## 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 (*LG _{pl}*) beams, where the amount OAM is given by

*lh̄*per photon [1]. These beams are solutions of the paraxial equation written in cylindrical coordinates and can be decomposed in terms of Hermite-Gauss beams (

*HG*) [2].

_{mn}*HG*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.

_{mn}On the other hand, it is well known that light beams also may carry spin angular momentum, that can be associated with a circular polarization state of the field. The spin angular momentum is given by +*h̄* for a right-hand circularly polarized light beam and -*h̄* for a left-hand circularly polarized beam. A left-hand circularly polarized light beam can be expressed as a superposition of a horizontal and a vertical polarized light beam possessing a relative phase delay of *π*/2. An analogous decomposition is found in the case of light beams with OAM; for example, *LG*
_{0}±_{1} beams can be decomposed as *HG*
_{10}±*iHG*
_{01}. Based on this similarity, Padgett and Courtial [3] proposed a geometrical representation for light beams with OAM. This representation is called “sphere of first-order modes.” Later, it was demonstrated by Agarwal [4], that these geometric representations are related to the *SU*(2) group of the transformations for both polarization states and transverse modes.

Alternatively, one can describe light beams with OAMby using high-order Bessel beams [5], which are solutions of the Helmholtz equation in cylindrical coordinates and present nondiffracting properties [6]. 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], and parabolic beams [8], 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 $B{B}_{l}(z,\rho ,\varphi )={e}^{i{k}_{z}z}{J}_{l}\left({k}_{\perp}\rho \right){e}^{il\varphi}$, where *k _{z}* and

*k*

_{⊥}are the modulus of the longitudinal and transversal components of the wavevector, respectively.

*J*(

_{l}*k*

_{⊥}

*ρ*) is the Bessel function of

*l*order and the OAM is equal to

*lh̄*per photon [5]. Writing the azimuthal phase factor in terms of the Cartesian coordinates

*x*=

*ρ*cos

*ϕ*and

*y*=

*ρ*sin

*ϕ*, and by using the recurrence relation of the Bessel functions ${J}_{l-1}\left(\xi \right)+{J}_{l+1}\left(\xi \right)=\frac{2l}{\xi}{J}_{l}\left(\xi \right)$ we obtain.

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

Let us define

$$H{B}_{01}=\left[{J}_{0}\left({k}_{\perp}\rho \right)+{J}_{2}\left({k}_{\perp}\rho \right)\right]{H}_{0}\left(x\right){H}_{1}\left(y\right),$$

where *H _{n}*(

*x*) and

*H*(

_{n}*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

$$B{B}_{-1}=-{k}_{\perp}{e}^{i{k}_{z}z}\left(H{B}_{10}-iH{B}_{01}\right).$$

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], 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.

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

The next step is to verify the decomposition of the produced Bessel beam. Now, the AB arm of the ABCD interferometer is blocked, while the EF arm of the DEFG interferometer is unblocked. When the *BB*
_{1} enters the interferometer each orthogonal transverse component of the beam can be observed in a different output port, because the interferometer is working as a transverse mode beam-splitter. Rotating the glass plate placed in the EF arm is possible to adjust the relative phase between the transverse components. The intensity profiles at the output ports are shown in Fig. 4 for the *HB*
_{10} beam (a) and the *HB*
_{01} beam (b), reproducing the theoretical result shown in Eq. 4.

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]. 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.

## 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.

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