Abstract

A general fast numerical algorithm for chirp transforms is developed by using two fast Fourier transforms and employing an analytical kernel. This new algorithm unifies the calculations of arbitrary real-order fractional Fourier transforms and Fresnel diffraction. Its computational complexity is better than a fast convolution method using Fourier transforms. Furthermore, one can freely choose the sampling resolutions in both x and u space and zoom in on any portion of the data of interest. Computational results are compared with analytical ones. The errors are essentially limited by the accuracy of the fast Fourier transforms and are higher than the order 10-12 for most cases. As an example of its application to scalar diffraction, this algorithm can be used to calculate near-field patterns directly behind the aperture, 0z<d2/λ. It compensates another algorithm for Fresnel diffraction that is limited to z>d2/λN [J. Opt. Soc. Am. A 15, 2111 (1998)]. Experimental results from waveguide-output microcoupler diffraction are in good agreement with the calculations.

© 2000 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. A. E. Siegman, Lasers (Mill Valley, Calif., 1986).
  2. S. A. Collins, “Lens-system diffraction integral written in forms of matrix optics,” J. Opt. Soc. Am. 60, 1168–1177 (1970).
    [CrossRef]
  3. S. Abe, J. T. Sheridan, “Optical operations on wave functions as the Abelian subgroup of the special affine Fourier transformation,” Opt. Lett. 19, 1801–1803 (1994).
    [CrossRef]
  4. S. Abe, J. T. Sheridan, “Almost-Fourier and almost-Fresnel transformations,” Opt. Commun. 113, 385–388 (1995).
    [CrossRef]
  5. S. Abe, J. T. Sheridan, “Generalization of the fractional Fourier transformation to an arbitrary linear lossless transformation: an operator approach,” J. Phys. A 27, 4179–4187 (1994).
    [CrossRef]
  6. X. Deng, Y. Li, D. Fan, Y. Qiu, “Propagation of paraxial flattened Gaussian beams in a general optical system,” Opt. Commun. 140, 226–230 (1997).
    [CrossRef]
  7. A. Katzir, A. C. Livanos, J. B. Shellan, A. Yariv, “Chirped gratings in integrated optics,” IEEE J. Quantum Electron. QE-13, 296–304 (1977).
    [CrossRef]
  8. G. W. Forbes, “Validity of the Fresnel approximation in the diffraction of collimated beams,” J. Opt. Soc. Am. A 13, 1816–1826 (1996).
    [CrossRef]
  9. B. Bihari, J. Gan, L. Wu, Y. Liu, S. Tang, R. T. Chen, “Optical clock distribution in supercomputers using polyimide-based waveguides,” in Optoelectronic Interconnects VI, J. P. Bristow, S. Tang, eds., Proc. SPIE3632, 123–133 (1999).
    [CrossRef]
  10. J. W. Goodman, F. I. Leonberger, S. Y. Kung, R. A. Athale, “Optical interconnections for VLSI systems,” Proc. IEEE 72, 850–866 (1984).
    [CrossRef]
  11. P. Cinato, K. C. Young, “Optical interconnections within multichip modules,” Opt. Eng. 32, 852–860 (1993).
    [CrossRef]
  12. J. Jahns, “Planar integrated free-space optics,” in Micro-Optics: Elements, Systems and Applications, H. P. Herzig, ed. (Taylor & Francis, London, UK, 1997), pp. 179–198; W. Singer, K. H. Brenner, “Stacked micro-optical systems,” pp. 199–221.
  13. V. Namias, “The fractional order Fourier transform and its application to quantum mechanics,” J. Inst. Math. Appl. 25, 241–265 (1980).
    [CrossRef]
  14. C. Kopp, P. Meyrueis, “Near-field Fresnel diffraction: improvement of a numerical propagator,” Opt. Commun. 158, 7–10 (1998).
    [CrossRef]
  15. S. B. Tucker, J. Ojeda-Castañeda, W. T. Cathey, “Matrix description of near-field diffraction and the fractional Fourier transform,” J. Opt. Soc. Am. A 16, 316–322 (1999).
    [CrossRef]
  16. V. Arizón, J. Ojeda-Castañeda, “Fresnel diffraction of substructured gratings: matrix description,” Opt. Lett. 20, 118–120 (1995).
    [CrossRef]
  17. X. Deng, Y. Li, D. Fan, Y. Qiu, “A fast algorithm for fractional Fourier transforms,” Opt. Commun. 138, 270–274 (1997).
    [CrossRef]
  18. F. J. Marinho, L. M. Bernardo, “Numerical calculation of fractional Fourier transforms with a single fast-Fourier-transform algorithm,” J. Opt. Soc. Am. A 15, 2111–2116 (1998).
    [CrossRef]
  19. H. Haman, J. L. de Bougrenet de la Tocnaye, “Efficient Fresnel-transform algorithm based on fractional Fresnel diffraction,” J. Opt. Soc. Am. A 12, 1920–1931 (1995).
    [CrossRef]
  20. B. W. Dickinson, K. Steigletz, “Eigenvectors and functions of the discrete Fourier transform,” IEEE Trans. Acoust. Speech Signal Process. ASSP-30, 25–31 (1982).
    [CrossRef]
  21. S. C. Pei, M. H. Yeh, “Improved discrete fractional Fourier transform,” Opt. Lett. 22, 1047–1049 (1997).
    [CrossRef] [PubMed]
  22. G. S. Agarwal, R. Simon, “A simple relation of fractional Fourier transform and relation to harmonic oscillator Green’s function,” Opt. Commun. 110, 23–26 (1994).
    [CrossRef]
  23. H. M. Ozaktas, D. Mendlovic, “Fourier transforms of fractional order and their optical implementation,” Opt. Commun. 101, 163–169 (1993).
    [CrossRef]
  24. A. W. Lohmann, “A fake zoom lens for fractional Fourier experiments,” Opt. Commun. 115, 437–443 (1995).
    [CrossRef]
