## Helico-conical optical beams: a product of helical and conical phase fronts

Optics Express, Vol. 13, Issue 5, pp. 1749-1760 (2005)

http://dx.doi.org/10.1364/OPEX.13.001749

Acrobat PDF (645 KB)

### Abstract

Helico-conical optical beams, different from higher-order Bessel beams, are generated with a parallel-aligned nematic liquid crystal spatial light modulator (SLM) by multiplying helical and conical phase functions leading to a nonseparable radial and azimuthal phase dependence. The intensity distributions of the focused beams are explored in two- and three-dimensions. In contrast to the ring shape formed by a focused optical vortex, a helico-conical beam produces a spiral intensity distribution at the focal plane. Simple scaling relationships are found between observed spiral geometry and initial phase distributions. Observations near the focal plane further reveal a cork-screw intensity distribution around the propagation axis. These light distributions, and variations upon them, may find use for optical trapping and manipulation of mesoscopic particles.

© 2005 Optical Society of America

## 1. Introduction

3. L. Allen, M.J. Padgett, and M. Babiker
, “The Orbital Angular Momentum of Light,” in *Progress in Optics*39,
E. Wolf, ed. (Elsevier, Amsterdam, 1999). [CrossRef]

4. K. T. Gahagan and G. A. Swartzlander Jr., “Simultaneous trapping of low-index and high-index microparticles observed with an optical-vortex trap,” J. Opt. Soc. Am. B **16**, 533–537 (1999). [CrossRef]

5. P. J. Rodrigo, V. R. Daria, and J. Glückstad, “Real-time interactive optical micromanipulation of a mixture of high- and low-index particles,” Opt. Express12, 1417–1425 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-7-1417. [CrossRef] [PubMed]

6. V. R. Daria, P. J. Rodrigo, and J. Glückstad, “Dynamic array of dark optical traps,” Appl. Phys. Lett. **84**, 323–325 (2004). [CrossRef]

7. N. B. Simpson, K. Dholakia, L. Allen, and M. J. Padgett, “Mechanical equivalence of spin and orbital angular momentum of light: an optical spanner,” Opt. Lett. **22**, 52–54 (1997). [CrossRef] [PubMed]

8. J. E. Curtis and D. G. Grier, “Structure of optical vortices,” Phys. Rev. Lett. **90**, 133901 (2003). [CrossRef] [PubMed]

9. K. Ladavac and D. G. Grier, “Microoptomechanical pumps assembled and driven by holographic optical vortex arrays,” Opt. Express12, 1144–1149 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-6-1144. [CrossRef] [PubMed]

10. P. A. Prentice, M. P. MacDonald, T. G. Frank, A. Cuschieri, G. C. Spalding, W. Sibbett, P.A. Campbell, and K. Dholakia, “Manipulation and filtration of low-index particles with holographic Laguerre-Gaussian optical trap arrays,” Opt. Express12, 593–600 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-4-593. [CrossRef] [PubMed]

11. M. J. Padgett and L. Allen, “The Poynting vector in Laguerre-Gaussian laser modes,” Opt. Commun. **121**, 36–40 (1995). [CrossRef]

12. J. A. Davis, E. Carcole, and D. M. Cottrell, Nondiffracting interference patterns generated with programmable spatial light modulators, Appl. Opt. **35**, 599–602 (1996). [CrossRef] [PubMed]

13. N. Chattrapiban, E. A. Rogers, D. Cofield, W. T. Hill III, and R. Roy, “Generation of nondiffracting Bessel beams by use of a spatial light modulator,” Opt. Lett. **28**, 2183–2185 (2003). [CrossRef] [PubMed]

*iψ*), where the phase function is

_{ℓ}*ℓ*, is an integer that determines the number of 2

*π*-phase shifts that occur across one revolution of the azimuthal angle,

*θ*. The sign of ℓ determines the handedness of the helix. While numerous methods are available to introduce a helical phase to an incident TEM

