Abstract

Spiral phase plates for the generation of Laguerre–Gaussian (LG) beam with non-null radial index were designed and fabricated by electron beam lithography on polymethylmethacrylate over glass substrates. The optical response of these phase optical elements was theoretically considered and experimentally measured, and the purity of the experimental beams was investigated in terms of LG modes contributions. The far-field intensity pattern was compared with theoretical models and numerical simulations, whereas interferometric analyses confirmed the expected phase features of the generated beams. The high quality of the output beams confirms the applicability of these phase plates for the generation of high-order LG beams.

© 2014 Optical Society of America

In the last decades a considerable interest has been shown in the generation of coherent beams carrying phase singularities [1,2]. In particular Laguerre–Gaussian (LG) beams have gained increasing attention, especially after Allen et al. [3] demonstrated that these modes carry orbital angular momentum (OAM) of light. LG modes form a complete set of orthonormal solutions of the paraxial wave equation whose complex amplitude, in cylindrical coordinates, is given by [4]

ψp(r,φ,z)=Cplw(2rw)||er2w2Lp||(2r2w2)eiφeikz+ikr22Rei(2p+||+1)ζ(z),
where p and are the radial and azimuthal indices, respectively, Cp=[2p!/π(p+||)!]1/2 is a normalization factor, w(z)=w0(1+z2/zR2)1/2 is the beam radius, R(z)=(z2+zR2)/z is the phase front curvature, ζ(z)=tan1(z/zR) is the Gouy phase shift, zR=π·w02/λ being the Rayleigh range and w0 being the beam waist, and Lp|| is the associated Laguerre polynomial [5]. The azimuthal index , corresponding to the topological charge of the embedded singularity, indicates the number of twists of the helical wavefront within a wavelength and represents the amount of OAM, in units of , carried by each single photon. The index p represents the number of radial nodes on a plane perpendicular to the direction of propagation and is thus related to the distribution of the intensity pattern in p+1 concentric rings around the central dark zone of the phase singularity (when ||1).

The first method for LG beam generation was demonstrated by Beijersbergen et al. [6] with the conversion of Hermite–Gaussian beams by means of a couple of cylindrical lenses. Intensive interest in LG beams has required the development of simple and reliable methods for their generation from conventional laser beams, e.g., Gaussian beams. One of the most widespread approaches is based on the use of the so-called fork-holograms, holographic structures encoding the interference pattern of the considered LG beam with the generating reference wave, e.g., a plane wave or a Gaussian beam. The interference pattern presents the structure of a grating with one or several central dislocations corresponding to the amount of topological charge [7,8]. Other groups exploit refractive optical elements, the so-called spiral phase plates (SPP), for the shaping of a Gaussian beam into an OAM-carrying optical vortex beam. These SPPs are helicoidal transmission optical elements that look like a spiral staircase with a certain number of steps that progressively build the total phase gap in air [9],

2π=2πλ(nSPP1)h,
after a complete rotation of the azimuthal angle φ, h being the height of the spiral, nSPP the refractive index of the SPP material, and λ the impinging wavelength.

Since SPPs are transmission phase elements, they do not alter the direction of propagation and moreover show a higher efficiency; therefore they are much more preferred especially in applications such as optical coronagraphy [10], spiral-phase contrast microscopy [11], and stimulated emission depletion microscopy [12] to improve both contrast and resolution of images.

So far, however, azimuthally higher and radially lowest-order LG beams have been extensively studied, while radially higher-order LG beams have not attracted much attention. Specifically, a few groups described the generation of high-order LG beams with the use of spatial light modulators [13] or fabricated fork-holograms [14,15]. Multi-ringed LG beams can be holographically generated with phase patterns involving radial phase discontinuities; however, this is technically difficult, especially when the azimuthal and radial mode indices become larger. In this case, the holographic phase patterns involve more phase discontinuities to cause nonideal diffraction of light.

In this Letter, we extend the use of SPPs for the generation of high-order LG beams with nonzero radial index and we present the work of design, fabrication, and optical characterization of such phase optical elements.

To design SPPs for the control of the radial index p, we started considering some basic properties of LG beams and the relations between far-field and near-field of a given phase element.

In the paraxial and Fraunhofer approximations, the far-field of an illuminated optical element is described as the Fourier transform (FT) of its transmission function multiplied by the impinging illumination [16].

It is worth noting that an LG beam with radial and azimuthal indices p and , described by the complex function ψp, is eigenfunction of the FT with eigenvalue λp=(1)p·(i) [17].

Therefore, since the inverse FT of an LG beam of radial index p and topological charge , is an LG beam again with the same indices, the transmission function of the generating optical element should be exactly the complex function ψp of the generated beam in the far-field. On the other hand, since we are looking for a phase-type element, by considering only the phase term of the complex function ψp, we get the following expression for the transmission function of such a SSP:

Up(r,φ)=eiφ·sign[Lp||(2r2w02)].

