## Instability of higher-order optical vortices analyzed with a multi-pinhole interferometer |

Optics Express, Vol. 20, Issue 20, pp. 22961-22975 (2012)

http://dx.doi.org/10.1364/OE.20.022961

Acrobat PDF (3638 KB)

### 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

*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

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 **45**, 8185–81891992. [CrossRef] [PubMed]

3. J. P. Torres and L. Torner, *Twisted Photons* (John Wiley, 2011). [CrossRef]

*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

4. J. F. Nye and M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. Lond. A **336**, 165–190 (1974). [CrossRef]

6. M. V. Berry and M. R. Dennis, “Knotted and linked phase singularities in monochromatic waves,” Proc. R. Soc. Lond. A **457**, 2251–2263 (2001). [CrossRef]

7. A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature **412**, 313–316 (2001). [CrossRef] [PubMed]

9. J. Wang, J.-Y. Yang, I.M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital
angular momentum multiplexing,” Nat. Photonics **6**, 488–496 (2012). [CrossRef]

10. J. Keller, A. Schönle, and S.W. Hell, “Efficient fluorescence inhibition patterns for RESOLFT
microscopy,” Opt. Express **15**, 3361–3371 (2007). [CrossRef]

11. G. Foo, D. M. Palacios, and G. A. Swartzlander Jr., “Optical vortex coronagraph,” Opt. Lett. **30**, 3308–3310 (2005). [CrossRef]

12. E. Serabyn, D. Mawet, and R. Burruss, “An image of an exoplanet separated by two diffraction beamwidths from a star,” Nature **464**, 1018–1020 (2010). [CrossRef] [PubMed]

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 **45**, 8185–81891992. [CrossRef] [PubMed]

13. S. S. R. Oemrawsingh, J. A. W. van Houwelingen, E. R. Eliel, J. P. Woerdman, E. J. K. Verstegen, J. G. Kloosterboer, and G. W. ’t Hooft, “Half-integral spiral phase plates for optical wavelengths,” J. Opt. A **6**, S228–S290 (2004). [CrossRef]

14. N. R. Heckenberg, R. McDuff, C. P. Smith, and A. G. White, “Generation of optical phase singularities by computer-generated holograms,” Opt. Lett. **17**, 221–223 (1992). [CrossRef] [PubMed]

5. I. V. Basistiy, V. Yu. Bazhenov, M. S. Soskin, and M. V. Vasnetsov, “Optics of light beams with screw dislocations,” Opt. Commun. **103**, 422–428 (1993). [CrossRef]

15. V. Y. Bazhenov, M. S. Soskin, and M. V. Vasnetsov, “Screw dislocations in light wavefronts,” J. Mod. Opt. **39**, 985–990 (1992). [CrossRef]

*ℓ*| > 1. In many experimental investigations (e.g., [5

5. I. V. Basistiy, V. Yu. Bazhenov, M. S. Soskin, and M. V. Vasnetsov, “Optics of light beams with screw dislocations,” Opt. Commun. **103**, 422–428 (1993). [CrossRef]

16. M. S. Soskin, V. N. Gorshkow, M. V. Vasnetsov, J. T. Malos, and N. R. Heckenberg, “Topological charge and angular momentum of light beams carrying optical vortices,” Phys. Rev. A **56**, 4064–4075 (1997). [CrossRef]

17. T. Ando, N. Matsumoto, Y. Ohtake, Y. Takiguchi, and T. Inoue, “Structure of optical singularities in coaxial superpositions of Laguerre-Gaussian modes,” J. Opt. Soc. Am. A **27**, 2602–2612 (2010). [CrossRef]

18. A. Kumar, P. Vaity, and R. P. Singh, “Crafting the core asymmetry to lift the degeneracy of
optical vortices,” Opt. Express **19**, 6182–6190 (2011). [CrossRef]

19. K. O’Holleran, M. J. Padgett, and M. R. Dennis, “Topology of optical vortex lines formed by the interference of three, four, and five plane waves,” Opt. Expr. **14**, 3039–3044 (2006). [CrossRef]

20. M. R. Dennis, “Rows of optical vortices from elliptically perturbing a high-order beam,” Opt. Lett. **31**, 1325–1327 (2006). [CrossRef] [PubMed]

21. G. C. G. Berkhout and M. W. Beijersbergen, “Method for probing the orbital angular momentum of optical vortices in electromagnetic waves from astronomical objects,” Phys. Rev. Lett. **101**, 100801 (2008). [CrossRef] [PubMed]

23. C.-S. Guo, S.-J. Yue, and G.-X. Wei, “Measuring the orbital angular momentum of optical vortices using a multipinhole plate,” Appl. Phys. Lett. **94**, 231104 (2009). [CrossRef]

## 2. Experimental setup and first results

### 2.1. Experimental setup

24. E. G. Churin, J. Hossfeld, and T. Tschudi, “Polarization configurations with singular point former by computer generated holograms,” Opt. Commun. **99**, 13–17 (1993). [CrossRef]

*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

5. I. V. Basistiy, V. Yu. Bazhenov, M. S. Soskin, and M. V. Vasnetsov, “Optics of light beams with screw dislocations,” Opt. Commun. **103**, 422–428 (1993). [CrossRef]

*±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.

