Loudon, R. & Baxter, C. Contribution of John Henry Poynting to the understanding of radiation pressure. Proc. Roy. Soc. A 468, 1825â1838 (2012). 3. Ashkin, A.
Direct measurement of the extraordinary optical momentum using a nano-cantilever Massimo Antognozzi1,2, Stephen Simpson1,3, Robert Harniman1,4, Jorden Senior1, Rosie Hayward1, Heinrich Hoerber1, Mark R. Dennis1, Aleksandr Y. Bekshaev5, Konstantin Y. Bliokh6, and Franco Nori6,7 1
H.H. Wills Physics Laboratory, University of Bristol, Bristol BS8 1TL, UK Centre for Nanoscience and Quantum Information, University of Bristol, Bristol, UK 3 Institute of Scientific Instruments of the ASCR, Brno, Czech Republic 4 School of Chemistry, University of Bristol, Bristol BS8 1TL, UK 5 I.I. Mechnikov National University, Odessa, 65082 Ukraine 6 Center for Emergent Matter Science, RIKEN, Wako-shi, Saitama 351-0198, Japan 7 Physics Department, University of Michigan, Ann Arbor, Michigan 48109-1040, USA 2
Radiation pressure has been known since Kepler’s observation that a comet’s tail is always oriented away from the sun, and in the past centuries this phenomenon stimulated remarkable discoveries in electromagnetism, quantum physics and relativity [1–3]. In modern terms, the pressure of light is associated with the momentum of photons, which plays a crucial role in a variety of systems, from atomic [4–7] to astronomical [8,9] scales. Experience from these cases leads us to assume that the direction of the optical momentum and the radiation-pressure force are naturally aligned with the propagation of light, i.e., its wavevector. Here we report the direct observation of an extraordinary optical momentum and force directed perpendicular to the wavevector, and proportional to the optical spin (i.e., degree of circular polarization). This transverse spin-dependent optical force, a few orders of magnitude weaker than the usual radiation pressure, was recently predicted for evanescent waves [10] and other structured fields [11]. Fundamentally, it can be associated with the enigmatic “spin momentum”, introduced by Belinfante in field theory 75 years ago [12–14]. We measure this unusual transverse momentum using a nano-cantilever with extremely low compliance (capable of femto-Newton resolution), which is immersed in an evanescent optical field directly above the total-internal-reflecting glass surface. Such sensors, perpendicular to a substrate, have already shown an extreme force resolution in various systems [15–19]. Our findings revisit fundamental momentum properties of light, while the experimental technique opens the way for precision measurements of fine optical forces in structured fields at subwavelength scales. Since Euler’s studies of classical sound waves [2], the wave momentum is naturally associated with the propagation direction of the wave, i.e., the normal to wavefronts or the wavevector. This idea was mathematically formulated by de Broglie for quantum matter waves: p = k . In both classical and quantum cases, the wave momentum can be measured via the pressure force on an absorbing or scattering detector. In agreement with this, Maxwell claimed in his celebrated electromagnetic theory that “there is a pressure in the direction normal to the waves” [2]. However, pioneering works by Poynting introduced the electromagnetic momentum density as a cross product of the electric and magnetic field vectors: P ∝ E × B [3,20]. Unlike the straightforward de Broglie formula, the Poynting momentum is not obviously associated with the wavevector k . It is indeed aligned with the wavevector in the ideal case of a homogeneous plane electromagnetic wave. However, in more complicated yet typical cases of structured optical fields (e.g., interference, optical vortices, or near fields [21,22]) the direction of P can significantly differ from the wavevector directions [10,11]. Notably, the origin of the above discrepancy between the Poynting momentum and wavevector lies within the framework of relativistic field theory (see Supplementary 1
Information). The conserved momentum of the electromagnetic field is associated with the translational symmetry of spacetime via Noether’s theorem [13,23]. Applied to the electromagnetic field Lagrangian, this theorem produces the so-called canonical momentum density P can . In the quantum-field framework, the canonical momentum generates spatial translations of the field, in the same way as the de Broglie formula is associated with the operator pˆ = −i∇ generating translations of the wavefunction. Therefore, the canonical momentum density of monochromatic optical fields is naturally associated with the local wavevector k loc of the wave electric field, which is determined by the phase gradient normal to the wavefront [10,11,21–23]. However, resolving fundamental difficulties with the canonical stress-energy tensor (which is non-symmetric and gauge-dependent), in 1940 Belinfante added a “virtual” contribution to get this to agree with the usual electromagnetic stress-energy tensor (symmetric and gauge-invariant) [12–14,23]. In monochromatic optical fields, assuming the Coulomb gauge, Belinfante’s 1 addition to the electromagnetic momentum is a solenoidal edge current P spin = ∇ × S produced 2 by the intrinsic spin angular momentum density S (i.e., the oriented ellipticity of the polarization) of the field. Due to its solenoidal nature, this spin momentum does not transport energy, and is usually considered as unobservable per se. In contrast to P can , Belinfante’s spin momentum is determined by the circular polarization and inhomogeneity of the field rather than by its wavevector [10–14]. Thus, the well-known Poynting vector represents a sum of qualitatively-different canonical and spin contributions: P can + P spin = P . Moreover, it is Belinfante’s spin momentum that is responsible for the difference between the local propagation and Poynting-vector directions in structured light.
Figure 1. Canonical and spin momenta of light in an evanescent wave. (a) The evanescent wave is generated by the total internal reflection of a polarized plane wave at the glass-air interface. It carries longitudinal canonical momentum determined by its wavevector, and also exhibits transverse spin momentum, which is determined by the degree of circular polarization (helicity) of the field [10]. (b) The longitudinal canonical momentum produces the well-known radiation pressure in light-matter interactions, while the transverse spin momentum exerts a weak helicitydependent force orthogonal to the propagation direction of light.
