## Natural quasy-periodic binary structure with focusing property in near field diffraction pattern

Optics Express, Vol. 18, Issue 12, pp. 12526-12536 (2010)

http://dx.doi.org/10.1364/OE.18.012526

Acrobat PDF (1430 KB)

### Abstract

A naturally-inspired phase-only diffractive optical element with a circular symmetry given by a quasi-periodic structure of the phyllotaxis type is presented in this paper. It is generated starting with the characteristic parametric equations which are optimal for the golden angle interval. For some ideal geometrical parameters, the diffracted intensity distribution in near-field has a central closed ring with almost zero intensity inside. Its radius and intensity values depend on the geometry or non-binary phase distribution superposed onto the phyllotaxis geometry. Along propagation axis, the transverse diffraction patterns from the binary-phase diffractive structure exhibit a self-focusing behavior and a rotational motion.

© 2010 OSA

## 1. Introduction

1. A. J. Caley, M. J. Thomson, J. Liu, A. J. Waddie, and M. R. Taghizadeh, “Diffractive optical elements for high gain lasers with arbitrary output beam profiles,” Opt. Express **15**(17), 10699–10704 (2007), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-17-10699. [CrossRef] [PubMed]

2. G. S. Khan, K. Mantel, I. Harder, N. Lindlein, and J. Schwider, “Design considerations for the absolute testing approach of aspherics using combined diffractive optical elements,” Appl. Opt. **46**(28), 7040–7048 (2007), http://www.opticsinfobase.org/abstract.cfm?URI=ao-46-28-7040. [CrossRef] [PubMed]

3. H. Angelskår, I.-R. Johansen, M. Lacolle, H. Sagberg, and A. S. Sudbø, “Spectral uniformity of two- and four-level diffractive optical elements for spectroscopy,” Opt. Express **17**(12), 10206–10222 (2009), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-12-10206. [CrossRef] [PubMed]

4. G. Mínguez-Vega, O. Mendoza-Yero, J. Lancis, R. Gisbert, and P. Andrés, “Diffractive optics for quasi-direct space-to-time pulse shaping,” Opt. Express **16**(21), 16993–16998 (2008), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-21-16993. [CrossRef] [PubMed]

6. C. Iemmi, J. Campos, J. C. Escalera, O. López-Coronado, R. Gimeno, and M. J. Yzuel, “Depth of focus increase by multiplexing programmable diffractive lenses,” Opt. Express **14**(22), 10207–10219 (2006), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-22-10207. [CrossRef] [PubMed]

7. K. Kimura, S. Hasegawa, and Y. Hayasaki, “Diffractive spatiotemporal lens with wavelength dispersion compensation,” Opt. Lett. **35**(2), 139–141 (2010), http://www.opticsinfobase.org/ol/abstract.cfm?URI=ol-35-2-139. [CrossRef] [PubMed]

8. O. Mendoza-Yero, G. Mínguez-Vega, M. Fernández-Alonso, J. Lancis, E. Tajahuerce, V. Climent, and J. A. Monsoriu, “Optical filters with fractal transmission spectra based on diffractive optics,” Opt. Lett. **34**(5), 560–562 (2009), http://www.opticsinfobase.org/ol/abstract.cfm?URI=ol-34-5-560. [CrossRef] [PubMed]

9. N. Ferralis and R. D. Diehl, “Diffraction from one- and two-dimensional quasicrystalline gratings,” Am. J. Phys. **72**(9), 1241–1246 (2004), http://dx.doi.org/10.1119/1.1758221. [CrossRef]

11. N. D. Lai, J. H. Lin, and C. C. Hsu, “Fabrication of highly rotational symmetric quasi-periodic structures by multiexposure of a three-beam interference technique,” Appl. Opt. **46**(23), 5645–5648 (2007), http://www.opticsinfobase.org/ao/abstract.cfm?URI=ao-46-23-5645. [CrossRef] [PubMed]

12. J. Duparré, P. Dannberg, P. Schreiber, A. Bräuer, and A. Tünnermann, “Artificial apposition compound eye fabricated by micro-optics technology,” Appl. Opt. **43**(22), 4303–4310 (2004), http://www.opticsinfobase.org/ao/abstract.cfm?URI=ao-43-22-4303. [CrossRef] [PubMed]

16. S.-P. Simonaho and R. Silvennoinen, “Sensing of wood density by laser light scattering pattern and diffractive optical element based sensor,” J. Opt. Technol. **73**(3), 170–174 (2006), http://www.opticsinfobase.org/JOT/abstract.cfm?URI=JOT-73-3-170. [CrossRef]

