Abstract

Aperiodic and fractal optical elements are proving to be promising candidates in image-forming devices. In this paper, we analyze the diffraction patterns of Fibonacci gratings (FbGs), which are prototypical examples of aperiodicity. They exhibit novel characteristics such as redundancy and robustness that keep their imaging characteristics intact even when there is significant loss of information. FbGs also contain fractal signatures and are characterized by a fractal dimension. Our study suggests that aperiodic gratings may be better than their fractal counterparts in technologies based on such architectures. We also identify the demarcating features of aperiodic and fractal diffraction, which have been rather fuzzy in the literature so far.

© 2014 Optical Society of America

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References

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  10. F. Gimenez, J. A. Monsoriu, W. D. Furlan, and A. Pons, “Fractal photon sieve,” Opt. Express 14, 11958–11963 (2006).
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  11. J. A. Monsoriu, C. J. Zapata-Rodriguez, and W. D. Furlan, “Fractal axicons,” Opt. Commun. 263, 1–5 (2006).
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    [CrossRef]
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    [CrossRef]
  33. D. Damanik, M. Embree, A. Gorodetski, and S. Tcheremchantsev, “The fractal dimension of the spectrum of the Fibonacci Hamiltonian,” Commun. Math. Phys. 280, 499–516 (2008).
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2014 (1)

2013 (3)

2012 (1)

2011 (1)

2010 (1)

J. Zhang, Y. Cao, and J. Zheng, “Fibonacci quasi-periodic superstructure fiber Bragg gratings,” Optik 121, 417–421 (2010).
[CrossRef]

2008 (2)

D. Damanik, M. Embree, A. Gorodetski, and S. Tcheremchantsev, “The fractal dimension of the spectrum of the Fibonacci Hamiltonian,” Commun. Math. Phys. 280, 499–516 (2008).
[CrossRef]

N. V. Grushina, P. V. Korolenko, and S. N. Markova, “Special features of the diffraction of light on optical Fibonacci gratings,” Moscow Univ. Phys. Bull. 63, 123–126 (2008).
[CrossRef]

2007 (1)

2006 (2)

F. Gimenez, J. A. Monsoriu, W. D. Furlan, and A. Pons, “Fractal photon sieve,” Opt. Express 14, 11958–11963 (2006).
[CrossRef]

J. A. Monsoriu, C. J. Zapata-Rodriguez, and W. D. Furlan, “Fractal axicons,” Opt. Commun. 263, 1–5 (2006).
[CrossRef]

2004 (2)

N. Ferralis, A. W. Szmodis, and R. D. Diehl, “Diffraction from one- and two-dimensional quasicrystalline gratings,” Am. J. Phys. 72, 1241–1246 (2004).
[CrossRef]

J. A. Monsoriu, G. Saavedra, and W. D. Furlan, “Fractal zone plates with variable lacunarity,” Opt. Express 12, 4227–4234 (2004).
[CrossRef]

2003 (1)

1999 (1)

X. Yang, Y. Liu, and X. Fu, “Transmission properties of light through the Fibonacci-class multilayers,” Phys. Rev. B 59, 4545–4548 (1999).
[CrossRef]

1996 (2)

C. A. Guerin and M. Holschneider, “Scattering on fractal measures,” J. Phys. A 29, 7651–7667 (1996).
[CrossRef]

D. A. Hamburger-Lidar, “Elastic scattering by deterministic and random fractals: self-affinity of the diffraction spectrum,” Phys. Rev. E 54, 354–370 (1996).
[CrossRef]

1994 (2)

Z.-X. Wen and Z.-Y. Wen, “Some properties of the singular words of the Fibonacci word,” Eur. J. Combin. 15, 587–598 (1994).
[CrossRef]

W. Gellermann, M. Kohmoto, B. Sutherland, and P. C. Taylor, “Localization of light waves in Fibonacci dielectric multilayers,” Phys. Rev. Lett. 72, 633–636 (1994).
[CrossRef]

1993 (1)

J. M. Dubois, “The applied physics of quasicrystals,” Phys. Scr. T49, 17–23 (1993).
[CrossRef]

1989 (1)

B. Dubuc, J. F. Quiniou, C. Roques-Carmes, C. Tricot, and S. W. Zucker, “Evaluating the fractal dimension of profiles,” Phys. Rev. A 39, 1500–1512 (1989).
[CrossRef]

