## Abstract

Higher-order optical vortices are inherently unstable in the sense that they tend to split up in a series of vortices with unity charge. We demonstrate this vortex-splitting phenomenon in beams produced with holograms and spatial light modulators and discuss its generic and practically unavoidable nature. To analyze the splitting phenomena in detail, we use a multi-pinhole interferometer to map the combined amplitude and phase profile of the optical field. This technique, which is based on the analysis of the far-field interference pattern observed behind an opaque screen perforated with multiple pinholes, turns out to be very robust and can among others be used to study very ’dark’ regions of electromagnetic fields. Furthermore, the vortex splitting provides an ultra-sensitive measurement method of unwanted scattering from holograms and other phase-changing optical elements.

© 2012 OSA

## 1. Introduction

Optical phase vortices or singularities are points where the optical phase is undefined, because the phase varies over *ℓ ·* 2*π* around these points, where *ℓ* determines the topological charge of the vortex. Such vortices appear for instance in the center of Laguerre-Gauss (LG) laser modes, which are an example of general orbital angular momentum (OAM) beams that possess an OAM of *ℓ h̄* per photon [1–3]. Although vortices with |*ℓ*| > 1 can exist in theory, it was found very early by Nye and Berry that such higher-order vortices are unstable under realistic conditions [4–6]. This, on one side, does not prevent application of higher-order OAM beams for instance in (quantum) communication [7–9], since such applications dominantly exploit the phase information at bright regions in the field. On the other side, there are numerous applications that rely on the perfectly dark region of (higher-order) singularities such as super-resolution microscopy [10] and stellar vortex coronagraphy [11, 12].

OAM fields can be produced either by mode conversion of Hermite-Gaussian laser modes [1] or by imprinting the helical phase onto a fundamental Gaussian beam. The latter method is more popular due to its flexibility and can be performed directly, with spiral phase plates [13], or diffractively with fork holograms [14], either lithographically defined [5,15] or programmed into a spatial light modulator (SLM). Higher-order OAM beams can simply be produced by choosing |*ℓ*| > 1. In many experimental investigations (e.g., [5, 16, 17]) it was found that experimental imperfections, such as a coherent background, lead to very strong splitting of the vortices, comparable to splittings appearing in astigmatic beams [18] or vortex rows [19, 20].

We aim to study here the fine structure of the dark core of high quality higher order OAM beams, produced with state of the art methods: either by using high grade dielectric fork holograms or with spatial light modulators. To analyze the dark region, we use two methods. First, we use a circular aperture centered on the OAM beam to block the bright ring, and then image the central core onto a CCD. Interference is used to determine the local phase. The second method is based on the analysis of the far-field interference pattern observed behind an opaque screen perforated with multiple pinholes [21–23]. We describe the implemented analysis technique in detail and discuss its practical limits.

## 2. Experimental setup and first results

#### 2.1. Experimental setup

The experimental setup that we used to generate and detect optical vortex beams is depicted as Fig. 1. We start with a Helium-Neon laser (*λ* = 633 nm) that emits a Gaussian beam with a measured divergence angle *θ*_{0} = 0.66 mrad and a corresponding beam waist *w*_{0} = *λ*/*πθ*_{0} = 305 *μ*m and Rayleigh length
${z}_{0}=\pi {w}_{0}^{2}/\lambda =46.2\hspace{0.17em}\text{cm}$. Knowing the divergence angle of the laser and the position of its waist, we placed lens *L*_{1} in a *f* − *f* geometry to illuminate our fork hologram with a Gaussian beam with a flat wavefront. A second lens *L*_{2} in another *f* − *f* system, images (a selected order of) the far-field diffraction pattern of the hologram onto what we call the pinhole plane. We analyze this diffraction pattern either with optical system A or B (see figure caption and text below for details). By choosing lenses of different focal length and adjusting the *f* − *f* geometry, we had full control over the size *w*^{′} = *f*_{1}*θ*_{0} of the beam at the hologram, and the size *w* = *λf*_{2}/(*πw*^{′}) of its Fourier image at the pinhole plane.

