Abstract

Sampling rules for numerically calculating ultrashort pulse fields are discussed. Such pulses are not monochromatic but rather have a finite spectral distribution about some central (temporal) frequency. Accordingly, the diffraction pattern for many spectral components must be considered. From a numerical implementation viewpoint, one may ask how many of these spectral components are needed to accurately calculate the pulse field. Using an analytical expression for the Fresnel diffraction from a 1-D slit, we examine this question by varying the number of contributing spectral components. We show how undersampling the spectral profile produces erroneous numerical artifacts (aliasing) in the spatial–temporal domain. A guideline, based on graphical considerations, is proposed that determines appropriate sampling conditions. We show that there is a relationship between this sampling rule and a diffraction wave that emerges from the aperture edge; comparisons are drawn with boundary diffraction waves. Numerical results for 2-D square and circular apertures are presented and discussed, and a potentially time-saving calculation technique that relates pulse distributions in different z planes is described.

© 2008 Optical Society of America

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References

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  1. C. J. R. Sheppard, “Generalized Bessel pulse beams,” J. Opt. Soc. Am. A 19, 2218-2222 (2002).
    [CrossRef]
  2. A. E. Kaplan, “Diffraction-induced transformation of near-cycle and subcycle pulses,” J. Opt. Soc. Am. B 15, 951-956 (1998).
    [CrossRef]
  3. R. W. Ziolkowski, “Propagation characteristics of ultrawide-bandwidth pulsed Gaussian beams,” J. Opt. Soc. Am. A 9, 2021-2030 (1992).
    [CrossRef]
  4. G. P. Agrawal, “Spectrum-induced changes in diffraction of pulsed optical beams,” Opt. Commun. 157, 52-56 (1998).
    [CrossRef]
  5. M. Gu, “Three-dimensional image formation in confocal microscopy under ultrashort laser pulse illumination,” J. Mod. Opt. 42, 747-762 (1995).
    [CrossRef]
  6. M. Gu, Advanced Optical Imaging Theory, 1st ed. (Springer-Verlag, 2000).
  7. M. A. Porras, “Ultrashort pulsed Gaussian light beams,” Phys. Rev. E 58, 1086-1093 (1998).
    [CrossRef]
  8. C. F. R. Caron, “Free-space propagation of ultrashort pulses: spacetime couplings in Gaussian pulse beams,” J. Mod. Opt. 45, 1881-1892 (1999).
    [CrossRef]
  9. C. J. R. Sheppard, “Bessel pulse beams and focus wave modes,” J. Mod. Opt. 18, 2594-2600 (2001).
  10. C. J. R. Sheppard and X. Gan, “Free-space propagation of femto-second light pulses,” Opt. Commun. 133, 1-6 (1997).
    [CrossRef]
  11. M. Gu and X. S. Gan, “Fresnel diffraction by circular and serrated apertures illuminated with an ultrashort pulsed-laser beam,” J. Opt. Soc. Am. A 13, 771-778 (1996).
    [CrossRef]
  12. A. Gürtler, C. Winnewisser, H. Helm, and P. Uhd Jepsen, “Terahertz pulse propagation in the near field and the far field,” J. Opt. Soc. Am. A 17, 74-83 (2000).
    [CrossRef]
  13. M. Lefrancois and S. F. Pereira, “Time evolution of the diffraction pattern of an ultrashort laser pulse,” Opt. Express 11, 1114-1122 (2003).
    [CrossRef] [PubMed]
  14. G. Girieud and S. F. Pereira, “Interference or not: analysis of Young's experiment for a single cycle pulse,” J. Eur. Opt. Soc. Rapid Publ. 1, 06016-1 (2006).
    [CrossRef]
  15. S. P. Veetil, C. Vijayan, D. K. Sharma, H. Schimmel, and F. Wyrowski, “Sampling rules in the frequency domain for numerical propagation of ultrashort pulses through linear dielectrics,” J. Opt. Soc. Am. B 23, 2227-2236 (2006).
    [CrossRef]