1987 (2)

M. Kohmoto, B. Sutherland, and K. Iguchi, “Localization in optics: quasiperiodic media,” Phys. Rev. Lett. 58, 2436–2438 (1987).
[CrossRef]

Y. Liu and R. Riklund, “Electronic properties of perfect and nonperfect one-dimensional quasicrystals,” Phys. Rev. B 35, 6034–6042 (1987).
[CrossRef]

1986 (3)

C. Allain and M. Cloitre, “Optical diffraction on fractals,” Phys. Rev. B 33, 3566–3569 (1986).
[CrossRef]

D. Levine and P. J. Steinhardt, “Quasicrystals. I. Definition and structure,” Phys. Rev. B 34, 596–616 (1986).
[CrossRef]

J. P. Lu, T. Odagaki, and J. L. Birman, “Properties of one-dimensional quasilattices,” Phys. Rev. B 33, 4809–4817 (1986).
[CrossRef]

1984 (1)

D. Levine and P. J. Steinhardt, “Quasicrystals: a new class of ordered structures,” Phys. Rev. Lett. 53, 2477–2480 (1984).
[CrossRef]

1968 (1)

1964 (1)

Allain, C.

C. Allain and M. Cloitre, “Optical diffraction on fractals,” Phys. Rev. B 33, 3566–3569 (1986).
[CrossRef]

Banerjee, V.

Birman, J. L.

J. P. Lu, T. Odagaki, and J. L. Birman, “Properties of one-dimensional quasilattices,” Phys. Rev. B 33, 4809–4817 (1986).
[CrossRef]

Born, M.

M. Born, Principles of Optics (Pergamon, 1980).

Calatayud, A.

Cao, Y.

J. Zhang, Y. Cao, and J. Zheng, “Fibonacci quasi-periodic superstructure fiber Bragg gratings,” Optik 121, 417–421 (2010).
[CrossRef]

Cloitre, M.

C. Allain and M. Cloitre, “Optical diffraction on fractals,” Phys. Rev. B 33, 3566–3569 (1986).
[CrossRef]

Damanik, D.

D. Damanik, M. Embree, A. Gorodetski, and S. Tcheremchantsev, “The fractal dimension of the spectrum of the Fibonacci Hamiltonian,” Commun. Math. Phys. 280, 499–516 (2008).
[CrossRef]

Diehl, R. D.

N. Ferralis, A. W. Szmodis, and R. D. Diehl, “Diffraction from one- and two-dimensional quasicrystalline gratings,” Am. J. Phys. 72, 1241–1246 (2004).
[CrossRef]

Dubois, J. M.

J. M. Dubois, “The applied physics of quasicrystals,” Phys. Scr. T49, 17–23 (1993).
[CrossRef]

Dubuc, B.

B. Dubuc, J. F. Quiniou, C. Roques-Carmes, C. Tricot, and S. W. Zucker, “Evaluating the fractal dimension of profiles,” Phys. Rev. A 39, 1500–1512 (1989).
[CrossRef]

Dunlap, R. A.

R. A. Dunlap, The Golden Ratio and Fibonacci Numbers (World Scientific, 1997).

Embree, M.

D. Damanik, M. Embree, A. Gorodetski, and S. Tcheremchantsev, “The fractal dimension of the spectrum of the Fibonacci Hamiltonian,” Commun. Math. Phys. 280, 499–516 (2008).
[CrossRef]

Ferralis, N.

N. Ferralis, A. W. Szmodis, and R. D. Diehl, “Diffraction from one- and two-dimensional quasicrystalline gratings,” Am. J. Phys. 72, 1241–1246 (2004).
[CrossRef]

Ferrando, V.

Fu, X.

X. Yang, Y. Liu, and X. Fu, “Transmission properties of light through the Fibonacci-class multilayers,” Phys. Rev. B 59, 4545–4548 (1999).
[CrossRef]

Furlan, W. D.

Gao, N.

Gellermann, W.

W. Gellermann, M. Kohmoto, B. Sutherland, and P. C. Taylor, “Localization of light waves in Fibonacci dielectric multilayers,” Phys. Rev. Lett. 72, 633–636 (1994).
[CrossRef]

Gerritsen, H. J.

Gimenez, F.