2
The above structure of the electromagnetic momentum has traditionally been regarded as an abstract field-theory construction. However, recently some of us argued [10] that one of the simplest inhomogeneous optical fields, a single evanescent wave, offers a unique opportunity to investigate, simultaneously and independently, the canonical and spin momenta of light in the laboratory environment, see Fig. 1. Considering the total internal reflection of a polarized plane wave at the glass-air interface, the canonical momentum density in the evanescent field in the air is proportional to its longitudinal wavevector: P can ∝ kz z . At the same time, the Poynting vector in an evanescent wave has an unusual transverse component, first noticed by Fedorov 60 years ago [24]. Remarkably, this component is proportional to the degree of circular polarization κk spin y . Here k is the vacuum (helicity) σ and has a pure Belinfante’s spin origin: P⊥ = P⊥ ∝ σ kz wavenumber, kz > k , and κ = kz2 − k 2 is the parameter of the vertical exponential decay of the evanescent wave amplitude ∝ exp ( −κ x ) . Thus, if the spin momentum and Poynting vector are observable physical quantities, this should lead to an extraordinary helicity-dependent light pressure, which is orthogonal to the propagation direction (wavevector) of the evanescent wave. Here we present a direct measurement of the transverse helicity-dependent momentum and force in an evanescent wave, using lateral molecular force microscopy (LMFM) [18]. Importantly, the above two kinds of optical momentum manifest themselves very differently in light-matter interactions (see Fig. 1b). The usual radiation pressure is produced by the canonical momentum [10,11,22,23,25–27] (even though it is often attributed to the Poynting vector [28]), and the corresponding force is always longitudinal, i.e., aligned with the wave propagation: Fpress ∝ P can . In turn, the transverse spin momentum, in agreement with its “virtual” nature, can produce only very weak force vanishing in the dipole-interaction approximation: F⊥spin ∝ P⊥spin , F spin F press [10,11]. In our experiment, we were able to isolate the weak transverse force as
LMFM uses a strongly anisotropic probe, which is highly sensitive to the optical force along one axis (Fig. 2). Namely, we used a planar dielectric nano-cantilever, which represents an ideal sensor for the force component normal to its plane [15–19]. Recently, there have been significant breakthroughs in the manufacturing of such highly compliant cantilevers, which are now truly nano-scale devices with femto-Newton sensitivity [17,19]. Mounting such a cantilever in the ( x, z ) plane of the evanescent wave (Fig. 1), one can measure the transverse y -component of the optical force. We emphasize that the force we measure is neither the z -directed radiationpressure force [1–7,25–27], nor the gradient x -directed force used for optical trapping [4–6,28], but a novel type of optical force orthogonal to both the propagation and inhomogeneity directions. The experimental setup shown in Fig. 2a is based on the LMFM described in [19]. The red laser 1 (wavelength λ = 2π k −1 = 660 nm) generates an evanescent field at the glass-air interface via an objective-based total internal reflection system. The polarization state of this field is controlled by a rotating quarter waveplate (QWP). Rotation of the QWP in the range of angles −45° ≤ φ ≤ 45° drives the polarization between opposite spin states, left-handed and right-handed circular polarizations with σ = −1 and σ = 1 , as it is shown on the Poincaré sphere in Fig. 2b. The cantilever with a spring constant γ 1.2 ⋅10 −6 N/m is manufactured from ultra-low stress silicon nitride (refractive index n = 2.3 ); it has thickness d 50 nm, width w 1000 nm, and length l 120µm . It is vertically mounted in the evanescent field above the glass cover-slip. Deflections of the cantilever, Δ , caused by optical forces, are registered using a detection system based on a non-interferometric scattered evanescent wave (SEW) method [19]. The SEW system involves the green laser 2 (wavelength 561 nm), and it allows measuring, with a 1 nm resolution, the cantilever deflections Δ as well as its vertical position. The intensity of the evanescent field produced by the red laser 1 is “on-off” modulated in time (TTL-modulation) to generate an 3
intermittent force field. This allows us to isolate optical forces produced by the laser 1 on the constant background of other forces (e.g., from the imaging laser 2) (Fig. 3a).
Figure 2. Lateral molecular force microscope probing optical forces in an evanescent field. (a) The LMFM setup. The red laser 1, with the modulated intensity and polarization controlled by the rotating quarter waveplate (QWP), produces the evanescent field to be probed. The green laser 2 images the position of the cantilever probing the evanescent field of laser 1. (b) Variations of the polarization state of the laser-1 field, caused by rotations of the QWP in the range of angles −45° ≤ φ ≤ 45° , are represented by the black curve on the Poincaré sphere. The values φ = −45° , 0° , and 45° correspond to the left-hand circular (L), horizontal linear (S), and right-hand circular (R) polarizations, respectively. (c) Top view of the cantilever orientation. The cantilever plane forms an angle θ 1 with the propagation z -axis to simultaneously measure the strong longitudinal radiation pressure and the weak transverse spin-momentum force (cf., Fig. 1). Importantly, in our experiments, the cantilever plane is not perfectly aligned with the ( x, z ) plane of the evanescent field, but constitutes a small angle θ 1 with this plane, as shown in Fig. 2c. Therefore, it actually measures a mixture of the transverse spin momentum and longitudinal canonical momentum, which are weighted by cosθ 1 and sin θ 1 , respectively. Thus, the effect of the transverse spin momentum is significantly enhanced over the radiation pressure. The different natures of the canonical and spin momenta enable us to separate the two mixed contributions in the measured force. Namely, the transverse spin momentum (force) is proportional to the wave helicity σ , and, therefore, it is an odd function of the QWP angle φ . In contrast, the longitudinal canonical momentum (radiation-pressure force) is independent of the spin, and, hence, is an even function of φ . (The radiation pressure varies with φ due to the difference in the interaction of the vertically- and horizontally-polarized waves with a vertical cantilever, see Supplementary Information.) Thus, measuring the optical force F (θ ,φ ) normal to the cantilever plane, we extract the spin-momentum and radiation-pressure contributions as 4
F (θ ,φ ) − F (θ ,−φ ) F (θ ,φ ) + F (θ ,−φ ) and Fz press (φ ) = . In our measurements, the angle 2 cosθ 2sin θ of the cantilever was estimated to be θ 5° 0.09 rad. Fyspin (φ ) =
Figure 3. Longitudinal and transverse optical forces in an evanescent wave. (a) (Right) Typical cantilever-position trace, recorded at φ = 45° and θ 5° , while the laser-1 intensity is TTL-modulated at 1 Hz. (Left) The histogram of the position distribution shows two Gaussian-like distributions separated by a distance Δ (θ ,φ ) . This measures the optical force acting on the cantilever: F (θ ,φ ) = γ Δ (θ ,φ ) . (b,c) The longitudinal radiation-pressure force Fz press (φ ) − Fz press ( 0 ) and transverse spindependent force Fyspin (φ ) , which are retrieved from the φ -even and φ -odd parts of
the force F (θ ,φ ) . The experimental results are compared with the results of exact numerical simulations and theoretical estimations based on a simplified Mie-particle model (see Supplementary Information). The results of our measurements are presented in Fig. 3. Figure 3a shows an example of the cantilever-position signal (detected via SEW by laser 2) varying in time due to the intermittent force produced by the laser-1 evanescent field. The distance Δ (θ ,φ ) between the centroids of the two Gaussian-like distributions, corresponding to the “on” and “off” laser 1, is a measure of the optical force: F (θ ,φ ) = γ Δ (θ ,φ ) . To improve the resolution and average out thermal fluctuations, we accumulated two distributions over several “on-off” cycles. The longitudinal radiation-pressure Fz press (φ ) and transverse spin-dependent force Fyspin (φ ) , extracted from the measurements of F (θ ,φ ) , are depicted in Figures 3b,c. These are the central results of our work. For clarity, we have neglected the φ -independent forces (e.g., the one produced by intensity gradients), and plotted the longitudinal force Fz press (φ ) with respect to its value at φ = 0 . Most importantly, the results presented in Fig. 3c clearly show the presence of the transverse 5