  25. S. Liu, J. Xu, Y. Zhang, L. Chen, C. Li, “General optical implementations of fractional Fourier transforms,” Opt. Lett. 20, 1053–1055 (1995).
    [CrossRef] [PubMed]
  26. L. M. Bernardo, O. D. D. Soares, “Fractional Fourier transform and imaging,” J. Opt. Soc. Am. A 11, 2622–2626 (1994).
    [CrossRef]
  27. P. Pellat-Finet, “Fresnel diffraction and the fractional Fourier transform,” Opt. Lett. 19, 1388–1390 (1994).
    [CrossRef] [PubMed]
  28. R. G. Dorsch, A. W. Lohmann, Y. Bitran, D. Mendlovic, H. M. Ozaktas, “Chirp filtering in the fractional Fourier domain,” Appl. Opt. 33, 7599–7602 (1994).
    [CrossRef] [PubMed]
  29. D. Mendlovic, H. M. Ozaktas, “Fractional Fourier transforms and their optical implementations: I,” J. Opt. Soc. Am. A 10, 1875–1881 (1993).
    [CrossRef]
  30. H. M. Ozaktas, D. Mendlovic, “Fractional Fourier transforms and their optical implementations. II,” J. Opt. Soc. Am. A 10, 2522–2531 (1993).
    [CrossRef]
  31. S. Granieri, O. Trabocchi, E. E. Sicre, “Fractional Fourier transform applied to spatial filtering in the Fresnel domain,” Opt. Commun. 119, 275–278 (1995).
    [CrossRef]
  32. H. M. Ozaktas, B. Barshan, D. Mendlovic, L. Onural, “Convolution, filtering, and multiplexing in fractional Fourier domains and their relation to chirp and wavelet transforms,” J. Opt. Soc. Am. A 11, 547–559 (1994).
    [CrossRef]
  33. D. Mendlovic, Y. Bitran, R. G. Dorsh, A. W. Lohmann, “Optical fractional correlation: experimental results,” J. Opt. Soc. Am. A 12, 1665–1670 (1995).
    [CrossRef]
  34. D. Mendlovic, Z. Zalevsky, R. G. Dorsh, Y. Bitran, A. W. Lohmann, H. M. Ozaktas, “New signal representation based on the fractional Fourier transform: definitions,” J. Opt. Soc. Am. A 12, 2424–2431 (1995).
    [CrossRef]
  35. H. M. Ozaktas, D. Mendlovic, “Fractional Fourier optics,” J. Opt. Soc. Am. A 12, 743–751 (1995).
    [CrossRef]
  36. P. Pellat-Finet, G. Bonnet, “Fractional-order Fourier transform and Fourier optics,” Opt. Commun. 111, 141–154 (1994).
    [CrossRef]
  37. A. I. Zayed, “On the relationship between Fourier transform and fractional Fourier transform,” IEEE Signal Process. Lett. 3, 310–311 (1996).
    [CrossRef]
  38. A. W. Lohmann, B. H. Soffer, “Relationships between the Radon–Wigner and fractional Fourier transforms,” J. Opt. Soc. Am. A 11, 1789–1801 (1994).
    [CrossRef]
  39. S. Roose, B. Brichau, E. W. Stijns, “An efficient interpolation algorithm for Fourier and diffractive optics,” Opt. Commun. 97, 312–318 (1993).
    [CrossRef]
  40. L. R. Rabiner, B. Gold, Theory and Applications of Digital Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1975).
  41. M. Sypek, “Light propagation in the Fresnel region: new numerical approach,” Opt. Commun. 116, 43–48 (1995).
    [CrossRef]
  42. P. A. Béleuger, “Beam propagation and the ABCD ray matrix,” Opt. Lett. 16, 196–198 (1991).
    [CrossRef]
  43. A. Yariv, Optical Electronics (CBC College Publishing, New York, 1985), Chap. 2, pp. 17–52.
  44. Y. B. Karasik, “Expression of the kernel of a fractional Fourier transform in elementary functions,” Opt. Lett. 19, 769–771 (1994).
    [CrossRef] [PubMed]
  45. For κ=2n the kernel becomes a Dirac delta function, BFrFT(2n)(u, x)=δ[u-(-1)nx], and the transform is straightforward and needs no further calculation.
  46. To clarify the later results and be self-consistent, we will adopt the following definition of Fourier transform in the discrete form: Given f(x), its Fourier transform is  g(u)≡F{f(x)}=∫-∞∞f(x)exp(-2πixu)dx, which could be numerically approximated by g(u)=F{f(x)}≈gk=∑l=0Nx-1fn exp[-2πi(n-Nx/2)(k-Nu/2)δxδu]δx. We have assumed that the Fourier transform will map f(x) from x∈[-(Nxδx)/2,+(Nxδx)/2] to g(u) in the domain u∈[-(Nuδu)/2,+(Nuδu)/2]. If gk is given by a standard DFT or FFT, however, the mapped domain will be u∈[-1/(2δx),+1/(2δx)] owing to the sampling condition δxδu≡1/Nx.
  47. A. W. Lohmann, R. G. Dorsch, D. Mendlovic, Z. Zalevshy, C. Ferria, “Space–bandwidth product of optical signals and systems,” J. Opt. Soc. Am. A 13, 470–473 (1996).
    [CrossRef]
  48. I. S. Granshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1985).
  49. H. M. Ozaktas, H. Urey, “Space–bandwidth product of conventional Fourier transforming systems,” Opt. Commun. 105, 1–6 (1994).
  50. L. Austander, F. A. Grünbaum, “The Fourier transform and the inverse Fourier transform,” Inverse Probl. 5, 149–164 (1989).
    [CrossRef]
  51. W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C: the Art of Scientific Computing (Cambridge U. Press, Cambridge, UK, 1992).
  52. The width of the mask used to fabricate the 10-µm-thick waveguide is 50 µm. Owing to the highly isotropic etching, the final width of the polyimide waveguide can be varied in the range of 60∼70 µm depending on the precise control of experiment environments. In our simulation, we used a typical value of 65 µm.
  53. F. Depasse, M. A. Paesler, D. Courjon, J. M. Vigoureux, “Huygens–Fresnel principle in the near field,” Opt. Lett. 20, 234–236 (1995).
    [CrossRef] [PubMed]
  54. A. Yariv, Optical Electronics (CBC College Publishing, New York, 1985).