17. R. T. Lee and G. S. Smith, “Detailed electromagnetic simulation for the structural color of butterfly wings,” Appl. Opt. **48**(21), 4177–4190 (2009), http://www.opticsinfobase.org/ao/abstract.cfm?URI=ao-48-21-4177. [CrossRef] [PubMed]

18. I. R. Hooper, P. Vukusic, and R. J. Wootton, “Detailed optical study of the transparent wing membranes of the dragonfly Aeshna cyanea,” Opt. Express **14**(11), 4891–4897 (2006), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-11-4891. [CrossRef] [PubMed]

19. A. R. Parker and H. E. Townley, “Biomimetics of photonic nanostructures,” Nat. Nanotechnol. **2**(6), 347–353 (2007). [CrossRef]

22. R. V. Jean, “Mathematical modelling in phyllotaxis: The state of the art,” Math. Biosci. **64**(1), 1–27 (1983). [CrossRef]

23. S. S. Liaw, “Phyllotaxis: Its geometry and dynamics,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics **57**(4), 4589–4593 (1998). [CrossRef]

27. C. Nisoli, “Spiraling solitons: A continuum model for dynamical phyllotaxis of physical systems,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. **80**(2), 026110 (2009). [CrossRef] [PubMed]

28. A. Murthy, “Mathematics of nature and nature of mathematics” Chapter I, http://www.scribd.com/doc/21990600/Maths-of-Nature-and-Nature-of-Maths-Chapter-1

29. M. Mihailescu, A. M. Preda, D. Cojoc, E. Scarlat, and L. Preda, “Diffraction pattern from a phyllotaxis type arrangement,” Opt. Lasers Eng. **46**(11), 802–809 (2008). [CrossRef]

## 2. Diffractive structures design

*r*, disposed in spirals as shown in the Fig. 1. The numbers of spirals in direct and inverse trigonometric sense are two consecutive Fibonacci numbers. Here, in the design of PtDOE, it was considered only the direct trigonometric sense. The location of the n

^{th}CE is given by the Cartesian coordinates

30. H. Vogel, “A better way to construct the sunflower head,” Math. Biosci. **44**(3–4), 179–189 (1979). [CrossRef]

*n*

^{th}CE and the centre of the PtDOE,

*c*is a constant scaling parameter linked with the filling factor, named here the filling interval.

*n*). For a different angle interval

*φ*, this total number

*n*is different, in the same area (see Fig. 1). For the optimum value

*n*, can be increased, through the decreases in the

*c*and

*r*values (compare Fig. 1 b and c). For the same filling interval

*c*, the total number

*n*varies with the angle interval. There are different angles where the total number

*n*is bigger than the number for the ideal arrangement (see Fig. 1b), but to avoid the overlapping of CE one must decrease their radius

*r*(see Fig. 1d). In these conditions, in order to keep the same radius

*r*, one must increase, also, the filling interval,

*c*(see Fig. 1e), which leads to a drastic decrease in the value of

*n*. When the values of the angle interval are far from the golden angle, the arrangement is also far from the ideal one with many overlapping CEs at the same

*r*(see Fig. 1f).

*φ*was modified from the optimum value (Fig. 1b and c) to different values (Fig. 1 a, d, e, f) with major changes in the CE positions and in the filling factor value for the same area. At the ideal angle interval

*c*=14 and

*r*=12 and the other is shown in the Fig. 1c for

*c*=8 and

*r*=6. If one needs to increase the value of the CE radius,

*r*, he must increase also the filling interval

*c*, in order to avoid the overlapping.

*r*of the CE. The case when some random defects (see Fig. 2a ) are present in the PtDOE structure (like black CEs) is also investigated. The value,

*ϒ*, calculated like a ratio between the absent CEs and the present ones, is employed to estimate the defects influence in the diffracted intensity.

*m*is a non zero integer called the topological charge and

*ϕ*is the azimuthal angle.

