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Optics Express

Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 21, Iss. 8 — Apr. 22, 2013
  • pp: 10234–10239
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Twin axial vortices generated by Fibonacci lenses

Arnau Calatayud, Vicente Ferrando, Laura Remón, Walter D. Furlan, and Juan A. Monsoriu  »View Author Affiliations


Optics Express, Vol. 21, Issue 8, pp. 10234-10239 (2013)
http://dx.doi.org/10.1364/OE.21.010234


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Abstract

Optical vortex beams, generated by Diffractive Optical Elements (DOEs), are capable of creating optical traps and other multi-functional micromanipulators for very specific tasks in the microscopic scale. Using the Fibonacci sequence, we have discovered a new family of DOEs that inherently behave as bifocal vortex lenses, and where the ratio of the two focal distances approaches the golden mean. The disctintive optical properties of these Fibonacci vortex lenses are experimentally demonstrated. We believe that the versatility and potential scalability of these lenses may allow for new applications in micro and nanophotonics.

© 2013 OSA

1. Introduction

In photonics technology, Diffrative Optical Elements (DOEs) have found a large number of new aplications in many different areas, covering the whole electromagnetic spectrum from X-ray Microscopy [1

1. A. Sakdinawat and Y. Liu, “Soft-x-ray microscopy using spiral zone plates,” Opt. Lett. 32, 2635–2637 (2007) [CrossRef] [PubMed] .

], to THz Imaging [2

2. A. Siemion, A. Siemion, M. Makowski, J. Suszek, J. Bomba, A. Czerwinski, F. Garet, J.-L. Coutaz, and M. Sypek, “Diffractive paper lens for terahertz optics,” Opt. Lett. 37, 4320–4322 (2012) [CrossRef] [PubMed] .

]. Difractive lenses such as conventional Fresnel zone plates, are essential in many focusing and image forming systems but they have inherent limitations. Fractal zone plates are a new type of multifocal diffractive lenses that have been proposed to overcome some of these limitations, mainly under polychromatic illumination [3

3. G. Saavedra, W. D. Furlan, and J. A. Monsoriu, “Fractal zone plates,” Opt. Lett. 28, 971–973 (2003) [CrossRef] [PubMed] .

, 4

4. J. A. Davis, S. P. Sigarlaki, J. M. Craven, and M. L. Calvo, “Fourier series analysis of fractal lenses: theory and experiments with a liquid-crystal display,” Appl. Opt. 45, 1187–1192 (2006) [CrossRef] [PubMed] .

]. In fact, it was shown that these lenses, generated with the fractal Cantor set have an improved behavior, especially under wide band illumination [5

5. W. D. Furlan, G. Saavedra, and J. A. Monsoriu, “White-light imaging with fractal zone plates,” Opt. Lett. 32, 2109–2111 (2007) [CrossRef] [PubMed] .

].

DOEs have been also designed to generate optical vortices. Optical vortices are high value optical traps because in addition to trap microparticles they are capable to set these particles into rotation due to its inherent orbital angular moment [6

6. F. S. Roux, “Distribution of angular momentum and vortex morphology in optical beams,” Opt. Commun. 242, 45–55 (2004) [CrossRef] .

, 7

7. G. Gbur and T. D. Visser, “Phase singularities and coherence vortices in linear optical systems,” Opt. Commun. 259, 428–435 (2006) [CrossRef] .

]. These special optical beams have been used, for exemple, as actuators and testers in micromechanical systems [8

8. A. Bishop, T. Nieminen, N. Heckenberg, and H. Rubinsztein-Dunlop, “Optical application and measurement of torque on microparticles of isotropic nonabsorbing material,” Phys. Rev. A 68, 033802 (2003) [CrossRef] .

]. Arrays of optical vortices have shown the ability to assemble and drive mesoscopic optical pumps in microfluidic systems [9

9. K. Ladavac and D. G. Grier, “Microoptomechanical pumps assembled and driven by holographic optical vortex arrays,” Opt. Express 12, 1144–1149 (2004) [CrossRef] [PubMed] .

].

Among the several methods that have been proposed for optical vortices generation, spiral phase plates [10

10. W. M. Lee, X.-C. Yuan, and W. C. Cheong, “Optical vortex beam shaping by use of highly efficient irregular spiral phase plates for optical micromanipulation,” Opt. Lett. 29, 1796–1798 (2004) [CrossRef] [PubMed] .

