2 The Fresnel Zone Plate: Theoretical Background

Transcription

1 1 Introduction One of te most interesting topics of study in te field of optics as become te optical vortex and te existence of pase singularities in propagating beams of ligt. First predicted in 1992 by Allen et al., an optical vortex is a beam of ligt were te pase canges wit te azimutal angle, te pase cange being 2πν, were ν is te strengt of te vortex [1]. Because of tis pase cange, an optical vortex as some very unique properties. One of te caracteristic properties of suc a vortex is te existence of a pase singularity in te center. Matematically, te azimutal angle, and ence te pase of te ligt wave, becomes undefined at te center of te beam. In order to retain consistency in pysics, te amplitude of te wave at te center of te beam must (and does) go to zero to prevent te formation of a pysical singularity, even toug, colloquially, te term singularity is used. Optical vortices are not just gimmicks confined to te laser labs, owever. Teir unique properties make tem extremely useful in a variety of applications. Optical tweezers/spanners, for example, use vortices to trap particles effectively and noninvasively and ave become extremely useful in biological applications for te manipulation of cells [2]. Optical vortices are also expected to become extremely useful in quantum computing [3], in searcing for exoplanets [4], and in producing solar coronagraps [5]. Traditionally, optical vortices ave been created using spatial ligt modulators, mode converters, pase plates or forked diffraction gratings [6, 7]. Tese metods ave been used extensively to create and analyze optical vortices, mostly of integer strengt and, to some lesser extent, of fractional strengt. Eac metod as its own advantages but also as drawbacks, suc as requiring extremely precise alignment, being expensive, or being relatively inefficient in te case of te forked grating. A simple metod of creating optical vortices is troug te use of spiral zone plates, wic are based on te same principles as te Fresnel zone plate. Tese metods ave been discussed and used for creating and studying vortices of integer order [8, 9]. In tis work, we will explore te possibility of extending tis idea to make zone plates for vortices of fractional order. A first step will be to reproduce some of te 1

2 known results on vortices of integer order. Using te formulae for spiral zones [8, 9], wic are temselves based on te properties of te Fresnel integral, we were able to grap te spirals and poto-reduce tem to make te diffraction gratings. It was ten possible to verify experimentally tat first order vortices could be produced from tese. We were able to modify te formulae for te spiral zones for integer vortices (by including an appropriate discontinuity) to produce diffraction patterns for fractional vortices. Experimentally, tese do seem to give fractional vortices because, as in te integer case, caracteristic forked interference patterns could be obtained using a Mac-Zender interferometer. Furtermore, measured intensity profiles were compared to te profiles produced by evaluating te corresponding Fresnel integral using Matematica. 2 Te Fresnel Zone Plate: Teoretical Background Introduced by Augustin-Jean Fresnel in te 1800s, te Fresnel zone plate is a pattern tat diffracts ligt in suc a way as to focus ligt. It consists of an alternating series of dark and ligt rings designed so tat only ligt tat will constructively interfere at te focal point will be allowed to pass troug te plate. Tis means tat te ligt coming from te inner and outer edges of any zone will be no more tan π out of pase wen tey reac te focal point. Ligt tat is more tan π out of pase will be blocked by te dark rings, preventing destructive interference. In order for te Fresnel zone plate to acieve constructive interference at te focal point, te zones tat compose it must ave very specific radii so tat te pat lengt from te center of eac ligt zone is an integer multiple of te wavelengt. In addition, eac zone must ave a very specific radial widt in order to ensure tat tere is no more tan a alf-wavelengt difference in pat lengt between te edges of te zone. Tis means tat from te first zone boundary, wic we sall call r 0, to te second zone boundary, r 1, te difference of te pat lengts to te focal point sould be λ/2, were λ is te wavelengt of te ligt. If we consider r 0 to be at te very center of te plate, ten te pat lengt to te focal point is f. We can terefore find a general equation for te radius of te zone boundaries so tat d n, te distance from te n t boundary to te focal point is λ/2 less tan te distance d n+1, te distance from te 2

