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Optics Express

Optics Express

  • Editor: Michael Duncan
  • Vol. 14, Iss. 11 — May. 29, 2006
  • pp: 4577–4582
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Hermite-Bessel beams and the geometrical representation of nondiffracting beams with orbital angular momentum

W. C. Soares, D. P. Caetano, and J. M. Hickmann  »View Author Affiliations


Optics Express, Vol. 14, Issue 11, pp. 4577-4582 (2006)
http://dx.doi.org/10.1364/OE.14.004577


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Abstract

We investigate theoretically and experimentally the decomposition of high-order Bessel beams in terms of a new family of nondiffracting beams, referred as Hermite-Bessel beams, which are solutions of the Helmholtz equation in Cartesian coordinates. Based on this decomposition we develop a geometrical representation of first-order Bessel beams, equivalent to the Poincaré sphere for the polarization states of light and implement an unitary transformation within our geometrical representation using linear optical elements.

© 2006 Optical Society of America

1. Introduction

Usually, light beams possessing orbital angular momentum (OAM) are described by Laguerre-Gauss (LGpl) beams, where the amount OAM is given by lh̄ 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 85, 8185–8189 (1992). [CrossRef]

]. These beams are solutions of the paraxial equation written in cylindrical coordinates and can be decomposed in terms of Hermite-Gauss beams (HGmn) [2

2. E. Abramochkin and V. Volostnikov, “Beam transformations and nontransformed beams,” Opt. Commun. 83, 123–135 (1991). [CrossRef]

]. HGmn beams are also solutions of the paraxial equation, but, in contrast to the former, they are derived in Cartesian coordinates and do not carry any OAM.

Alternatively, one can describe light beams with OAMby using high-order Bessel beams [5

5. K. Volke-Sepulveda, V. Garcés-Chéz, S. Chávez-Cerda, J. Arlt, and K. Dholakia, “Orbital angular momentum of a high-order Bessel light beam,” J. Opt. B: Quantum and Semiclass. Opt. 4, S82–S89 (2002). [CrossRef]

], which are solutions of the Helmholtz equation in cylindrical coordinates and present nondiffracting properties [6

6. J. Durnin “Exact solutions for nondiffracting beams.I The scalar theory,” J. Opt. Soc. Am. A4, 651–654(1987); J. Durnin, J. J. Miceli Jr., and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett.58, 1499–1501 (1987). [CrossRef]

]. Recently, it has been demonstrated that Bessel beams are one of many solution families for the Helmholtz equation that possesses propagation invariance properties. Other families are obtained solving the Helmholtz equation in Cartesian, elliptic cylindrical, and parabolic coordinates corresponding to plane waves, Mathieu beams [7

7. J. C. Gutiérrez-Vega, M. D. Iturbe-Castillo, and S. Chávez-Cerda“Alternative formulation for invariant optical fields: Mathieu beams,” Opt. Lett.25, 1493–1495 (2000); J. C. Gutiérrez-Vega, M. D. Iturbe-Castillo, G. A. Ramírez, E. Tepichín, R. M. Rodríguez-Dagnino, S. Chávez-Cerda, and G.H.C. New“Experimental demonstration of optical Mathieu beams,” Opt. Commun.195, 35–40 (2001); S. Chávez-Cerda, M. J. Padgett, I. Allison, G. H. C. New, J. C. Gutiérrez-Vega, A. T. O’Neil, I. MacVicar, and J. Courtial, “Holographic generation and orbital angular momentum of high-order Mathieu beams,” J. Opt. B: Quantum Semiclass. Opt.4, S52–S57 (2002). [CrossRef]

], and parabolic beams [8

8. M. A. Bandres, J. C. Gutiérrez-Vega, and S. Chávez-Cerda“Parabolic nondiffracting optical wave fields,” Opt. Lett.29, 44–46 (2004); C. López-Mariscal, M. A. Bandres, J. C. Gutiérrez-Vega, and Sabino Chávez-Cerda, “Observation of parabolic nondiffracting optical fields,” Opt. Express13, 2364–2369 (2005). [CrossRef] [PubMed]

], respectively. In this work, the decomposition of high-order Bessel beams is investigated both theoretical and experimentally in order to find a equivalent decomposition to the LG beams. Based on this decomposition, a geometrical representation equivalent to the sphere of first-order modes is developed. We also have investigated unitary transformations within the equivalent sphere using linear optical elements, analogous to a polarization state rotation.