Goodman, J. W.

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

Gorodetski, A.

D. Damanik, M. Embree, A. Gorodetski, and S. Tcheremchantsev, “The fractal dimension of the spectrum of the Fibonacci Hamiltonian,” Commun. Math. Phys. 280, 499–516 (2008).
[CrossRef]

Grushina, N. V.

N. V. Grushina, P. V. Korolenko, and S. N. Markova, “Special features of the diffraction of light on optical Fibonacci gratings,” Moscow Univ. Phys. Bull. 63, 123–126 (2008).
[CrossRef]

Guerin, C. A.

C. A. Guerin and M. Holschneider, “Scattering on fractal measures,” J. Phys. A 29, 7651–7667 (1996).
[CrossRef]

Hamburger-Lidar, D. A.

D. A. Hamburger-Lidar, “Elastic scattering by deterministic and random fractals: self-affinity of the diffraction spectrum,” Phys. Rev. E 54, 354–370 (1996).
[CrossRef]

Hannan, W. J.

Holschneider, M.

C. A. Guerin and M. Holschneider, “Scattering on fractal measures,” J. Phys. A 29, 7651–7667 (1996).
[CrossRef]

Iguchi, K.

M. Kohmoto, B. Sutherland, and K. Iguchi, “Localization in optics: quasiperiodic media,” Phys. Rev. Lett. 58, 2436–2438 (1987).
[CrossRef]

Kohmoto, M.

W. Gellermann, M. Kohmoto, B. Sutherland, and P. C. Taylor, “Localization of light waves in Fibonacci dielectric multilayers,” Phys. Rev. Lett. 72, 633–636 (1994).
[CrossRef]

M. Kohmoto, B. Sutherland, and K. Iguchi, “Localization in optics: quasiperiodic media,” Phys. Rev. Lett. 58, 2436–2438 (1987).
[CrossRef]

Korolenko, P. V.

N. V. Grushina, P. V. Korolenko, and S. N. Markova, “Special features of the diffraction of light on optical Fibonacci gratings,” Moscow Univ. Phys. Bull. 63, 123–126 (2008).
[CrossRef]

Leith, E. N.

Levine, D.

D. Levine and P. J. Steinhardt, “Quasicrystals. I. Definition and structure,” Phys. Rev. B 34, 596–616 (1986).
[CrossRef]

D. Levine and P. J. Steinhardt, “Quasicrystals: a new class of ordered structures,” Phys. Rev. Lett. 53, 2477–2480 (1984).
[CrossRef]

Liu, Y.

X. Yang, Y. Liu, and X. Fu, “Transmission properties of light through the Fibonacci-class multilayers,” Phys. Rev. B 59, 4545–4548 (1999).
[CrossRef]

Y. Liu and R. Riklund, “Electronic properties of perfect and nonperfect one-dimensional quasicrystals,” Phys. Rev. B 35, 6034–6042 (1987).
[CrossRef]

Loewen, E.

C. Palmer and E. Loewen, Diffraction Grating Handbook (Newport Corporation, 2005).

Lu, J. P.

J. P. Lu, T. Odagaki, and J. L. Birman, “Properties of one-dimensional quasilattices,” Phys. Rev. B 33, 4809–4817 (1986).
[CrossRef]

Markova, S. N.

N. V. Grushina, P. V. Korolenko, and S. N. Markova, “Special features of the diffraction of light on optical Fibonacci gratings,” Moscow Univ. Phys. Bull. 63, 123–126 (2008).
[CrossRef]

Monsoriu, J. A.

Odagaki, T.

J. P. Lu, T. Odagaki, and J. L. Birman, “Properties of one-dimensional quasilattices,” Phys. Rev. B 33, 4809–4817 (1986).
[CrossRef]

Palmer, C.

C. Palmer and E. Loewen, Diffraction Grating Handbook (Newport Corporation, 2005).

Pons, A.

Quiniou, J. F.

B. Dubuc, J. F. Quiniou, C. Roques-Carmes, C. Tricot, and S. W. Zucker, “Evaluating the fractal dimension of profiles,” Phys. Rev. A 39, 1500–1512 (1989).
[CrossRef]

Ramberg, E. G.

Remon, L.

Riklund, R.

