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 belowϕ, y = thatr sin levelϕ, one sets gets of aarg function(x + σiy form)m = aσ mfamilyϕ 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, ϕ, z) = qLG(, rϕ , z )= (2) 2 2 m 2 n h 2io Re(A) 22+ Im(A) r|m | exp Re(B) 2r exp i mσϕ + arg(A) Im(B2) r (2) Re()A+ Im() Ar ⋅· exp( −− Re() Br ⋅⋅· ) exp· { imσϕ + 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 constant of the constant phase 0 phase and π 0form and a π complete form a completeFermat’s Fermat’sspiral (both spiral branches (both plotted branches with plotted diﬀerent with colors), different (b) colors) tangent, ( lineb) tangentϕ = 0; ( linec) Tangent φ = 0; line(c) Tϕangent= 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 mmϕ −⋅ 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 | | coordinates,coordinates,AN(( rr coscosϕϕϕ ,, rr sinsin ϕ ) )= = Φ Φ, is, isa acurve curve given given by by a aformula: formula:

22 2 2 rr2 ==a 2ϕϕϕ +=+ bb = aa(2 ϕ ++ ba/b/a)2 (4(4)) where where p a== m/ ImB andbbA==(argA −ΦΦ)/ /Im B B (5(5)) am/ Im| B and| ( − ) 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 ϕ = 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: ϕϕα==⋅+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

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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 r22kr⋅  (9) 0 ⋅ ⋅ −LG(r ⋅, ϕ, z) − = ϕφ + + +Φ + E0022exp exp i m kz G  wwz() m wz2() 2Rz2 ()  (9) w0 r | | r k r E0 exp exp i mϕ + · + kz + ΦG + 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, ΦG is Gouyk phase, and φ0 is1 initial initial phase. By comparing Equations (2) and (9), one reads that ImkB = and Re1 B = . phase. By comparing Equations (2) and (9), one reads that ImB = and ReB = . Applying2 2R(z) 2Rz( ) w(z)2 wz( ) EquationsApplying (5)Equations and (7), (5) one and gets (7), the one expression gets the expression for the radius for the of curvature radius of curvature of the wavefront of the wavefront in terms of thein parameterterms of theα :parameter α: k R = k (10) R = 2mα (10) 2mα The value of β indicates an angle of tangent line at radial coordinate r = 0 (see Figure1). InThe the value next paragraphs,of β indicates we an describe angle of a tangent procedure line forat radial recovering coordinate the parameters r = 0 (see Figure encoding 1). Fermat’s spiralIn (α , theβ in next Equation paragraphs (6)), first,, we from describe numerical a procedure phase maps, for andrecovering then from the the parameters phase maps encoding obtained inFermat’s the experiment. spiral (α, β in Equation (6)), first, from numerical phase maps, and then from the phase maps obtained in the experiment. 3. Numerical Fitting Procedure 3. Numerical Fitting Procedure To demonstrate how the spiral equation can be found and verify the fitting procedure, we started from numericallyTo demonstrate generated how phasethe spiral maps. equation In the firstcan step,be found the complex and verify amplitude the fitting of the procedure, vortex beam we is calculatedstarted from using numerically Equation (1),generated from which phase the maps. phase In term the hasfirst been step extracted., the complex amplitude of the vortexThe be phaseam is valuescalculated plotted using modulo-2 Equationπ (1),are from presented which inthe Figure phase3 terma as ahas surface been extracted. plot. One can see The phase values plotted modulo-2π are presented in Figure 3a as a surface plot. One can see the 2π phase jump (from 3.14 rad to 3.14 rad) which forms a spiral line. It is one of many isophase the 2π phase jump (from− −3.14 rad to 3.14 rad) which forms a spiral line. It is one of many isophase lines on the map but it is the easiest to find. This phase jump line can be treated as an edge and can be lines on the map but it is the easiest to find. This phase jump line can be treated as an edge and can extracted using Roberts method [14] for edge detection used in image processing. The results of this be extracted using Roberts method [14] for edge detection used in image processing. The results of step are presented in Figure3b. In this way, we obtain Cartesian coordinates of points forming a spiral this step are presented in Figure 3b. In this way, we obtain Cartesian coordinates of points forming a spiral (Figure 3b) which, next, have to be transformed into polar coordinates (radial radius r and azimuth angle φ).

