OSA's Digital Library

Journal of the Optical Society of America A

Journal of the Optical Society of America A

| OPTICS, IMAGE SCIENCE, AND VISION

  • Editor: Franco Gori
  • Vol. 31, Iss. 6 — Jun. 1, 2014
  • pp: 1295–1302
« Show journal navigation

Propagation of an arbitrary vortex pair through an astigmatic optical system and determination of its topological charge

Salla Gangi Reddy, Shashi Prabhakar, A. Aadhi, J. Banerji, and R. P. Singh  »View Author Affiliations


JOSA A, Vol. 31, Issue 6, pp. 1295-1302 (2014)
http://dx.doi.org/10.1364/JOSAA.31.001295


View Full Text Article

Acrobat PDF (951 KB)





Browse Journals / Lookup Meetings

Browse by Journal and Year


   


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools

Share
Citations

Abstract

We embed a pair of vortices with different topological charges in a Gaussian beam and study its evolution through an astigmatic optical system, a tilted lens. The propagation dynamics are explained by a closed-form analytical expression. Furthermore, we show that a careful examination of the intensity distribution at a predicted position past the lens can determine the charge present in the beam. To the best of our knowledge, our method is the first noninterferometric technique to measure the charge of an arbitrary vortex pair. Our theoretical results are well supported by experimental observations.

© 2014 Optical Society of America

1. INTRODUCTION

Optical vortices have drawn considerable attention in science and engineering due to their dark core and helical wavefront. An optical vortex of order l centered at the origin (r=0) has a field distribution of the form E(r)exp(ilϕ). The distribution is such that the field intensity tends to zero as r0, and the phase shift in one cycle around the origin is 2lπ, where l is an integer. The azimuthal mode index l, also called topological charge of the vortex, has a physical meaning in that the vortex carries an orbital angular momentum (OAM) of l per photon [1

1. L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992). [CrossRef]

]. This angular momentum can be imparted to microscopic particles in order to manipulate them optically [2

2. K. T. Gahagan and G. A. Swartzlander, “Optical vortex trapping of particles,” Opt. Lett. 21, 827–829 (1996). [CrossRef]

4

4. D. G. Grier, “A revolution in optical manipulation,” Nature 424, 810–816 (2003). [CrossRef]

]. In recent years, the OAM of light has also found applications in classical [5

5. N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, and S. Ramachandran, “Terabit-scale orbital angular momentum mode division multiplexing in fibers,” Science 340, 1545–1548 (2013). [CrossRef]

] as well as quantum communication [6

6. L. Allen, S. M. Barnett, and M. J. Padgett, Optical Angular Momentum (IOP, 2003).

,7

7. G. M. Terriza, J. P. Torres, and L. Torner, “Twisted photons,” Nat. Phys. 3, 305–310 (2007). [CrossRef]

]. These applications have led to considerable interest in the generation and study of optical vortices both in free space [8

8. L. Allen, M. J. Padgett, and M. Babiker, “The orbital angular momentum of light,” Prog. Opt. 39, 291–372 (1999). [CrossRef]

10

10. A. V. Carpentier, H. Michinel, J. R. Salgueiro, and D. Olivieri, “Making optical vortices with computer-generated holograms,” Am. J. Phys. 76, 916–921 (2008). [CrossRef]

] and in guided media [11

11. J. Scheuer and M. Orenstein, “Optical vortices crystals: spontaneous generation in nonlinear semiconductor microcavities,” Science 285, 230–233 (1999). [CrossRef]

,12

12. R. M. Jenkins, J. Banerji, and A. R. Davies, “The generation of optical vortices and shape preserving vortex arrays in hollow multimode waveguides,” J. Opt. A 3, 527–532 (2001). [CrossRef]

].

Since the topological charge of a vortex determines its OAM, an accurate measurement of the topological charge is an essential and important task. There are a number of methods to determine the charge of an optical vortex and its sign [20

20. D. P. Ghai, S. Vyas, P. Senthilkumaran, and R. S. Sirohi, “Detection of phase singularity using a lateral shear interferometer,” Opt. Lasers Eng. 46, 419–423 (2008). [CrossRef]

27

27. P. Vaity, J. Banerji, and R. P. Singh, “Measuring the topological charge of an optical vortex by using a tilted convex lens,” Phys. Lett. A 377, 1154–1156 (2013). [CrossRef]

]. However, there is no method to measure the net charge for a beam with a vortex pair. We show that the intensity distribution of such a beam at a predicted position beyond a tilted lens can provide information about the net charge present in it. Our technique is quite simple and can be realized easily in the laboratory just by tilting a lens.

Our work can be useful in trapping two metal nanoparticles where the distance between them can be varied by simply changing the position of the vortices in the beam [28

28. M. Dienerowitz, M. Mazilu, P. J. Reece, T. F. Krauss, and K. Dholakia, “Optical vortex trap for resonant confinement of metal nanoparticles,” Opt. Express 16, 4991–4999 (2008). [CrossRef]

]. Trapping the two metal particles in two dark regions of the beam avoids overheating. One can also fine-tune the field and intensity correlations of the scattered light using these kind of structures [29

29. A. Kumar, J. Banerji, and R. P. Singh, “Hanbury BrownTwiss-type experiments with optical vortices and observation of modulated intensity correlation on scattering from rotating ground glass,” Phys. Rev. A 86, 013825 (2012). [CrossRef]

,30

30. A. Kumar, J. Banerji, and R. P. Singh, “Intensity correlation properties of high-order optical vortices passing through a rotating ground-glass plate,” Opt. Lett. 35, 3841–3843 (2010). [CrossRef]

