Abstract

The product of the spatial and spatial frequency extents of a wave field has proven useful in the analysis of the sampling requirements of numerical simulations. We propose that the ratio of these quantities is also illuminating. We have shown that the distance at which the so-called “direct method” becomes more efficient than the so-called “spectral method” for simulations of Fresnel transforms may be written in terms of this space–bandwidth ratio. We have proposed generalizations of these algorithms for numerical simulations of general ABCD systems and derived expressions for the “transition space–bandwidth ratio,” above which the generalization of the spectral method is the more efficient algorithm and below which the generalization of the direct method is preferable.

© 2011 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
  13. L. Ahrenberg, A. J. Page, B. M. Hennelly, J. B. McDonald, and T. J. Naughton, “Using commodity graphics hardware for real-time digital hologram view-reconstruction,” J. Display Technol. 5, 111–119 (2009).
    [CrossRef]

2010

2009

2008

A. Koç, H. M. Ozaktas, C. Candan, and M. A. Kutay, “Digital computation of linear canonical transforms,” IEEE Trans. Signal Process. 56, 2383–2394 (2008).
[CrossRef]

J. J. Healy, B. M. Hennelly, and J. T. Sheridan, “An additional sampling criterion for the linear canonical transform,” Opt. Lett. 33, 2599–2601 (2008).
[CrossRef] [PubMed]

2007

S. G. Johnson and M. Frigo, “A modified split-radix FFT with fewer arithmetic operations,” IEEE Trans. Signal Process. 55, 111–119 (2007).
[CrossRef]

2006

A. Stern, “Sampling of linear canonical transformed signals,” Signal Process. 86, 1421–1425 (2006).
[CrossRef]

D. P. Kelly, B. M. Hennelly, W. T. Rhodes, and J. T. Sheridan, “Analytical and numerical analysis of linear optical systems,” Opt. Eng. 45, 088201 (2006).
[CrossRef]

2005

2001

J.-J. Ding, “Research of fractional Fourier transform and linear canonical transform,” Ph.D. thesis (National Taiwan University, 2001).

1997

D. Mendlovic, Z. Zalevsky, and N. Konforti, “Computation considerations and fast algorithms for calculating the diffraction integral,” J. Mod. Opt. 44, 407–414 (1997).
[CrossRef]

1996

Ahrenberg, L.

Candan, C.

A. Koç, H. M. Ozaktas, C. Candan, and M. A. Kutay, “Digital computation of linear canonical transforms,” IEEE Trans. Signal Process. 56, 2383–2394 (2008).
[CrossRef]

Ding, J.-J.

J.-J. Ding, “Research of fractional Fourier transform and linear canonical transform,” Ph.D. thesis (National Taiwan University, 2001).

Dorsch, R. G.

Ferreira, C.

Frigo, M.

S. G. Johnson and M. Frigo, “A modified split-radix FFT with fewer arithmetic operations,” IEEE Trans. Signal Process. 55, 111–119 (2007).
[CrossRef]

Healy, J. J.

Hennelly, B. M.

Johnson, S. G.

S. G. Johnson and M. Frigo, “A modified split-radix FFT with fewer arithmetic operations,” IEEE Trans. Signal Process. 55, 111–119 (2007).
[CrossRef]

Kelly, D. P.

D. P. Kelly, B. M. Hennelly, W. T. Rhodes, and J. T. Sheridan, “Analytical and numerical analysis of linear optical systems,” Opt. Eng. 45, 088201 (2006).
[CrossRef]

Koç, A.

A. Koç, H. M. Ozaktas, C. Candan, and M. A. Kutay, “Digital computation of linear canonical transforms,” IEEE Trans. Signal Process. 56, 2383–2394 (2008).
[CrossRef]

Konforti, N.

D. Mendlovic, Z. Zalevsky, and N. Konforti, “Computation considerations and fast algorithms for calculating the diffraction integral,” J. Mod. Opt. 44, 407–414 (1997).
[CrossRef]

Kutay, M. A.

A. Koç, H. M. Ozaktas, C. Candan, and M. A. Kutay, “Digital computation of linear canonical transforms,” IEEE Trans. Signal Process. 56, 2383–2394 (2008).
[CrossRef]

Lohmann, A. W.

McDonald, J. B.

Mendlovic, D.

D. Mendlovic, Z. Zalevsky, and N. Konforti, “Computation considerations and fast algorithms for calculating the diffraction integral,” J. Mod. Opt. 44, 407–414 (1997).
[CrossRef]

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

Naughton, T. J.

Ojeda-Castañeda, J.

M. E. Testorf, B. M. Hennelly, and J. Ojeda-Castañeda, Phase-Space Optics: Fundamentals and Applications (McGraw-Hill, 2009).

