Measuring the topological chargeof optical vortices with an axicon

Yujing Han* and Guanghui ZhaoDepartment of Physics, Heze University, Heze 274015, China

*Corresponding author: yjhan1981@yahoo.com.cn

Received March 23, 2011; revised April 21, 2011; accepted April 21, 2011;posted April 22, 2011 (Doc. ID 144610); published May 26, 2011

We analyzed the spatial spectrum of the diffraction intensity pattern of an ideal Bessel beam and found an implicitrule that the number of the bright rings in the spatial spectrum is equal to the topological charge of the Bessel beam.The radius of the bright and dark rings has some relation with the topological charge and can be determined ac-curately. It provides us with a new way for measuring the topological charge of an optical vortex through its dif-fraction intensity pattern after an axicon. The results of simulation coincide with the theory. © 2011 OpticalSociety of AmericaOCIS codes: 050.4865, 080.4298, 260.6042, 270.5585.

Optical vortices are the optical fields with helical wave-fronts and well-defined orbital angular momentum [1].The orbital angular momentum is associated with thefields’ phase distribution expðilφÞ, and l is the topologicalcharge of the optical vortex. Each photon of the opticalvortex has the orbital angular momentum of lℏ. The op-tical vortices can be produced by computer-generatedholograms [2], spiral phase plates [3], and a spatial lightmodulator [4]. They can be applied in quantum optics in-formation processing [5,6], free-space communications[7], and optical micromanipulations [8,9] for their orbitalangular momentum properties. Measuring the topologi-cal charge of optical vortices has become an interestingresearch field.Over the past years, researchers have measured the to-

pological charge of the optical vortex by interference ofthe optical vortex with its mirror image [10,11] or anotherreference wave [12,13]. Sztul et al. [14,15] introduced adouble-slit interference to measure the helical phasestructure of a given wavefront. Recently, Berkhout et al.[16] proposed a method for probing the orbital angularmomentum of optical vortices by using a multipinhole in-terferometer, and Guo et al. [17] demonstrated that thephase of an optical vortex passing through a multipinholeplate can be directly extracted from the Fourier transformof a single diffraction intensity pattern. Further, Guo et al.[18] proposed a method for measuring the topologicalcharge of an optical vortex by replacing the multipinholeplate with an annular aperture and found that the numberof bright rings in the spatial spectrum of a diffraction in-tensity pattern is equal to the topological charge of an op-tical vortex; and Berkhout et al. [19] presented a newmethod for sorting orbital angular momentum states oflight by the use of two static optical elements that can con-vert an optical vortex in tilted plane waves. Finally,Hickmann et al. [20,21] showed that a triangular apertureilluminated with a vortex beam creates a truncated latticediffraction pattern that identifies the charge of the vortex.In this Letter, we propose a new method for measuring

the topological charge of an optical vortex by analyzingthe spatial spectrum of the intensity pattern of an idealBessel beam. A higher-order Bessel beam contains anoptical vortex. It has nondiffracting properties, and itscomplex amplitude can be expressed as

ulðr;φÞ ¼ JlðβrÞ expðilφÞ; ð1Þ

where ðr;φÞ represents polar coordinates, l is the topo-logical charge, and Jl is the Bessel function of the lth or-der. The intensity distribution of the higher-order Besselbeam shown in Eq. (1) can be written as

IlðrÞ ¼ jJlðβrÞj2 ¼ J2l ðβrÞ: ð2Þ

The Fourier transform of Eq. (2) can be expressed asfollows:

gðρ; θÞ ¼ FT½J2l ðβrÞ� ¼

1λf

Z∞

0J2l ðβrÞrdr

Z2π

0

× exp

�−j

2πrρλf cosðφ − θÞ

�dφ

¼ 1λf

Z∞

0J2l ðβrÞJ0

�2πrρλf

�rdr; ð3Þ

where ðρ;φÞ are the polar coordinate on the spatial spec-tral plane, FT indicates the Fourier transform operator, λis the wavelength, f is the focal length of the Fourier lens,and J0 is the Bessel function of the zero order.

