Correcting vortex splitting in higher order vortex beams

Correcting vortex splitting in higherorder vortex beams

Richard Neo,1 Shiaw Juen Tan,2,3 Xavier Zambrana-Puyalto,2,3 SergioLeon-Saval,1 Joss Bland-Hawthorn,1 and Gabriel Molina-Terriza2,3,∗1Institute of Photonics and Optical Science (IPOS), School of Physics, University of Sydney,

NSW 2006, Australia2ARC Centre of Excellence for Engineered Quantum Systems (EQuS), Macquarie University,

NSW 2109, Australia3Department of Physics and Astronomy, Macquarie University, NSW 2109, Australia

∗[email protected]

Abstract: We demonstrate a general method for the first order com-pensation of singularity splitting in a vortex beam at a single plane. Bysuperimposing multiple forked holograms on the SLM used to generatethe vortex beam, we are able to compensate vortex splitting and generatebeams with desired phase singularities of order � = 0, 1, 2, and 3 in oneplane. We then extend this method by application of a radial phase, in orderto simultaneously compensate the observed vortex splitting at two planes(near and far field) for an � = 2 beam.

© 2014 Optical Society of America

OCIS codes: (080.4865) Optical vortices; (090.1000) Aberration compensation; (070.6120)Spatial light modulators; (230.6120) Spatial light modulators.

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#206010 - $15.00 USD Received 5 Feb 2014; revised 6 Apr 2014; accepted 10 Apr 2014; published 17 Apr 2014(C) 2014 OSA 21 April 2014 | Vol. 22, No. 8 | DOI:10.1364/OE.22.009920 | OPTICS EXPRESS 9920

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1. Introduction

The control of the angular momentum of a paraxial beam using its spatial properties has stimu-lated substantial research with applications in many areas of physics for example: spatial modemultiplexing for telecommunications [1], optical spanners [2], quantum optics [3, 4] and as-tronomy [5–9]. For many of these applications (e.g. vortex coronography [10]) it is required tocontrol the phase singularities, also called optical vortices, of the paraxial field. The control ofphase singularities in a paraxial beam is motivated because in this regime, the phase singulari-ties of the beam are directly associated with the density of orbital angular momentum (OAM)that the beam carries. Outside the paraxial approximation, the split of the total angular momen-tum into orbital and spin components becomes problematic. Nevertheless, in some applicationsthe control of the phase singularities in the paraxial regime determines the local structure of afocused beam. This is of paramount importance to control the interaction of light and matter atthe nanoscale [11].

Paraxial beams with a well defined value of OAM, �, have a cylindrically symmetric in-tensity profile. They present an optical vortex of charge � at the centre, i.e. the phase of thebeam twists around the centre of the beam by 2π� radians. Thus, the intensity at the positionof the singularity is zero. Unfortunately, implementing beams with higher order singularitiesis challenging. It has been observed that, due to any uncontrolled breaking of the symmetry,higher order phase singularities split into sets of lower order singularities [12, 13]. Typically,the splitting results in |�| singularities of order ±1 depending on the sign of �. The splittingof the central null of a vortex beam suggests coherent interference occurring between multipleco-propagating modes, either due to noise or imperfections of the set-up. Clearly, the resultingfield will be a superposition of different OAM modes [14–18].

In this article we show a very simple and effective technique in order to compensate for sucheffects when producing angular momentum states. In order to compensate for the interference,we generate multiple collinear beams each with some orbital angular momentum. Proper con-trol of the weighting and relative phases using a phase-only spatial light modulator allows usto generate optical beams with stable high order phase singularities. This has previously beendemonstrated using non-collinear beams for the case �= 2 [19]. The method of producing sev-


eral interfering copropagating modes is also used in the generation of “interference vortices”in particle trapping [20]. By tuning both the relative amplitude and phase of these beams, wecan combine them such that they destructively interfere at a certain plane with the highest or-der OAM impurities in our original beam, thus increasing the purity of the generation of thedesired OAM state. Other more complex ways of correcting this effect rely on aberration com-pensation methods based on fitting successive Zernike polynomials to each order of wavefrontcorrections [21].