The hologram that we use is a binary-phase computer-generated hologram, comprising a regular grating with edge dislocation (see ref. [24] and references therein). At a carrier period of 50 *μ*m, the opening angle between consecutive diffraction orders is 12.7 mrad. Just like a normal grating, a dislocation or fork hologram diffracts the incident beam in multiple orders. It does, however, add a spiral phase pattern or screw dislocation of the form exp(*iℓϕ*) to the *ℓ*-th diffraction order [5]. Although the binary phase step was optimized for a wavelength of 800 nm, the hologram also operates well at a wavelength of 633 nm. This is easily shown by writing the binary phase step as exp(*±iϕ*) = cos(*ϕ*) *± i*sin(*ϕ*). The cos(*ϕ*)-term corresponds to a uniform transmission, observable as the zero-th order non-diffracted beam. The *±i*sin(*ϕ*)term corresponds an ideal binary grating, with phase step *π*, albeit with reduced diffraction efficiency.

In the rest of this paper we will analyze the far-field diffraction patterns of the dominant odd-order beams diffracted from this hologram, and compare the obtained results with theoretical expectations and with vortex beams produced with a different (SLM-based) technique. We will use three different tools of analysis: *(i)* intensity measurements, *(ii)* interference measurements, *(iii)* multi-pinhole interferometry. The first two tools are more-or-less standard and discussed in subsection (2.2) below. Multi-pinhole interferometry is a new and powerful tool that deserves its own Section (3).

#### 2.2. First results

Figure 2 shows the measured intensity profiles for the *ℓ* = 1*,*3*,*5 diffraction order of our fork hologram. All intensity profiles are (rotationally) averaged over the azimuthal angle and normalized to a peak intensity of 1. This figure demonstrates two important aspects of vortex beams. First of all, it shows that the spatial extent of the central dark region increases rapidly with the order of the beam. Indeed, we expect the intensity in the central dark region of a pure Laguerre-Gauss beam with vortex change *ℓ* to vary as *I*(*r*) ∝ |*r|*^{2}* ^{ℓ}*, while the same scaling applies to the Kummer beams that we actually generate (see below and ref. [25]). As a second observation we note that the intensity tails of the diffraction patterns are much longer than expected for pure

*p*= 0 Laguerre-Guassian beams of the form

*r*and

*ϕ*are the radial and azimuthal transverse coordinates with respect to the beam axis,

*w*is the beam width,

*ℓ*and

*p*are the azimuthal and radial quantum number. These long tails proof that the diffracted beam is not a pure Laguerre-Gauss beam but a superposition of various

*ℓ, p*modes with fixed

*ℓ*but different

*p*values. Propagation of the field

*u*(

*r*,

*ϕ*) =

*u*

_{0}exp(−

*r*

^{2}

*/*(

*w*

^{′})

^{2})exp(

*±iϕ*) to the far-field actually yields the confluent hypergeometric or Kummer function [25], which can be expressed as the difference of two Bessel functions:

Here, the tilde indicates that we now consider the far-field (= Fourier transform) and *I _{m}* is the

*m*th-order modified Bessel function. The three theoretical curves in Fig. 2 represent the squared modulus of the

*Kummer beam*field function in Eq. (2); they are based on the calculated beam waist

*w*= 407

*μ*m and contain no fit parameters, apart from a normalization in the vertical direction.

Next, we wanted to zoom into the dark central region of the intensity profiles. Being interested in the splitting of higher-order vortices, and expecting this splitting to be much smaller than the typical size of the dark region, we wanted to screen the light coming from the brighter ring. This goal was achieved by introducing a pinhole followed by a relay system to image the dark region of the vortex on the CCD camera (see detection system A in Fig. 1). Typical pinholes diameters in our experiment are 400 – 800 *μ*m, depending on the topological charge of the observed main vortex and on the size of the beam in the pinhole plane.

Figures 3(a) and 3(c) show the (dark center of the) intensity patterns measured for the diffraction orders *ℓ* = +3 and *ℓ* = −3. These figures clearly show that the dark center of such beams presents *ℓ* split vortices, instead of a single higher-order vortex. The color scale is normalized with respect to the peak of the rotationally averaged intensity profile of the full beam, as depicted in Fig. 2. With the additional pinhole, which blocks the brightest parts of the beam, we easily observe intensity features that are a factor 10^{−3} − 10^{−4} below the peak intensity and that would otherwise be impossible to see due to the limited dynamic range of CCDs.