spin-dependent optical force, which is orthogonal to both the propagation and decay directions of the evanescent wave. This confirms the presence and observability of the enigmatic Belinfante’s spin momentum, which so far has been considered as “virtual”. Moreover, the measurements in Fig. 3b,c show that the spin momentum is indeed almost “invisible”, as it produces a force several orders of magnitude weaker than the radiation-pressure force from the canonical momentum. In turn, this proves that the Poynting vector, which has been used in optics for many decades, does not present a single meaningful momentum of light, but rather a sum of two independent contributions of different nature and properties [10,11]. To verify our theoretical interpretation of the experimental measurements, we performed numerical simulations and analytical model calculations of optical forces on a matter probe in the evanescent field. Numerical simulations were performed using the coupled-dipole method, which models the cantilever as an assembly of interacting point particles. The simulations confirm the presence of the longitudinal radiation-pressure and transverse spin-dependent forces, which are in an excellent agreement with the experimental results, Figs. 3b,c. The fitting of the two curves involves only one common scaling factor. The φ -dependences and the scale-free ratio Fyspin / Fz press of the forces are perfectly reproduced within the experimental accuracy (see Supplementary Information for more details). Moreover, the same longitudinal and transverse forces, including their φ -dependences, are obtained within a greatly simplified model of a spherical Mie particle interacting with the field [10]. Despite the enormous difference between the particle and cantilever geometries, choosing a suitable particle’s radius ( r 25 nm, i.e., ~ d / 2 in our case) results in a good agreement between the calculated and experimentallymeasured forces, Figs. 3b,c. Importantly, the particle model provides analytical expressions for the forces, which confirm their direct proportionality to the canonical and spin momentum densities in optical fields: Fz press ∝ Pz can and Fyspin ∝ Py = Pyspin (see Supplementary Information). The numerical simulations also enabled us to investigate dependences of the above two forces on the shape of the cantilever (see Supplementary Figure S4). In particular, varying the cantilever width w (i.e., its area) we found that the longitudinal force Fz press grows near-linearly with w , which reflects its usual radiation-pressure nature related to the planar surface of the cantilever. In contrast to this, the transverse force Fyspin approximately saturates after w reaches few wavelengths. This means that the helicity-dependent force associated with the Belinfante spin momentum is not a pressure force, but rather an edge effect related to wave diffraction on the vertical edges of the cantilever. Indeed, one can show analytically that the transverse force vanishes for an infinite lamina without edges aligned with the ( x, z ) -plane: Fyspin = 0 . This is in extreme contrast to the infinite radiation pressure for the same lamina in the
( y, z ) -plane:
Fz = ∞ . This proves that the spin momentum does not exert the usual radiation pressure on planar objects. Nonetheless, it can be detected (as we do in this work) due to its weak interaction with the edges of finite-size probes. Thus, our results re-examine one of the most basic properties of light: its momentum. In contrast to numerous previous studies, which involved the radiation pressure in the direction of propagation of light or trapping forces along the intensity gradients, we have observed, orthogonal to both of these directions, the extraordinary optical momentum and force. Remarkably, the transverse Belinfante’s momentum and force are determined by the spin (circular polarization) of light rather than by its wavevector. Our results demonstrate that the canonical and spin momenta, forming the Poynting vector within field theory, manifest themselves very differently in interactions with matter. This offers a new paradigm for numerous studies and applications involving optical momentum and its manifestations in light-matter interactions [4–7]. Furthermore, the interplay between the canonical and Belinfante–Poynting momenta is closely related to fundamental quantum and field-theory problems, such as “quantum weak measurements of photon trajectories” [22,29], “local superluminal propagation of light” press
6
[22,26,27], and the “proton spin crisis” in quantum chromodynamics [30]. Last but not least, the LMFM technique used in our experiment offers a new platform for precision direction-resolved measurements of optical momenta and forces in structured light fields at subwavelength scales.
References 1. Jones, R.V. Pressure of radiation. Nature 171, 1089–1093 (1953). 2. Worrall, J. The pressure of light: the strange case of the vacillating ‘crucial experiment’. Stud. Hist. Phil. Sci. 13, 133–171 (1982). 3. Loudon, R. & Baxter, C. Contribution of John Henry Poynting to the understanding of radiation pressure. Proc. Roy. Soc. A 468, 1825–1838 (2012). 4. Phillips, W.D. Laser cooling and trapping of neutral atoms. Rev. Mod. Phys. 70, 721–741 (1998). 5. 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). 6. Grier, D.G. A revolution in optical manipulation. Nature 424, 810–816 (2003). 7. Aspelmeyer, M., Kippenberg, T.J. & Marquardt, F. Cavity optomechanics. Rev. Mod. Phys. 86, 1391–1452 (2014). 8. Kippenhahn, R., Weigert, A. & Weiss, A. Stellar Structure and Evolution (Springer, 1990). 9. Wright, J.L. Space Sailing (Taylor & Francis, 1992). 10. Bliokh, K.Y., Bekshaev, A.Y. & Nori, F. Extraordinary momentum and spin in evanescent waves. Nature Commun. 5, 3300 (2014). 11. Bekshaev, A.Y., Bliokh, K.Y. & Nori, F. Transverse spin and momentum in two-wave interference. Phys. Rev. X 5, 011039 (2015). 12. 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). 13. Soper, D.E. Classical Field Theory (Wiley, 1976). 14. Ohanian, H.C. What is spin? Am. J. Phys. 54, 500–505 (1986). 15. Sidles, J.A., Garbini, J.L., Bruland, K.J., Rugar, D., Züger, O., Hoen, S. & Yannoni, C.S. Magnetic resonance force microscopy. Rev. Mod. Phys. 67, 249–265 (1995). 16. Rugar, D., Budakian, R., Mamin, H.J. & Chui, B.W. Single spin detection by magnetic resonance force microscopy. Nature 430, 329–332 (2004). 17. Lee, D.-W., Kang, J.-H., Gysin, U., Rast, S., Meyer, E., Despont, M. & Gerber, C. Fabrication and evaluation of single-crystal silicon cantilevers with ultra-low spring constants. J. Micromech. Microeng. 15, 2179–2183 (2005). 18. Scholz, T., Vicary, J.A., Jeppesen, G.M., Ulcinas, A., Hörber, J.K.H. & Antognozzi, M. Processive behaviour of kinesin observed using micro-fabricated cantilevers. Nanotechnology 22, 095707 (2011). 19. Vicary, J.A., Ulcinas, A., Hörber, J.K.H. & Antognozzi, M. Micro-fabricated mechanical sensors for lateral molecular-force microscopy. Ultramicroscopy 111, 1547–1552 (2011). 20. Jackson, J.D. Classical Electrodynamics 3rd ed. (Wiley, 1999). 21. Berry, M.V. Optical currents. J. Opt. A: Pure Appl. Opt. 11, 094001 (2009). 22. 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). 23. Bliokh, K.Y., Bekshaev, A.Y. & Nori, F. Dual electromagnetism: helicity, spin, momentum, and angular momentum. New J. Phys. 15, 033026 (2013). 24. Fedorov, F.I. To the theory of total reflection. Doklady Akademii Nauk SSSR 105, 465–468 (1955) [translated and reprinted in J. Opt. 15, 014002 (2013)]. 25. Ashkin, A. & Gordon, J.P. Stability of radiation-pressure particle traps: an optical Earnshaw theorem. Opt. Lett. 8, 511–513 (1983). 7
26. Huard, S. & Imbert, C. Measurement of exchanged momentum during interaction between surface-wave and moving atom. Opt. Commun. 24, 185–189 (1978). 27. Barnett, S.M. & Berry, M.V. Superweak momentum transfer near optical vortices. J. Opt. 15, 125701 (2013). 28. Bowman, R.W. & Padgett, M.J. Optical trapping and binding. Rep. Prog. Phys. 76, 026401 (2013). 29. Kocsis, S., Braverman, B., Ravets, S., Stevens, M.J., Mirin, R.P., Shalm, L.K. & Steinberg, A.M. Observing the average trajectories of single photons in a two-slit interferometer. Science 332, 1170–1173 (2011). 30. Leader, E. & Lorce, C. The angular momentum controversy: What’s it all about and does it matter? Phys. Rep. 541, 163–248 (2014). Acknowledgements. This research was supported by Ministry of Education, Youth and Sports of the Czech Republick (project LO1212), RIKEN iTHES Project, MURI Center for Dynamic Magneto-Optics via the AFOSR (grant number FA9550-14-1-0040), and Grant-in-Aid for Scientific Research (A). M.A. would like to thank Nick and Susan Woollacott who kindly funded the equipment used in this research, as well as Adrian Crimp, Dave Engledew, Josh Hugo, and Pete Dunton for their essential technical support.