Ozaktas, H. M.

A. Koç, H. M. Ozaktas, C. Candan, and M. A. Kutay, “Digital computation of linear canonical transforms,” IEEE Trans. Signal Process. 56, 2383–2394 (2008).
[CrossRef]

Page, A. J.

Rhodes, W. T.

D. P. Kelly, B. M. Hennelly, W. T. Rhodes, and J. T. Sheridan, “Analytical and numerical analysis of linear optical systems,” Opt. Eng. 45, 088201 (2006).
[CrossRef]

Sheridan, J. T.

Stern, A.

A. Stern, “Sampling of linear canonical transformed signals,” Signal Process. 86, 1421–1425 (2006).
[CrossRef]

Testorf, M. E.

M. E. Testorf, B. M. Hennelly, and J. Ojeda-Castañeda, Phase-Space Optics: Fundamentals and Applications (McGraw-Hill, 2009).

Zalevsky, Z.

D. Mendlovic, Z. Zalevsky, and N. Konforti, “Computation considerations and fast algorithms for calculating the diffraction integral,” J. Mod. Opt. 44, 407–414 (1997).
[CrossRef]

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

IEEE Trans. Signal Process.

A. Koç, H. M. Ozaktas, C. Candan, and M. A. Kutay, “Digital computation of linear canonical transforms,” IEEE Trans. Signal Process. 56, 2383–2394 (2008).
[CrossRef]

S. G. Johnson and M. Frigo, “A modified split-radix FFT with fewer arithmetic operations,” IEEE Trans. Signal Process. 55, 111–119 (2007).
[CrossRef]

J. Display Technol.

J. Mod. Opt.

D. Mendlovic, Z. Zalevsky, and N. Konforti, “Computation considerations and fast algorithms for calculating the diffraction integral,” J. Mod. Opt. 44, 407–414 (1997).
[CrossRef]

J. Opt. Soc. Am. A

Opt. Eng.

D. P. Kelly, B. M. Hennelly, W. T. Rhodes, and J. T. Sheridan, “Analytical and numerical analysis of linear optical systems,” Opt. Eng. 45, 088201 (2006).
[CrossRef]

Opt. Lett.

Signal Process.

A. Stern, “Sampling of linear canonical transformed signals,” Signal Process. 86, 1421–1425 (2006).
[CrossRef]

Other

M. E. Testorf, B. M. Hennelly, and J. Ojeda-Castañeda, Phase-Space Optics: Fundamentals and Applications (McGraw-Hill, 2009).

J.-J. Ding, “Research of fractional Fourier transform and linear canonical transform,” Ph.D. thesis (National Taiwan University, 2001).

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Figures (1)

Fig. 1
Fig. 1

Plots of N DM LCT / N (thick, continuous curve) and N SM LCT / N (thick, dashed curve) for the randomly chosen LCT parameters A = 0.6 , B = 2 , C = 0.17 , and D = 1.1 .

Tables (1)

Tables Icon

Table 1 Number of Samples Required by the DM and the SM to Calculate the Fresnel Transform of a Rectangular Aperture of Width 0.02 m , Bandwidth 50000 m 1 for Various Propagation Distances for Light of Wavelength 500 nm

Equations (13)

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( A B C D ) = ( 1 λ z 0 1 ) .
( 1 0 1 / λ f 1 ) .
( M 0 0 1 / M ) .
( 0 1 1 0 ) .
( 1 λ z 0 1 ) = ( 1 0 1 λ z 1 ) ( λ z 0 0 1 λ z ) ( 0 1 1 0 ) ( 1 0 1 λ z 1 ) .
( A B C D ) = ( 1 0 D B 1 ) ( B 0 0 1 B ) ( 0 1 1 0 ) ( 1 0 A B 1 ) .
( 1 λ z 0 1 ) = ( 0 1 1 0 ) ( 1 0 λ z 1 ) ( 0 1 1 0 ) .
( A B C D ) = ( 1 0 C A 1 ) ( A 0 0 1 A ) ( 0 1 1 0 ) ( 1 0 B A 1 ) ( 0 1 1 0 ) .
N SM Fresnel = SBP ( 1 + λ z SBR ) .
N DM Fresnel = SBP ( max { ( 1 + SBR λ z ) , ( 1 + λ z SBR ) } ) .
z < Z t = SBR λ .
N DM LCT = SBP ( max { | 1 ± A B SBR | , | ( A SBR ± B 1 SBR ) ( C SBR ± D 1 SBR ) | } ) .
N SM LCT = SBP ( max { | 1 ± B A 1 SBR | , | ( A SBR ± B 1 SBR ) ( C SBR ± D 1 SBR ) | } ) .

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