Spotlight Summary by Giovanni Piredda
Orbital angular momentum in noncollinear second-harmonic generation by off-axis vortex beams
Light beams can carry “orbital” angular momentum, that is, an angular momentum directed along the beam’s propagation. This angular momentum is in general associated with an azimuthal dependence of the phase of the type exp(il?), associated with helical phase fronts. The Poynting vector, which in free space is proportional to the momentum density of the electromagnetic field, is perpendicular to the phase front and therefore circulates around the direction of propagation: its azimuthal component produces the angular momentum parallel to the propagation direction. If the parameter l is an integer number, the phase singularity at the center of the beam corresponds to a “screw dislocation” of the wave front (an “optical vortex”).
Light’s orbital angular momentum has several important applications: for example, one can use it to transfer angular momentum to material objects, such as particles trapped in a beam focus (thus acting as “light spanners”); one can use the spatial structure of these beams (whose intensity is zero in correspondence to phase dislocations) for microscopy; and one can use them to realize entanglement experiments. In the case of entanglement a relevant fact is that the transverse spatial modes of a light beam form an infinite-dimensional space, and a basis for this space can be expressed in terms of beams carrying integer orbital angular momentum (one such basis is the one formed by the Laguerre–Gaussian modes); in practice this provides an occasion to test the foundation of quantum mechanics in a system with many degrees of freedom (limited by the maximum angular momentum that it is possible to impart to a light beam).
In this context, beams with fractional angular momentum are particularly significant, as they can be formed by a superposition of beams with integer angular momentum up to very high l. One possible realization of a beam with fractional angular momentum is a vortex beam with noninteger l; the phase discontinuity in this case corresponds to a mixed screw-edge dislocation (the intensity must be zero on the entire half line starting from the center of the beam on which the phase has a jump).
The paper of Bovino et al. deals with these fractional-angular-momentum beams and their manipulation with the use of nonlinear optics. In particular, the authors show numerically how two of these beams with opposite orbital angular momentum, combined in a nonlinear crystal in a noncollinear second-harmonic generation scheme, produce beams with zero angular momentum, thus satisfying momentum conservation in the nonlinear interaction. The angular momentum of the incident beam is varied continuously with a simple technique, and the output beam’s angular momentum is verified to be zero for each input configuration; in addition, the authors compare the intensity distributions resulting from their numerical findings with experimental results.
The authors do not measure interferometrically the transverse phase of the beams, leaving this (a nontrivial task, given the difficulty of performing interferometry experiments with ultrashort pulses) as an open possibility for experimental investigation. Another possible development inspired from this work is the nonlinear combination of orbital-angular-momentum-carrying beams with completely different beam shapes: for example, a vortex Bessel beam with a vortex Laguerre–Gaussian beam. It would be interesting to verify the conservation of orbital angular momentum for this more complicated case as well, as it would show that the conservation law does not depend on the details of the interacting beams.
In conclusion, in their detailed study, Bovino et al. show some interesting manipulations of beams carrying orbital angular momentum that should extend the “collection” of beams with which researchers are familiar.
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Light’s orbital angular momentum has several important applications: for example, one can use it to transfer angular momentum to material objects, such as particles trapped in a beam focus (thus acting as “light spanners”); one can use the spatial structure of these beams (whose intensity is zero in correspondence to phase dislocations) for microscopy; and one can use them to realize entanglement experiments. In the case of entanglement a relevant fact is that the transverse spatial modes of a light beam form an infinite-dimensional space, and a basis for this space can be expressed in terms of beams carrying integer orbital angular momentum (one such basis is the one formed by the Laguerre–Gaussian modes); in practice this provides an occasion to test the foundation of quantum mechanics in a system with many degrees of freedom (limited by the maximum angular momentum that it is possible to impart to a light beam).
In this context, beams with fractional angular momentum are particularly significant, as they can be formed by a superposition of beams with integer angular momentum up to very high l. One possible realization of a beam with fractional angular momentum is a vortex beam with noninteger l; the phase discontinuity in this case corresponds to a mixed screw-edge dislocation (the intensity must be zero on the entire half line starting from the center of the beam on which the phase has a jump).
The paper of Bovino et al. deals with these fractional-angular-momentum beams and their manipulation with the use of nonlinear optics. In particular, the authors show numerically how two of these beams with opposite orbital angular momentum, combined in a nonlinear crystal in a noncollinear second-harmonic generation scheme, produce beams with zero angular momentum, thus satisfying momentum conservation in the nonlinear interaction. The angular momentum of the incident beam is varied continuously with a simple technique, and the output beam’s angular momentum is verified to be zero for each input configuration; in addition, the authors compare the intensity distributions resulting from their numerical findings with experimental results.
The authors do not measure interferometrically the transverse phase of the beams, leaving this (a nontrivial task, given the difficulty of performing interferometry experiments with ultrashort pulses) as an open possibility for experimental investigation. Another possible development inspired from this work is the nonlinear combination of orbital-angular-momentum-carrying beams with completely different beam shapes: for example, a vortex Bessel beam with a vortex Laguerre–Gaussian beam. It would be interesting to verify the conservation of orbital angular momentum for this more complicated case as well, as it would show that the conservation law does not depend on the details of the interacting beams.
In conclusion, in their detailed study, Bovino et al. show some interesting manipulations of beams carrying orbital angular momentum that should extend the “collection” of beams with which researchers are familiar.
Article Reference
Orbital angular momentum in noncollinear second-harmonic generation by off-axis vortex beams
- J. Opt. Soc. Am. B 28(11) 2806-2811 (2011)
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