_{00}beam [12

12. J. A. Davis, E. Carcole, and D. M. Cottrell, Nondiffracting interference patterns generated with programmable spatial light modulators, Appl. Opt. **35**, 599–602 (1996). [CrossRef] [PubMed]

18. C. Rotschild, S. Zommer, S. Moed, O. Hershcovitz, and S. G. Lipson, “Adjustable spiral phase plate,” Appl. Opt. **43**, 2397–2399 (2004). [CrossRef] [PubMed]

8. J. E. Curtis and D. G. Grier, “Structure of optical vortices,” Phys. Rev. Lett. **90**, 133901 (2003). [CrossRef] [PubMed]

*ψ*, the optical beam may also have some radial phase dependence. For example, with the generation of higher-order Bessel beams the phase-carrying term can be written as exp[

_{ℓ}*iψ*(

*r,θ*)]

*r*being a normalization factor of the radial coordinate,

_{o}*r*[12

12. J. A. Davis, E. Carcole, and D. M. Cottrell, Nondiffracting interference patterns generated with programmable spatial light modulators, Appl. Opt. **35**, 599–602 (1996). [CrossRef] [PubMed]

19. K. Crabtree, J. A. Davis, and I. Moreno, “Optical processing with vortex-producing lenses,” Appl. Opt. **43**, 1360–1367 (2004). [CrossRef] [PubMed]

*r*and

*θ*:

*K*is a constant that takes a value of either 1 or 0. In contrast to Eq. (2), which can be seen as a sum of a helical phase and a conical phase (such as from an axicon), Eq. (3) appears as a product of such functions. Consequently, the complex exponential cannot be separated into radial and azimuthal terms. We shall refer to Eq.(3) as a helico-conical phase, not to be confused with the phase in Eq. (2) for higher-order Bessel beams.

*πℓ*, equivalent to a phase step of zero, for integral values of

*ℓ*. Interestingly, optical vortices with fractional values of

*ℓ, i.e*. discontinuous phase surfaces, have recently been described analytically [20

20. M. V. Berry, “Optical vortices evolving from helicoidal integer and fractional phase steps,” J. Opt. A: Pure Appl. Opt. **6**, 259–268 (2004). [CrossRef]

21. J. Leach, E. Yao, and M. J. Padgett, “Observation of the vortex structure of a non-integer vortex beam,” New J. Phys.6, 71 (2004), http://www.iop.org/EJ/abstract/1367-2630/6/1/071. [CrossRef]

22. S. H. Tao, W. M. Lee, and X. Yuan, “Experimental study of holographic generation of fractional Bessel beams” Appl. Opt. **43**, 122–126 (2003). [CrossRef]

*r*and the phase becomes continuous around an azimuthal circuit at

*r=r*.

_{o}## 2. Experiment setup

*f*configuration with the SLM and CCD at the front and back focal planes, respectively. Video and images are captured from the CCD by a personal computer through a PCI video capture card.

*π*in 230 discrete steps. The 480×480 array of 40-µm pixels are optically addressed to minimize dead-space on the SLM surface. Phase masks are drawn as grey-level images on a personal computer and directed via the VGA output to the SLM. Typical images used as phase masks are presented in Fig. 2(a) and Fig. 2(b) corresponding to Eq. (3) with

*K*=1 and

*K*=0, respectively.

*r*is chosen to coincide with the radius of CA. Note that the encoded phases are wrapped in 2

_{o}*π*. Figures 2(c) and 2(d) show the mixed helical and conical shapes of the corresponding unwrapped phase profiles.

## 3. Intensity distribution at the focal plane

*I*(

*ρ,ϕ*)=|

*u*(

*ρ,ϕ*)|

^{2}with

*circ*(

*r/r*) is the

_{o}*Circle*function [23] used to describe the circular aperture stop, and

*ψ*(

*r,θ*) is defined as in Eq. (3). For simplicity, we assume that the area of the expanded Gaussian beam enclosed by radius

*r*can be approximated by a plane wave. As emphasized earlier, because of the chosen form of

_{o}*ψ*(

*r,θ*), the double integral is not separable in

*r*and

*θ*. It is thus most convenient to evaluate Eq. (4) numerically with an FFT algorithm.