2. Theoretical discussion

To understand the method proposed in this paper, let us first consider a paraxial mode with welldefined OAM of order �:

U�(ρ ,φ ,z) =C�(ρ ,z)× exp(i�φ) (1)

where (ρ ,φ ,z) represent the cylindrical space coordinates and C�(ρ ,z) represents the radialprofile. As we are only interested in the spatial properties of the field, this model would bevalid for any polarization as long as it remains constant across the beam. An arbitrary beam (E)can be written as a superposition of OAM modes [22]:

E = C�(ρ ,z)exp(i�φ)+∞

∑n=1

Cn(ρ ,z)exp(i(�nφ +ϑn)), (2)

where in this case we are singling out the � component in which we are interested. Cn, �n andϑn are the amplitude, OAM and relative phase of each OAM mode respectively. The additional�n terms can be thought of as undesired interference components. Experimentally, when pro-ducing a mode with a well defined �, the contribution from the other unwanted modes decaysvery quickly with increasing values of |�− �n|. Let us now assume that the term with �1 is thedominant aberration, and hence we can disregard the higher orders. Our method for compen-sation is to introduce an additional collinear beam with OAM �′, weighting C′ and phase ϑ ′,allowing us to approximate E from Eq. (2) as E ′:

E ′ ≈ C(ρ ,z)exp(i�φ)+C1(ρ ,z)exp(i(�1φ +ϑ1)+C′(ρ ,z)exp(i(�′φ +ϑ ′)) (3)

Thus by matching C′(ρ ,z) = C1(ρ ,z), �′ = �1 and ϑ ′ = ϑ1 +π we can remove the dominantaberration in our beam by destructive interference. In this paper we use a phase-only spatiallight modulator [23] to generate the third term for compensation in Eq. (3). In this situation, theability to control the radial profile C′(ρ ,z) is rather limited, which as we will see will imposesome restrictions to our ability to fully compensate the vortex beam. Future approaches couldmake use of pseudo-amplitude modulation using a phase-only SLM as demonstrated in [24].

2.1. Hologram generation

Our aim is to remove the undesired modes by using a slightly modified pattern in the SLM.We achieved this by modifying the phase hologram to generate a second collinear diffractingbeam with �′. This second beam is weighted by a real constant α ′ with a phase relative to thefirst beam of ϑ ′. We are then free to tune α ′ and ϑ ′ to destructively interfere with the dominantunwanted OAM mode �1 as outlined previously. The total phase function displayed on the SLMΨSLM is given below:

ΨSLM = arg[exp(i[�φ ])+α ′ exp(i[�′φ +ϑ ′])

](4)

where α ′ and ϑ ′ are the relative amplitude and phase of the mode with OAM of �′ correspondingto �1 and � is the desired OAM of the hologram. ΨSLM is then weighted by the amplitude of


the combined exponentials within the arg operator in Eq. (4) in order to approximate intensitymodulation with our phase-only SLM. We combine this phase hologram with a grating term toseparate out our desired beam into the first diffraction order.

The beam after the SLM in the first diffraction order therefore has the following form:

E(ρ ,φ ,z) = G(ρ ,z)exp

[iarctan

(sin(�φ)+α ′ sin(�′φ +ϑ ′)cos(�φ)+α ′ cos(�′φ +ϑ ′)

)](5)

where G(ρ ,z) is the Gaussian function.Using a phase-only SLM we can only access the phase of the field. Thus the OAM spectrum

of the output of the SLM will not be exactly a mode with OAM � plus the OAM of the newcollinear beam �′. The complex amplitude of each OAM mode can be calculated from thefollowing expression:

C�(ρ ,z) =∫ 2π

0E(ρ ,φ ,z)exp(−i�φ)dφ (6)

Figures 1(b) and 1(c) show the original and modified hologram patterns with the followingparameters: � = 2, �′ = 0, α ′ = 0.19 and ϑ ′ = 4.5 radians whereas Fig. 1(a) gives the OAMspectrum of the beam after acquiring the phase from the modified hologram shown in Fig. 1(c).These values of �′, α ′ and ϑ ′ of which exemplify a modified hologram in Fig. 1(c) are actu-ally the optimal ones for the correction found after applying the method described in Sec.5.As demonstrated in Fig. 1(a), additional L-G modes with other OAM (� = 4 in this case) areinevitably generated from our hologram.