In the case of p=0, we get the usual form U0=eiφ. For p0 instead, it presents a number of p radial phase discontinuities of π, in correspondence to the change of sign of the associated Laguerre polynomial.

By illuminating the SPP with a Gaussian beam u(i)=a·exp(r2/w12), the far-field χ(ρ,θ) is given by the FT of the product between the incident beam u(i) and the transmission function Up [16]:

χ(ρ,θ)=FT[c0u(i)Upl]=c002πdφ0+u(i)Upleirρcos(θφ)rdr,
where the radial coordinate ρ has been scaled by f/k, k=2π/λ, and c0 includes the multiplicative constant 1/(iλf), f being the focal length of the optical system. After applying the convolution theorem [18] we get
χ(ρ,θ)=FT[c0u(i)eiφ]*FT[sign[Lp||(2r2w02)]],
where the result of the first FT is the Kummer beam [19]:
FT[c0u(i)eiφ]=c0iπ3/22w12eiθeη2/2η[I12(η22)I+12(η22)],
where η=ρw1/2 and In are modified Bessel functions of the first kind. The second term is given by
FT[sign[Lp||(2r2w02)]]=k=1p2rk(1)kJ1(rkρ)ρδ(ρ)2π,
{rk} being the zeroes of the associated Laguerre polynomial and Jn Bessel functions of the first kind.

Electron beam lithography (EBL) represents a powerful tool to generate this kind of structures, because of the possibility to realize continuous surface profiles, high flexibility in the element’s design, and good optical quality of the fabricated reliefs [20,21]. 3D profiles can be generated modulating the local dose distribution, inducing different dissolution rates in the polymer exposed, giving rise to different resist thicknesses at the end of the development process. A dose correction to compensate for proximity effect is required to obtain a good shape definition, especially at the phase steps and at the central anomaly of the spiral phase mask. A JEOL JBX-6300FS EBL machine was used to fabricate the SPPs [22]. The 3D-spiral phase mask profiles were written on a 2.0 μm thick poly (methyl methacrylate) (PMMA) resist layer. As a substrate, we chose a 1.1 mm thick ITO coated soda lime float glass slide. For the considered wavelength of the laser beam (λ=632.8nm), PMMA refractive index results n=1.489 from spectroscopic ellipsometry analysis; therefore according to Eq. (2) the total height of the spiral as a function of the topological charge is h=·1294.1nm.

Electron-beam exposures were carried out at 100 keV, in high resolution mode, with a beam probe current of 100 pA and a beam diameter of 2 nm. Single- and multi-step SPPs have been realized, with different and p, with 256 phase steps in a period (Fig. 1). Careful inspection of exposure parameters, along with accurate proximity effect correction and optimization of developing conditions, has allowed obtaining SPPs with superior optical quality.

 

Fig. 1. SEM micrograph of PMMA SSP for the generation of LG51 beam.

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The fabricated samples were characterized with an optical setup assembled with commercial components mounted on an optical table. The Gaussian beam (λ=632.8nm, beam waist w0=240μm, power 0.8 mW) emitted by a HeNe laser source (HNR008R, Thorlabs) was resized by a lens of focal length f1 placed between the laser and the SPP at such a distance that the beam illuminates the SPP with the prescribed waist size and flat wavefront. The far-field intensity distribution was recorded with a CCD camera (AxioCam MRc 5, Zeiss, 2584×1936 pixels, 3×12-bit RGB color depth) placed at the back-focal plane of a second lens of focal length f2.

Figure 2 exhibits some results of the characterization process for the SPP sample of radial index p=3 and topological charge =1. For the given phase pattern of the optical element [Fig. 2(a)], the far-field was obtained by implementing in MATLAB environment a custom code exploiting commercial algorithm for fast Fourier transform (FFT) calculation. The result is reported in Fig. 2(b) in a false-color scale where the brightness indicates the field amplitude and the color exhibits its phase. The intensity pattern measured on the camera [Fig. 2(d)] well reproduces the simulated intensity distribution; however, a more significant comparison is given by Fig. 2(c), where an experimental cross section on a plane perpendicular to the beam axis and crossing the central singularity reproduces the numerical trend well. While the far-field trend perfectly reproduces the theoretical LG mode close to the central singularity, this overlap is less accurate in correspondence to the outer ring. A stronger overlap could be achieved by illuminating the SSP with a top-hat input beam [13].

 

Fig. 2. Generation of LG31 beam. (a) Phase pattern of the corresponding SSP. (b) Simulation of the far-field. Brightness and colors refer, respectively, to intensity and phase of the field. (c) Comparison between a cross section of the experimental far-field intensity (blue dots), numerical computation of the far-field intensity (blue solid line) and theoretical profile of a LG31 beam (red solid line). (d) Experimental far-field intensity, normalized to the maximal value.

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We estimated the purity of the generated modes by using an amplitude overlap integral along a selected radial direction on the plane perpendicular to the beam axis in the far-field, calculated as the scalar product between the measured χexp and the normalized theoretical distributions ψp (Eq. 1):

γp=ψp|χexpχexp|χexp.