8
SUPPLEMENTARY INFORMATION
1. Calculations of the fields and momentum densities 1.1. Canonical and spin momenta: from relativistic field theory to optical fields The canonical energy-momentum tensor for the free-space Maxwell field follows from the space-time translation symmetry of the field Lagrangian and Noether’s theorem [13]. Using standard relativistic notation with the Einstein summation rule, the canonical energy-momentum tensor reads
(
)
αβ Tcan = ∂α Aγ F βγ −
1 αβ γδ g F Fγδ , 4
(S1)
where Aα is the electromagnetic four-potential, F αβ is the field tensor, and gαβ is the Minkowski spacetime metric tensor. The tensor (1) is gauge-dependent (due to the presence of Aα ) and non-symmetric. Nonetheless, it is this tensor that corresponds to the generators of spatial translations for the electromagnetic field. In 1940, Belinfante suggested a symmetrisation procedure to “improve” tensor (S1), i.e., to make it gauge-invariant and symmetric [12,13]. He added the following total-divergence term (constructed from the spin tensor) to the canonical energy-momentum tensor: αβ Tspin = − ∂γ ( Aα F βγ ) ,
(S2)
The resulting symmetric energy-momentum tensor (also known as the Belinfante energymomentum tensor) is αβ αβ T αβ = Tcan + Tspin = F αγ F βγ −
1 αβ γδ g F Fγδ . 4
(S3)
This tensor (S3) is typically considered as meaningful in field theory, because it is gaugeinvariant and is naturally coupled to gravity [13]. In turn, the Belinfante spin-correction term (S2) is usually regarded as “virtual”, because it does not contribute to the energy-momentum conservation law, energy transport, and integral momentum of a localized field [13,14,23]. The momentum density of a free electromagnetic field is given by the T i 0 ≡ P i components of the energy-momentum tensor. In this manner, the canonical, spin, and Poynting momentum densities are obtained from Eqs. (S1)–(S3) as
P can = E ⋅ ( ∇ ) A , P spin = − ( E ⋅∇ ) A , P = P can + P spin = E × B ,
(S4)
where E and B are the electric and magnetic fields, whereas A is the vector-potential. Note that in Eqs. (S4) and in what follows we use Berry’s notation X ⋅ ( ∇) Y ≡ X i ∇Yi [21] and natural electrodynamical units ε 0 = µ0 = c = 1 . Although the canonical and spin momenta are gauge-dependent, there are several strong indications that in a number of situations the experimentally-measured quantities correspond to the canonical quantities taken in some particular gauge. Recently, this was actively discussed in relation to optical experiments with laser fields [10,11,22,23,26,27,29,31–35] and QED experiments detecting gluon and quark spin and orbital contributions to the proton spin [30]. In
1
optical experiments, the measured quantities correspond to the Coulomb gauge, i.e., A 0 = 0 and ∇ ⋅ A = 0 , and hereafter we assume this gauge. We are interested in monochromatic optical fields, which can be described by complex time-independent field amplitudes: E ( r,t ) → Re ⎡⎣ E ( r ) e−iω t ⎤⎦ , B ( r,t ) → Re ⎡⎣ B ( r ) e−iω t ⎤⎦ ,
A (r, t ) → Re ⎡⎣A (r ) e−iωt ⎤⎦ , where ω is the frequency. Here we use the same letters E , B , and A for the complex field amplitudes, and imply only these complex fields in what follows. Importantly, the corresponding vector-potential amplitude (in the Coulomb gauge) becomes simply proportional to the electric field amplitude [23]: A ( r ) = −iω −1E ( r ) . Substituting these equations into Eqs. (S4), and performing time averaging over the ω -oscillations, we obtain expressions for the canonical, spin, and Poynting momentum densities in a generic optical field [21,23]: P can =
1 1 1 Im ⎡⎣ E∗ ⋅ ( ∇ ) E ⎤⎦ , P spin = ∇ × Im ⎡⎣ E∗ × E ⎤⎦ , P = P can + P spin = Re ( E∗ × B ) . (S5) 2ω 4ω 2
Here we use the same letters P can , P spin , and P for time-averaged optical momenta densities, and consider only these quantities in what follows. Notably, the canonical momentum in Eqs. (S5) is proportional to the local expectation value of the canonical momentum operator pˆ = −i∇ , and is proportional to the phase gradient or local wavevector of the field [21–23,27]: P can ∝ Re ⎡⎣ E∗ ⋅ ( pˆ ) E ⎤⎦ ∝ k loc E . 2
(S6)
In turn, the spin momentum in Eqs. (S5) represents a solenoidal edge current, which is generated by the intrinsic spin angular momentum in the field:
1 P spin = ∇ × S , 2
S=
()
1 Im ⎡⎣ E∗ × E ⎤⎦ ∝ E∗ ⋅ Sˆ E , 2ω
(S7)
where Sˆ is the vector of spin-1 matrices [21,22,33,35]. Equations (S5)–(S7) reveal different physical origins and meanings of the two optical momenta constituting the Poynting vector. Note that throughout this work we use the “standard” formalism for the canonical and spin quantities, which is based on the electric field [13,22,23,33]. There is also a “dual-symmetric” formalism, where all quantities are symmetrized with respect to similar electric and magnetic field contributions [10,11,23,33]. While the dual-symmetric approach is more natural for freespace fields, in our experiments the probes are sensitive only to the electric field, and, therefore, the standard “electric-biased” formalism is more suitable. 1.2. Fields and momenta in an evanescent optical wave To calculate the fields and momenta in the evanescent field investigated in our experiment, we need to describe several transformations of the optical field on its way from the laser to the probe (cantilever). The schematic of the experiment is shown in Figures 1 and 2 of the main text. We describe the complex electric field of the laser-1 beam in its accompanying frame XyZ , where the Z -axis is directed along the beam and the y -axis is the transverse (horizontal) axis. Thus, the s and p polarizations correspond to the y and X linear polarizations, respectively. Neglecting the longitudinal Z -component in the paraxial laser field, the laser 1 emits the s polarized field E0 described by the following Jones-vector form:
2
⎛ E0 X ⎜ ⎜⎝ E0 y
⎞ ⎛ 0 ⎞ exp ( ikZ ) . ⎟ = E0 ⎜ ⎟⎠ ⎝ 1 ⎟⎠
(S8)
Here E0 is the electric-field amplitude, and k = 2π λ is the wavenumber of the laser 1. Next, the field (S8) is transmitted through the quarter-wave plate (QWP) with its fast axis forming an angle φ with the y -axis. The QWP used in our experiment was designed for the nominal wavelength λ0 = 633 nm, while the laser 1 wavelength was λ = 660 nm. Therefore, the fast-axis-polarized component acquired the following phase shift with respect to the orthogonally-polarized component: 2δ =
λ0 π . λ 2
(S9)
As a result, the beam field E1 after the QWP is described by the following Jones-matrix transformation [36]: ⎛ E1X ⎜ ⎜⎝ E1y
⎞ ⎛ sin 2φ + cos 2φ e−2iδ − sin φ cosφ (1− e−2iδ ) ⎜ ⎟= ⎟⎠ ⎜ − sin φ cosφ (1− e−2iδ ) sin 2φ e−2iδ + cos 2φ ⎝
⎞⎛ E ⎟ ⎜ 0X ⎟ ⎜⎝ E0 y ⎠
⎞ ⎟. ⎟⎠
(S10)
Substituting here Eq. (S8) and omitting the common −iexp ( −iδ ) factor, we obtain ⎛ E1X ⎜ ⎜⎝ E1y
⎞ ⎛ ⎞ sin δ sin 2φ ⎟ = E0 ⎜ ⎟ exp ( ikZ ) . ⎟⎠ i cos δ − sin δ cos 2 φ ⎝ ⎠
(S11)
Next, the beam enters the glass (a high-NA objective in our case, see Fig. S5 below) and undergoes a total internal reflection at the glass-air interface. The generation of the evanescent field and its properties at such interface are described in detail in the Supplementary Materials of [10]. Following that work, we represent the field (S11) inside the glass in three-dimensional form: ⎛ E1X ⎜ E1 ≡ ⎜ E1y ⎜ E ⎝ 1Z
⎞ ⎟ E0 ⎟= 2 1+ m1 ⎟ ⎠
⎛ 1 ⎞ ⎜ m ⎟ exp ( in kZ ) , 1 ⎜ 1 ⎟ ⎜⎝ 0 ⎟⎠
(S12)
where
m1 =
E1y i cos δ − sin δ cos 2φ = E1X sin δ sin 2φ
(S13)
is the complex polarization parameter, and n1 = 1.5 is the refractive index of the glass. The plane-wave field (S12) impinges the glass-air interface with the angle of incidence α , n1 sin α > 1 ( α 47.5° in our experiment). The evanescent wave field E generated in the air can be written as (see Supplementary Materials of [10]) ⎛ Ex ⎞ ⎜ ⎟ E E ≡ ⎜ Ey ⎟ = 2 1+ m ⎜ E ⎟ ⎝ z ⎠
⎛ 1 ⎜ mk k z ⎜ ⎜⎝ −iκ kz
where 3
⎞ ⎟ ⎟ exp ( ikz z − κ x ) , ⎟⎠
(S14)
kz = kn1 sin α ≡ k cosh ϑ ,
κ = kz2 − k 2 ≡ k sinh ϑ ,
(S15)
are the propagation and exponential-decay wavevector parameters of the evanescent wave, which are expressed via the hyperbolic angle ϑ . (In our experiment the propagation wavelength 2π kz−1 597 nm and decay scale of the evanescent wave κ −1 222 nm.) In Eq. (S14) we used the ( x, y, z ) coordinates shown in Figures 1 and 2, and introduced the following amplitude and polarization parameters: 2
E=
k 1+ m T p E0 , kz 1+ m1 2
m=
Ts m1 , Tp
(S16)
which involve the Fresnel transmission coefficients [20] Ts =
2n1 cos α 2n1 cos α , Tp = . n1 cos α + i sinh ϑ cos α + i n1 sinh ϑ
(S17)
Equations (S13)–(S17) completely describe the evanescent wave electric field E (r ) that we probe in the experiment. Substituting this field into the general equations (S5)–(S7), we obtain the canonical and spin momentum densities in the evanescent wave: P can =
P spin =
(
W ω
W ⎛ κ2⎞ W ⎛ sinh 2ϑ ⎞ kz ⎜ 1+ τ 2 ⎟ z = k ⎜ cosh ϑ + τ z . ω kz ⎠ ω ⎝ cosh ϑ ⎟⎠ ⎝
⎛ κ2 κk ⎞ W − z + σ y⎟= ⎜ k k z z ⎝ ⎠ ω
⎛ sinh 2ϑ sinh ϑ ⎞ k⎜− z+ σ y . cosh ϑ ⎟⎠ ⎝ cosh ϑ
(S18)
(S19)
)
1 1 2 2 2 E + B = E exp ( −2κ x ) is the energy density of the field, y and z are the 4 2 unit vectors of the corresponding axes, whereas
Here W =
2
1− m τ= 2 1+ m
and
σ=
2 Im m 2 1+ m
(S20)
are the first and the third Stokes parameters of the evanescent field [10]. In particular, the third Stokes parameter σ is the degree of circular polarization, i.e., helicity, which determines the longitudinal z -directed spin angular momentum of light. Substituting Eqs. (S13)–(S17) into Eq. (S20), we express the Stokes parameters of the evanescent wave as a function of the varying QWP angle φ and other parameters:
τ= σ =2
sin 2 2φ − ( cos 2 2φ + cot 2δ ) ( cosh 2ϑ − cos 2α )
sin 2 2φ + ( cos 2 2φ + cot 2δ ) ( cosh 2ϑ − cos 2α )
,
sin 2φ ( cosh ϑ sin α cot δ − sinh ϑ cos α cos 2φ ) sin 2 2φ + ( cos 2 2φ + cot 2δ ) ( cosh 2ϑ − cos 2α )
.
(S21)
Importantly, the parameters τ (φ ) and σ (φ ) are, respectively, even and odd functions of φ , which enables us to separate the τ - and σ -dependent effects in our φ -dependent measurements. Note that the energy density W (φ ) is also an even function of φ :
4
W ∝ E0
2
⎞ 4 cot 2α sin 2δ ⎛ sin 2 2φ + cos 2 2φ + cot 2δ ⎟ , 2 2 2 ⎜ ⎠ ( n1 − 1) ⎝ cosh ϑ − cos α
(S22)
which does not affect the above features of the polarization dependences. Equations (S18) and (S19) show the presence of two kinds of optical momenta in the evanescent wave. The first one is the longitudinal τ -dependent canonical momentum density Pz can , proportional to the wavevector component kz [shown in the cyan frames in (S18)]. As kz > k = ω / c , this momentum density exceeds ω / c per photon. Therefore, the canonical momentum in an evanescent wave produces anomalously-high radiation pressure, which has been experimentally measured using light-atom interactions [22,26]. Second, the evanescent wave exhibits the spin momentum, which has a σ -dependent component Pyspin , perpendicular to both the propagation and decay directions of the wave [shown in the magenta frames in (S19)]. This helicity-dependent transverse momentum is the main subject of our work. Note that the longitudinal component of the spin momentum in Eq. (S19), Pz spin , produces only very weak helicity-independent correction to the strong radiation pressure from the canonical momentum, and, hence, can be ignored. Furthermore its magnitude has a smallness of ∝ κ 2 when κ k . This longitudinal spin momentum is associated with another intriguing phenomenon: the transverse helicity-independent spin in evanescent waves [10,37–40].