Fig. 1. (a) OAM spectrum calculated for � = 2, �′ = 0, α ′ = 0.19, ϑ ′ = 4.5 radians. (b) Phasehologram for beam of � = 2. (c) Phase hologram for beam of � = 2, and �′ and weightingand relative phase corresponding to (a). The center of both holograms in (b) and (c) areexpanded to highlight the differences.

3. Experimental setup

The experimental setup for compensation of the OAM impurities in a beam is shown in Fig. 2.The source of the initial OAM beam was a 632 nm HeNe (Thorlabs) which was expanded by thearrangement of lenses L1 and L2 in order to maximize illumination of the pixels on the spatiallight modulator (SLM). A half-wave plate (HWP) aligned the polarization of the field incidenton the SLM. The angle between the incident beam and the SLM was minimized. A CambridgeCorrelators SDE1024 SLM displayed the desired phase-only hologram (see Eq. (4)) to producediffracting beams with non-zero OAM, separated from the zeroth order. An iris (SF1) selectedonly the first diffraction order. The phase contrast of the SLM (grey level to output phase) wasvaried [23] in order to maximise the diffraction efficiency into the first diffraction order.


Fig. 2. Experimental setup used to generate a beam with arbitrary OAM, �, and subse-quently compensate the observed separation of nulls in the near and far-field. The lens sys-tem consisting of lenses L3 and L4 magnifies and images the far-field spot onto the CCD.A typical image is given by inset (a). Lenses L3 and L5 magnify and image the near-fieldi.e. the plane at the SLM onto the CCD. A typical image is shown in inset (b). Note that theentire beam profile in (b) cannot be observed with the CCD. The focal lengths of the lensesare as follows: L1 = 25.4 mm, L2 = 75 mm, L3 = 500 mm, L4 = 35 mm, L5 = 125 mm.HWP = Half wave plate, SLM = spatial light modulator, BS = beam splitter, ND = neutraldensity filter, SF = spatial filter, CCD = charge-coupled device, HeNe = Helium-Neon laser.

To simultaneously observe both the near-field and far-field profile of the vortex beam, a 50:50beam splitter (BS) separated the beam into two paths. We define the near-field as the plane justafter the SLM display, while the far-field is the Fourier transform of the near-field plane. LensesL3 and L4 in the upper path imaged the Fourier transform of the beam after the SLM onto theCCD. Inset (a) in Fig. 2 is a typical image of the far-field distribution for a beam generated withOAM � = 2. Lenses L3 and L5 in the lower path in Fig. 2 combined to image the beam justafter the SLM onto the CCD. This imaging system had a total magnification of 2.89 in order tosufficiently resolve the central null in the near-field image. Figure 2(b) is a typical image of thenear-field intensity distribution of the same beam in Fig. 2(a). It should be noted that with thetotal magnification of the L3, L5 imaging system, the entire near-field image was larger thanthe pixel display of the CCD. SF2 was used to select only the central region of the near-fieldimage.

4. Vortex splitting

Using the setup in Fig. 2, we generated beams using phase holograms constructed from thephase of beams with OAM � = 0, 1, 2 and 3. The near- and far-field intensity distributions ofthe resulting beams were imaged to observe the effect of the aberrations specific to our systemfor beams of varying OAM. These results are summarized in Fig. 3. By observing the effect ofthe aberrations on the structure of the beam for different �, we were able to deduce the �′ of thefirst order correction to apply (Eq. (3)) for each beam of OAM � in Section 5.

Figures 3(a)–3(d) shows the far-field intensity profile of beams with an OAM of � = 0, 1, 2,and 3 generated with the above setup (Fig. 2) in the absence of any compensation. The samebeam was imaged in the near-field for � = 0, 1, 2 and 3 in Figs. 3(e)–3(h). Figure 3 clearlyshows the influence of aberrations on vortex beams with various OAM ranging from � = 0-3.