Although the results in Fig. 3(a) and Fig. 3(c) show with high precision the splitting of the main vortex, these intensity pictures do not contain any information about the phase of optical field. This information can be obtained by adding a weak reference beam in order to create interference with the main singular beam at the pinhole plane [5, 26]. The reference beam is obtained by introducing a glass wedge behind the dislocation hologram to reflect part of the fundamental Gaussian beam from the 0^{th} diffraction order. The interference pattern appearing in the pinhole is detected with system A in Fig. 1.

Figures 3(b) and 3(d) show the experimental results of this interference experiment. The white dotted lines serve to guide the eye towards the vortices. At the same position where we previously observed dark spots, we now observe fork-shaped interference fringes. The information on the polarity of the vortices is carried by the orientation of the forks. As expected from considerations about topological charge conservation, the results shows the presence of three positive single-charged vortices in the *ℓ* = +3 case, and three negative single-charged vortices in the *ℓ* = −3 case. Note that the described interference is severely limited: for a clear observation of the vortices one needs a high spatial resolution (i.e., many interference fringes), whereas these fringes can only be properly imaged if they are not too numerous. Further, the strongly varying intensity around the split-up singularities makes observation of high-visibility interference fringes over a large scale (Fig. 3) very challenging.

## 3. Multi-pinhole interferometry

#### 3.1. Working principle

As a third and very promising technique to probe the spatial structure of an optical field, and in particular the structure of its optical vortices, we introduce the multi-pinhole interferometer (MPI) [21–23]. This device, which is sketched as system B in Fig. 1, combines a series of *N* small holes in an opaque screen, spaced equidistantly on a circle, with an imaging system that observes the far-field diffraction pattern of light passing through these holes. For *N* ≥ 3 and odd *N*, a Fourier decomposition of this diffraction pattern allows one to uniquely determine the optical amplitudes *E _{m}* at each of the individual holes [21, 22], apart from an overall phase factor, and the associated OAM components of the sampled field.

Figure 4 shows the front end of our multi-pinhole interferometer. The *N* = 7 holes are distributed evenly on a circle with radius *a* and define a heptagon. When the hole diameter *b* ≪ *a* is small in relation to the investigated spatial structures, the optical field has an approximately constant value *E _{m}* in each of the holes and the far-field diffraction pattern resembles that of

*N*point sources. The finite hole diameter

*b*merely modifies this pattern by limiting the emission angle of the individual holes and thereby the angular range over which the interference pattern can be observed.

The data obtained with the multi-pinhole interferometer is analyzed as follows. We start from the measured far-field diffraction pattern, of which Fig. 5(a) shows a typical example. Next we perform a 2-D inverse Fourier transform of this intensity pattern. Figure 5(b) shows the result in a false-color scale where the brightness indicates the Fourier amplitude and the color indicates its phase. Being the Fourier transform of an *intensity* pattern, this Fourier image corresponds to the *autocorrelation function* of the optical field at the pinholes. Hence, it consists of a series of *N*(*N* − 1) peaks, associated with interference terms of the form
${E}_{i}{E}_{j}^{*}$, and a central peak, associated with the average field [23]. For our analysis, we will only use the interference terms
${E}_{i}{E}_{j}^{*}$ with fixed *i* [the case *i* = 1 is indicated by the heptagon in Fig. 5(b)]. In the Appendix, we describe how further analysis allows us to recover the optical fields *E _{m}* =

*A*exp (

_{m}*iϕ*), apart from an overall phase factor.

_{m}The multi-pinhole interferometer is an ideal probe for the orbital angular momentum contents of the incident field, because the *N* samples that it takes are evenly distributed on a circle. A simple Fourier decomposition therefore already yields the sampled OAM modal amplitudes

*λ*

_{0}coefficient equals the average optical field, while the

*λ*coefficients are the amplitudes of the various OAM components of the sampled field. Equation 3 shows why the MPI is ideally suited for the study of optical vortices. When the MPI is centered at a pure vortex and when its radius

_{ℓ}*a*is small enough, the average field

*λ*

_{0}disappears and only one

*λ*component will be non-zero. The

_{ℓ}*ℓ*value thereof is the vortex charge, while its amplitude

*λ*is proportional to the field strength around the vortex.