  16. S. P. Veetil, H. Schimmel, F. Wyrowski, and C. Vijayan, “Wave optical modelling of focusing of an ultra short pulse,” J. Mod. Opt. 53, 2187-2194 (2006).
    [CrossRef]
  17. R. Gase, “Ultrashort-pulse measurements applying generalized time-frequency distribution functions,” J. Opt. Soc. Am. B 14, 2915-2920 (1997).
    [CrossRef]
  18. C. Dorrer and I. Kang, “Complete temporal characterization of short optical pulses by simplified chronocyclic tomography,” Opt. Lett. 28, 1481-1483 (2003).
    [CrossRef] [PubMed]
  19. J. O'Hara and D. Grischkowsky, “Quasi-synthetic phased-array terahertz imaging,” J. Opt. Soc. Am. B 21, 1178-1191 (2004).
    [CrossRef]
  20. J. W. Goodman, Introduction to Fourier Optics, 3rd ed. (Roberts & Co., 2005).
  21. E. Hecht, Optics, 2nd edition (Addison-Wesley, 1989).
  22. D. P. Kelly, B. M. Hennelly, A. Grün, J. Darmo, and K. Unterrainer, “Fast numerical algorithm for ultrashort THz pulse diffraction,” Proc. SPIE 6697, 66970Q (2007).
    [CrossRef]
  23. B. M. Hennelly and J. T. Sheridan, “Generalizing, optimizing, and inventing numerical algorithms for the fractional Fourier, Fresnel, and linear canonical transforms,” J. Opt. Soc. Am. A 22, 917-927 (2005).
    [CrossRef]
  24. B. M. Hennelly and J. T. Sheridan, “Optical encryption and the space bandwidth product,” Opt. Commun. 247, 291-305 (2005).
    [CrossRef]
  25. D. P. Kelly, B. M. Hennelly, W. T. Rhodes, and J. T. Sheridan, “Analytical and numerical analysis of linear optical systems,” Opt. Eng. (Bellingham) 45, 088201 (2006).
    [CrossRef]
  26. D. P. Kelly, J. T. Sheridan, and W. T. Rhodes, “Finite-aperture effects for Fourier transform systems with convergent illumination. Part I: 2-D system analysis,” Opt. Commun. 263, 171-179 (2006).
    [CrossRef]
  27. R. N. Bracewell, The Fourier Transform and Its Applications (McGraw-Hill, 1965).
  28. M. Unser, “Sampling--50 years after Shannon,” Proc. IEEE 88, 569-587 (2000).
    [CrossRef]
  29. A. Stern and B. Javidi, “Sampling in the light of Wigner distribution,” J. Opt. Soc. Am. A 21, 360-366 (2004).
    [CrossRef]
  30. A. Stern and B. Javidi, “Sampling in the light of Wigner distribution: errata,” J. Opt. Soc. Am. A 21, 2038-2038 (2004).
    [CrossRef]
  31. K. B. Wolf, D. Mendlovic, and Z. Zalevsky, “Generalized Wigner function for the analysis of superresolution systems,” Appl. Opt. 37, 4374-4379 (1998).
    [CrossRef]
  32. D. Mendlovic and A. W. Lohmann, “Space-bandwidth product adaptation and its application to superresolution: fundamentals,” J. Opt. Soc. Am. A 14, 558-562 (1997).
    [CrossRef]
  33. D. Mendlovic, A. W. Lohmann, and Z. Zalevsky, “Space-bandwidth product adaptation and its application to superresolution: examples,” J. Opt. Soc. Am. A 14, 563-567 (1997).
    [CrossRef]
  34. G. B. Thomas, Jr., and R. L. Finney, Calculus (Addision-Wesley, 1996).
  35. S. Wolf, The Mathematica Book, 4th ed. (Cambridge U. Press, 1999).
  36. A. W. Lohmann, Optical Information Processing, S.Sinzinger, ed. (Universitätsverlag Ilmenau, 2006).
  37. J. Stamnes, “Waves, rays and the method of stationary phase,” Opt. Express 10, 740-751 (2002).
    [PubMed]
  38. J. B. Keller, “Geometrical theory of diffraction,” J. Opt. Soc. Am. 52, 116-130 (1962).
    [CrossRef] [PubMed]
  39. Z. L. Horvath and Zs. Bor, “Diffraction of short pulses with boundary diffraction wave theory,” Phys. Rev. E 63, 026601 (2001).
    [CrossRef]
  40. C. Rydberg and J. Bengtsson, “Efficient numerical representation of the optical field for the propagation of partially coherent radiation with a specified spatial and temporal coherence function,” J. Opt. Soc. Am. A 23, 1616-1625 (2006).
    [CrossRef]