]. The beam considered in the present article contains two separate vortices with arbitrary topological charges and is different from a beam containing a single vortex with a high topological charge considered earlier [31

31. A. Kumar, P. Vaity, and R. P. Singh, “Crafting the core asymmetry to lift the degeneracy of optical vortices,” Opt. Express 19, 6182–6190 (2011). [CrossRef]

34

34. M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, and J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. 96, 123–132 (1993). [CrossRef]

]. Vortices being generic to all the waves [35

35. J. F. Nye and M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. London, Ser. A 336, 165–190 (1974). [CrossRef]

], the present study also can be extended to acoustic [36

36. J.-L. Thomas and R. Marchiano, “Pseudo angular momentum and topological charge conservation for nonlinear acoustical vortices,” Phys. Rev. Lett. 91, 244302 (2003). [CrossRef]

] and matter waves [37

37. S. Prabhakar, R. P. Singh, S. Gautam, and D. Angom, “Annihilation of vortex dipoles in an oblate Bose-Einstein condensate,” J. Phys. B 46, 125302 (2013).

].

2. THEORY

Consider a pair of optical vortices embedded in a Gaussian beam, one with topological charge ϵ1m(ϵ1=±1) located at x1=x0, y1=0 and another with topological charge ϵ2n(ϵ2=±1) at x1=x0, y1=0. The net charge of a considered vortex pair is ϵ1m+ϵ2n. The complex field distribution of the vortex pair at the waist plane of the host Gaussian beam, with waist size w0, is given by
E1(x1,y1)=(x1+x0+iϵ1y1)m(x1x0+iϵ2y1)n×exp[(x12+y12w02)].
(1)

The tilted lens is placed at a distance z0 from the waist plane. The vortex passes through the lens and travels a further distance z. The overall ray transfer matrix Mtot is given by [27

27. P. Vaity, J. Banerji, and R. P. Singh, “Measuring the topological charge of an optical vortex by using a tilted convex lens,” Phys. Lett. A 377, 1154–1156 (2013). [CrossRef]

]
Mtot=(ABC/fD),
(2)
where A, B, C, and D are 2×2 diagonal matrices with diagonal elements given by aj, bj, cj, and dj, respectively. Explicitly,
c1=secθ,c2=cosθ,aj=1zcj/f,dj=1z0cj/f,bj=z0+zdj,j=1,2.
(3)
Next we define two column vectors, r1, r2, so that their transposes are given by row vectors, riT=(xi,yi), i=1,2. The field E2(x2,y2) at a distance z past the lens is given by the generalized Huygens–Fresnel integral [38

38. A. E. Siegman, Lasers (University Science Books, 1986).

]:
E2(x2,y2)=i/λ|B|1/2dx1dy1E1(x1,y1)e(iπ/λ)ϕ(r1,r2),
(4)
where |B|=|b1b2| is the determinant of B and
ϕ(r1,r2)=r1TB1Ar1+r2TDB1r22r1TB1r2=x12a1/b1+y12a2/b2+x22d1/b1+y22d2/b22(x1x2/b1+y1y2/b2).
(5)

The integration over x1 and y1 is carried out by writing E1(x1,y1) as
E1(x1,y1)=limt0t0[mtmntnexp{f(t,t)}],
(6a)
f(t,t)=t(x1+x0+iϵ1y1)+t(x1x0+iϵ2y1)x12+y12w02.
(6b)
Using the definition of the Hermite polynomial and a recurrence relation,
Hn(x)=ntnexp(2xtt2)|t=0,
(7a)
djdxjHn(x)=2jn!(nj)!Hnj(x),
(7b)
we finally get
E2(x2,y2)=kw1w2(i/2)m+n+1γm+n(b1b2)1/2×exp[(β1x22+β2y22)]Fm,n(x2,y2),
(8a)
Fm,n(x2,y2)=j=0min(m,n)(mj)(nj)Δjj!Hmj[f1(x2,y2)]Hnj[f2(x2,y2)],
(8b)
where k=2π/λ,
1wj2=1w02+ikaj2bj,
(9a)
γ=(w12w22)1/2,
(9b)
Δ=2(w12w22ϵ1ϵ2)/γ2,
(9c)
αj=kwj22bj,
(9d)
βj=(kwj2bj)2+ikdj2bj,
(9e)
and
[f1(x2,y2)f2(x2,y2)]=1γ[α1x2+i(ϵ1α2y2x0)α1x2+i(ϵ2α2y2+x0)]=1γ[ϕ1(x2,y2)ϕ2(x2,y2)].
(10)

The sum Fm,n can be evaluated formally as follows. We introduce the two-variable Hermite–Kampé de Fériet polynomials Hn(x,y) as [39

39. S. Khan, M. A. Pathan, N. A. M. Hassan, and G. Yasmin, “Implicit summation formulae for Hermite and related polynomials,” J. Math. Anal. Appl. 344, 408–416 (2008). [CrossRef]

]
Hn(x,y)=n!r=0[n/2]xn2ryr(n2r)!r!,
(12)
in terms of which the classical Hermite polynomials Hn(x) are given by
Hn(x)=Hn(2x,1).
(13)
Next we consider the four-variable two-index one-parameter Hermite polynomials Hm,n(x,z;y,w|τ) defined as [39

39. S. Khan, M. A. Pathan, N. A. M. Hassan, and G. Yasmin, “Implicit summation formulae for Hermite and related polynomials,” J. Math. Anal. Appl. 344, 408–416 (2008). [CrossRef]

,40

40. G. Dattoli, “Hermite-Bessel and Laguerre-Bessel functions: a byproduct of the monomiality principle,” in Advanced Special Functions and Applications (Melfi, 1999), pp. 147–164.