] stands out, mainly because they provide high energy efficiency. Spiral phase plates have been recently combined with Fractal Zone Plates to produce a sequence of focused optical vortices along the propagation direction [11

11. S. H. Tao, X.-C. Yuan, J. Lin, and R. E. Burge, “Sequence of focused optical vortices generated by a spiral fractal zone plate,” Appl. Phys. Lett. 89, 031105 (2006) [CrossRef] .

, 12

12. W. D. Furlan, F. Giménez, A. Calatayud, and J. A. Monsoriu, “Devils vortex-lenses,” Opt. Express 17, 21891–21896 (2009) [CrossRef] [PubMed] .

].

2. Fibonacci vortex lenses design

A FVL is defined as a pure phase diffractive element whose phase distribution is given by ΦFVL(ζ, θ0) = mod2πa(θ0) + Φj(ζ)]. It combines the azimuthal phase variation that characerizes a vortex lens, i.e. Φa = 0, where m is a non zero integer called the topological charge [18

18. H. T. Dai, Y. J. Liu, and X. W. Sun, “The focusing property of the spiral Fibonacci zone plate,” in Optical Components and Materials IX, S. Jiang, M. J. F. Digonnet, and J. C. Dries, eds., Proc. SPIE8257, 82570T1 (2012) [CrossRef] .

] and θ0 is the azimuthal angle about the optical axis at the pupil plane, with the radial phase distribution that is generated through the Fibonacci sequence in the following way:

Starting with two elements (seeds) F0 = 0 and F1 = 1, the Fibonacci numbers, Fj = {0, 1, 1, 2, 3, 5, 8, 13, 21,...}, are obtained by the sequential application of the following rule: Fj+1 = Fj + Fj−1, (j = 0, 1, 2,...). The golden mean, or golden ratio, is defined as the limit of the ratio of two consecutive Fibonacci numbers:
φ=limjFj/Fj1=(1+5)/2.
(1)

Fig. 1 Bottom: Phase distributions of FVLs based on the Fibonacci sequence S8, with different topological charges. Top: The equivalent periodic lenses with the same number of zones.

3. Focusing properties

To study the focusing properties of FVLs we have computed the irradiance provided by the transmittance of this lens, t(ζ, θ0) = q(ζ)exp[imθ0], when it is illuminated by a plane wave of wavelength λ. Within the Fresnel approximation the irradiance function is given by:
I(u,v,θ)=u2|0102πt(ζ,θ0)exp(i2πuζ)exp[i4πuvζ1/2cos(θθ0)]dζdθ0|2,
(2)
where u = a2/2λz is the dimensionless reduced axial coordinate [3

3. G. Saavedra, W. D. Furlan, and J. A. Monsoriu, “Fractal zone plates,” Opt. Lett. 28, 971–973 (2003) [CrossRef] [PubMed] .

] and v = r/a is the normalized transverse coordinate. Note that the reduced axial coordinate contains the full dependende of the irradiance with λ. Replacing t(ζ, θ0) and taking into account that
02πexp(imθ0)exp[i4πuvζ1/2cos(θθ0)]dθ0=2πexp[im(θ+π2)]Jm(4πuvζ1/2),
(3)
Eq. (2) is reduced to
I(u,v)=4π2u2|01q(ζ)exp(i2πuζ)Jm(4πuvζ1/2)dζ|2,
(4)
being Jm() the Bessel function of the first kind of order m.

By using Eq. (4) we have computed the irradiances provided by the lenses shown in Fig. 1. The integrals were numerically evaluated applying the Simpson’s rule using a step length 1/2000. The result is shown in Fig. 2. It can be seen that FVLs produce a twin foci whose positions are given by the Fibonacci numbers. In this case, for S8 FVLs, the first focus is located at u1 = 13 = Fj−1 = 1/p1 and the other one at u2 = 21 = Fj = 1/p2. Thus, the ratio of the focal distances satisfies u2/u1φ. Note also that, for non-null values of the topological charge, each focus is a vortex, thus, in general, a pair of doughnut shaped foci is generated by a FVL. Comparing the diffraction patterns provided by FVLs with different topological charges it can be verified that the diameter of the doughnuts increases with the topological charge as those produced by conventional vortex lenses [19

19. J. Swartzlander, “Peering into darkness with a vortex spatial filter,” Opt. Lett. 26, 497–499 (2001) [CrossRef] .

, 20

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

].

Fig. 2 Evolution of the transverse irradiance for S8 based FVLs with different topological charges and their periodic equivalent lenses.