*(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

*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}

*, while the same scaling applies to the Kummer beams that we actually generate (see below and ref. [25*

^{ℓ}25. L. Janicijevic and S. Topuzoski, “Fresnel and Fraunhofer diffraction of a Gaussian laser beam by fork-shaped gratings,” J. Opt. Soc. Am. A **25**, 2659–2669 (2008). [CrossRef]

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

*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

25. L. Janicijevic and S. Topuzoski, “Fresnel and Fraunhofer diffraction of a Gaussian laser beam by fork-shaped gratings,” J. Opt. Soc. Am. A **25**, 2659–2669 (2008). [CrossRef]

*I*is the

_{m}*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.

*μ*m, depending on the topological charge of the observed main vortex and on the size of the beam in the pinhole plane.

*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.

**103**, 422–428 (1993). [CrossRef]

26. V. G. Denisenko, A. Minovich, A. S. Desyatnikov, W. Krolikowski, M. S. Soskin, and Y. S. Kivshar, “Mapping phases of singular scaler light fields,” Opt. Lett. **35**, 89–91 (2008). [CrossRef]

^{th}diffraction order. The interference pattern appearing in the pinhole is detected with system A in Fig. 1.

*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

21. G. C. G. Berkhout and M. W. Beijersbergen, “Method for probing the orbital angular momentum of optical vortices in electromagnetic waves from astronomical objects,” Phys. Rev. Lett. **101**, 100801 (2008). [CrossRef] [PubMed]

23. C.-S. Guo, S.-J. Yue, and G.-X. Wei, “Measuring the orbital angular momentum of optical vortices using a multipinhole plate,” Appl. Phys. Lett. **94**, 231104 (2009). [CrossRef]

*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*at each of the individual holes [21

_{m}21. G. C. G. Berkhout and M. W. Beijersbergen, “Method for probing the orbital angular momentum of optical vortices in electromagnetic waves from astronomical objects,” Phys. Rev. Lett. **101**, 100801 (2008). [CrossRef] [PubMed]

22. G. C. G. Berkhout and M. W. Beijersbergen, “Using a multipoint interferometer to measure the orbital angular momentum of light in astronomy,” J. Opt. A **11**, 094021 (2009). [CrossRef]

*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*in each of the holes and the far-field diffraction pattern resembles that of

_{m}*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.

*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

23. C.-S. Guo, S.-J. Yue, and G.-X. Wei, “Measuring the orbital angular momentum of optical vortices using a multipinhole plate,” Appl. Phys. Lett. **94**, 231104 (2009). [CrossRef]

*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}*N*samples that it takes are evenly distributed on a circle. A simple Fourier decomposition therefore already yields the sampled OAM modal amplitudes With this definition, the

*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

*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*).

*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

*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*= 0 for

_{ℓ}*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 at a distance

*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.

*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 where

*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*(

*d*

^{2}),

*λ*

_{1}= 2

*Day*

_{0},

*λ*

_{2}=

*Da*

^{2}, and

*λ*= 0 for

_{ℓ}*ℓ*≠ {0,1,2}, making For large vortex splitting

*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

*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).

27. M. R. Sharpe and D. Irish, “Stray light in diffraction grating monochromators,” Opt. Acta **25**, 861–893 (1978). [CrossRef]

*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.

*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.

11. G. Foo, D. M. Palacios, and G. A. Swartzlander Jr., “Optical vortex coronagraph,” Opt. Lett. **30**, 3308–3310 (2005). [CrossRef]

12. E. Serabyn, D. Mawet, and R. Burruss, “An image of an exoplanet separated by two diffraction beamwidths from a star,” Nature **464**, 1018–1020 (2010). [CrossRef] [PubMed]

*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

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

*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

_{m}**94**, 231104 (2009). [CrossRef]

*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}*×*) by adding zeros and apply a 2D Hann window before performing the Fourier transform. The mentioned peaks in the Fourier spectrum yield the 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: where we note that

*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}*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*}, we rewrite the components

_{N}*S*as and invert this relation to We finally solve this set of equations numerically by an iterative algorithm. To retrieve of the optical phases {

_{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

## 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 |

2. | L. Allen, S. M. Barnett, and M. J. Padgett, |

3. | J. P. Torres and L. Torner, |

4. | J. F. Nye and M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. Lond. A |

5. | I. V. Basistiy, V. Yu. Bazhenov, M. S. Soskin, and M. V. Vasnetsov, “Optics of light beams with screw dislocations,” Opt. Commun. |