]
Hm,n(x,z;y,w|τ)=s=0min(m,n)τss!(ms)(ns)Hms(x,z)Hns(y,w).
(14)
It is then easy to show that
Fm,n=Hm,n(2f1,1;2f2,1|Δ),
(15)
which has the following generating function:
exp[(u2+v2)+2(f1u+f2v)+Δuv]=m,n=0umvnm!n!Hm,n(2f1,1;2f2,1|Δ).
(16)

A. Determination of Net Topological Charge

As noted earlier [27

27. P. Vaity, J. Banerji, and R. P. Singh, “Measuring the topological charge of an optical vortex by using a tilted convex lens,” Phys. Lett. A 377, 1154–1156 (2013). [CrossRef]

], the modulations due to the Hermite polynomial become most prominent when w2=w1*. This happens at a certain value z=zc. To determine zc and also the distance z0 between the waist plane and the lens, we impose the following conditions:
ka12b1|z=zc=ka22b2|z=zc=1w02.
(17)
Solving Eqs. (17) and introducing the Rayleigh range zR=kw02/2, we get
z0=zR(1+2fcosθzRsin2θ)1/2zc=zR(1+cos2θ)+z0sin2θ2(zR/f)cosθsin2θ.
(18)
The first equality in Eq. (17) ensures that w2=w1* at z=zc [see Eq. (9a)], whereas the last equality makes many expressions appearing in Eqs. (8)–(10) considerably simpler at z=zc. Thus, at z=zc,
Δ={2ifϵ1ϵ2=1,2iifϵ1ϵ2=1;(w12w22)=w022(exp(iπ/4)exp(iπ/4));γ=w0exp(iπ/4);f1=δ1x2ϵ1δ2y2+(x0/w0)exp(iπ/4),f2=δ1x2ϵ2δ2y2(x0/w0)exp(iπ/4),
(19)
where
δj=kw022bi.
(20)

1. Vortices with Topological Charges of the Same Sign

Suppose ϵ1=ϵ2=1. Then, f1=θ+θ0 and f2=θθ0, where
θ=δ1x2δ2y2,θ0=(x0/w0)exp(iπ/4).
(21)
Note that the dependence on x2 and y2 is in the form θ only.

For a small separation between the vortices, one can expand the Hermite polynomials appearing in Eq. (8) as functions of x0/w0 by using the formula
Hn(x+y)=Hn(x)+2nyHn1(x)+O(y2).
(22)
Substituting in Eq. (8) and using the summation rule [41

41. W. Magnus, F. Oberhettinger, and R. P. Soni, Formulas and Theorems for Special Functions of Mathematical Physics (Springer-Verlag, 1966).

]
r=0min(m,n)(2)rr!(mr)(nr)Hmr(x)Hnr(x)=Hm+n(x),
(23)
we get
Fm,n=Hm+n(θ)+2θ0(mn)Hm+n1(θ)+O(θ02).
(24)
For ϵ1=ϵ2=1, θ will change to θ+=δ1x2+δ2y2 in the above expressions.

2. Vortices with Topological Charges of Opposite Signs

Suppose ϵ1=1, ϵ2=1. In this case,
f1=θ+θ0,f2=θ+θ0.
(25)
Note that in this case, the dependence on x2 and y2 is in the form θ±=δ1x2±δ2y2.

B. Propagation Dynamics Away from z=zc

As |zzc| increases, the absolute value of |γ| falls off rapidly, and the modulations, due to the Hermite polynomials, fade away quickly. Using the limiting form limγ0Hm(x/γ)=(2x/γ)m, we can write E2(x2,y2) in terms of incomplete two-variable Hermite polynomials hm,n(x,y|τ), which are defined as [39

39. S. Khan, M. A. Pathan, N. A. M. Hassan, and G. Yasmin, “Implicit summation formulae for Hermite and related polynomials,” J. Math. Anal. Appl. 344, 408–416 (2008). [CrossRef]

,42

42. G. Dattoli, “Incomplete 2D Hermite polynomials: properties and applications,” J. Math. Anal. Appl. 284, 447–454 (2003). [CrossRef]

]
hm,n(x,y|τ)=m!n!j=0min(m,n)τjxmjynjj!(mj)!(nj)!={m!τmxnmLm(nm)(xy/τ),n>m,n!τnymnLn(mn)(xy/τ),m>n.
(27)
Thus E2(x2,y2) reduces to
E2(x2,y2)=kw1w2im+n+12(b1b2)1/2exp[(β1x22+β2y22)]×{m!τmϕ1nmLm(nm)(ϕ1ϕ2/τ),n>m,n!τnϕ2mnLn(mn)(ϕ1ϕ2/τ),m>n,
(28)
where ϕj are as in Eq. (10) and τ=(w12w22ϵ1ϵ2)/2.

C. Propagation Dynamics When the Lens Is Not Tilted (θ=0)

For isopolar vortices (ϵ1ϵ2=1), only the j=0 term survives in Eq. (30), and we get
E2(x2,y2)=ikw22bexp[β(x22+y22)]Δ1mΔ2n.
(32)
For dipolar vortices (ϵ1ϵ2=1), we use Eq. (27) to get
E2(x2,y2)=ikw22bexp[β(x22+y22)]×{m!w2mΔ1nmLm(nm)(Δ1Δ2/w2),n>m,n!w2nΔ2mnLn(mn)(Δ1Δ2/w2),m>n.
(33)
In what follows, we will experimentally demonstrate the validity of our theoretical results.

