, x and y are unit vectors of the corresponding axes, the complex number m determines the polarization state with s ¼ 1 2Imm 2 ½ 1; 1 being the helicity þ j m j 2 (ellipticity of polarization), k ¼ o/c is the wave number, and throughout the paper we imply monochromatic ﬁelds with omitted exp( iot) factor.

The momentum and spin AM in wave (1) can be characterized by the corresponding spatial densities

w w ð2Þ p ¼ k z ; s ¼ s z ;

o o where w ¼ go|A| 2 is the energy density, and we use Gaussian units with g ¼ (8po) 1 . In agreement with the quantummechanical picture of photons 4 , the momentum p is determined by the wave vector k z and is independent of polarization. At the same time, the spin s is proportional to the polarization helicity s and is also collinear with the wave vector.

The optical momentum and spin densities can be measured experimentally by placing a small absorbing particle in the ﬁeld and observing its linear and spinning motion 15–19 . Naturally, the radiation force and torque (with respect to the particle’s centre) quantify the momentum and AM transfer to the particle and are proportional to the densities (2) in the plane wave (1): Fpp and Tps (refs 19–25), Fig. 1.

The above picture is simple and intuitively clear in the plane-wave case, but complex, spatially-inhomogeneous ﬁelds require a more

ARTICLE NATURE COMMUNICATIONS | DOI: 10.1038/ncomms4300 Note that these quantities are naturally split into electric and O magnetic ﬁeld contributions: p O ¼ p O

e þ p m and s ¼ s e þ s m . It is important to emphasize that although the Poynting vector p is usually considered in optics as a single momentum density of light 2 , it actually represents the sum of two quantities p O and p S , with drastically different physical meanings and properties. Below we uncover the contrasting manifestations of the canonical and spin momenta.

The canonical (orbital) and spin parts of the momentum density (3), p ¼ p O þ p S , generate, respectively, the orbital and spin parts of the AM density: j ¼ l þ s (refs 26,27,32,33). The orbital AM density is l ¼ r p O , and this is an extrinsic origindependent quantity. At the same time, the spin AM density s, equation (4), is intrinsic (origin-independent). Nonetheless, its integral value is R determined by the circulation of the spin R momentum: S ¼ sdV ¼ r p S dV, where integration by parts should be performed 28–33 .

The orbital momentum density p O is naturally proportional to the local phase gradient (wave vector) in the ﬁeld 22 . In contrast, Belinfante’s spin momentum p S is a rather enigmatic quantity 26–33 . On the one hand, the spin momentum provides the physical origin of the spin AM of quantum particles. On the other hand, it is usually considered as an auxiliary ‘virtual’ quantity, which cannot be observed per se. Indeed, the spin momentum represents a solenoidal current, which R does not contribute to the energy transport (= . p S ¼ 0 and p S dV ¼ 0), and only generates spin AM. Consider, for instance, the elliptically polarized electromagnetic plane wave (1). This ﬁeld carries only the longitudinal orbital momentum density: p O / k z , while the spin momentum vanishes: p S ¼ 0. In this case, what generates the spin AM density s / s z ? This known paradox 26,34,35 is resolved by representing the zero transverse momentum as an array of inﬁnitely small loops of circulating spin momentum in the (x,y) plane 26,30–32 , see Fig. 1b. Currents from the neighbouring loops cancel each other, but at the same time they provide non-zero circulation along any ﬁnite closed loop, that is, non-zero spin AM along the z axis. The formal integral relation between p S and s does not work here, because a plane wave is an unbounded state; the introduction of a boundary (for example, Gaussian intensity distribution in the transverse plane) immediately produces a non-zero boundary spin current p S a0 with the integral circulation yielding the spin S (refs 28,30–32). Thus, Belinfante’s spin momentum is similar to the boundary magnetization current or topological quantum-Hall current in solid-state systems (multiple current loops are produced there by electron orbitals), whereas the spin AM is analogous to the bulk magnetization in such systems.