3 (n + 1) t boundary to te focal point. i.e., d n = f + nλ 2 = rn 2 + f 2 r n = fnλ + n2 λ 2 4 (1) Wile (1) allows us to create a Fresnel zone plate, wat is more important for us is an explanation of te teory beind te plate, especially since we use te teory to modify te Fresnel zone plate into a spiral zone plate. We must first begin wit te wave teory of ligt in wic te complex amplitude of any beam of ligt can be represented by Φ(d, t) = (A/d) exp(i(kd ωt)), were Φ is te complex amplitude of te beam as a function of d, te distance propagated, and t, te time. A is a constant related to te intensity of te ligt, k is 2π/λ, te wavenumber, and ω is te circular frequency. In order to simplify tis considerably, we can assume tat we will be using a steady beam wic makes te time-dependence irrelevant, so we can use te time-independent form, Φ(d) = A d exp (ikd) (2) We can now take tis equation and integrate it over te openings in te plate, and ten sum over all te openings, to obtain te amplitude of te wave reacing te screen as a function of te radius on te screen, Φ(r) = rings R2 R 1 dx dy ( A d From ere, we need to take (3) and rewrite it in terms of r, d = (x x ) 2 + (y y ) 2 + (z z ) 2 ) exp (ikd) (3) = 2 + (x x ) 2 + (y y ) [ (x x ) 2 + (y y ) 2] [ r 2 + r 2 2rr cos(θ θ ) ] (4) 2 were is te distance between te grating and te screen. Eventually, we will take to be te focal distance f. 3

4 Now tat we ave written and expanded d in terms of r, te radius on te screen, r, te radius on te grating, we can go aead wit te integration, Φ = rings A exp (ik) R2 R 1 dx dy exp [ ik [ r 2 + r 2 2rr cos(θ θ ) ]] (5) 2 Tis formula is a good approximation to (3) assuming is muc bigger tan r, r. At tis point, we must rewrite dx dy in terms of r and azimutal angle θ. Tis can be done by converting te area element of dx dy from cartesian coordinates to polar coordinates giving us, Based on (6), te formula for te amplitude becomes: Φ = rings A exp (ik) R2 R 1 dr r dx dy = r dθ dr (6) 2π 0 dθ exp [ ik [ r 2 + r 2 2rr cos(θ θ ) ]] (7) 2 Because we know tat te pattern produced will be symmetrical around te z axis, we can replace θ θ by te single angle φ. Correspondingly, dθ dφ, wic will allow us to simplify te integral even furter, Φ = A exp (ik) R2 2π [ ik [ dr r dφ exp r 2 + r 2 2rr cos φ ]] rings R = A exp (ik) R2 2π [ ] [ ] [ ] ik ik dr r dφ exp rings R r2 exp 2 r 2 exp ikrr cos φ Te φ integration can be done, producing a Bessel function, Terefore, Φ = rings 2π 0 2π A exp (ik) dθ exp [ ix cos θ] = 2πJ 0 ( x ) 0 [ ] ( ) dφ exp ikrr cos φ krr = 2πJ 0 R2 R 1 [ ] [ ] ( ) ik ik krr dr r exp 2 r2 exp 2 r 2 2πJ 0 (8) We can use (8) in combination wit (1) to model te intensity profiles for any zone plate wit te number of zones set by te number of summations done in (8), see figure 1. 4

5 Figure 1: Te cross sectional intensity pattern for a zone plate of 30 rings. Note: Units are arbitrary. It is also interesting to look at te intensity along te optical axis as a function of d, te distance from te zone plate to te screen. Wat we see is tat, as predicted, tere are secondary focal points at f/3, f/5, f/7, etc. See figure Figure 2: Te on-axis intensity produced on te screen d meters away from a zone plate wit a primary focal point of one meter. Note: Intensity is in arbitrary units. 3 Te Spiral Zone Plate 3.1 Forming Integer Order Vortices We can now modify te Fresnel zone plate to obtain a spiral zone plate; tis spiral plate sould give te beam an azimutal pase dependence wic sould create an optical vortex. In order to do tis, we need a grating were te pat lengt of ligt from te grating to te screen must vary wit te azimutal angle, wic means 5