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(Figure3b) which, next, have to be transformed into polar coordinates (radial radius r and azimuth Photonicsangle ϕ 20).20, 7, x FOR PEER REVIEW 4 of 11

(c)

FigureFigure 3. 3. (a(a) ) Phase Phase distribution distribution of of the the Laguerre Laguerre–Gauss–Gauss ( (LG)LG) beam beam (m (m == 1),1), phase phase values values plotted plotted modulomodulo-2-2ππ.. Edge Edge in in this this figure ﬁgure is is 2π 2π phasephase jump jump;; ( (bb)) P Pointsoints assigned assigned to to the the 2π 2π phasephase jump jump detected detected byby Roberts Roberts algorithm algorithm for for edge edge detection; detection; (c (c) )S Spiralpiral line line from from (b (b)) expresse expressedd in in polar polar coordinates. coordinates. UnwrappingUnwrapping is is needed needed to to get get the the correct correct values values of of azimuth azimuth angle angle φϕ..

InIn the the next next step step of of calculations, calculations, the the received received points points are are sorted sorted according according to to radial radial radii, radii, and and then then unwrappingunwrapping isis necessarynecessary to to assign assign the the proper proper value value of the of azimuthal the azimut anglehal toangle the radialto the radius. radial Withoutradius. unwrapping, an azimuth angle ranges from π to π, which is not the case for the spiral coordinates Without unwrapping, an azimuth angle ranges− from −π to π, which is not the case for the spiral coordinates(Figure3c). ( IncreasingFigure 3c). theIncreasing azimuth the angle azimuth enables angle a recognition enables a recognition of subsequent of subsequent twirls of the twirls spiral. of theNow, spiral. coordinates Now, coordinates are set in proper are set order in proper in order order to reconstructin order to areconstruct curve expressed a curve by expressed Equation by (6). EquationIn order to(6). determine In order theto determine values of parametersthe values αof andparametersβ, we use α aand nonlinear β, we use least a squaresnonlinear method least squareswithout method normalization without [ 15normalization]. The nonlinear [15]. leastThe n squaresonlinear is least the formsquares of leastis the squaresform of analysisleast squares used to fit a set of mk observations with a model that is nonlinear in “n” unknown parameters (mk n). analysis used to fit a set of mk observations with a model that is nonlinear in “n” unknown≥ parametersThe basis of (mk the method≥ n). The is basis to approximate of the method the modelis to approximate by a linear one the and model to refine by a thelinear parameters one and to by refinesuccessive the param iterations.eters by successive iterations. WeWe simulatedsimulated numerically numerically a beam a beam with anwith optical an vortexoptical with vortex the following with the optical following characteristics: optical values A and B (Equation (1)) for this beam were equal to A = 0.0177 + 0.0797i and B = 0.8139 + characteristics:h i values A and B (Equation (1)) for this beam were− equalh ito A =−+0.0177 0.0797i and 8.9525i 1 , λ = 0.633 10 3[mm] and correspondingly α = 8.9525 1 and β = 1.3521. The results mm2 1 − mm2 1 = + · −3 α = Bof a0.8139 spiral recovering 8.9525i  are2 presented, λ =0.633 in Figure ⋅ 10[ mm4. Figure] and4a showscorrespondingly two spirals, one spiral8.9525 extracted2 from and mm mm numerically generated helical wavefronts (dots), and a second spiral (circles) given by Equation (6) β = 1.3521. The results of a spiral recovering are presented in Figure 4. Figure 4a shows two spirals, with the recovered values of α and β. Values of parameters are fitted (with 95% confidence bounds) as one spiral extractedh i from numerically generated helical wavefronts (dots), and a second spiral α = 8.95 0.01 1 and β = 1.35 0.1. Figure4b displays a dependence between the radial radius (circles) given± bymm Equation2 (6) with ±the recovered values of α and β. Values of parameters are fitted and azimuthal angle for both spirals. The orientation (determined by the tangent line) of the calculated 1 (with 95% confidence bounds) as α = 8.95 ± 0.01 and β = 1.35 ± 0.1. Figure 4b displays a 2 mm dependence between the radial radius and azimuthal angle for both spirals. The orientation (determined by the tangent line) of the calculated spiral was β = 77.3 ± 0.7°, while the radius of the

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Photonics 2020, 7, 102 5 of 11 Photonicswavefront 2020, 7curvature, x FOR PEER of REVIEW the beam (Equation (9)) is R = 357 mm. One sees that, in this way, we are5 of able11 to recover the spiral characteristics. spiralwavefront was βcurvature= 77.3 of0.7 the, while beam the(Equation radius of(9) the) is R wavefront = 357 mm. curvature One sees ofthat the, in beam this way (Equation, we are (9)) able is ± ◦ Rto= recover357 mm. the One spiral sees characteristics. that, in this way, we are able to recover the spiral characteristics.

