Controlled Non-Canonical Vortices from Higher Order Fractional Screw Dislocations SATYAJIT MAJI AND MARUTHI M. BRUNDAVANAM* Department of Physics, Indian Institute of Technology Kharagpur, Kharagpur-721302, West Bengal, India. *Corresponding author:
[email protected] Received XX Month XXXX; revised XX Month, XXXX; accepted XX Month XXXX; posted XX Month XXXX (Doc. ID XXXXX); published XX Month XXXX
A pair of stable non-canonical scalar vortices of same charge is generated experimentally across the crosssection of an optical beam using computer generated hologram with higher order fractional screw dislocation. Non-canonical nature of the generated vortices is identified by the gradient of phase around each vortex and also the crossing angle between the zero contours of the real and imaginary parts of the optical field. The anisotropy of the vortices is controlled by the fractional order of the computer generated hologram. The behavior of the rate of change of phase around each individual vortex is found to be different from the earlier reports on non-canonical vortices. The observed experimental results are qualitatively explained based on the effect of non-localized phases of the generated vortices. The generated beams with nested pair of non-canonical vortices can be of importance in accelerated rotation of the beams and in optical micro-manipulation. © 2017 Optical Society of America OCIS codes:(050.4865) Optical vortices; (260.6042) Singular optics;(090.2890) Holographic optical elements.
It is now well known that optical vortex (OV) beams have intensity null due to the singularity in phase at the point of screw wavefront dislocation [1]. Around the singularity, the phase varies from zero to an integral multiple of 2π and the integral multiple is called topological charge or order of the OV beam. OV beams with integer order screw dislocations have linear phase variation with azimuthal angle as the gradients of the real and imaginary parts of the vortex fields are in quadrature at the isolated phase singularities [2]. They are considered as canonical vortices possessing circular symmetry in their field structure around the singularity [1]. Asymmetric OV beams also have been realized and studied as non-canonical optical vortices [3-6]. The dynamic trajectory of the non-canonical vortex has significant deviations from their canonical counterpart of the same topological charge [3, 5]. The anisotropic energy flow around the non-canonical vortex due to asymmetric intensity and phase morphology [6] makes noncanonical OV beams a suitable candidate for accelerated rotation of beams [7]. Non-canonical OV beams have been realized experimentally through deformation of a canonical vortex by astigmatism [2], asymmetric illumination of a Gaussian beam on a
vortex producing object [8] and perturbing higher order integer OV beam [9]. The mentioned methods are found to have less control over the morphology of the non-canonical vortices. In the present paper, we have experimentally generated a pair of stable non-canonical scalar vortices nested in a single beam using computer generated hologram (CGH) with higher order fractional screw dislocation (FSD) illuminated by a Gaussian beam with finite wavefront curvature. The advantage of employing this holographic approach is that the envisaged beam is easily isolated in the first order diffraction from the other higher modes as they are separated in different diffraction orders. It is shown here that the morphology of the generated non-canonical vortices especially the anisotropy of the phase structure around the vortices can be modified by varying the orders of the FSD. The results are compared when CGH with integer order screw dislocation is used. It is also observed that the nature of gradient of phase (GoP) variation around each individual vortex is deviated from that of the non-canonical vortices mentioned in the literature [6]. The observed deviation of GoP is explained by employing a theoretical model considering the non-localized effect of vortex phases around the generated pair of vortices. The optical field for an integer charge (m) vortex due to the diffraction of a Gaussian beam from an integer order screw dislocation at its waist plane is given as [10, 11]
π
um ( ρ , φ , z ) =
2
(−i )|m|+1 exp ( imϕ ) exp(
ik 2 zR ρ ) × z − iz R 2z
A exp(− A) I |m|−1 ( A) − I |m|+1 ( A) . (1) 2 2 Here Iν denotes the modified Bessel function, z R is the Rayleigh 2
kz R ρ . length and A = z 4(1 − i z R ) z The optical field due to an incident Gaussian beam with its waist plane on the CGH containing a screw dislocation of fractional order α with a single phase discontinuity line can be expressed in terms of superposition of integer charge vortex fields as [12],
uα ( ρ , φ , z ) =
exp[i ( z + πα )]sin(πα )
π
um ( ρ , φ , z ) . m =−∞ (α − m) ∞
(2)
If the CGH is placed away from the waist plane of the incident Gaussian beam, i.e. the incident beam has a finite radius of curvature ( Rc ) on the CGH, then in Eq. (1), ρ , z and the overall
field u m gets modified m by a facctor f = 1 + z Rc [11].