6 tat we will need a spiral. Specifically, we will use an Arcimedian spiral were te radius, r, is directly dependent on azimutal and angle θ, giving us te basic equation: r = nθ. Tis, owever, cannot be used directly since te pattern on diffraction grating must ave te specific widt of one Fresnel zone. To do tis we must go back to (7). Te pase of te beam is given by te value of te exponential part after integration. Let us rewrite some of te variables in (7) in terms of λf as, r = v λf r = u λf Tis allows us to now rewrite a simplified form of (7) in terms of te new variables (also assume tat = f). Note: terms suc as [A exp(ik))/] and exp [(ikr 2 )/(2)] ave been dropped because tey can be taken out of te integration and contribute a constant overall pase. Ψ = dθ dr r exp ( iπu 2 2iπ u v ) (9) In order to get a vortex, we must modify tis so tat tere is a factor of exp(iθν) in te final result. Consider te term exp(iθ ν). Tis can be rewritten as exp (iν(θ θ)) exp(iνθ). Te last part of tis term as only a θ-dependence and can be taken outside te integral, giving us a θ-dependent amplitude on te screen, wile θ θ becomes te variable of te angular integration. Tis means tat exp (iπu 2 ) must ave a factor of exp(iθ ν) somewere in it, so tat we can take u 2 = νθ /π + w 2, were w is now te radial variable. Terefore, we can rewrite (9) as: Ψ = dθ dw w exp(iνθ ) exp(iπw 2 ) exp( 2iπ u v) (10) In order to find w 2, we must consider te restrictions on u 2. First, we must consider u as lying between u 1 and u 2, te inner and outer radii of eac Fresnel zone. In order to prevent destructive interference at te focal point, eac zone must ave a widt wic gives a pat difference of no more tan a alf-wavelgnt. Tis means tat u 2 2 u 2 1 = 1, so tat tere is exactly π pase sift on te screen as one goes radially across te zone, yielding te maximum possible intensity for te vortex formed. 6

7 At tis point, we will define a variable n tat will become advantageous wen dealing wit fractional zone plates. Because fractional vortices ave a discontinuous azimutal pase cange, te spiral zone plates used to make tem will also ave a discontinuity. Tis means tat eac ring will be continuous from 0 to 2π, but tere will be a jump to te next ring. Terefore, n will be defined as te ring number, te value of wic will tell us wic ring is being graped. To create a spiral zone plate wit 20 rings, for example, we will grap u 1 and u 2 for all integer values of n from 0 to 19. Tis definition of n places anoter restriction on u 2. For an integer order zone plate, u 2 (θ + 2π, n) = u 2 (θ, n + 1). Tis ensures tat for an integer order vortex te spiral formed is continuous. Terefore: Using u 2 = (νθ)/π + w 2, we find, u 2 1(θ + 2π, n) = u 2 1(θ, n + 1) u 2 2(θ + 2π, n) = u 2 2(θ, n + 1) w 2 1(n + 1) w 2 1(n) = 2ν w 2 2(n + 1) w 2 2(n) = 2ν wit te solution w 2 1(n) = 2νn + c 1, w 2 2(n) = 2νn + c 2 were c 1 and c 2 are constants. Tus, u 2 1 = νθ π + 2νn + c 1 u 2 2 = νθ π + 2νn + c 2 Applying te restriction u 2 2 u 2 1 = 1, gives c 2 c 1 = 0 wit te solution c 2 = c 0, c 1 = c 0 1 for some constant c 0. For te sake of simplicity, we will set c 0 to 0. Finally, tis leads to, u 2 1 = νθ π + 2νn 1 u 2 2 = νθ π + 2νn (11) We can now use tese equations to create spiral zone plates for any integer order vortex. Tere is one final adjustment we can make, owever. For vortices iger 7