Figure 4. Results of spiral recovering (m = 1), (a) Comparison of two spirals, numerically generated (circles) and fitted (points), recovered spiral equation is ϕ =8.95r 2 + 1.35 ; (b) Comparison of radial Figureradius 4.4. and ResultsResults azimuthal ofof spiralspiral angle recoveringrecovering of both spirals (m(m == 1),. (a)) ComparisonComparison ofof twotwo spirals,spirals, numericallynumerically generatedgenerated (circles)(circles) andand fittedfitted (points), recovered spiral spiral equation equation is is ϕ ==8.958.95r22 + + 1.351.35;(; (b)b) ComparisonComparison ofof radialradial radiusFor the andand LG azimuthalazimuthal beam with angleangle higher ofof bothboth topological spirals.spirals. charges, fitting is more complicated, but can be done. There are more 2π phase jumps lines forming Fermat’s spirals in the phase distribution. In these cases,For the the points LG beam found with with higher the Roberts topological method charges charges, must, fitting fittingbe separated is more into complicated, separate spirals. but can If bethe done. points Thereare converted arearemore more 2 πto2πphase polarphase jumps coordinates jumps lines lines forming and forming arranged Fermat’s Fermat’s in spirals an spirals increasing in the in phasethe order phase distribution. of distribution. the radius, In these Inthen cases,thes, thee thecases,particular points the points found spirals found with can the withbe Robertsseparated the Roberts method based method muston the must be second separated be separated coordinate into separateinto, the separate azimuth spirals. spirals. angle If the If φ pointsthe. Figures points are 5 convertedareand converted 6 present to polar to the polar coordinates results coordinates of the and algorithm arranged and arranged infor an topological increasing in an increasing charge order ofs, order them = radius, 3of and the then,m radius, = 4 the, respectively particularthen, the , spiralsparticulartogether can spiralswith be separated the can fitted be based separated parameters on the based second α and on coordinate, βthe. One second can the coordinatesee azimuth that the, anglethe values azimuthϕ. of Figures parameter angle5 and φ.6 Figuresα present are kept 5 theandalmost results 6 present constant of the the algorithm forresults the givenof for the topological topologicalalgorithm charges, forcharge topological mwhile= 3 andthe charge β m value=s,4, m respectively,s =change 3 and, mpointing together= 4, respectively out with that the the, fittedtogethersubsequent parameters with spirals the αfittedand are rotatedβparameters. One by can an seeα angleand that β of the. One119 values° can± 2° see ofin parameter thethat case the of values αmare = 3 keptofand parameter 90 almost° ± 2° constantin α theare case kept for of thealmostm given= 4 . constant topological for the charge given while topological the β values charge change, while pointingthe β value outs thatchange the, subsequent pointing out spirals that arethe rotated by an angle of 119 2 in the case of m = 3 and 90 2 in the case of m = 4. subsequent spirals are rotated◦ ± ◦by an angle of 119° ± 2° in the◦ ± case◦ of m = 3 and 90° ± 2° in the case of m = 4.

FigureFigure 5. 5.Results Results of of fitting fitting Fermat’s Fermat’s spirals spirals to to the the LG LG beam beam with with topological topological charge charge m m= 3.= 3 Parameters. Parameters ofof subsequent subsequent spirals spirals are are (95% (95% confidence confidence bounds). bounds). ( 1(1)) Spiral Spiral 1, 1,α α= = 4.6664.666 (4.659,(4.659, 4.672)4.672) (1(1/mm/mm22) and β βFigure== −1.1721.172 5. Results (−1.179, ( 1.179, of − fitting1.164)1.164);; Fermat’s(2 ()2 Spiral) Spiral spirals 2, 2, αα = to=4.589 4.589the LG(4.583, (4.583, beam 4.595) 4.595)with ( topological1/mm (1/mm2)2 and) and charge β β= =3.083 3.083m = (3.076, 3 (3.076,. Parameters 3.09); 3.09); (3 ) − − − (of3Spiral) subsequent Spiral 3, 3, αα = = 4.653spirals4.653 (4.647, (4.647,are (95% 4.659) 4.659) confidence (1/mm (1/mm2) 2 andbounds)) and β β= 0.944=. (0.9441) Spiral (0.937, (0.937, 1 ,0.951) α 0.951). = 4..666 (4.659, 4.672) (1/mm2) and β = −1.172 (−1.179, −1.164); (2) Spiral 2, α = 4.589 (4.583, 4.595) (1/mm2) and β = 3.083 (3.076, 3.09); (3) InSpiral the 3 experimental, α = 4.653 (4.647, part, 4.659) which (1/mm is described2) and β = 0.944 in the (0.937, next 0.951) section,. the vortex beam is generated by the spiral phase plate (SPP). In this case, the complex amplitude is described by the sum of two special Bessel functions ([16], Equation (22)). We numerically generate phase maps and intensity distribution within such beam and notice that the spiral line can be divided into two parts, i.e., one Fermat’s spiral fits the internal part of the beam (Figure7a) while the second Fermat’s spiral fits the external areas of the beam (Figure7b). In the experiments, we examined only the central area of the beam (inside