*K*=1. The strong correspondence between numerical and experimental results is emphasized by the composite image in Fig. 3(c). The intensity distribution at the focal plane appears quite distinctly as a spiral, specifically an arithmetic spiral of points with radial position,

*ρ*, proportional to azimuthal angle,

_{ϕ}*ϕ*. An arithmetic spiral,

*ρ*=0.13

_{ϕ}*ϕ*, is superimposed on the results in Fig. 3(c) to highlight the basic geometry of the intensity distribution.

20. M. V. Berry, “Optical vortices evolving from helicoidal integer and fractional phase steps,” J. Opt. A: Pure Appl. Opt. **6**, 259–268 (2004). [CrossRef]

22. S. H. Tao, W. M. Lee, and X. Yuan, “Experimental study of holographic generation of fractional Bessel beams” Appl. Opt. **43**, 122–126 (2003). [CrossRef]

20. M. V. Berry, “Optical vortices evolving from helicoidal integer and fractional phase steps,” J. Opt. A: Pure Appl. Opt. **6**, 259–268 (2004). [CrossRef]

21. J. Leach, E. Yao, and M. J. Padgett, “Observation of the vortex structure of a non-integer vortex beam,” New J. Phys.6, 71 (2004), http://www.iop.org/EJ/abstract/1367-2630/6/1/071. [CrossRef]

*m*. Accumulating the contributions from each segment around the azimuthal angle thus draws an arithmetic spiral on the Fourier plane. Moreover, following Eq. (5), we would expect this spiral to scale linearly with

_{θ}*ℓ/r*. We shall explore this point further later.

_{o}*K*=0 may also be decomposed into radial segments with slopes

*m*. We might then expect the intensity distribution to be the same as in the previous case. Figures 4(a) and 4(b) however, are clearly different from Figs. 3(a) and 3(b). The most obvious point is an intensity peak forming a compact head on the spiral that is displaced from the origin of the Fourier plane when

_{θ}*K*=0. Figure 4(c) also reveals that, close to the origin, the intensity distribution deviates from an arithmetic spiral drawn in red.

*K*=1. The complete phase function across a single radial segment as earlier considered is

*K*=0,

*ℓθ*is constant within each radial segment but varies between segments, it effectively introduces an azimuthally varying phase-offset in

*ψ*that is not present in

_{K=1}*ψ*

_{K=0}. We then interpret the results Figs. 4(a) and 4(b) as indicating that the coherent superposition of diffracted waves from each segment interfere destructively close to the origin when

*K*=0. When

*K*=1, such coherent superposition is compromised by relative phase shifts,

*ℓθ*, between radial segments, leading to a fully formed arithmetic spiral at the focal plane.

*π*for the wavelength used (632.8 nm), but peaks at 1.8

*π*. Such limitations can be mitigated by techniques that map the initial phase function to a projection optimal for the limited range of the device [24

24. I. Moreno, C. Iemmi, A. Marquez, J. Campos, and M. J. Yzuel, “Modulation light efficiency of diffractive lenses displayed in a restricted phase-mostly modulation display,” Appl. Opt. **43**, 6278–6284 (2004). [CrossRef] [PubMed]

21. J. Leach, E. Yao, and M. J. Padgett, “Observation of the vortex structure of a non-integer vortex beam,” New J. Phys.6, 71 (2004), http://www.iop.org/EJ/abstract/1367-2630/6/1/071. [CrossRef]

*i.e*. large

*ℓ/r*-values, are encoded.