8 tan order 1, te distance between u 2 (n) and u 1 (n + 1), te region tat will be made dark, will not correspond to a pase sift of π, but will instead correspond to a pase sift of 3π for a 2 nd order vortex, 5π for a 3 rd order vortex, etc. Tis means tat we can still introduce extra spirals tat will constructively interfere wit te first spiral, leading to a iger overall intensity on te screen. To do tis we will introduce l, te spiral number, wic ranges from 0 to ν 1. Te rewritten equations now look like, u 2 1 = νθ π u 2 2 = νθ π + 2νn 1 2l + 2νn 2l We can also rewrite tese equations back in terms of r in a format tat can be easily graped, r 1 = r 2 = ( νθ λf π ( νθ λf π ) + 2νn 1 2l ) + 2νn 2l Using Matematica and Adobe Potosop we were able to grap tese functions and properly sade tem in to produce poto-reducible negatives, see figures 3 and 4. (12) Equations (12) can be related to te ones used by Heckenberg, et al. [8], producing essentially te same spiral patterns. Te advantage of our equations, owever, is tat Figure 3: Spiral pattern and zone plate negative for vortex order 1 wit focal lengt one meter and n from zero to ten. All units are in meters. 8

9 Figure 4: Zone plate negatives for vortex orders 2 and 3 wit l = 0, 1 and l = 0, 1, 2, respectively. In bot cases n goes from zero to ten. tey can be easily adapted to produce a fractional zone plate, as discussed in te next section. 3.2 Forming Fractional Order Vortices In order to make fractional order vortices from (13), we must first make a few modifications to te equations. First, we will remove te 2k term for simplicity, since te value of ν is no longer integral. We must also make modifications to te 2νn term.tis requirement arises from te jump tat is created by te n term as one goes from θ = 0 of one cycle to θ = 0 of te next cycle. u 2 (0, n + 1) u 2 (0, n) = 2ν Tis means tat te pase of te ligt coming from eac at te focal point will be exp(ik2πν). Tis is fine if ν is of integer value. If ν is not of integer value, owever, tis is a problem since te interference will not be completely constructive. To fix tis, we can simply remove ν from te 2νn term, giving us a final solution, ( ) νθ r 1 = λf π + 2n 1 ( ) νθ r 2 = λf π + 2n (13) 9

10 Using Matematica and Potosop, we can once again produce negatives of te gratings, see figures 5 and 6. Notice tat we no longer get a continuous spiral; te zones for fractional angular rotation must ave a discontinuity. 3.3 Simulating te Spiral Plates Using te same metods tat we used for te Fresnel zone plates, wit a few adjustments, we can simulate te patterns produced by te spiral zone plates. In order to facilitate tis, we must go back to (7) and derive a simpler expression for te am Figure 5: Spiral pattern and zone plate negative for vortex order 1.5. Focal lengt is one meter and n goes from zero to ten. All units are in meters. Figure 6: Zone plate negatives for vortex orders 1.3 and 1.7, respectively. In bot cases n goes from zero to ten. 10

