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

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2010 (1)

2009 (5)

2008 (2)

2007 (6)

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 .
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[CrossRef] [PubMed]

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]

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

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).

2006 (3)

2004 (2)

2003 (2)

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]

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]

1999 (1)

1998 (1)

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

1997 (1)

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]

1996 (1)

1992 (1)

S. Douady and Y. Couder, “Phyllotaxis as a physical self-organized growth process,” Phys. Rev. Lett. 68(13), 2098–2101 (1992).
[CrossRef] [PubMed]

1991 (1)

L. S. Levitov, “Fibonacci numbers in botany and physics: Phyllotaxis,” J. Exp. Theor. Phys. Lett. 54(9), 546–550 (1991).

1983 (1)

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

1979 (1)

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

Abookasis, D.

Andrés, P.

Angelskår, H.

Balázs, J.

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]

Bálint, Z.

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]

Bernasconi, G. P.

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]

Biró, L. P.

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]

Boissonade, J.

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]

Bräuer, A.

Caley, A. J.

Campos, J.

Climent, V.

Cojoc, D.

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]

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).

Couder, Y.

S. Douady and Y. Couder, “Phyllotaxis as a physical self-organized growth process,” Phys. Rev. Lett. 68(13), 2098–2101 (1992).
[CrossRef] [PubMed]

Dannberg, P.

Di Fabrizio, E.

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).

Diehl, R. D.

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]

Dorsch, R. G.

Douady, S.

S. Douady and Y. Couder, “Phyllotaxis as a physical self-organized growth process,” Phys. Rev. Lett. 68(13), 2098–2101 (1992).
[CrossRef] [PubMed]

Duparré, J.

Escalera, J. C.

Fernández-Alonso, M.

Ferralis, N.

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]

Ferrari, E.

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).

Garbin, V.

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).

García, J.

Gimeno, R.

Gisbert, R.

Harder, I.

Hasegawa, S.

Hayasaki, Y.

Hooper, I. R.

Horváth, Z. E.

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]

Hsu, C. C.

Iemmi, C.

Jean, R. V.

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

Johansen, I.-R.

Kertész, K.

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]

Khan, G. S.

Kilby, G. R.

Kimura, K.

Kiricsi, I.

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]

Lacolle, M.

Lai, N. D.

Lancis, J.

Lee, R. T.

Levitov, L. S.

L. S. Levitov, “Fibonacci numbers in botany and physics: Phyllotaxis,” J. Exp. Theor. Phys. Lett. 54(9), 546–550 (1991).

Liaw, S. S.

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

Lin, J. H.

Lindlein, N.

Liu, J.

López-Coronado, O.

Lousse, V.

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]

Mait, J. N.

Mantel, K.

Márk, G. I.

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]

Mas, D.

Méhn, D.

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]

Mendoza-Yero, O.

Mihailescu, M.

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]

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]

Mínguez-Vega, G.

Monsoriu, J. A.

Moradi, A. R.

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).

Nisoli, C.

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]

Parker, A. R.

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

Petcu, A. C.

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]

Preda, A. M.

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]

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]

Preda, L.

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]

Radtke, D.

Ressler, E. K.

Rosen, J.

Sagberg, H.

Scarlat, E.

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]

Schreiber, P.

Schwider, J.

Shoop, B. L.

Silvennoinen, R.

Simonaho, S.-P.

Smith, G. S.

Sobetkii, A.

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]

Sudbø, A. S.

Taghizadeh, M. R.

Tajahuerce, E.

Thomson, M. J.

Townley, H. E.

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

Tünnermann, A.

Vértesy, Z.

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]

Vigneron, J. P.

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]

Vogel, H.

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

Vukusic, P.

Waddie, A. J.

Wagner, T. D.

Wootton, R. J.

Yzuel, M. J.

Zeitner, U. D.

