The interference pattern of a Laguerre Gaussian beam in a double slit
experiment is reported. Whereas a typical laser beam phase front is planar, a
Laguerre Gaussian beam exhibits a wave front that is twisting along the
direction of propagation. This leads to a distorted interference pattern. The
topological charge also called the order of the twisted beam can be then
readily and simply determined. More precisely, the naked eye resolution of the
distortion shift of the interference pattern directly informs about the number
of twists made as well as on the sign of the twist. These results are in very
good agreement with theoretical calculations that offer a general description
of the double slit interference with twisted beams.Comment: 11 pages, 4 figure

Emile, Janine

Emile, Olivier

English

arXiv

13 pagesInternational audienceA wide range of diffractive elements have been used to evaluate the topological charge of Laguerre-Gaussian beams. Here, we show theoretically and experimentally that this charge can be simply and readily measured from the interference pattern in Young's double-slit experiment. It can be evaluated from the twisting order of the interference. The results are confronted with previously published studies. The potentialities of the method are then compared with existing techniques

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Young’s double-slit interference pattern from a twistedbeamOlivier Emile, Janine EmileTo cite this version:Olivier Emile, Janine Emile. Young’s double-slit interference pattern from a twisted beam. Ap-plied Physics B - Laser and Optic, Springer Verlag, 2014, 117 (1), pp.487-491. <10.1007/s00340-014-5859-1>. <hal-01065119>HAL Id: hal-01065119https://hal.archives-ouvertes.fr/hal-01065119Submitted on 2 Oct 2014HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.L’archive ouverte pluridisciplinaire HAL, estdestine´e au de´poˆt et a` la diffusion de documentsscientifiques de niveau recherche, publie´s ou non,e´manant des e´tablissements d’enseignement et derecherche franc¸ais ou e´trangers, des laboratoirespublics ou prive´s.Young’s double slit interference pattern from a twistedbeam.Olivier Emile1 and Janine Emile21LPL, URU 435 Universite´ de Rennes I, 35042 Rennes Cedex, France.∗2IPR, UMR CNRS 6251, Universite´ de Rennes I, 35042 Rennes Cedex, France.(Dated: May 12, 2014)AbstractA wide range of diffractive elements have been used to evaluate the topological chargeof Laguerre-Gaussian beams. Here we show theoretically and experimentally that thischarge can be simply and readily measured from the interference pattern in Young’sdouble slit experiment. It can be evaluated from the twisting order of the interference.The results are confronted with previously published studies. The potentialities of themethod are then compared with existing techniques.PACS numbers: 42.50.Tx, 42.87.Bg, 42.50.Xa, 42.25.Hz∗ Corresponding author; olivier.emile@univ-rennes1.fr1I. INTRODUCTIONYoung’s double slit is may be one of the most popular and fascinating experi-ment in physics. As R. Feynman said: ”It is a phenomenon which is impossible toexplain in any classical way and which has in it the heart of quantum mechanics”[1]. Curiously, when a phase difference is added in between the two interferingpaths, like in an Aharonov-Bohm type experiment [2], the interference pattern isshifted. However, along a single slit, the wave encounters exactly the same phaseshift and thus the interference fringes remain straight lines. On the other hand, anew category of waves, called ”twisted waves” appeared in the 1990s, which phasedistribution is not uniform in a plane perpendicular to the direction of propagation[3, 4]. Since its first observation in optics, it has then found many applicationsin various domains including microwaves [5], atom optics [6], quantum cryptog-raphy [7], telecommunications [8? ], astronomy [9], biophysics [10], acoustics [11]and electron beams [12]. Besides, these waves are known to produce diffraction orinterference patterns as well [13? –19]. However, researchers in the field mainlyuse the diffraction by a triangle aperture [20–22], or transformations with cylin-drical or tilted spherical lens [23, 24], to characterize the beam whereas the doubleslit experiment is hardly ever implemented although it seems easy to settle up andhandle. The aim of this letter is to seek an analytical expression of the interferencepattern of a Laguerre-Gaussian (LG) beam in a double slit experiment, to compareit with experimental observations and to investigate practical applications in thecharacterization of LG beams.II. THEORETICAL CONSIDERATIONSFrom a theoretical point of view, the phase ψ of the twisted beam, on a planeperpendicular to the direction of propagation, is not uniform as for usual planewaves. It varies from 0 to 2lpi as one makes one complete turn around the direction2of propagation [4]. l is called the Topological Charge (TC) of the beam. Thephase