um ( ρ , φ , z ) =
1 ρ z um , φ , . f f f
(3)
Replacing u m in Eq. (2) using g Eq. (3), the opttical field of an OV O beeam generated by diffraction of a Gaussian beam with finiite cu urvature from a fractional f order dislocation can be written as
uα ( ρ , φ , z ) =
exp[i ( z + πα )]sin( ) πα )
π
um ( ρ , φ , z ) . (4) m =−∞ (α − m ) ∞
From Eq. (4), it i can be conclud ded that in the fiield of the OV wiith fraactional topolog gical charge α , the t dominant co ontribution is fro om OV V fields of neareest integers. Thee contributions from f other integger ch harge OV fields decrease d rapidly y as (α − m ) inccreases. Fig. 1(b) schematically showss the experimen ntal setup used for f th he present stud dies. From a He-Ne H laser, a linearly l polarized Gaaussian beam off light with a finitte curvature is allowed a to incideent on n a spatial light modulator (SLM M) from Holoey ye (LC-R 1080) on o wh hich CGHs with integer and fracctional screw disslocations (show wn in Fig 1 (a)) are displayed. d The CGHs are prepareed numerically by b caalculating the intterference patteern of a vortex beam b with a plan ne waave and withou ut any loss of geeneralization, th he direction of the th sin ngle discontinuiity is taken along positive x axiss (horizontal). The firrst order diffraccted beam is thee envisaged beaam containing the th paair of vortices. Itt is isolated, colllimated and theen interfered wiith th he non-diffracted d and expanded central Gaussian n beam acting ass a reeference beam. The T resultant in nterference patteerns are recorded byy a charge couplled device (CCD D) and are used to t reconstruct the th inttensity and ph hase structure of the generatted beams usin ng Fo ourier fringe anaalysis [13].
respecttively. The high her order fractio onal charges 1.6 6 and 1.8 are chosen n as only the ch harges above h half integer resu ult in a new n the beam crosss-section at th he far field in vortex (NV) nested in addition to the existingg initial vortex ((IV) [14, 15]. Th his is evident he number of cro ossing points of the zero contou urs of real and from th imaginaary part of the reconstructed ffield within the beam crosssection n. Solid (magentaa) lines and dottted (green) liness on top of the intensitty profiles rep present the zeero contours of real and imaginaary parts respeectively of the ggenerated opticaal fields. The corresp ponding simulatted fields usingg Eq. (4) (wheree the twenty most siignificant termss are taken out of the infinite ssum) and the corresp ponding real and d imaginary zerro contours are sshown in Fig. 1(i), (j) and (k) which ssupport the expeerimental patterrns. It iss clearly seen fro om Fig. 1(g), (h)), (j) and (k) thatt the position of the N NV is angularly d deviated from th he position of thee partial edge dislocattion of the CGH H. The angular shift is quantifi fied from the 4 (measured position ns of the two vvortices and is ffound to be π/4 clockw wise as origin is in the top left ccorner as depiccted in Fig.1). This ob bserved angularr shift in positiion of the NV iis due to the dynam mic Gouy phase sshift experienceed by the opticaal beam with phase for a charrge ‘m’ vortex propaggation [16]. The dynamic Gouy p field is (m+1) tan-1(z/zzR), where zR is the Rayleigh raange and z is the disttance from waisst position [16]. It is clearly undeerstood from Eq. (4)) that for fractiional charges in n between 1.0 and 2.0, the dominaant contribution ns are from O OV fields of neaarest integer chargess i.e. 1.0 and 2.0. In our case, thee SLM is kept att z = zR, which impliess that the Gouy phase for m = 1 1.0 and 2.0 is π π/2 and 3π/4 respecttively for z = zR tto infinity after the SLM. As thee Gouy phase differen nce between thee charges is π/4 4, the position off the vortex is at an an ngle π/4 in clockkwise direction ffrom the edge diislocation. To understand th he nature of tthe generated vortices, the ng angles betweeen the real and imaginary zero contours are crossin estimatted by the anglees between the ttangents to the zzero contours at the vvortex positions. The crossing aangles for the charges 