4. Experimental results

For the experimental study of the properties of FVLs, we implemented the experimental setup shown in Fig 3. The proposed lenses were recorded on a Liquid Crystal in a Silicon SLM (Holoeye PLUTO, 8-bit gray-level, pixel size 8μm and resolution equal to 1920 × 1080 pixels), calibrated for a 2π phase shift at λ = 633nm operating in phase-only modulation mode. The procedure to compensate the wavefront distorsions caused by the lack of flatness of the SLM and the other optical components was detailed elsewhere [21

21. A. Calatayud, W. D. Furlan, and J. A. Monsoriu, “Experimental generation and characterization of devils vortex-lenses,” Appl. Phys. B 106, 915–919 (2012) [CrossRef] .

]. In addition to the diffractive lenses, a linear phase carrier was modulated on the SLM to avoid the noise originated by the specular reflection (zero order diffraction) and the pixelated structure of the SLM (higher diffraction orders). This linear phase is compensated by tilting the SLM and a pin-hole (PH) is used to filter all diffraction orders except the first one. Then at the L3 lens focal plane (exit pupil) a rescaled image of the desired lens pupil is achieved. A collimated beam (He-Ne Laser λ = 633nm) impinges onto the SLM and the diffracted field is captured and registered with a microscope (10x Zeiss Plan-Apochromat objective) attached to a CCD camera (EO-1312M 1/2” CCD Monochrome USB Camera, 8-bit gray-level, pixel pitch of 4.65μm and 1280 × 1024 pixels). The microscope and the CCD are mounted on a translation stage (Thorlabs LTS 300, Range: 300mm and 5μm precision) along the optical axis.

Fig. 3 Experimental setup used for studying the focusing properties of FVLs.

The experimental and computed irradiances produced by a S8 FVL with topological charge m = 6 along the optical axis (using the dimensionless reduced axial coordinate) are shown in Fig. 4. As predicted by the theoretical analysis, the axial localization of the focal rings depends on the Fibonacci numbers Fj and Fj−1, and such focal distances satisfy the following relationship: f1/f2 = Fj/Fj−1φ.

Fig. 4 Experimental and computed transverse irradiance evolution along the optical axis provided by the S8 FVL with m = 6 and a = 1.1mm.

Figure 5 shows the experimental and the numerically simulated transverse irradiance at the focal planes for the same FVL. Interestingly, the diameter of the focal rings, Δ, which also depends on the topological charge of the FVL, satisfies a similar rule i.e. Δ12φ (see Fig. 5). Thus, FVLs are capable of generating twin axial vortices with different, but perfectly established, diameter of the central core.

Fig. 5 Experimental and numerical transverse irradiance at the focal planes provided by the S8 based on FVL with m = 6 and a = 1.1mm. The intensity profiles along the white, dotted lines are plotted (bottom) together with the numerical results for comparison.

5. Conclusions

Sumarizing, a new type of bifocal vortex lenses has been introduced, whose design is based on the Fiboonacci sequence. It was found that a FVL produces a twin optical vortices along the axial coordinate. The positions of both foci depend on the two incommensurable periods of the Fibonacci sequence in which the FVL is based on. The radii of these twin vortices increase with the topological charge of the vortex lens, but always their ratio approaches the golden mean. The 3D distribution of the diffracted field provided by FVLs has been tested experimentally using a SLM. An excellent agreement between the experimental results and the theoretical predictions has been demonstrated.

Acknowledgments

We acknowledge the financial support from Ministerio de Economía y Competitividad (grant FIS2011-23175), Generalitat Valenciana (grant PROMETEO2009-077), and Universitat Politècnica de València ( SP20120569), Spain. L.R. acknowledges a fellowship of “Fundación CajaMurcia”, Spain.

References and links

1.

A. Sakdinawat and Y. Liu, “Soft-x-ray microscopy using spiral zone plates,” Opt. Lett. 32, 2635–2637 (2007) [CrossRef] [PubMed] .

2.

A. Siemion, A. Siemion, M. Makowski, J. Suszek, J. Bomba, A. Czerwinski, F. Garet, J.-L. Coutaz, and M. Sypek, “Diffractive paper lens for terahertz optics,” Opt. Lett. 37, 4320–4322 (2012) [CrossRef] [PubMed] .

3.

G. Saavedra, W. D. Furlan, and J. A. Monsoriu, “Fractal zone plates,” Opt. Lett. 28, 971–973 (2003) [CrossRef] [PubMed] .

4.

J. A. Davis, S. P. Sigarlaki, J. M. Craven, and M. L. Calvo, “Fourier series analysis of fractal lenses: theory and experiments with a liquid-crystal display,” Appl. Opt. 45, 1187–1192 (2006) [CrossRef] [PubMed] .