Probing the momentum and spin densities with small particles. Having the above theoretical picture, let us consider measurements of the momentum and spin densities in an electromagnetic ﬁeld. As we mentioned in the Introduction, a small absorbing particle immersed in the ﬁeld can be employed as a natural meter of these quantities. Calculating the radiation force and torque on a dipole Rayleigh particle with equal electric and magnetic polarizabilities, a e ¼ a m ¼ a, one can show that they are determined by the canonical momentum and spin densities 19–25 (see Supplementary Note 3):

Here w ¼ 2 g o jEj 2 þ jHj 2 ¼ w e þ w m is the energy density of the ﬁeld. The ﬁrst term in the ﬁrst equation (6) is the gradient force, while the second term is the radiation pressure force. Together they ‘measure’ the imaginary and real parts of the complex canonical ~ y ð r Þ ^ momentum: ~ p O ð r Þ ¼ c p ~ c ð r Þ ¼ p O ðrÞ i=wðrÞ=2o, which is proportional to the quantum weak value of the photon’s

the Poynting vector p that represents the physicallymeaningful momentum density of light that appears in experiments (p ¼ p O only in linearly polarized paraxial ﬁelds and plane waves) 19,33 . In particular, the orbital and spinning motions of probe particles in circularly-polarized vortex beams 15–18 originate exactly due to the force from the azimuthal component of p O and the torque from the longitudinal component of the spin density s. Furthermore, the quantum-mechanical resonant momentum transfer from light to a two-level atom is also determined by the canonical momentum density 37,38 . Finally, a remarkable recent experiment 36 , which realized quantum weak measurements of the local momentum of photons, also measured p O (ref. 19). Thus, the spin momentum p S turns out to be indeed ‘virtual’, that is, non-observable for weak-interaction measurements.

Note that we considered an ‘ideal’ particle with equal electric and magnetic polarizabilities. In reality, local light–matter inter4|a m |. This is actions usually have electric character, and |a e |4 because of the fundamental electric–magnetic (dual) asymmetry of matter, which breaks the intrinsic dual symmetry of the free space Maxwell equations 33,39,40 . In this case, the particle will ‘measure’ only the electric parts of the momentum and spin densities (5):

25,33) (see Supplementary Note 3). The electric and magnetic contributions to the local dynamical characteristics of light are O equivalent in paraxial propagating ﬁelds 22 : p O

e ’ p m , s e Cs m , but they can differ signiﬁcantly in other cases.

Extraordinary momentum and spin in a single evanescent wave. We are now in a position to consider the main subject of the present study: evanescent waves. A single evanescent wave propagating along the z axis and decaying in the x40 half-space can formally be obtained via a rotation of the propagating plane wave (1) by an imaginary angle iW about the y axis 41 . In doing so, we obtain the electric evanescent-wave ﬁeld:

A k k E ¼ q ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ x þ m y i z exp ð ik z z kx Þ: ð7Þ k

z k z 1 þ j m j 2

Here k z ¼ k cosh W4k is the longitudinal wave number, whereas k ¼ k sinh W is the exponential decay rate, so that the complex x . Substituting ﬁeld (7), with the wave vector is k ¼ k z z þ ik corresponding magnetic wave ﬁeld (see Supplementary Note 2 and Supplementary Fig. 2), into equations (3) and (5), we calculate the canonical momentum, spin momentum and spin AM densities in the evanescent wave:

w w k 2 kk ð8Þ z þ s y ; p O ¼ k z z ; p S ¼ o o k

z k z w k k s ¼ ð9Þ s z þ y ; o k

z k z where w ¼ go|A| 2 exp( 2kx) is the spatially-inhomogeneous energy density of the wave.

Equations (8) and (9) reveal remarkable peculiarities of the momentum and spin in evanescent waves and represent the key analytical point of our study. First, note that the evanescent wave (7) possesses longitudinal canonical momentum p O z / k z 4k, which exceeds the momentum of a plane wave with the same local intensity. Divided by the energy density w, this momentum NATURE COMMUNICATIONS | 5:3300 | DOI: 10.1038/ncomms4300 | www.nature.com/naturecommunications

3 ARTICLE NATURE COMMUNICATIONS | DOI: 10.1038/ncomms4300 s y p(k/k z )w turns out to be helicity independent! This is in sharp contrast to propagating waves and photons, equation (2).

The transverse momentum and spin appear due to the two features of the evanescent ﬁeld (7). The ﬁrst one is the imaginary longitudinal component of the ﬁeld polarization: i ð k=k z Þ z ^ . This induces a rotation of the ﬁelds in the propagation (x,z) plane (see Fig. 2a,d), and generates the spin s y / Im E z E x þ H z H x independently of s. Recently, we described such spin for surface plasmon-polaritons 42 , and it was shown that the imaginary longitudinal ﬁeld component plays an important role in optical coupling processes 44,45 . The second feature is the inhomogeneous intensity wpexp( 2kx). This inhomogeneity destroys the cancellation of the spin momentum loops in the (x,y) plane, which results in the non-zero transverse Belinfante’s spin momentum: p S y ¼ @ x s z =2 6¼ 0, see Fig. 2e. Note that although equation (8) and Fig. 2b,e show unidirectional spin momentum in the x40 half-space, the accurate consideration of the interface x ¼ 0 and medium in the xo0 half-space ensures R the vanishing of the integral spin momentum, in agreement with p S dV ¼ 0 (ref. 42).

