Abstract

We introduce circular Fibonacci gratings (CFGs) that combine the concept of circular gratings and Fibonacci structures. Theoretical analysis shows that the diffraction pattern of CFGs is composed of fractal distributions of impulse rings. Numerical simulations are performed with two-dimensional fast Fourier transform to reveal the fractal behavior of the diffraction rings. Experimental results are also presented and agree well with the numerical results. The fractal nature of the diffraction field should be of great theoretical interest, and shows potential to be further developed into practical applications, such as in laser measurement with wideband illumination.

© 2011 Optical Society of America

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References

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  15. For example, in Fig. , the intensity of central peak A is 0.0033, and this is just the diffraction efficiency of this diffraction peak. To get the diffraction efficiency of other peaks, one should consider the circular symmetry of the system, and summarize the intensity over all the circumference.
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2010 (2)

2009 (1)

2008 (1)

2007 (2)

2004 (1)

2003 (2)

2001 (1)

O. Trabocchi, S. Granieri, and W. D. Furlan, “Optical propagation of fractal fields. Experimental analysis in a single display,” J. Mod. Opt. 48, 1247–1253 (2001).
[CrossRef]

1998 (1)

A. D. Jaggard and D. L. Jaggard, “Cantor ring diffractals,” Opt. Commun. 158, 141–148 (1998).
[CrossRef]

1997 (1)

1993 (1)

G. Y. Oh and M. H. Lee, “Band-structural and Fourier-spectral properties of one-dimensional generalized Fibonacci lattices,” Phys. Rev. B 48, 12465–12477 (1993).
[CrossRef]

Amidror, I.

Calatayud, A.

Chen, Z.

Chung, P. S.

Dobson, K.

Doh, K. B.

Furlan, W. D.

Giménez, F.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics2nd ed. (McGraw-Hill, 1996).

Granieri, S.

O. Trabocchi, S. Granieri, and W. D. Furlan, “Optical propagation of fractal fields. Experimental analysis in a single display,” J. Mod. Opt. 48, 1247–1253 (2001).
[CrossRef]

Jaggard, A. D.

A. D. Jaggard and D. L. Jaggard, “Cantor ring diffractals,” Opt. Commun. 158, 141–148 (1998).
[CrossRef]

Jaggard, D. L.

A. D. Jaggard and D. L. Jaggard, “Cantor ring diffractals,” Opt. Commun. 158, 141–148 (1998).
[CrossRef]

Jia, J.

Lee, M. H.

G. Y. Oh and M. H. Lee, “Band-structural and Fourier-spectral properties of one-dimensional generalized Fibonacci lattices,” Phys. Rev. B 48, 12465–12477 (1993).
[CrossRef]

Liu, L.

Monsoriu, J. A.

Oh, G. Y.

G. Y. Oh and M. H. Lee, “Band-structural and Fourier-spectral properties of one-dimensional generalized Fibonacci lattices,” Phys. Rev. B 48, 12465–12477 (1993).
[CrossRef]

Poon, T.

Saavedra, G.

Schanda, J.

J. Schanda, Colorimetry: Understanding the CIE System(Wiley, 2007).

Senechal, M.

M. Senechal, Quasicrystals and Geometry (Cambridge University, 1995).

Sun, X.

Trabocchi, O.

O. Trabocchi, S. Granieri, and W. D. Furlan, “Optical propagation of fractal fields. Experimental analysis in a single display,” J. Mod. Opt. 48, 1247–1253 (2001).
[CrossRef]

Wen, F. J.

Wen, J. F.

Zhao, S.

Zhou, C.

Appl. Opt. (5)

J. Mod. Opt. (1)

O. Trabocchi, S. Granieri, and W. D. Furlan, “Optical propagation of fractal fields. Experimental analysis in a single display,” J. Mod. Opt. 48, 1247–1253 (2001).
[CrossRef]

J. Opt. Soc. Am. A (2)

Opt. Commun. (1)

A. D. Jaggard and D. L. Jaggard, “Cantor ring diffractals,” Opt. Commun. 158, 141–148 (1998).
[CrossRef]

Opt. Lett. (3)

Phys. Rev. B (1)

G. Y. Oh and M. H. Lee, “Band-structural and Fourier-spectral properties of one-dimensional generalized Fibonacci lattices,” Phys. Rev. B 48, 12465–12477 (1993).
[CrossRef]

Other (4)

J. W. Goodman, Introduction to Fourier Optics2nd ed. (McGraw-Hill, 1996).

M. Senechal, Quasicrystals and Geometry (Cambridge University, 1995).

For example, in Fig. , the intensity of central peak A is 0.0033, and this is just the diffraction efficiency of this diffraction peak. To get the diffraction efficiency of other peaks, one should consider the circular symmetry of the system, and summarize the intensity over all the circumference.