5.

W. D. Furlan, G. Saavedra, and J. A. Monsoriu, “White-light imaging with fractal zone plates,” Opt. Lett. 32, 2109–2111 (2007) [CrossRef] [PubMed] .

6.

F. S. Roux, “Distribution of angular momentum and vortex morphology in optical beams,” Opt. Commun. 242, 45–55 (2004) [CrossRef] .

7.

G. Gbur and T. D. Visser, “Phase singularities and coherence vortices in linear optical systems,” Opt. Commun. 259, 428–435 (2006) [CrossRef] .

8.

A. Bishop, T. Nieminen, N. Heckenberg, and H. Rubinsztein-Dunlop, “Optical application and measurement of torque on microparticles of isotropic nonabsorbing material,” Phys. Rev. A 68, 033802 (2003) [CrossRef] .

9.

K. Ladavac and D. G. Grier, “Microoptomechanical pumps assembled and driven by holographic optical vortex arrays,” Opt. Express 12, 1144–1149 (2004) [CrossRef] [PubMed] .

10.

W. M. Lee, X.-C. Yuan, and W. C. Cheong, “Optical vortex beam shaping by use of highly efficient irregular spiral phase plates for optical micromanipulation,” Opt. Lett. 29, 1796–1798 (2004) [CrossRef] [PubMed] .

11.

S. H. Tao, X.-C. Yuan, J. Lin, and R. E. Burge, “Sequence of focused optical vortices generated by a spiral fractal zone plate,” Appl. Phys. Lett. 89, 031105 (2006) [CrossRef] .

12.

W. D. Furlan, F. Giménez, A. Calatayud, and J. A. Monsoriu, “Devils vortex-lenses,” Opt. Express 17, 21891–21896 (2009) [CrossRef] [PubMed] .

13.

J. A. Monsoriu, A. Calatayud, L. Remón, W. D. Furlan, G. Saavedra, and P. Andrés, “Zone plates generated with the Fibonacci sequence,” in Proceedings of EOS Topical Meeting on Diffractive Optics , pp. 151–152 (2010).

14.

J. A. Monsoriu, A. Calatayud, L. Remón, W. D. Furlan, G. Saavedra, and P. Andrés, “Bifocal Fibonacci diffractive lenses,” IEEE Photon. J. (to be published), DOI: [CrossRef] .

15.

E. Maciá, “Exploiting aperiodic designs in nanophotonic devices,” Rep. Prog. Phys. 75, 1–42 (2012) [CrossRef] .

16.

Y. Sah and G. Ranganath, “Optical diffraction in some Fibonacci structures,” Opt. Commun. 114, 18–24 (1995) [CrossRef] .

17.

N. Gao, Y. Zhang, and C. Xie, “Circular Fibonacci gratings,” Appl. Opt. 50, G142–G148 (2011) [PubMed] .

18.

H. T. Dai, Y. J. Liu, and X. W. Sun, “The focusing property of the spiral Fibonacci zone plate,” in Optical Components and Materials IX, S. Jiang, M. J. F. Digonnet, and J. C. Dries, eds., Proc. SPIE8257, 82570T1 (2012) [CrossRef] .

19.

J. Swartzlander, “Peering into darkness with a vortex spatial filter,” Opt. Lett. 26, 497–499 (2001) [CrossRef] .

20.

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

21.

A. Calatayud, W. D. Furlan, and J. A. Monsoriu, “Experimental generation and characterization of devils vortex-lenses,” Appl. Phys. B 106, 915–919 (2012) [CrossRef] .

OCIS Codes
(050.1940) Diffraction and gratings : Diffraction
(050.1970) Diffraction and gratings : Diffractive optics
(050.4865) Diffraction and gratings : Optical vortices

ToC Category:
Diffraction and Gratings

History
Original Manuscript: February 28, 2013
Revised Manuscript: April 5, 2013
Manuscript Accepted: April 12, 2013
Published: April 18, 2013

Virtual Issues
Vol. 8, Iss. 5 Virtual Journal for Biomedical Optics

Citation
Arnau Calatayud, Vicente Ferrando, Laura Remón, Walter D. Furlan, and Juan A. Monsoriu, "Twin axial vortices generated by Fibonacci lenses," Opt. Express 21, 10234-10239 (2013)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-8-10234