Y. Liu and R. Riklund, “Electronic properties of perfect and nonperfect one-dimensional quasicrystals,” Phys. Rev. B 35, 6034–6042 (1987).
[CrossRef]

Roques-Carmes, C.

B. Dubuc, J. F. Quiniou, C. Roques-Carmes, C. Tricot, and S. W. Zucker, “Evaluating the fractal dimension of profiles,” Phys. Rev. A 39, 1500–1512 (1989).
[CrossRef]

Saavedra, G.

Senthilkumaran, P.

Sharma, M. K.

Steinhardt, P. J.

D. Levine and P. J. Steinhardt, “Quasicrystals. I. Definition and structure,” Phys. Rev. B 34, 596–616 (1986).
[CrossRef]

D. Levine and P. J. Steinhardt, “Quasicrystals: a new class of ordered structures,” Phys. Rev. Lett. 53, 2477–2480 (1984).
[CrossRef]

Sutherland, B.

W. Gellermann, M. Kohmoto, B. Sutherland, and P. C. Taylor, “Localization of light waves in Fibonacci dielectric multilayers,” Phys. Rev. Lett. 72, 633–636 (1994).
[CrossRef]

M. Kohmoto, B. Sutherland, and K. Iguchi, “Localization in optics: quasiperiodic media,” Phys. Rev. Lett. 58, 2436–2438 (1987).
[CrossRef]

Szmodis, A. W.

N. Ferralis, A. W. Szmodis, and R. D. Diehl, “Diffraction from one- and two-dimensional quasicrystalline gratings,” Am. J. Phys. 72, 1241–1246 (2004).
[CrossRef]

Taylor, P. C.

W. Gellermann, M. Kohmoto, B. Sutherland, and P. C. Taylor, “Localization of light waves in Fibonacci dielectric multilayers,” Phys. Rev. Lett. 72, 633–636 (1994).
[CrossRef]

Tcheremchantsev, S.

D. Damanik, M. Embree, A. Gorodetski, and S. Tcheremchantsev, “The fractal dimension of the spectrum of the Fibonacci Hamiltonian,” Commun. Math. Phys. 280, 499–516 (2008).
[CrossRef]

Tricot, C.

B. Dubuc, J. F. Quiniou, C. Roques-Carmes, C. Tricot, and S. W. Zucker, “Evaluating the fractal dimension of profiles,” Phys. Rev. A 39, 1500–1512 (1989).
[CrossRef]

Upatnieks, J.

Verma, R.

Wang, G. P.

Wen, Z.-X.

Z.-X. Wen and Z.-Y. Wen, “Some properties of the singular words of the Fibonacci word,” Eur. J. Combin. 15, 587–598 (1994).
[CrossRef]

Wen, Z.-Y.

Z.-X. Wen and Z.-Y. Wen, “Some properties of the singular words of the Fibonacci word,” Eur. J. Combin. 15, 587–598 (1994).
[CrossRef]

Wu, K.

Xie, C.

Yang, X.

X. Yang, Y. Liu, and X. Fu, “Transmission properties of light through the Fibonacci-class multilayers,” Phys. Rev. B 59, 4545–4548 (1999).
[CrossRef]

Zapata-Rodriguez, C. J.

J. A. Monsoriu, C. J. Zapata-Rodriguez, and W. D. Furlan, “Fractal axicons,” Opt. Commun. 263, 1–5 (2006).
[CrossRef]

Zhang, J.

J. Zhang, Y. Cao, and J. Zheng, “Fibonacci quasi-periodic superstructure fiber Bragg gratings,” Optik 121, 417–421 (2010).
[CrossRef]

Zhang, Y.

Zheng, J.

J. Zhang, Y. Cao, and J. Zheng, “Fibonacci quasi-periodic superstructure fiber Bragg gratings,” Optik 121, 417–421 (2010).
[CrossRef]

Zucker, S. W.