6. | M. V. Berry and M. R. Dennis, “Knotted and linked phase singularities in monochromatic waves,” Proc. R. Soc. Lond. A |

7. | A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature |

8. | H. Di Lorenzo Pires, H. C. B. Florijn, and M. P. van Exter, “Measurement of the spiral spectrum of entangled two-photon states,” Phys. Rev. Lett. |

9. | J. Wang, J.-Y. Yang, I.M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital
angular momentum multiplexing,” Nat. Photonics |

10. | J. Keller, A. Schönle, and S.W. Hell, “Efficient fluorescence inhibition patterns for RESOLFT
microscopy,” Opt. Express |

11. | G. Foo, D. M. Palacios, and G. A. Swartzlander Jr., “Optical vortex coronagraph,” Opt. Lett. |

12. | E. Serabyn, D. Mawet, and R. Burruss, “An image of an exoplanet separated by two diffraction beamwidths from a star,” Nature |

13. | S. S. R. Oemrawsingh, J. A. W. van Houwelingen, E. R. Eliel, J. P. Woerdman, E. J. K. Verstegen, J. G. Kloosterboer, and G. W. ’t Hooft, “Half-integral spiral phase plates for optical wavelengths,” J. Opt. A |

14. | N. R. Heckenberg, R. McDuff, C. P. Smith, and A. G. White, “Generation of optical phase singularities by computer-generated holograms,” Opt. Lett. |

15. | V. Y. Bazhenov, M. S. Soskin, and M. V. Vasnetsov, “Screw dislocations in light wavefronts,” J. Mod. Opt. |

16. | M. S. Soskin, V. N. Gorshkow, M. V. Vasnetsov, J. T. Malos, and N. R. Heckenberg, “Topological charge and angular momentum of light beams carrying optical vortices,” Phys. Rev. A |

17. | T. Ando, N. Matsumoto, Y. Ohtake, Y. Takiguchi, and T. Inoue, “Structure of optical singularities in coaxial superpositions of Laguerre-Gaussian modes,” J. Opt. Soc. Am. A |

18. | A. Kumar, P. Vaity, and R. P. Singh, “Crafting the core asymmetry to lift the degeneracy of
optical vortices,” Opt. Express |

19. | K. O’Holleran, M. J. Padgett, and M. R. Dennis, “Topology of optical vortex lines formed by the interference of three, four, and five plane waves,” Opt. Expr. |

20. | M. R. Dennis, “Rows of optical vortices from elliptically perturbing a high-order beam,” Opt. Lett. |

21. | G. C. G. Berkhout and M. W. Beijersbergen, “Method for probing the orbital angular momentum of optical vortices in electromagnetic waves from astronomical objects,” Phys. Rev. Lett. |

22. | G. C. G. Berkhout and M. W. Beijersbergen, “Using a multipoint interferometer to measure the orbital angular momentum of light in astronomy,” J. Opt. A |

23. | C.-S. Guo, S.-J. Yue, and G.-X. Wei, “Measuring the orbital angular momentum of optical vortices using a multipinhole plate,” Appl. Phys. Lett. |

24. | E. G. Churin, J. Hossfeld, and T. Tschudi, “Polarization configurations with singular point former by computer generated holograms,” Opt. Commun. |

25. | L. Janicijevic and S. Topuzoski, “Fresnel and Fraunhofer diffraction of a Gaussian laser beam by fork-shaped gratings,” J. Opt. Soc. Am. A |

26. | V. G. Denisenko, A. Minovich, A. S. Desyatnikov, W. Krolikowski, M. S. Soskin, and Y. S. Kivshar, “Mapping phases of singular scaler light fields,” Opt. Lett. |

27. | M. R. Sharpe and D. Irish, “Stray light in diffraction grating monochromators,” Opt. Acta |

28. | M. R. Dennis and J. B. Götte, “Topological aberration of optical vortex beams and singularimetry of dielectric interfaces,” pre-print (2012), arXiv:1205.6457. |

29. | W. Löffler, A. Aiello, and J. P. Woerdman, “Observation of OAM sidebands due to optical reflection,” pre-print (2012), arXiv:1204.4003 (PRL, in print). |

**OCIS Codes**

(050.1970) Diffraction and gratings : Diffractive optics

(050.4865) Diffraction and gratings : Optical vortices

(260.6042) Physical optics : Singular optics

**ToC Category:**

Physical Optics

**History**

Original Manuscript: July 23, 2012

Revised Manuscript: September 14, 2012

Manuscript Accepted: September 16, 2012

Published: September 21, 2012

**Citation**

F. Ricci, W. Löffler, and M.P. van Exter, "Instability of higher-order optical vortices analyzed with a multi-pinhole interferometer," Opt. Express **20**, 22961-22975 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-20-22961

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