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Optical vortex is an intriguing mode of light in the sense that it is a line of phase singularity where the complex scalar field vanishes.  There have been lots of  publications on optical vortices from theory to applications such as in optical micromanipulation.  These vortices abound specially in coherent fields.

In some applications, these vortices obscure information in the field that they are  in.  Exact determination of their positions and subsequent corrections of the phase of the beam is therefore necessary.

Maallo and Almoro of the National Institute of  Physics, UP Diliman, writing in Optics Letters [1], “developed an algorithm based on the axial behavior of retrieved phase maps to demonstrate a technique for the detection and correction of an optical vortex”.

The algorithm is fairly straight forward (see figure 1). After reconstruction of the wavefront, the field is propagated at different z, the total length of propagation is one wavelength and the number of planes that are obtained within this propagation length is k.  The length of different z is therefore controlled by the number of planes k.  On every plane, the standard deviation of a sub image/portion of the phase map (l x l matrix) is obtained and that that standard deviation is assigned to the central pixel position. These are then mapped to form a matrix of dimension  N- l + 1, where N is the size of the element of original phase map (matrix).  The final image is obtained by plotting the maximum difference between the standard deviation of the same pixel position of the different planes.

Figure 1. Algorithm to determine the exact vortex position (Courtesy, Almoro, Optics Letters).

How does this algorithm exactly detect the position of the vortex?

Simple.  The eye of the vortex does not change its position at a propagation length of one wavelength.  The value of the standard deviation at that point does not change compared to other positions. Hence, it will appear as a black dot as shown in Figure 2 (b).

Once the position of the vortex in determined, the correction of the phase by adding a phase filter equivalent to but opposite in sign, follows.

Figure 2. Different k's give different resolution (Courtesy, Almoro, Private Comm).

Although the algorithm works nicely with a lone optical vortex.  It still needs to be tested for multiple vortices and elliptical vortices.

As an aside, wouldn’t it be easier to look at the intersection of the zeroes of the real and complex components of the fields to determine exactly the position of a vortex as just what Roux did in the mid 90’s [2] as the field was already reconstructed anyway? Just a thought.

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[1] Maallo, A., &Almoro, P. (2011). Numerical correction of optical vortex using a wrapped phase map analysis algorithm Optics Letters, 36 (7) DOI: 10.1364/OL.36.001251

[2] Roux, F. (1995). Dynamical behavior of optical vortices Journal of the Optical Society of America B, 12 (7) DOI: 10.1364/JOSAB.12.001215

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