For � = 0 in Fig. 3(a), aberrations in the beam have distorted the intensity profile into anellipse. Observing the uncompensated � = 1 beam in Fig. 3(b), the annular intensity distribution(dark red) lacks circular symmetry with the two lobes of high intensity separated by a line oflow intensity resulting in a central null which is elliptical. We can tell by direct observation that


Fig. 3. (a)-(d) Far-field images of beams with OAM � = 0, 1, 2 and 3 in the absence of anycompensation. The configuration of lenses for these images was L3 and L4 (Fig. 2). (e)-(f) Near-field images of the same beams are given for � = 0, 1, 2 and 3. The configurationused for these images was L3 and L5 (see Fig. 2). From these images (a)-(d) and (e)-(h), wecan see that the effect of aberrations on a vortex beam is to induce a splitting of the centralsingularity and a deformation of the intensity profile. This is consistent with the existenceof another mode with �1 = �− 2. In all images a logarithmic intensity scale is used. Thephysical scale used in all images is the same 490px×490px (3mm×3mm).

the intensity modulation observed in the Fig. 3(b) is due to interference between componentswith � = 1 with �1 = −1, where �1 denotes the undesired OAM in the beam with the highestweighting. Considering the two OAM modes above of phase �φ and �1φ , destructive interfer-ence occurs whenever φ − (−φ) = nπ (for odd n), resulting in the line of low intensity. φ isdefined in Section 2 as the azimuthal angle in polar coordinates. Calculations also show that theelliptical intensity profile in Fig. 3(a) is consistent with interference between the desired �= 0mode and a component of �1 = −2. A comparison of calculated and experimental far-field in-tensity profiles for a pure mode with � = 0 and a mode consisting of (� = 0) and (�1 = −2) isgiven in Fig. 4.

The intensity profile in Fig. 4 was calculated for the following expression: G(r,φ)e0φ +0.19G(r,φ)e−2φ where G(r,φ) is a Gaussian profile in polar coordinates (r,φ ) and we haveset α1 = 0.19, � = 0 and �1 = −2. The value of α1 was chosen based on the value of α ′ de-termined experimentally after applying the method described in Sec. 5. We neglect the phasedifference ϑ1 as it only contributes a rotation to the elliptical profile in Fig. 4(b). In our cal-culation we also assume that the functional form of the field amplitude for the �1 componentis identical to within a scaling constant to the electric field amplitude of the � = 0 component.Our observations of the compensated near-field beam profile in the following section show thatthis is not necessarily the case. It is the difference in field profiles between the � and �1 compo-nents which likely results in the differences between the calculated and experimental intensityprofiles, however from our calculations we can see that an elliptical intensity profile associatedwith a �= 0 Gaussian beam is qualitatively consistent with the existence of an additional �1=-2component.

In Figs. 3(c) and 3(d) the splitting of the central null into |�| separate nulls is apparent. For


Fig. 4. Calculated far-field images of a beam with OAM � = 0 (a) and � = 0+ �1 = −2(b) and experimentally measured far-field images of our � = 0 beam uncompensated (c)and with compensation (d). Calculated images have identical scaling in z. In all images alogarithmic scale is used.

� = 2, the central vortex has been split into two in the far-field in Fig. 3(c). This can be shownto be consistent with interference between a � = 2 and �1 = 0 mode. Let us first recognize thatthe expression which gives rise to the singularity rm exp imφ (the rm term is contained withinthe function C(ρ ,z) in Eq. (2)), may be written as (x+ iy)m. We can then express the electricfield at some plane due to our combination of �= 2 and �1 = 0 modes as (x+ iy)2+A to withinsome factor, where A is a normalized constant. By finding the intersections of lines of zero realand imaginary components for this expression we find the corresponding singularities in thefield. From this analysis it can be seen that the addition of the constant A due to the �1 = 0mode results in a splitting of the central � = 2 singularity. For the case of the uncompensated� = 3 beam in the far-field in Fig. 3(d), the nulls are distributed along a horizontal line whilethere is still a null at the center of the beam. This central null of the total field implies thatthe � = 3 component and the unknown �1 component must both possess a singularity at theircentre and hence �1 �= 0. We also know that singularities occur at positions at which the electricfield amplitude is zero, and so we can associate the position of singularities with regions ofdestructive interference between the � and �1 modes. Thus in Fig. 3(d), destructive interferencemust be occurring along the line of singularities. From our previous analysis of Fig. 3(b), weknow that a single straight line of destructive interference is a consequence of interferencebetween �= 1 and �=−1 components. In the case of Fig. 3(d), �= 3, and hence �1 = 1 in orderto maintain the relative phase difference for a straight line of singularities such that nulls occurat φ = nπ