yields the superluminal local group velocity in evanescent waves: v gz ¼ ck z /k4c (refs 19,42). Although the Poynting vector corresponds to subluminal propagation, S 2 p z ¼ p

Oz þ p z / ð k =k z Þok, it is the canonical momentum that represents the observable momentum density. In particular, the momentum transfer via the radiation force (6) F z / p O z to a dipole particle in the evanescent wave (7) will be larger than k per photon 43 . Such ‘super-momentum’ transfer was observed by Huard and Imbert 37 in the resonant Doppler coupling with a moving atom. In terms of the quantum weak measurements paradigm, the ‘super momentum’ p O z represents a weak value of the photon momentum with the post selection in a ‘forbidden’ zone unreachable for propagating waves (for example, beyond a totally reﬂecting interface) 19 .

However, what is much more intriguing in equations (8) and (9) is the presence of the transverse y-components of the momentum and spin in the wave (7) propagating solely within the (x,z) plane. Moreover, here the momentum p S y / s ð kk=k z Þw becomes proportional to the helicity s, while the spin

ARTICLE NATURE COMMUNICATIONS | DOI: 10.1038/ncomms4300 Relation to the Fedorov-Imbert controversy. Here we should make a historical digression and note that an example of the transverse helicity-dependent momentum in an evanescent ﬁeld was ﬁrst found by F.I. Fedorov in 1955 (ref. 46). Incidentally, this discovery caused a half-century-long controversy in the physics of light reﬂection and refraction. Analysing the total internal reﬂection of a polarized plane wave, Fedorov found a helicitydependent transverse component of the Poynting vector in the transmitted evanescent ﬁeld. Fedorov concluded that ‘the lateral energy ﬂux should lead to a speciﬁc light pressure’ and (by an analogy with the Goos–Hänchen effect) that ‘the reﬂected beam in the case of total reﬂection must be displaced in the lateral direction’. Later, C. Imbert indeed observed such helicitydependent beam shift experimentally 47 , and the effect is now known as the Imbert–Fedorov transverse shift or spin Hall effect of light (for a review, see ref. 48).

The 50 years after Fedorov’s ﬁnding brought about numerous controversies about this effect. Finally, only recently an accurate theoretical description was given 49,50 , which was followed by a precise experiment 51 using quantum weak measurements (see also refs 52,53). Remarkably, the current theory of the spin Hall effect of light is completely unrelated to evanescent waves and their transverse momentum. Indeed, the helicity-dependent beam shift arises from the interference of multiple plane waves in the beam, taking into account the geometric-phase effect, that is, the spin-orbit interaction of light 48 . This shift occurs in partial reﬂection or refraction, focusing, scattering and other optical phenomena without any evanescent waves.

Thus, curiously, Fedorov predicted two fundamental helicitydependent but unrelated phenomena (transverse momentum and beam shift) using a ﬁctitious connection between them. Now, from equations (3) and (8), we can conclude that Fedorov’s transverse momentum is an example of Belinfante’s spin momentum, which does not transport energy and therefore cannot shift the ﬁeld.

Interaction of the evanescent wave with Mie particles. The evanescent ﬁeld (7) represents an exceptional conﬁguration with a pure spin momentum without any orbital part in the transverse y-direction. This offers a unique opportunity to investigate this fundamental ﬁeld-theory quantity per se. Equation (6) show that the spin momentum does not appear in the dipole interaction with small point-like particles. But does this result hold true for larger particles and higher-order interactions? To address this question, we examine the interaction of the evanescent optical ﬁeld with ﬁnite-size Mie particles.

The Mie scattering theory provides an exact solution for the plane-wave interaction with an isotropic spherical particle. Using the Maxwell stress tensor, one can calculate the ﬂux of the momentum and AM through a sphere enclosing the particle, and thereby determine the force and torque acting on the particle 54 . Recently, we developed and successfully tested an extension of the Mie theory (based on complex-angle rotation of the standard theory), which describes the scattering of the incident evanescent wave (7) 41 (see also refs 55,56). Using this exact semi-analytical method, we calculate the radiation force and torque acting on the particle of radius a, complex permittivity e p , and permeability m p , immersed in the evanescent ﬁeld (7).

