Divergence of optical vortex beams
Salla Gangi Reddy, Chithrabhanu P, Shashi Prabhakar, Ali Anwar, J. Banerji, R. P. Singh
aa r X i v : . [ phy s i c s . op ti c s ] M a y Divergence of optical vortex beams
Salla Gangi Reddy, Chithrabhanu P, Shashi Prabhakar, Ali Anwar, J. Banerji, and R. P. Singh
Physical Research Laboratory, Navarangpura, Ahmedabad, India-380 009. ∗ (Dated: September 14, 2018)We show, both theoretically and experimentally, that the propagation of optical vortices in free space can beanalyzed by using the width ( w ( z ) ) of the host Gaussian beam and the inner and outer radii of the vortex beamat the source plane ( z =
0) as defined in
Optics Letters . We also studied the divergence ofvortex beams, considered as the rate of change of inner or outer radius with the propagation distance, and foundthat it varies with the order in the same way as that of the inner and outer radii at zero propagation distance.These results may be useful in designing optical fibers for orbital angular momentum modes that play a crucialrole in quantum communication.
I. INTRODUCTION
Optical vortices or phase singular beams are well knowndue to their orbital angular momentum (OAM) [1, 2]. ThisOAM can be used as an information carrier which enhancesthe bandwidth as its modes have an infinite dimensional or-thonormal basis [3]. It has been shown that communicationis possible with radio wave vortices [4] and demonstrated ex-perimentally for wireless communication [5]. To use differentOAM states for information processing, one should have bothmultiplexing and de-multiplexing setups for these modes andthe same has been made with the use of fibers specially de-signed for OAM modes [6–8]. For implementing these proto-cols with vortices, one should design fibers that support OAMmodes in which case one should know the intensity distribu-tion of these OAM modes as well as their divergence.The spatial distribution of vortex beams has been studiedunder strong focussing conditions [9, 10] for which the po-sition of maxima varies linearly with the order rather thansquare root of the order. The divergence of the vortex beamsin free space propagation was studied theoretically by usingthe position of the radial variance of intensity [11]. Recently,it has been shown theoretically that the divergence varies asthe square root of the order if they are generated by usingthe pure mode converter [12] and it varies linearly [13] withthe order if they are generated by using diffractive optical el-ements such as spatial light modulators [14]. We have stud-ied the spatial profile of vortices using two novel and mea-surable parameters (the inner and outer radii) and have giventhe analytical expressions [15]. In this article, we study thedivergence of vortices using these parameters experimentallyas well as theoretically. We show that the inner and outerradii vary linearly with the propagation distance beyond theRayleigh range. Furthermore, we show that the vortex at theplane of observation depends simply on the width of the hostGaussian beam in that plane ( w ( z ) ) and on the inner and outerradii at z =
0. To the best of our knowledge, this is the firstexperimental study on the divergence of optical vortex beamsthat is in close agreement with our theoretical predictions. ∗ [email protected] II. THEORY
We start with the field distribution of a vortex of order m ,embedded in a Gaussian host beam of width w , as E m ( r ) = ( x + iy ) m exp (cid:18) − x + y w (cid:19) (1)and its intensity I m ( r ) = r m exp (cid:18) − r w (cid:19) , r = x + y . (2) -300 -150 0 150 3000.00.20.40.60.81.0 I Max /e r r N o r m a li ze d I n t e n s it y Pixel Number r I Max
FIG. 1. (Color Online) The intensity distribution (left) and its lineprofile (right) for an optical vortex beam of order 1.
This intensity distribution is shown in Fig. 1. Here, we havedefined two parameters for a vortex beam: inner and outerradii ( r , r ). These are the radial distances at which the inten-sity falls to 1 / e ( . ) of the maximum intensity at r = r (say). Here, r is the point closer to the origin or the centerand r is the point farther from the center, the outer region ofthe beam. The distances r i ( i = , ,
2) can be obtained as fol-lows. For the sake of convenience, we set w =
1, that is, w is the unit of measuring radial distances. The inner and outerradii of the bright ring are given by [15] r ( ) = ( m + . − √ q m ) / / √ , (3a) r ( ) = ( m + . + √ q m ) / / √ , (3b) q m = ( m + . ) − m exp ( − . / m ) . These radii correspond to the source plane ( z =
0) at whichthe vortices are being generated. The variation of inner andouter radii with propagation can be studied by propagatingthe vortex beam through free space. The field distribution of avortex beam after propagating through an ABCD optical sys-tem is [16] E m ( u , v ) = (cid:18) ikw B (cid:19) m + ( u + iv ) m exp (cid:18) − u + v w (cid:19) (4)where 1 w = w + ikA B , (5a)1 w = (cid:18) w k B (cid:19) + ikD B . (5b)The corresponding intensity distribution is I m ( u , v ) = a m r m exp ( − r b ) (6)where a m = k w w ∗ B , r = u + v , b = w + w ∗ . (7)Here, a m is a constant multiplicative factor and will not dis-turb the intensity pattern. Defining r = r / b , the above equa-tion can be written as F m = I m a m b m = r m exp ( − r ) . (8)This equation is similar to Eq. 2 and have the solutions r and r which are the same as that of r ( ) and r ( ) of Eq. 3.Now the inner and outer radii of the vortex beam at a particularpropagation distance are given by r = r ( z ) = b r ( ) and r = r ( z ) = b r ( ) where r ( ) and r ( ) are the inner andouter radii at z =
0. The expression for b is b = w A (cid:18) + B k A w (cid:19) / . (9)For free space propagation ( A = B = z ), the above equa-tion becomes b = w (cid:18) + z k w (cid:19) / = w (cid:18) + z z R (cid:19) / = w ( z ) (10)where z R = p w / l is the Rayleigh range and w ( z ) is the widthof the host Gaussian beam at distance z . Now, the inner andouter radii as a function of z can be written as r ( z ) = w ( z ) r ( ) , r ( z ) = w ( z ) r ( ) . (11) This relation shows that the defined parameters can de-scribe the vortex beams as their propagation is similar to thehost Gaussian beam.Now, the divergence of optical vortices has been definedas the rate of change of inner and outer radii r i , i = , z and is given by d im ( z ) = ¶ r i ( z ) ¶ z = w o r i ( ) (cid:0) z + z R ) (cid:1) / zz R , i = , . (12)The divergence depends mainly on the corresponding radii at z =
0. At large z ( z >> z R ), the divergence is constant and isgiven by d im = w o r i ( ) z R . (13)These results are same for vortex beams either they are gen-erated by using mode converter or diffractive optical elementsas we consider only the intensity distribution [13].
