1. INTRODUCTION

Optical vortices (OVs) in light beams are tightly bound to phase dislocations (or singularities): due to the continuous spatial nature of a field, the presence of these defects implies the vanishing of the field’s amplitude in the singularity; in some types of dislocations, the phase circulates around the singularity and creates a vortex [1]. Nye and Berry [2] used the term “phase dislocation” to define the locus of the zero amplitude of a field. Similarly to crystallography, phase dislocations can be classified in edge, screw, and mixed screw edge [3]. The interest in OVs was raised because fields in which they are included show a helical wavefront structure—developing around the screw dislocation line—entailing the presence of an orbital angular momentum (OAM). This is an important feature that can be exploited in several applications, from optical tweezers, to the generation of N-dimensional quantum states (quNits) hence to a large potential information capacity.

A screw wave dislocation can be defined by means of the integer topological charge Q, which represents the winding number of the phase and can be found by means of a circulation integral around the dislocation line:

As mentioned above, beams carrying optical vortices are characterized by an OAM per photon of $\ell \hslash$ [4, 5]. There are several ways of generating optical vortex beams, and these include cylindrical lenses [5], computer-generated holograms (fork holograms) [6], spatial light modulators, and spiral phase plates (SPPs) [7, 8]. The latter are transparent optical devices that force a field, collinear with the optical axis, to experience an azimuthally dependent optical path; therefore, they can be treated by means of a transmission function $\exp (il\theta )$, where θ is the azimuthal polar coordinate and l is the (integer) phase’s winding number or topological charge: when a Gaussian beam hits an SPP, the outgoing field can be modeled by Laguerre–Gaussian (LG) modes or by a superposition of them:

Of course it is possible to obtain beams with noninteger OAM by enclosing in the beam a mixed screw-edge dislocation [8] or, as will be shown later, by displacing the SPP with respect to the incident beam’s axis [9, 10].

Among nonlinear optical interactions, second-harmonic (SH) and parametric processes concerning fields carrying optical vortices have attracted interest because they grant the possibility of manipulating the dynamics of the phase singularities [11, 12, 13, 14, 15, 16, 17]. This means that, together with the frequency doubling, the SH-generated field shows phase defects correlated to the ones present in the pump beam. The properties of OAM in nonlinear interactions of OV beams have been widely investigated in collinear SH processes where an on-axis vortex pump beam impinged on quadratic nonlinear crystal [11, 12]. In [11], collinear SHG was studied using LG beams as the pump field and it was shown that the SH field generated from an ${\mathrm{LG}}_{n0}$ (n is an integer) doubles its OAM, thus verifying the momentum conservation. However, there are no studies known to the authors on noncollinear SHG involving fractional OAM beams, and only a little attention was placed on parametric processes with fields carrying off-axis vortices [13].

In this paper we present an experiment of noncollinear second-harmonic generation (SHG), where two femtosecond input beams, at the fundamental frequency (FF) of $830\mathrm{nm}$, carrying a fractional OAM, impinge on a Type I nonlinear crystal of BBO [18]. The noninteger angular momentum (AM) is obtained by means of a fractional SPP, which impresses on the incident field a mixed screw-edge dislocation; the latter is then displaced from the beam’s axis. The “specular” configuration of the pumping beams, i.e., each beam carries the same OAM but with the opposite sign, allows obtaining an SH field with different patterns, all having a total OAM equal to zero.

2. HALF-INTEGER SPIRAL PHASE PLATE

It is known that light beams can carry AM. This can be split in two independent components, the spin angular momentum (SAM), bound to the polarization, and the OAM generated by the tangential component of the Poynting vector, thus dependent on the spatial field distribution. In terms of AM per photon, SAM gives a contribution of $\pm \hslash$ if the field is circularly polarized (0 otherwise). In our analysis we consider linearly polarized fields in order to neglect the spin contribution.

Because we are considering paraxial beams, which means that OAM must be constant during propagation, we can refer to the transverse plane at any fixed z [4]. In order to calculate the intrinsic OAM, i.e., not dependent on the choice of the axis, a suitable reference system that guarantees a zero transverse momentum has to be chosen [19]. We are dealing with off-axis OVs. Thus, the choice of a z axis coincident with the propagation axis is not so obvious: the right choice should be fixing the origin in the center of mass of the vortex beam; however, the vorticity of the field is introduced by the SPP on the input Gaussian beam. Hence, it is possible to choose the z axis parallel to the propagation axis of the beam and to place the origin at the center of the beam in the near field [20].