J. Schanda, Colorimetry: Understanding the CIE System(Wiley, 2007).

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Figures (6)

Fig. 1
Fig. 1

(a) Fibonacci sequences for the stages S = 0 , 1, 2, 3, and 4. (b) Structure of CFGs. Here A and B are zone structures with width D A and D B . The ratio between D A and D B is the golden ratio τ, and the groove width is w = D B / 2 . Note the horizontal coordinate is the radius r.

Fig. 2
Fig. 2

(a) Structure of a CFG. The white and black areas correspond to 0 and π phase delay. (b) Simulated diffraction pattern of the CFG. In the simulation, D A = 8.81 μm , D B = 5.44 μm , w = 2.72 μm , R = 256 μm , and every pixel in the diffraction pattern corresponds to λ z / 1024 μm in the observation plane. Only the central 400   pixel × 400   pixel is shown to emphasize the bright rings visually. (c) 1D profile of the diffraction intensity along the central line illustrated in (b). The whole simulation range (1024 pixels) is shown in this figure.

Fig. 3
Fig. 3

1D diffraction profile for CFGs at four stages of (a)  S = 1 (solid curve) and S = 2 (dotted curve), and (b)  S = 3 (solid curve) and S = 4 (dotted curve).

Fig. 4
Fig. 4

(a) Microscopy image of the fabricated CFG. (b) Experimental setup. (c) Normalized diffraction pattern recorded by CCD camera. The zeroth-order diffraction is saturated for clear vision of the impulse rings. The laser beam power utilized is 6 μW , and the diffraction intensity is normalized for the summation over all pixels to be 1. (d) 1D profile of the diffraction pattern along the central line. The locations of the five labeled peaks continue to follow the golden section rule, as shown in Table 1.

Fig. 5
Fig. 5

Diffraction pattern of CFGs with linewidth ratios w / D B of (a) 0.2, (b) 0.4, (c) 0.6, and (d) 0.8. Other parameters of the CFGs are D A = 8.81 μm , D B = 5.44 μm , and R = 256 μm .

Fig. 6
Fig. 6

Diffraction pattern of CFGs with ridge phase shifts of (a)  π / 3 , (b)  π / 2 , (c)  4 π / 3 , and (d)  3 π / 2 . Other parameters of the CFGs are D A = 8.81 μm , D B = 5.44 μm , w = 2.72 μm , and R = 256 μm .

Tables (2)

Tables Icon

Table 1 Position Relation of the Five Diffraction Peaks (e.g., A E ¯ Means the Distance Between the Two Peaks A and E)

Tables Icon

Table 2 Chromaticity Parameters of the Two Representative Points I and II for a CFG and a Conventional CG

Equations (11)

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A A B , B A .
H 0 [ g ( r ) ] = 0 g ( r ) J 0 ( q r ) r d r .
sin ( 2 π f r ) H 0 f π ( f + q ) 3 δ ( 1 / 2 ) ( f 2 q 2 ) ,
g ( r ) = G ( f ) exp ( i 2 π f r ) d f = 2 i 0 G ( f ) sin ( 2 π f r ) d f ,
G ( f ) = g ( r ) exp ( i 2 π f r ) d r = 2 i 0 g ( r ) sin ( 2 π f r ) d r .
g ( r ) = { [ n δ ( r r n ) ] * rect ( r w ) } rect ( r R 1 2 ) ,
G ( f ) = [ ( n exp ( i 2 π f r n ) ) · w sinc ( w f ) ] * [ R sinc ( R f ) exp ( i π f R n ) ] .
I ( ρ λ z , φ ) = I ( ξ λ z , η λ z ) = | 1 λ z g ( x , y ) exp ( i 2 π ξ λ z x ) exp ( i 2 π η λ z y ) d x d y | 2 = | 1 λ z FT { g ( x , y ) } | 2 ,
ρ u , v = λ z D B 1 + τ 2 ( u τ + v ) , ( u , v Z ) .
ρ = ρ u 1 , v 1 + ( τ 1 ) ρ u 2 , v 2 τ = ( τ 1 ) ρ u 1 , v 1 + ( 2 τ ) ρ u 2 , v 2 ( because     1 τ = τ 1 ) = λ z D B 1 + τ 2 [ ( τ 1 ) ( u 1 τ + v 1 ) + ( 2 τ ) ( u 2 τ + v 2 ) ] = λ z D B 1 + τ 2 [ ( u 1 u 2 ) τ 2 + ( u 1 + v 1 + 2 u 2 v 2 ) τ + ( v 1 + 2 v 2 ) ] = λ z D B 1 + τ 2 [ ( v 1 + u 2 v 2 ) τ + ( u 1 v 1 u 2 + 2 v 2 ) ] ( because     τ 2 = τ + 1 ) = ρ v 1 + u 2 v 2 , u 1 v 1 u 2 + 2 v 2 ,
ρ = ( τ 1 ) ρ u 1 , v 1 + ρ u 2 , v 2 τ = ρ v 2 + u 1 v 1 , u 2 v 2 u 1 + 2 v 1 ,

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