writes ψ(θ) = lθ, where θ is the usual polar coordinate (see figure 1). θ isrelated to the coordinate z along the vertical direction by the following relation:tan(θ) = a/z, where 2a is the distance between the slits. Then the phase differenceδψ between the two paths of the double slit experiment, at a given height z, writes,δψ = ψ(θ)−ψ(−θ) = 2lθ = 2l tan−1(a/z). The intensity variation I(x) due to theinterference between the paths along the horizontal axis can be written as [1]I(x) = I0 cos2(2pixa/(λD) + δψ) (1)where D is the distance between the slits and the screen used for observation. Theinterference pattern in the x direction varies as2pixa/(λD) + 2l tan−1(a/z) (2)In particular, this means that as z tends towards +∞ (i.e. in the upper zoneof the laser beam where θ = 0), the interference fringes should correspond to theusual pattern of a plane wave. The two interfering beams have the same phase onthe double slit. Using exactly the same reasoning, as z tends towards −∞ (i.e.in the bottom zone of the laser beam where θ = pi) the interference fringes alsocorrespond to the usual pattern of a plane wave. The phase difference equals 2lpi.Actually equations 1 and 2 offer a general description of the double slit inter-ference with a LG beam. This theoretical result is in disagreement with what hasbeen found previously [15], where, from numerical calculations, a pi phase differ-ence between the top and the bottom of the slits is predicted (see relation (4) andfigure 2d [15]). For a correct description, one has to first take into account thephase difference δψ between the interfering paths at the top and at the bottomseparately (see figure 1). Second, one has to compare the resulting δψ betweenthe top and the bottom of the slits. It unambiguously equals 2pi.3θ2azδφ=lpi δφ     2lpiδφ=2lθIδφ     0Dl=1FIG. 1. Schematic of the phase of the LG beam impinging on a double slit experiment.2a: distance between the slits, D distance between the slits and a screen. Two corre-sponding points at the same height have a phase difference equal to δψ = 2lθ. This leadsto a twisted interference pattern.III. EXPERIMENTS AND RESULTSFrom an experimental point of view, let us consider a typical double slit ex-periment like the one that can be found in textbooks [1]. We replace the usuallight source by a twisted laser beam (see figure 2). The twisted beam is here aLG beam, but the experiment could be implemented for any twisted beam. It isgenerated from the fundamental beam of a red He-Ne laser (λ = 633 nm, MellesGriot). The beam passes through a vortex phase plate [25] (RC Photonics) with4TC that can be chosen from l = 0 to l = 3 and a telescope (final beam waist0.7 mm), before impinging on the double slit experiment. The slits are 3 cm longand 70 µm large. The distance between them is 2a = 300 µm. Pictures of theinterference patterns are taken on a screen at a distance D = 4 m with a camera.VPP DScreenHe-Ne laser λ= 633 nm2axzYoung slitsFIG. 2. Experimental set up. VPP: Vortex Phase Plate. The zoom of the VPP showsa variable thickness that induces a phase variation in order to generate a LG beam.Figure 3 presents several photographs of the interference pattern for variousvalues of l. In figure 3a (l = 0), one recognizes the usual interference pattern ofthe typical double slit experiment using plane wave sources. In particular, thefringes are straight lines. The diffraction pattern has the same symmetry as thediffracting object. It has a cylindrical symmetry for a diffracting hole and hasa Cartesian symmetry for diffracting slits [26]. However, for a twisted beam, forexample for l = 1 (see figure 3b), the interference pattern is twisted. The fringesare not straight lines any more. They follow a 2 tan−1(a/z) variation, as expected.The interferences, at the bottom zone of the laser beam, have been shifted by5exactly one interference order compared with the ones corresponding to the upperzone of the laser beam (see green arrows on the figure). This is a very unusualbehaviour of the interference pattern in a double slit experiment. Moreover, as lis further increased, the twist of the fringes becomes more and more important.For l = 2, the fringes at the bottom zone (see figure 3c) have been shifted by twointerference orders compared to the ones at the upper zone, whereas for l = 3 (seefigure 3d), they correspond to three interference orders. Analogously, for l = n,they should correspond to n orders.IV. DISCUSSIONActually this experiment is a quick and easy to handle way to determine theTC of a twisted beam. The twist of the fringes can be readily seen on a screenwith the naked eye. One has only to count the number of twists from the top ofthe interference pattern to the bottom, following the whole interference pattern.However, one cannot determine the TC of the beam when considering the top andthe bottom of the fringes only, since the bottom fringes are shifted exactly by aninteger number order of