Since the topological charge is conserved, for a given * we limited the decomposition into the set of functions {ψp*} of different p and constant =*. In Fig. 3 the efficiency values are reported for SPPs of =1 and radial index in the range p={0,,5} for a decomposition into LG functions with the same topological charge and radial index up to 9. The plot exhibits a high generation efficiency of the dominant mode for the fabricated SPPs, with out-of-diagonal values below 5% and greater than 0.5% for no more than a few p-values closed to the highest-efficiency one. While it is well known that, for p=0, the radius of maximum field is proportional to [23], no experimental data have been presented in the literature concerning the behavior of the inner-ring radius as a function of p. In Fig. 4 we reported the experimental data of the inner-radius of maximum field RM, normalized to the beam waist radius, for increasing values of the radial index and fixed =1. The experimental data well reproduce the theoretical trend of analytical LG modes ψp1; moreover the linear dependence of 1/RM2 on the radial index suggests, for a given p and in case of =1, the behavior of RM as (p+1)1/2.

 

Fig. 3. Efficiency γp of LGp generation for different SSPs encoding LG beams with radial number pSPP from 0 to 5, fixed azimuth number =1. Diagonal efficiency terms: γ01=0.93, γ11=0.91, γ21=0.88, γ31=0.86, γ41=0.84, γ51=0.81.

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Fig. 4. Experimental data of the first-ring radius RM, normalized to w0/2 (blue dots), and of the corresponding value 1/RM2 (red squares), as a function of the radial number p for fixed azimuth number =1. Comparison with the theoretical trends, given by interpolation of data points calculated on LG beams (solid lines). In particular, the fit with a linear trend 1/RM2=k·(p+1)+q gives k=1.18, q=2e2.

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We confirmed the phase structures of the generated beams by observing the interference patterns of the output beams and a plane reference wave with a Mach–Zehnder interferometer system. Figure 5 displays typical interference patterns of the output beams generated by SSPs with phase patterns of LG31 [Figs. 5(a) and 5(b)] and LG32 [Fig. 5(c)] modes, respectively. In Fig. 5, we can observe fork-like patterns of fringes corresponding to azimuthal mode index of the observed beams as well as radial phase discontinuities of half a period corresponding to the radial mode index p, which reflect the phase properties of LG beams.

 

Fig. 5. Experimental results. (a) Intensity of the pattern obtained from interference of a reference wave with a generated LG31 beam and (b) details of phase dislocations. (c) Detail of the interference pattern for an LG32 beam. Central dislocations confirm the presence of a phase singularity along the axis of the beam. Shifts of half-period between the interference fringes of consecutive rings confirm the π phase-shift because of the change of sign in the associated Laguerre polynomial.

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The usual SPPs are designed for a certain wavelength; hence they induce topological-charge dispersion for short pulses or broadband sources, leading to a superposition of contributions with different values of . Solutions presented in the literature, e.g., [24], could be considered in order to extend the bandwidth and assure integer phase steps of the singularity and π phase shift between adjacent annular rings of the SPP.

In conclusion, the possibility of fabricating SSPs for the generation of high-order Laguerre–Gauss beams has been considered and investigated. The far-field intensity and phase patterns are in good accordance with the expected distributions and present a remarkable quality with a high-purity level of the dominant LG term. These optical elements could find interest and applications not only in astronomy and microscopy, as above mentioned, but also in other fields where optical beams carrying OAM are exploited, such as optical trapping and tweezing [25], encoding of optical quantum information [26], and telecommunications [27].

References

1. J. F. Nye and M. V. Berry, Proc. R. Soc. A 336, 165 (1974). [CrossRef]  

2. P. Coullet, L. Gil, and F. Rocca, Opt. Commun. 73, 403 (1989). [CrossRef]  

3. L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, Phys. Rev. A 45, 8185 (1992). [CrossRef]  

4. J. P. Torres and L. Torner, Twisted Photons (Wiley, 2011).

5. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, 1972).

6. M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, and J. P. Woerdman, Opt. Commun. 96, 123 (1993). [CrossRef]  

7. N. R. Heckenberg, R. Mc Duff, C. P. Smith, H. Rubinsztein-Dunlop, and M. J. Wegener, Opt. Quantum Electron. 24, S951 (1992). [CrossRef]  

8. Z. S. Sacks, D. Rozas, and G. A. Swartzlander, J. Opt. Soc. Am. B 15, 2226 (1998). [CrossRef]  

9. G. A. Turnbull, D. A. Robertson, G. M. Smith, L. Allen, and M. J. Padgett, Opt. Commun. 127, 183 (1996). [CrossRef]  

10. E. Mari, G. Anzolin, F. Tamburini, M. Prasciolu, G. Umbriaco, A. Bianchini, C. Barbieri, and F. Romanato, Opt. Express 18, 2339 (2010). [CrossRef]  

