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Characterization of the Vortex Beam by Fermat's Spiral

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Characterization of the Vortex Beam by Fermat's Spiral hv photonics Article Characterization of the Vortex Beam by Fermat’s Spiral Ewa Fr ˛aczek 1,*, Agnieszka Popiołek-Masajada 2 and Sławomir Szczepaniak 1 1 Department of Telecommunications and Teleinformatics, Wroclaw University of Science and Technology, 50-370 Wroclaw, Poland; [email protected] 2 Faculty of Fundamental Problems of Technology, Wroclaw University of Science and Technology, 50-370 Wroclaw, Poland; [email protected] * Correspondence: [email protected] Received: 16 October 2020; Accepted: 3 November 2020; Published: 5 November 2020 Abstract: In this paper, we characterize the helical beam structure through an analysis of the spiral character of the phase distribution inside a light beam. In particular, we show that a line connected with the 2π phase jump in the Laguerre–Gauss beam can be described by a Fermat’s spiral. We propose a numerical fitting method to determine the parameters of a spiral equation for the phase distribution of the helical beam. Next, we extend the procedure to a vortex beam created by the spiral phase plate and apply it to experimental phase maps, which allows us to recover the phase shift introduced into the object beam in the optical vortex scanning microscope. Keywords: optical vortices; Fermat’s spiral; phase recover; interferometry 1. Introduction Structured light with non-Gaussian intensity profiles and with spatially variant phase distributions is getting attention in many fields of optical science and technology. A special case of such beams is a vortex beam, which features a helical shaped wavefront and annular intensity distribution [1,2]. A specific example of this kind of beam is, for example, the Laguerre–Gauss (LG) beam [3]. During propagation, the phase of this beam rotates around a line, where the phase value is undetermined. At the screen, the line is reduced to a point, which is usually called the vortex point. One of the interesting physical features of the vortex beam is its donut-shaped intensity distribution. In the intensity pattern, one can observe a dark area (the vortex core) surrounded by a bright ring [4]. Another unique aspect of this beam is the phase distribution. Its contour lines reveal a spiral character. Many methods have been used to characterize a vortex beam regarding its intensity profile. The bright ring of the ideal vortex beam is circular, with a diameter that encapsulates information of the beam size [5]. In the case of optical imperfections, the intensity profile loses its spherical symmetry, changing, for example, into an elliptical one [6,7]. There are also methods to characterize a vortex beam by looking at the correlation between the measured and pure (ideal) modes [8] or to identify order modes by its orbital angular momentum spectra [9]. In this paper, we propose a new way for characterizing a vortex beam through the analysis of its phase distribution contour. We show that the phase profile of the LG beam can be described by a Fermat’s spiral with parameters that determine the curvature of the beam wavefront, as well as its spiral orientation. Moreover, we propose a procedure for recovering parameters of the spiral from the phase map. We also show that this concept of beam description can be applied in the case of a helical beam created by the spiral phase plate which is used in the optical vortex scanning microscope [10,11]. In the experimental part, we apply this new way of beam description to recover the phase shift introduced into the object beam in the optical vortex scanning microscope. Photonics 2020, 7, 102; doi:10.3390/photonics7040102 www.mdpi.com/journal/photonics Photonics 2020, 7, 102 2 of 11 Photonics 2020, 7, x FOR PEER REVIEW 2 of 11 2. Laguerre–Gauss Beam and a Fermat’s Spiral in the Phase Profile 2. LaguerreThe complex–Gauss amplitude Beam and of a theFermat’s Laguerre–Gauss Spiral in the beam Phase (mode Profile LG0m* ) can be expressed in Cartesian coordinates by Equation (1): The complex amplitude of the Laguerre–Gauss beam (mode LG0m*) can be expressed in Cartesian coordinates by Equation (1): LG(x, y) = A (x + σiy)m exp B x2 + y2 (1) · m · − 22 LG(,) x y=⋅+ A( xσ iy) ⋅exp( − B( x + y )) (1) where σ = 1, and A and B belong to the complex numbers and m is topological charge, which measures ± wherethe phase σ = change ±1, and in A one and close B belong trip around to the the complex axis (m =numbers1, 2, 3, ::: and). Figurem is 1topological presents a contourcharge, plotwhich of measuresthe arg(LG the(x ,phasey)). It change is shown in below one close that leveltrip around sets of athe function axis (m form = 1, a 2, family 3, …) of. Figure Fermat’s 1 presents spirals [12 a]. contourFirst, by plot passing of the to polararg(LG coordinates ( x , y )) . It is xshown= r cos below', y = thatr sin level', one sets gets of aarg function(x + σiy form)m = aσ mfamily' by theof Fermat’sMoivre’s spirals formula. [12 Using]. First,arg (byexp passing(z)) = Imtoz ,polarx2 + y2coordinates= r2, and thexr= additioncosϕ , ruleyr= forsinϕ arguments, one gets of arg(complexx+=σ iy numbers )m σϕ m [13 by] onethe obtains: Moivre’s formula. Using arg(exp(zz ))= Im , xyr222+=, and the addition rule for arguments of complex numbers [13] one obtains: LG(r, ', z) = qLG(, rϕ , z )= (2) 2 2 m 2 n h 2io Re(A) 22+ Im(A) rjm j exp Re(B) 2r exp i mσϕ + arg(A) Im(B2) r (2) Re()A+ Im() Ar ⋅· exp( −− Re() Br ⋅⋅· ) exp· { imσϕ + arg() A − Im()− Br ⋅·} FigureFigure 1. 