## 3. Near-field diffraction patterns from the PtDOEs

*λ*, normally illuminating the structure of the PtDOE situated at

*z*from the object (the output plane), can be expressed using the 2-D convolution operator [31]:where

32. B. L. Shoop, T. D. Wagner, J. N. Mait, G. R. Kilby, and E. K. Ressler, “Design and analysis of a diffractive optical filter for use in an optoelectronic error-diffusion neural network,” Appl. Opt. **38**(14), 3077–3088 (1999), http://www.opticsinfobase.org/ao/abstract.cfm?URI=ao-38-14-3077. [CrossRef]

*η*, is calculated like a ratio between the intensity distributed in the CCR at a given

*z*, and the whole diffracted intensity (at the same

*z*) from the PtDOE structure, considered as a phase-only element.

33. J. García, D. Mas, and R. G. Dorsch, “Fractional-Fourier-transform calculation through the fast-Fourier-transform algorithm,” Appl. Opt. **35**(35), 7013–7018 (1996), http://www.opticsinfobase.org/ao/abstract.cfm?URI=ao-35-35-7013. [CrossRef] [PubMed]

*φ*= 137.51°

*n*= 323

*c*= 17

*r*= 11, and

*z*between 60.5mm and 68.5mm with a step of 0.5mm (View the movie Media 1, Fig. 4 ). As one can see in Fig. 4, the CCR from the diffraction pattern is clockwise rotated with 23.25° ± 0.01° when it is propagated along 0.8cm.

*φ*=137.51°

*n*=357

*c*=16

*r*=11. The propagation in the Fresnel approximation is simulated between

*η*, has the following values: a) 12.69%, b) 12.97%, c) 13.62%, d) 13.24%.

*r*, at different filling intervals,

*c*. As one can see, the minimum values of the CCR diameter decrease with

*c*and the in focus image appears at smaller values of the axial coordinate,

*z*. These dependencies are in concordance with the Fig. 6.

*c*, but different

*r*, is presented in Fig. 8 . The CE radius has an insignificant influence on the position or the diameter of the CCR, at the same filling interval,

*c*. In conclusion, one can say that the self focusing property of the PtDOE generated after phyllotaxis geometry is given by the arrangement in spirals of the CEs, with the curvature depending on the filling interval,

*c*.

*η*, is summarized in the Table 1 . For accidentally defects, randomly distributed in proportion which is lower than 5%, the diffraction efficiency isn't significantly affected.

## 4. Non-binary phase distribution onto the PtDOEs structure

*c*=8,

*n*=1414,

*r*=6 and a vortex with the topological charge

*m*=34. The intensity and the uniformity in the CCR increase, besides the case when is considered only the PtDOE structure. The intensity distribution after CCR is also in concentric rings which have a clockwise motion when it propagates along the optical axis (see the blue and the red marks from Fig. 9). In this case, the diffraction efficiency values are different from frame to frame, with the mean value of 15%.

## 5. Conclusion

## Acknowledgements**:**

## References and links

1. | A. J. Caley, M. J. Thomson, J. Liu, A. J. Waddie, and M. R. Taghizadeh, “Diffractive optical elements for high gain lasers with arbitrary output beam profiles,” Opt. Express |

2. | G. S. Khan, K. Mantel, I. Harder, N. Lindlein, and J. Schwider, “Design considerations for the absolute testing approach of aspherics using combined diffractive optical elements,” Appl. Opt. |

3. | H. Angelskår, I.-R. Johansen, M. Lacolle, H. Sagberg, and A. S. Sudbø, “Spectral uniformity of two- and four-level diffractive optical elements for spectroscopy,” Opt. Express |

4. | G. Mínguez-Vega, O. Mendoza-Yero, J. Lancis, R. Gisbert, and P. Andrés, “Diffractive optics for quasi-direct space-to-time pulse shaping,” Opt. Express |

5. | A. R. Moradi, E. Ferrari, V. Garbin, E. Di Fabrizio, and D. Cojoc, “Strength control in multiple optical traps generated by means of diffractive optical elements,” J. Opt. Adv. Mat. RC |

6. | C. Iemmi, J. Campos, J. C. Escalera, O. López-Coronado, R. Gimeno, and M. J. Yzuel, “Depth of focus increase by multiplexing programmable diffractive lenses,” Opt. Express |

7. | K. Kimura, S. Hasegawa, and Y. Hayasaki, “Diffractive spatiotemporal lens with wavelength dispersion compensation,” Opt. Lett. |

8. | O. Mendoza-Yero, G. Mínguez-Vega, M. Fernández-Alonso, J. Lancis, E. Tajahuerce, V. Climent, and J. A. Monsoriu, “Optical filters with fractal transmission spectra based on diffractive optics,” Opt. Lett. |