2. Theory

Let us start with a high-order Bessel beam given by BBl(z,ρ,ϕ)=eikzzJl(kρ)eilϕ, where kz and k are the modulus of the longitudinal and transversal components of the wavevector, respectively. Jl(k ρ) is the Bessel function of l order and the OAM is equal to lh̄ per photon [5

5. K. Volke-Sepulveda, V. Garcés-Chéz, S. Chávez-Cerda, J. Arlt, and K. Dholakia, “Orbital angular momentum of a high-order Bessel light beam,” J. Opt. B: Quantum and Semiclass. Opt. 4, S82–S89 (2002). [CrossRef]

]. Writing the azimuthal phase factor in terms of the Cartesian coordinates x=ρ cosϕ and y=ρ sinϕ, and by using the recurrence relation of the Bessel functions Jl1(ξ)+Jl+1(ξ)=2lξJl(ξ) we obtain.

BBl(z,ρ,ϕ)=kl2leikzz[Jl1(kρ)+Jl+1(kρ)]1(kρ)l1[x+iy]l
(1)

In the particular case where l=±1, Eq. 1 yields

BB±1(z,ρ,ϕ)=±k2eikzz{[J0(kρ)+J2(kρ)]x±i[J0(kρ)+J2(kρ)]y}.
(2)

Let us define

HB10=[J0(kρ)+J2(kρ)]H1(x)H0(y)
HB01=[J0(kρ)+J2(kρ)]H0(x)H1(y),
(3)

where Hn(x) and Hn(y) are Hermite polynomials of degree n. It is possible to demonstrate that HB 10 and HB 01 are solutions of the Helmholtz equation in Cartesian coordinates. For this reason, HB 10 and HB 01 will be referred as Hermite-Bessel beams. Therefore, Eq. 2 can be written as

BB1=keikzz(HB10+iHB01)
BB1=keikzz(HB10iHB01).
(4)

This result shows that a first-order Bessel beam can be decomposed as a superposition of the so-called Hermite-Bessel (HB) beams with a π/2 phase difference, which is analogous to the decomposition of LG 0±1 beams in terms of HG 01 and HG 10 beams. It is worth noting that this decomposition was obtained in terms of different solutions of the Helmholtz equation in Cartesian coordinates rather than plane waves, as could be expected based on the decomposition of the LG beams. Our result also indicates that the HB beams form a new family of nondiffracting beams, since they can be expressed as a superposition of Bessel beams. If the phase difference between the HB beams is 0 or π we obtain light beams which are analogous to the HG beams at +45° and -45°, respectively. This procedure can be extended for orders larger than 1 using Eq. 1 to obtain the decomposition of high-order Bessel beams in terms of the corresponding high-order HB beams.

Using theses results we are able to construct a geometrical representation describing light beams possessing OAM in terms of nondiffracting beams, namely BB and HB beams, equivalent to the Poincaré sphere for representing different polarization states of light and the first-order modes sphere for representing light beams possessing OAM in terms of LG and HG. Figure 1 illustrates such representation, where we are representing the BB and HB beams by density (a) and tridimensional (b) plots of their intensity profiles. In the density plot a gray scale bitmap represents the transverse profile of the intensity, while in the tridimensional plot we are representing the behavior of the functions that describe the beams. As we can see, at the top and the bottom of the sphere we are representing the BB beams, while at the halfway between these two points we are representing the different HB beams. If compared with the Poincaré sphere, in this representation HB10 is equivalent to vertical polarization, HB01 is equivalent to horizontal polarization, BB1 is equivalent to left-handed circular polarization, and BB-1 is equivalent to right-handed circular polarization. Simliar to Poincaré sphere and sphere of firstorder modes, each point on the surface of the sphere represents an angular momentum state of the light beam and can be written in terms of a linear superposition of the Hermite-Bessel beams

