Orbital angular momentum of mixed vortex beams

Transcription

1 Orbital angular momentum of mixed vortex beams Z. Bouhal *, V. Kollárová, P. Zemánek, ** T. ižmár ** Department of Optis, Palaký University, 17. listopadu 5, 77 7 Olomou, Ceh Republi ** Institute of Sientifi Instruments, ademy of Sienes of the Ceh Republi, Královopolská 147, Brno, Ceh Republi BSTRCT The orbital angular momentum (OM) of the single vortex beam depends on its power and wavefront heliity. In the paper, this relation is generalied for mixed vortex beams omposed of several oaxial vorties with different topologial harges. The presented interferene law indiates interferene effets of the OM resulting in loal spatial gradients of the OM density. Desription of the OM of mixed vortex beams is used for demonstration of a possibility to tune the OM density of a omposite vortex field without hanging topologial harges or intensity distribution. Experimental realiation of the OM tuning is disussed for interferene of two foused vortex beams generated by means of a spiral phase mask. Keywords: optial vorties, orbital angular momentum, spatial light modulator 1. INTRODUCTION During reent years, the peuliar properties assoiated with phase disloations of light have exited an extensive optial researh resulting in very promising theoretial results, experiments and physial appliations. partiular attention has been foused on the so-alled optial vorties. They represent a speial kind of fields possessing helial wavefronts with spiral phase singularities 1. The vorties nested in well loalied laser beams an exhibit interesting spatial effets depending on the propagation properties of the used host beam. The best known examples of spatial evolution are looping, knotting and braiding of vorties, their expansion and rotation, and attration or repulsion of vortex pairs. The helial phase struture of vortex beams is assoiated with a spiral flow of the eletromagneti energy induing the OM. Though the magnitude of light torque is only 1-18 Nm for a vortex beam with the power of several milliwatts, it an be suessfully transferred to miropartiles in trapping experiments. This is a reason why the optial vorties are onsidered to be a prospetive propeller for Miro Eletro Mehanial Systems. So far, the OM has been mainly examined and experimentally proved for single vortex beams. In the paper, the OM of omposite vortex beams representing a oherent superposition of vorties with different topologial harges is examined. In this ase, the OM density exhibits interferene effets resulting in its nonhomogeneous spatial distribution. In the simplest ase of two oaxial vorties, the interferene of the OM an be suppressed and the total OM an be tuned adusting the relative amplitudes of the superposed vorties 3. The experimental verifiation of the effet based on appliation of a spiral phase mask is proposed and disussed.. ORBITL NGULR MOMENTUM OF SINGLE VORTEX BEM In a salar approximation, the omplex amplitude U of a monohromati optial beam arrying a single vortex an be onveniently expressed in the irular ylindrial oordinates ( r,, ) as U ( r,, ) u( r,, ) exp[ i( r,, )], (1) where u and are real funtions representing amplitude and phase of the beam, respetively and an osillating temporal term has been omitted. This general form inludes free-spae propagation of Bessel, Bessel-Gauss or Laguerre-Gauss beams and remains valid also for foused vortex beams realied by means of a spiral phase mask. In the ase of a single vortex beam, the phase inludes the term m, where m is a non-ero integer alled the topologial harge of the vortex. The OM of the vortex beam is aused by the spiral flow of the eletromagneti energy expressed by the non-ero aimuthal omponent of the Poynting vetor. In the vetorial theory, the Poynting vetor S and the energy volume density W are defined by vetors of the eletromagneti field. In the framework of the salar theory, they an be approximated by * bouhal@prfnw.upol. 15th Ceh-Polish-Slovak Conferene on Wave and Quantum spets of Contemporary Optis, edited by Miroslav Miler, Dagmar Senderáková, Miroslav Hrabovský, Pro. of SPIE Vol. 669, 6697, (7) X/7/$18 doi: / Pro. of SPIE Vol