9. | N. Ferralis and R. D. Diehl, “Diffraction from one- and two-dimensional quasicrystalline gratings,” Am. J. Phys. |

10. | M. Mihailescu, A. M. Preda, A. Sobetkii, and A. C. Petcu, “Fractal-like diffractive arrangement with multiple focal points”, Opto-Electr, Rev. |

11. | N. D. Lai, J. H. Lin, and C. C. Hsu, “Fabrication of highly rotational symmetric quasi-periodic structures by multiexposure of a three-beam interference technique,” Appl. Opt. |

12. | J. Duparré, P. Dannberg, P. Schreiber, A. Bräuer, and A. Tünnermann, “Artificial apposition compound eye fabricated by micro-optics technology,” Appl. Opt. |

13. | L. P. Biró, Z. Bálint, K. Kertész, Z. Vértesy, G. I. Márk, Z. E. Horváth, J. Balázs, D. Méhn, I. Kiricsi, V. Lousse, and J. P. Vigneron, “Role of photonic-crystal-type structures in the thermal regulation of a Lycaenid butterfly sister species pair,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. |

14. | D. Radtke, J. Duparré, U. D. Zeitner, and A. Tünnermann, “Laser lithographic fabrication and characterization of a spherical artificial compound eye,” Opt. Express |

15. | J. Rosen and D. Abookasis, “Seeing through biological tissues using the fly eye principle,” Opt. Express |

16. | S.-P. Simonaho and R. Silvennoinen, “Sensing of wood density by laser light scattering pattern and diffractive optical element based sensor,” J. Opt. Technol. |

17. | R. T. Lee and G. S. Smith, “Detailed electromagnetic simulation for the structural color of butterfly wings,” Appl. Opt. |

18. | I. R. Hooper, P. Vukusic, and R. J. Wootton, “Detailed optical study of the transparent wing membranes of the dragonfly Aeshna cyanea,” Opt. Express |

19. | A. R. Parker and H. E. Townley, “Biomimetics of photonic nanostructures,” Nat. Nanotechnol. |

20. | P. Prusinkiewicz, and A. Lindenmayer, “The Alogirthmic Beauty of Plants” (Springer-Verlag, New York, 1990) |

21. | R. O. Erickson, “The geometry of phyllotaxis” in |

22. | R. V. Jean, “Mathematical modelling in phyllotaxis: The state of the art,” Math. Biosci. |

23. | S. S. Liaw, “Phyllotaxis: Its geometry and dynamics,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics |

24. | L. S. Levitov, “Fibonacci numbers in botany and physics: Phyllotaxis,” J. Exp. Theor. Phys. Lett. |

25. | S. Douady and Y. Couder, “Phyllotaxis as a physical self-organized growth process,” Phys. Rev. Lett. |

26. | G. P. Bernasconi and J. Boissonade, “Phyllotactic order induced by symmetry breaking in advanced Turing patterns,” Phys. Lett. |

27. | C. Nisoli, “Spiraling solitons: A continuum model for dynamical phyllotaxis of physical systems,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. |

28. | A. Murthy, “Mathematics of nature and nature of mathematics” Chapter I, http://www.scribd.com/doc/21990600/Maths-of-Nature-and-Nature-of-Maths-Chapter-1 |

29. | M. Mihailescu, A. M. Preda, D. Cojoc, E. Scarlat, and L. Preda, “Diffraction pattern from a phyllotaxis type arrangement,” Opt. Lasers Eng. |

30. | H. Vogel, “A better way to construct the sunflower head,” Math. Biosci. |

31. | J. W. Goodman, “Introduction to Fourier optics”, Mc Graw-Hill Book Company, 1968 |

32. | B. L. Shoop, T. D. Wagner, J. N. Mait, G. R. Kilby, and E. K. Ressler, “Design and analysis of a diffractive optical filter for use in an optoelectronic error-diffusion neural network,” Appl. Opt. |

33. | J. García, D. Mas, and R. G. Dorsch, “Fractional-Fourier-transform calculation through the fast-Fourier-transform algorithm,” Appl. Opt. |