11 plitude. Notice tat, because of te angular dependency in (12), (13), we no longer ave angular symmetry. Because of asymmetry, we cannot simply cange θ θ into φ nor can we cange dθ into dφ. Instead, we will rewrite some of te variables in order to reduce te expression. A exp (ik) 2π R2 [ ik [ Φ = dθ dr r exp r 2 + r 2 (2xr cos θ + 2yr sin θ ) ]] R 1 2 rings = A exp [ik( + r2 /2)] 0 rings 2π 0 dθ R2 R 1 dr r H(x, y, r, θ ) (14) were r 2 = x 2 + y 2 and [ ikr H(x, y, r, θ 2 ) = exp 2 ik ] (xr cos θ + yr sin θ ) (15) Here x and y are te cartesian coordinates on te screen and R 1 and R 2 are te inner and outer radii of eac ring as a function of θ. We notice tat equation (14) is muc more complicated tan (8), since tere is no way for us to simplify any part of te integral into a Bessel function or someting similar. Because tere is no analytic solution, eac integration as to be done numerically and takes a tremendous amount of computer time. In order to speed up te process we can use an approximation for te integration over r. We carry out a partial integration over r, using te factor of r and te first term in te exponential in H, to get R2 R 1 dr r H(x, y, r, θ ) = ik [ ] H(x, y, R 2, θ ) H(x, y, R 1, θ ) R2 + dr (x cos θ + y sin θ )H(x, y, r, θ ) R 1 (16) Because of te (x cos θ +y sin θ ) factor in te second term, wen x and y are small, te entire second term becomes small. Tus, te first term serves as a good approximation of te amplitude near te center of te screen. Terefore, for most calculations, we can drop te second term and use only te first, wic as no integration over r, making tis calculation muc simpler. (We ave cecked by te full numerical integration of (14) in some sample cases tat te first term in (16) is a good approximation for te first few diffraction maxima near te center.) 11

12 Te equation for te amplitude tat we will use to simulate te fraction spiral zone plates can now be written as: Φ A exp [ik( + r2 /2)] rings 2π 0 dθ [H(x, y, R 2, θ ) H(x, y, R 1, θ )] (17) 4 Te Experimental Setup In order to test tese spiral zone plates we used a Helium Neon laser wit a wavelengt of nm. Tis beam was first collimated and ten passed troug a series of attenuators before passing troug te grating. A convex lens followed by a concave lens was used to make te beam sligtly divergent so tat te vortex would form furter away tan te one meter focal lengt of te plate. At tis extended focal point, we placed a igly divergent lens wic prevents furter interference of te vortex and expands te image. By using tis metod of expanding te beam, as opposed to te conventional tecnique of using a single convex lens at te focal point, we can prevent te vortex from going troug any foci. For fractional vortices, wic are unstable, going troug foci caused te vortices to completely come apart. In addition to tis, we built a Mac-Zender interferometer around te setup. Figure 7: Te experimental setup. Note: Not to scale. 12

13 Tis consisted of a beam splitter tat splits off alf of te beam before te grating. Tis new beam is ten aligned so tat it is almost parallel to te vortex beam and te two beams are ten recombined to form te interference pattern. Te zone plates temselves were produced using a combination of Matematica and Adobe Potosop and were printed on a laser printer at 1200dpi. Tese printouts were ten potograped using a Canon film SLR camera and te negatives were developed. Tese negatives were ten used as our diffractive elements. 5 Results Using te previously described setup and te previously explained equations, we created a series of spiral zone plates wit focal lengts of one meter and varying values for te vortex strengt, ν. Using te CCD camera we were able to take intensity measurements along certain axes for eac vortex and compare tem to te corresponding simulated intensities. We were also able to simulate te pase distributions around te central axis and sow tat te simulated vortex was indeed a vortex. Unfortunately, we were not able to experimentally measure te pase distribution of te formed vortex due to lack of equipment. 5.1 First Order Vortices Te first order plates were successful in producing first order vortices at te focal point. Te vortices sowed a good correlation between te experimentally measured intensity and te simulated intensity and te interference pattern did indeed sow a fork at te center of te vortex, see figures 8, Second Order Vortices Te second order vortices were a little more problematic tan te first order ones. We were able to get some correlation between te measured intensity profiles and te simulated ones, but one axis turned out to be muc dimmer tan expected, even toug te two graps matced eac oter in sape. Te interference pattern sowed 13

