Abstract

The construction of fractal generalized zone plates from a set of periodic diffractive optical elements with circular symmetry is proposed. This allows us, for instance, to increase the number of foci of a conventional fractal zone plate while keeping the self-similarity property within the axial irradiance. The focusing properties of these fractal diffractive optical elements for points not only along but also in the close vicinity of the optical axis are investigated. In both cases analytical expressions for the irradiance are derived. Numerical simulations of the energetic efficiency of fractal generalized zone plates under plane wave illumination are carried out. In addition, some effects on the axial irradiance caused by variations in the area of their transparent rings are shown.

© 2009 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. H. Takayasu, Fractals in Physical Science (Manchester University, 1990).
  2. G. Saavedra, W. D. Furlan, and J. A. Monsoriu, “Fractal zone plates,” Opt. Lett. 28, 971-973 (2003).
    [CrossRef] [PubMed]
  3. J. A. Monsoriu, G. Saavedra, and W. D. Furlan, “Fractal zone plates with variable lacunarity,” Opt. Express 12, 4227-4234 (2004).
    [CrossRef] [PubMed]
  4. G. S. Waldman, “Variations on the Fresnel zone plate,” J. Opt. Soc. Am. 56, 215-218 (1966).
    [CrossRef]
  5. M. Bottema, “Fresnel zone-plate diffraction patterns,” J. Opt. Soc. Am. 59, 1632-1638 (1969).
    [CrossRef]
  6. J. A. Davis, L. Ramirez, J. A. R. Martín-Romo, T. Alieva, and M. L. Calvo, “Focusing properties of fractal zone plates: experimental implementation with a liquid-crystal display,” Opt. Lett. 29, 1321-1323 (2004).
    [CrossRef] [PubMed]
  7. H.-T. Dai, X. Wang, and K.-S. Xu, “Focusing properties of fractal zone plates with variable lacunarity: experimental studies based on liquid crystal on silicon,” Chin. Phys. Lett. 22, 2851-2854 (2005).
    [CrossRef]
  8. 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]
  9. W. D. Furlan, G. Saavedra, and J. A. Monsoriu, “White-light imaging with fractal zone plates,” Opt. Lett. 32, 2109-2111 (2007).
    [CrossRef] [PubMed]
  10. O. Mendoza-Yero, G. Mínguez-Vega, J. Lancis, and V. Climent, “Focusing and spectral characteristics of periodic diffractive optical elements with circular symmetry under femtosecond pulsed illumination,” J. Opt. Soc. Am. A 24, 3600-3605 (2007).
    [CrossRef]
  11. A. Sakdinawat and Y. Liu, “Phase contrast soft x-ray microscopy using Zernike zone plates,” Opt. Express 16, 1559-1564 (2008).
    [CrossRef] [PubMed]
  12. S. Wang and X.-C. Zhang, “Tomographic imaging with a terahertz binary lens,” Appl. Phys. Lett. 82, 1821-1823 (2003).
    [CrossRef]
  13. O. Mendoza-Yero, G. Mínguez-Vega, J. Lancis, E. Tajahuerce, and V. Climent, “Spectral analysis of femtosecond pulse diffraction through binary diffractive optical elements: theory and experiment,” Opt. Express 16, 2541-2546 (2008).
    [CrossRef] [PubMed]
  14. K. Shi, S. Yin, and Z. Liu, “Wavelength division scanning for two-photon excitation fluorescence imaging,” J. Microsc. 223, 83-87 (2006).
    [CrossRef] [PubMed]
  15. R. Zheng, L. Jiang, and M. Feldman, “Properties of zone plates used for lithography,” J. Vac. Sci. Technol. B 24, 2844-2847 (2006).
    [CrossRef]
  16. S. Cavalieri, L. Fini, E. Sali, and R. Buffa, “Enhancement of harmonic generation by Fresnel-lensing effects,” Opt. Lett. 31, 1298-1300 (2006).
    [CrossRef] [PubMed]
  17. X. Ni, C. Wang, X. Liang, M. Alrubaiee, and R. R. Alfano, “Fresnel diffraction supercontinuum generation,” IEEE J. Sel. Top. Quantum Electron. 10, 1229-1231 (2004).
    [CrossRef]
  18. J. A. Monsoriu, W. D. Furlan, G. Saavedra, and F. Giménez, “Devil's lenses,” Opt. Express 15, 13858-13864 (2007).
    [CrossRef] [PubMed]
  19. W. Dong, N. Li-Gang, C. Qi-Dai, W. Rui, and S. Hong-Bo, “High efficiency multilevel phase-type fractal zone plates,” Opt. Lett. 33, 2913-2915 (2008).
    [CrossRef]
  20. 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]

2008 (3)

2007 (3)

2006 (5)

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]

K. Shi, S. Yin, and Z. Liu, “Wavelength division scanning for two-photon excitation fluorescence imaging,” J. Microsc. 223, 83-87 (2006).
[CrossRef] [PubMed]