1999 (1)

1998 (2)

1997 (3)

X. Deng, Y. Li, D. Fan, Y. Qiu, “A fast algorithm for fractional Fourier transforms,” Opt. Commun. 138, 270–274 (1997).
[CrossRef]

X. Deng, Y. Li, D. Fan, Y. Qiu, “Propagation of paraxial flattened Gaussian beams in a general optical system,” Opt. Commun. 140, 226–230 (1997).
[CrossRef]

S. C. Pei, M. H. Yeh, “Improved discrete fractional Fourier transform,” Opt. Lett. 22, 1047–1049 (1997).
[CrossRef] [PubMed]

1996 (3)

1995 (11)

H. M. Ozaktas, D. Mendlovic, “Fractional Fourier optics,” J. Opt. Soc. Am. A 12, 743–751 (1995).
[CrossRef]

D. Mendlovic, Y. Bitran, R. G. Dorsh, A. W. Lohmann, “Optical fractional correlation: experimental results,” J. Opt. Soc. Am. A 12, 1665–1670 (1995).
[CrossRef]

H. Haman, J. L. de Bougrenet de la Tocnaye, “Efficient Fresnel-transform algorithm based on fractional Fresnel diffraction,” J. Opt. Soc. Am. A 12, 1920–1931 (1995).
[CrossRef]

D. Mendlovic, Z. Zalevsky, R. G. Dorsh, Y. Bitran, A. W. Lohmann, H. M. Ozaktas, “New signal representation based on the fractional Fourier transform: definitions,” J. Opt. Soc. Am. A 12, 2424–2431 (1995).
[CrossRef]

V. Arizón, J. Ojeda-Castañeda, “Fresnel diffraction of substructured gratings: matrix description,” Opt. Lett. 20, 118–120 (1995).
[CrossRef]

F. Depasse, M. A. Paesler, D. Courjon, J. M. Vigoureux, “Huygens–Fresnel principle in the near field,” Opt. Lett. 20, 234–236 (1995).
[CrossRef] [PubMed]

S. Liu, J. Xu, Y. Zhang, L. Chen, C. Li, “General optical implementations of fractional Fourier transforms,” Opt. Lett. 20, 1053–1055 (1995).
[CrossRef] [PubMed]

M. Sypek, “Light propagation in the Fresnel region: new numerical approach,” Opt. Commun. 116, 43–48 (1995).
[CrossRef]

S. Granieri, O. Trabocchi, E. E. Sicre, “Fractional Fourier transform applied to spatial filtering in the Fresnel domain,” Opt. Commun. 119, 275–278 (1995).
[CrossRef]

A. W. Lohmann, “A fake zoom lens for fractional Fourier experiments,” Opt. Commun. 115, 437–443 (1995).
[CrossRef]

S. Abe, J. T. Sheridan, “Almost-Fourier and almost-Fresnel transformations,” Opt. Commun. 113, 385–388 (1995).
[CrossRef]

1994 (11)

S. Abe, J. T. Sheridan, “Generalization of the fractional Fourier transformation to an arbitrary linear lossless transformation: an operator approach,” J. Phys. A 27, 4179–4187 (1994).
[CrossRef]

G. S. Agarwal, R. Simon, “A simple relation of fractional Fourier transform and relation to harmonic oscillator Green’s function,” Opt. Commun. 110, 23–26 (1994).
[CrossRef]

P. Pellat-Finet, G. Bonnet, “Fractional-order Fourier transform and Fourier optics,” Opt. Commun. 111, 141–154 (1994).
[CrossRef]

H. M. Ozaktas, H. Urey, “Space–bandwidth product of conventional Fourier transforming systems,” Opt. Commun. 105, 1–6 (1994).

H. M. Ozaktas, B. Barshan, D. Mendlovic, L. Onural, “Convolution, filtering, and multiplexing in fractional Fourier domains and their relation to chirp and wavelet transforms,” J. Opt. Soc. Am. A 11, 547–559 (1994).
[CrossRef]

A. W. Lohmann, B. H. Soffer, “Relationships between the Radon–Wigner and fractional Fourier transforms,” J. Opt. Soc. Am. A 11, 1789–1801 (1994).
[CrossRef]

L. M. Bernardo, O. D. D. Soares, “Fractional Fourier transform and imaging,” J. Opt. Soc. Am. A 11, 2622–2626 (1994).
[CrossRef]

R. G. Dorsch, A. W. Lohmann, Y. Bitran, D. Mendlovic, H. M. Ozaktas, “Chirp filtering in the fractional Fourier domain,” Appl. Opt. 33, 7599–7602 (1994).
[CrossRef] [PubMed]

Y. B. Karasik, “Expression of the kernel of a fractional Fourier transform in elementary functions,” Opt. Lett. 19, 769–771 (1994).
[CrossRef] [PubMed]

P. Pellat-Finet, “Fresnel diffraction and the fractional Fourier transform,” Opt. Lett. 19, 1388–1390 (1994).
[CrossRef] [PubMed]

S. Abe, J. T. Sheridan, “Optical operations on wave functions as the Abelian subgroup of the special affine Fourier transformation,” Opt. Lett. 19, 1801–1803 (1994).
[CrossRef]

1993 (5)

D. Mendlovic, H. M. Ozaktas, “Fractional Fourier transforms and their optical implementations: I,” J. Opt. Soc. Am. A 10, 1875–1881 (1993).
[CrossRef]

H. M. Ozaktas, D. Mendlovic, “Fractional Fourier transforms and their optical implementations. II,” J. Opt. Soc. Am. A 10, 2522–2531 (1993).
[CrossRef]

S. Roose, B. Brichau, E. W. Stijns, “An efficient interpolation algorithm for Fourier and diffractive optics,” Opt. Commun. 97, 312–318 (1993).
[CrossRef]

P. Cinato, K. C. Young, “Optical interconnections within multichip modules,” Opt. Eng. 32, 852–860 (1993).
[CrossRef]

H. M. Ozaktas, D. Mendlovic, “Fourier transforms of fractional order and their optical implementation,” Opt. Commun. 101, 163–169 (1993).
[CrossRef]

1991 (1)

1989 (1)

L. Austander, F. A. Grünbaum, “The Fourier transform and the inverse Fourier transform,” Inverse Probl. 5, 149–164 (1989).
[CrossRef]

1984 (1)

J. W. Goodman, F. I. Leonberger, S. Y. Kung, R. A. Athale, “Optical interconnections for VLSI systems,” Proc. IEEE 72, 850–866 (1984).
[CrossRef]

1982 (1)

B. W. Dickinson, K. Steigletz, “Eigenvectors and functions of the discrete Fourier transform,” IEEE Trans. Acoust. Speech Signal Process. ASSP-30, 25–31 (1982).
[CrossRef]

1980 (1)

V. Namias, “The fractional order Fourier transform and its application to quantum mechanics,” J. Inst. Math. Appl. 25, 241–265 (1980).
[CrossRef]

1977 (1)

A. Katzir, A. C. Livanos, J. B. Shellan, A. Yariv, “Chirped gratings in integrated optics,” IEEE J. Quantum Electron. QE-13, 296–304 (1977).
[CrossRef]

1970 (1)

Abe, S.