3. EXPERIMENT

The experimental setup is shown in Fig. 1. Suitable phase masks for creating vortex pairs are produced by using a computer-generated holography (CGH) technique [10

10. A. V. Carpentier, H. Michinel, J. R. Salgueiro, and D. Olivieri, “Making optical vortices with computer-generated holograms,” Am. J. Phys. 76, 916–921 (2008). [CrossRef]

] and sent to a spatial light modulator (SLM) (Holoeye LCR 2500) via computer. The SLM is illuminated by an intensity-stabilized He–Ne laser (Spectra-Physics, Model 117A) of power 1 mW and wavelength 632.8 nm to produce the desired vortex pair. The vortex pair is selected with an aperture (A) and passed through a spherical biconvex lens of focal length 50 cm, which is tilted by an angle of 6°. The tilting of the lens has been done with a rotational stage with least count of 0.1°. The aperture is at a distance of z1=90cm in front of the SLM. We use the method described in [43

43. R. S. Sirohi, A Course of Experiments with He-Ne Lasers (New Age International, 1991).

] to find that the Gaussian laser beam hosting the selected vortex pair has a beam waist 0.186 mm at a virtual point, which is at a distance of z2=60.8cm behind the SLM. The distance between the lens and the aperture is z3=245cm. Thus the total distance traveled by the vortex pair from the waist plane to the lens is z0=z1+z2+z3=395.8cm. The resultant intensity patterns are recorded by a CCD camera (MediaCybernetics, Evolution VF cooled Color Camera) placed at a distance z past the lens.

Fig. 1. Experimental setup for the determination of the net charge of an arbitrary vortex pair embedded in a Gaussian beam.

We start the experiment by taking the intensity distributions of a pair of isopolar vortices and the corresponding interference patterns at two planes as shown in Fig. 2. One of the planes is at 96 cm from the SLM, the nearest plane where the diffraction orders can be separated with an aperture and another after freely propagating a distance 147 cm from the SLM. In our experiments, where intensity distributions have been recorded after the tilted lens, the aberrations due to the SLM have been neglected, as they will be insignificant compared with the astigmatism introduced by the lens.

Fig. 2. Intensity distributions of a vortex pair embedded in a Gaussian beam and the corresponding interferograms at 96 cm (top) and 147 cm (bottom) from the SLM with the orders (left) m=n=1; (right) m=n=2.

4. INTENSITY PATTERN AT z=zc AND DETERMINATION OF NET TOPOLOGICAL CHARGE

In this section, we determine the net topological charge of the vortex pair from its intensity distribution at z=zc. The predicted value of zc from Eq. (18) is 57.2 cm, which is close to the experimentally observed value of 56.3 cm. In the intensity patterns, with reference to Eq. (1), the vortex on the left (x1=x0) has a charge ϵ1m and the vortex on the right (x1=x0) has a charge ϵ2n. The corresponding figure is labeled as (ϵ1m,ϵ2n).

A. Vortices with Topological Charges of the Same Sign

Figure 3 shows the theoretical (first two rows) and experimental (last two rows) images for the intensity patterns of a pair of vortices with the same sign (ϵ1=ϵ2=1) but different magnitudes m and n with the separation parameter set at x0=0.1w0.

Fig. 3. Theoretical (first two rows) and experimental (last two rows) results for the intensity patterns of a vortex pair with topological charges of the same sign, at z=zc for x0=0.1w0.

For small separation x0, these patterns can be explained by Eq. (24). Since the first term in Eq. (24) is the leading term, one can obtain the net charge m+n by noting there are m+n+1 bright stripes in the intensity distribution. These stripes are parallel to one another and lie along a line that is neither horizontal nor vertical but tilted in a clockwise direction almost along a diagonal, as the dependence on x and y is through a single variable θ=δ1x2δ2y2 and δ1δ2. However, interference with the second term will lead to a slightly asymmetric distribution of brightness among the stripes. As is clear from the second term in Eq. (24), this asymmetry depends on the difference between the magnitude of charges and the separation between them. Additionally, when the vortices swap their positions as in (4,1) and (1,4), the lower half of the pattern becomes the mirror image of the upper half and vice versa. For m=n as in (2,2), (3,3), and (4,4), the two halves have identical intensity patterns. In this special case, the net charge is even and the charge of each vortex is half of the net charge. If the charge of each vortex were negative (ϵ1=ϵ2=1), then θ would be replaced by θ+=δ1x2+δ2y2 and the bright stripes would be tilted in an anticlockwise fashion (not shown).

Figure 4 shows theoretical results corresponding to asymmetry of the intensity distributions (at z=zc) with the separation between two vortices and the difference between their topological charges. This asymmetry increases with the separation, provided there is a difference in their topological charges as shown in Fig. 4 (top). These results are for m=4, n=1 at separations from 0.1w0 to 0.5w0. If there is no difference in the charges of the two vortices (m=n=2), then there is no asymmetry in lobes as shown in Fig. 4 (bottom). The corresponding experimental results are given in Fig. 5. A line profile of the intensity along the center of the lobes as shown in Fig. 6 makes it more clear.