fringes. Besides, in order to obtain such patterns, theremust be some light impinging on the various zones of the slits. In particular, sincethe center of the laser beam is a vortex, one has to adapt the distance 2a betweenthe slits to obtain a good and usable pattern.Let us be more quantitative and define a minimum and a maximum distance2a between the slits to determine the TC of the beam. There are two criteriaconcerning: (i) the visibility of the fringes, (ii) the zone of the beam probed. For(i), from the naked eye, the intensity of a fringe should not vary more than a factorof 5 to be clearly detected. If one considers a l = 1 LG beam with a waist w, thebeam intensity I(r) (r being the polar coordinate) is proportional to6l=1l=2 l=3l=02 310a) b)c) d)FIG. 3. Twisted interference patterns for beams with a) l = 0; b) l = 1; c) l = 2; d)l = 3. The twist corresponds to l bright fringes of the interference pattern. The greenarrows indicate the fringe order shift.I(r) ∼ r2w2exp−2r2w2(3)The beam intensity is maximum for r = w√l/√2. The intensity of a singlefringe F (z) follows the same variation as I(r), taking r =√a2 + z2. The minimumintensity of the fringe is for z = 0 (θ = pi/2). The minimum distance 2a betweenthe slits, corresponding to F0/5 is 2w/5. The maximum intensity F0 corresponds7to z = 0.95w/√2. As z increases further, the intensity of the fringe decreases andthe intensity F0/5 corresponds to z = 2w/√2 (θ = 0.14 rad). The phase variationof the beam is thus probed in the region 0.14 < θ < pi − 0.14. Let us move to the(ii) criterium. We assume that the TC of the beam could be easily identified ifwe probe more than 80% of the beam, i.e. 0.35 < θ < pi − 0.35. Then criterium(i), applied to θ = 0.35 rad, leads to a < w/2. Thus, practically, the distance 2abetween the slits should be such as w/5 < a < w/2 in order the TC of the LGbeam to be easily determined. In our case, w = 700 µm and a = 150 µm, fulfill thetwo criteria. Besides, for a = w/5, the height of the slits should be at least morethan 4w/√2 ≃ 3w (θ = 0.14 rad) not to truncate the beam. This last argumentabout the height of the slits may be the reason why previous works incorrectlyevidence a smaller twist of the fringes [15]. Finally, the beam must be centered onthe middle of the slits.So far, we have shown that the interference pattern of a double slit experimentallows to precisely determine the TC of a twisted beam. One may thus wonderwhether the sign of this charge could be also fixed. Actually, following equation2, the sign of the twist of the pattern should be reversed when changing thesign of the TC. Experimentally, let us reverse the orientation of the vortex phaseplate (see figure 2) so as to reverse the sign of l. Figure 4 shows the interferencepattern for l = 1 and l = −1. The sign of the twist of the fringes is reversed,as expected. It can be determined unambiguously. Could this technique be usedto precisely determine non integer TC [27–29]? It seems difficult since the lightintensity distribution is non symmetrical. This would scramble the interferencepattern. Nevertheless, when the slits are aligned with the light discontinuity, thesymmetry is restored and the non integer TC could be estimated.Several interferential techniques have been proposed to determine the TC of thebeam. The most popular one is perhaps the diffraction by a triangular aperture[16, 20–22] that needs to count the number of diffracted spots. However, for l > 78l=+1 l=−1a) b)FIG. 4. Twisted interference patterns for two opposite signs of l. The interferences areshifted in opposite directions (see blue arrows).it is sometimes hard to determine the number of spots. There are similar tech-niques using more complicated apertures such as hexagonal aperture [30], annularaperture [31? ], or multi points interferometer [32]. One can mention the inter-ference of the LG beam with itself that needs a biprism and a lens [13, 18], thediffraction by an edge [33] and by a single slit [34], or even with a double angularslit [35], where the relationship between the pattern and l is not straightforward.The image in the focal plane of a cylindrical lens [23] or equivalently with a tiltedspherical lens [24] has also been performed. Nevertheless, Young’s double slit ex-periment is indeed easy to settle and use. The measurements could be performedwith the naked eye. Even with high TC, or with partially coherent beams [? ] orwith tiny spots [19] the experiment would be of practical use. The calculations areelementary and the interpretation of the experimental results is straightforward.Besides, the sign of the TC could be easily determined.9V. CONCLUSIONThis easy-to-settle experiment enables to precisely measure, with the naked eye,the value and the sign of the topological charge of a twisted beam. 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Young's double slit interference pattern from a twisted beam

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