1.0, 1.6 and 1.8 8 are found to bee ηi (i = 0, 1, 2) ≈ 9 900, 1120, 1170 at IV and ξi (i = 1, 0 0 2) ≈ 10 05 , 95 at NV rrespectively. Th he deviation of the crossing angles from the orthoggonality denotee that the generrated vortices due to the FSDs are n non-canonical in n nature. In ordeer to confirm our find dings further, th he deviation anggle from the ortthogonality is investiggated from the gradient of the real and imagin nary parts of the field d. The deviation n angle (ψ ) fro om orthogonalityy is related to the anissotropy of the vo ortex and can bee represented in n terms of the field vaariable, u ( = uα o or um ) as [4],
u− 2 − u + ψ = cos 2 u +u + − −1
Fig g. 1. (Color onlin ne) Experimenta al setup for the generation g of beaam wiith a pair of non-canonical vortice es. The CGHs con ntaining integer an nd fraactional screw disslocation displayed on the SLM arre shown in (a). The T recorded interfero ogram, reconstru ucted intensity and a phase for the t ch harge 1.6 OV field d are shown in (c)), (d) and (e) resp pectively. On top of the experimental ((f), (g), (h)) and d simulation ((i)), (j), (k)) intensity prrofile for charge 1.0 ((f), (i)), 1.6 6 ((g), (j)) and 1.8 1 ((h), (k)), zeero co ontours of real an nd imaginary pa arts of the opticaal fields are show wn wiith solid (magentta) lines and dotte ed (green) lines respectively. r
One recorded d interferogram corresponding to charge 1.6 OV O fieeld and the cross-sectional intensity and d phase proffile reeconstructed from m it is shown in Fig. 1 (c), (d) an nd (e) respectiveely. Figgs. 1(f), (g) and d (h) correspon nd to the intensity profiles of the th beeams generated d from CGHs with charge 1.0, 1 1.6 and 1.8 1
, 2
2
wh here, u± = ∂± u( x, y) and ∂ ± =
(5)
1 ∂ ∂ ± . 2 ∂x ∂y
ngularity. The Here alll the values aree calculated at tthe points of sin values oof ψ using Eq. ((5) are found to be 240 and 250 at the IV and NV of ccharge 1.6 OV fieeld respectively and they are fou und to be 310 and 160 at the IV and N NV of charge 1.8 8 OV field respecctively. These values cconfirm the non n-canonical natu ure of the vorticees. Thee values of devviation angles also represent th he degree of anisotrropy of the vortiices. The more tthe deviation angle, the more is the an nisotropy. As th he value of the an ngle of crossing iis found to be differen nt for both IV an nd NV of chargee 1.6 and 1.8, it is concluded
th hat the vortices are of differen nt non-canonicaal strength. So, by b co ontrolling the fraactional charge the t non-canoniccal strength of the th initial vortex can be b controlled. The T off-axial vorttices generated by b peerturbing any higher order inteeger charge vorrtex are also no oncaanonical but aree equivalent difffering only in th he translational or ro otational position n [9]. But in this case of a beam m evolving from ma higgher order fracctional screw diislocation, the point p vortices are a no on-equivalent ass their morpholo ogies can vary grreatly. To investigatee the anisotropy more quantitatiively, the points at a constant distance around eaach vortex are considered. Th he onstant distance around the poin nt singularity in the OV beam wiith co ch harge 1.0 is cho osen where the intensity falls off to 1/e2 of the th m maximum intensiity and is found to be 230±5 μm m. Same distancces arre considered around a each vortex v due to the t higher ord der fraactional chargess. The circles sh hown in Fig. 2((a) around the IV (d dashed line) and d NV (solid line)) of the OV beam m with charge 1.6 1 reepresent locus of o the points un nder consideratiion. The intensiity alo ong each circlee is observed to o be non-uniforrm and followss a m modified elliptic pattern as sho own in Fig. 2(b b). It can also be b ob bserved that thee major axes off the modified ellipses e are nearrly orrthogonal to eaach other imply ying that the orrientations of the th an nisotropy in thee intensity of the t vortices aree orthogonal. Th he orrthogonality of the anisotropy y makes the tw wo non-canoniccal vo ortices stable witthout changing their t trajectory in i the far field [4 4].