Figure 3a,c shows the schematic of the corresponding experiment using the total internal reﬂection at a glass prism. For this numerical experiment, we use parameters corresponding to real experiments manipulating particles with evanescent ﬁelds (for example see refs 57–62). Namely, we consider radiation with the wavelength l ¼ 650 nm, a gold particle (e p ¼ 12.2 þ 3i, m p ¼ 1) in water (e ¼ 1.77, m ¼ 1), and near-critical total internal reﬂection (the angle of incidence is y ¼ 51° ¼ y c þ 1.5°) from the interface between heavy ﬂint glass (e 1 ¼ 3.06, m 1 ¼ 1) and water. Calculations of the corresponding wave ﬁelds and characteristics are given in the Supplementary Notes 2 and 3. The resulting force and torque components (normalized by F 0 ¼ a 2 |A 1 |/4p and T 0 ¼ F 0 /k, with A 1 being the amplitude of the incident wave in the glass), as functions of the dimensionless particle radius ka are shown in Fig. 3b,d (see also Supplementary Fig. 4 and Supplementary Table 1). These are the main numerical results of our work, which offer several new experiments for the detection of extraordinary spin and momentum properties of evanescent waves.

Figure 3b depicts the radiation torque components for righthand and left-hand circularly-polarized waves (m ¼ ± i, s ¼ ± 1). While the longitudinal torque T z from the usual spin s z ﬂips with the sign of s, the transverse torque T y is helicityindependent and present even in the linear-polarization Imm ¼ 0 case. This conﬁrms the presence of the transverse helicityindependent spin AM (8) s y in the evanescent ﬁeld and its transfer to the particle. For small Rayleigh particles, kao o1, the torque is described by the dipole approximation (6). Due to the strong dual (electric–magnetic) asymmetry of the gold, the torque appears mostly from the electric part of the spin (5): TCg 1 Im(a e )s e , where the electric polarizability is proportional to the particle’s volume: a e p(ka) 3 , while the magnetic polarizability is small: a m p(ka) 5 C0 (refs 63,64). Therefore, the transverse torque is maximal for the TM-mode with m ¼ 0 and s y ¼ s ey , and minimal for the TE-mode with m ¼ N and s y ¼ s my (see Supplementary Notes 2 and 3).

The dual asymmetry results in another remarkable effect. Namely, for waves linearly diagonally-polarized at ± 45° (m ¼ ± 1), a vertical radiation torque T x appears, which is proportional to the degree of diagonal polarization 2 ½ 1; 1 . This torque signals the presence of the w ¼ 1 2Rem þ j m j 2 vertical electric spin component:

kk w s ex ¼ s mx ¼ w 2 ; ð10Þ 2k

z o which arises from the diagonal-electric-ﬁeld rotation in the (y,z)- plane (see equation (7) and Supplementary Note 2). Importantly, the total vertical spin vanishes in (9), because the electric and magnetic ﬁelds rotate in opposite directions: s x ¼ s ex þ s mx ¼ 0. Nonetheless, the dual-asymmetric gold particle unveils the electric vertical spin (10), as shown in Fig. 3b.

Figure 3d shows the longitudinal and transverse components of the radiation force for circular (m ¼ ± i, s ¼ ± 1) and diagonal (m ¼ w ¼ ± 1) polarizations. The longitudinal force F z represents the radiation pressure (mostly polarization-independent) from the orbital momentum p O z . Akin to the torque, it exhibits an effective electric-dipole interaction F ’ g 1 Im ð a e Þp O e in the Rayleigh regime kao o1. The transverse force vanishes in this regime, F y ¼ 0, which conﬁrms the ‘virtual’ character of the spin momentum.

Nonetheless, a non-zero helicity-dependent transverse force arises for larger Mie particles with kaB1. This force originates from the higher-order interaction between electric- and magneticinduced dipoles, and in the quadratic dipole–dipole approximation it can be written as 63,64 (see Supplementary Note 3)

3 k ~ F ¼ g 1 Re a e a m Re~ p þ Im a e a m Im~ p : ð11Þ

3 Here we introduced the complex Poynting momentum: ~ ¼ gk ð E H Þ, with Re~ p ¼ p ¼ p O þ p S being the usual p Poynting vector and Im~ p / Im ð E H Þ characterizing an alternating ﬂow of the so-called ‘stored energy’ 2 . Alongside with the transverse s-dependent spin momentum (8) Re p ~ y ¼ p S y / s ð kk=k z Þw, the evanescent wave (7) possesses a NATURE COMMUNICATIONS | 5:3300 | DOI: 10.1038/ncomms4300 | www.nature.com/naturecommunications