Fig. 4. Theoretical results for the intensity patterns of a vortex pair with topological charges of the same sign with varying separation at z=zc, (top) m=4, n=1; (bottom) m=n=2.
Fig. 5. Experimental images corresponding to Fig. 4.
Fig. 6. Line profiles of intensity distributions along the center of the lobes corresponding to Fig. 5 at x0=0.4w0, (left) m=4, n=1; (right) m=n=2.

Fig. 7. Theoretical (first two rows) and experimental (last two rows) results for the intensity patterns of an off-axis vortex of charge 2, at z=zc for different values of x0 as labeled in the figures.

B. Vortices with Topological Charges of Opposite Signs

Figure 8 shows the theoretical (first two rows) and experimental (last two rows) images corresponding to opposite singularities (ϵ1ϵ2=1) for separation parameter x0=0.1w0 and topological charges as shown in the images.

Fig. 8. Theoretical (first two rows) and experimental (last two rows) results for the intensity patterns of a vortex pair with topological charges of opposite signs, at z=zc for x0=0.1w0.

For small values of m and n, these patterns can be explained by expanding the Hermite polynomials in power series. The calculation would be long and tedious. Instead, we make the following empirical observation. If mn and m2, n2 the pattern has a rectangular razor-blade structure, which is tilted clockwise (anticlockwise) if the net charge is positive (negative). On closer observation, we note that there are m bright spots on two parallel sides and n bright spots on the remaining two parallel sides. Thus, for vortex dipoles (m=n), the pattern is square with its corners in the east, west, north, and south directions, each side having m=n bright spots. Clearly, for vortices of opposite signs with m2 and n2, we can determine the individual charges as well [see, for example, figures for (2,4) and (4,2)].

As far as we know, Fig. 8 represents the first optical realization of the four-variable two-index one-parameter Hermite polynomials Hm,n(x,z;y,w|τ) modulated by an elliptical Gaussian beam [see Eqs. (8), (15), and (26)].

In Fig. 9, we show the evolution of a dipole vortex of charge (2,2) as a function of separation between the two vortices. For small separation, the intensity distribution is symmetric in both the transverse directions. As the separation is increased, the pattern becomes asymmetric. When the separation parameter x0 becomes negative, the vortex is identically described as (2,2) with x0=|x0| and the pattern flips vertically.

Fig. 9. Theoretical (top row) and experimental (bottom row) results for the intensity patterns of a dipole vortex of charge (2,2), at z=zc for different values of x0 as labeled in the figures.

We have also studied the variation in the intensity distribution at a particular plane (z=56.3cm corresponding to the experimentally observed value of zc for a tilt angle θ=6°) by changing the tilt angle of the lens. As z0 and zc depend on θ, the bright or dark lobes with highest contrast are obtained only at θ=6° in this case. The theoretical as well as experimental results have been shown in Figs. 10 and 11, respectively. Using Eq. (32), we have also obtained the intensity patterns when the lens is not tilted to highlight the dramatic impact a tilted lens makes.

Fig. 10. Theoretical results for the intensity patterns of a vortex pair with topological charges (top) m=n=2; (bottom) m=2, n=3, at z=zc corresponding to the tilt angle θ=6°. As the tilt angle moves away from 6°, the sharpness of the patterns decreases, as expected.
Fig. 11. Experimental images corresponding to Fig. 10 at z=zc.

5. PROPAGATION DYNAMICS AWAY FROM z=zc

As we move away from the point z=zc, the modulations due to the Hermite polynomials disappear quickly. The propagation dynamics is now governed by Eq. (28). The theoretical and corresponding experimental intensity patterns for various values of z are shown in Figs. 12 and 13, respectively. The intensity patterns are, in general, elliptical. Far away from zc, all patterns become circularly symmetric as α1α2 and β1β2 [27

27. P. Vaity, J. Banerji, and R. P. Singh, “Measuring the topological charge of an optical vortex by using a tilted convex lens,” Phys. Lett. A 377, 1154–1156 (2013). [CrossRef]

].

Fig. 12. Theoretical intensity patterns of a vortex pair of different charges (as given on the top) at various values of the propagation distance z (as given on the left).
Fig. 13. Experimental images corresponding to Fig. 12.

6. CONCLUSIONS

We have studied, analytically and experimentally, the propagation of a Gaussian beam carrying a vortex pair of arbitrary topological charges through a tilted lens. We have also demonstrated a method to find the net topological charge of the vortex pair. Moreover, for vortices with topological charges of opposite signs and magnitudes m2, n2, we can determine the individual charges as well. Our method is easy to implement in the laboratory, as it needs just a single tilted lens, except the tilt has to be small for paraxial approximation to be valid. Vortices being generic to all the waves, this study can be extended for other systems such as acoustic and matter waves.

REFERENCES

1.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992). [CrossRef]

2.

K. T. Gahagan and G. A. Swartzlander, “Optical vortex trapping of particles,” Opt. Lett. 21, 827–829 (1996). [CrossRef]

3.

H. He, M. E. J. Friese, N. R. Heckenberg, and H. R. Dunlop, “Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity,” Phys. Rev. Lett. 75, 826–829 (1995). [CrossRef]

4.

D. G. Grier, “A revolution in optical manipulation,” Nature 424, 810–816 (2003). [CrossRef]

5.

N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, and S. Ramachandran, “Terabit-scale orbital angular momentum mode division multiplexing in fibers,” Science 340, 1545–1548 (2013). [CrossRef]

6.

L. Allen, S. M. Barnett, and M. J. Padgett, Optical Angular Momentum (IOP, 2003).

7.