**OCIS Codes**

(050.1940) Diffraction and gratings : Diffraction

(050.1970) Diffraction and gratings : Diffractive optics

**ToC Category:**

Diffraction and Gratings

**History**

Original Manuscript: March 4, 2010

Revised Manuscript: April 5, 2010

Manuscript Accepted: April 6, 2010

Published: May 27, 2010

**Virtual Issues**

Vol. 5, Iss. 10 *Virtual Journal for Biomedical Optics*

**Citation**

Mona Mihailescu, "Natural quasy-periodic binary structure
with focusing property
in near field diffraction pattern," Opt. Express **18**, 12526-12536 (2010)

http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=oe-18-12-12526

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### References

- A. J. Caley, M. J. Thomson, J. Liu, A. J. Waddie, and M. R. Taghizadeh, “Diffractive optical elements for high gain lasers with arbitrary output beam profiles,” Opt. Express 15(17), 10699–10704 (2007), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-17-10699 . [CrossRef] [PubMed]
- G. S. Khan, K. Mantel, I. Harder, N. Lindlein, and J. Schwider, “Design considerations for the absolute testing approach of aspherics using combined diffractive optical elements,” Appl. Opt. 46(28), 7040–7048 (2007), http://www.opticsinfobase.org/abstract.cfm?URI=ao-46-28-7040 . [CrossRef] [PubMed]
- H. Angelskår, I.-R. Johansen, M. Lacolle, H. Sagberg, and A. S. Sudbø, “Spectral uniformity of two- and four-level diffractive optical elements for spectroscopy,” Opt. Express 17(12), 10206–10222 (2009), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-12-10206 . [CrossRef] [PubMed]
- G. Mínguez-Vega, O. Mendoza-Yero, J. Lancis, R. Gisbert, and P. Andrés, “Diffractive optics for quasi-direct space-to-time pulse shaping,” Opt. Express 16(21), 16993–16998 (2008), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-21-16993 . [CrossRef] [PubMed]
- A. R. Moradi, E. Ferrari, V. Garbin, E. Di Fabrizio, and D. Cojoc, “Strength control in multiple optical traps generated by means of diffractive optical elements,” J. Opt. Adv. Mat. RC 1(4), 158–161 (2007).
- C. Iemmi, J. Campos, J. C. Escalera, O. López-Coronado, R. Gimeno, and M. J. Yzuel, “Depth of focus increase by multiplexing programmable diffractive lenses,” Opt. Express 14(22), 10207–10219 (2006), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-22-10207 . [CrossRef] [PubMed]
- K. Kimura, S. Hasegawa, and Y. Hayasaki, “Diffractive spatiotemporal lens with wavelength dispersion compensation,” Opt. Lett. 35(2), 139–141 (2010), http://www.opticsinfobase.org/ol/abstract.cfm?URI=ol-35-2-139 . [CrossRef] [PubMed]
- O. Mendoza-Yero, G. Mínguez-Vega, M. Fernández-Alonso, J. Lancis, E. Tajahuerce, V. Climent, and J. A. Monsoriu, “Optical filters with fractal transmission spectra based on diffractive optics,” Opt. Lett. 34(5), 560–562 (2009), http://www.opticsinfobase.org/ol/abstract.cfm?URI=ol-34-5-560 . [CrossRef] [PubMed]
- N. Ferralis and R. D. Diehl, “Diffraction from one- and two-dimensional quasicrystalline gratings,” Am. J. Phys. 72(9), 1241–1246 (2004), http://dx.doi.org/10.1119/1.1758221 . [CrossRef]
- M. Mihailescu, A. M. Preda, A. Sobetkii, and A. C. Petcu, “Fractal-like diffractive arrangement with multiple focal points”, Opto-Electr, Rev. 17(4), 330–337 (2009). [CrossRef]
- N. D. Lai, J. H. Lin, and C. C. Hsu, “Fabrication of highly rotational symmetric quasi-periodic structures by multiexposure of a three-beam interference technique,” Appl. Opt. 46(23), 5645–5648 (2007), http://www.opticsinfobase.org/ao/abstract.cfm?URI=ao-46-23-5645 . [CrossRef] [PubMed]
- J. Duparré, P. Dannberg, P. Schreiber, A. Bräuer, and A. Tünnermann, “Artificial apposition compound eye fabricated by micro-optics technology,” Appl. Opt. 43(22), 4303–4310 (2004), http://www.opticsinfobase.org/ao/abstract.cfm?URI=ao-43-22-4303 . [CrossRef] [PubMed]