_{o}## 4. Analysis by local spatial frequency

*ξ,ζ*). Strictly speaking, frequency components at the Fourier plane cannot be associated to particular spatial coordinates in the front focal plane. However, for an object field with slowly varying amplitude,

*a*(

*x, y*), and phase,

*ψ*̄(

*x, y*), functions, an approximate mapping can be provided by local spatial frequencies defined as

25. O. Bryngdahl, “Geometrical transformations in optics,” J. Opt. Soc. Am. **64**, 1092–1099 (1974). [CrossRef]

26. J. Cederquist and A. M. Tai, “Computer-generated holograms for geometric transformations,” Appl. Opt. **23**, 3099–3104 (1984). [CrossRef] [PubMed]

27. J. E. Curtis and D. G. Grier, “Modulated optical vortices,” Opt. Lett. **28**, 872–874 (2003). [CrossRef] [PubMed]

*∂ψ/∂θ*, the radius of a focused optical vortex is modulated into various Lissajous curves while still imparting orbital angular momentum to trapped particles. Local spatial frequency analysis possibly provides a basis for the observed pattern modulation.

*K*=1 we arrive at

*ξ′, ζ′*) as a spot diagram following Eq. (11). The density of points is representative of intensity observable at the focal plane. In the figure, we observe that some points have accumulated in a spiral. The local spatial frequencies forming the spiral appear to be associated with points on the object plane closest to the circular boundary. Taking the limit as

*r→r*, we find

_{o}*ϕ*=tan

^{-1}(

*v/u*)→

*θ*. The dense collection of points thus forms an arithmetic spiral on the focal plane that scales linearly with

*ℓr*, as suggested earlier by Eq. (5).

_{o}*K*=0 yields

*r*. This results in a very compact distribution of points into a spiral as shown in Fig. 6(b), and an observable dislocation of the spiral’s head from the origin of the focal plane as opposed to when

*K*=1. In agreement with results from the numerical simulation and experiment in Fig. 4(c), Eq. (12) does not yield a strictly arithmetic spiral. Further, the resulting pattern also scales linearly with

*ℓ/r*.

_{o}*r*may also be inferred from the

_{o}*similarity theorem*of Fourier transforms [23]. A normalization of radial coordinates in the input plane,

*r/r*, translates to an expansion of radial coordinates in the Fourier plane,

_{o}*r*. Scaling with

_{o}ρ*ℓ*is seen as analogous to the scaling behavior observed between optical vortices and their corresponding topological charge [8

8. J. E. Curtis and D. G. Grier, “Structure of optical vortices,” Phys. Rev. Lett. **90**, 133901 (2003). [CrossRef] [PubMed]

*ℓ/r*-scaling.

_{o}*ρ*is measured along

_{ϕ}*ϕ=π*on images generated with varying permutations of

*ℓ*(5, 10, 20, 40, 60, 80, and 100) and

*r*(3.0 mm, 4.5 mm, and 5.5 mm).

_{o}## 5. Three-dimensional intensity distribution near the focal plane

*u*(

*ρ,ϕ*) is decomposed into plane wave components by a Fourier transform, then multiplied by the quadratic phase factor associated with free-space propagation for a distance,

*z*. An inverse Fourier transform of the result yields the desired distribution at the defocused plane. Intensity values extracted from these field distributions for different

*z*are stacked to form a three-dimensional representation of the light distribution near the focal plane.

*z*<0 (approaching the focal plane) and z>0 (beyond the focal plane). An interesting feature for

*z*<0 is the apparent rotation of the intensity profile as the beam propagates towards the focal plane. The irregular-spiral intensity pattern also appears to reverse in handedness as it evolves in the region prior to the focal plane. Propagation for

*z*>0, meanwhile, is dominantly characterized by a dilation of the intensity distribution from the focal plane and remains nearly invariant in structure. These phenomena are seen more clearly with the pseudo-color representation of the numerical results in Fig. 9. It is perhaps worth noting that the propagation dynamics are reversed when the conjugate phase functions are considered, for example if

*ℓ*is chosen to be negative.