_{ℓ}#### 3.2. OAM-resolved spatial images

After this short detour, we return to the analysis of higher-order vortex beam. Detection system B in Fig. 1 allows us to scan the MPI through the full beam of any diffraction order. At each position (*x*,*y*), we record the far-field intensity pattern and apply the Fourier analysis mentioned above, and some tricks described in the appendix, to extract the sample OAM co-efficients *λ _{ℓ}* (

*x*,

*y*). We finally convert these to powers

*P*= |

_{ℓ}*λ*(

_{ℓ}*x*,

*y*)|

^{2}and normalize these to ∑

*P*(

_{ℓ}*x*,

*y*) = 1 such that

*P*(

_{ℓ}*x*,

*y*) is the relative power in the

*ℓ*-th OAM mode at position (

*x,y*).

Figure 6 shows the OAM maps *P _{ℓ}* (

*x*,

*y*) that we measured while scanning an MPI (with

*b*= 20

*μ*m and

*a*= 100

*μ*m) through the

*ℓ*= 3 and

*ℓ*= −3 diffraction order of the hologram, respectively. First of all, note the high quality of the obtained results. The MPI enables a clear OAM decomposition even in the darkest region of the diffracted beam: the fundamental

*P*

_{0}component dominates practically everywhere, apart from three regions where

*P*

_{0}→ 0 and

*P*

_{±1}dominates instead. Both figures (a) and (b) show the presence of

*ℓ*=

*±*1 vortices only, demonstrating how the higher-order vortex that we tried to generate has split into single-charged vortices. In both cases, the intensity at the vortices becomes practically zero, as can easily be seen by comparing Fig. 6 with Fig. 3. Also note that all higher-order coefficients

*P*with

_{ℓ}*ℓ*≠ {0,1} are small. The strongest high-order component is found in the top figure, where we observed

*P*

_{2}≈ 0.45 when the MPI is positioned right in between two closely-spaced

*ℓ*= 1 vortices. This value is observed when the MPI has sufficient overlap with both

*ℓ*= 1 vortices (see below for a discussion of the associated spatial resolution). We will save our discussion on the origin of the observed splitting for Section 4. To set the stage for that discussion, we will first present additional evidence on the practically unavoidable nature of vortex splitting.

Figure 7 shows the measured OAM-maps for the *ℓ* = 5 diffraction order from the fork hologram. We again observe that the expected higher-order (*ℓ* = 5) vortex splits into a series of vortices of unity charge arranged approximately on a circle. The observed splitting is more prominent and more symmetric than for the |*ℓ*| = 3 case, presumably due to the larger *ℓ* value (to be discussed in the next section).

#### 3.3. Spatial resolution of MPI

As a final experiment, we present a simple measurement of the spatial resolution obtainable with a scanning MPI, if only to set the stage for future applications of this device. This spatial resolution is naturally limited by its radius *a*. In order to quantify this statement, we calculate the expected signal from a pure *ℓ* = 1 optical vortex field *E*(*x*,*y*) = *C*(*x* + *iy*) probed with an MPI positioned at (*x*_{0}, *y*_{0}). Fourier decomposition of the sampled complex field yields the modal amplitudes *λ*_{0} = *C*(*x*_{0} + *iy*_{0}), *λ*_{1} = *Ca*, and *λ _{ℓ}* = 0 for

*ℓ*≠ {0,1}. The spatial resolution of the MPI for observation of a single (

*ℓ*= 1) vortex is best characterized by the fraction of the optical power

*P*≡ |

_{ℓ}*λ*|

_{ℓ}^{2}in the

*ℓ*= 1 mode, which is

Figure 8 shows the measured spatial resolution of the MPI, by displaying the normalized OAM powers *P*_{0} and *P*_{1} for a cross section through a pure *ℓ* = 1 vortex (2D data not shown) and a cross section through one of the unity-charge vortices present in the dark center of the *ℓ* = −3 diffracted beam (2D data in Fig. 6). These four curves were fitted simultaneously to the Lorentzian profile of Eq. (4) and its complement *P*_{0} = 1 − *P*_{1}, using the width *a* as the only fitting parameter. To our surprise, the fitted width of *a ≈* 72*±*3 *μ*m was significantly below the expected MPI radius of 100 *μ*m. We do not yet understand why.