2 .This can also be verified by finding the zero crossing of the real and imaginary components in

the expression (x+ iy)3+A(x+ iy). The intensity distribution for �= 2 and 3 are both elliptical.A beam with OAM of � = 4 was also generated and observed that the central vortex splitalong a horizontal line of 4 separate vortices, however this data was not included in this paper.Comparing the desired OAM � and the highest interfering OAM component �1 deduced fromFigs. 3(a)–3(d), the relationship between the two is �1 = �−2. This relationship is independentof the value of � imposed by the SLM on the input beam. Applying different values of � willshift the maximum in the OAM spectrum to the appropriate value, while the non-zero OAMbandwidth arising from aberrations along the optical path remains unaffected. From this weconclude that the highest contributing OAM impurities have OAM values of �1 = �−2.

The question arises from where does this interference component originate? Let us consider


the interference mode and consider the linear distribution of vortices for � = 3 in the far-fieldshown in Fig. 3(d). This symmetry breaking implies a perturbation which lacks cylindrical sym-metry. In fact rows of singularities have been shown theoretically [15] and experimentally [25]to arise from both small elliptical perturbations to and astigmatic transformations of LG modesrespectively. The origin of the elliptical perturbation is possibly due to a non-normal angle ofincidence between the output of the HeNe and the SLM [26]. Under the effect of both typesof perturbations, the original LG mode is best approximated as either Ince-Gauss, or Hermite-Laguerre-Gauss modes respectively [15] which have singularities distributed along a horizontalline.

The typical near-field distribution of the vortex beams just after the SLM are shown inFigs. 3(e)–3(h). Figures 3(e)–3(h) indicate that in the near-field, just after the plane of theSLM, vortex splitting is occurring. It is interesting to note that in the near-field, the distributionof nulls for the case � = 3 has symmetry in the azimuthal coordinate of the plane, compared toits far-field distribution. In order to resolve the structure associated with the core of the beam inthe near-field, the imaging system (L3 and L5 in Fig. 2) magnified the beam profile in the planeof the SLM.

5. Compensation

Fig. 5. To construct the phase hologram used to generate two collinear beams with OAM� = 2 and �′ = 0, we calculate the complex amplitude of each individual beam, add themand extract the resulting phase profile (ψSLM). Parameters α ′ and ϑ ′ control the relativeweighting and phase between the beams. When illuminated with a Gaussian beam, theoutput is a �′ = 0 and a � = 2 beam. Although not additive, displayed are the hologramsassociated with the �′ = 0 component, � = 2 component and the final hologram used tocreate the two collinear beams.

We apply compensation to the output of the SLM by holographically generating an additionalcollinear beam with a weighting (α ′), relative phase (ϑ ′) and orbital angular momentum (�′). Aschematic describing the construction of phase-holograms for the generation of multiple beamsis given in Fig. 5.

We now describe in detail the method for finding the optimum parameters to correct for thesplitting of the central null. Once the value of �1 has been deduced (�1 = �−2), we incorporatea hologram of a beam with OAM �′ = �1 into the original hologram (as in Fig. 5). Specificallywe performed this for a desired OAM of � = 2. We then fix the weighting (α ′) at some arbi-trary value, and vary the phase ϑ ′, relative to the original hologram while looking at the effecton the singularities in the far-field beam profile. We observe the rotation of the vortices in thefar-field as the angles at which destructive interference occur are shifted by ϑ ′/(�− �1) rad.When the distance between the vortices has been minimized, the phase (ϑ ′) is then fixed, whilethe weighting (α ′) is varied, again aiming to minimize the distance between the vortices for the