5 ARTICLE NATURE COMMUNICATIONS | DOI: 10.1038/ncomms4300

Figure 3 | Numerically calculated forces and torques on a Mie particle in an evanescent ﬁeld. (a,c) Schematics of a proposed experiment (see also Supplementary Fig. 3). A polarized propagating wave undergoes total internal reﬂection at the glass–water interface, thereby generating the evanescent wave (7) in water. A gold spherical particle of radius a is placed in water on the glass surface, and its observable linear and spinning motion is proportional to the forces and torques exerted by the evanescent wave. (b,d) Normalized torque and force components for circular ( m ¼ ± i, s ¼ ± 1) and diagonal (m ¼ w ¼ ± 1) polarizations versus the particle size ka (the Rayleigh dipole region ka51 is indicated by the orange circles). (b) The longitudinal helicity-dependent torque T z ps z ps indicates the usual spin. The transverse torque is helicity independent and signals the appearance of the transverse spin (9): T y ps y p(k/k z )w. The vertical torque for the diagonal polarizations w ¼ ± 1 appears because of the strong electric–magnetic (dual) asymmetry of the particle. It is caused by the non-zero electric part (10) of the (zero-net) vertical spin density: T x / s ex / w kk=2k z 2 w, s ex ¼ s mx . (d) The orbital momentum density (8) produces mostly polarization-independent longitudinal radiation pressure force: F z / p O z / k z 4k. At the same time, the transverse force F ~ y vanishes in the Rayleigh region, but becomes non-zero for larger Mie particles with kaB1, see equation (11). This transverse force has the helicity-dependent part proportional to the transverse spin momentum (8): F ~ y / p S y / s ð kk=k z Þw, and also the w-dependent part proportional to the ‘imaginary’ Poynting momentum (12) ~ F y / Im~ p y / w ð kk=k z Þw.

w-dependent transverse imaginary Poynting momentum (see Supplementary Note 3)

Im ~ p y ¼ w kk w : k

z o ð12Þ These two transverse momenta determine two contributions to the transverse dipole–dipole force (11): F ~ y / Re a e a m p S y þ Im a e a m Im ~ p y . In complete agreement with this, numerical calculations in Fig. 3d show s-dependent and w-dependent transverse radiation forces on Mie particles with kaB1, which o1. This proves the presence vanish as F ~ y pa e a m p(ka) 8 at kao and observability of the transverse Belinfante’s spin momentum in the evanescent optical ﬁeld.

Detailed analysis and calculations of all torque and force components for all basic polarizations m ¼ 0,N, ± i, ± 1 can be 6 found in the Supplementary Note 3, Supplementary Figs 4 and 5, and Supplementary Table 1.

Discussion To summarize, we have found that a single evanescent electromagnetic wave offers a rich and highly non-trivial structure of the local momentum and spin distributions. In sharp contrast to standard photon properties, evanescent waves carry helicityindependent transverse spin and helicity-dependent transverse momentum. Moreover, the transverse momentum turns out to be a fundamental spin momentum introduced by Belinfante in ﬁeld theory and ﬁrst remarked in optics (as an unusual Poynting vector) by Fedorov. We have examined the measurements of the extraordinary spin and momentum in the evanescent ﬁeld by analysing its interaction with a probe particle. Analytical NATURE COMMUNICATIONS | 5:3300 | DOI: 10.1038/ncomms4300 | www.nature.com/naturecommunications

ARTICLE NATURE COMMUNICATIONS | DOI: 10.1038/ncomms4300 evaluations and exact numerical simulations based on parameters of typical optical-manipulation experiments show that the transverse helicity-independent spin (and also the vertical electric spin for diagonal polarizations) can be detected straightforwardly via the radiation torque exerted on an absorbing small particle. At the same time, the Belinfante–Fedorov’s spin momentum does not exert the standard optical pressure in the dipole approximation, which conﬁrms its ‘virtual’ character (in contrast to Fedorov’s interpretation). Nonetheless, it appears detectable (in contrast to the ﬁeld-theory interpretation) via a helicitydependent transverse optical force from the higher-order interaction with Mie particles. Thus, an exceptional evanescentwave structure with pure spin transverse momentum offers a unique opportunity for the direct observation of this fundamental ﬁeld-theory quantity, which was previously considered as ‘virtual’.