- L. P. Biró, Z. Bálint, K. Kertész, Z. Vértesy, G. I. Márk, Z. E. Horváth, J. Balázs, D. Méhn, I. Kiricsi, V. Lousse, and J. P. Vigneron, “Role of photonic-crystal-type structures in the thermal regulation of a Lycaenid butterfly sister species pair,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 67(2), 021907 (2003). [CrossRef] [PubMed]
- D. Radtke, J. Duparré, U. D. Zeitner, and A. Tünnermann, “Laser lithographic fabrication and characterization of a spherical artificial compound eye,” Opt. Express 15(6), 3067–3077 (2007), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-6-3067 . [CrossRef] [PubMed]
- J. Rosen and D. Abookasis, “Seeing through biological tissues using the fly eye principle,” Opt. Express 11(26), 3605–3611 (2003), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-11-26-3605 . [CrossRef] [PubMed]
- S.-P. Simonaho and R. Silvennoinen, “Sensing of wood density by laser light scattering pattern and diffractive optical element based sensor,” J. Opt. Technol. 73(3), 170–174 (2006), http://www.opticsinfobase.org/JOT/abstract.cfm?URI=JOT-73-3-170 . [CrossRef]
- R. T. Lee and G. S. Smith, “Detailed electromagnetic simulation for the structural color of butterfly wings,” Appl. Opt. 48(21), 4177–4190 (2009), http://www.opticsinfobase.org/ao/abstract.cfm?URI=ao-48-21-4177 . [CrossRef] [PubMed]
- I. R. Hooper, P. Vukusic, and R. J. Wootton, “Detailed optical study of the transparent wing membranes of the dragonfly Aeshna cyanea,” Opt. Express 14(11), 4891–4897 (2006), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-11-4891 . [CrossRef] [PubMed]
- A. R. Parker and H. E. Townley, “Biomimetics of photonic nanostructures,” Nat. Nanotechnol. 2(6), 347–353 (2007). [CrossRef]
- P. Prusinkiewicz and A. Lindenmayer, “The Alogirthmic Beauty of Plants” (Springer-Verlag, New York, 1990)
- R. O. Erickson, “The geometry of phyllotaxis” in The growth and functioning of leaves J. E. Dale and F. L. Milthrope, ed. pages 53–88. (University Press, Cambridge, 1983).
- R. V. Jean, “Mathematical modelling in phyllotaxis: The state of the art,” Math. Biosci. 64(1), 1–27 (1983). [CrossRef]
- S. S. Liaw, “Phyllotaxis: Its geometry and dynamics,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 57(4), 4589–4593 (1998). [CrossRef]
- L. S. Levitov, “Fibonacci numbers in botany and physics: Phyllotaxis,” J. Exp. Theor. Phys. Lett. 54(9), 546–550 (1991).
- S. Douady and Y. Couder, “Phyllotaxis as a physical self-organized growth process,” Phys. Rev. Lett. 68(13), 2098–2101 (1992). [CrossRef] [PubMed]
- G. P. Bernasconi and J. Boissonade, “Phyllotactic order induced by symmetry breaking in advanced Turing patterns,” Phys. Lett. 232(3–4), 224–230 (1997). [CrossRef]
- C. Nisoli, “Spiraling solitons: A continuum model for dynamical phyllotaxis of physical systems,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 80(2), 026110 (2009). [CrossRef] [PubMed]
- A. Murthy, “Mathematics of nature and nature of mathematics” Chapter I, http://www.scribd.com/doc/21990600/Maths-of-Nature-and-Nature-of-Maths-Chapter-1
- M. Mihailescu, A. M. Preda, D. Cojoc, E. Scarlat, and L. Preda, “Diffraction pattern from a phyllotaxis type arrangement,” Opt. Lasers Eng. 46(11), 802–809 (2008). [CrossRef]
- H. Vogel, “A better way to construct the sunflower head,” Math. Biosci. 44(3–4), 179–189 (1979). [CrossRef]
- J. W. Goodman, Introduction to Fourier optics,( Mc Graw-Hill Book Company, 1968)
- B. L. Shoop, T. D. Wagner, J. N. Mait, G. R. Kilby, and E. K. Ressler, “Design and analysis of a diffractive optical filter for use in an optoelectronic error-diffusion neural network,” Appl. Opt. 38(14), 3077–3088 (1999), http://www.opticsinfobase.org/ao/abstract.cfm?URI=ao-38-14-3077 . [CrossRef]
- J. García, D. Mas, and R. G. Dorsch, “Fractional-Fourier-transform calculation through the fast-Fourier-transform algorithm,” Appl. Opt. 35(35), 7013–7018 (1996), http://www.opticsinfobase.org/ao/abstract.cfm?URI=ao-35-35-7013 . [CrossRef] [PubMed]

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