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References

  1. A. Sakdinawat and Y. Liu, “Soft-x-ray microscopy using spiral zone plates,” Opt. Lett.32, 2635–2637 (2007). [CrossRef] [PubMed]
  2. A. Siemion, A. Siemion, M. Makowski, J. Suszek, J. Bomba, A. Czerwinski, F. Garet, J.-L. Coutaz, and M. Sypek, “Diffractive paper lens for terahertz optics,” Opt. Lett.37, 4320–4322 (2012). [CrossRef] [PubMed]
  3. G. Saavedra, W. D. Furlan, and J. A. Monsoriu, “Fractal zone plates,” Opt. Lett.28, 971–973 (2003). [CrossRef] [PubMed]
  4. J. A. Davis, S. P. Sigarlaki, J. M. Craven, and M. L. Calvo, “Fourier series analysis of fractal lenses: theory and experiments with a liquid-crystal display,” Appl. Opt.45, 1187–1192 (2006). [CrossRef] [PubMed]
  5. W. D. Furlan, G. Saavedra, and J. A. Monsoriu, “White-light imaging with fractal zone plates,” Opt. Lett.32, 2109–2111 (2007). [CrossRef] [PubMed]
  6. F. S. Roux, “Distribution of angular momentum and vortex morphology in optical beams,” Opt. Commun.242, 45–55 (2004). [CrossRef]
  7. G. Gbur and T. D. Visser, “Phase singularities and coherence vortices in linear optical systems,” Opt. Commun.259, 428–435 (2006). [CrossRef]
  8. A. Bishop, T. Nieminen, N. Heckenberg, and H. Rubinsztein-Dunlop, “Optical application and measurement of torque on microparticles of isotropic nonabsorbing material,” Phys. Rev. A68, 033802 (2003). [CrossRef]
  9. K. Ladavac and D. G. Grier, “Microoptomechanical pumps assembled and driven by holographic optical vortex arrays,” Opt. Express12, 1144–1149 (2004). [CrossRef] [PubMed]
  10. W. M. Lee, X.-C. Yuan, and W. C. Cheong, “Optical vortex beam shaping by use of highly efficient irregular spiral phase plates for optical micromanipulation,” Opt. Lett.29, 1796–1798 (2004). [CrossRef] [PubMed]
  11. S. H. Tao, X.-C. Yuan, J. Lin, and R. E. Burge, “Sequence of focused optical vortices generated by a spiral fractal zone plate,” Appl. Phys. Lett.89, 031105 (2006). [CrossRef]
  12. W. D. Furlan, F. Giménez, A. Calatayud, and J. A. Monsoriu, “Devils vortex-lenses,” Opt. Express17, 21891–21896 (2009). [CrossRef] [PubMed]
  13. J. A. Monsoriu, A. Calatayud, L. Remón, W. D. Furlan, G. Saavedra, and P. Andrés, “Zone plates generated with the Fibonacci sequence,” in Proceedings of EOS Topical Meeting on Diffractive Optics, pp. 151–152 (2010).
  14. J. A. Monsoriu, A. Calatayud, L. Remón, W. D. Furlan, G. Saavedra, and P. Andrés, “Bifocal Fibonacci diffractive lenses,” IEEE Photon. J. (to be published), DOI: . [CrossRef]
  15. E. Maciá, “Exploiting aperiodic designs in nanophotonic devices,” Rep. Prog. Phys.75, 1–42 (2012). [CrossRef]
  16. Y. Sah and G. Ranganath, “Optical diffraction in some Fibonacci structures,” Opt. Commun.114, 18–24 (1995). [CrossRef]
  17. N. Gao, Y. Zhang, and C. Xie, “Circular Fibonacci gratings,” Appl. Opt.50, G142–G148 (2011). [PubMed]
  18. H. T. Dai, Y. J. Liu, and X. W. Sun, “The focusing property of the spiral Fibonacci zone plate,” in Optical Components and Materials IX, S. Jiang, M. J. F. Digonnet, and J. C. Dries, eds., Proc. SPIE8257, 82570T1 (2012). [CrossRef]
  19. J. Swartzlander, “Peering into darkness with a vortex spatial filter,” Opt. Lett.26, 497–499 (2001). [CrossRef]
  20. J. E. Curtis and D. G. Grier, “Structure of optical vortices,” Phys. Rev. Lett.90, 133901 (2003). [CrossRef] [PubMed]
  21. A. Calatayud, W. D. Furlan, and J. A. Monsoriu, “Experimental generation and characterization of devils vortex-lenses,” Appl. Phys. B106, 915–919 (2012). [CrossRef]

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