It is well known that reflection of a Gaussian light beam (TEM(00)) by a planar dielectric interface leads to four beam shifts when compared to the geometrical-optics prediction. These are the spatial Goos-Hanchen (GH) shift, the angular GH shift, the spatial Imbert-Fedorov (IF) shift, and the angular IF shift. We report here, theoretically and experimentally, that endowing the beam with orbital angular momentum leads to coupling of these four shifts; this is described by a 4 x 4 mixin

We report the first measurement of the spin Hall effect of light (SHEL) on an air-metal interface. The SHEL is a polarization-dependent out-of-plane shift on the reflected beam. For the case of metallic reflection with a linearly polarized incident light, both the spatial and angular variants of the shift are observed and are maximum for -45°/45° polarization, but zero for pure s and p polarization. For an incoming beam with circular polarization states however, only the spatial out-of-plane shift is present.

One procedure widely used to detect the velocity of a moving object is by using the Doppler effect. This is the perceived change in frequency of a wave caused by the relative motion between the emitter and the detector, or between the detector and a reflecting target. The relative movement, in turn, generates a time-varying phase which translates into the detected frequency shift. The classical longitudinal Doppler effect is sensitive only to the velocity of the target along the line-of-sight between the emitter and the detector (longitudinal velocity), since any transverse velocity generates no frequency shift. This makes the transverse velocity undetectable in the classical scheme. Although there exists a relativistic transverse Doppler effect, it gives values that are too small for the typical velocities involved in most laser remote sensing applications. Here we experimentally demonstrate a novel way to detect transverse velocities. The key concept is the use of structured light beams. These beams are unique in the sense that their phases can be engineered such that each point in its transverse plane has an associated phase value. When a particle moves across the beam, the reflected light will carry information about the particle's movement through the variation of the phase of the light that reaches the detector, producing a frequency shift associated with the movement of the particle in the transverse plane.

That the speed of light in free space c is constant has been a pillar of modern physics since the derivation of Maxwell and in Einstein’s postulate in special relativity. This has been a basic assumption in light’s various applications. However, a physical beam of light has a finite extent such that even in free space it is by nature dispersive. The field confinement changes its wavevector, hence, altering the light’s group velocity vg. Here, we report the subluminal vg and consequently the dispersion in free space of Laguerre-Gauss (LG) beam, a beam known to carry orbital angular momentum. The vg of LG beam, calculated in the paraxial regime, is observed to be inversely proportional to the beam’s divergence θ0, the orbital order ℓ and the radial order p. LG beams of higher orders travel relatively slower than that of lower orders. As a consequence, LG beams of different orders separate in the temporal domain along propagation. This is an added effect to the dispersion due to field confinement. Our results are useful for treating information embedded in LG beams from astronomical sources and/or data transmission in free space.

Puentes, Hermosa, and Torres Reply: In a preceding Comment [1] to our Letter [2] concerning weak measurements with orbital-angular-momentum (OAM) pointer states [2], Pan and Panigrahi show that the real and imaginary parts of higher-order weak values can be made accessible using Gaussian pointer states. In [2], our goal is to show that by using pointer states embedded with OAM it is possible to extract higher-order weak values where pointer states with a Gaussian shape cannot. As a consequence, our work put forward a new tool for weak amplification schemes, i.e., the use of pointer states with more general spatial forms that could be advantageous in some scenarios. We did not intend to claim that a pointer state with orbital angular momentum was needed to extract higherorder weak values in general. We clarify our point with this Reply.In our Letter we consider a specific Hamiltonian of interaction of the form H ¼ g A AP x þ g B BP y , where A, B are operators, (P x , P y ) are the pointer momentum operators, conjugate to the pointer position operators (X, Y), and g A;B are coupling constants. Moreover, we calculate the two-dimensional pointer displacement hXYi, corresponding to a specific measurement. We emphasize that we consider a particular interaction and a specific measurement, which are generally dictated by the physical system under investigation.The result, Eq. (4) of our Letter, shows that pointer states with OAM (l ¼ AE1) can be used to retrieve the imaginary part of the higher-order weak values hA 2 i w and hB 2 i w , whereas this is not possible with pointer states with no OAM (l ¼ 0). In addition, Eq. (9) of our Letter shows, by means of a specific example with B ¼ 0, that hXYi ¼ 0 for Gaussian pointer states. On the contrary, a pointer state with l ¼ AE1 allows us to extract the imaginary part of the weak value hA 2 i w . Pan and Panigrahi show that by measuring hX 2 i one can obtain the real part of the weak value hA 2 i w using a Gaussian pointer [see Eq. (4) of [1]]. However, in order to access its imaginary part, they are forced to use a different interaction Hamiltonian, and a measurement observable involving noncommuting operators. The work of Pan and Panigrahi proves the strength of our proposal: One