1. ((aa)) Equiphase lineslines fromfrom Equation Equation (1) (1) form form a seta set of spirals,of spirals, each each spiral spiral can becan viewed be viewed as single as singlebranches branches of Fermat’s of Fermat’s spirals spirals (case m(case= 1); m (b =) Lines1); (b) of Lines the constantof the constant phase 0 phase and π 0form and aπ complete form a completeFermat’s Fermat’s spiral (both spiral branches (both plottedbranches with plotted different with colors), different (b) colors) tangent, ( lineb) tangent' = 0; (linec) Tangent φ = 0; line(c) T'angent= b2 line/a. ϕ = −ba2 . − FromFrom the the above above equation equation we we get: get: 2 ANAN(x( x, ,y y) ) == arg A ++σϕσ mm' −⋅ ImIm BB r r (mod2(mod2 2π )π ) (3(3)) − · IfIf σ ==sgn(Imsgn(ImB ) B, )then, then σσ// ImImBBB== 1/1 Im/ Im.B Hence,. Hence, a level a level set set of of the the fu functionnction AN AN in in polar polar j j coordinates,coordinates,AN(( rr coscosϕϕ' ,, rr sinsin ' ) )= = Φ F, is, isa acurve curve given given by by a aformula: formula: 22 2 2 rr2 ==a 2'ϕϕ +=+ bb = aa(2 ' ++ ba/b/a)2 (4(4)) where where p a== m/ ImB andbbA==(argA −ΦF)/ /Im B B (5(5)) am/ Imj B andj ( − ) TheThe curve curve described described by by Equation Equation (3) (3) is is a a Fermat’s Fermat’s spiral spiral with with a a tangent tangent line line at at r r == 00 equal equal to to 2 ' = b/2a . Two cases of tangent lines are presented in Figure1b,c. Equation (4) can be rewritten as: ϕ = −ba−/ . Two cases of tangent lines are presented in Figure 1b,c. Equation (4) can be rewritten as: 'ϕα==⋅+α rr22 + ββ (6(6)) · which is a more convenient form to perform the numerical fitting. Here, which is a more convenient form to perform the numerical fitting. Here, 1 α = 12 (7) α =a (7) a2 b β = − (8) 2b. β = a . (8) −a2 Figure 2 presents the exemplary phase maps calculated using Equation (3) and printed in grayscale, for different values of parameters α (Equation (7)) and β (Equation (8)). The phase distribution reveals a characteristic 2π jump, which we call the phase step line. As stated above, it Photonics 2020, 7, 102 3 of 11 Figure2 presents the exemplary phase maps calculated using Equation (3) and printed in grayscale, for different values of parameters α (Equation (7)) and β (Equation (8)). The phase distribution reveals Photonics 2020, 7, x FOR PEER REVIEW 3 of 11 a characteristic 2π jump, which we call the phase step line. As stated above, it forms a Fermat’s spiral.forms Figure a Fermat’s2a,b presentsspiral. Figure the phase 2a,b presents maps with the the phase same map values with of parameterthe same valueα and of parameter different β αvalues, and showingdifferent that β values,β indicates showing spiral that orientation. β indicates Figure spiral2b,c orientation. shows the Figure same 2b,β valuec shows and the diff sameerent βα valuevalues. Here,and different one can see α values that α. isHere, responsible one can forsee spiral that α density. is responsible Thus, spiralfor spiral density density depends. Thus on, spiral the parameter density αdependsalone, while on the the parameter parameter α βalone,affects while the the spiral parameter orientation. β affects the spiral orientation. FigureFigure 2. 2.Phase Phase distributiondistribution obtained by by using using Equation Equation (2 (2)) for for m m= 1= and1 and different different values values of α of andα andβ. 2 2 2 β(.(a)a α) α= 13.9= 13.9 (1/mm (1/mm) and2) and β = 0.77;β = 0.77;(b) α (=b 13.9) α =(1/mm13.9 (1) /andmm β2 )= and−0.79;β =(c) α0.79; = 5.26 (c )(1/mmα = 5.26) and (1 /βmm = −20.79.) and − β = 0.79. Equations− (1) and (2) can be compared to the LG equation expressed by the physical parameter of theEquations Gaussian (1) beam and (2)[3]:can be compared to the LG equation expressed by the physical parameter of the Gaussian beamLG [3(,]: rϕ , z )= m w r r22kr⋅ (9) 0 ⋅ ⋅ −LG(r ⋅, ', z) − = ϕφ + + +Φ + E0022exp exp i m kz G wwz() m wz2() 2Rz2 () (9) w0 r j j r k r E0 exp exp i m' + · + kz + FG + φ0 w w(z)2 w(z)2 2R(z) · · − 2 · − 2 z zR where w0 is a beam waist, wz()= w0 1 + 2 is transverse beam dimension, Rz()= z 1 + 2 is r zR 2 z z2 zR w w(z) = w + 2 R(z) = z + where 0 is a beam waist, 0 1 2 is transverseπ ⋅ w beam dimension, 1 z2 is radius zR 0 radius of curvature of the beam wavefront, zR = is Rayleigh range, ϕG is Gouy phase, and ϕ0 is π w2 λ = · 0 of curvature of the beam wavefront, zR λ is Rayleigh range, FG is Gouyk phase, and φ0 is1 initial initial phase.
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