By using the equations in the Handbook of Mathema-tical Functions [22], Eq. (3) can be further expressed as

gðρ;θÞ ¼ 1

2πβ2λf

×cos

hð2lþ 1Þarcsin

�πρβλf

�iþ πρ

βλf sin½ð2lþ 1Þarcsinð πρβλfÞ�πρβλf

h1−

�πρβλf

�2i1=2

¼ 1

2πβ2λf

cosnð2lþ 1Þarcsin

�πρβλf

�− arcsin

h πρβλf

ð1þð πρβλfÞ2Þ1=2io

πρβλf

h1 −

�πρβλf

�4i1=2 :

ð4ÞThe Bessel functions in Eq. (3)can take only the real

value for their properties. This means the value of gðρ; θÞcan only take the real value. It can be seen that theπρ=βλf can only take the value range from 0 to 1. The

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function in the cosine function on the numerator ofEq. (4) can take the value range from 0 to ðlþ 1=4Þπ.Figure 1 shows the spatial spectrum of intensity distri-

bution of ideal Bessel beams. Figure 1(a) represents theBessel beam with second order and (b) represents theBessel beam with fifth order. Figures 1(a) and 1(b) takethe absolute value. From Fig. 1 we can see the spatialspectrums have the central bright spots, some brightrings, and dark rings. The more attractive rule is thatthe numbers of the bright rings or dark rings are equalto the topological charge of the Bessel beam.Based on Eq. (4), the radius of the dark rings can be

calculated as

ð2lþ 1Þ arcsin� πρβλf

�− arcsin

264

πρβλf�

1þ�

πρβλf

�2�1=2

375

¼�nþ 1

2

�π; ð5Þ

where n is an integer number more than or equal to 0. ncan take the integer numbers ranging from 0 to l − 1 inEq. (5). The radius of bright rings can be calculated as

ð2lþ 1Þ arcsin� πρβλf

�− arcsin

264

πρβλf�

1þ�

πρβλf

�2�1=2

375

¼ nπ: ð6Þn can take the integer numbers ranging from 1 to l inEq. (6). From Eqs. (5) and (6), we can calculate the radiusof the bright and dark rings accurately.From the above analysis, we can measure the topolo-

gical charge of a high-order Bessel beam by counting thenumber of bright rings or dark rings on the spatial spec-trum. It is well known that the Bessel beam can be trans-formed from a common optical vortex by an axicon. Theaxicon can be used to generate Bessel beams [23] for op-tical trapping, wavefront sensing [24], and so on. It is aconical lens with the phase function of expð−iκrÞ.Figure 2 shows the schematic of the experiment setup

for recording the diffraction intensity pattern of a Besselbeam. An optical vortex transmitted an axicon andformed a Bessel beam. The CCD is set at the suitableplace for recording the diffraction intensity pattern ofthe Bessel beam.Here we assume the optical vortices have the complex

amplitude function of

uðr;φÞ ¼ C

� ffiffiffi2

pr

ω

�lexp

�−

r2

ω2

�expðilφÞ; ð7Þ

where C is a constant, and ω is the beam waist. In the si-mulation, we take κ ¼ 10 π=mm, ω ¼ 2mm, and λ ¼632:8nm. Figures 3 and 4 show the simulation resultsof optical vortices with the function of Eq. (7). Figure 3shows the results of simulation with l ¼ 2; (a1)–(a3)shows the normalized diffraction intensity pattern at z ¼200mm, 1000mm, and 1600mm separately. The imagesshown in Fig. 3 (a1)–(a3) have the same size and havethe gray scale of 28. (b1)–(b3) show the spatial spectrumof (a1)–(a3) separately. Figure 3 (a2) is the intensity dis-tribution of an ideal Bessel beam and (b2) has the samepattern as Fig. 1(a). Figures 3 (a1) and (a3) have the cen-tral weak Bessel distribution. The outer intensity ofFigs. 3 (a1) and (a3) formed the central bright spots inFigs. 3 (b1) and (b3). The patterns shown in Figs. 3 (b1)and (b3) reflect the superposition of the bright spots andFig. 1 (a). Figures 3 (b1) and (b3) have similarities ofFig. 3 (b2).

Figure 4 shows the results of simulation with l ¼ 5.(a1)–(a3) show the normalized diffraction intensity

Fig. 1. Spatial spectrum of intensity distribution of idealBessel beams at (a) second order and (b) fiftth order.

Fig. 2. Schematic of the experiment setup.

Fig. 3. Results of simulation with l ¼ 2. (a1)–(a3) Normalizeddiffraction intensity pattern at z ¼ 200mm, 1000mm, and1600mm. (b1)–(b3) Spatial spectrum of (a1)–(a3).

