Evolution of the vortex and the asymmetrical parts of orbital angular momentum in separable first-order optical systems

Abstract

We analyze the evolution of the vortex and the asymmetrical parts of orbital angular momentum during its propagation through separable first-order optical systems. We find that the evolution of the vortex part depends on only parameters a x , a y , b x , and b y of the ray transformation matrix and that isotropic systems with the same ratio b͞a produce the same change of the vortex part of the orbital angular momentum. Finally, it is shown that, when light propagates through an optical fiber with a quadratic refractive-index profile, the vortex part of the orbital angular momentum cannot change its sign more than four times per period. © 2004 Optical Society of America OCIS codes: 070.2580, 070.4690, 030.5630, 120.4820, 140.3300. During the past decade the concept of orbital angular momentum (OAM) has been applied for the description of coherent optical vortex beams. 1,2In recent publications 3,4 it was suggested that OAM be decomposed into two parts: the asymmetrical OAM and the vortex OAM. The first part describes an astigmatic beam but with a smooth wave front, while the second is related to the singularity of the wave front (screw dislocations).In this Letter we study the evolution of the asymmetrical and the vortex OAM of linearly polarized, partially coherent beams during their propagation through separable first-order optical, or ABCD, systems. It was recently reported that some partially coherent fields also exhibit vortex behavior. 5 Taking into account that the concept of OAM can be generalized to the case of partially coherent beams, 6 one can apply the results of this study to both the completely coherent and the partially coherent cases.We base the def inition of OAM on the moments of the Wigner distribution (WD). 7The WD W ͑x, u; y, v͒ represents partially coherent light in a combined spacespatial-frequency domain, the so-called phase space, where u is the spatial-frequency variable associated with space variable x and v is the spatial-frequency variable associated with space variable y. The treatment in this Letter is based on the normalized moments of the WD, where the normalization is with respect to total energy E of the signal: E R R R R W ͑x, u; y, v͒dxdudydv. These normalized moments m pqrs of the WD are thus def ined byWe restrict ourselves to the ten second-order moments, p 1 q 1 r 1 s 2, assuming moreover, without loss of generality, that the four f irst-order moments are zero.OAM L of an optical beam can be expressed in terms of second-order moments of its WD as [Eq. (3) To study the evolution of the WD moments in separable f irst-order systems, we note that the input-output relationship between the WD W in ͑x, u; y, v͒ at the input plane and the WD W out ͑x, u; y, v͒ at the output plane of a separable first-order optical system reads as 10,11 W out ͑x, u; y, v͒ W in ͑d x x 2 b x u, 2c x x 1 a x u; d y y 2 b y v, 2c y y 1 a y v͒. This relationship is based on an affine transformation of the WD described by the so-called ray transformation matrix, 12 or the ABCD matrix. From the sympl...

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