S. Abe, J. T. Sheridan, “Almost-Fourier and almost-Fresnel transformations,” Opt. Commun. 113, 385–388 (1995).
[CrossRef]

S. Abe, J. T. Sheridan, “Generalization of the fractional Fourier transformation to an arbitrary linear lossless transformation: an operator approach,” J. Phys. A 27, 4179–4187 (1994).
[CrossRef]

S. Abe, J. T. Sheridan, “Optical operations on wave functions as the Abelian subgroup of the special affine Fourier transformation,” Opt. Lett. 19, 1801–1803 (1994).
[CrossRef]

Agarwal, G. S.

G. S. Agarwal, R. Simon, “A simple relation of fractional Fourier transform and relation to harmonic oscillator Green’s function,” Opt. Commun. 110, 23–26 (1994).
[CrossRef]

Arizón, V.

Athale, R. A.

J. W. Goodman, F. I. Leonberger, S. Y. Kung, R. A. Athale, “Optical interconnections for VLSI systems,” Proc. IEEE 72, 850–866 (1984).
[CrossRef]

Austander, L.

L. Austander, F. A. Grünbaum, “The Fourier transform and the inverse Fourier transform,” Inverse Probl. 5, 149–164 (1989).
[CrossRef]

Barshan, B.

Béleuger, P. A.

Bernardo, L. M.

Bihari, B.

B. Bihari, J. Gan, L. Wu, Y. Liu, S. Tang, R. T. Chen, “Optical clock distribution in supercomputers using polyimide-based waveguides,” in Optoelectronic Interconnects VI, J. P. Bristow, S. Tang, eds., Proc. SPIE3632, 123–133 (1999).
[CrossRef]

Bitran, Y.

Bonnet, G.

P. Pellat-Finet, G. Bonnet, “Fractional-order Fourier transform and Fourier optics,” Opt. Commun. 111, 141–154 (1994).
[CrossRef]

Brichau, B.

S. Roose, B. Brichau, E. W. Stijns, “An efficient interpolation algorithm for Fourier and diffractive optics,” Opt. Commun. 97, 312–318 (1993).
[CrossRef]

Cathey, W. T.

Chen, L.

Chen, R. T.

B. Bihari, J. Gan, L. Wu, Y. Liu, S. Tang, R. T. Chen, “Optical clock distribution in supercomputers using polyimide-based waveguides,” in Optoelectronic Interconnects VI, J. P. Bristow, S. Tang, eds., Proc. SPIE3632, 123–133 (1999).
[CrossRef]

Cinato, P.

P. Cinato, K. C. Young, “Optical interconnections within multichip modules,” Opt. Eng. 32, 852–860 (1993).
[CrossRef]

Collins, S. A.

Courjon, D.

de Bougrenet de la Tocnaye, J. L.

Deng, X.

X. Deng, Y. Li, D. Fan, Y. Qiu, “A fast algorithm for fractional Fourier transforms,” Opt. Commun. 138, 270–274 (1997).
[CrossRef]

X. Deng, Y. Li, D. Fan, Y. Qiu, “Propagation of paraxial flattened Gaussian beams in a general optical system,” Opt. Commun. 140, 226–230 (1997).
[CrossRef]

Depasse, F.

Dickinson, B. W.

B. W. Dickinson, K. Steigletz, “Eigenvectors and functions of the discrete Fourier transform,” IEEE Trans. Acoust. Speech Signal Process. ASSP-30, 25–31 (1982).
[CrossRef]

Dorsch, R. G.

Dorsh, R. G.

Fan, D.

X. Deng, Y. Li, D. Fan, Y. Qiu, “A fast algorithm for fractional Fourier transforms,” Opt. Commun. 138, 270–274 (1997).
[CrossRef]

X. Deng, Y. Li, D. Fan, Y. Qiu, “Propagation of paraxial flattened Gaussian beams in a general optical system,” Opt. Commun. 140, 226–230 (1997).
[CrossRef]

Ferria, C.

Flannery, B. P.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C: the Art of Scientific Computing (Cambridge U. Press, Cambridge, UK, 1992).

Forbes, G. W.

Gan, J.

B. Bihari, J. Gan, L. Wu, Y. Liu, S. Tang, R. T. Chen, “Optical clock distribution in supercomputers using polyimide-based waveguides,” in Optoelectronic Interconnects VI, J. P. Bristow, S. Tang, eds., Proc. SPIE3632, 123–133 (1999).
[CrossRef]

Gold, B.

L. R. Rabiner, B. Gold, Theory and Applications of Digital Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1975).

Goodman, J. W.

J. W. Goodman, F. I. Leonberger, S. Y. Kung, R. A. Athale, “Optical interconnections for VLSI systems,” Proc. IEEE 72, 850–866 (1984).
[CrossRef]

Granieri, S.

S. Granieri, O. Trabocchi, E. E. Sicre, “Fractional Fourier transform applied to spatial filtering in the Fresnel domain,” Opt. Commun. 119, 275–278 (1995).
[CrossRef]

Granshteyn, I. S.

I. S. Granshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1985).

Grünbaum, F. A.

L. Austander, F. A. Grünbaum, “The Fourier transform and the inverse Fourier transform,” Inverse Probl. 5, 149–164 (1989).
[CrossRef]

Haman, H.

Jahns, J.

J. Jahns, “Planar integrated free-space optics,” in Micro-Optics: Elements, Systems and Applications, H. P. Herzig, ed. (Taylor & Francis, London, UK, 1997), pp. 179–198; W. Singer, K. H. Brenner, “Stacked micro-optical systems,” pp. 199–221.

Karasik, Y. B.

Katzir, A.

