Characterization of a pair of superposed vortex beams having different winding numbers via diffraction from a quadratic curved-line grating
Saifollah Rasouli, Pouria Amiri, Victor V. Kotlyar, A. A. Kovalev
Abstract
In a recent study, we have reported a simple, efficient, and robust method that is based on diffraction in an amplitude parabolic-line linear grating for determination of the topological charge (TC), <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>l</mml:mi> </mml:math> , of an optical vortex beam [ J. Opt. Soc. Am. B 37 , 2668 ( 2020 ) JOBPDE 0740-3224 10.1364/JOSAB.398143 ]. Here, we present a demonstration of the application of that method for characterization of a pair of superposed vortex beams having different winding numbers. It is shown that, when two vortex beams, described by Laguerre–Gaussian beams with winding numbers <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>l</mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:mrow> </mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>l</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:mrow> </mml:math> and radial index <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>p</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:math> , impinge on-axis and collinearly on a diffraction grating having a quadratic curvature on its lines, with a simple analysis of the resulted diffraction patterns at the zero and first order, the TCs and their signs can be determined. The zero-order diffraction pattern shows an interference pattern of the beams. For close values of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>l</mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:mrow> </mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>l</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:mrow> </mml:math> , it has a petal-like pattern in which the number of spots is equal to <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>l</mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>−</mml:mo> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>l</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> </mml:math> . It is also found that the first-order diffraction pattern depending to the signs of the beams’ TCs shows two different forms. If <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>l</mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:mrow> </mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>l</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:mrow> </mml:math> have the same signs, the first-order diffraction pattern is only a set of elongated intensity spots. When the signs of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>l</mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:mrow> </mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>l</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:mrow> </mml:math> are opposite, the resulted pattern is a <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mo stretchy="false">(</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>l</mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:mrow> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:math> by <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mo stretchy="false">(</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>l</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:mrow> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:math> slanted checkered-like matrix of bright spots. In addition, in this work, we use a simple, novel, and initiative method to generate and combine on-axis and collinearly a pair of vortex beams. Finally, a supporting theoretical study is presented that fully confirms the experimental results and simulation of propagation.