Fig g.2. (Color online e) (a) The circles (of radius 230 μm m) centered on the t po oint singularities on top of the in ntensity of the ch harge 1.6 OV beaam (th hicknesses have been b exaggerated d for visual claritty): Circle with th hin daashed (red) line around IV and thick th solid (blue) line around NV V is sh hown. Along the circles, c (b) intensity variation and (c) phase variation wiith azimuthal ang gle are shown: ciircles (red) are daata for IV; trianglles (b blue) are for NV an nd in (c) black lin ne shows phase of 1.0 OV.
The phase variation along th he circles is show wn in Fig. 2(c) for f th he IV and NV by b open circles and open trian ngles data poin nts reespectively and compared c to thee phase variation n around the OV V of un nit charge show wn by the line. The T linear ramp p is reminiscent of th he uniform linearr phase variation n in case of cano onical vortices an nd th he superposed oscillatory variation stands fo or acceleration of ph hase around these anisotropic vortices. To qualitativeely elaborate thee morphology, the t GoP along the th cirrcles in Fig. 2(aa) is investigateed. The GoP alo ong the azimuth hal direction θ around d a single non-caanonical OV is giiven by [6]
dϕ 1 = . 2 2 dθ a cos θ + b sin θ + c sin θ cos θ
(6)
Heere a, b and c are a constants ch haracterizing th he anisotropy off a sin ngle non-canoniical vortex and iss given by [6], a = 1 + q2 2 ( q1q2 − q3 ) , b = q12 + q3 2 ( q1q2 − q3 ) and
(
)
c = ( q1 + q2 q3 ) ( q1q2 − q3 ) .
(
)
Here q1 = I y I x , q2 = Rx I x and q3 = Ry I x wherre Rx and Ry are thee gradient of the real part of thee field along x an nd y direction I and I and x y part of the field d. y are that of the imaginary Froom the reconstru ucted field, GoP is estimated along the circles around d the IV and N NV shown in Fiig. 2(a). The ob bserved GoP around d the IV (open ciircles) and NV (o open triangles) of charge 1.6 with azzimuthal angle is shown as p polar plot in F Fig. 3(a). The calculatted values of a, b, c are -8.0, -4.0 0, 5.6 respectiveely for IV and 1.4, 2.7 7, -2.3 respectiveely for NV. The n nature of GoP iss observed to be deviiated from the p previous studiess [6] for a single anisotropic vortex which follows E Eq. (6) and sho own as dotted liine and solid line in Fig. 3(a). To eexplain this deeviation in GoP,, a model is propossed here consid dering the effeect of non-locaalized vortex phases around the two o generated non--canonical vorticces.
Fig.3. (C Color online) Gra radient of Phase around the vortiices of charge 1.6; Exp perimental data p points are shown n for IV by (red) ccircles and for NV by ((blue) triangles. (a) GoP around IV (dotted (red)) line) and NV (solid (b blue) line) calcullated using Eq. ((6). The variation ns as given by our model are shown in n (b). To project tthe contributionss of the vortex ns are shown in n (c) and (d) ass a linear plot separattely, the variation where tthe black line sho ows the plot of E Eq. (7). The contrribution of the primaryy vortex is sho own in star poiints and contrib bution of the secondaary vortex is sho own by square po oints. For (c) IV iss primary, NV is secon ndary and for (d) NV is primary, IV V is secondary.
