Hopf index and the helicity of elliptically polarized twisted light
Koray Köksal, M. Babiker, V. E. Lembessis, J. Yuan
Abstract
Here, we describe a systematic derivation of the general form of the optical helicity density of ellipticaly polarized paraxial Laguerre–Gaussian modes <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal">L</mml:mi> <mml:mi mathvariant="normal">G</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>ℓ</mml:mi> <mml:mo>,</mml:mo> <mml:mi>p</mml:mi> <mml:mo>,</mml:mo> <mml:mi>σ</mml:mi> </mml:mrow> </mml:msub> </mml:mrow> </mml:math> . The treatment incorporates the contributions of the longitudinal field components for both the paraxial electric <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mtext mathvariant="bold">E</mml:mtext> </mml:mrow> </mml:mrow> </mml:math> and magnetic <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mtext mathvariant="bold">B</mml:mtext> </mml:mrow> </mml:mrow> </mml:math> fields, which satisfy Maxwell’s self-consistency condition in the sense that <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mtext mathvariant="bold">E</mml:mtext> </mml:mrow> </mml:mrow> </mml:math> is derivable from <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mtext mathvariant="bold">B</mml:mtext> </mml:mrow> </mml:mrow> </mml:math> and vice versa. Contributions to the helicity density to leading order in <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 class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>k</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> <mml:msubsup> <mml:mi>w</mml:mi> <mml:mn>0</mml:mn> <mml:mn>2</mml:mn> </mml:msubsup> <mml:msup> <mml:mo stretchy="false">)</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>−</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> </mml:mrow> </mml:math> (where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>k</mml:mi> </mml:math> is the axial wavenumber and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>w</mml:mi> <mml:mn>0</mml:mn> </mml:msub> </mml:mrow> </mml:math> the beam waist) include terms proportional to optical spin <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>σ</mml:mi> </mml:math> and topological charge <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>ℓ</mml:mi> </mml:math> , as well as a spin-orbit <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>σ</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> <mml:mi>ℓ</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> </mml:math> term. However, evaluations of the space integrals leading to the total helicity confirm that the space integral of the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>ℓ</mml:mi> </mml:math> -dependent term in the density (which is due entirely to the longitudinal fields) vanishes identically for all <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>ℓ</mml:mi> </mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>p</mml:mi> </mml:math> , so that, in general, only <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>σ</mml:mi> </mml:math> determines the Hopf index, with the optical vortex <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal">L</mml:mi> <mml:mi mathvariant="normal">G</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>ℓ</mml:mi> <mml:mi>p</mml:mi> </mml:mrow> </mml:msub> </mml:mrow> </mml:math> character featuring only in the action constant.