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the bright intensity ring), therefore, the part of the spiral included, only in this area, has been taken Photonicsinto account. 2020, 7, x FOR PEER REVIEW 6 of 11

Figure 6. Results of fitting Fermat’s spirals to the LG beam with topological charge m = 4. Parameters of subsequent spirals are (95% confidence bounds). (1) Spiral 1, α = 3.497(3.492, 3.502) (1/mm2) and β = 2.635(2.629, 2.64); (2) Spiral 2, α = 3.447 (3.442, 3.452) (1/mm2) and β = −2.045 (−2.051,−2.039); (3) Spiral 3, α = 3.454 (3.449, 3.459) (1/mm2) and β = −0.4907 (−0.4966, −0.4849); (4) Spiral 4, α = 3.505 (3.499, 3.51) (1/mm2) and β = 1.048 (1.042, 1.054).

In the experimental part, which is described in the next section, the vortex beam is generated by the spiral phase plate (SPP). In this case, the complex amplitude is described by the sum of two special Bessel functions ([16], Equation (22)). We numerically generate phase maps and intensity distributionFigureFigure 6. withinResultsResults suchof of fitting ﬁtting beam Fermat’s Fermat’s and notice spirals thatto the the LG spiral beam linewith can topological be divided charge into m ==two 44.. Parameters Parameters parts, i.e., one Fermat’sofof subsequent spiral fits spirals spirals the internal are are (95% (95% part confidence conﬁdence of the bounds)beam bounds). (Figure. (1 (1) )Spiral Spiral 7a) 1while, 1, αα == 3.497(3.492, the3.497(3.492, second 3.502) 3.502)Fermat’s (1/mm (1/mm spiral2) 2and) and fitsβ the external=β =2.635(2.629, 2.635(2.629,areas of the 2.64);2.64) beam; (2)( 2( SpiralFigure) Spiral 2, 7b)α =. 2In,3.447 theα (3.442,experiments= 3.447 3.452) (3.442, (1, we/mm examine3.452)2) and β(d1/mm= only2.0452 )the and ( central 2.051, β = −2.0452.039);area of the − − − beam(−2.051,−(3 (inside) Spiral2.039) 3,theα =bright; 3.454(3) Spiral intensity (3.449, 3 3.459), ring),α (1=/ mm 3.454therefore,2) and(3.449,β the= part0.49073.459) of( the0.4966,(1/mm spiral2) 0.4849);and included β (=4 )−0.4907, Spiral only in4, α (−0.4966,this= 3.505 area , has − − − been−(3.499, 0.4849)taken 3.51)into; (4) (1Spiralaccount./mm 24), andα = 3.505β = 1.048 (3.499, (1.042, 3.51) 1.054).(1/mm2) and β = 1.048 (1.042, 1.054).

FigureFigure 7.7. NumericalNumerical phasephase profileprofile ofof thethe spiralspiral beambeam createdcreated byby thethe spiralspiral phasephase plateplate plottedplotted onon thethe backgroundbackground ofof thethe intensityintensity distribution.distribution. SpiralSpiral lineline cancan bebe describeddescribed byby twotwo Fermat’sFermat’s spirals.spirals. ((aa)) OneOne spiralspiral fittingfitting to to the the central central area area of of the the beam; beam; (b) ( Ab) second A second spiral spiral fitting fitting to the to external the external areas ofareas the of beam. the Recoveredbeam. Recovered spiral equations spiral equations are also are provided. also provided.