2007

D. P. Kelly, B. M. Hennelly, A. Grün, J. Darmo, and K. Unterrainer, “Fast numerical algorithm for ultrashort THz pulse diffraction,” Proc. SPIE 6697, 66970Q (2007).
[CrossRef]

2006

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

D. P. Kelly, J. T. Sheridan, and W. T. Rhodes, “Finite-aperture effects for Fourier transform systems with convergent illumination. Part I: 2-D system analysis,” Opt. Commun. 263, 171-179 (2006).
[CrossRef]

C. Rydberg and J. Bengtsson, “Efficient numerical representation of the optical field for the propagation of partially coherent radiation with a specified spatial and temporal coherence function,” J. Opt. Soc. Am. A 23, 1616-1625 (2006).
[CrossRef]

G. Girieud and S. F. Pereira, “Interference or not: analysis of Young's experiment for a single cycle pulse,” J. Eur. Opt. Soc. Rapid Publ. 1, 06016-1 (2006).
[CrossRef]

S. P. Veetil, C. Vijayan, D. K. Sharma, H. Schimmel, and F. Wyrowski, “Sampling rules in the frequency domain for numerical propagation of ultrashort pulses through linear dielectrics,” J. Opt. Soc. Am. B 23, 2227-2236 (2006).
[CrossRef]

S. P. Veetil, H. Schimmel, F. Wyrowski, and C. Vijayan, “Wave optical modelling of focusing of an ultra short pulse,” J. Mod. Opt. 53, 2187-2194 (2006).
[CrossRef]

2005

2004

2003

2002

2001

Z. L. Horvath and Zs. Bor, “Diffraction of short pulses with boundary diffraction wave theory,” Phys. Rev. E 63, 026601 (2001).
[CrossRef]

C. J. R. Sheppard, “Bessel pulse beams and focus wave modes,” J. Mod. Opt. 18, 2594-2600 (2001).

2000

1999

C. F. R. Caron, “Free-space propagation of ultrashort pulses: spacetime couplings in Gaussian pulse beams,” J. Mod. Opt. 45, 1881-1892 (1999).
[CrossRef]

1998

1997

1996

1995

M. Gu, “Three-dimensional image formation in confocal microscopy under ultrashort laser pulse illumination,” J. Mod. Opt. 42, 747-762 (1995).
[CrossRef]

1992

1962

Agrawal, G. P.

G. P. Agrawal, “Spectrum-induced changes in diffraction of pulsed optical beams,” Opt. Commun. 157, 52-56 (1998).
[CrossRef]

Bengtsson, J.

Bor, Zs.

Z. L. Horvath and Zs. Bor, “Diffraction of short pulses with boundary diffraction wave theory,” Phys. Rev. E 63, 026601 (2001).
[CrossRef]

Bracewell, R. N.

R. N. Bracewell, The Fourier Transform and Its Applications (McGraw-Hill, 1965).

Caron, C. F. R.

C. F. R. Caron, “Free-space propagation of ultrashort pulses: spacetime couplings in Gaussian pulse beams,” J. Mod. Opt. 45, 1881-1892 (1999).
[CrossRef]

Darmo, J.

D. P. Kelly, B. M. Hennelly, A. Grün, J. Darmo, and K. Unterrainer, “Fast numerical algorithm for ultrashort THz pulse diffraction,” Proc. SPIE 6697, 66970Q (2007).
[CrossRef]

Dorrer, C.