In total, this work offers four novel experiments for the detection of extraordinary momentum and spin properties of a single evanescent wave (red and green curves in Fig. 3). These proposed experiments could detect the following optical torques and forces. First, the transverse helicity-independent torque T y indicating the transverse spin s y (Fig. 3b, equations (6) and (9)). Second, the vertical diagonal-polarization-dependent torque T x exerted on a dual-asymmetric (for example, electric-dipole) particle and caused by the vertical electric spin s ex (Fig. 3b and equation (10)). Third, the transverse helicitydependent force F ~ y produced by the transverse spin momentum p S y (Fig. 3d, equations (8) and (11)). Fourth, the transverse diagonal-polarization-dependent force F ~ y , which is associated with the transverse imaginary Poynting vector Im ~ p y (Fig. 3d, equations (11) and (12)).

These results add a distinct chapter in the physics of momentum and spin of classical and quantum ﬁelds, and offers a variety of non-trivial light-matter interaction effects involving evanescent ﬁelds.

References

1. Loudon, R. & Baxter, C. Contribution of John Henry Poynting to the

understanding of radiation pressure. Proc. Roy. Soc. A 468, 1825–1838 (2012).

2. Jackson, J. D. Classical Electrodynamics 3rd edn (Wiley, 1999). 3. Allen, L., Barnett, S. M. & Padgett, M. J. Optical Angular Momentum (IoP

Publishing, 2003).

4. Berestetskii, V. B., Lifshitz, E. M. & Pitaevskii, L. P. Quantum Electrodynamics

(Pergamon, 1982).

5. Cohen-Tannoudji, C., Dupont-Roc, J. & Grynberg, G. Atom-Photon

Interactions (Wiley-VCH, 2004).

6. Brevik, I. Experiments in phenomenological electrodynamics and the

electromagnetic energy-momentum tensor. Phys. Rep. 52, 133–201 (1979).

7. Pfeifer, R. N. C., Nieminen, T. A., Heckenberg, N. R. & Rubinsztein-Dunlop, H.

Momentum of an electromagnetic wave in dielectric media. Rev. Mod. Phys. 79, 1197–1216 (2007).

8. Chu, S. The manipulation of neutral particles. Rev. Mod. Phys. 70, 685–706

(1998).

9. Phillips, W. D. Laser cooling and trapping of neutral atoms. Rev. Mod. Phys. 70,

721–741 (1998).

10. Ashkin, A. History of optical trapping and manipulation of small-neutral

particle, atoms, and molecules. IEEE J. Sel. Top. Quantum Electron. 6, 841–856 (2000).

11. Grier, D. G. A revolution in optical manipulation. Nature 424, 810–816

(2003).

12. Dienerowitz, M., Mazilu, M. & Dholakia, K. Optical manipulation of

nanoparticles: a review. J. Nanophoton. 2, 021875 (2008).

13. Bowman, R. W. & Padgett, M. J. Optical trapping and binding. Rep. Prog. Phys.

76, 026401 (2013).

14. Aspelmeyer, M., Kippenberg, T. J. & Marquardt, F. Cavity optomechanics.

Preprint at http://www.arxiv.org/abs/1303.0733 (2013).

15. Friese, M. E. J., Nieminen, T. A., Heckenberg, N. R. & Rubinsztein-Dunlop, H.

Optical alignment and spinning of laser-trapped microscopic particles. Nature 394, 348–350 (1998).

16. O’Neil, A. T., MacVicar, I., Allen, L. & Padgett, M. J. Intrinsic and extrinsic

nature of the orbital angular momentum of a light beam. Phys. Rev. Lett. 88, 053601 (2002).

17. Curtis, J. E. & Grier, D. G. Structure of optical vortices. Phys. Rev. Lett. 90,

133901 (2003).

18. Garcés-Chavéz, V. et al. Observation of the transfer of the local angular

momentum density of a multiringed light beam to an optically trapped particle. Phys. Rev. Lett. 91, 093602 (2003).

19. Bliokh, K. Y., Bekshaev, A. Y., Kofman, A. G. & Nori, F. Photon trajectories,

anomalous velocities, and weak measurements: a classical interpretation. New J. Phys. 15, 073022 (2013).

20. Ashkin, A. & Gordon, J. P. Stability of radiation-pressure particle traps: an

optical Earnshaw theorem. Opt. Lett. 8, 511–513 (1983).

21. Chaumet, P. C. & Nieto-Vesperinas, M. Time-averaged total force on a dipolar

sphere in an electromagnetic ﬁeld. Opt. Lett. 25, 1065–1067 (2000).