B. Dubuc, J. F. Quiniou, C. Roques-Carmes, C. Tricot, and S. W. Zucker, “Evaluating the fractal dimension of profiles,” Phys. Rev. A 39, 1500–1512 (1989).
[CrossRef]

Am. J. Phys. (1)

N. Ferralis, A. W. Szmodis, and R. D. Diehl, “Diffraction from one- and two-dimensional quasicrystalline gratings,” Am. J. Phys. 72, 1241–1246 (2004).
[CrossRef]

Appl. Opt. (2)

Commun. Math. Phys. (1)

D. Damanik, M. Embree, A. Gorodetski, and S. Tcheremchantsev, “The fractal dimension of the spectrum of the Fibonacci Hamiltonian,” Commun. Math. Phys. 280, 499–516 (2008).
[CrossRef]

Eur. J. Combin. (1)

Z.-X. Wen and Z.-Y. Wen, “Some properties of the singular words of the Fibonacci word,” Eur. J. Combin. 15, 587–598 (1994).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Phys. A (1)

C. A. Guerin and M. Holschneider, “Scattering on fractal measures,” J. Phys. A 29, 7651–7667 (1996).
[CrossRef]

Moscow Univ. Phys. Bull. (1)

N. V. Grushina, P. V. Korolenko, and S. N. Markova, “Special features of the diffraction of light on optical Fibonacci gratings,” Moscow Univ. Phys. Bull. 63, 123–126 (2008).
[CrossRef]

Opt. Commun. (1)

J. A. Monsoriu, C. J. Zapata-Rodriguez, and W. D. Furlan, “Fractal axicons,” Opt. Commun. 263, 1–5 (2006).
[CrossRef]

Opt. Express (5)

Opt. Lett. (4)

Optik (1)

J. Zhang, Y. Cao, and J. Zheng, “Fibonacci quasi-periodic superstructure fiber Bragg gratings,” Optik 121, 417–421 (2010).
[CrossRef]

Phys. Rev. A (1)

B. Dubuc, J. F. Quiniou, C. Roques-Carmes, C. Tricot, and S. W. Zucker, “Evaluating the fractal dimension of profiles,” Phys. Rev. A 39, 1500–1512 (1989).
[CrossRef]

Phys. Rev. B (5)

C. Allain and M. Cloitre, “Optical diffraction on fractals,” Phys. Rev. B 33, 3566–3569 (1986).
[CrossRef]

X. Yang, Y. Liu, and X. Fu, “Transmission properties of light through the Fibonacci-class multilayers,” Phys. Rev. B 59, 4545–4548 (1999).
[CrossRef]

D. Levine and P. J. Steinhardt, “Quasicrystals. I. Definition and structure,” Phys. Rev. B 34, 596–616 (1986).
[CrossRef]

J. P. Lu, T. Odagaki, and J. L. Birman, “Properties of one-dimensional quasilattices,” Phys. Rev. B 33, 4809–4817 (1986).
[CrossRef]

Y. Liu and R. Riklund, “Electronic properties of perfect and nonperfect one-dimensional quasicrystals,” Phys. Rev. B 35, 6034–6042 (1987).
[CrossRef]

Phys. Rev. E (1)

D. A. Hamburger-Lidar, “Elastic scattering by deterministic and random fractals: self-affinity of the diffraction spectrum,” Phys. Rev. E 54, 354–370 (1996).
[CrossRef]

Phys. Rev. Lett. (3)

M. Kohmoto, B. Sutherland, and K. Iguchi, “Localization in optics: quasiperiodic media,” Phys. Rev. Lett. 58, 2436–2438 (1987).
[CrossRef]

D. Levine and P. J. Steinhardt, “Quasicrystals: a new class of ordered structures,” Phys. Rev. Lett. 53, 2477–2480 (1984).
[CrossRef]

W. Gellermann, M. Kohmoto, B. Sutherland, and P. C. Taylor, “Localization of light waves in Fibonacci dielectric multilayers,” Phys. Rev. Lett. 72, 633–636 (1994).
[CrossRef]

Phys. Scr. (1)

J. M. Dubois, “The applied physics of quasicrystals,” Phys. Scr. T49, 17–23 (1993).
[CrossRef]

Other (5)

R. A. Dunlap, The Golden Ratio and Fibonacci Numbers (World Scientific, 1997).

A. Monnerot-Dumaine, “The Fibonacci word fractal,” 2009, http://hal.archives-ouvertes.fr/hal-00367972/fr .

C. Palmer and E. Loewen, Diffraction Grating Handbook (Newport Corporation, 2005).

M. Born, Principles of Optics (Pergamon, 1980).

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

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Figures (8)

Fig. 1.
Fig. 1.

(a) FbGs for generations n=5, 6, 7, and 8, and (b) their complements.