14 Figure 8: Te on screen intensity pattern for te first order optical vortex followed by te interference pattern for te same vortex. Notice te forked pattern in te center Figure 9: Te first diagram sows a good correlation between te experimental intensity measurements (black points) and te teoretical model (blue line) for te first order optical vortices. Te second diagram sows te teoretical pase of te beam as a function θ. Clearly te pase is dependent on its azimutal angle. not one second order fork but two first order forks, implying tat te singularities ad not converged and tat te second order vortex was not really second order, but was instead composed of two first order singularities. Similarly, te tird order plates produced tree first order singularities. 5.3 Fractional Order Vortices Te fractional order vortex results were very similar to te second order results as far as te intensity patterns were concerned. Te vertical axis matced te predicted intensity profile rater well, but te oter axis was muc dimmer even toug it still matced te predicted profile in sape. Te interference pattern yielded a series of 14

15 Figure 10: Te diagram on te left is te on screen intensity for te experimentally produced second order vortex. Te diagram on te left is te corresponding interference pattern sowing not on second order fork, but two first order forks Figure 11: Te first diagram is a comparison of te measured and simulated intensities along te orizontal axis, wile te second diagram is te corresponding grap for te vertical axis of te second order vortex. forks, as is expected for fractional vortices [10] and some of tese forks can actually be confirmed troug our simulations. Here we will present results from a spiral plate for ν = 1.4 order. Wile we did measurements on fractional plates of orders 1.25, 1.4, 1.5, 1.6, and 1.75, te 1.4 order vortex is representative of te series so only results for tis plate are given ere, see figures 12, 13, and

16 Figure 12: Te diagram on te left is te on screen intensity pattern for te 1.4 order vortex and te diagram on te rigt is te simulated intensity pattern. Figure 13: Te diagram on te left is te experimental interference pattern, wile te diagram on te rigt is te simulated interference pattern for te 1.4 order vortices. Note te correlation between te primary forks (red) and te secondary diffraction forks (green). 6 Discussion Our results on te integer vortex, particularly for ν = 1, reproduce te known results in [8, 9] giving us confidence in te metod. In addition to te caracteristic forked pattern in te Mac-Zender interferometer, te intensity profiles produced by te vortices also matced wat was obtained by te diffraction integral via Matematica rater well. Overall, we were able to obtain rater decent results for te creation of te fractional order vortices as well. Tese too gave us forked patterns wit te interferometer 16

17 Figure 14: Te first diagram sows a good correlation between te experimental intensity measurements (black points) and te teoretical model (blue line) along te vertical (fractional) axis in te vertical direction. Te second diagram is intensity profile along te orizontal axis. Te beam appears to be muc weaker tan expected. and potograps of tese vortices were very similar to fractional vortices produced using angular pase plates [11]. Togeter, tese provide good evidence tat we did in fact form fractional optical vortices even toug te completely unambiguous identification of tese patterns as fractional vortices may require more detailed comparisons and more precise production. Altoug tere were some intensity problems along some of te axes, tese may not be inerent problems wit te fractional zone plates temselves, but may instead be problems wit te experiment. One major source of error could be te diffractive elements tat were used. Wile using film to produce te diffraction gratings provided te best quality to cost ratio, te film was far from perfect. Ideally, te diffractive element sould block all ligt from te dark zones and transmit all ligt from te clear zones. Te film, unfortunately, was a ligt brown color overall, not dark enoug to block all of te ligt from te dark zones and not transparent enoug to transmit all of te ligt from te clear zones. Additionally, te film used ad a resolution of eiter 200 or 400 ISO. Altoug tis is good for potograpy, it is not ideal for optics. We would ave preferred to use film of 100 or 50 ISO, but tis film was not readily available. Anoter problem wit te film was tat it was very flexible. Wen placed into te constructed older, it is very probable tat te film underwent small distortions. Tese distortions can create a pase difference in te ligt as it passes troug, wic 17