A. Katzir, A. C. Livanos, J. B. Shellan, A. Yariv, “Chirped gratings in integrated optics,” IEEE J. Quantum Electron. QE-13, 296–304 (1977).
[CrossRef]

Kopp, C.

C. Kopp, P. Meyrueis, “Near-field Fresnel diffraction: improvement of a numerical propagator,” Opt. Commun. 158, 7–10 (1998).
[CrossRef]

Kung, S. Y.

J. W. Goodman, F. I. Leonberger, S. Y. Kung, R. A. Athale, “Optical interconnections for VLSI systems,” Proc. IEEE 72, 850–866 (1984).
[CrossRef]

Leonberger, F. I.

J. W. Goodman, F. I. Leonberger, S. Y. Kung, R. A. Athale, “Optical interconnections for VLSI systems,” Proc. IEEE 72, 850–866 (1984).
[CrossRef]

Li, C.

Li, Y.

X. Deng, Y. Li, D. Fan, Y. Qiu, “A fast algorithm for fractional Fourier transforms,” Opt. Commun. 138, 270–274 (1997).
[CrossRef]

X. Deng, Y. Li, D. Fan, Y. Qiu, “Propagation of paraxial flattened Gaussian beams in a general optical system,” Opt. Commun. 140, 226–230 (1997).
[CrossRef]

Liu, S.

Liu, Y.

B. Bihari, J. Gan, L. Wu, Y. Liu, S. Tang, R. T. Chen, “Optical clock distribution in supercomputers using polyimide-based waveguides,” in Optoelectronic Interconnects VI, J. P. Bristow, S. Tang, eds., Proc. SPIE3632, 123–133 (1999).
[CrossRef]

Livanos, A. C.

A. Katzir, A. C. Livanos, J. B. Shellan, A. Yariv, “Chirped gratings in integrated optics,” IEEE J. Quantum Electron. QE-13, 296–304 (1977).
[CrossRef]

Lohmann, A. W.

Marinho, F. J.

Mendlovic, D.

A. W. Lohmann, R. G. Dorsch, D. Mendlovic, Z. Zalevshy, C. Ferria, “Space–bandwidth product of optical signals and systems,” J. Opt. Soc. Am. A 13, 470–473 (1996).
[CrossRef]

D. Mendlovic, Z. Zalevsky, R. G. Dorsh, Y. Bitran, A. W. Lohmann, H. M. Ozaktas, “New signal representation based on the fractional Fourier transform: definitions,” J. Opt. Soc. Am. A 12, 2424–2431 (1995).
[CrossRef]

D. Mendlovic, Y. Bitran, R. G. Dorsh, A. W. Lohmann, “Optical fractional correlation: experimental results,” J. Opt. Soc. Am. A 12, 1665–1670 (1995).
[CrossRef]

H. M. Ozaktas, D. Mendlovic, “Fractional Fourier optics,” J. Opt. Soc. Am. A 12, 743–751 (1995).
[CrossRef]

R. G. Dorsch, A. W. Lohmann, Y. Bitran, D. Mendlovic, H. M. Ozaktas, “Chirp filtering in the fractional Fourier domain,” Appl. Opt. 33, 7599–7602 (1994).
[CrossRef] [PubMed]

H. M. Ozaktas, B. Barshan, D. Mendlovic, L. Onural, “Convolution, filtering, and multiplexing in fractional Fourier domains and their relation to chirp and wavelet transforms,” J. Opt. Soc. Am. A 11, 547–559 (1994).
[CrossRef]

H. M. Ozaktas, D. Mendlovic, “Fractional Fourier transforms and their optical implementations. II,” J. Opt. Soc. Am. A 10, 2522–2531 (1993).
[CrossRef]

H. M. Ozaktas, D. Mendlovic, “Fourier transforms of fractional order and their optical implementation,” Opt. Commun. 101, 163–169 (1993).
[CrossRef]

D. Mendlovic, H. M. Ozaktas, “Fractional Fourier transforms and their optical implementations: I,” J. Opt. Soc. Am. A 10, 1875–1881 (1993).
[CrossRef]

Meyrueis, P.

C. Kopp, P. Meyrueis, “Near-field Fresnel diffraction: improvement of a numerical propagator,” Opt. Commun. 158, 7–10 (1998).
[CrossRef]

Namias, V.

V. Namias, “The fractional order Fourier transform and its application to quantum mechanics,” J. Inst. Math. Appl. 25, 241–265 (1980).
[CrossRef]

Ojeda-Castañeda, J.

Onural, L.

Ozaktas, H. M.

Paesler, M. A.

Pei, S. C.

Pellat-Finet, P.

P. Pellat-Finet, “Fresnel diffraction and the fractional Fourier transform,” Opt. Lett. 19, 1388–1390 (1994).
[CrossRef] [PubMed]

P. Pellat-Finet, G. Bonnet, “Fractional-order Fourier transform and Fourier optics,” Opt. Commun. 111, 141–154 (1994).
[CrossRef]

Press, W. H.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C: the Art of Scientific Computing (Cambridge U. Press, Cambridge, UK, 1992).

Qiu, Y.

X. Deng, Y. Li, D. Fan, Y. Qiu, “Propagation of paraxial flattened Gaussian beams in a general optical system,” Opt. Commun. 140, 226–230 (1997).
[CrossRef]

X. Deng, Y. Li, D. Fan, Y. Qiu, “A fast algorithm for fractional Fourier transforms,” Opt. Commun. 138, 270–274 (1997).
[CrossRef]

Rabiner, L. R.

L. R. Rabiner, B. Gold, Theory and Applications of Digital Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1975).

Roose, S.

S. Roose, B. Brichau, E. W. Stijns, “An efficient interpolation algorithm for Fourier and diffractive optics,” Opt. Commun. 97, 312–318 (1993).
[CrossRef]

Ryzhik, I. M.

I. S. Granshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1985).

Shellan, J. B.

A. Katzir, A. C. Livanos, J. B. Shellan, A. Yariv, “Chirped gratings in integrated optics,” IEEE J. Quantum Electron. QE-13, 296–304 (1977).
[CrossRef]

Sheridan, J. T.