11. C. Maurer, A. Jesacher, S. Furhapter, S. Bernet, and M. Ritsch-Marte, J. Microsc. 230, 134 (2008). [CrossRef]  

12. D. Wildanger, J. Buckers, V. Westphal, S. W. Hell, and L. Kastrup, Opt. Express 17, 16100 (2009). [CrossRef]  

13. N. Matsumoto, T. Ando, T. Inoue, Y. Ohtake, N. Fukuchi, and T. Hara, J. Opt. Soc. Am. A 25, 1642 (2008). [CrossRef]  

14. J. Arlt, K. Dholakia, L. Allen, and M. J. Padgett, J. Mod. Opt. 45, 1231 (1998). [CrossRef]  

15. M. Granata, C. Buy, R. Ward, and M. Barsuglia, Phys. Rev. Lett. 105, 231102 (2010). [CrossRef]  

16. M. Born and E. Wolf, Principles of Optics (Pergamon, 1986).

17. V. A. Soifer, Methods for Computer Design of Diffractive Optical Elements (Wiley, 2002).

18. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1996).

19. G. Anzolin, F. Tamburini, A. Bianchini, and C. Barbieri, Phys. Rev. A 79, 033845 (2009). [CrossRef]  

20. E. B. Kley, Microelectron. Eng. 34, 261 (1997). [CrossRef]  

21. 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, Appl. Opt. 43, 688 (2004). [CrossRef]  

22. M. Prasciolu, F. Tamburini, G. Anzolin, E. Mari, M. Melli, A. Carpentiero, C. Barbieri, and F. Romanato, Microelectron. Eng. 86, 1103 (2009). [CrossRef]  

23. M. J. Padgett and L. Allen, Opt. Commun. 121, 36 (1995). [CrossRef]  

24. G. A. Swartlander, Opt. Lett. 31, 2042 (2006). [CrossRef]  

25. A. T. O’Neil and M. J. Padgett, Opt. Commun. 193, 45 (2001). [CrossRef]  

26. A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, Nature 412, 313 (2001). [CrossRef]  

27. N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, and S. Ramachandran, Science 340, 1545 (2013). [CrossRef]  

References

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  1. J. F. Nye and M. V. Berry, Proc. R. Soc. A 336, 165 (1974).
    [CrossRef]
  2. P. Coullet, L. Gil, and F. Rocca, Opt. Commun. 73, 403 (1989).
    [CrossRef]
  3. L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, Phys. Rev. A 45, 8185 (1992).
    [CrossRef]
  4. J. P. Torres and L. Torner, Twisted Photons (Wiley, 2011).
  5. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, 1972).
  6. M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, and J. P. Woerdman, Opt. Commun. 96, 123 (1993).
    [CrossRef]
  7. N. R. Heckenberg, R. Mc Duff, C. P. Smith, H. Rubinsztein-Dunlop, and M. J. Wegener, Opt. Quantum Electron. 24, S951 (1992).
    [CrossRef]
  8. Z. S. Sacks, D. Rozas, and G. A. Swartzlander, J. Opt. Soc. Am. B 15, 2226 (1998).
    [CrossRef]
  9. G. A. Turnbull, D. A. Robertson, G. M. Smith, L. Allen, and M. J. Padgett, Opt. Commun. 127, 183 (1996).
    [CrossRef]
  10. E. Mari, G. Anzolin, F. Tamburini, M. Prasciolu, G. Umbriaco, A. Bianchini, C. Barbieri, and F. Romanato, Opt. Express 18, 2339 (2010).
    [CrossRef]
  11. C. Maurer, A. Jesacher, S. Furhapter, S. Bernet, and M. Ritsch-Marte, J. Microsc. 230, 134 (2008).
    [CrossRef]
  12. D. Wildanger, J. Buckers, V. Westphal, S. W. Hell, and L. Kastrup, Opt. Express 17, 16100 (2009).
    [CrossRef]
  13. N. Matsumoto, T. Ando, T. Inoue, Y. Ohtake, N. Fukuchi, and T. Hara, J. Opt. Soc. Am. A 25, 1642 (2008).
    [CrossRef]
  14. J. Arlt, K. Dholakia, L. Allen, and M. J. Padgett, J. Mod. Opt. 45, 1231 (1998).
    [CrossRef]
  15. M. Granata, C. Buy, R. Ward, and M. Barsuglia, Phys. Rev. Lett. 105, 231102 (2010).
    [CrossRef]
  16. M. Born and E. Wolf, Principles of Optics (Pergamon, 1986).
  17. V. A. Soifer, Methods for Computer Design of Diffractive Optical Elements (Wiley, 2002).
  18. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1996).
  19. G. Anzolin, F. Tamburini, A. Bianchini, and C. Barbieri, Phys. Rev. A 79, 033845 (2009).
    [CrossRef]
  20. E. B. Kley, Microelectron. Eng. 34, 261 (1997).
    [CrossRef]
  21. 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, Appl. Opt. 43, 688 (2004).
    [CrossRef]
  22. M. Prasciolu, F. Tamburini, G. Anzolin, E. Mari, M. Melli, A. Carpentiero, C. Barbieri, and F. Romanato, Microelectron. Eng. 86, 1103 (2009).
    [CrossRef]
  23. M. J. Padgett and L. Allen, Opt. Commun. 121, 36 (1995).
    [CrossRef]
  24. G. A. Swartlander, Opt. Lett. 31, 2042 (2006).
    [CrossRef]
  25. A. T. O’Neil and M. J. Padgett, Opt. Commun. 193, 45 (2001).
    [CrossRef]
  26. A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, Nature 412, 313 (2001).
    [CrossRef]
  27. N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, and S. Ramachandran, Science 340, 1545 (2013).
    [CrossRef]