R. Zheng, L. Jiang, and M. Feldman, “Properties of zone plates used for lithography,” J. Vac. Sci. Technol. B 24, 2844-2847 (2006).
[CrossRef]

S. Cavalieri, L. Fini, E. Sali, and R. Buffa, “Enhancement of harmonic generation by Fresnel-lensing effects,” Opt. Lett. 31, 1298-1300 (2006).
[CrossRef] [PubMed]

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]

2005 (1)

H.-T. Dai, X. Wang, and K.-S. Xu, “Focusing properties of fractal zone plates with variable lacunarity: experimental studies based on liquid crystal on silicon,” Chin. Phys. Lett. 22, 2851-2854 (2005).
[CrossRef]

2004 (3)

2003 (2)

S. Wang and X.-C. Zhang, “Tomographic imaging with a terahertz binary lens,” Appl. Phys. Lett. 82, 1821-1823 (2003).
[CrossRef]

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

1969 (1)

1966 (1)

Alfano, R. R.

X. Ni, C. Wang, X. Liang, M. Alrubaiee, and R. R. Alfano, “Fresnel diffraction supercontinuum generation,” IEEE J. Sel. Top. Quantum Electron. 10, 1229-1231 (2004).
[CrossRef]

Alieva, T.

Alrubaiee, M.

X. Ni, C. Wang, X. Liang, M. Alrubaiee, and R. R. Alfano, “Fresnel diffraction supercontinuum generation,” IEEE J. Sel. Top. Quantum Electron. 10, 1229-1231 (2004).
[CrossRef]

Bottema, M.

Buffa, R.

Burge, R. E.

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]

Calvo, M. L.

Cavalieri, S.

Climent, V.

Craven, J. M.

Dai, H.-T.

H.-T. Dai, X. Wang, and K.-S. Xu, “Focusing properties of fractal zone plates with variable lacunarity: experimental studies based on liquid crystal on silicon,” Chin. Phys. Lett. 22, 2851-2854 (2005).
[CrossRef]

Davis, J. A.

Dong, W.

Feldman, M.

R. Zheng, L. Jiang, and M. Feldman, “Properties of zone plates used for lithography,” J. Vac. Sci. Technol. B 24, 2844-2847 (2006).
[CrossRef]

Fini, L.

Furlan, W. D.

Giménez, F.

Hong-Bo, S.

Jiang, L.

R. Zheng, L. Jiang, and M. Feldman, “Properties of zone plates used for lithography,” J. Vac. Sci. Technol. B 24, 2844-2847 (2006).
[CrossRef]

Lancis, J.

Liang, X.

X. Ni, C. Wang, X. Liang, M. Alrubaiee, and R. R. Alfano, “Fresnel diffraction supercontinuum generation,” IEEE J. Sel. Top. Quantum Electron. 10, 1229-1231 (2004).
[CrossRef]

Li-Gang, N.

Lin, J.

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]

Liu, Y.

Liu, Z.

K. Shi, S. Yin, and Z. Liu, “Wavelength division scanning for two-photon excitation fluorescence imaging,” J. Microsc. 223, 83-87 (2006).
[CrossRef] [PubMed]

Martín-Romo, J. A. R.

Mendoza-Yero, O.

Mínguez-Vega, G.

Monsoriu, J. A.

Ni, X.

X. Ni, C. Wang, X. Liang, M. Alrubaiee, and R. R. Alfano, “Fresnel diffraction supercontinuum generation,” IEEE J. Sel. Top. Quantum Electron. 10, 1229-1231 (2004).
[CrossRef]

Qi-Dai, C.

Ramirez, L.

Rui, W.

Saavedra, G.

Sakdinawat, A.

Sali, E.

Shi, K.

K. Shi, S. Yin, and Z. Liu, “Wavelength division scanning for two-photon excitation fluorescence imaging,” J. Microsc. 223, 83-87 (2006).
[CrossRef] [PubMed]

Sigarlaki, S. P.

Tajahuerce, E.

Takayasu, H.

H. Takayasu, Fractals in Physical Science (Manchester University, 1990).

Tao, S. H.

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]

Waldman, G. S.

Wang, C.

X. Ni, C. Wang, X. Liang, M. Alrubaiee, and R. R. Alfano, “Fresnel diffraction supercontinuum generation,” IEEE J. Sel. Top. Quantum Electron. 10, 1229-1231 (2004).
[CrossRef]

Wang, S.

S. Wang and X.-C. Zhang, “Tomographic imaging with a terahertz binary lens,” Appl. Phys. Lett. 82, 1821-1823 (2003).
[CrossRef]

Wang, X.

H.-T. Dai, X. Wang, and K.-S. Xu, “Focusing properties of fractal zone plates with variable lacunarity: experimental studies based on liquid crystal on silicon,” Chin. Phys. Lett. 22, 2851-2854 (2005).
[CrossRef]

Xu, K.-S.