3. Experiment

An experiment has been performed to confirm the decomposition showed in Eq. 4. The experimental setup is depicted in Fig. 2. First, a laser beam is passed through a spatial filter, in order to be used as a plane wave beam. After, it enters into two Mach-Zehnder interferometers, referred as ABCD interferometer and DEFG interferometer. Note that the DEFG interferometer is a non-symmetrical interferometer, since it has an additional reflection in one of the arms which is provided by the penta prism. This kind of non-symmetrical interferometer has been used as a transverse-mode beam splitter [10

10. H. Sasada and M. Okamoto, “Transverse-mode beam splitter of a light beam and its application to quantum cryptography,” Phys. Rev. A 68, 012323 (2003). [CrossRef]

], equivalent to a polarizing beam splitter. The light beams emerging from the output ports of these interferometers are detected by a charge-coupled device (CCD) connected to a lab computer by a video digitalizing card.

Fig. 1. The Poincaré-equivalent sphere for representing first order cylindrical and Cartesian nondifracting light beams: a) BB and HB modes are represented by a density plot of their intensity profiles; b) BB and HB modes are represented by a tridimensional plot of their intensity.
Fig. 2. Experimental setup, where M1, M2, and M3 are mirrors; BS1, BS2, BS3 and BS4 are beam-splitters; and SF is a spatial filter.

The hologram showed in the experimental setup is a computer generated hologram [9

9. A. Vasara, J. Turunen, and A. T. Friberg, “Realization of general nondiffracting beams with computer-generated holograms,” J. Opt. Soc. Am. A 6, 1748–1754 (1989). [CrossRef] [PubMed]

], which produces a first-order Bessel beam (l=+1). The intensity profile of the generated beam is shown in Fig. 3(a). The measurement was performed by blocking the AB arm of the ABCD interferometer and the EF arm of the DEFG interferometer. To demonstrate that this beam presents OAM, the phase structure was measured by unblocking the AB arm of the ABCD interferometer. The resultant interferogram of the Bessel beam with a plane wave is shown in Fig. 3(b). Note that there is a bifurcation instead of parallel fringes in the central region of the pattern. This bifurcation is a signature of a light beam possessing OAM corresponding to l=+1,as we expect for a BB 1 beam.

Fig. 3. a) Intensity profile of a first order Bessel beam with l=+1; b) Interference pattern of the first order Bessel beam with l=+1.

Fig. 4. Intensity profile measured at the output ports of the DEFG interferometer. a) HB 10 beam; b) HB 01 beam. In the multimedia file it is shown a movie where the output ports of the DEFG interferometer are measured by rotating the glass plate.

Using the produced BB 1 beam as a initial state, we proceed to implement an unitary transformation within our sphere of nondiffracting beams showed in Fig. 1. The idea is to transform the initial state into BB -1 beam, which is equivalent to changing a left circular polarization state to a right circular polarization state. This kind of transformation can be done by using a pair of cylindrical lenses [11

11. 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 (1993). [CrossRef]

]. The experiment was realized by blocking the EF arm of the DEFG interferometer and inserting a pair of cylindrical lenses in the DG arm. First we measured the intensity profile of the beam after crossing the cylindrical lenses while blocking the AB arm of the ABCD interferometer. The result is shown in Fig. 5(a), indicating that the beam has a first-order Bessel beam profile. Unblocking the AB arm of the ABCD interferometer allowed us to measure the interference pattern in order to verify the signal of the OAM. The result is shown in Fig. 5(b). As we can see the bifurcation has an inverted orientation with respect to the BB 1 shown in Fig. 3(b), demonstrating that l=-1 for this beam. Therefore, our unitary transformation was successfully applied.