18 is very problematic in an experiment tat is primarily a measurement of te pase of te ligt. Furtermore, in [12], it is sown tat several modes besides te desired one, can be generated by a Gaussian beam passing troug glass plates. It is possible tat tis mode contamination can also appen wit our zone plates and may explain te splitting of te second order vortex into two first order ones. In te future, it would be interesting to use a Sack-Hartmann wavefront sensor to directly measure te pase of te formed vortices. Unfortunately, tis equipment was not available to us for te experiment. Better results migt also be obtained troug te use of precision etcing to create te zone plates. Te glass would prevent te plate from flexing or getting distorted and one could be sure tat tere would be no imperfections in te pattern. Strictly speaking, tere are no beams of ligt wit fractional orbital angular momentum. Berry [10] as argued tat te fractional vortex must be interpreted as an infinite sum of integer order vortices placed at specific positions. Wit improved imaging of te forks in te interference pattern for fractional vortices, it may be possible to verify tat teir positions agree wit te teory. Te spiral zone plates tat we used were all indirectly based on te Arcimedian spiral. It would be interesting to see if vortices could be formed using a spiral oter tan te Arcimedian, suc as a Cotes spiral or oter spirals tat involve iger powers of θ. Along a similar note, it would be interesting to see if zone plates could be constructed tat could form knotted singularities [13, 14, 15] or oter interesting singular structures. 18

19 References [1] L. Allen, W. Beijersbergen, R. C. Spreeuw, and J. P. Woerdman, Orbital angular momentum of ligt and te transformation of Laguerre-Gaussian laser modes, Pys. Rev. A 45, 8185 (1992). [2] N. B. Simpson, D. McGloin, K. Dolakia, L. Allen and M. J. Padgett, Optical tweezers wit increased axial trapping efficiency, J. Mod. Optics 45, 1943 (1998). [3] H. H. Arnaut and G. A. Barbosa, Orbital and intrinsic angular momentum of single potons and entangled pairs of potons generated by parametric downconversion, Pys. Rev. Lett. 85, 286 (2000). [4] G. Foo, D. M. Palacios, and A. Swartzlander Jr., Optical vortex coronagrap, Opt. Lett. 30, 3008 (2005). [5] G. A. Swartzlander Jr., E. L. Ford. R. S. Abdul-Malik, L. M. Close, M. A. Peters, D. M. Palacios and D. W. Wilson, Astronomical demonstration of an optical vortex coronagrap, Opt. Exxpress 16, (2008). [6] I. V. Basistiy, V. A. Pas ko, V. V. Slyusar, M. S. Soskin and M. V. Vasnetsov, Syntesis and analysis of optical vortices wit fractional topological carges, J. Opt. A: Pure Appl. Opt. 6, S166 (2004). [7] G. F. Brand, Pase singularities in beams, Am. J. Pys. 67, 55 (1999). [8] N. R. Heckenberg, R. McDuff, C. P. Smit and A. G. Wite, Generation of optical pase singularities by computer generated olograms, Opt. Lett. 17, 221 (1992). [9] S. Anand and H. C. Kandpal, An experimental study of anomalous beaviour of spectra near pase singularity, Optics Commun. 232, 69 (2004) [10] M. V. Berry, Optical vortices evolving from elicoidal integer and fractional pase steps, J. Opt. A: Pure Appl. Opt. 6, 259 (2004). [11] K. Crabtree, J. A. Davis and I. Moreno, Optical processing wit vortexproducing lenses, Appl. Opt. 43, 1360 (2004). 19

20 [12] Y. Yosikawa and H. Sasada, Versatile generation of optical vortices based on paraxial mode expansion, J. Opt. Soc. Am. A 19, 2127 (2002). [13] M. V. Berry and M. R. Dennis, Knotted and linked pase singularities in monocromatic waves, Proc. R. Soc. A 457, 2251 (2001). [14] M. V. Berry and M. R. Dennis, Knotting and unknotting of pase singularities: Helmoltz waves, paraxial waves and waves in spacetime, J. Pys. A: Mat Gen. 34, 8877 (2001). [15] J. Leac, M. R. Dennis, J. Courtial and M. J. Padgett, Vortex knots in ligt, New J. Pys. 7, 55 (2005). 20