Direct measurements of the extraordinary optical momentum and transverse spin-dependent force using a nano-cantilever

Abstract

Radiation pressure is associated with the momentum of light 1,2 , and it plays a crucial role in a variety of physical systems 3-6 . It is usually assumed that both the optical momentum and the radiation-pressure force are naturally aligned with the propagation direction of light, given by 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 (degree of circular polarization). Such an optical force was recently predicted for evanescent waves 7 and other structured fields 8 . It can be associated with the 'spin-momentum' part of the Poynting vector, introduced by Belinfante in field theory 75 years ago 9-11 . We measure this unusual transverse momentum using a femtonewton-resolution nano-cantilever immersed in an evanescent optical field above the total internal reflecting glass surface. Furthermore, the measured transverse force exhibits another polarization-dependent contribution determined by the imaginary part of the complex Poynting vector. By revealing new types of optical forces in structured fields, our findings revisit fundamental momentum properties of light and enrich optomechanics.Since Euler's studies of classical sound waves, the wave momentum has been naturally associated with the propagation direction of the wave, that is, the normal to wavefronts, or the wavevector. This idea was mathematically formulated by de Broglie for quantum matter waves: p = k, where p is the momentum, k is the wavevector and is the reduced Planck constant. In both classical and quantum cases, the wave momentum can be measured by means of 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' 1 . However, pioneering works by Poynting introduced the electromagnetic momentum density as a cross product of the electric and magnetic field vectors 2,12 :P∝ E × B. 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 simplest case of a homogeneous plane electromagnetic wave. However, in more complicated yet typical cases of structured optical fields 13,14 (for example, interference, optical vortices, or near fields) the direction of P can differ from the wavevector directions 7,8 .Notably, the origin of this discrepancy between the Poynting momentum and wavevector lies within the framework of relativistic field theory (Supplementary Information). The conserved momentum of the electromagnetic field is associated with the translational symmetry of spacetime through Noether's theorem 10,15 . 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...

Influence of evanescent wave on birefringent microplates

“…Uncommonness of the properties of evanescent waves generated by a linearly polarized probing wave manifests itself, in part, in the occurrence of the vertical spin, as it has recently been shown [15,16], see also earlier relevant papers [17][18][19]. In a recent paper [11], direct measurement of an extraordinary optical momentum and helicity-dependent force directed perpendicularly to the wavevector being proportional to the ellipticity of the local polarization of the probing beam has been reported. Such an optical force takes place for evanescent waves and other structured fields being associated with the spin-momentum part of the Poynting vector.…”

“…Other feasibilities are realized in the case of evanescent waves. As it has been shown in recent papers [11,12], evanescent electromagnetic waves offer a rich and highly non-trivial structure of the local momentum and spin distributions. The transverse spin momentum turns out to be the fundamental spin momentum introduced by Belinfante in field theory [13].…”

Linear and angular momenta in tightly focused vortex segmented beams of light (Invited Paper)

“…In general, the discussed field distributions with their focal points of zero field intensity and transverse AM might become relevant for stimulated emission depletion [38] and optical manipulation experiments [1][2][3][4]13,34] . On the experimental side, such VSBs can be generated experimentally by utilizing spatial light modulators [39] or Pancharatman-Berry phase optical elements [40][41][42] .…”

“…A deep understanding of this type of light-matter interaction is particularly important in highly confined and/or structured electromagnetic fields [6][7][8] . Recently, this led to a growing interest in the theoretical description [5,[9][10][11] and measurement [12,13] of the individual constituents of LM and AM occurring in various nano-optical systems. In general, it is convenient to define these dynamical quantities locally, when complex three-dimensional field distributions are considered.…”

“…The spin LM density p s , sometimes referred to as Belinfante's spin momentum density [5,[12][13][14][15] , is defined by the curl of the spin AM density s,…”