Another criterion for the spatial resolution attainable with a scanning MPI might be its ability to resolve neighboring vortices. For this we consider the spatial mapping of two *ℓ* = 1 vortices that are formed by admixing a uniform background field *E*_{0} to an pure *ℓ* = 2 vortex *E*(*x,y*) = *D*(*x* + *iy*)^{2} such that the total field is

*x*

_{0},

*y*

_{0}) can again be decomposed relatively easily into its OAM components. For displacements along the (

*x*

_{0}= 0) line that connects the two vortices, this calculation yields

*λ*

_{0}=

*D*( ${y}_{0}^{2}$ −

*d*

^{2}),

*λ*

_{1}= 2

*Day*

_{0},

*λ*

_{2}=

*Da*

^{2}, and

*λ*= 0 for

_{ℓ}*ℓ*≠ {0,1,2}, making

*d*>

*a*, this relative modal intensity resembles the sum of two well-resolved Lorentzian shapes. For smaller splitting

*d*<

*a*, Eq. (6) changes into a combination of two Lorentzian-like shapes with somewhat smaller peak values at

*y*

_{0}≈ ±

*a*and a prominent central minimum

*P*

_{1}= 0 at

*y*

_{0}= 0, accompanied by a large value for

*P*

_{2}as the MPI now encloses both vortices. An intriguing property of the MPI analysis is that it seems to enhance the spatial resolution; although the spatial width of the measured

*P*(

_{i}*x*

_{0},

*y*

_{0}) profiles is naturally limited by radius

*a*of the MPI, a potential lack of symmetry in these profiles makes even small vortex splittings prominently visible.

## 4. Discussion

A higher-order vortex tends to split in multiple vortices with unity charge when the vortex beam interferes with a coherent optical background. This explanation was first introduced by Soskin et al. [5], who considered the interference of the simple *ℓ*-th order vortex beam described by Eq. (1) with a general Gaussian reference beam. We will simplify their analysis to the case where the two beams have identical waist parameters and where the reference beam is weak. The higher-order vortex is then expected to split up in a series of |*ℓ*| = 1 vortices spaced uniformly on a ring with radius *r*_{0} with

*I*and

_{b}*I*are the peak intensities of the reference beam and the vortex beam, respectively. Our earlier statement that vortex splitting is omnipresent and practically unavoidable is based on the large exponent 2|

_{ℓ}*ℓ*| in Eq. (7). Let’s for instance consider the expected splitting of an

*ℓ*= 5 vortex, for which this exponent is 10. Even if the background intensity is as low as 10

^{−5}, this higher-order vortex will split in a series of unity vortices spaced on a ring with radius

*r*

_{0}/

*w*≈ 0.36. A reduction of the background intensity by another impressive factor of 10

^{−5}will reduce this radius only by a factor $\sqrt{10}$ to

*r*

_{0}

*/w ≈*0.11. This explains our earlier remark that “vortex splitting is practically unavoidable”.

Let us compare this simple theory with our experimental results. A fit of the full intensity profiles yields beam waists of *w* = 406 *μ*m and *w* = 403 *μ*m for the *ℓ* = +3 and *ℓ* = −3 beam, respectively. On the other hand, the splitting of the three vortices in Fig. 3 yields a splitting radius *r*_{0}, calculated as the average distance of these vortices to their center of mass, of *r*_{0} = 137 *μ*m for the *ℓ* = +3 case and *r*_{0} = 176 *μ*m for *ℓ* = −3. Substitution into Eq. (7) shows that these splitting can be explained by a coherent background with a relative intensity of only *I _{b}*/

*I*≈ 2.0

_{ℓ}*×*10

^{−3}and 9.2

*×*10

^{−3}, respectively. These values are close to the intensity that we actually observe in the center of these beams (see Fig. 3). The vortices in the

*ℓ*= 5 beam, displayed as Fig. 7, exhibit a more prominent splitting with an approximately splitting radius as large as 560

*μ*m. A similar analysis as above yields an estimate

*I*/

_{b}*I*≈ 2.4

_{ℓ}*×*10

^{−3}when we increase the beam waist to

*w*= 900

*μ*m to accommodate for the change in illumination used in this experiment. The observation that higher-order vortices are more fragile and result in a larger splitting radius is a natural consequence of the exponent 2|

*ℓ*| in Eq. (7). This observation is confirmed by measurement on the splitting of high

*ℓ*vortices produced with a spatial-light modulator (see Fig. 9 below).