22. Berry, M. V. Optical currents. J. Opt. A: Pure Appl. Opt. 11, 094001 (2009). 23. Ruffner, D. B. & Grier, D. G. Comment on ‘Scattering forces from the curl of

the spin angular momentum of a light ﬁeld’. Phys. Rev. Lett. 111, 059301 (2013).

24. Canaguier-Durand, A., Cuche, A., Cyriaque, G. & Ebbesen, T. W. Force and

torque on an electric dipole by spinning light ﬁelds. Phys. Rev. A 88, 033831 (2013).

25. Liberal, I., Ederra, I., Gonzalo, R. & Ziolkowski, R. W. Electromagnetic force

density in electrically and magnetically polarizable media. Phys. Rev. A 88, 053808 (2013).

26. Soper, D. E. Classical Field Theory (Wiley, 1976). 27. Belinfante, F. J. On the current and the density of the electric charge, the

energy, the linear momentum and the angular momentum of arbitrary ﬁelds. Physica 7, 449–474 (1940).

28. Ohanian, H. C. What is spin? Am. J. Phys. 54, 500–505 (1986). 29. Salesi, G. & Recami, E. A velocity ﬁeld and operator for spinning particles in

(nonrelativistic) quantum mechanics. Found. Phys. 28, 763–773 (1998).

30. Mita, K. Virtual probability current associated with the spin. Am. J. Phys. 68,

259–264 (2000).

31. Witting, C. Photon and electron spins. J. Phys. Chem. A 113, 15320–15327

(2009).

32. Bekshaev, A., Bliokh, K. Y. & Soskin, M. Internal ﬂows and energy circulation

in light beams. J. Opt. 13, 053001 (2011).

33. Bliokh, K. Y., Bekshaev, A. Y. & Nori, F. Dual electromagnetism: helicity, spin,

momentum, and angular momentum. New J. Phys. 15, 033026 (2013).

34. Allen, L. & Padgett, M. J. Response to Question #79. Does a plane wave carry

spin angular momentum? Am. J. Phys. 70, 567–568 (2002).

35. Zambrini, R. & Barnett, S. M. Local transfer of optical angular momentum to

matter. J. Mod. Opt. 52, 1045–1052 (2005).

36. Kocsis, S. et al. Observing the average trajectories of single photons in a two-slit

interferometer. Science 332, 1170–1173 (2011).

37. Huard, S. & Imbert, C. Measurement of exchanged momentum during

interaction between surface-wave and moving atom. Opt. Commun. 24, 185–189 (1978).

38. Barnett, S. M. & Berry, M. V. Superweak momentum transfer near optical

vortices. J. Opt. 15, 125701 (2013).

39. Fernandez-Corbaton, I., Zambrana-Puyalto, X. & Molina-Terriza, G. Helicity

and angular momentum: a symmetry-based framework for the study of light–matter interactions. Phys. Rev. A 86, 042103 (2012).

40. Cameron, R. P., Barnett, S. M. & Yao, A. M. Optical helicity, optical spin, and

related quantities in electromagnetic theory. New J. Phys. 14, 053050 (2012).

41. Bekshaev, A. Y., Bliokh, K. Y. & Nori, F. Mie scattering and optical forces from

evanescent ﬁelds: a complex-angle approach. Opt. Express 21, 7082–7095 (2013).

42. Bliokh, K. Y. & Nori, F. Transverse spin of a surface polariton. Phys. Rev. A 85,

061801(R) (2012).

43. Nieto-Vesperinas, M. & Saenz, J. J. Optical forces from an evanescent wave on a

magnetodielectric small particle. Opt. Lett. 35, 4078–4080 (2010).

44. Gorodetski, Y. et al. Weak measurements of light chirality with a plasmonic slit.

Phys. Rev. Lett. 109, 013901 (2012).

45. Junge, C., O’Shea, D., Volz, J. & Rauschenbeutel, A. Strong coupling between

single atoms and nontransversal photons. Phys. Rev. Lett. 110, 213604 (2013).

46. Fedorov, F. I. To the theory of total reﬂection. Dokl. Akad. Nauk. SSSR. 105,

465–468 (1955) [translated and reprinted in J. Opt. 15, 014002 (2013)].

47. Imbert, C. Calculation and experimental proof of the transverse shift induced

by total internal reﬂection of a circularly polarized light beam. Phys. Rev. D 5, 787–796 (1972).