S. Abe, J. T. Sheridan, “Almost-Fourier and almost-Fresnel transformations,” Opt. Commun. 113, 385–388 (1995).
[CrossRef]

S. Abe, J. T. Sheridan, “Generalization of the fractional Fourier transformation to an arbitrary linear lossless transformation: an operator approach,” J. Phys. A 27, 4179–4187 (1994).
[CrossRef]

S. Abe, J. T. Sheridan, “Optical operations on wave functions as the Abelian subgroup of the special affine Fourier transformation,” Opt. Lett. 19, 1801–1803 (1994).
[CrossRef]

Sicre, E. E.

S. Granieri, O. Trabocchi, E. E. Sicre, “Fractional Fourier transform applied to spatial filtering in the Fresnel domain,” Opt. Commun. 119, 275–278 (1995).
[CrossRef]

Siegman, A. E.

A. E. Siegman, Lasers (Mill Valley, Calif., 1986).

Simon, R.

G. S. Agarwal, R. Simon, “A simple relation of fractional Fourier transform and relation to harmonic oscillator Green’s function,” Opt. Commun. 110, 23–26 (1994).
[CrossRef]

Soares, O. D. D.

Soffer, B. H.

Steigletz, K.

B. W. Dickinson, K. Steigletz, “Eigenvectors and functions of the discrete Fourier transform,” IEEE Trans. Acoust. Speech Signal Process. ASSP-30, 25–31 (1982).
[CrossRef]

Stijns, E. W.

S. Roose, B. Brichau, E. W. Stijns, “An efficient interpolation algorithm for Fourier and diffractive optics,” Opt. Commun. 97, 312–318 (1993).
[CrossRef]

Sypek, M.

M. Sypek, “Light propagation in the Fresnel region: new numerical approach,” Opt. Commun. 116, 43–48 (1995).
[CrossRef]

Tang, S.

B. Bihari, J. Gan, L. Wu, Y. Liu, S. Tang, R. T. Chen, “Optical clock distribution in supercomputers using polyimide-based waveguides,” in Optoelectronic Interconnects VI, J. P. Bristow, S. Tang, eds., Proc. SPIE3632, 123–133 (1999).
[CrossRef]

Teukolsky, S. A.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C: the Art of Scientific Computing (Cambridge U. Press, Cambridge, UK, 1992).

Trabocchi, O.

S. Granieri, O. Trabocchi, E. E. Sicre, “Fractional Fourier transform applied to spatial filtering in the Fresnel domain,” Opt. Commun. 119, 275–278 (1995).
[CrossRef]

Tucker, S. B.

Urey, H.

H. M. Ozaktas, H. Urey, “Space–bandwidth product of conventional Fourier transforming systems,” Opt. Commun. 105, 1–6 (1994).

Vetterling, W. T.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C: the Art of Scientific Computing (Cambridge U. Press, Cambridge, UK, 1992).

Vigoureux, J. M.

Wu, L.

B. Bihari, J. Gan, L. Wu, Y. Liu, S. Tang, R. T. Chen, “Optical clock distribution in supercomputers using polyimide-based waveguides,” in Optoelectronic Interconnects VI, J. P. Bristow, S. Tang, eds., Proc. SPIE3632, 123–133 (1999).
[CrossRef]

Xu, J.

Yariv, A.

A. Katzir, A. C. Livanos, J. B. Shellan, A. Yariv, “Chirped gratings in integrated optics,” IEEE J. Quantum Electron. QE-13, 296–304 (1977).
[CrossRef]

A. Yariv, Optical Electronics (CBC College Publishing, New York, 1985).

A. Yariv, Optical Electronics (CBC College Publishing, New York, 1985), Chap. 2, pp. 17–52.

Yeh, M. H.

Young, K. C.

P. Cinato, K. C. Young, “Optical interconnections within multichip modules,” Opt. Eng. 32, 852–860 (1993).
[CrossRef]

Zalevshy, Z.

Zalevsky, Z.

Zayed, A. I.

A. I. Zayed, “On the relationship between Fourier transform and fractional Fourier transform,” IEEE Signal Process. Lett. 3, 310–311 (1996).
[CrossRef]

Zhang, Y.

Appl. Opt. (1)

IEEE J. Quantum Electron. (1)

A. Katzir, A. C. Livanos, J. B. Shellan, A. Yariv, “Chirped gratings in integrated optics,” IEEE J. Quantum Electron. QE-13, 296–304 (1977).
[CrossRef]

IEEE Signal Process. Lett. (1)

A. I. Zayed, “On the relationship between Fourier transform and fractional Fourier transform,” IEEE Signal Process. Lett. 3, 310–311 (1996).
[CrossRef]

IEEE Trans. Acoust. Speech Signal Process. (1)

B. W. Dickinson, K. Steigletz, “Eigenvectors and functions of the discrete Fourier transform,” IEEE Trans. Acoust. Speech Signal Process. ASSP-30, 25–31 (1982).
[CrossRef]

Inverse Probl. (1)

L. Austander, F. A. Grünbaum, “The Fourier transform and the inverse Fourier transform,” Inverse Probl. 5, 149–164 (1989).
[CrossRef]

J. Inst. Math. Appl. (1)

V. Namias, “The fractional order Fourier transform and its application to quantum mechanics,” J. Inst. Math. Appl. 25, 241–265 (1980).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (13)

A. W. Lohmann, R. G. Dorsch, D. Mendlovic, Z. Zalevshy, C. Ferria, “Space–bandwidth product of optical signals and systems,” J. Opt. Soc. Am. A 13, 470–473 (1996).
[CrossRef]

G. W. Forbes, “Validity of the Fresnel approximation in the diffraction of collimated beams,” J. Opt. Soc. Am. A 13, 1816–1826 (1996).
[CrossRef]

H. M. Ozaktas, D. Mendlovic, “Fractional Fourier optics,” J. Opt. Soc. Am. A 12, 743–751 (1995).
[CrossRef]

D. Mendlovic, Y. Bitran, R. G. Dorsh, A. W. Lohmann, “Optical fractional correlation: experimental results,” J. Opt. Soc. Am. A 12, 1665–1670 (1995).
[CrossRef]

H. Haman, J. L. de Bougrenet de la Tocnaye, “Efficient Fresnel-transform algorithm based on fractional Fresnel diffraction,” J. Opt. Soc. Am. A 12, 1920–1931 (1995).
[CrossRef]