Finney, R. L.

G. B. Thomas, Jr., and R. L. Finney, Calculus (Addision-Wesley, 1996).

Gan, X.

C. J. R. Sheppard and X. Gan, “Free-space propagation of femto-second light pulses,” Opt. Commun. 133, 1-6 (1997).
[CrossRef]

Gan, X. S.

Gase, R.

Girieud, G.

G. Girieud and S. F. Pereira, “Interference or not: analysis of Young's experiment for a single cycle pulse,” J. Eur. Opt. Soc. Rapid Publ. 1, 06016-1 (2006).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, 3rd ed. (Roberts & Co., 2005).

Grischkowsky, D.

Grün, A.

D. P. Kelly, B. M. Hennelly, A. Grün, J. Darmo, and K. Unterrainer, “Fast numerical algorithm for ultrashort THz pulse diffraction,” Proc. SPIE 6697, 66970Q (2007).
[CrossRef]

Gu, M.

M. Gu and X. S. Gan, “Fresnel diffraction by circular and serrated apertures illuminated with an ultrashort pulsed-laser beam,” J. Opt. Soc. Am. A 13, 771-778 (1996).
[CrossRef]

M. Gu, “Three-dimensional image formation in confocal microscopy under ultrashort laser pulse illumination,” J. Mod. Opt. 42, 747-762 (1995).
[CrossRef]

M. Gu, Advanced Optical Imaging Theory, 1st ed. (Springer-Verlag, 2000).

Gürtler, A.

Hecht, E.

E. Hecht, Optics, 2nd edition (Addison-Wesley, 1989).

Helm, H.

Hennelly, B. M.

D. P. Kelly, B. M. Hennelly, A. Grün, J. Darmo, and K. Unterrainer, “Fast numerical algorithm for ultrashort THz pulse diffraction,” Proc. SPIE 6697, 66970Q (2007).
[CrossRef]

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

B. M. Hennelly and J. T. Sheridan, “Optical encryption and the space bandwidth product,” Opt. Commun. 247, 291-305 (2005).
[CrossRef]

B. M. Hennelly and J. T. Sheridan, “Generalizing, optimizing, and inventing numerical algorithms for the fractional Fourier, Fresnel, and linear canonical transforms,” J. Opt. Soc. Am. A 22, 917-927 (2005).
[CrossRef]

Horvath, Z. L.

Z. L. Horvath and Zs. Bor, “Diffraction of short pulses with boundary diffraction wave theory,” Phys. Rev. E 63, 026601 (2001).
[CrossRef]

Javidi, B.

Kang, I.

Kaplan, A. E.

Keller, J. B.

Kelly, D. P.

D. P. Kelly, B. M. Hennelly, A. Grün, J. Darmo, and K. Unterrainer, “Fast numerical algorithm for ultrashort THz pulse diffraction,” Proc. SPIE 6697, 66970Q (2007).
[CrossRef]

D. P. Kelly, J. T. Sheridan, and W. T. Rhodes, “Finite-aperture effects for Fourier transform systems with convergent illumination. Part I: 2-D system analysis,” Opt. Commun. 263, 171-179 (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. (Bellingham) 45, 088201 (2006).
[CrossRef]

Lefrancois, M.

Lohmann, A. W.

Mendlovic, D.

O'Hara, J.

Pereira, S. F.

G. Girieud and S. F. Pereira, “Interference or not: analysis of Young's experiment for a single cycle pulse,” J. Eur. Opt. Soc. Rapid Publ. 1, 06016-1 (2006).
[CrossRef]

M. Lefrancois and S. F. Pereira, “Time evolution of the diffraction pattern of an ultrashort laser pulse,” Opt. Express 11, 1114-1122 (2003).
[CrossRef] [PubMed]

Porras, M. A.

M. A. Porras, “Ultrashort pulsed Gaussian light beams,” Phys. Rev. E 58, 1086-1093 (1998).
[CrossRef]

Rhodes, W. T.