far-field image. The minimum vortex separations were qualitatively selected by comparing theCCD images at different α ′ and ϑ ′. This method was sufficient for the purposes of the paper, butmore precise algorithms can be implemented by calculating the separation distances betweenthe vortices. This process is iterated as necessary until the separated nulls have been collapsedback to a single central null. At this point the collinear beam generated by the SLM (�′) and the�1 component of the original beam are out of phase by π radians, and destructive interferencehas been maximized at the observation plane of the CCD by having matched α ′ of the addi-tional collinear beam to the α1 of the unwanted interference component. These compensationparameters are valid for all values of � imposed by the SLM. These are however, only a firstorder correction. For larger values of �, higher order corrections must be applied (i.e. additionalholograms with ϑn+1, αn+1, �n+1).

Fig. 6. Far-field images of beams with OAM quantum numbers of � = 0, 1, 2 and 3 (a)-(d)with compensation applied as detailed in the text. The deformation of the intensity profilesare decreased compared to Figs. 3(a) and 3(b). Vortex splitting associated with �= 3 and 4(c,d) has been reduced relative to Fig. 3. Near-field images of the same beams are given for� = 0, 1, 2 and 3 (e)-(f). In all images a logarithmic scale is used.

5.1. Post-compensation

In order to test our proposed method of compensation, we generated beams with OAM � = 0, 1,2 and 3 (using the same setup in Fig. 2), with the addition of the compensation terms mentionedabove and compared them with our original beams from Fig. 3. The resulting far-field andnear-field images are presented in Figs. 6(a)–6(d) and Figs. 6(e)–6(h) respectively for � = 0–3.Comparing with Fig. 3, introduction of our proposed method has yielded improvements to thesingularity splitting and the induced ellipticity due to aberrations in the far-field.

For � = 0 in Fig. 6(a), the intensity distribution has decreased in ellipticity compared to theuncompensated beam in Fig. 3(a). This is also the case for � = 1, which presents a more evendistribution of intensity over the beam profile relative to without compensation. This increasein symmetry is consistent with the removal of the unwanted �′ = -2 and �′ = -1 componentspresent in the � = 0 and � = 1 beams respectively. The most pronounced improvements in thebeam profile occur in Fig. 6(c) for � = 2 where the two previously split singularities in Fig. 3(c)have been reformed into a central null as a result of the compensation. Comparing The far-field


compensation of the � = 3 beam shown in Fig. 6(d) resulted in a reduction in the separationof the 3 nulls compared with Fig. 3(d) to the extent that each singularity could no longer befully resolved. For both � = 2 and � = 3, we successfully removed the singularity splittingdue to aberration, by destructive interference of holographically generated �′ and inherent �1

components at the plane of the CCD. The absence of circular symmetry in the singularitieshowever, implies that there exist higher order OAM components that have not been accountedfor. For � = 1-3 in Figs. 6(b)–6(d), after the removal of the most significant unwanted OAMmode there still exists an asymmetry in the beam profile, with intensity concentrated on thelower right hand side of the singularity. This arises from a slight misalignment of the beamfrom the hologram. In fact this asymmetry can also be seen in the uncompensated beams inFig. 3.

Comparison of compensated with uncompensated intensity profiles in the near-field,Figs. 3(e)–3(h) and Figs. 6(e)–6(h) respectively, shows an increase in the separation of thenulls for � = 2 and 3 upon addition of the collinear compensating beam, while the nulls in the� = 3 case have rearranged along a straight line to resemble the uncompensated far-field pro-files. This is a reflection of our inability to fully control the radial profile of the compensationterm. Thus, in this case what we have achieved is C′(δρ ,zFF)≈C1(δρ ,zFF), with zFF the farfield plane and δρ the radial points close to the center of the beam ρ = 0. However, in generalC′(ρ ,z) �= C1(ρ ,z). Even though for most applications this can be enough to probe the localproperties of the higher order optical vortex, in the next section we provide a method to allowa better control of the radial profile of the compensating mode.