2. Calculations of optical forces on dielectric probes 2.1. Estimations of optical forces using a dipole-particle model Having the evanescent field (S13)–(S17) and its momentum properties (S18)–(S22), we now calculate how these properties reveal themselves in interactions with matter probes immersed in the evanescent wave. The momentum transfer in light-matter interactions produces optical forces, which we calculate below. We first consider the simplest analytical model of a small spherical isotropic dielectric particle of a radius r λ and permittivity ε [10]. In the lowest orders in kr 1 , such a particle is characterized by the complex electric polarizability χ e :
Re χ e = r 3
ε −1 , ε +2
Im χ e =
2 3 2 k ( Re χ e ) , 3
(S23)
where the small imaginary part, Im χ e Re χ e , originates from the radiation-friction effects [41–43]. The lowest-order magnetic polarizability of the particle is
χ m = k 2r 5
ε −1 , 30
(S24)
so that Re χ m Re χ e and Im χ m 0 . In the leading-order electric-dipole approximation, the optical force is given by [10,25,42– 44]: F ∝ Re ( χ e ) ∇We +
1 ω Im ( χ e ) P can . 2 Fpress
5
(S25)
Here the first term is the gradient force F grad involving the electric energy density We = E / 4 , 2
and the second term is the radiation pressure force F press proportional to the canonical momentum density P can . In the evanescent wave under consideration, the gradient force is directed along the vertical x -axis of the exponential decay, i.e., along the normal to the interface, while the radiation pressure is associated with the longitudinal z -direction of propagation and the canonical momentum density (S18). Calculating the next-order correction to the electric-dipole force (S25), one can obtain a weak force, which originates from the dipole-dipole interaction between the electric and magnetic polarizabilities of the particle [10,43]:
ω k3 ⎛ ⎞ δF ∝ − Re ( χ e χ m* ) ⎜ P can + P spin ⎟ . spin ⎝ ⎠ 3 F
(S26)
The first term proportional to P can is only a small correction to the radiation-pressure force in Eq. (S25), while the term proportional to P spin includes a weak transverse y -directed force F spin associated with the Belinfante spin momentum. Remarkably, for dielectric particles Re ( χ e χ m* ) > 0 , and this force is negative, i.e., antiparallel to the spin-momentum direction. Note
that even for non-absorbing dielectric particles, k 3 Re ( χ e χ m* ) ∝ k 5 r 8 and Im χ e ∝ k 3r 6 , so that
δ F / F ∝ ( kr ) 1 . 2
In this work, we measure the longitudinal radiation-pressure force Fz = Fz press ∝ Pz can and the transverse helicity-dependent force δ Fy = Fyspin ∝ Pyspin . Importantly, since the canonical and spin momentum densities (S18) and (S19) are, respectively, even and odd functions of the QWP angle φ , the corresponding radiation-pressure force in Eq. (S25) and the transverse force in Eq. (S26) share the same properties. Note that, in addition to the radiation-pressure force depending on the first Stokes parameter τ and the transverse spin-momentum force proportional to the third Stokes parameter σ (helicity), in the generic case the weak transverse force (S26) also has a contribution proportional to Im ( χ e χ m* ) and the second Stokes parameter [10]. However, this force is extremely small in our case due to the smallness of Im ( χ e χ m* ) , and,
hence, can be neglected. The above small-particle equations can be used as an analytical toy model for the description of optical forces on the cantilever in the experiment. Of course, the cantilever and particle differ drastically in shape, and one can expect that the particle model can only produce rough estimations of the experimentally-measured forces. To compare forces exerted on a small spherical particle and the cantilever, we set the dielectric constant of the particle as that of the cantilever, ε = n2 = 5.3 , and also involve two fitting parameters in the particle model. First, the radius of the particle, r , is the main free parameter of the model. Second, since the field amplitude is unknown, and we cannot expect that the model will correctly reproduce absolute values of the forces, we introduce the scaling coefficient K between the calculated and measured quantities. As we are interested in the φ -dependences of the forces, we scale the difference between the maximum and minimum values of the radiation-pressure force Fz press (φ ) :
{max ⎡⎣ F
z
press
(φ )⎤⎦ − Fz press ( 0 )}measured = K {max ⎡⎣ Fz press (φ )⎤⎦ − Fz press ( 0 )}calculated .
(S27)
Importantly, being determined from the data for the longitudinal force component, the same scaling factor K is then applied to the calculated transverse spin-dependent force component:
6
{F
spin y
(φ )}measured = K {Fyspin (φ )}calculated ,
(S28)
so that the ratio between quantities (S27) and (S28) is scale-independent and uses only the particle radius r as the fitting parameter. The results of the comparison between the analytical particle-model calculations and experimentally-measured forces is shown in Fig. S1. Using the scaling relation (S27), the calculated longitudinal radiation-pressure force (S25), Fz press (φ ) − Fz press ( 0 ) , is in very good agreement with the measured one, independently of the particle radius r . At the same time, the calculated transverse helicity-dependent force (S26), Fyspin (φ ) , depends on the radius and shows the best agreement with the experiment for r 25 nm. Note that this radius approximately corresponds to the half-thickness of the cantilever in our case: r ~ d / 2 . The polarization φ dependences of both longitudinal and transverse particle-model forces are in a good agreement with the experimentally-measured forces (up to a small φ -shift of the transverse-force maximum). This proves that the measured forces can indeed be associated with the presence of the longitudinal canonical and transverse spin momenta, as in Eqs. (S25) and (S26), and their polarization dependences are largely independent of the probe (unless we introduce additional specific properties of the probe, such as chirality [34,45]).
Figure S1. Longitudinal (a) and transverse (b) optical forces in an evanescent wave, calculated using an analytical small-particle model (S23)–(S26) (curves) and measured in the experiment (markers “+”). The fitting procedure includes the scaling (S27) and (S28) and the fitting of the particle radius r . The particle model nicely reproduces the main features of the polarization φ -dependences of the two forces associated with the canonical and spin momenta in the evanescent wave.
It is worth noticing that in the isotropic spherical-particle model, the longitudinal radiationpressure force has the same order of magnitude for the p- and s-polarizations of light ( τ = 1 and τ = −1 ). In contrast, for the cantilever with a highly-anisotropic vertical shape, the radiation pressure becomes very small for the horizontally s-polarized light. This difference between the particle and cantilever probes is not seen in our plots because we scale the variations of the radiation-pressure with respect to the s-polarized light ( φ = 0 ), Eq. (S27).