48. Bliokh, K. Y. & Aiello, A. Goos–Hänchen and Imbert–Fedorov beam shifts: an

overview. J. Opt. 15, 014001 (2013).

49. Bliokh, K. Y. & Bliokh, Y. P. 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).

NATURE COMMUNICATIONS | 5:3300 | DOI: 10.1038/ncomms4300 | www.nature.com/naturecommunications

7 ARTICLE NATURE COMMUNICATIONS | DOI: 10.1038/ncomms4300

50. Bliokh, K. Y. & Bliokh, Y. P. Polarization, transverse shift, and angular

momentum conservation laws in partial reﬂection and refraction of an electromagnetic wave packet. Phys. Rev. E 75, 066609 (2007).

51. Hosten, O. & Kwiat, P. Observation of the spin Hall effect of light via weak

measurements. Science 319, 787 (2008).

52. Aiello, A. & Woerdman, J. P. Role of beam propagation in Goos–Hänchen and

Imbert–Fedorov shifts. Opt. Lett. 33, 1437–1439 (2008).

53. Dennis, M. R. & Götte, J. B. The analogy between optical beam shifts and

quantum weak measurements. New J. Phys. 14, 073013 (2012).

54. Bohren, C. F. & Huffman, D. R. Absorption and Scattering of Light by Small

Particles (Wiley, 1983).

55. Almaas, E. & Brevik, I. Radiation forces on a micrometer-sized sphere in an

evanescent ﬁeld. J. Opt. Soc. Am. B 12, 2429–2438 (1995).

56. Zvyagin, A. V. & Goto, K. Mie scattering of evanescent waves by a dielectric

sphere: comparison of multipole expansion and group-theory methods. J. Opt. Soc. Am. A 15, 3003–3008 (1998).

57. Kawata, S. & Tani, T. Optically driven Mie particles in an evanescent ﬁeld along

a channeled waveguide. Opt. Lett. 21, 1768–1770 (1996).

58. Ng, L. N., Zevras, M. N., Wilkinson, J. S. & Luff, B. J. Manipulation of colloidal

gold nanoparticles in the evanescent ﬁeld of a channel waveguide. Appl. Phys. Lett. 76, 1993–1995 (2000).

59. Volpe, G., Quidant, R., Badenes, G. & Petrov, D. Surface plasmon radiation

forces. Phys. Rev. Lett. 96, 238101 (2006).

60. Gaugiran, S., Getin, S., Fideli, J. M. & Derouard, J. Polarization and particle size

dependence of radiative forces on small metallic particles in evanescent optical ﬁelds. Evidences for either repulsive or attractive gradient forces. Opt. Express 15, 8146–8156 (2007).

61. Wang, K., Schonbrun, E. & Crozier, K. B. Propulsion of gold nanoparticles

with surface plasmon polaritons: evidence of enhanced optical force from near-ﬁeld coupling between gold particle and gold ﬁlm. Nano. Lett. 9, 2623–2629 (2009).

62. Yang, A. H. J., Lerdsuchatawanich, T. & Erickson, D. Forces and transport

velocities for a particle in a slot waveguide. Nano Lett. 9, 1182–1188 (2009).

63. Nieto-Vesperinas, M., Saenz, J. J., Gomez-Medina, R. & Chantada, L. Optical

forces on small magnetodielectric particles. Opt. Express 18, 11428–11443 (2010).

64. Bekshaev, A. Y. Subwavelength particles in an inhomogeneous light ﬁeld:

Optical forces associated with the spin and orbital energy ﬂows. J. Opt. 15, 044004 (2013).

Acknowledgements This work was partially supported by the RIKEN iTHES Project, MURI Center for Dynamic Magneto-Optics, JSPS-RFBR contract no. 12-02-92100, Grant-in-Aid for Scientiﬁc Research (S), MEXT Kakenhi on Quantum Cybernetics, and the JSPS via its FIRST program.

Author contributions All authors contributed equally to this work.

Additional information Supplementary Information accompanies this paper at http://www.nature.com/ naturecommunications

Competing ﬁnancial interests: The authors declare no competing ﬁnancial interests.

Reprints and permission information is available online at http://npg.nature.com/ reprintsandpermissions/ How to cite this article: Bliokh, K. Y. et al. Extraordinary momentum and spin in evanescent waves. Nat. Commun. 5:3300 doi: 10.1038/ncomms4300 (2014).

NATURE COMMUNICATIONS | 5:3300 | DOI: 10.1038/ncomms4300 | www.nature.com/naturecommunications