2013 (1)

N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, and S. Ramachandran, Science 340, 1545 (2013).
[CrossRef]

2010 (2)

2009 (3)

G. Anzolin, F. Tamburini, A. Bianchini, and C. Barbieri, Phys. Rev. A 79, 033845 (2009).
[CrossRef]

D. Wildanger, J. Buckers, V. Westphal, S. W. Hell, and L. Kastrup, Opt. Express 17, 16100 (2009).
[CrossRef]

M. Prasciolu, F. Tamburini, G. Anzolin, E. Mari, M. Melli, A. Carpentiero, C. Barbieri, and F. Romanato, Microelectron. Eng. 86, 1103 (2009).
[CrossRef]

2008 (2)

N. Matsumoto, T. Ando, T. Inoue, Y. Ohtake, N. Fukuchi, and T. Hara, J. Opt. Soc. Am. A 25, 1642 (2008).
[CrossRef]

C. Maurer, A. Jesacher, S. Furhapter, S. Bernet, and M. Ritsch-Marte, J. Microsc. 230, 134 (2008).
[CrossRef]

2006 (1)

2004 (1)

2001 (2)

A. T. O’Neil and M. J. Padgett, Opt. Commun. 193, 45 (2001).
[CrossRef]

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, Nature 412, 313 (2001).
[CrossRef]

1998 (2)

J. Arlt, K. Dholakia, L. Allen, and M. J. Padgett, J. Mod. Opt. 45, 1231 (1998).
[CrossRef]

Z. S. Sacks, D. Rozas, and G. A. Swartzlander, J. Opt. Soc. Am. B 15, 2226 (1998).
[CrossRef]

1997 (1)

E. B. Kley, Microelectron. Eng. 34, 261 (1997).
[CrossRef]

1996 (1)

G. A. Turnbull, D. A. Robertson, G. M. Smith, L. Allen, and M. J. Padgett, Opt. Commun. 127, 183 (1996).
[CrossRef]

1995 (1)

M. J. Padgett and L. Allen, Opt. Commun. 121, 36 (1995).
[CrossRef]

1993 (1)

M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, and J. P. Woerdman, Opt. Commun. 96, 123 (1993).
[CrossRef]

1992 (2)

N. R. Heckenberg, R. Mc Duff, C. P. Smith, H. Rubinsztein-Dunlop, and M. J. Wegener, Opt. Quantum Electron. 24, S951 (1992).
[CrossRef]

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, Phys. Rev. A 45, 8185 (1992).
[CrossRef]

1989 (1)

P. Coullet, L. Gil, and F. Rocca, Opt. Commun. 73, 403 (1989).
[CrossRef]

1974 (1)

J. F. Nye and M. V. Berry, Proc. R. Soc. A 336, 165 (1974).
[CrossRef]

’t Hooft, G. W.

Abramowitz, M.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, 1972).

Allen, L.

J. Arlt, K. Dholakia, L. Allen, and M. J. Padgett, J. Mod. Opt. 45, 1231 (1998).
[CrossRef]

G. A. Turnbull, D. A. Robertson, G. M. Smith, L. Allen, and M. J. Padgett, Opt. Commun. 127, 183 (1996).
[CrossRef]

M. J. Padgett and L. Allen, Opt. Commun. 121, 36 (1995).
[CrossRef]

M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, and J. P. Woerdman, Opt. Commun. 96, 123 (1993).
[CrossRef]

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, Phys. Rev. A 45, 8185 (1992).
[CrossRef]

Ando, T.

Anzolin, G.

E. Mari, G. Anzolin, F. Tamburini, M. Prasciolu, G. Umbriaco, A. Bianchini, C. Barbieri, and F. Romanato, Opt. Express 18, 2339 (2010).
[CrossRef]

G. Anzolin, F. Tamburini, A. Bianchini, and C. Barbieri, Phys. Rev. A 79, 033845 (2009).
[CrossRef]

M. Prasciolu, F. Tamburini, G. Anzolin, E. Mari, M. Melli, A. Carpentiero, C. Barbieri, and F. Romanato, Microelectron. Eng. 86, 1103 (2009).
[CrossRef]

Arlt, J.

J. Arlt, K. Dholakia, L. Allen, and M. J. Padgett, J. Mod. Opt. 45, 1231 (1998).
[CrossRef]

Barbieri, C.