D. Mendlovic, Z. Zalevsky, R. G. Dorsh, Y. Bitran, A. W. Lohmann, H. M. Ozaktas, “New signal representation based on the fractional Fourier transform: definitions,” J. Opt. Soc. Am. A 12, 2424–2431 (1995).
[CrossRef]

H. M. Ozaktas, B. Barshan, D. Mendlovic, L. Onural, “Convolution, filtering, and multiplexing in fractional Fourier domains and their relation to chirp and wavelet transforms,” J. Opt. Soc. Am. A 11, 547–559 (1994).
[CrossRef]

A. W. Lohmann, B. H. Soffer, “Relationships between the Radon–Wigner and fractional Fourier transforms,” J. Opt. Soc. Am. A 11, 1789–1801 (1994).
[CrossRef]

L. M. Bernardo, O. D. D. Soares, “Fractional Fourier transform and imaging,” J. Opt. Soc. Am. A 11, 2622–2626 (1994).
[CrossRef]

S. B. Tucker, J. Ojeda-Castañeda, W. T. Cathey, “Matrix description of near-field diffraction and the fractional Fourier transform,” J. Opt. Soc. Am. A 16, 316–322 (1999).
[CrossRef]

F. J. Marinho, L. M. Bernardo, “Numerical calculation of fractional Fourier transforms with a single fast-Fourier-transform algorithm,” J. Opt. Soc. Am. A 15, 2111–2116 (1998).
[CrossRef]

D. Mendlovic, H. M. Ozaktas, “Fractional Fourier transforms and their optical implementations: I,” J. Opt. Soc. Am. A 10, 1875–1881 (1993).
[CrossRef]

H. M. Ozaktas, D. Mendlovic, “Fractional Fourier transforms and their optical implementations. II,” J. Opt. Soc. Am. A 10, 2522–2531 (1993).
[CrossRef]

J. Phys. A (1)

S. Abe, J. T. Sheridan, “Generalization of the fractional Fourier transformation to an arbitrary linear lossless transformation: an operator approach,” J. Phys. A 27, 4179–4187 (1994).
[CrossRef]

Opt. Commun. (12)

X. Deng, Y. Li, D. Fan, Y. Qiu, “Propagation of paraxial flattened Gaussian beams in a general optical system,” Opt. Commun. 140, 226–230 (1997).
[CrossRef]

C. Kopp, P. Meyrueis, “Near-field Fresnel diffraction: improvement of a numerical propagator,” Opt. Commun. 158, 7–10 (1998).
[CrossRef]

S. Abe, J. T. Sheridan, “Almost-Fourier and almost-Fresnel transformations,” Opt. Commun. 113, 385–388 (1995).
[CrossRef]

S. Granieri, O. Trabocchi, E. E. Sicre, “Fractional Fourier transform applied to spatial filtering in the Fresnel domain,” Opt. Commun. 119, 275–278 (1995).
[CrossRef]

S. Roose, B. Brichau, E. W. Stijns, “An efficient interpolation algorithm for Fourier and diffractive optics,” Opt. Commun. 97, 312–318 (1993).
[CrossRef]

X. Deng, Y. Li, D. Fan, Y. Qiu, “A fast algorithm for fractional Fourier transforms,” Opt. Commun. 138, 270–274 (1997).
[CrossRef]

G. S. Agarwal, R. Simon, “A simple relation of fractional Fourier transform and relation to harmonic oscillator Green’s function,” Opt. Commun. 110, 23–26 (1994).
[CrossRef]

H. M. Ozaktas, D. Mendlovic, “Fourier transforms of fractional order and their optical implementation,” Opt. Commun. 101, 163–169 (1993).
[CrossRef]

A. W. Lohmann, “A fake zoom lens for fractional Fourier experiments,” Opt. Commun. 115, 437–443 (1995).
[CrossRef]

M. Sypek, “Light propagation in the Fresnel region: new numerical approach,” Opt. Commun. 116, 43–48 (1995).
[CrossRef]

P. Pellat-Finet, G. Bonnet, “Fractional-order Fourier transform and Fourier optics,” Opt. Commun. 111, 141–154 (1994).
[CrossRef]

H. M. Ozaktas, H. Urey, “Space–bandwidth product of conventional Fourier transforming systems,” Opt. Commun. 105, 1–6 (1994).

Opt. Eng. (1)

P. Cinato, K. C. Young, “Optical interconnections within multichip modules,” Opt. Eng. 32, 852–860 (1993).
[CrossRef]

Opt. Lett. (8)

Proc. IEEE (1)

J. W. Goodman, F. I. Leonberger, S. Y. Kung, R. A. Athale, “Optical interconnections for VLSI systems,” Proc. IEEE 72, 850–866 (1984).
[CrossRef]

Other (11)

B. Bihari, J. Gan, L. Wu, Y. Liu, S. Tang, R. T. Chen, “Optical clock distribution in supercomputers using polyimide-based waveguides,” in Optoelectronic Interconnects VI, J. P. Bristow, S. Tang, eds., Proc. SPIE3632, 123–133 (1999).
[CrossRef]

J. Jahns, “Planar integrated free-space optics,” in Micro-Optics: Elements, Systems and Applications, H. P. Herzig, ed. (Taylor & Francis, London, UK, 1997), pp. 179–198; W. Singer, K. H. Brenner, “Stacked micro-optical systems,” pp. 199–221.

L. R. Rabiner, B. Gold, Theory and Applications of Digital Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1975).

A. E. Siegman, Lasers (Mill Valley, Calif., 1986).

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C: the Art of Scientific Computing (Cambridge U. Press, Cambridge, UK, 1992).

The width of the mask used to fabricate the 10-µm-thick waveguide is 50 µm. Owing to the highly isotropic etching, the final width of the polyimide waveguide can be varied in the range of 60∼70 µm depending on the precise control of experiment environments. In our simulation, we used a typical value of 65 µm.

For κ=2n the kernel becomes a Dirac delta function, BFrFT(2n)(u, x)=δ[u-(-1)nx], and the transform is straightforward and needs no further calculation.