7
2.2. Numerical calculations of the forces for the cantilever For accurate numerical simulations of the interaction between the evanescent optical field and the cantilever, we employ the Coupled Dipole Method (CDM) [46,47]. In this method, continuous matter objects are decomposed into cubic arrays of point polarizable dipoles (cells) coupled through the electromagnetic dipole interaction tensor. When exposed to an external field, each cell feels not only the incident field but also the field scattered by all the other cells in the structure. Algebraically, this results in a large set of linear equations whose solution yields the polarization of each cell. Our implementation of this method for the simulation of the mechanical action of light is discussed elsewhere [48], and we use the Quasi-Minimal Residual Method (QMR) [49] with matrix-vector multiplication accelerated by fast Fourier transforms [50]. The total (incident plus scattered) field outside the matter object is then given by the sum of the incident field with all of the radiating dipole fields. As the numerical lattice is refined, the total field converges to the continuum result. This approach lends itself well to structures of extreme geometry, such as the cantilever used in our experiment. Once the total field is known, the time-averaged optical force acting on each cell can be found by calculating the flux of the Maxwell stress tensor [44,47,48,51,52]. In our simulations, the incident evanescent field is given by Eqs. (S13)–(S17), while the cantilever is modeled as a dielectric lamina with permittivity ε = n2 = 5.3 . If not otherwise stated, we use the following geometric parameters of the cantilever: thickness d = 50 nm, width w = 1000 nm, and infinite vertical length l → ∞ . Convergence of the model has been achieved by (i) refining the numerical lattice, (ii) increasing the vertical length l of the cantilever. The resulting model uses N ~ 2.5 ⋅10 6 cells and l = 500 nm (this value approximately equals to the double decay length 2κ −1 444 nm, and further increase of l practically does not affect the simulation results). The plane of the cantilever is rotated by a small angle θ 5° with respect to the z -axis in the ( y, z ) -plane, see Fig. 2c. The numerically-calculated force F (φ ) = F (φ ) n is directed approximately along the normal n to the cantilever surface and describes the “integral” mechanical action, including both the spin-dependent Fyspin (φ ) and the radiation pressure
Fz press (φ ) contributions. We separate these contributions using their oddness and evenness with respect to φ . As explained in the main text, together with projections of these forces on the corresponding axes, this yields Fyspin (φ ) =
F (φ ) − F ( −φ ) , 2 cosθ
Fz press (φ ) =
F (φ ) + F ( −φ ) . 2sin θ
(S29)
Figures S2 and S3 show the longitudinal radiation-pressure force Fz press (φ ) − Fz press ( 0 ) and
transverse helicity-dependent force Fyspin (φ ) obtained from the numerical simulations described above and from the experimental measurements. Since the field intensity is unknown, a single scaling parameter is used to multiply all the simulation data. Evidently, experimentally-measured forces are in excellent agreement with numerical simulations. To examine the sensitivity of the two forces to the parameters of the problem, in Figs. S2 and S3 we depict multiple curves corresponding to different values of the cantilever refractive index n and thickness d . One can see that the transverse spin-dependent force is weakly sensitive to small variations of the refractive index (unlike the radiation-pressure force), while both forces are equally scaled with small variations of the cantilever thickness.
8
Figure S2. Longitudinal (a) and transverse (b) optical forces acting on the cantilever in the evanescent wave, calculated numerically using the Coupled Dipoles Method (curves) and measured in the experiment (markers “+”). Numerically-simulated curves are shown for different values of the refractive index of the cantilever, n .
Figure S3. Same as in Fig. S2 but for different values of the cantilever thickness d .
Furthermore, by varying the cantilever width w in the numerical simulations, we investigated the dependences of the two forces on the area of the cantilever. These results are shown in Fig. S4. The longitudinal force Fz press grows near-linearly with w , which reflects its usual radiation-pressure nature related to the planar surface of the cantilever. In contrast to this, the transverse force Fyspin approximately saturates after w reaches few wavelengths. This means that the helicity-dependent force associated with the Belinfante spin momentum is not a pressure force, but rather an edge effect related to wave diffraction on the vertical edges of the cantilever. Indeed, one can show analytically that the transverse force vanishes for an infinite lamina without edges aligned with the ( x, z ) -plane: Fyspin = 0 . This is in extreme contrast to the infinite radiation pressure for the same lamina in the ( y, z ) -plane: Fz press = ∞ . In other words, this proves that the Belinfante spin momentum is indeed “virtual”, and it does not exert the usual radiation pressure on planar objects. Nonetheless, it can be detected (as we do this in the present work) due to its weak interaction with the edges of finite-size probes.
9
Figure S4. Numerically-calculated longitudinal (a) and transverse (b) optical forces acting on the cantilever in the evanescent wave versus the cantilever width w . The linear growth of the longitudinal force Fz press ( w ) reveals its radiation-pressure nature (proportional to the area of the cantilever surface). In contrast, the saturation of the transverse force Fyspin ( w ) indicates that it is produced by the field diffraction on the vertical edges of the cantilever.
3. Measurements of optical forces using a nano-cantilever Here we describe details of the experimental setup and measurements. In addition to the force measurements described in the main text, we performed a number of extra measurements of the field and cantilever properties in order to determine their parameters and control the system. 3.1. Characterization of the laser field The experimental setup is shown in Figure 2a. Two laser beams, orthogonal to each other, undergo a total internal reflection in a high numerical aperture (NA) objective lens and produce concentric evanescent areas above the glass cover slip of the objective (Fig. S5). The red laser 1 (Cube from Coherent Inc.) operates with a wavelength λ = 660 nm and power 50 mW. This radiation, with the polarization controlled using the QWP, is the source of the evanescent field under consideration. The green laser 2 (Versa-lase from Vortran Laser Technology Inc.) has a wavelength λ ′ = 561 nm and power 50 mW; its radiation is s-polarized at the glass-air interface where the evanescent field is formed. This radiation is used for measuring the cantilever position. The evolution of the laser beams in the high-NA objective and generation of the evanescent wave is illustrated in Figure S5 (the laser-1 beam is shown). The input off-axial laser beam is focused in the back focal plane of the high-NA objective lens. The off-axis refraction at the objective lens forms a collimated beam propagating at the refraction angle α , which depends on the displacement of the incident beam. At a certain off-axis distance, the condition for total internal reflection at the cover-slip surface is reached (for refractive index n1 = 1.5 , the critical angle of incidence is α c = 41.5° ), and the evanescent field is generated in the air above the surface. 10
Figure S5. Formation of the evanescent field by the total internal reflection of the off-axis laser-1 beam in the high-NA objective covered by a glass slip. The laser-2 beam forms a similar concentric evanescent field in the orthogonal ( x, y ) -plane. In this technique, we cannot measure the angle of incidence α directly, but it can be determined from the exponential decay length κ −1 of the evanescent wave (i.e., the x -distance above the glass surface where the field amplitude drops by a factor of e−1 ). We measured the decay length by measuring the scattered intensity of the laser-1 radiation as the cantilever tip is moved away from the surface, Fig. S6. Fitting the data with an exponential function results in a decay length κ −1 = ( 222 ± 3) nm. According to Eqs. (S14) and (S15), this corresponds to the angle of incidence α = 47.5° ± 0.3° .
Figure S6. Measurement of the exponential decay of the evanescent-wave intensity (laser 1 here) above the glass surface. The polarization state of the laser-1 field is controlled by the rotating quarter waveplate (QWP) (Fig. 2a). Its orientation is varied in the range of angles −45° ≤ φ ≤ 45° . Here φ = 0 corresponds to the linear s-polarization of the beam. Because the QWP is designed for the λ0 = 633 nm radiation, it does not produce a perfect circular polarization for the λ = 660 nm 11
light, see Eqs. (S9)–(S11). Figure S7 shows the measured near-circular polarizations of the beam after passing the QWP with φ = ±45° . This ellipticity of the polarization is taken into account during the data processing, see Eqs. (S9)–(S13).
