Introduction
Under the paraxial approximation, electromagnetic waves in free space are governed by paraxial wave equation. In rectangular coordinates, the solutions to the paraxial equation are Hermite-Gaussian (HG) modes; while in cylindrical coordinates, the solutions are Laguerre-Gaussian (LG) modes.
In 1992, Allen et al presented the result that the photons of Laguerre-Gaussian mode possess a well defined orbital angular momentum (OAM) of M=lh, where l is the azimuthal index, h is the reduced Planck constant. As shown in their article, the realization of the existence of the orbital angular momentum was inspired by the analogy between quantum mechanics and paraxial optics, and the formula is obtained through the integration of the angular momentum density over the whole beam profile of the Laguerre-Gaussian mode. Besides, a method of measuring the mechanical torque of OAM by use of a suspended mode convertor is suggested in their article. Since its publication, this article has triggered considerable research activities in the area relating to orbital angular momentum.
Here a view point from the connection between orbital angular momentum and rotational frequency shift (RFS) is presented, which can be drawn from a thought experiment based on the interference patterns between Laguerre-Gaussian beams and Gaussian beams—a manifestation of the rotational frequency shift resulted from the transfer of orbital angular momentum. From this view point, the existence of the orbital angular momentum of the Laguerre-Gaussian mode is a natural result from its azimuthal dependent item of exp(jlφ), and the formula M=lh can be readily obtained through a simple algebra process.
The suspended mode convertor approach for the measurement of mechanic torque arising from OAM suggested by Allen et al has to deal with considerable difficulty in the measurement of the feeble mechanical torque. However, from above mentioned point of view, this difficulty can be circumvented by measuring the rotational frequency shift instead.
The article shown below elucidates this view point in detail.
the_view_point_from_the_connection_between_orbital_angular_momentum_and_rotational_frequency_shift_.pdf | |
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