G. M. Terriza, J. P. Torres, and L. Torner, “Twisted photons,” Nat. Phys. 3, 305–310 (2007). [CrossRef]

8.

L. Allen, M. J. Padgett, and M. Babiker, “The orbital angular momentum of light,” Prog. Opt. 39, 291–372 (1999). [CrossRef]

9.

S. N. Khonina, V. V. Kotlyar, R. V. Skidanov, V. A. Soifer, P. Laakkonen, and J. Turunen, “Gauss–Laguerre modes with different indices in prescribed diffraction orders of a diffractive phase element,” Opt. Commun. 175, 301–308 (2000). [CrossRef]

10.

A. V. Carpentier, H. Michinel, J. R. Salgueiro, and D. Olivieri, “Making optical vortices with computer-generated holograms,” Am. J. Phys. 76, 916–921 (2008). [CrossRef]

11.

J. Scheuer and M. Orenstein, “Optical vortices crystals: spontaneous generation in nonlinear semiconductor microcavities,” Science 285, 230–233 (1999). [CrossRef]

12.

R. M. Jenkins, J. Banerji, and A. R. Davies, “The generation of optical vortices and shape preserving vortex arrays in hollow multimode waveguides,” J. Opt. A 3, 527–532 (2001). [CrossRef]

13.

V. Yu. Bazhenov, M. S. Soskin, and M. V. Vasnetsov, “Screw dislocations in light wavefronts,” J. Mod. Opt. 39, 985–990 (1992). [CrossRef]

14.

G. Indebetouw, “Optical vortices and their propagation,” J. Mod. Opt. 40, 73–87 (1993). [CrossRef]

15.

F. S. Roux, “Spatial evolution of the morphology of an optical vortex dipole,” Opt. Commun. 236, 433–440 (2004). [CrossRef]

16.

F. S. Roux, “Canonical vortex dipole dynamics,” J. Opt. Soc. Am. B 21, 655–663 (2004). [CrossRef]

17.

M. Chen and F. S. Roux, “Accelerating the annihilation of an optical vortex dipole in a Gaussian beam,” J. Opt. Soc. Am. A 25, 1279–1286 (2008). [CrossRef]

18.

Z. Chen, J. Pu, and D. Zhao, “Tight focusing properties of linearly polarized Gaussian beam with a pair of vortices,” Phys. Lett. A 375, 2958–2963 (2011). [CrossRef]

19.

H. Chen, Z. Gao, H. Yang, F. Wang, and X. Huang, “Propagation of a pair of vortices through a tilted lens,” Optik 124, 4201–4205 (2013). [CrossRef]

20.

D. P. Ghai, S. Vyas, P. Senthilkumaran, and R. S. Sirohi, “Detection of phase singularity using a lateral shear interferometer,” Opt. Lasers Eng. 46, 419–423 (2008). [CrossRef]

21.

J. Arlt, K. Dholakia, L. Allen, and M. J. Padgett, “The production of multiringed Laguerre-Gaussian modes by computer-generated holograms,” J. Mod. Opt. 45, 1231–1237 (1998). [CrossRef]

22.

A. Kumar, S. Prabhakar, P. Vaity, and R. P. Singh, “Information content of optical vortex fields,” Opt. Lett. 36, 1161–1163 (2011). [CrossRef]

23.

P. Vaity and R. P. Singh, “Topological charge dependent propagation of optical vortices under quadratic phase transformation,” Opt. Lett. 37, 1301–1303 (2012). [CrossRef]

24.

S. Prabhakar, A. Kumar, J. Banerji, and R. P. Singh, “Revealing the order of a vortex through its intensity record,” Opt. Lett. 36, 4398–4400 (2011). [CrossRef]

25.

J. Hickmann, E. Fonseca, W. Soares, and S. Chvez-Cerda, “Unveiling a truncated optical lattice associated with a triangular aperture using lights orbital angular momentum,” Phys. Rev. Lett. 105, 053904 (2010). [CrossRef]

26.

A. Mourka, J. Baumgartl, C. Shanor, K. Dholakia, and E. M. Wright, “Visualization of the birth of an optical vortex using diffraction from a triangular aperture,” Opt. Express 19, 5760–5771 (2011). [CrossRef]

27.

P. Vaity, J. Banerji, and R. P. Singh, “Measuring the topological charge of an optical vortex by using a tilted convex lens,” Phys. Lett. A 377, 1154–1156 (2013). [CrossRef]

28.

M. Dienerowitz, M. Mazilu, P. J. Reece, T. F. Krauss, and K. Dholakia, “Optical vortex trap for resonant confinement of metal nanoparticles,” Opt. Express 16, 4991–4999 (2008). [CrossRef]

29.

A. Kumar, J. Banerji, and R. P. Singh, “Hanbury BrownTwiss-type experiments with optical vortices and observation of modulated intensity correlation on scattering from rotating ground glass,” Phys. Rev. A 86, 013825 (2012). [CrossRef]

30.

A. Kumar, J. Banerji, and R. P. Singh, “Intensity correlation properties of high-order optical vortices passing through a rotating ground-glass plate,” Opt. Lett. 35, 3841–3843 (2010). [CrossRef]

31.

A. Kumar, P. Vaity, and R. P. Singh, “Crafting the core asymmetry to lift the degeneracy of optical vortices,” Opt. Express 19, 6182–6190 (2011). [CrossRef]

32.

A. Y. Bekshaev, M. S. Soskin, and M. V. Vasnetsov, “Transformation of higher-order optical vortices upon focussing by an astigmatic lens,” Opt. Commun. 241, 237–247 (2004). [CrossRef]

33.