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Supplementary Material (3)

» Media 1: MOV (2217 KB)     
» Media 2: MOV (2069 KB)     
» Media 3: MOV (3504 KB)     

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Figures (11)

Fig. 1
Fig. 1

PtDOEs generated with different geometrical parameters a) φ=137.55° n=459 c=14 r=10, b) φ=137.51° n=467 c=14 r=12, c) φ=137.51° n=1414 c=8 r=6, d) φ=137.45° n=470 c=14 r=9, e) φ=137.45° n=401 c=15 r=11, f) φ=130.00° n=458 c=14 r=10

Fig. 2
Fig. 2

a) Random defects and non-binary phase map superposed onto the PtDOE with different geometrical parameters a) and b) φ=137.51° n=467 c=14 r=10, c) φ=137.51° n=467 c=14 r=11 m=34

Fig. 3
Fig. 3

Diffraction patterns at z = 40.0mm when the PtDOEs was generated with the following geometrical parameters a) φ=137.55° n=459 c=14 r=10, b) φ=137.51° n=467 c=14 r=12, c) φ=137.51° n=1414 c=8 r=6, d) φ=137.45° n=470 c=14 r=9, e) φ=137.45° n=401 c=15 r=11, f) φ=130.00° n=458 c=14 r=10

Fig. 4
Fig. 4

The transverse diffraction patterns from the PtDOE generated with the parameters φ=137.51° n=323 c=17 r=11, at a) z = 60.5mm and b) z = 68.5mm (Media 1)

Fig. 5
Fig. 5

The transverse diffraction patterns from a PtDOE generated with the parameters φ=137.51° n=357 c=16 r=11 at a) z = 55.0mm, b) z = 56.8mm, c) z = 58.6mm (Media 2)

Fig. 6
Fig. 6

Transverse diffraction patterns from the PtDOE generated with different geometrical parameters and the focalized CCRs are at different axial coordinate a) φ=137.51 c=8, n=1414, r=6, z = 34.5mm b) φ=137.51, c=13, n=535, r=10, z = 39.4mm, c) φ=137.51, c=14, n=467, r=11, z = 44.3mm d) φ=137.51, c=15, n=412, r=11, z = 50.7mm

Fig. 7
Fig. 7

The diameter evolution of the CCR when the PtDOE was generated with the same CE radius, r=11, but different filling intervals, c

Fig. 8
Fig. 8

The diameter evolution of the CCR when the PtDOE structures was generated with the same filling interval c=15, but different values for r

Fig. 9
Fig. 9

The transverse diffraction patterns at a) z = 54.0mm, b) z = 56.0mm, c) z = 58.0mm, d) z = 68.0mm, e) z = 70.0mm, f) z = 72.0mm when a vortex is superposed on the PtDOE

Fig. 10
Fig. 10

Diffractive pattern from the PtDOE when an inverse vortex is superposed (Media 3)

Fig. 11
Fig. 11

The transverse diffraction pattern at z = 41.8mm when the PtDOE structure was generated with the geometrical parameters: φ=137.51°, c=14, n=467, r=11, and a non-binary phase distribution of the lens type is superposed on each CE

Tables (1)

Tables Icon

Table 1 The influence of the defect number in the diffraction efficiency for a PtDOE generated withφ=137.51° n=467 c=14 r=10

Equations (8)

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x n = c n cos ( n φ ) , y n = c n sin ( n φ )
t ( x , y ) = exp ( i π Φ ( x , y ) )
t n b ( x , y ) = exp ( i π Φ ( x , y ) + Φ n b ( x , y ) )
U ( x 0 , y 0 , z ) = t ( x , y , 0 ) h ( x 0 , y 0 , x , y , z ) ,
h ( x , y , z ) = exp ( i k z ) i λ z exp { i k 2 z [ x 2 + y 2 ] } ,
U ( x 0 , y 0 , z ) = exp ( i k z ) i λ z exp [ i k 2 z ( x 0 2 + y 0 2 ) ] + + U ' ( x , y , 0 ) exp [ i 2 π ( x 0 x + y 0 y ) ] d x d y
U ' ( x , y , 0 ) = t ( x , y , 0 ) exp [ i k 2 z ( x 2 + y 2 ) ] ,
N Δ x Δ x 0 = λ z and M Δ y Δ y 0 = λ z

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