Remarkable laser beams formed by computer-generated optical elements: properties and applications V. A. Soifer, V. V. Kotlyar, S. N. Khonina, R.V. Skidanov, Image Processing Systems Institute, Russian Academy of Sciences, Samara State Aerospace University, 151 Molodogvardejskaya, Samara 443001, Russia ABSTRACT We deduce and study an analytical expression for Fresnel diffraction of a plane wave by a spiral phase plate (SPP) with an arbitrary-order phase singularity and an analytical expression for far-field diffraction of the Gaussian beam by a SSP with n-th order. The SPP was implemented as a 32-level "helical" microrelief on a low-contrast resist using an electron beam. Experiments of diffraction of the Gaussian beam by SPPs with the second-order singularity and of trapping polystyrene beads using computer-generated optical elements are discussed. Keywords: diffractive optical elements, spiral phase plates, optical microparticle manipulation 1. INTRODUCTION The optical vortices, also referred to as phase singularities or wavefront dislocations were for the first time discussed in Ref. 1 . The light field with phase singularity is characterized by the presence of an isolated intensity zero. Such fields can be generated with the SPP whose transmittance is defmed as exp(inço), where ço is the polar angle. In Ref. 2 a relationship to describe Fresnel diffraction of a plane wave by a SPP with the first-order phase singularity (n=1) has been derived. The diffraction of the Gauss-Laguerre (GL) mode of order (m0,n) was analyzed, with the Laguerre polynomial being L(x) = L(x) = 1 The Fresnel integral of the GL mode (O,n) was calculated in Ref. 5 through a hyper-geometric function. In Ref. 6 diffraction and interference of two GL modes, namely, the modes (0,0) (Gaussian beam) and (O,n), was studied theoretically and experimentally. For the first time, a relationship for diffraction of the Gaussian beam by a SPP with phase singularity was deduced in Ref.7. Subsequently, this relationship was modified and studied numerically in Ref. 8. In Ref. 9 the relationship similar to that derived in Ref.7 was derived for the Gaussian beam that passed through the out-of-waist phase singularity. Interference between two similar Gaussian beams propagated after the SPPs with different-order phase singularities was discussed in Ref. 10. Note that the analytical relations derived in Refs. 7-10 have been investigated only numerically. In this paper, as distinct from Ref. 2, we deduce and study an analytical expression for Fresnel diffraction of a plane wave by a SPP with an arbitrary-order phase singularity. Estimates for the optical vortex radius that depends on the singularity order n (also termed topological charge, or order of the dislocation 34) are derived. Also, an analytical expression for Fresnel diffraction ofthe Gaussian beam by a SPP with n-th order singularity7'° is analyzed. The far-field intensity distribution at z—>oo is derived. The behavior of the Gaussian beam after a SPP with second-order singularity (n=2) is studied in more detail. The parameters ofthe light beams generated numerically using the Fresnel transform and via analytical formulae are in good agreement. Experiments of diffraction of the Gaussian beam by SPPs with the second-order singularity are discussed. The phase singularity was implemented as a 32-level "helical" microrelief on a low-contrast resist using an electron beam. Experimental results on trapping polystyrene beads using DOEs are discussed. 2. DifFRACTION OF A PLANE BEAM BY A SPP In Ref. 2 diffraction of a plane beam by a SPP with the first-order phase singularity (n1) was considered. Let us consider Fresnel diffraction of a plane wave by a SPP with the nth-order (arbitrary) singularity. The complex transmittance of this SPP is given by Remarkable laser beams formed by computer-generated optical elements : properties and applications V. A. Soifer, V. V. Kotlyar, S. N. Khonina, R.V. Skidanov, Image Processing Systems Institute, Russian Academy of Sciences, Samara State Aerospace University, 15 1 Molodogvardejskaya, Samara 443001, Russia ABSTRACT We deduce and study an analytical expression for Fresnel diffraction of a plane wave by a spiral phase plate (SPP) with an arbitrary-order phase singularity and an analytical expression for far-field diffraction of the Gaussian beam by a SSP with n-th order. The SPP was implemented as a 32-level "helical" microrelief on a low-contrast resist using an electron beam. Experiments of diffraction of the Gaussian beam by SPPs with the second-order singularity and of trapping polystyrene beads using computer-generated optical elements are discussed. Keywords: diffractive optical elements, spiral phase plates, optical microparticle manipulation 1. INTRODUCTION The optical vortices, also referred to as phase singularities or wavefront dislocations were for the first time discussed in Ref. 1 . The light field with phase singularity is characterized by the presence of an isolated intensity zero. Such fields can be generated with the SPP whose transmittance is defmed as exp(inço), where ço is the poiar angle. In Ref 2 a relationship to describe Fresnel diffraction of a plane wave by a SPP with the first-order phase singularity (n=1) has been derived. The diffraction of the Gauss-Laguerre (GL) mode of order (m=O,n) was analyzed, with the Laguerre polynomial being L,(x) = L(x) = 1 The Fresnel integral of the GL mode (O,n) was calculated in Ref 5 through a hyper-geometric function. In Ref 6 diffraction and interference of two GL modes, namely, the modes (0,0) (Gaussian beam) and (O,n), was studied theoretically and experimentally. For the first time, a relationship for diffraction of the Gaussian beam by a SPP with phase singularity was deduced in Ref7. Subsequently, this relationship was modified and studied numerically in Ref 8. In Ref 9 the relationship similar to that derived in Ref7 was derived for the Gaussian beam that passed through the out-of-waist phase singularity. Interference between two similar Gaussian beams propagated after the SPPs with different-order phase singularities was discussed in Ref 10. Note that the analytical relations derived in Refs. 7-10 have been investigated only numerically. In this paper, as distinct from Ref 2, we deduce and study an analytical expression for Fresnel diffraction of a plane wave by a SPP with an arbitrary-order phase singularity. Estimates for the optical vortex radius that depends on the singularity order n (also termed topological charge, or order of the dislocation 34) are derived. Also, an analytical expression for Fresnel diffraction ofthe Gaussian beam by a SPP with n-th order singularity7'° is analyzed. The far-field intensity distribution at z—>oo is derived. The behavior of the Gaussian beam after a SPP with second-order singularity (n=2) is studied in more detail. The parameters ofthe light beams generated numerically using the Fresnel transform and via analytical formulae are in good agreement. Experiments of diffraction of the Gaussian beam by SPPs with the second-order singularity are discussed. The phase singularity was implemented as a 32-level "helical" microrelief on a low-contrast resist using an electron beam. Experimental results on trapping polystyrene beads using DOEs are discussed. 2. DIFFRACTION OF A PLANE BEAM BY A SPP In Ref 2 diffraction of a plane beam by a SPP with the first-order phase singularity (n1) was considered. Let us consider Fresnel diffraction of a plane wave by a SPP with the nth-order (arbitrary) singularity. The complex transmittance of this SPP is given by Holography 2005: Intl. Conf. on Holography, Optical Recording, and Processing of Information, Yury Denisyuk, Ventseslav Sainov, Elena Stoykova, Eds., Proceedings of SPIE Vol. 6252, 62521B, (2006) · 0277-786X/06/$15 · doi: 10.1117/12.677054 Proc. of SPIE Vol. 6252 62521B-1 Downloaded from SPIE Digital Library on 30 Dec 2009 to 89.186.234.25. Terms of Use: http://spiedl.org/terms