In this way, following a general treatment, the AM density can be defined as $L=\mathbf{r}\times \mathbf{P}$, where P is the linear momentum density of the beam. If we consider a linearly polarized field propagating along z with a transverse distribution $u(r,\theta ,z)$

It can be seen that the z component of L depends only on the azimuthal component of the linear momentum vector [21]. Choosing the transverse plane at $z=0$, by integrating the momentum density L over the transverse plane and dividing by the field energy per unit length, the OAM per photon—in ℏ units—can be obtained as follows:

Oemrawsingh et al. [20] observed that displacing a fork hologram from the beam’s axis would result in a decrement of the OAM of the field. In a very similar fashion, the OAM can be tuned by moving an SPP off the axis: the momentum obtained from the fractional SPP can be varied between $1/2$ and 0 through the displacement of the SPP itself. The output momentum thus results smaller, following the Gaussian law presented in [20] $Q*\exp (-2r_v^2/w_0^2)$, where Q is the topological charge impressed by the device, $r_v$ is the displacement, and $w_0$ is the spot size of the incident beam. In this way the OAM can be tuned to a well-defined value. The decompositions of Eqs. (2, 6) are still valid to model an off-axis dislocation and to evaluate its OAM.

3. NONLINEAR INTERACTION

3A. Experiment

We consider a noncollinear SHG scheme in which two fractional helical beams generated by an SPP impinge with opposite OAMs on a nonlinear crystal (Type I BBO) with small angles of incidence. An SH signal that carries information on the input phase’s dislocations is generated. We followed the evolution of the SH field’s spatial distribution as the SPP is displaced from the beam’s axis and evaluated the OAM of the SH generated beam. Subsequently, we compared the experimental results with the numerical ones obtained by using the modal expansion of Eq. (2).

Figure 1 shows the experimental setup for the noncollinear SHG interaction, together with the one used for displacing the SPP (inset). In our experiment, the output of a mode-locked femtosecond Ti:sapphire laser system tuned at $\lambda =830\mathrm{nm}$ ($76\mathrm{MHz}$ repetition rate, $130\mathrm{fs}$ pulse width), collimated and spatially filtered by a $50\mathrm{\mu m}$ pinhole, hits an SPP, with a spot- size of about $500\mathrm{\mu m}$. The distance between the lens and the SPP is $50\mathrm{cm}$, so as the distance from the lens to the BBO crystal. The SPP [22] was designed to operate at a wavelength of $830\mathrm{nm}$ introducing a fractional topological charge, thus an OAM per photon in ℏ units, of $1/2$. The center of the SPP is dominated by the finite size of the height anomaly of the order of $50\mathrm{\mu m}$, the SPP diameter is about $8\mathrm{mm}$.

The generated vortex beam is split in two beams of about the same intensity and reflected by mirrors M1 and M2; the tightly focused beams intersect in the focus region with an angle of $7°$ with respect to one another, on the BBO crystal of $1\mathrm{mm}$ length. The SPP is mounted on a $XY$ translational stage in order to perform a full scan of the device. Therefore, the phase plate was displaced from the beam’s propagation axis both in the horizontal and vertical directions, in a $8\mathrm{mm}\times 8\mathrm{mm}$ range, with a $100\mathrm{\mu m}$ step. The displacement of the SPP causes the horizontal movement (besides the vertical one) of the mixed screw-edge dislocations, enclosed in the fields reflected by the mirrors M1 and M2, in opposite directions; because in this way one of the FF fields experiences an even number of reflections while the other an odd number, fields with opposite values of OAM impinge on the input plane of the crystal.

At the output plane of the crystal, the two FF fields come out with their collinear SH signals with the same angle that they had at the input plane, while the outgoing noncollinear SH signal propagates along the bisector of the total incidence angle. The latter is finally filtered and collected by a CCD camera about $30\mathrm{cm}$ away from the BBO crystal. The noncollinear SHG occurs under Type I phase-matching conditions.

An example of experimental results is shown in Fig. 2, where the horizontal displacement of the SPP gives rise to an SH field with different configurations. The experimental results (bottom row) come from the horizontal scan in the central area of the SPP, i.e., for $y=0$. The upper row is the near-field distribution calculated numerically, and the middle row is the far-field SHG distribution (see next section). We attribute the slight difference among experiments and numerical calculations to the finite size of the height anomaly of the SPPs.

At the output plane of the crystal, we expect an SH field that carries zero OAM, namely the sum of the incident angular momenta [15, 16].

3B. Numerical Simulations

Numerical results are obtained by modeling the input fields by using Eq. (2) both for the on- and off-axis cases. The coefficients $C_{l,p}$ were calculated numerically using a $200\times 200$ mesh and integrating over a spatial range equal to 5 times the beam waist.