Figure S7. The measured near-circular polarizations of the laser-1 beam after the QWP is rotated by φ = ±45° . The ratio between the principal axes of the ellipses is 0.9.
3.2. Optical detection of the cantilever position The cantilever, interacting with the evanescent laser-1 and laser-2 fields, is placed right above the cover-slip surface (Fig. 2a). It is perpendicular to the surface with an accuracy of 1º and is adjusted to be in the centre of the Gaussian-like area of the evanescent fields (diameter ~ 50µm ) within ± 2µm . The cantilever deflection is measured using the scattered evanescent wave (SEW) detection system [19] involving the laser-2 radiation. Note that the laser-2 field does not interfere with the laser-1 radiation and does not affect the cantilever interaction with the probed laser-1 field. Furthermore, the reflected laser-2 beam is the only signal reaching the quadrant photodetector, because two sets of filters are used to stop the reflected laser-1 beam (Fig. 2a). We have checked that the cross-talk between the two lasers at the detector is smaller than 0.1%. We monitor the distance between the cantilever tip and the cover-slip surface using the total intensity of the scattered laser-2 light (the SUM signal produced by the photodetector). This distance is kept constant during the measurements by using a negative feedback loop. We adjust the bending direction of the cantilever (i.e., normal to its plane) to be almost perpendicular to the propagation ( x, z ) plane of the laser-1 beam: θ 0 (Fig. 2c). The actual small angle θ 1 of the cantilever with respect to the ( x, z ) plane can be directly measured (within ± 2° ) by observing the signal of the scattered laser-2 radiation on the quadrant photodetector. 3.3. Characterization of the nano-cantilever We used different cantilevers to measure optical forces described in the paper. The force magnitude is determined from the cantilever deflection once the stiffness of the cantilever is
12
known. We determined the stiffness (spring constant) of the cantilever using the following two methods. (i) We measured the thermal power spectral density (PSD) of the cantilever, S ( f ) (where
f is the frequency), see Fig. S8 for an example of one such measurement. The measured PSD was fitted using the equation for the over-damped harmonic oscillator [53]: S( f ) =
kBT , b π ( fc2 + f 2 ) 2
(S30)
where k B is the Boltzmann constant, T is the absolute temperature, fc = γ / ( 2π b ) is the corner frequency of the PSD, and b is the damping term. The resulting spring constant for the case shown in Fig. S8 is γ = (1.2 ± 0.1) ⋅10 −6 N/m. (ii) By using the method based on the equipartition theorem [54], the spring constant can be obtained once the mean square displacement x 2 of the thermally activated cantilever is measured. In this case, the spring constant is γ = kBT / x 2 . For the cantilever used in Fig. S8, this yields the spring constant γ = (1.3 ± 0.1) ⋅10−6 N/m.
Thus, the two methods provide very similar results, so that we are confident that the spring constant can be determined within 10% of its actual value.
Figure S8. Experimentally-measured PSD of the thermally driven cantilever and its fitting curve obtained using the over-damped harmonic oscillator model (S30).
3.4. Optical force measurements Knowing the evanescent field properties, as well as the position and the stiffness of the cantilever, we can measure the desired optical forces from the cantilever deflections caused by the laser-1 field. The intensity of the laser-1 beam is “on-off” modulated in time (TTL modulation) with a frequency of 1 Hz (see Fig. 3a). The deflection of the cantilever, Δ , as well as the modulation signal are recorded at different QWP angles φ , from –45º to +45º in steps of around 8º. Thus, one complete set of measurements consists of 11 points (approximately 90º/8º), each measuring the position of the cantilever for 30 s. We did not notice any significant drift in the instrument during the 15 min necessary to complete a set of measurements.
13
The data are collected at 64 KHz but decimated by a factor of 600 to remove any cross correlation in the measurements. For each angle of the QWP orientation, the recorded trace is split into two sets of positions: one measured when the laser is “on” and the other one measured when the laser is “off”. Two distributions of positions are generated with their relative mean and the standard error of the mean, as shown in Fig. 3a. The difference of the two mean values is the total displacement Δ (φ ) caused by the evanescent wave from laser 1. The deflection is finally multiplied by the spring constant to obtain the optical force normal to the cantilever plane: F (φ ) = γ Δ (φ ) . To remove inessential force contributions, which do not vary with the QWP orientation (e.g., the gradient force emerging because the cantilever position is not exactly in the centre of the evanescent area, the possible mechanical action of the laser-2 radiation, etc.), the final data are offset to zero when the beam is s-polarized, i.e., when φ = 0 . The results of the measurements of the optical force F (φ ) are depicted in Fig. S9. Most importantly, the dependence F (φ ) is not symmetric with respect to φ → −φ . This allows us to separate the φ -even and φ -odd components of the total force, which correspond to the longitudinal radiation-pressure force Fz press (φ ) and transverse helicity-dependent force Fyspin (φ ) , respectively, see Eqs. (S29). The measurement data for these two retrieved forces are shown in Figs. 3b,c. The measured forces are in very good agreement with numerically-calculated and theoretically-modelled forces from Figs. S1–S3. Thus, in our experiment and calculations, we have carefully traced and verified at all stages the appearance of the longitudinal and transverse optical forces. This allows us to unambiguously associate these measured forces with the longitudinal canonical momentum P can , Eqs. (S6) and (S18), and transverse Belinfante spin momentum P⊥spin , Eqs. (S7) and (S19), in the evanescent wave [10].
Figure S9. Measurements of the total optical force normal to the cantilever plane versus the QWP angle. The fitting curve is mirrored with respect to the vertical axis (dashed line) to emphasize the asymmetry of the experimental data with respect to φ → −φ . The φ -odd part of the measured force F (φ ) corresponds to the helicitydependent transverse force (S26) associated with the Belinfante spin momentum (S19) (see Figs. 3b,c).
14
3.5. Additional controls We note that the above measurements required an extremely low-noise environment without air currents around the probe. Therefore, air conditioning was switched off and a double enclosure was constructed around the LMFM unit. We also checked that the thermally-induced deflection due to asymmetric illumination of the cantilever was negligible. The measurement protocol described above was repeated with the cantilever rotated by 180°, and it provided the same results within the experimental accuracy. These results confirm earlier findings [55,56] that the optical pressure on the cantilever is predominant over the photothermal effect. Finally, we estimated the longitudinal torque Tz on the cantilever produced by the spin angular momentum Sz of the elliptically-polarized evanescent wave [10]. In principle, such torques could also cause a deflection proportional to the helicity of light. The ratio between the displacements due to the linear force Fyspin and the torque Tz can be estimated as [57]
( Δ )F ( Δ )T
y
z
2Fyspin = l, 3Tz
(S31)
where l is the vertical length of the cantilever. A very conservative approximation, where Pyspin and Sz are used in place of Fyspin and Tz , yields the ratio (S31) of ~350, confirming the negligible effect of the torque.
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