E. Mari, G. Anzolin, F. Tamburini, M. Prasciolu, G. Umbriaco, A. Bianchini, C. Barbieri, and F. Romanato, Opt. Express 18, 2339 (2010).
[CrossRef]

G. Anzolin, F. Tamburini, A. Bianchini, and C. Barbieri, Phys. Rev. A 79, 033845 (2009).
[CrossRef]

M. Prasciolu, F. Tamburini, G. Anzolin, E. Mari, M. Melli, A. Carpentiero, C. Barbieri, and F. Romanato, Microelectron. Eng. 86, 1103 (2009).
[CrossRef]

Barsuglia, M.

M. Granata, C. Buy, R. Ward, and M. Barsuglia, Phys. Rev. Lett. 105, 231102 (2010).
[CrossRef]

Beijersbergen, M. W.

M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, and J. P. Woerdman, Opt. Commun. 96, 123 (1993).
[CrossRef]

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, Phys. Rev. A 45, 8185 (1992).
[CrossRef]

Bernet, S.

C. Maurer, A. Jesacher, S. Furhapter, S. Bernet, and M. Ritsch-Marte, J. Microsc. 230, 134 (2008).
[CrossRef]

Berry, M. V.

J. F. Nye and M. V. Berry, Proc. R. Soc. A 336, 165 (1974).
[CrossRef]

Bianchini, A.

Born, M.

M. Born and E. Wolf, Principles of Optics (Pergamon, 1986).

Bozinovic, N.

N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, and S. Ramachandran, Science 340, 1545 (2013).
[CrossRef]

Buckers, J.

Buy, C.

M. Granata, C. Buy, R. Ward, and M. Barsuglia, Phys. Rev. Lett. 105, 231102 (2010).
[CrossRef]

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P. Coullet, L. Gil, and F. Rocca, Opt. Commun. 73, 403 (1989).
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C. Maurer, A. Jesacher, S. Furhapter, S. Bernet, and M. Ritsch-Marte, J. Microsc. 230, 134 (2008).
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P. Coullet, L. Gil, and F. Rocca, Opt. Commun. 73, 403 (1989).
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M. Granata, C. Buy, R. Ward, and M. Barsuglia, Phys. Rev. Lett. 105, 231102 (2010).
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N. R. Heckenberg, R. Mc Duff, C. P. Smith, H. Rubinsztein-Dunlop, and M. J. Wegener, Opt. Quantum Electron. 24, S951 (1992).
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Hell, S. W.

Huang, H.

N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, and S. Ramachandran, Science 340, 1545 (2013).
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C. Maurer, A. Jesacher, S. Furhapter, S. Bernet, and M. Ritsch-Marte, J. Microsc. 230, 134 (2008).
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E. B. Kley, Microelectron. Eng. 34, 261 (1997).
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N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, and S. Ramachandran, Science 340, 1545 (2013).
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Mair, A.

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, Nature 412, 313 (2001).
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Mari, E.

E. Mari, G. Anzolin, F. Tamburini, M. Prasciolu, G. Umbriaco, A. Bianchini, C. Barbieri, and F. Romanato, Opt. Express 18, 2339 (2010).
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M. Prasciolu, F. Tamburini, G. Anzolin, E. Mari, M. Melli, A. Carpentiero, C. Barbieri, and F. Romanato, Microelectron. Eng. 86, 1103 (2009).
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Matsumoto, N.

Maurer, C.

C. Maurer, A. Jesacher, S. Furhapter, S. Bernet, and M. Ritsch-Marte, J. Microsc. 230, 134 (2008).
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Mc Duff, R.

N. R. Heckenberg, R. Mc Duff, C. P. Smith, H. Rubinsztein-Dunlop, and M. J. Wegener, Opt. Quantum Electron. 24, S951 (1992).
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Melli, M.

M. Prasciolu, F. Tamburini, G. Anzolin, E. Mari, M. Melli, A. Carpentiero, C. Barbieri, and F. Romanato, Microelectron. Eng. 86, 1103 (2009).
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Nye, J. F.

J. F. Nye and M. V. Berry, Proc. R. Soc. A 336, 165 (1974).
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O’Neil, A. T.

A. T. O’Neil and M. J. Padgett, Opt. Commun. 193, 45 (2001).
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Oemrawsingh, S. S. R.

Ohtake, Y.

Padgett, M. J.

A. T. O’Neil and M. J. Padgett, Opt. Commun. 193, 45 (2001).
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J. Arlt, K. Dholakia, L. Allen, and M. J. Padgett, J. Mod. Opt. 45, 1231 (1998).
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G. A. Turnbull, D. A. Robertson, G. M. Smith, L. Allen, and M. J. Padgett, Opt. Commun. 127, 183 (1996).
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M. J. Padgett and L. Allen, Opt. Commun. 121, 36 (1995).
[CrossRef]

Prasciolu, M.