Fig. 5. a) Intensity profile of the BB 1 beam after the passage by the pair of cylindrical lens; b) Interference pattern of the of the BB 1 beam after the passage by the pair of cylindrical lens. [Media 1]

4. Conclusion

In summary, we have demonstrated that Bessel beams can be decomposed in terms of a new class of nondiffracting beams that are solutions of the Helmholtz equation in Cartesian coordinates, named by us as Hermite-Bessel beams. With this decompostion we developed a SU(2) structure to represent light beams with OAM in terms of nondiffracting beams. Finally, we demonstrated an unitary transformation which transforms a BB 1 into a BB -1 using linear optical elements. Besides the fundamental contribution on the description of nondiffracting light beams possessing orbital angular momentum, our findings may have important applications in the field of free-space quantum key distribution. In a similar way to the protocol using polarization, the orbital angular momentum would be used to create entangled photon states but with the very advantageous nondiffracting property of Bessel beams.

Acknowledgments

The authors thank the financial support from Instituto do Milênio de Informação Qûantica, CAPES, CNPq, FAPEAL, PADCT, Nanofoton, and ANP-CTPETRO.

References and links

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 85, 8185–8189 (1992). [CrossRef]

2.

E. Abramochkin and V. Volostnikov, “Beam transformations and nontransformed beams,” Opt. Commun. 83, 123–135 (1991). [CrossRef]

3.

M. J. Padgett and J. Courtial, “Poincaré-sphere equivalent for light beams containing orbital angular momentum,” Opt. Lett. 24, 430–432 (1999). [CrossRef]

4.

G. S. Agarwal, “SU(2) structure of the Poincaré sphere for light beams with orbital angular momentum,“J. Opt. Soc. Am. B 16, 2914–2916 (1999). [CrossRef]

5.

K. Volke-Sepulveda, V. Garcés-Chéz, S. Chávez-Cerda, J. Arlt, and K. Dholakia, “Orbital angular momentum of a high-order Bessel light beam,” J. Opt. B: Quantum and Semiclass. Opt. 4, S82–S89 (2002). [CrossRef]

6.

J. Durnin “Exact solutions for nondiffracting beams.I The scalar theory,” J. Opt. Soc. Am. A4, 651–654(1987); J. Durnin, J. J. Miceli Jr., and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett.58, 1499–1501 (1987). [CrossRef]

7.

J. C. Gutiérrez-Vega, M. D. Iturbe-Castillo, and S. Chávez-Cerda“Alternative formulation for invariant optical fields: Mathieu beams,” Opt. Lett.25, 1493–1495 (2000); J. C. Gutiérrez-Vega, M. D. Iturbe-Castillo, G. A. Ramírez, E. Tepichín, R. M. Rodríguez-Dagnino, S. Chávez-Cerda, and G.H.C. New“Experimental demonstration of optical Mathieu beams,” Opt. Commun.195, 35–40 (2001); S. Chávez-Cerda, M. J. Padgett, I. Allison, G. H. C. New, J. C. Gutiérrez-Vega, A. T. O’Neil, I. MacVicar, and J. Courtial, “Holographic generation and orbital angular momentum of high-order Mathieu beams,” J. Opt. B: Quantum Semiclass. Opt.4, S52–S57 (2002). [CrossRef]

8.

M. A. Bandres, J. C. Gutiérrez-Vega, and S. Chávez-Cerda“Parabolic nondiffracting optical wave fields,” Opt. Lett.29, 44–46 (2004); C. López-Mariscal, M. A. Bandres, J. C. Gutiérrez-Vega, and Sabino Chávez-Cerda, “Observation of parabolic nondiffracting optical fields,” Opt. Express13, 2364–2369 (2005). [CrossRef] [PubMed]

9.

A. Vasara, J. Turunen, and A. T. Friberg, “Realization of general nondiffracting beams with computer-generated holograms,” J. Opt. Soc. Am. A 6, 1748–1754 (1989). [CrossRef] [PubMed]

10.