*K*=0. Rotation of the intensity profile, simultaneous with a switching in the direction of the tail of the spiral, is observed while approaching the focus. This, again, is followed by dilation of the intensity pattern beyond the focal plane. However, the particular intensity profile projected by the phase distribution when

*K*=0 leads to a more interesting dynamics in the region

*z*<0. Following the intensity profile carefully, we can observe that the head of the spiral follows a cork-screw path around the optical axis. Again, the use of a pseudo-color representation more prominently presents this behavior with the numerical results in Fig. 11. With sufficiently intense input illumination, one could imagine particles being guided along the path of this intensity “hot spot”.

28. J. Courtial, “Self-imaging beams and the Guoy effect,” Opt. Commun. **151**, 1–4 (1998). [CrossRef]

*i.e*. the rotation and the switching of handedness by the intensity patterns, with either the Rayleigh range (

*z*), or the Gouy phase shift (arctan(

_{R}*z/z*)). The latter in particular is a key parameter in the propagation of LG modes. This result would also suggest a large set of LG component modes forming the helico-conical beams.

_{R}## 6. Conclusion

*ℓ/r*, of the encoded phase profiles.

_{o}*K*=0, where the projected intensity distribution forms an intensity hot spot that follows a cork-screw path around the propagation axis.

## References and links

1. | M. V. Berry
, “Much ado about nothing: optical dislocation lines (phase singularities, zeros, vortices…),” in |

2. | M.S. Soskin and M.V. Vasnetsov
, “Singular Optics,” in |

3. | L. Allen, M.J. Padgett, and M. Babiker
, “The Orbital Angular Momentum of Light,” in |

4. | K. T. Gahagan and G. A. Swartzlander Jr., “Simultaneous trapping of low-index and high-index microparticles observed with an optical-vortex trap,” J. Opt. Soc. Am. B |

5. | P. J. Rodrigo, V. R. Daria, and J. Glückstad, “Real-time interactive optical micromanipulation of a mixture of high- and low-index particles,” Opt. Express12, 1417–1425 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-7-1417. [CrossRef] [PubMed] |

6. | V. R. Daria, P. J. Rodrigo, and J. Glückstad, “Dynamic array of dark optical traps,” Appl. Phys. Lett. |

7. | N. B. Simpson, K. Dholakia, L. Allen, and M. J. Padgett, “Mechanical equivalence of spin and orbital angular momentum of light: an optical spanner,” Opt. Lett. |

8. | J. E. Curtis and D. G. Grier, “Structure of optical vortices,” Phys. Rev. Lett. |

9. | K. Ladavac and D. G. Grier, “Microoptomechanical pumps assembled and driven by holographic optical vortex arrays,” Opt. Express12, 1144–1149 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-6-1144. [CrossRef] [PubMed] |

10. | P. A. Prentice, M. P. MacDonald, T. G. Frank, A. Cuschieri, G. C. Spalding, W. Sibbett, P.A. Campbell, and K. Dholakia, “Manipulation and filtration of low-index particles with holographic Laguerre-Gaussian optical trap arrays,” Opt. Express12, 593–600 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-4-593. [CrossRef] [PubMed] |

11. | M. J. Padgett and L. Allen, “The Poynting vector in Laguerre-Gaussian laser modes,” Opt. Commun. |

12. | J. A. Davis, E. Carcole, and D. M. Cottrell, Nondiffracting interference patterns generated with programmable spatial light modulators, Appl. Opt. |

13. | N. Chattrapiban, E. A. Rogers, D. Cofield, W. T. Hill III, and R. Roy, “Generation of nondiffracting Bessel beams by use of a spatial light modulator,” Opt. Lett. |

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. | M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, and J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. |

16. | G.-H. Kim, J.-H. Jeon, K.-H Ko, H.-J. Moon, J.-H. Lee, and J.-S. Chang, “Optical vortices produced with a nonspiral phase plate,” Appl. Opt. |

17. | G. Biener, A. Niv, V. Kleiner, and E. Hasman, “Formation of helical beams by use of Pancharatnam-Berry phase optical elements,” Opt. Lett. |

18. | C. Rotschild, S. Zommer, S. Moed, O. Hershcovitz, and S. G. Lipson, “Adjustable spiral phase plate,” Appl. Opt. |