E. Mari, G. Anzolin, F. Tamburini, M. Prasciolu, G. Umbriaco, A. Bianchini, C. Barbieri, and F. Romanato, Opt. Express 18, 2339 (2010).
[CrossRef]

M. Prasciolu, F. Tamburini, G. Anzolin, E. Mari, M. Melli, A. Carpentiero, C. Barbieri, and F. Romanato, Microelectron. Eng. 86, 1103 (2009).
[CrossRef]

Ramachandran, S.

N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, and S. Ramachandran, Science 340, 1545 (2013).
[CrossRef]

Ren, Y.

N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, and S. Ramachandran, Science 340, 1545 (2013).
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C. Maurer, A. Jesacher, S. Furhapter, S. Bernet, and M. Ritsch-Marte, J. Microsc. 230, 134 (2008).
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Robertson, D. A.

G. A. Turnbull, D. A. Robertson, G. M. Smith, L. Allen, and M. J. Padgett, Opt. Commun. 127, 183 (1996).
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Rocca, F.

P. Coullet, L. Gil, and F. Rocca, Opt. Commun. 73, 403 (1989).
[CrossRef]

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E. Mari, G. Anzolin, F. Tamburini, M. Prasciolu, G. Umbriaco, A. Bianchini, C. Barbieri, and F. Romanato, Opt. Express 18, 2339 (2010).
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M. Prasciolu, F. Tamburini, G. Anzolin, E. Mari, M. Melli, A. Carpentiero, C. Barbieri, and F. Romanato, Microelectron. Eng. 86, 1103 (2009).
[CrossRef]

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N. R. Heckenberg, R. Mc Duff, C. P. Smith, H. Rubinsztein-Dunlop, and M. J. Wegener, Opt. Quantum Electron. 24, S951 (1992).
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Sacks, Z. S.

Smith, C. P.

N. R. Heckenberg, R. Mc Duff, C. P. Smith, H. Rubinsztein-Dunlop, and M. J. Wegener, Opt. Quantum Electron. 24, S951 (1992).
[CrossRef]

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G. A. Turnbull, D. A. Robertson, G. M. Smith, L. Allen, and M. J. Padgett, Opt. Commun. 127, 183 (1996).
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V. A. Soifer, Methods for Computer Design of Diffractive Optical Elements (Wiley, 2002).

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L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, Phys. Rev. A 45, 8185 (1992).
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M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, 1972).

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Swartzlander, G. A.

Tamburini, F.

E. Mari, G. Anzolin, F. Tamburini, M. Prasciolu, G. Umbriaco, A. Bianchini, C. Barbieri, and F. Romanato, Opt. Express 18, 2339 (2010).
[CrossRef]

G. Anzolin, F. Tamburini, A. Bianchini, and C. Barbieri, Phys. Rev. A 79, 033845 (2009).
[CrossRef]

M. Prasciolu, F. Tamburini, G. Anzolin, E. Mari, M. Melli, A. Carpentiero, C. Barbieri, and F. Romanato, Microelectron. Eng. 86, 1103 (2009).
[CrossRef]

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J. P. Torres and L. Torner, Twisted Photons (Wiley, 2011).

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J. P. Torres and L. Torner, Twisted Photons (Wiley, 2011).

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N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, and S. Ramachandran, Science 340, 1545 (2013).
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G. A. Turnbull, D. A. Robertson, G. M. Smith, L. Allen, and M. J. Padgett, Opt. Commun. 127, 183 (1996).
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Umbriaco, G.

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M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, and J. P. Woerdman, Opt. Commun. 96, 123 (1993).
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A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, Nature 412, 313 (2001).
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Ward, R.

M. Granata, C. Buy, R. Ward, and M. Barsuglia, Phys. Rev. Lett. 105, 231102 (2010).
[CrossRef]

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N. R. Heckenberg, R. Mc Duff, C. P. Smith, H. Rubinsztein-Dunlop, and M. J. Wegener, Opt. Quantum Electron. 24, S951 (1992).
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A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, Nature 412, 313 (2001).
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Wildanger, D.

Willner, A. E.

N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, and S. Ramachandran, Science 340, 1545 (2013).
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Woerdman, J. P.

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, Appl. Opt. 43, 688 (2004).
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M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, and J. P. Woerdman, Opt. Commun. 96, 123 (1993).
[CrossRef]

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, Phys. Rev. A 45, 8185 (1992).
[CrossRef]

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Pergamon, 1986).

Yue, Y.

N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, and S. Ramachandran, Science 340, 1545 (2013).
[CrossRef]

Zeilinger, A.