WeWe useuse the the algorithm algorithm described described above above to to recover recover the the uniform uniform phase phase shift shift which which is introduced is introduced into theinto vortex the vortex beam. beam. The phase The phase shift inshift the in vortex the vortex beam manifestsbeam manifests itself in itself the rotationin the rotation of the phase of the profile. phase Figureprofile.8 Figurepresents 8 thepresents spiral the lines spiral (plotted lines over(plotted the intensityover the distribution)intensity distribution) in the case in of the four case di offferent four φ π valuesdifferent of values phase of0 .phase The phase ϕ0. The is increasedphase is increased subsequently subsequently by the amount by the ofamount/5, causing of π/5, thecausing rotation the of the 2π jump line, and the angle by which the spiral rotates corresponds to the introduced phase rotationFigure of 7.the Numerical 2π jump phase line, andprofile the of angle the spiral by which beam creatthe spiraled by therotates spiral corresponds phase plate plotted to the introducedon the shift. As mentioned earlier, the parameter β from the spiral equation describes the spiral orientation, phasebackground shift. As ofmentioned the intensity earlier, distribution. the parameter Spiral line canβ from be described the spiral by two equation Fermat’s describes spirals. (a )the One spiral so its value changes when the spiral rotates. Table1 shows the results of the numerical fitting of the orientation,spiral fitting so its to valuethe central changes area whenof the thebeam; spiral (b) Arotates. second Table spiral 1fitting shows to ththee externalresults ofareas the ofnumerical the α β Fermat’sfittingbea ofm. spiralthe Recovered Fermat’s parameters. spiral spiral equations One parameters. can are see also that One provided.value can see is keptthat almostα value constant, is kept almost while constant,changes indicatewhile β thechanges rotation indicate of the spiral.the rotation By calculating of the spiral. the di ffByerence calculating between the the differenceβ values inbetween particular the cases, β values we can in recoverparticularWe theuse cases, valuethe algorithmwe of can the introducedreco describedver the value phase above of shift to the recover∆ introducedφ, which the uniform corresponds phase shift phase Δϕ well shift, which with which thecorresponds is known introduced value well intointroducedwith the the vortex known into beam. thevalue beam. The introduced phase shift into in the the beam. vortex beam manifests itself in the rotation of the phase profile. Figure 8 presents the spiral lines (plotted over the intensity distribution) in the case of four different values of phase ϕ0. The phase is increased subsequently by the amount of π/5, causing the rotation of the 2π jump line, and the angle by which the spiral rotates corresponds to the introduced phase shift. As mentioned earlier, the parameter β from the spiral equation describes the spiral orientation, so its value changes when the spiral rotates. Table 1 shows the results of the numerical fitting of the Fermat’s spiral parameters. One can see that α value is kept almost constant, while β changes indicate the rotation of the spiral. By calculating the difference between the β values in particular cases, we can recover the value of the introduced phase shift Δϕ, which corresponds well with the known value introduced into the beam.

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Figure 8. Spiral lines plotted over the intensity distribution in the case where different phase shifts Figure 8. Spiral lines plotted over the intensity distribution in the case where different diﬀerent phase shifts have been introduced into the vortex beam showing rotation of the phase map. have been introduced into the vortexvortex beam showing rotation ofof thethe phasephase map.map.

TableTable 1. IntroducedIntroduced and recovered phase shifts inin thethe cases shownshown inin FigureFigure6 6.. IntroducedIntroduced∆φ Δϕ(rad) (rad) φϕ00+ +π π/5/5 (0.62) (0.62) ϕ0φ +0 2+π2/5π/ 5(1.26) (1.26) ϕ0 +φ 30π+/53 π(1.88)/5 (1.88) Introduced Δϕ (rad) φ0 ϕ0 ϕ0 + π/5 (0.62) ϕ0 + 2π/5 (1.26) ϕ0 + 3π/5 (1.88) Introduced ∆φ ( ) ϕ0 36 72 107 Introduced Δϕ◦ (°) 36°◦ 72° ◦ 107° ◦ 2 2 α (1/αmm (1/mm) 2) 19.119.0.41 ± 0.4 18.818.8 ±0.7 0.7 18.8 18.8 ± 0.70.7 18.9 18.9 ± 0.7 0.7 α (1/mm ) 19.± 1 ± 0.4 18.8± ± 0.7 18.8 ± ±0.7 18.9 ± 0.7± β 1.93 0.04 1.34 0.05 0.71 0.06 0.09 0.06 β − −1.93± ± 0.04 −−1.34± ± 0.05 −0.71− ± ±0.06 −0.09− ± 0.06± Recovered ∆φ (rad) - 0.59 0.05; 1.22 0.06; 1.8 0.06; Recovered Δϕ (rad) - 0.59± ± 0.05; 1.22 ± ±0.06; 1.8 ± 0.06;± Recovered ∆φ (◦) - 34◦ 3◦ 70◦ 3.5◦ 105◦ 3.5◦ Recovered Δϕ (°) - 34°± ± 3° 70° ± ±3.5° 105° ± 3.5°± 4. Experimental Results 4. Experimental Results The algorithm described above was tested on experimental data. Figure9 shows the scheme The algorithm described above was tested on experimental data. Figure 9 shows the scheme of of the experimental setup of the optical vortex scanning microscope [17]. In the object arm of the the experimental setup of the optical vortex scanning microscope [17]. In the object arm of the interferometer, the Gaussian beam from the He-Ne laser (λ = 0.633 µm) passes through the spiral phase interferometer, the Gaussian beam from the He-Ne laser (λ = 0.633 μm) passes through the spiral plate (SPP) which introduces helical wavefront into the beam [18]. Then, this beam is focused by the phase plate (SPP) which introduces helical wavefront into the beam [18]. Then, this beam is focused microscope objective ( 20, NA = 0.4) into the sample plane. The spot size of the vortex beam reveals a by the microscope objective× (×20, NA = 0.4) into the sample plane. The spot size of the vortex beam donut shape (see Figure9) and its size can be described by the bright ring diameter, which in our case reveals a donut shape (see Figure 9) and its size can be described by the bright ring diameter, which was about 2.8 µm. Through the imaging system (NA = 0.4, magniﬁcation 200 ) a spot was imaged in our case was about 2.8 μm. Through the imaging system (NA = 0.4, magnification× 200×) a spot was on the CCD camera and its size has been enlarged to about 0.6 mm. The object beam, together with imaged on the CCD camera and its size has been enlarged to about 0.6 mm. The object beam, the reference beam, creates on the CCD, the interferogram. When the vortex is present in the beam, together with the reference beam, creates on the CCD, the interferogram. When the vortex is present the interference fringes split forming the characteristic fork pattern. in the beam, the interference fringes split forming the characteristic fork pattern.