H.-T. Dai, X. Wang, and K.-S. Xu, “Focusing properties of fractal zone plates with variable lacunarity: experimental studies based on liquid crystal on silicon,” Chin. Phys. Lett. 22, 2851-2854 (2005).
[CrossRef]

Yin, S.

K. Shi, S. Yin, and Z. Liu, “Wavelength division scanning for two-photon excitation fluorescence imaging,” J. Microsc. 223, 83-87 (2006).
[CrossRef] [PubMed]

Yuan, X. C.

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]

Zhang, X.-C.

S. Wang and X.-C. Zhang, “Tomographic imaging with a terahertz binary lens,” Appl. Phys. Lett. 82, 1821-1823 (2003).
[CrossRef]

Zheng, R.

R. Zheng, L. Jiang, and M. Feldman, “Properties of zone plates used for lithography,” J. Vac. Sci. Technol. B 24, 2844-2847 (2006).
[CrossRef]

Appl. Opt. (1)

Appl. Phys. Lett. (2)

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]

S. Wang and X.-C. Zhang, “Tomographic imaging with a terahertz binary lens,” Appl. Phys. Lett. 82, 1821-1823 (2003).
[CrossRef]

Chin. Phys. Lett. (1)

H.-T. Dai, X. Wang, and K.-S. Xu, “Focusing properties of fractal zone plates with variable lacunarity: experimental studies based on liquid crystal on silicon,” Chin. Phys. Lett. 22, 2851-2854 (2005).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron. (1)

X. Ni, C. Wang, X. Liang, M. Alrubaiee, and R. R. Alfano, “Fresnel diffraction supercontinuum generation,” IEEE J. Sel. Top. Quantum Electron. 10, 1229-1231 (2004).
[CrossRef]

J. Microsc. (1)

K. Shi, S. Yin, and Z. Liu, “Wavelength division scanning for two-photon excitation fluorescence imaging,” J. Microsc. 223, 83-87 (2006).
[CrossRef] [PubMed]

J. Opt. Soc. Am. (2)

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

J. Vac. Sci. Technol. B (1)

R. Zheng, L. Jiang, and M. Feldman, “Properties of zone plates used for lithography,” J. Vac. Sci. Technol. B 24, 2844-2847 (2006).
[CrossRef]

Opt. Express (4)

Opt. Lett. (5)

Other (1)

H. Takayasu, Fractals in Physical Science (Manchester University, 1990).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (5)

Fig. 1
Fig. 1

Steps for construction of a FraGZP with N = 2 and S = 2 from GZPs with ε = 4 . (a) geometric bars, (b) rotation process, (c) pupil in r 2 , and (d) pupil in r.

Fig. 2
Fig. 2

Characteristic irradiance profiles of FraGZPs (top and middle rows) and GZPs (bottom row).

Fig. 3
Fig. 3

Normalized off-axis irradiance profiles of FraGZPs (top) and GZPs (bottom) given in the middle and bottom rows of Fig. 2.

Fig. 4
Fig. 4

Energy efficiency of FraGZPs (thin curves) and GZPs (thick curves) on the planes z = z 5 (for ε = 2 ), z = z 8 (for ε = 3 ), and z = z 10 (for ε = 4 ). In all cases S = 2 .

Fig. 5
Fig. 5

Normalized on-axis irradiance of FraGZPs as a function of the normalized variable u with (a) ε = 2 , (b) ε = 3 , (c) ε = 4 . In all cases N = 2 and S = 2 .

Equations (4)

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

I FraGZP ( u , N , S , ε ) = I GZP ( u , N , S , ε ) i = 1 S 1 sin 2 ( N ε π u ( ε N ( ε 1 ) ) i ) sin 2 ( ε π u ( ε N ( ε 1 ) ) i ) ,
I GZP ( u , N , S , ε ) = 4 sin 2 ( π u ( ε N ( ε 1 ) ) S ) sin 2 ( M ε π u ( ε N ( ε 1 ) ) S ) sin 2 ( ε π u ( ε N ( ε 1 ) ) S ) .
I ( u , R ) = { m = 1 M [ J 0 ( 4 π u r i m R a 2 ) cos ( 2 π u r i m 2 a 2 ) J 0 ( 4 π u r o m R a 2 ) cos ( 2 π u r o m 2 a 2 ) ] } 2 + { m = 1 M [ J 0 ( 4 π u r i m R a 2 ) sin ( 2 π u r i m 2 a 2 ) J 0 ( 4 π u r o m R a 2 ) sin ( 2 π u r o m 2 a 2 ) ] } 2 .
E FraGZP E FraZP = 2 N S [ ε N ( ε 1 ) ] S 2 N S ( 2 N 1 ) S > 0 .

Metrics