Being coherent, the background must necessarily originate from scattering. The most likely cause of this scattering are imperfections in the fork hologram. Imperfection in regular gratings, such as groove depth, groove spacing, and phase step errors or surface roughness, are known to create stray light in monochromators [27] and the same phenomena will naturally occur in our hologram. In this paper, we do not want to study the origin of this scattering in further detail, but we do want to stress that the measurement of vortex splitting provides an ultra-sensitive measurement method of unwanted scattering from the hologram or phase plate.

The (lack of) symmetry in the observed vortex splitting provides additional information on the nature of the scattering. The interference of a higher-order vortex beam with a uniform or Gaussian background should result in a series of |*ℓ*| = 1 vortices spaced equidistantly on a ring. Any deviation from this rotational symmetry demonstrates that the background is not uniform. Furthermore, for a pure binary phase grating we expect the *ℓ* and −*ℓ* diffraction orders to be perfect mirror images of each other. Although there is some mirror symmetry between the *ℓ* = 3 and *ℓ* = −3 images in Fig. 3, the images differ enough to argue that large-scale phase errors in the binary-phase hologram must be present.

One might think that the observed vortex splitting could be caused by misalignment, but this is not the case. We checked this by displacing the fork hologram in the transverse direction with respect to the beam. As a result, the dark region in the far-field of the *ℓ*-th order diffracted beam moved out of center towards the edge (not shown). However, the spatial structure of the split vortices hardly changed during this process. Vortex splitting thus proves to be quite robust.

To generalize our result, we have also studied the far-field diffraction from a spatial light modulator (SLM) that was programmed to produce a pure high-order vortex beam. We use a Hamamatsu X10468-07 liquid crystal on silicon modulator (pixel size 20*μ*m, phase errors *<* 0.05*λ*) that operates in reflection. The programmed hologram is a high-efficiency blazed fork grating with a blaze angle of 3 mrad. The imaging geometry is identical to that used with the fork hologram, but the Gaussian beam size at the SLM was somewhat smaller, as we now use an *f*_{1} = 75 cm lens instead of *f*_{1} = 1 m, making the far-field somewhat wider. Figure 9 show the observed intensity pattern for the *ℓ* = 1,2,3,4, and 5 vortex beams generated by our spatial light modulator. We again observe a splitting of the higher-order vortex in unity-charge vortices. A quantitative analysis of the observed splitting, based on Eq. (7), yields an estimated background intensity *I _{b}*/

*I*= 2 − 5

_{ℓ}*×*10

^{−3}for the

*ℓ*= 2 − 5 images, without any specific trend. The most likely explanation for the required coherent background is scattering caused by the unavoidable pixilation of the SLM. To check this, we numerically simulated far-field diffraction patterns for different pixel sizes and found that the vortex splitting disappears (for such an otherwise perfect hologram)

*only*for pixilation below 200 nm. For 20

*μ*m pixel size, we obtain good agreement with experimental data (Fig. 9).

True higher-order vortices have, to our knowledge, never been observed in natural speckle patterns, despite extensive numerical and experimental searches. Our findings support this observation, as only a very special plane-wave spectrum, which resembles precisely that of a higher-order vortex without any other plane-wave components, would achieve such a goal. We consider it very unlikely that any natural stochastic process can produce such a field. Additionally, it has very recently been shown that even simple optical reflection splits up vortices [28, 29].

Several applications rely on the specific structure in the dark center of vortex modes. Probably the most prominent one is the use of higher-order vortex phase masks in optical vortex coronagraphy [11], which allows direct imaging of exoplanets [12]. Here, the dark center of the diffraction pattern from the phase element is used to remove the unwanted bright light of the central star from the telescope. Our findings suggest that the observation and minimization of vortex splitting is a very suitable method to assess and optimize the quality of the phase elements in use.

Through observation of the splitting of high-order vortices we effectively presented an ultra-sensitive method to probe the intensity of coherent background light, as well as the potential admixture of other (*ℓ* ≠ 0) modes. This method thus can be used to determine the quality of mode convertors, such as fork gratings and spatial light modulators. If a nearly perfect mode converter is available, it can also be used to characterize the OAM mode purity of a light source, such as a laser. As the vortex splitting observed in the *ℓ*-th diffraction order enables one to quantify the background intensity in that beam, even if it is very weak, it also yields the amplitude of the −*ℓ* OAM component in the original light source with high accuracy.