Figure 9. SchemeScheme of of the the experimental experimental setup setup.. Bs1, Bs1, Bs2, Bs2, and and Bs3 Bs3,, beam beam splitters splitters;; SPP SPP,, spiral spiral phase phase plate plate;; MOMO,, microscope objective. Interferogram and scheme of phase extraction are shown in thethe inset.inset.

The standard Fourier technique [19] was applied to recover the phase of the object beam. To do this, the Fourier transform is applied to the detected interferogram. In its spectrum, the first order is extracted and shifted to zero order position to remove the carrier frequency. Next, the inverse

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The standard Fourier technique [19] was applied to recover the phase of the object beam. To do this,Photonics the Fourier2020, 7, x transformFOR PEER REVIEW is applied to the detected interferogram. In its spectrum, the first order8 of is 11 extracted and shifted to zero order position to remove the carrier frequency. Next, the inverse Fourier transformFourier transform is applied. is applied. This recovers This recover the complexs the complex amplitude amplitude of the detected of the detected beam, in be whicham, in phase which distributionphase distribution is coded. is Figurecoded. 10Figurea displays 10a displays the exemplary the exemplary phase maps phase of maps the central of the partcentral of thepart vortex of the beamvortex obtained beam obtained from the from experiment. the experiment. Figure 10 Figureb plots 10b the plots polar the coordinates polar coordinates of the localized of the localized 2π phase 2π jumpphase line jump (points) line (points) together together with the with fitted th linee fitted of the line equation. of the equation. Figure 10 cFigure compares 10c compares the fitted Fermat’sthe fitted spiralFermat’s (circles) spiral with (circles) the experimental with the experimental points (crosses). points Values (crosses). of coe Valuesfficients, of coefficients, fitted with 95% fitted confidence with 95% h i 2 bounds,confidence are equalbounds, to αare= equal20.0 to0.04 α =1/20.0mm ±2 0.04and 1/β mm= 2.32 and 0.04.β =−±2.32 0.04 ± − ± .

FigureFigure 10. 10.(a )(a The) The phase phase map map obtained obtained from from the experiment; the experiment (b) Polar; (b) coordinates Polar coordinates of the edge of positionthe edge extractedposition fromextracted (a) and from the fitted(a) and line; the (c) Fermat’sfitted line; spiral (c) fitted Fermat’s to the spiralexperimental fitted todata-crosses the experimental denote thedata 2π-crossesjump line denote from the (a), 2 circlesπ jump indicate line from the (a fitted), circles spiral. indicate the fitted spiral.