As tthe vortex phasse is non-localized, when two n non-canonical vorticess exist simultan neously in a hostt beam with a seeparation, the phase aat any point acrross the field can n be estimated aas the sum of the phaases of the indivvidual vortices i.ee. ϕ = ϕ1 + ϕ 2 . Here ϕ1 and
ϕ 2 aree the individual phases of IV annd NV respectiveely when the two voortices are indep pendent. Two ccoordinates are chosen with their origins coincid ding with thee two point singularities corresp ponding to IV and NV (see F Fig. 2(a)). If th he two noncanoniccal vortices arre independen nt, the GoP aaround each individu ual vortex follow ws Eq. (6) wherre the phase arou und only one of the vvortices (either ϕ1 or ϕ 2 ) is conssidered and GoP P is estimated with reespect to the azim muthal angle around it ( θ for ϕ1 and θ ′ for
ϕ 2 ). Inn-order to estimaate the gradientt of the total phaase ( ϕ ) with respectt to one (primarry) vortex, the gradient of phasee of the other (second dary) vortex aro ound the primarry vortex needs tto be known. Thiis is achieved d by a coord dinate transforrmation and ′ θ expresssing in termss of θ . Considerring IV as the prrimary vortex
an nd NV as the seccondary vortex, the GoP of the total phase alon ng th he circle around IV I is calculated to t be, d ϕ dϕ1 dϕ 2 dϕ1 dϕ 2 dθ ′ = + = + . dθ dθ dθ d θ d θ ′ dθ
=
1 a1 cos θ + b1 sin θ + c1 siin θ .cos θ 2
2
+
1 − d coss(θ − θo )
(7)
a2 ( cos θ − d c ) + b2 ( sin θ − d s ) + c2 ( sin θ − d s )( cos θ − d c ) 2
2
Th he constants d, dc and ds are defined d as
.
d = r0 r ,
d c = d cos θ 0 = (r0 r ) cos θ 0 an nd d s = d sin θ 0 = (r0 r ) sin θ 0 wh here r is the rad dius of the circlle around the primary vortex, r0 an nd θ 0 are respeectively the disttance and angullar position of the th seecondary vortexx with respecct to the prim mary vortex. Th he m measured value of o r0 is 950±5 μm μ that gives thee value of d as 4.1 4 an nd the value of θ 0 is π/4. The GoP of the totaal phase along the th cirrcle around NV V can also be calculated consid dering NV as the th prrimary and IV as a the secondary y vortex. In this case the value of
θ 0′ that gives thee angular positiion of IV with respect to NV, is fou und to be 5π/4 4. The plot of Eq. (7) using expeerimental valuess is sh hown in Fig. 3(b b) which qualittatively follows the experimenttal daata. To project the t contributionss of the two terrms in Eq. (7), the t vaariation of GoP are shown as a linear plot in Fig.. 3(c) taking IV as prrimary vortex an nd in Fig. 3(d) tak king NV as prim mary vortex. The GoP variiation due to th he first term off Eq. (7) which is identical to Eq. (6) correspondss to the case where wh the primaary vo ortex is treated d as independeent non-canoniccal vortex and is sh hown as star poiints in Fig. 3(c) and 3(d). The modification m to the th Go oP around the primary p vortex due d to the preseence of secondaary vo ortex which is giiven by the seco ond term in Eq. (7) is representted ass square points. The position off the minimum in the plot show wn byy squares in Fig. 3(c) and 3(d) corresponds c to the t value of θo an nd θoˊ respectively. The T second term m in Eq. (7) is also found to be b sw witching betweeen positive and negative valuess with increase in th he azimuthal angle. This changee in the sign off the modificatio on terrm is attributeed to reversal in the sense of the azimuth hal direction with reespect to the secondary s vorttex when GoP is esstimated around d the primary vo ortex. The modeel can be similarrly ad dopted to explaiin the results obtained o with otther higher ord der FSSD for example charge c 1.8.
Fig g. 4. (Color onlin ne) Experimental variation of optiical current (a) an nd vo orticity (b) aroun nd the IV (shown n in circle (red)) and NV (shown in triiangle (blue)) of 1.6 1 OV field.