The near-field shape of the intensity profile of the FF beams is shown in Figs. 3a, 3b, when the input Gaussian beam is located at the center of the SPP, thus generating an OAM per photon of $1/2$.

Considering a phase-matched interaction, neglecting walk-off and under paraxial approximation [11, 15], the near-field noncollinear SH intensity is given by

The phase of the FF and SH fields is shown in Fig. 4, where the presence of phase singularities is evident. It can be noticed that the SH signal [Fig. 4c] carries both the phase singularities of the incident fields, but the helical phase winds in opposite directions.

This result, which comes out from the simplistic model adopted [11] is not surprising. In fact, in the nonlinear interaction, the phases of the FFs are added; thus, the singularities compose in order to cancel each other.

In order to perform a comparison among experimental result and modeling, far-field calculations of the SH signal have been carried out; in this way, still under paraxial approximation, the intensity of the SH field in a plane transverse to the noncollinear SH field’s propagation direction is given by

3C. Orbital Angular Momentum Calculations

We used both Eqs. (5, 6) to estimate numerically the OAM of the fields. The OAM of the SH signal was numerically calculated for each step of the displacement by using Eq. (5). As expected, the OAM of the generated field is conserved [11], producing a beam with an AM equal to the sum of the individual OAMs of the two input beams: as a consequence, ${\ell }_{\mathrm{SH}}$ is found to be always 0. Obviously, because the dislocation is not purely a screw type, in the generated field only the screw components of the singularity cancel, while the edge dislocations of both fields, which do not possess a direction, remain unaffected, thus creating the nodal lines that can be seen in the figure above.

The behavior of the OAM of the SH as a function of the SPP’s displacement is presented in Fig. 6 together with the AM of the FFs. As pointed out before, the value of the OAM of the SH is always zero whichever is the displacement, namely regardless of the FF’s momentum.

The zero OAM of the SH field can also be put in evidence by calculating the Poynting vector; Fig. 7 shows the Poynting vector of the SH field obtained when the displacement of the SPP is zero [as shown in Fig. 3c]: it can be seen that the arrows are always directed along the same direction; thus, the Poynting vector does not rotate, implying a zero AM.

4. CONCLUSIONS

This work investigated the process of noncollinear SHG involving beams carrying phase singularities generated by a half-integral SPP. The focus was posed on the possibility of reaching, starting from fields with a certain OAM, different spatial configurations, all characterized by the absence of OAM, simply by moving the SPP from the beam’s axis.

Experimental results were obtained by performing a transverse scan of the SPP and by letting interact fields with opposite AM. These results were then successfully compared to numerical simulations and it was found that the OAM of the generated SH field is always zero and is independent on the displacement of the SPP. Furthermore, the transverse component of the Poynting vector of the SH beam was calculated in order to evidence that the energy of the considered field does not circulate, confirming then the absence of the OAM in all the spatial configuration obtained.

ACKNOWLEDGMENTS

We thank J. P. Woerdman and Eric Eliel at Huygens Laboratory, Leiden, for producing and providing the SPPs used in our work.

 

Fig. 1 Experimental setup of noncollinear SHG. The collimated beam impinges on the SPP, and then it is split and focused on the nonlinear crystal with relative angles of $\pm 3.5°$ (total $7°$). Bottom-left corner, scan of the Gaussian beam performed by moving the SPP in the horizontal direction.

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Fig. 2 Noncollinear SHG signal obtained by displacing the SPP along the x axis at $y=0$, for six different x displacement values. (a) Numerically simulated near field; (b) numerically simulated far field; and (c) experimental results.

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Fig. 3 Numerically estimated near-field intensity profiles of the FF beams, (a) and (b), and of the noncollinear SH beam; in FF beams the singularity shifts in opposite directions. The SH beam’s profile carries the singularities of both the FF beams, maintaining their original orientation.

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Fig. 4 Phase plots of the electric field for the FFs, (a) and (b), and for the SH (c), corresponding to the configuration of fields given in Fig. 3. The phase changes from $-\pi$ (dark blue), to 0 (light green), to π (dark red).

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Fig. 5 Frame of the far-field movie of the SHG signal as a function of the SPP’s displacement; the top frame corresponds to zero displacement ($x=0$) (Media 1).

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Fig. 6 OAM of pump and SH fields with the SPP’s displacement. As the device is moved from the beam’s axis, both the FF beams [blue (upper) and black (lower) lines] reduce the absolute value of their OAM from the initial values of $\pm 0.5$. The SH’s OAM [red (middle) line] is always the sum of the other two, that is, zero.

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Fig. 7 Transverse Poynting vector of the SH field generated in the case of on-axis SPP. The absence of rotation of the arrows implies that there is no screw dislocation; therefore, there is no OAM.

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