2 quantities S and W whih an be expressed by means of the gradient of the salar amplitude and fulfil relation formally idential with the energy onservation law, S W t. For a monohromati field with the angular frequeny, we an write * * S i( U U UU ). () Taking into aount the time averaged quantities, the angular momentum J of the beam an be written as a vetorial produt of the position vetor r and the linear momentum p, J r S /, where propagation of the beam in vauum with a phase veloity was assumed. If the beam propagates along the -axis, its OM density is given by the - omponent of J, J rs /. By means of () we obtain * i * U U. J U U (3) * The density of the OM of the vortex field (1) then depends on the beam intensity I UU and the wavefront heliity, J I. (4) The total OM arried by the beam L then an be obtained as L J rdrd. In the ase of a single vortex beam, the wavefront heliity is unambiguously defined by the topologial harge m as m and the intensity I is rotationally symmetrial. In this ase, the total OM an be expressed by means of the beam power P as mp L. (6) 3. TUNING OF THE ORBITL NGULR MOMENTUM possibility to tune the OM an be simply demonstrated by interferene of two oaxial vortex beams U and U B with the topologial harges m and m and amplitudes uand Bu, respetively, U U U B u( r, )[ exp( im i ) B exp( im ib )], (7) where, B, and B are real onstants enabling adusting of relative amplitudes and phase offsets of vortex beams. The omplex amplitude of the interferene field (7) an be rewritten in the form (1) used with u u B B os[m( )], (8) B artg{ Ctg[ m( )]}, ( ) / B C ( B) /( B) and the ondition m( ) /, / where, must be fulfilled. It was shown 3 by numerial analysis of (4) used with (8) and (9) that the hanges of the intensity I u and the phase heliity / aused by a hange of C anel out to give a onstant loal OM density at a given radius r. Here, this effet is quantified by means of analytial alulations. The OM density J and the total OM L related to the interferene field (7) an be written as J ( B ) u m, (5) (9) (1) Pro. of SPIE Vol

3 ( B ) L Pm, ( B ) where P is the total power of the interferene field given by a sum of powers of the superposed vortex beams, (11) P ( B ) u ( r) rdr. (1) s is obvious, the OM density (1) remains rotationally symmetrial even if the intensity of the interferene field given by (8) is aimuthally modulated. Furthermore, both the OM densityj and the total OM L an be tuned adusting relative amplitudes of the interfering vortex beams and B. s follows from (6), the ratios of the OM and the power have the same magnitude and opposite signs for superimposed vortex beams with the opposite topologial harges m and m, L / P m / L / P B B. If we introdue the relative amplitude of the vortex beams as Q / B, the ratio of the total OM and the total power arried by the interferene field (7) an be written as L m ( Q 1). P ( Q 1) (13) The ratio of the OM and the power an be tuned in the interval L m m, P. (14) The marginal values are obtained by swithing of the single vortex beams B, ( Q ) and, ( Q ). For vortex beams of equal amplitudes Q 1 the OM of the interferene field vanishes. s is obvious, by adusting amplitudes of two oaxial vortex beams with opposite topologial harges, the total OM of the resulting field an be tuned, reversed or aneled without hange of the topology of superimposed vortex beams. By the tuning of the total OM, its spatial distribution remains rotationally symmetrial. 4. ORBITL NGULR MOMENTUM OF VORTEX SUPERPOSITION mixed vortex field is omprehended as a superposition of J oaxial vortex beams with rotationally symmetrial amplitudes u, different helial phases and amplitude weighting fators a, J U a u ( r, ) exp[ i ( r,, )]. (15) 1 The omplex amplitude of the mixed vortex field (15) an be expressed in the form (1) and its OM density alulated applying (4). lternatively, the OM density an be obtained by (3) used with the omplex amplitude given by the series (15). In this ase, the OM density an be expressed as a sum J JVB J I, (16) where J VB is a ontribution of separate vortex beams and J I represents an interferene term. They an be written in the form VB J a u 1, J (17) J J p J I aapuup p p os. (18) 1 The helial wavefronts of the separate vortex beams are unambiguously defined by the topologial harges, ( r,, ) m ( r, ). In this ase, we an simply show that the interferene term J I does not ontribute to the total OM of the mixed field L. It is given by the sum of the OM of the superposed vortex beams and an be written as Pro. of SPIE Vol