6. Near-field compensation

While we have demonstrated control of the singularities of an optical field, reforming the splitsingularities in the far-field, this is apparently achieved at the cost of separating the nulls of thebeam in the near-field. In the paraxial regime, the far-field intensity distribution F , is related tothe near-field f , by a Fourier transform (and vice versa):

F(kx,ky) =∫ ∞

−∞

∫ ∞

−∞f (x,y)ei(kxx+kyy)dxdy (7)

Considering the centre of the beam at F(0,0), Eq. (7) becomes:

F(0,0) =∫ ∞

−∞

∫ ∞

−∞f (x,y)dxdy (8)

which is the average of the field f (x,y). Thus by modulating the field f (x,y) such that it’s meanis zero, we can create a null at the origin F(kx,ky). By manipulating the amplitude of the fieldsuch that the mean value of the field is zero we can force the center of the beam to have zero ora minimal amplitude to minimize the vortex splitting observed.

We implement this by introducing an additional phase ψr(r) to the expression from whichwe calculate our holograms (Eq. (4)) in order to modulate the field of the compensation (�′)component generated by the SLM:

ΨSLM = arg[exp(i[�φ ])+α ′ exp(i[�′φ +ϑ ′+ψr(r)])

](9)

Where ψr(r) is a step function in the radius of the beam varying from 0 to π radians:

ψr =

{0 if r ≤ r0

π if r > r0(10)


This radial phase modulation maintains the relative phase relationship among each term inEq. (3) in the vicinity of the singularities, and hence preserves the effects of the compensationdemonstrated in Section 5. By inverting the sign of the field within a radius of r0 we can find ar0 at which the mean value of the field is zero for the �′ collinear beam. To perform this experi-mentally, we apply the far-field compensation using the parameters determined in the previoussection for the case � = 2, splitting the vortex singularities in the near-field. We then modifythe hologram according to Eq. (9) and vary r0, minimizing the separation of the singularitiesassociated with the near-field profile of the beam.

Fig. 7. Far-field (a) and near-field (b) images of a beam with OAM � = 2 in the absence ofany compensation. Far-field (c) and near-field (d) images of an � = 2 beam compensatedby the removal of the �1 component by destructive interference with a collinear beam of�′. Far-field (e) and near-field (f) images of a � = 2 beam with the same compensationas applied in (c),(d) but with a radial phase (Eq. (10)) applied to the �′ = 0 compensationcomponent. For the radial phase, r0 = 10 pixels (90 μm). The result of this additional radialphase is that the separation of the vortices in the near-field has been reduced from (c) to (e).In all images a logarithmic scale is used.

The results of our radial phase modulation for the � = 2 beam are described in Fig. 7 andcompared with the previously compensated and uncompensated beams �= 2 beams from. Fig-ures 7(a) and 7(b) and Figs. 7(c) and 7(d) are control images of the far-field and near-fielddistribution of nulls without and with the far-field compensation from Figs. 3(c) and 3(g) andFigs. 6(c) and 6(g) respectively. The increase in vortex splitting between Figs. 7(a) and 7(c)(without and with compensation) is visible. From the near-field image, we observe that whilethere still exists a distortion in the intensity of the near-field (two lobes of higher intensitycircling the central nulls), the distance between the nulls has been reduced to distances whichappear comparable to the original vortex splitting occurring in the near-field. With this compen-sation, we are successful in reducing the splitting of the two nulls to below that of the originaluncompensated beam in Fig. 7(e), while retaining the central singularity in the far-field.


7. Conclusion

We propose a simple method for the compensation of mode impurities in a OAM beam gen-erated by holographic mode conversion. By introducing an additional OAM mode and varyingthe relative phase and weighting, we can remove the highest order OAM impurity at a singleplane by destructive interference. This is performed with a phase-only SLM, removing the in-tensity distribution associated with interference between multiple OAM modes for � = 0 and 1,while collapsing the multiple split singularities associated with interference among higher order� modes, � = 2,3. We achieved this by interference with a collinear beam of OAM �′ = �− 2.We then applied a phase with a radial step-function between 0 and π , and by varying the radiusat which the step occurs, we were able to simultaneously reform the nulls in the near-field whileretaining our compensation in the far field for OAM of � = 2.

Acknowledgments

This work was supported by the Australian Research Council (ARC) Future Fellowship(FT1110924) scheme.


Correcting vortex splitting in higher order vortex beams

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