ToTo demonstrate demonstrate how how this this vortex vortex beam beam description description can becan used be inused optical in optical measurement, measurement, the phase the samplephase issample introduced is introduced into the objectinto the beam object (at beam a sample (at a plane). sample When plane). the When vortex the beam vortex passes beam through passes thethrough transparent the transparent sample, its sa phasemple, distributionits phase distribution changes. Aschanges. was mentioned As was mentioned in the preceding in the preceding section, insection, the case in of the uniform case phaseof uniform change phase within change the whole within beam, the the whole phase beam, profile the rotates, phase manifesting profile rotates the , rotationmanifest ofing the the spiral rotation and this of rotationthe spiral can and be this read rotation from the can parameter be readβ from. In our the experiment, parameter β the. In tiny, our transparentexperiment sample, the tiny, had transparent the shape ofsample the square had the step shape of high of the 200 square nm and step size of 2highµm. 20 As0 thenm sampleand size transverse2 μm. As dimensionsthe sample weretransverse a little dimensions bit smaller thanwere the a little vortex bit beamsmaller diameter than the and vortex because beam it was diameter hard toand localize because it perfectly it was hard in theto localize beam center, it perfectly it was in mounted the beam oncenter, the motorizedit was mounted XYZ tableon the in motorized order to positionXYZ table the samplein order with to nanometerposition the precision. sample The with sample nanometer was shifted precision. gradually The with sample the step was 0.05 shiftedµm. Whengradually the step with was the within step 0.05 the beam,μm. When the additional the step was phase within shift equalthe beam, to ∆ φthe= additional2π (n 1) hphase 0.9 radshift λ − · (h is the step height2π equal 200 nm in our case, sample was a PMMA resist applied to the saphire equal to ∆=φ (nh −1) ⋅≅ 0.9rad (h is the step height equal 200 nm in our case, sample was a substrates, n 1.48λ ) was expected, which should cause rotation of the spiral by the amount of ∆φ, as wasPMMA explained resist inapplied the previous to the section.saphire Figuresubstrates, 11 shows n ≅ 1.48 the) ideawas ofexpected, the experiment which should and obtained cause rotation results. Forof the each spiral sample by the position, amount the of interferogram Δϕ, as was explained was detected, in the phaseprevious map section. recovered, Figure and 11 parametersshows the idea of Fermat’sof the experiment spiral fitted and (Figure obtained 11a,b). results. Figure For11c showseach sample the change position, of the the fitted interferogram value β while was the detected, sample wasphase moved map acrossrecovered the beam., and parameters The reference of Fermat’s phase value spiral (in fitted the case (Figure where 11a,b the). stepFigure was 11c outside shows the the beam) was calculated for a sequence of measurements and equaled 2.51 0.15 rad (uncertainty was change of the fitted value β while the sample was moved across the beam.± The reference phase value evaluated(in the case on thewhere base the of thestep series was ofoutside measurements the beam) taken was when calculated the object for wasa sequence outside theof m vortexeasurements beam). Whileand equal the phaseed 2.5 object1 ± 0.15 was rad gradually (uncertainty moved was through evaluated the vortexon the beam,base of the the spiral series was of rotatedmeasurements and β reached the mean value 3.37 0.15 rad when the whole phase step was inside the beam. Therefore, the taken when the object was± outside the vortex beam). While the phase object was gradually moved spiral rotated by an angle of 0.86 0.15 rad, which corresponded well with the value expected from through the vortex beam, the spiral± was rotated and β reached the mean value 3.37 ± 0.15 rad when objectthe whole geometry. phase step was inside the beam. Therefore, the spiral rotated by an angle of 0.86 ± 0.15 rad, which corresponded well with the value expected from object geometry.

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FigureFigure 11.11. ((aa,,bb)) SchematicSchematic presentationpresentation of thethe experiment,experiment, samplesample with thethe phasephase stepstep waswas shiftedshifted throughthrough thethe focusedfocused vortexvortex beambeam causingcausing rotationrotation ofof thethe spiralspiral proﬁleprofile inin thethe casecase wherewhere thethe stepstep waswas insideinside thethe beam; beam; ( c()c Values) Values of of the the parameter parameterβ correspond β correspond well well with with the expected the expected phase phase shift. Theshift lower. The solidlower line solid indicates line indicatesϕ0 level φ0 (2.51 level rad) (2.51 and rad) the and upper the upper solid line solid indicates line indicates 3.37 rad. 3.37 rad.