A. Y. Bekshaev and A. I. Karamoch, “Astigmatic telescopic transformation of a high-order optical vortex,” Opt. Commun. 281, 5687–5696 (2008). [CrossRef]

34.

M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, and J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. 96, 123–132 (1993). [CrossRef]

35.

J. F. Nye and M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. London, Ser. A 336, 165–190 (1974). [CrossRef]

36.

J.-L. Thomas and R. Marchiano, “Pseudo angular momentum and topological charge conservation for nonlinear acoustical vortices,” Phys. Rev. Lett. 91, 244302 (2003). [CrossRef]

37.

S. Prabhakar, R. P. Singh, S. Gautam, and D. Angom, “Annihilation of vortex dipoles in an oblate Bose-Einstein condensate,” J. Phys. B 46, 125302 (2013).

38.

A. E. Siegman, Lasers (University Science Books, 1986).

39.

S. Khan, M. A. Pathan, N. A. M. Hassan, and G. Yasmin, “Implicit summation formulae for Hermite and related polynomials,” J. Math. Anal. Appl. 344, 408–416 (2008). [CrossRef]

40.

G. Dattoli, “Hermite-Bessel and Laguerre-Bessel functions: a byproduct of the monomiality principle,” in Advanced Special Functions and Applications (Melfi, 1999), pp. 147–164.

41.

W. Magnus, F. Oberhettinger, and R. P. Soni, Formulas and Theorems for Special Functions of Mathematical Physics (Springer-Verlag, 1966).

42.

G. Dattoli, “Incomplete 2D Hermite polynomials: properties and applications,” J. Math. Anal. Appl. 284, 447–454 (2003). [CrossRef]

43.

R. S. Sirohi, A Course of Experiments with He-Ne Lasers (New Age International, 1991).

OCIS Codes
(050.0050) Diffraction and gratings : Diffraction and gratings
(050.4865) Diffraction and gratings : Optical vortices

ToC Category:
Diffraction and Gratings

History
Original Manuscript: February 6, 2014
Revised Manuscript: April 14, 2014
Manuscript Accepted: April 22, 2014
Published: May 22, 2014

Citation
Salla Gangi Reddy, Shashi Prabhakar, A. Aadhi, J. Banerji, and R. P. Singh, "Propagation of an arbitrary vortex pair through an astigmatic optical system and determination of its topological charge," J. Opt. Soc. Am. A 31, 1295-1302 (2014)
http://www.opticsinfobase.org/josaa/abstract.cfm?URI=josaa-31-6-1295