4 where L J 1 m P, (19) P a u rdr denotes the power of separate vortex beams. The amplitude weights of superposed vortex beams determine the total OM and strongly influene its spatial distribution through the interferene term (18). The interferene of the OM of the separate modes results in its spatial gradients important for trapping experiments. mplitudes and phases of the superposed vortex beams enable spatial shaping of the OM density to a predetermined form. possibility to tune the OM by adusting amplitude weights of two vortex modes is now learly evident from a general form of the interferene law for the OM (16) - (18). Considering interferene of two vortex modes with the amplitude weights a1 and a and the topologial harges m1 and m ( m 1 < m ), Eqs. (17) and (18) an be simplified to the form J VB a1 u1 m1 aum, () J I m1 m a1u 1au os[ m1 m 1 ]. (1) For vortex beams having topologial harges of equal magnitudes but opposite signs m1 m m, the interferene of the OM vanishes, and the OM density J JVB is idential with (1) (used with denotation a1 and a B ). 5. PROPOSL OF EXPERIMENT The mixed vortex fields omposed of several vortex beams with a well defined heliity enable spatial shaping of the OM and are of partiular interest in trapping experiments. n understanding of suh experiments requires ompliated numerial models enabling analysis of real experimental onditions. In this paper, the simplest ase of interferene of two vortex beams realied by means of Mah-Zehnder interferometer is examined. n expanded laser beam is direted into two arms where the spiral phase masks M1 and M harateried by the topologial harges m1 and m are plaed (Fig.1). The reated beams with helial phases are then ombined and foused by a mirosope obetive O. To simulate the experiment, the vortex fousing must be solved. We assume a spiral phase mask with the topologial harge m adaent to a lens with the foal length f. The impinging laser beam has a Gaussian profile with the waist radius w. The beam waist is plaed at the lens. Spiral Phase Mask (SPM) M 1 SPM 1 BS 1 O SPM LSER M BS Fig.1 Interferene of two foused vortex beams realied by means of the Mah Zehnder interferometer (M - mirror, BS -beam splitter, SPM - spiral phase mask, O - mirosope obetive). Pro. of SPIE Vol

5 dopting paraxial approximation, the foused vortex beam has been desribed analytially for an arbitrary distane behind the lens so that the defousing of interfering beams an be inluded in our analysis. If we onfine to the field distribution at the foal plane of the lens f, the omplex amplitude of the foused vortex beam with the topologial harge m an be written as where U r, u r exp[ im i] r, () u r u w w f 3/ r r exp w f I 1 r w I 1 m1 m1 f r w f, (3) kr r, (4) f f w f, (5) w (a) (b) normalied OM density y - oordinate () x - oordinate Fig. Interferene of two oaxial preisely foused vortex beams with the topologial harges m 1 =1 and m = -1 and with the ratio of amplitudes Q = /B = 1/. (a) intensity spot, (b) aimuthal omponent of the Poynting vetor and () spatial distribution of the OM density at the foal plane. Pro. of SPIE Vol