19. | K. Crabtree, J. A. Davis, and I. Moreno, “Optical processing with vortex-producing lenses,” Appl. Opt. |

20. | M. V. Berry, “Optical vortices evolving from helicoidal integer and fractional phase steps,” J. Opt. A: Pure Appl. Opt. |

21. | J. Leach, E. Yao, and M. J. Padgett, “Observation of the vortex structure of a non-integer vortex beam,” New J. Phys.6, 71 (2004), http://www.iop.org/EJ/abstract/1367-2630/6/1/071. [CrossRef] |

22. | S. H. Tao, W. M. Lee, and X. Yuan, “Experimental study of holographic generation of fractional Bessel beams” Appl. Opt. |

23. | J. W. Goodman, |

24. | I. Moreno, C. Iemmi, A. Marquez, J. Campos, and M. J. Yzuel, “Modulation light efficiency of diffractive lenses displayed in a restricted phase-mostly modulation display,” Appl. Opt. |

25. | O. Bryngdahl, “Geometrical transformations in optics,” J. Opt. Soc. Am. |

26. | J. Cederquist and A. M. Tai, “Computer-generated holograms for geometric transformations,” Appl. Opt. |

27. | J. E. Curtis and D. G. Grier, “Modulated optical vortices,” Opt. Lett. |

28. | J. Courtial, “Self-imaging beams and the Guoy effect,” Opt. Commun. |

**OCIS Codes**

(050.1970) Diffraction and gratings : Diffractive optics

(070.0070) Fourier optics and signal processing : Fourier optics and signal processing

(140.7010) Lasers and laser optics : Laser trapping

(230.6120) Optical devices : Spatial light modulators

**ToC Category:**

Research Papers

**History**

Original Manuscript: January 18, 2005

Revised Manuscript: February 6, 2005

Published: March 7, 2005

**Citation**

Carlo Alonzo, Peter John Rodrigo, and Jesper Glückstad, "Helico-conical optical beams: a product of helical and conical phase fronts," Opt. Express **13**, 1749-1760 (2005)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-13-5-1749