Sort:  Author  |  Year  |  Journal  |  Reset  

References

  1. L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992). [CrossRef]
  2. K. T. Gahagan and G. A. Swartzlander, “Optical vortex trapping of particles,” Opt. Lett. 21, 827–829 (1996). [CrossRef]
  3. H. He, M. E. J. Friese, N. R. Heckenberg, and H. R. Dunlop, “Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity,” Phys. Rev. Lett. 75, 826–829 (1995). [CrossRef]
  4. D. G. Grier, “A revolution in optical manipulation,” Nature 424, 810–816 (2003). [CrossRef]
  5. N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, and S. Ramachandran, “Terabit-scale orbital angular momentum mode division multiplexing in fibers,” Science 340, 1545–1548 (2013). [CrossRef]
  6. L. Allen, S. M. Barnett, and M. J. Padgett, Optical Angular Momentum (IOP, 2003).
  7. G. M. Terriza, J. P. Torres, and L. Torner, “Twisted photons,” Nat. Phys. 3, 305–310 (2007). [CrossRef]
  8. L. Allen, M. J. Padgett, and M. Babiker, “The orbital angular momentum of light,” Prog. Opt. 39, 291–372 (1999). [CrossRef]
  9. S. N. Khonina, V. V. Kotlyar, R. V. Skidanov, V. A. Soifer, P. Laakkonen, and J. Turunen, “Gauss–Laguerre modes with different indices in prescribed diffraction orders of a diffractive phase element,” Opt. Commun. 175, 301–308 (2000). [CrossRef]
  10. A. V. Carpentier, H. Michinel, J. R. Salgueiro, and D. Olivieri, “Making optical vortices with computer-generated holograms,” Am. J. Phys. 76, 916–921 (2008). [CrossRef]
  11. J. Scheuer and M. Orenstein, “Optical vortices crystals: spontaneous generation in nonlinear semiconductor microcavities,” Science 285, 230–233 (1999). [CrossRef]
  12. R. M. Jenkins, J. Banerji, and A. R. Davies, “The generation of optical vortices and shape preserving vortex arrays in hollow multimode waveguides,” J. Opt. A 3, 527–532 (2001). [CrossRef]
  13. V. Yu. Bazhenov, M. S. Soskin, and M. V. Vasnetsov, “Screw dislocations in light wavefronts,” J. Mod. Opt. 39, 985–990 (1992). [CrossRef]
  14. G. Indebetouw, “Optical vortices and their propagation,” J. Mod. Opt. 40, 73–87 (1993). [CrossRef]
  15. F. S. Roux, “Spatial evolution of the morphology of an optical vortex dipole,” Opt. Commun. 236, 433–440 (2004). [CrossRef]
  16. F. S. Roux, “Canonical vortex dipole dynamics,” J. Opt. Soc. Am. B 21, 655–663 (2004). [CrossRef]
  17. M. Chen and F. S. Roux, “Accelerating the annihilation of an optical vortex dipole in a Gaussian beam,” J. Opt. Soc. Am. A 25, 1279–1286 (2008). [CrossRef]
  18. Z. Chen, J. Pu, and D. Zhao, “Tight focusing properties of linearly polarized Gaussian beam with a pair of vortices,” Phys. Lett. A 375, 2958–2963 (2011). [CrossRef]
  19. H. Chen, Z. Gao, H. Yang, F. Wang, and X. Huang, “Propagation of a pair of vortices through a tilted lens,” Optik 124, 4201–4205 (2013). [CrossRef]
  20. D. P. Ghai, S. Vyas, P. Senthilkumaran, and R. S. Sirohi, “Detection of phase singularity using a lateral shear interferometer,” Opt. Lasers Eng. 46, 419–423 (2008). [CrossRef]
  21. J. Arlt, K. Dholakia, L. Allen, and M. J. Padgett, “The production of multiringed Laguerre-Gaussian modes by computer-generated holograms,” J. Mod. Opt. 45, 1231–1237 (1998). [CrossRef]
  22. A. Kumar, S. Prabhakar, P. Vaity, and R. P. Singh, “Information content of optical vortex fields,” Opt. Lett. 36, 1161–1163 (2011). [CrossRef]
  23. P. Vaity and R. P. Singh, “Topological charge dependent propagation of optical vortices under quadratic phase transformation,” Opt. Lett. 37, 1301–1303 (2012). [CrossRef]
  24. S. Prabhakar, A. Kumar, J. Banerji, and R. P. Singh, “Revealing the order of a vortex through its intensity record,” Opt. Lett. 36, 4398–4400 (2011). [CrossRef]
  25. J. Hickmann, E. Fonseca, W. Soares, and S. Chvez-Cerda, “Unveiling a truncated optical lattice associated with a triangular aperture using lights orbital angular momentum,” Phys. Rev. Lett. 105, 053904 (2010). [CrossRef]
  26. A. Mourka, J. Baumgartl, C. Shanor, K. Dholakia, and E. M. Wright, “Visualization of the birth of an optical vortex using diffraction from a triangular aperture,” Opt. Express 19, 5760–5771 (2011). [CrossRef]
  27. P. Vaity, J. Banerji, and R. P. Singh, “Measuring the topological charge of an optical vortex by using a tilted convex lens,” Phys. Lett. A 377, 1154–1156 (2013). [CrossRef]
  28. M. Dienerowitz, M. Mazilu, P. J. Reece, T. F. Krauss, and K. Dholakia, “Optical vortex trap for resonant confinement of metal nanoparticles,” Opt. Express 16, 4991–4999 (2008). [CrossRef]
  29. A. Kumar, J. Banerji, and R. P. Singh, “Hanbury BrownTwiss-type experiments with optical vortices and observation of modulated intensity correlation on scattering from rotating ground glass,” Phys. Rev. A 86, 013825 (2012). [CrossRef]
  30. A. Kumar, J. Banerji, and R. P. Singh, “Intensity correlation properties of high-order optical vortices passing through a rotating ground-glass plate,” Opt. Lett. 35, 3841–3843 (2010). [CrossRef]
  31. A. Kumar, P. Vaity, and R. P. Singh, “Crafting the core asymmetry to lift the degeneracy of optical vortices,” Opt. Express 19, 6182–6190 (2011). [CrossRef]
  32. A. Y. Bekshaev, M. S. Soskin, and M. V. Vasnetsov, “Transformation of higher-order optical vortices upon focussing by an astigmatic lens,” Opt. Commun. 241, 237–247 (2004). [CrossRef]
  33. A. Y. Bekshaev and A. I. Karamoch, “Astigmatic telescopic transformation of a high-order optical vortex,” Opt. Commun. 281, 5687–5696 (2008). [CrossRef]
  34. M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, and J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. 96, 123–132 (1993). [CrossRef]
  35. J. F. Nye and M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. London, Ser. A 336, 165–190 (1974). [CrossRef]
  36. J.-L. Thomas and R. Marchiano, “Pseudo angular momentum and topological charge conservation for nonlinear acoustical vortices,” Phys. Rev. Lett. 91, 244302 (2003). [CrossRef]
  37. S. Prabhakar, R. P. Singh, S. Gautam, and D. Angom, “Annihilation of vortex dipoles in an oblate Bose-Einstein condensate,” J. Phys. B 46, 125302 (2013).
  38. A. E. Siegman, Lasers (University Science Books, 1986).
  39. S. Khan, M. A. Pathan, N. A. M. Hassan, and G. Yasmin, “Implicit summation formulae for Hermite and related polynomials,” J. Math. Anal. Appl. 344, 408–416 (2008). [CrossRef]
  40. G. Dattoli, “Hermite-Bessel and Laguerre-Bessel functions: a byproduct of the monomiality principle,” in Advanced Special Functions and Applications (Melfi, 1999), pp. 147–164.
  41. W. Magnus, F. Oberhettinger, and R. P. Soni, Formulas and Theorems for Special Functions of Mathematical Physics (Springer-Verlag, 1966).
  42. G. Dattoli, “Incomplete 2D Hermite polynomials: properties and applications,” J. Math. Anal. Appl. 284, 447–454 (2003). [CrossRef]
  43. R. S. Sirohi, A Course of Experiments with He-Ne Lasers (New Age International, 1991).

Cited By

Alert me when this paper is cited

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.


« Previous Article  |  Next Article »

OSA is a member of CrossRef.

CrossCheck Deposited