6 and u, w f and I denote a onstant amplitude, the waist radius of a Gaussian envelope at the foal plane of the lens and the modified Bessel funtion, respetively. phase offset differs for even and odd topologial harges, kf for m m' and kf for m m' 1. Out of the foal plane of the lens, the arguments of the modified Bessel funtions beome omplex and additional phase terms appear in the field omplex amplitude. (a) (b) normalied OM density y - oordinate () x - oordinate Fig. 3 Interferene of two oaxial slightly defoused vortex beams with the topologial harges m 1 = 1 (negative defousing = - 5 m) and m = -1 (positive defousing = 5 m) and with equal amplitudes Q = /B = 1. (a) intensity spot, (b) aimuthal omponent of the Poynting vetor and () spatial distribution of the OM at the foal plane. possibility to tune both the OM density and the total OM by adusting amplitudes of two preisely foused vortex beams with the topologial harges m 1 1 and m 1 is demonstrated in Fig.. Though the intensity profile of the interferene field exhibits dependene on the aimuthal angle, both the aimuthal omponent of the Poynting vetor and the OM density possess rotational symmetry. Changing the ratio of the amplitudes of the vortex beams Q, the OM an be ompletely aneled or its sign an be reversed. This behaviour is learly evident from (3) used with U U 1 U, where U1 and U are the omplex amplitudes of the vortex beams given by ()-(5). The OM of interfering vortex beams with equal amplitudes and the topologial harges m1 m m has equal magnitudes and opposite signs so that the OM of the resulting field vanishes. In the ase of foused vortex beams this is true only if the beams are aurately foused and their foal regions oinide. If the beams are slightly defoused, the total OM is nonero and furthermore, the OM density has a speifi spatial distribution. The situation obtained for two vorties with the topologial harges m 1 1 and m 1 and equal amplitudes is illustrated in Fig. 3. The OM distribution results from the defousing of Pro. of SPIE Vol

7 interfering beams. While the waist of the beam with m 1 1is shifted 5 m behind the foal plane of the lens, the waist of other beam is plaed 5m in front of it. t the foal plane of the lens, the intensity of the resulting field has a spiral shape illustrated in Fig. 3a) while the aimuthal omponent of the Poynting vetor Fig. 3b) and the OM density Fig. 3) are rotationally symmetrial. s is obvious, in the ross-setion of the interferene field there are two annular regions where the eletromagneti energy irulates in reverse diretions and the OM density has opposite signs. If the defousing shifts of the interfering vortex beams are interhanged, the diretion of the energy flow in the annular regions is reversed and the OM density hanges its sign there. This situation is shown in Fig. 4. (a) (b) normalied OM density y - oordinate () x - oordinate Fig. 4 The same as in Fig. 3 but for a reversed defousing of interfering beams. In the mixed vortex field omposed of two vorties with topologial harges of different magnitudes, an interferene of the OM takes plae. It results in spatial gradients of the OM density depending on the topologial harges, amplitudes and phases of interfering vortex beams. In Fig. 5, an interferene of two aurately foused vortex beams with the topologial harges m 1 1 and m 4 and the ratio of amplitudes Q 3/ is shown. In this ase, the aimuthal omponent of the Poynting vetor and the OM density exhibit an aimuthal dependene aused by onstrutive and destrutive interferene. Suh effets enable shaping of the OM density potentially appliable to partile manipulation by laser radiation. 6. CONCLUSION In the paper, the OM density of the mixed vortex field omposed of two or several vortex beams with different topologial harges was examined. general form of the obtained interferene law for the OM density was used to explain an effet of the OM tuning by interferene of two vortex beams. The experiment enabling realiation of Pro. of SPIE Vol

8 (a) (b) normalied OM density y - oordinate () x - oordinate Fig. 5 Interferene of two oaxial exatly foused vortex beams with the topologial harges m 1 =1 and m = 4 and with the ratio of amplitudes Q = /B = 3/. (a) intensity spot, (b) aimuthal omponent of the Poynting vetor and () spatial distribution of the OM density at the foal plane. interferene of two foused vorties was proposed and desribed mathematially. The speifi distributions of the OM density promising for trapping experiments were explored and demonstrated. CKNOWLEDGMENTS This work was supported by the Researh proets Measurement and Information in optis MSM , Center of Modern Optis LC67 and proet FT-T/59 of the Ceh Ministry of Industry and Trade. REFERENCES 1. M. S. Soskin, M. V. Vasnetsov, Singular Optis, Progress in Optis vol. 4, Elsevier, msterdam 1.. L. llen, S. M. Barnett, M.J. Padgett, Optial ngular Momentum, Institut of Physis Publishing, Bristol Ch. H. J. Shmit, K. Uhrig, J. P. Spat, J. E. Curtis, Tuning the orbital angular momentum in optial vortex beams, Opt. Exp. 14, (6). Pro. of SPIE Vol