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, Nature 412, 313 (2001).
[CrossRef]

Appl. Opt. (1)

J. Microsc. (1)

C. Maurer, A. Jesacher, S. Furhapter, S. Bernet, and M. Ritsch-Marte, J. Microsc. 230, 134 (2008).
[CrossRef]

J. Mod. Opt. (1)

J. Arlt, K. Dholakia, L. Allen, and M. J. Padgett, J. Mod. Opt. 45, 1231 (1998).
[CrossRef]

J. Opt. Soc. Am. A (1)

J. Opt. Soc. Am. B (1)

Microelectron. Eng. (2)

M. Prasciolu, F. Tamburini, G. Anzolin, E. Mari, M. Melli, A. Carpentiero, C. Barbieri, and F. Romanato, Microelectron. Eng. 86, 1103 (2009).
[CrossRef]

E. B. Kley, Microelectron. Eng. 34, 261 (1997).
[CrossRef]

Nature (1)

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, Nature 412, 313 (2001).
[CrossRef]

Opt. Commun. (5)

M. J. Padgett and L. Allen, Opt. Commun. 121, 36 (1995).
[CrossRef]

A. T. O’Neil and M. J. Padgett, Opt. Commun. 193, 45 (2001).
[CrossRef]

G. A. Turnbull, D. A. Robertson, G. M. Smith, L. Allen, and M. J. Padgett, Opt. Commun. 127, 183 (1996).
[CrossRef]

M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, and J. P. Woerdman, Opt. Commun. 96, 123 (1993).
[CrossRef]

P. Coullet, L. Gil, and F. Rocca, Opt. Commun. 73, 403 (1989).
[CrossRef]

Opt. Express (2)

Opt. Lett. (1)

Opt. Quantum Electron. (1)

N. R. Heckenberg, R. Mc Duff, C. P. Smith, H. Rubinsztein-Dunlop, and M. J. Wegener, Opt. Quantum Electron. 24, S951 (1992).
[CrossRef]

Phys. Rev. A (2)

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, Phys. Rev. A 45, 8185 (1992).
[CrossRef]

G. Anzolin, F. Tamburini, A. Bianchini, and C. Barbieri, Phys. Rev. A 79, 033845 (2009).
[CrossRef]

Phys. Rev. Lett. (1)

M. Granata, C. Buy, R. Ward, and M. Barsuglia, Phys. Rev. Lett. 105, 231102 (2010).
[CrossRef]

Proc. R. Soc. A (1)

J. F. Nye and M. V. Berry, Proc. R. Soc. A 336, 165 (1974).
[CrossRef]

Science (1)

N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, and S. Ramachandran, Science 340, 1545 (2013).
[CrossRef]

Other (5)

M. Born and E. Wolf, Principles of Optics (Pergamon, 1986).

V. A. Soifer, Methods for Computer Design of Diffractive Optical Elements (Wiley, 2002).

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1996).

J. P. Torres and L. Torner, Twisted Photons (Wiley, 2011).

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, 1972).

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Figures (5)

Fig. 1.
Fig. 1.

SEM micrograph of PMMA SSP for the generation of LG51 beam.

Fig. 2.
Fig. 2.

Generation of LG31 beam. (a) Phase pattern of the corresponding SSP. (b) Simulation of the far-field. Brightness and colors refer, respectively, to intensity and phase of the field. (c) Comparison between a cross section of the experimental far-field intensity (blue dots), numerical computation of the far-field intensity (blue solid line) and theoretical profile of a LG31 beam (red solid line). (d) Experimental far-field intensity, normalized to the maximal value.

Fig. 3.
Fig. 3.

Efficiency γp of LGp generation for different SSPs encoding LG beams with radial number pSPP from 0 to 5, fixed azimuth number =1. Diagonal efficiency terms: γ01=0.93, γ11=0.91, γ21=0.88, γ31=0.86, γ41=0.84, γ51=0.81.

Fig. 4.
Fig. 4.

Experimental data of the first-ring radius RM, normalized to w0/2 (blue dots), and of the corresponding value 1/RM2 (red squares), as a function of the radial number p for fixed azimuth number =1. Comparison with the theoretical trends, given by interpolation of data points calculated on LG beams (solid lines). In particular, the fit with a linear trend 1/RM2=k·(p+1)+q gives k=1.18, q=2e2.

Fig. 5.
Fig. 5.

Experimental results. (a) Intensity of the pattern obtained from interference of a reference wave with a generated LG31 beam and (b) details of phase dislocations. (c) Detail of the interference pattern for an LG32 beam. Central dislocations confirm the presence of a phase singularity along the axis of the beam. Shifts of half-period between the interference fringes of consecutive rings confirm the π phase-shift because of the change of sign in the associated Laguerre polynomial.

Equations (8)

Equations on this page are rendered with MathJax. Learn more.

ψp(r,φ,z)=Cplw(2rw)||er2w2Lp||(2r2w2)eiφeikz+ikr22Rei(2p+||+1)ζ(z),
2π=2πλ(nSPP1)h,
Up(r,φ)=eiφ·sign[Lp||(2r2w02)].
χ(ρ,θ)=FT[c0u(i)Upl]=c002πdφ0+u(i)Upleirρcos(θφ)rdr,
χ(ρ,θ)=FT[c0u(i)eiφ]*FT[sign[Lp||(2r2w02)]],
FT[c0u(i)eiφ]=c0iπ3/22w12eiθeη2/2η[I12(η22)I+12(η22)],
FT[sign[Lp||(2r2w02)]]=k=1p2rk(1)kJ1(rkρ)ρδ(ρ)2π,
γp=ψp|χexpχexp|χexp.

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