Pottential applicatio ons of these beams in trappingg is expected due to multiple trappiing centers and the anisotropicc variation of which results in anisotropic mo omentum distrribution. This GoP w could b be expressed byy the distribution n of vorticity wh hich is curl of the lineear momentum aand a measure o of the momentu um transfer to matter [17], though th he exact amoun nt of momentum m transferred will dep epend on the traansfer mechaniism. The variatiion of optical currentt (OC) and vorticcity [17] around d the IV and NV o of 1.6 OV field is show wn in Fig. 4(a) and (b) respecctively where th he scales are normallized. The non n-uniform oscilllatory nature of vorticity indicatees that an asym mmetric micron n size low refrractive index absorbiing particle w will experience accelerated rotation when trapped d at the position ns of minimum in ntensity of thesee beams. In ssummary, we haave generated a pair of non-can nonical scalar vorticess using a CGH w with higher orderr fractional screw w dislocation illumin nated by a Gau ussian beam w with finite curv rvature. It is demon nstrated that thee anisotropy off the non-canon nical vortices can bee controlled viaa fractional chaarge of the CG GH. It is also observeed that the GoP P around each vortex is differeent from the behavioor of the non-canonical vortices observed p previously. A model is proposed to o explain the ob bserved anomally in GoP by consideering the effectt of non-localizeed phases of th he two noncanoniccal vortices. T The generated OV beams w with multiple anisotrropic vortices caan be used for aaccelerating anggular rotation and caan also have aadvantage in en nabling more control over trapped d micro-particlees in optical man nipulation appliccations. Fun nding. IIT Khaaragpur (IIT/SR RIC/PHY/VBC/2 2014-15/43); Departm tment of Scien nce and Techno ology, India (D DST/INSPIRE Facultyy Award/2012/P PH-62). Ack knowledgmentt. S. Maji would d like to thank A A. Roy for his help in experiments. RENCES REFER 1. M. R. Dennis, K. O’H Holleran, and M M. J. Padgett, Pro og. Opt. 53, 293 ((2009). 2. G. M Molina-Terriza, J. Recolons, J. P. Torres, L. Torn ner, and E. M. Wrigght, Phys. Rev. Leett. 87, 023902 (2001). 3. G. Molina-Terriza,, E. M. Wright, aand L. Torner, O Opt. Lett. 26, 163 ((2001). 4. F. SS. Roux, J. Opt. So oc. Am. B, 21, 66 64 (2004). 5. R. P P. Singh, S. R. Ch howdhury, Opt. Commun. 215, 231 (2003). 6. I. FFreund and V. Frreilikher, J. Opt. Soc. Am. A 14, 1 1902 (1997). 7. C. Schulze, F. S. R Roux, A. Dudley,, R. Rop, M. Dup parré, and A. 91, 043821 (20 015). Forbees, Phys Rev. A 9 8. V. V V. Kotlyar, A. A. Kovalev, and A.. P. Porfirev, Optt. Lett. 42, 139 ((2017). 9. I. FFreund, Opt. Com mmun. 159, 99 (1999). 10. V V. V. Kotlyar, A. A. Almazov, S. N. Khonina, V. A. Soifer, H. Elfstrrom, and J. Turu unen, J. Opt. Soc. Am. A 22, 849 (2005). 11. A A. Y. Bekshaev and A. I. Karaamoch, Opt. Commun. 281, 1366 6 (2008). 12. M M. V. Berry, J. Op pt. A: Pure Appll. Opt. 6, 259 (20 004). 13. M M. Takeda, H. Inaa, and S. Kobayaashi, J. Opt. Soc. Am. 72, 156 (1982 2). 14. J. Leach, E. Yao, aand M. J. Padgettt, New J. Phys. 6,, 71 (2004). 15. W W. M. Lee, X-C. Y Yuan, and K. Dh holakia, Opt. Co ommun. 239, 129 ((2004). 16. S.. M. Baumann, D D. M. Kalb, L. H.. MacMillan, and d E. J. Galvez, Opt. E Express 17, 981 18 (2009). 17. M M.V. Berry, J. Optt. A: Pure Appl. O Opt. 11, 094001 1 (2009).
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