H. Sasada and M. Okamoto, “Transverse-mode beam splitter of a light beam and its application to quantum cryptography,” Phys. Rev. A 68, 012323 (2003). [CrossRef]

11.

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 (1993). [CrossRef]

OCIS Codes
(030.4070) Coherence and statistical optics : Modes
(140.3300) Lasers and laser optics : Laser beam shaping

ToC Category:
Coherence and Statistical Optics

History
Original Manuscript: March 31, 2006
Revised Manuscript: May 8, 2006
Manuscript Accepted: May 9, 2006
Published: May 29, 2006

Citation
W. C. Soares, D. P. Caetano, and J. M. Hickmann, "Hermite-Bessel beams and the geometrical representation of nondiffracting beams with orbital angular momentum," Opt. Express 14, 4577-4582 (2006)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-11-4577


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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 85, 8185-8189 (1992). [CrossRef]
  2. E. Abramochkin and V. Volostnikov, "Beam transformations and nontransformed beams," Opt. Commun. 83, 123-135 (1991). [CrossRef]
  3. M. J. Padgett and J. Courtial, "Poincaré-sphere equivalent for light beams containing orbital angular momentum," Opt. Lett. 24, 430-432 (1999). [CrossRef]
  4. G. S. Agarwal, "SU(2) structure of the Poincaré sphere for light beams with orbital angular momentum," J. Opt. Soc. Am. B 16, 2914-2916 (1999). [CrossRef]
  5. K. Volke-Sepulveda, V. Garcés-Chéz, S. Chávez-Cerda, J. Arlt, and K. Dholakia,"Orbital angular momentum of a high-order Bessel light beam," J. Opt. B: Quantum and Semiclass. Opt. 4, S82-S89 (2002). [CrossRef]
  6. J. Durnin, "Exact solutions for nondiffracting beams.I The scalar theory," J. Opt. Soc. Am. A 4, 651-654(1987);J. Durnin, J. J. MiceliJr., and J. H. Eberly, "Diffraction-free beams," Phys. Rev. Lett. 58, 1499-1501 (1987). [CrossRef]
  7. J. C. Gutiérrez-Vega, M. D. Iturbe-Castillo, and S. Chávez-Cerda, "Alternative formulation for invariant optical fields: Mathieu beams," Opt. Lett. 25, 1493-1495 (2000);J. C. Gutiérrez-Vega, M. D. Iturbe-Castillo, G. A. Ramírez, E. Tepichín, R. M. Rodríguez-Dagnino, S. Chávez-Cerda, and G.H.C. New, "Experimental demonstration of optical Mathieu beams," Opt. Commun. 195, 35-40 (2001);S. Chávez-Cerda, M. J. Padgett, I. Allison, G. H. C. New, J. C. Gutiérrez-Vega, A. T. O’Neil, I. MacVicar, and J. Courtial, "Holographic generation and orbital angular momentum of high-order Mathieu beams," J. Opt. B: Quantum Semiclass. Opt. 4, 52-57 (2002). [CrossRef]
  8. M. A. Bandres, J. C. Gutiérrez-Vega, and S. Chávez-Cerda, "Parabolic nondiffracting optical wave fields," Opt. Lett. 29, 44-46 (2004);C. López-Mariscal, M. A. Bandres, and J. C. Gutiérrez-Vega, and Sabino Chávez-Cerda, "Observation of parabolic nondiffracting optical fields," Opt. Express 13, 2364-2369 (2005). [CrossRef] [PubMed]
  9. A. Vasara, J. Turunen, and A. T. Friberg, "Realization of general nondiffracting beams with computer-generated holograms," J. Opt. Soc. Am. A 6, 1748-1754 (1989). [CrossRef] [PubMed]
  10. H. Sasada and M. Okamoto, "Transverse-mode beam splitter of a light beam and its application to quantum cryptography," Phys. Rev. A 68, 012323 (2003). [CrossRef]
  11. 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 (1993). [CrossRef]

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