Shearograms of an optical phase singularity Devinder Pal Ghai a,b, * , P. Senthilkumaran a , R.S. Sirohi c a Department of Physics, Indian Institute of Technology Delhi, New Delhi 110 016, India b Adaptive Optics Group, Laser Science and Technology Centre, Metcalfe House, Delhi, India c Barkatullah University, Hoshangabad Road, Bhopal 462 026, India Received 16 July 2007; received in revised form 19 October 2007; accepted 2 November 2007 Abstract Shearograms are known to represent phase gradients but when vortices are present in the optical field, these do not represent true phase gradients. Phase gradients of an optical phase singularity are presented. A lateral shear interferometer is used for obtaining shear- ograms of optical fields with vortices. A diffractive phase element is used to generate vortices. It is shown that shearograms can be used in the detection of optical vortices. Shearogram of speckle field is also presented. Ó 2007 Elsevier B.V. All rights reserved. OCIS: 120.3180; 120.4290; 120.3940; 120.2650; 999.9999 (optical phase singularity) 1. Introduction Optical fields with phase singularities, also called optical vortices, have generated a lot of interest in recent years. Phase singularity in a two-dimensional optical field distri- bution, is characterized by an isolated dark spot, where phase is ambiguous and amplitude is zero [1,2]. The wave- front around the point of singularity has helicoidal phase profile. All the wavefronts that are separated by k from each other in a vortex free optical field get connected and a single helical structure fills the entire space, when an opti- cal field is infected with a vortex. The number of times the wavefront undergoes 2p phase variation in one complete rotation around the singular point gives the value of topo- logical charge m, which is an integer. Finding phase gradi- ent of an optical field is important, as it is related to lateral aberrations of optical systems [3] and to spatial frequency components [4,5] of the field. Gradient detection methods are useful in optical testing [6–8]. The local phase gradient estimation is an important parameter that is used in adap- tive optics [9,10] for real-time correction of aberrations by a deformable mirror [11,12]. We present here shearograms of optical vortex. Shearo- grams are space derivatives of phase distribution but when vortices are present they do not represent phase gradients. Normally, the spatial phase profile of optical vortices is revealed by interferometric methods, where fork or spiral fringes are formed. The prevalent vortex detection tech- niques [13–15] use two-beam interference in which separate test and reference beams are involved. The formation of shearograms, we present here, is a very simple but elegant, new technique to detect phase singularity. It is a robust, self-referencing technique, which makes use of a single optical element. Further, we present phase gradient of iso- tropic and anistropic vortices in Cartesian and polar coor- dinates, fringe formation in a shearogram, effect of magnitude of shear and defocus on fringes. Simulation and experimental results are presented and discussed. 2. Shear interferometer In a shear interferometer the test wavefront is amplitude split and spatially shifted in the transverse plane to produce 0030-4018/$ - see front matter Ó 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2007.11.006 * Corresponding author. Address: Department of Physics, Indian Institute of Technology Delhi, New Delhi 110 016, India. Tel.: +91 11 23811242. E-mail address: dpghai@yahoo.co.in (D.P. Ghai). www.elsevier.com/locate/optcom Available online at www.sciencedirect.com Optics Communications 281 (2008) 1315–1322