## 5. Conclusion

In conclusion, we have presented three methods to investigate the fine structure of high-quality optical vortex fields by analysis of the intensity and interference pattern and by using a novel multi-pinhole interferometer (MPI). Clearly, the use of the MPI results in the most accurate and robust characterization of the local vorticity. We have found that, independent on the production method of the vortices, higher-order vortices are very fragile, being vulnerable to any coherent background, and easily split up in unity-charge vortices. We have given theoretical arguments and experimental confirmation why vortex-splitup is practically unavoidable.

## 6. Appendix: OAM analysis with multi-pinhole interferometer

This appendix describes the data analysis that we apply to convert the far-field diffraction pattern *I*(*μ*, *ν*) into the optical fields *E _{m}* =

*A*exp (

_{m}*iϕ*) at the holes of the MPI. We take the work of Guo et al. [23] as a starting point and write this far-field diffraction pattern as

_{m}*ℱ*denotes the Fourier transform and circ(

*x*−

*x*,

_{m}*y*−

*y*) is the (disk-like) transmission function of the

_{m}*m*-th pinhole, located at position (

*x*,

_{m}*y*) in the pinhole plane.

_{m}As a first step in the analysis, we calculate the inverse Fourier transform of the measured intensity pattern

*P*(

_{mn}*x, y*) are convolutions of the transmission functions of holes

*m*and

*n*; these functions peak at position (

*x*−

_{m}*x*,

_{n}*y*−

_{m}*y*) and have identical cone-like shapes with a width related to the hole diameter

_{n}*b*. The complex field

*g*(

*x,y*) thus has

*N*(

*N*− 1) peaks arranged in a circularly symmetric configuration around a central maximum [see Fig. 5(b)]. We find the products ${E}_{m}{E}_{n}^{*}(m\ne n)$ from the complex amplitudes of these peaks, integrated over a finite area to improve the signal-to-noise ratio. We do not use the central peak.

In the experiment, we first center and resize the CCD image. To avoid artifacts associated with the (modest) cutoff of the image, we then increase the size of the image (by a factor 8*×*) by adding zeros and apply a 2D Hann window before performing the Fourier transform. The mentioned peaks in the Fourier spectrum yield the products ∝
${E}_{m}{E}_{n}^{*}={A}_{m}{A}_{n}\text{exp}[i({\phi}_{m}-{\phi}_{n})]$, which we separate in amplitude products *a _{mn}* =

*A*and phase differences

_{m}A_{n}*φ*=

_{mn}*ϕ*−

_{m}*ϕ*. We further combine these to two vectors

_{n}*S⃗*and

*P⃗*with the following components:

*P⃗*depends on both amplitudes and phases. This has the purpose of weighting the

*N*(

*N*− 1) phase differences with the brightness of the spot in the Fourier transformed image, which improves the data significantly. We also combine the single-pinhole amplitudes and phases in the vectors

*A⃗*= (

*A*

_{1},

*A*

_{2},...,

*A*) and Φ

_{N}*⃗*= (

*ϕ*

_{1},

*ϕ*

_{2},...,

*ϕ*).

_{N}The remaining task is now to express the measured/calculated *S⃗* and *P⃗* in terms of the original *A⃗* and Φ*⃗*, in order to invert the relation and extract the desired quantities. To retrieve the optical amplitudes {*A*_{1},..., *A _{N}*}, we rewrite the components

*S*as

_{m}*ϕ*

_{1},...,

*ϕ*} we simply need to solve

_{N}*P⃗*=

*M*Φ

*⃗*. Since M is a singular matrix, we first get rid of a global phase by setting

*ϕ*

_{1}= 0. The new (

*N*− 1) dimensional matrix

*M*can be easily inverted to obtain the remaining

_{eff}*N*− 1 phases of the input field via

## Acknowledgments

We gratefully acknowledge E.G. Churin for the production of the hologram, G.C.G. Berkhout and M.W. Beijersbergen for the development of the MPI, and G.C.G. Berkhout and J.P. Woerdman for fruitful discussions. We thank the “Stichting voor Fundamenteel Onderzoek der Materie (FOM)”, the European Union Commission within the 7th Framework Project No. 255914 (PHORBITECH) and NWO for financial support.

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