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### References

- M. V. Berry, �??Much ado about nothing: optical dislocation lines (phase singularities, zeros, vortices�?� ),�?? in Singular Optics, M.S. Soskin, ed., Proc. SPIE 3487, 1-5 (1998).
- M.S. Soskin and M.V. Vasnetsov, �??Singular Optics,�?? in Progress in Optics 42, E. Wolf, ed. (Elsevier, Amsterdam, 2001).
- L. Allen, M.J. Padgett and M. Babiker, �??The Orbital Angular Momentum of Light,�?? in Progress in Optics 39, E. Wolf, ed. (Elsevier, Amsterdam, 1999). [CrossRef]
- K. T. Gahagan and G. A. Swartzlander Jr., �??Simultaneous trapping of low-index and high-index microparticles observed with an optical-vortex trap,�?? J. Opt. Soc. Am. B 16, 533-537 (1999). [CrossRef]
- P. J. Rodrigo, V. R. Daria and J. Glückstad, �??Real-time interactive optical micromanipulation of a mixture of high- and low-index particles,�?? Opt. Express 12, 1417-1425 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-7-1417. [CrossRef] [PubMed]
- V. R. Daria, P. J. Rodrigo and J. Glückstad, �??Dynamic array of dark optical traps,�?? Appl. Phys. Lett. 84, 323-325 (2004). [CrossRef]
- N. B. Simpson, K. Dholakia, L.Allen and M. J. Padgett, �??Mechanical equivalence of spin and orbital angular momentum of light: an optical spanner,�?? Opt. Lett. 22, 52-54 (1997). [CrossRef] [PubMed]
- J. E. Curtis and D. G. Grier, �??Structure of optical vortices,�?? Phys. Rev. Lett. 90, 133901 (2003). [CrossRef] [PubMed]
- K. Ladavac and D. G. Grier, �??Microoptomechanical pumps assembled and driven by holographic optical vortex arrays,�?? Opt. Express 12, 1144-1149 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-6-1144. [CrossRef] [PubMed]
- P. A. Prentice, M. P. MacDonald, T. G. Frank , A. Cuschieri, G. C. Spalding, W. Sibbett, P.A. Campbell and K. Dholakia, �??Manipulation and filtration of low-index particles with holographic Laguerre-Gaussian optical trap arrays,�?? Opt. Express 12, 593-600 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-4-593. [CrossRef] [PubMed]
- M. J. Padgett and L. Allen, �??The Poynting vector in Laguerre-Gaussian laser modes,�?? Opt. Commun. 121, 36-40 (1995). [CrossRef]
- J. A. Davis, E. Carcole and D. M. Cottrell, "Nondiffracting interference patterns generated with programmable spatial light modulators," Appl. Opt. 35, 599-602 (1996). [CrossRef] [PubMed]
- N. Chattrapiban, E. A. Rogers, D.Cofield, W. T. Hill III and R. Roy, �??Generation of nondiffracting Bessel beams by use of a spatial light modulator,�?? Opt. Lett. 28, 2183-2185 (2003). [CrossRef] [PubMed]
- 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]
- M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen and J. P. Woerdman, �??Astigmatic laser mode converters and transfer of orbital angular momentum,�?? Opt. Commun. 96, 123-132 (1993). [CrossRef]
- G.-H. Kim, J.-H. Jeon, K.-H Ko, H.-J. Moon, J.-H. Lee and J.-S. Chang, �??Optical vortices produced with a nonspiral phase plate,�?? Appl. Opt. 36, 8614-8621 (1997). [CrossRef]
- G. Biener, A. Niv, V. Kleiner and E. Hasman, �??Formation of helical beams by use of Pancharatnam-Berry phase optical elements,�?? Opt. Lett. 27, 1875-1877 (2002). [CrossRef]
- C. Rotschild, S. Zommer, S. Moed, O. Hershcovitz and S. G. Lipson, �??Adjustable spiral phase plate,�?? Appl. Opt. 43, 2397-2399 (2004). [CrossRef] [PubMed]
- K. Crabtree, J. A. Davis and I. Moreno, �??Optical processing with vortex-producing lenses,�?? Appl. Opt. 43, 1360-1367 (2004). [CrossRef] [PubMed]
- M. V. Berry, �??Optical vortices evolving from helicoidal integer and fractional phase steps,�?? J. Opt. A: Pure Appl. Opt. 6, 259�??268 (2004). [CrossRef]
- J. Leach, E. Yao and M. J. Padgett, �??Observation of the vortex structure of a non-integer vortex beam,�?? New J. Phys. 6, 71 (2004), http://www.iop.org/EJ/abstract/1367-2630/6/1/071. [CrossRef]
- S. H. Tao, W. M. Lee and X. Yuan, "Experimental study of holographic generation of fractional Bessel beams" Appl. Opt. 43, 122-126 (2003). [CrossRef]
- J. W. Goodman, Introduction to Fourier Optics, Second Edition (McGraw-Hill, New York, 1996).
- I. Moreno, C. Iemmi, A. Marquez, J. Campos and M. J. Yzuel, �??Modulation light efficiency of diffractive lenses displayed in a restricted phase-mostly modulation display,�?? Appl. Opt. 43, 6278-6284 (2004). [CrossRef] [PubMed]
- O. Bryngdahl, �??Geometrical transformations in optics,�?? J. Opt. Soc. Am. 64, 1092-1099 (1974). [CrossRef]
- J. Cederquist and A. M. Tai, �??Computer-generated holograms for geometric transformations,�?? Appl. Opt. 23, 3099-3104 (1984). [CrossRef] [PubMed]
- J. E. Curtis and D. G. Grier, "Modulated optical vortices,�?? Opt. Lett. 28, 872-874 (2003). [CrossRef] [PubMed]
- J. Courtial, �??Self-imaging beams and the Guoy effect,�?? Opt. Commun. 151, 1-4 (1998). [CrossRef]

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