500 600 700 800 900 1000 11000.100.120.140.160.18 I nn e r R a d i u s ( m = ) Propagation distance ( z )
500 600 700 800 900 1000 11000.200.220.240.260.280.300.320.340.360.380.400.42 I nn e r R a d i u s ( m = ) Propagation distance ( z )
500 600 700 800 900 1000 11000.300.350.400.450.500.550.60 I nn e r R a d i u s ( m = ) Propagation distance ( z )
500 600 700 800 900 1000 11000.400.450.500.550.600.650.700.75 I nn e r R a d i u s ( m = ) Propagation distance ( z ) FIG. 2. (Color Online) The inner radius for the optical vortex beamsof orders 1–4 at different propagation distances. The experimentaldata (red dots) and the corresponding fitting (blue solid line) withEq. 11.
III. RESULTS AND DISCUSSION
We use an intensity stabilised He-Ne laser to generate opti-cal vortex beams using computer generated holography tech-nique. We introduce holograms corresponding to the vorticesof different orders to the spatial light modulator (SLM) (Holo-eye LCR-2500). We then allow the Gaussian laser beam to beincident normally on the SLM at the branch points of holo-gram which gives the vortex beam in the first diffraction order[15]. The vortices of different orders from 1 to 5 have beenrecorded at different propagation distances starting from 52cm to 107 cm in steps of 5 cm using an Evolution VF color
500 600 700 800 900 1000 11001.21.41.61.82.02.2 O u t e r R a d i u s ( m = ) Propagation distance ( z )
500 600 700 800 900 1000 11001.01.21.41.61.82.0 O u t e r R a d i u s ( m = ) Propagation distance ( z )
500 600 700 800 900 1000 11001.01.21.41.61.8 O u t e r R a d i u s ( m = ) Propagation distance ( z )
500 600 700 800 900 1000 11000.80.91.01.11.21.31.41.51.61.7 O u t e r R a d i u s ( m = ) Propagation distance ( z ) FIG. 3. (Color Online) The outer radius of optical vortices withorders 1–4 at different propagation distances. The experimental data(red dots) and the corresponding fitting (blue solid line) with Eq. 11. cooled CCD camera of pixel size 4.65 µ m. We have recorded20 images for a given order and at a given propagation dis-tance. These images have been further processed to determinethe inner and outer radii.Figures 2 and 3 show the variation of inner and outer radiiwith propagation distance for optical vortex beams of orders m = 1–4. Dots represent the experimental data and the solidlines represent the fitting with Eq. 11. From the figures, it is clear that the experimental results are in excellent agreementwith the theoretical predictions. The slope of these curvesat z >> z R has been taken as the divergence i.e. the rate ofchange of inner and outer radii. The experimental results areobtained by taking the average over 20 images and the errorbars are too small to be visible in the plot.Figure 4 shows the variation of divergence with order ofthe vortex. The rate of change of inner and outer radii are di-rectly proportional to their value at z =
0. The blue diamondsrepresent experimentally obtained divergence and the red dotsrepresent the corresponding results obtained from Eq. 13. Thevariation of divergence with order for the inner and outer radiiis the same as the variation of corresponding radius with orderat z = IV. CONCLUSION
In conclusion, we have described the intensity distributionof optical vortices by using two measurable parameters, theinner and outer radii. The propagation dynamics of vorticescan be easily described by its intensity distribution at thesource plane and the width of the host Gaussian beam at theplane of observation. We have also studied the rate of changeof inner and outer radii, divergence, and their variation withthe order. The experimental results are well supported by theanalytical results. These results may be useful in designingfibers specially for the orbital angular momentum modes. [1] A. T. O’Neil, I. MacVicar, L. Allen, and M. J. Padgett,
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