5. Conclusions 5. Conclusions WeWe proposedproposed a a new new method method for for characterizing characterizing avortex a vortex beam beam with with a topological a topological charge charge of one of andone higher.and higher. We focused We focused on the phaseon the distribution phase distribution and showed and that showed its contour that its lines contour could belines described could bybe adescribed Fermat’s by spiral a Fermat’s in the case spiral of thein the LG case beam. of Inthe addition, LG beam. we In showed addition, that w thee showed same couldthat the be donesame forcould the be central done partfor the of thecentral phase part profile of the of phase the Gaussian profile of beam the Gaussian passing through beam passing SPP. Fermat’s through spiral SPP. equationFermat’s wasspiral expressed equation bywas two expressed parameters, by two which parameters, could be which assigned could to thebe assigned physical characteristicsto the physical ofcharacteristics the beam-radius of the of thebeam wavefront-radius curvatureof the wavefront and the phasecurvature orientation. and the We phase proposed orientation. and tested We a procedureproposed and for fitting tested the a procedure spiral equation for fitting to the the phase spiral maps equation of a vortex to the beam. phase The maps procedure of a vortex consisted beam. π ofThe three procedure main stages. consist Ined the of firstthree step, main a characteristicstages. In the linefirst assigned step, a characteristic to the 2 phase line jump assigned was extractedto the 2π fromphase the jump phase was map, extracted in our casefrom with the thephase help map, of the in Roberts our case method with the for help edge of detection. the Roberts Next, method the found for pointsedge detection. were arranged Next, inthe the found order points of increasing were arranged values ofin radialthe order radii of and increasing azimuth values angles. of Inradial the thirdradii step,and azimuth a nonlinear angles. least In squares the third method step, withouta nonlinear normalization least squares was method used in without order to fitnormalization extracted points was intoused the in spiralorder equation. to fit extracted This procedure points into worked the wellspiral for equation. numerical, This as wellprocedure as for experimental worked well data. for numericalTo demonstrate, as well as thefor possibleexperimental application data. of this way of beam characterization, we showed, that the spiralTo demonstrate rotated in response the possible to the application transparent of object this way by the of anglebeam equalcharacterization to the introduced, we showed, phase shiftthat andthe spiral one of rotate the spirald in parametersresponse to couldthe transparent read this rotation. object by We the showed angle equal in [11 ]to that the the introduced accuracy ofphase the recoveredshift and one phase of shiftthe spiral using parameters OV could be could very read high this because rotation. optical We vortex showed could in [ be11] treated that the as accuracy a single fringeof the recovered surrounding phase the singularshift using point, OV c andould in be such very geometry, high because rotation optical could vor betex determined could be treated with high as a precision.single fringe If thesurrounding phase disturbance the singular is not point uniform, and withinin such the geometry beam, then, rotation the phase could undergoes be determined local changes,with high deforming precision. somehow If the phase the ideal disturbance Fermat’s spiral,is not which uniform happens within when the LG beam, beams then are generated,the phase forundergo example,es local by SLM.changes, Some deforming methods someh of SLMow correction the ideal haveFermat’s used spiral vortex, which structure happens as a when template. LG Distortedbeams are vortex generated has been, for compared example, withby SLM. an ideal Some one methods [20,21]. In of another SLM correction work [22], have deformation used vortex from thestructure ideal vortexas a template was used. D foristorted determining vortex has misalignment. been compared Thus, with the an description ideal one of[20,21 the ideal]. In another overall phasework distribution[22], deformation gives afrom new setthe of ideal tools vortex in data was analysis. used Wefor candetermining treat it as amisalignment kind of reference. Thus profile, the againstdescription which of wavefrontthe ideal overall deformation phase distribution could be determined. gives a new We set plan of tools to use in this data idea analysis. in our We further can work.treat it Inas oura kind experimental of reference part, profile we against considered which only wavefront m = 1 case, deformation as higher could order be vortices determined. are very We unstableplan to use structures this idea and in our split further into a work. constellation In our experimental of single-charged part,vortices we considered [8,23]. Furtheronly m research= 1 case, as is neededhigher order to show vortices their utility are very in experimental unstable structures conditions. and split into a constellation of single-charged vortices [8,23]. Further research is needed to show their utility in experimental conditions. Author Contributions: Conceptualization, E.F. and A.P.M.; methodology, E.F.; software, E.F.; validation, A.P.M., E.F. and S.S.; formal analysis, E.F and S.S.; investigation, A.P.M. and E.F.; resources, E.F. and A.P.M.; data curation, E.F.; writing—original draft preparation, A.P.M., E.F. and S.S.; writing—review and editing,

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Author Contributions: Conceptualization, E.F. and A.P.M.; methodology, E.F.; software, E.F.; validation, A.P.M., E.F. and S.S.; formal analysis, E.F and S.S.; investigation, A.P.M. and E.F.; resources, E.F. and A.P.M.; data curation, E.F.; writing—original draft preparation, A.P.M., E.F. and S.S.; writing—review and editing, A.P.M., E.F.; visualization, E.F.; supervision, E.F.; All authors have read and agreed to the published version of the manuscript. Funding: This research received no external funding. Acknowledgments: We acknowledge the support of this work by the Wroclaw University of Science and Technology statutory fund no. 8201003902/K34W04D03. Conﬂicts of Interest: The authors declare no conﬂict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, or in the decision to publish the results.

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