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pattern at z ¼ 400mm, 1000mm, and 2000mm sepa-rately. The images shown in Figs. 4 (a1)–(a3) have thesame size and have the gray scale of 28. (b1)–(b3) showthe spatial spectrum of (a1)–(a3) separately. Figures 4(a1) and (a3) have the central weak Bessel distribution.Figures 4 (b1)–(b3) have the same pattern as Fig. 1(b).From the simulation results shown in Figs. 3 and 4 we

can see the topological charge of an optical vortex can bemeasured by the experimental system we proposed. Thenumber of bright rings in the spatial spectrum is equal tothe topological charge of the optical vortex. The resultsof the simulation coincide with the above theory. In theexperiment the axicon can be implemented by a spatiallight modulator, and we can measure the topologicalcharge of the optical vortex effectively by analyzingthe spatial spectrum of captured images with a clearor unclear central Bessel distribution.In conclusion, we analyzed the spatial spectrum of the

diffraction intensity pattern of an ideal Bessel beam andfound an implicit rule that the number of the bright ringsin the spatial spectrum is equal to the topological chargeof the Bessel beam. The radius of the bright and darkrings has some relation with the topological charge. It

provides us with a newway for measuring the topologicalcharge of an optical vortex through its diffraction inten-sity pattern after an axicon. The results of the simulationand the theory demonstrate the feasibility of this method.

This work is supported in part by the National NaturalScience Foundation of China (NSFC) under 11074152and the Natural Science Foundation (NSF) of HezeUniversity under XY09WL02.

References

1. L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P.Woerdman, Phys. Rev. A 45, 8185 (1992).

2. K. T. Gahagan and G. A. Swartzlander Jr., Opt. Lett. 21,827 (1996).

3. N. Khonina, V. V. Kotlyar, M. V. Shinkarev, V. A. Soifer, andG. V. Uspleniev, J. Mod. Opt. 39, 1147 (1992).

4. M. Reicherter, T. Haist, E. U. Wagemann, and H. J. Tiziani,Opt. Lett. 24, 608 (1999).

5. A. Vaziri, J. W. Pan, T. Jennewein, G. Weihs, and A.Zeilinger, Phys. Rev. Lett. 91, 227902 (2003).

6. Z. Bouchal and R. Celechovsky, New J. Phys. 6, 131 (2004).7. G. Gibson, J. Courtial, M. Padgett, M. Vasnetsov, V. Pas’ko,

S. Barnett, and S. Franke-Arnold, Opt. Express 12,5448 (2004).

8. J. E. Curtis, B. A. Koss, and D. G. Grier, Opt. Commun. 207,169 (2002).

9. K. Ladavac and D. G. Grier, Opt. Express 12, 1144 (2004).10. M. Harris, C. A. Hill, P. R. Tapster, and J. M. Vaughan, Phys.

Rev. A 49, 3119 (1994).11. M. Padgett, J. Arit, N. Simpson, and L. Allen, Am. J. Phys.

64, 77 (1996).12. J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, and

J. Courtial, Phys. Rev. Lett. 88, 257901 (2002).13. J. Leach, J. Courtial, K. Skeldon, S. M. Barnett, S. Franke-

Arnold, and M. J. Padgett, Phys. Rev. Lett. 92, 013601(2004).

14. H. I. Sztul and R. R. Alfano, Opt. Lett. 31, 999 (2006).15. Z. Wang, Z. Zhang, and Q. Lin, J. Opt. A 11, 085702 (2009).16. G. C. G. Berkhout and M. W. Beijersbergen, Phys. Rev. Lett.

101, 100801 (2008).17. C.-S. Guo, S.-J. Yue, and G.-X. Wei, Appl. Phys. Lett. 94,

231104 (2009).18. C.-S.Guo, L.-L. Lu, andH.-T.Wang,Opt. Lett.34, 3686 (2009).19. G. C. G. Berkhout, M. P. J. Lavery, J. Courtial, M. W.

Beijersbergen, and M. J. Padgett, Phys. Rev. Lett. 105,153601 (2010).

20. J. M. Hickmann, E. J. S. Fonseca, W. C. Soares, and S.Chávez-Cerda, Phys. Rev. Lett. 105, 053904 (2010).

21. L. E. E. de Araujo and M. E. Anderson, Opt. Lett. 36,787 (2011).

22. M. Abramowitz and I. A. Stegun, Handbook of Mathemati-cal Functions (Dover, 1965).

23. V. Garcés-Chávez, D. McGloin, H. Melville, W. Sibbett, andK. Dholakia, Nature 419, 145 (2002).

24. B. Vohnsen, S. Castillo, and D. Rativa, Opt. Lett. 36,846 (2011).

Fig. 4. Results of simulation with l ¼ 5. (a1)–(a3) Normalizeddiffraction intensity pattern at z ¼ 400mm, 1000mm, and2000mm. (b1)–(b3) Spatial spectrum of (a1)–(a3).

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