To clarify the later results and be self-consistent, we will adopt the following definition of Fourier transform in the discrete form: Given f(x), its Fourier transform is  g(u)≡F{f(x)}=∫-∞∞f(x)exp(-2πixu)dx, which could be numerically approximated by g(u)=F{f(x)}≈gk=∑l=0Nx-1fn exp[-2πi(n-Nx/2)(k-Nu/2)δxδu]δx. We have assumed that the Fourier transform will map f(x) from x∈[-(Nxδx)/2,+(Nxδx)/2] to g(u) in the domain u∈[-(Nuδu)/2,+(Nuδu)/2]. If gk is given by a standard DFT or FFT, however, the mapped domain will be u∈[-1/(2δx),+1/(2δx)] owing to the sampling condition δxδu≡1/Nx.

A. Yariv, Optical Electronics (CBC College Publishing, New York, 1985).

A. Yariv, Optical Electronics (CBC College Publishing, New York, 1985), Chap. 2, pp. 17–52.

I. S. Granshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1985).

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 (8)

Fig. 1
Fig. 1

Schematic top view of the H-tree waveguide used in optical interconnections. The side view of one of its 48 microcouplers is illustrated at the right. Near-field diffraction patterns of the outcoupling are critical to the coupling of the optical signal to detectors in successive layers and hence to the performance of the optical interconnections.

Fig. 2
Fig. 2

(a) Errors ep=max[e(x)] of the ChT’s of a Gaussian function compared with the analytical transforms for different zoom factors ζzoom. (b) Typical error curve e(x) (ζzoom=0.5380).

Fig. 3
Fig. 3

(a) Errors max(e) of the ChT’s of a rect(x/a) function compared with the analytical transforms for different zoom factors ζzoom. (b) Typical error curve e(x)(ζzoom=0.5380).

Fig. 4
Fig. 4

Typical error curve of the FFT for (a) a Gaussian function, (b) rect(x/a).

Fig. 5
Fig. 5

Fresnel diffraction patterns of different zoom factors at different distances behind the aperture. (a) Layout of the optical system. Diffraction patterns for (b) ζzoom=0.325, l2=4500 µm and (c) ζzoom=0.250, l2=5000 µm.

Fig. 6
Fig. 6

First ten normalized eigenmodes in the 10 µm direction of the outcoupling points of the 1-to-48 fan-out H-tree polyimide waveguide. A slab waveguide model is used in the calculation because the width or the thickness of the waveguide is much larger than the wavelength.

Fig. 7
Fig. 7

Near-field outcoupling patterns from the waveguide as depicted in Fig. 1. Simulated results at (a) z=10-8 µm, ζzoom=1.000; (b) z=100 µm, ζzoom=1.120; (c) z=1000 µm, ζzoom=1.600; (d) z=5000 µm, ζzoom=4.000. Experimental images corresponding to (e) z=100 µm, (f) z=1000 µm, and (g) z=5000 µm.

Fig. 8
Fig. 8

FrFT’s of a rectangular aperture, ζzoom=1.0. (a) κ=0.1; (b) κ=0.4; (c) κ=0.7; (d) κ=0.9.

Equations (44)

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

gchp(u)Cp{f(x)}=-+f(x)Bchp(u, x)dx,
Bchp(u, x)A(p)exp[-2πil=1N(αlxl2+βlxlul+γlul2)],
u, xRN,
BFnTM(u, x)-iBλexpik2B(Ax2-2xu+Du2),
MABCD,det M=±1.
p(α, β, γ)=-k4πB(A,-2, D),
A(p)=-iBλ.
BFrFTκ(u, x)=exp[-i(ϕs-ϕ)/2](2π|sin(ϕ)|)1/2×expi(u2+x2)2×cot ϕ-ux csc ϕ,
κ2n,
p(α, β, γ)=-14π[cot(ϕ),-2 csc(ϕ), cot(ϕ)],
A(p)=exp[-i(ϕs-ϕ)/2](2π|sin(ϕ)|)1/2.
xnδxn-Nx2δx,n=0, 1, 2, Nx-1,
ukδuk-Nu2δu,k=0, 1, 2, Nu-1,
gkp=gchp(kδu)n=0Nx-1fnBDp(k, n)δx,
BDp(k, n)A(p)exp{-2πi[α(nδx)2+βnkδxδu+γ(kδu)2]},
gchp(u)g(v)F{fm(x)}=F{f(x)exp(-2πiαx2)},
δx12vm12βum.
gchp(u-c)g(v-βc)=F{fm(x)exp[2πiβcx]}.
nk=-12[(n-k)2-n2-k2],
BDp(k, n)=A(p)PnpQkpBconvolp(n-k),
Pnp=exp-2πiαδx2-β2δxδu(n)2,
Qkp=exp-2πiγδu2-β2δxδu(k)2,
Bconvolp(n-k)=exp[πiβ(n-k)2δxδu].
gkp=A(p)δxn=0Nx-1fn(m)Bconvolp(n-k)Qkp,
gkp=A(p){F-1{F{fn(m)}F{δxBconvolp(n)}}Qkp.
F{exp(-ax2)}=πa exp-(πu)2a,R{a}0,
Bˆconvolp(k)=F{δxBconvolp(n)}
=iδxβδu exp-πi u2βδxδu,Riβ0.
Bˆconvolp(k)=iδxβδu exp-πi 1βδxδukLx2,
kk-Lx2,k=0, 1, 2,, Lx-1.
gkp=A(p){F-1{F{fn(m)}Bˆconvolp(k)}Qkp}.
1βδxδu1Lx21.
Z{fn(m)}(n)
=00n<(Lx-Nx)2fn-(Lx-Nx)/2(m)(Lx-Nx)2n<(Lx+Nx)20(Lx+Nx)2nLx-1.
C(l)=F-{Z{fn(m)}}Bˆconvolp(k)},
l=0, 1, 2,, Lx-1.
gkp=A(p)QkpCk+Lx-Nu2,
ζzoom=δuδx,ζzoom>0,
e=|gkp-F{exp(-ax2)}|max(|gkp|)
udiffM(ux, uy)CM{rect(x/a, y/b)}=[CM{rect(x/a)}]tCM{rect(y/b)},
1βδxδuLx2=λB(Nxδx)2ζzoom(Lx/Nx)21,
(x2u2)1/214π.
g(u)F{f(x)}=-f(x)exp(-2πixu)dx,
g(u)=F{f(x)}gk=l=0Nx-1fn exp[-2πi(n-Nx/2)(k-Nu/2)δxδu]δx.

Metrics