Fig. 2.
Fig. 2.

Amplitude of the diffraction pattern for generations n=6, 7, and 8 of the FbG (left panel) and the corresponding complements (right panel).

Fig. 3.
Fig. 3.

(a) Scaled amplitudes for generations n=10, 11, and 12 of the FbG. A part of the secondary structure is magnified in the inset. (b) Scaled amplitude profiles (|An(f)|2) for generations n=1, 2, and 3 of a Cantor grating. (c) Comparison of the first secondary band of the diffraction pattern corresponding to the n=7 FbG and CFbG.

Fig. 4.
Fig. 4.

Evaluation of fractal dimension on a log–log plot from the integrated structure factor of the PB as a function of frequency for the FbG and its complement (CFbG). The slope of the best-fit line yields df0.88.

Fig. 5.
Fig. 5.

Reconstruction of the FbG (left panel) and the CFbG (right panel) for generation n=8. The top row shows the gratings. The middle row shows the reconstruction from the PB of the corresponding diffraction patterns. The bottom row shows the reconstruction from the first secondary band of the corresponding diffraction patterns.

Fig. 6.
Fig. 6.

Schematic of the “4f configuration” and the corresponding experimental arrangement.

Fig. 7.
Fig. 7.

Reconstructions for the FbG (left panel) and the CFbG (right panel) for n=7 using the “4f configuration.” The gratings displayed on the SLM are shown in the top row. The middle row shows the reconstructions using the entire diffraction pattern corresponding to the FbG or the CFbG as the case may be. The bottom row shows reconstructions from the secondary band of the appropriate diffraction pattern.

Fig. 8.
Fig. 8.

(a) Top row shows the FbG for generation n=8 and the next rows show two distinct disorder realizations with disorder D=50%. (b) CFbG for generation n=8 is shown in the top row and the next rows show two distinct disorder realizations with D=50%.

Tables (2)

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Table 1. Power Diffracted from the PB and Different Secondary Bands (SB) of the FbG, the Complement FbG (CFbG), and the Cantor Grating

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Table 2. Fractal Dimension df as a Function of Disorder in FbGs

Equations (27)

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G1F(x)=R1(x)*δ(x0),
G2F(x)=R2(x)*δ(x+a2),
G3F(x)=R3(x)*[δ(x+2a3)+δ(x2a3)],
G4F(x)=R4(x)*[δ(x+4a5)]+R42(x)*[δ(xa5)],
G4F(x)=R4(x)*[δ(x+4a5)+δ(x0)+δ(x2a5)].
xn(k)={[Nn2(i)±aFn1]/Fn,i=1,2,,Fn3,Nn3(i)/Fn,i=1,2,,Fn4.
GnF(x)=Rn(x)*Δn(x),n4,
G0F(x)=R0(x)*δ(x0),
G2F(x)=R2(x)*δ(xa2),
G3F(x)=R3(x)*δ(x0),
G4F(x)=R4(x)*[δ(x+2a5)+δ(x4a5)],
GnF(x)=Rn(x)*Δn(x),n5,
A(f)=FT{G(x)}=dxei2πfxG(x),
A1F(f)=2asinc(2πaf),
A2F(f)=asinc(πaf)eiπfa,
A3F(f)=2a3sinc(2πaf3)[ei4πfa/3+ei4πfa/3],
AnF(f)=FT{GnF(x)}=dxei2πfxGnF(x),n4=2aFnsinc(2πafFn)k=1Fn1ei2πfxn(k).
A0F(f)=2asinc(2πaf),
A2F(f)=asinc(πaf)eiπfa,
A3F(f)=2a3sinc(2πaf3),
A4F(f)=2a5sinc(2πaf5)[ei4πfa/5+ei8πfa/5].
AnF(f)=FT{GnF(x)}=dxei2πfxGnF(x),n5=2aFnsinc(2πafFn)k=1Fn2ei2πfxn(k).
AnF(F)(f)=An1F(F)(fϕ).
InF(f)=[2aFnsinc(2πafFn)]2|k=1Fn1ei2πfxn(k)|2,n4.
InF(f)=[2aFnsinc(2πafFn)]2|k=1Fn2ei2πfxn(k)|2,n5.
S(f)¯=1Δfff+ΔfdqS(q)fdf.
P¯=f1f2I(f)df,

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