JW4A.114.pdf Frontiers in Optics 2017 © OSA 2017 Conservation of Topological Charge During Diffraction Through a Triangular Aperture Jos´ e Carlos do Amaral Rocha, Jo˜ ao Paulo Amaral, Alcen´ ısio J. Jesus-Silva, Eduardo Jorge da Silva Fonseca Instituto de F´ ısica, UFAL, Cidade Universit´ aria, Macei´ o-AL, 57072-970, Brazil jcamrocha@gmail.com Abstract: We break the symmetry of non-generic singularities in Laguerre-Gauss fields by diffraction through a triangular aperture and studied their propagation. Conservation of topological charges is also observed. OCIS codes: 050.0050, 260.6042. 1. Introduction Not every dynamical variable in an optical system is well behaved. Surprisingly, the phase of optical fields, which give rise to interference fringes, may be ill-defined. If the amplitude of a wavefield is zero, we cannot associate to that point a phase value. In this case we say that the wavefield has a phase singularity at that specific spatial point [1]. Topological singularities are ubiquitous in many physical situations. For example, it was reported that liquid crystals may contain points where the director cannot be defined [2]. The phase of a complex wavefield in the transverse plane can increase or decrease by a 2π m, with m integer, in a complete turn around a point of zero intensity. We assign to these points a topological charge of ±m depending on the sense of rotation of the phase. If m = ±1 the singularities are stable, or generic, and remain unperturbed during propagation. On the other hand, if |m| > 1, these singularities are non-generic and, under very general conditions, unstable. Like subatomic particles, these singularities can interact between each other. Creation and annihilation of topological charges are possible and this, in turn, imply new conservation laws in optical fields. To verify these conservation laws, in this work, we numerically study the propagation of optical beams with embedded non-generic singularities diffracting through a triangular aperture. It was shown recently that a triangular aperture can be used to measure the topological charge (along with its sign) [3]. By using numerical techniques, we have observed creation and annihilation of optical singularities after passing the beam through the aperture along with the conservation of the net charge during propagation. 2. Theory and Results There are special classes of optical beams that have topological singularities embedded in them. One of these classes are the Laguerre-Gaussian beams, LG nm , which are solutions of the paraxial wave equation [∇ ⊥ + 2ik∂ z ]u(x, y , z)= 0, n + 1 is the number of maxima rings in the transverse plane and m is associated with the topological charge of the beam. In a linear medium, the diffraction of a complex field U can be calculated by the Rayleigh-Sommerfeld integral [4]: U (x 0 , y 0 , z 0 )= -  Ω U (x, y , 0) 2z 0 |r 0 - r| ×  ik - 1 |r 0 - r| exp(ik(|r 0 - r|) 4π |r 0 - r|  dxdy , (1) with U (x, y , 0) given by U (x, y , 0)= U 0     r w 0     |m| exp  - r 2 w 2 0  exp(imφ ) (2) where Ω is the region at z = 0 where U is non-zero, w 0 is the beam waist and m is the assigned topological charge. In our case Ω is the region of the triangular aperture. Using Fast-Fourier-Transform technique [5] to numerically evaluate this integral, we have propagated a Laguerre-Gaussian beam through a triangular aperture with sides equal to w 0  m/2 (Fig. 1). Our objective is to study the propagation dynamics of singularities throughout diffraction. It is possible to show that if the beam is propagating in free space its initially non-generic singularities remain stable and no new pairs are created nor annihilated while diffracting. While initially there is only one non-generic singularity