D. P. Kelly, J. T. Sheridan, and W. T. Rhodes, “Finite-aperture effects for Fourier transform systems with convergent illumination. Part I: 2-D system analysis,” Opt. Commun. 263, 171-179 (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. (Bellingham) 45, 088201 (2006).
[CrossRef]

Rydberg, C.

Schimmel, H.

Sharma, D. K.

Sheppard, C. J. R.

C. J. R. Sheppard, “Generalized Bessel pulse beams,” J. Opt. Soc. Am. A 19, 2218-2222 (2002).
[CrossRef]

C. J. R. Sheppard, “Bessel pulse beams and focus wave modes,” J. Mod. Opt. 18, 2594-2600 (2001).

C. J. R. Sheppard and X. Gan, “Free-space propagation of femto-second light pulses,” Opt. Commun. 133, 1-6 (1997).
[CrossRef]

Sheridan, J. T.

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

D. P. Kelly, J. T. Sheridan, and W. T. Rhodes, “Finite-aperture effects for Fourier transform systems with convergent illumination. Part I: 2-D system analysis,” Opt. Commun. 263, 171-179 (2006).
[CrossRef]

B. M. Hennelly and J. T. Sheridan, “Optical encryption and the space bandwidth product,” Opt. Commun. 247, 291-305 (2005).
[CrossRef]

B. M. Hennelly and J. T. Sheridan, “Generalizing, optimizing, and inventing numerical algorithms for the fractional Fourier, Fresnel, and linear canonical transforms,” J. Opt. Soc. Am. A 22, 917-927 (2005).
[CrossRef]

Stamnes, J.

Stern, A.

Thomas, G. B.

G. B. Thomas, Jr., and R. L. Finney, Calculus (Addision-Wesley, 1996).

Uhd Jepsen, P.

Unser, M.

M. Unser, “Sampling--50 years after Shannon,” Proc. IEEE 88, 569-587 (2000).
[CrossRef]

Unterrainer, K.

D. P. Kelly, B. M. Hennelly, A. Grün, J. Darmo, and K. Unterrainer, “Fast numerical algorithm for ultrashort THz pulse diffraction,” Proc. SPIE 6697, 66970Q (2007).
[CrossRef]

Veetil, S. P.

Vijayan, C.

Winnewisser, C.

Wolf, K. B.

Wolf, S.

S. Wolf, The Mathematica Book, 4th ed. (Cambridge U. Press, 1999).

Wyrowski, F.

Zalevsky, Z.

Ziolkowski, R. W.

Appl. Opt.

J. Eur. Opt. Soc. Rapid Publ.

G. Girieud and S. F. Pereira, “Interference or not: analysis of Young's experiment for a single cycle pulse,” J. Eur. Opt. Soc. Rapid Publ. 1, 06016-1 (2006).
[CrossRef]

J. Mod. Opt.

S. P. Veetil, H. Schimmel, F. Wyrowski, and C. Vijayan, “Wave optical modelling of focusing of an ultra short pulse,” J. Mod. Opt. 53, 2187-2194 (2006).
[CrossRef]

M. Gu, “Three-dimensional image formation in confocal microscopy under ultrashort laser pulse illumination,” J. Mod. Opt. 42, 747-762 (1995).
[CrossRef]

C. F. R. Caron, “Free-space propagation of ultrashort pulses: spacetime couplings in Gaussian pulse beams,” J. Mod. Opt. 45, 1881-1892 (1999).
[CrossRef]

C. J. R. Sheppard, “Bessel pulse beams and focus wave modes,” J. Mod. Opt. 18, 2594-2600 (2001).

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

D. Mendlovic and A. W. Lohmann, “Space-bandwidth product adaptation and its application to superresolution: fundamentals,” J. Opt. Soc. Am. A 14, 558-562 (1997).
[CrossRef]

D. Mendlovic, A. W. Lohmann, and Z. Zalevsky, “Space-bandwidth product adaptation and its application to superresolution: examples,” J. Opt. Soc. Am. A 14, 563-567 (1997).
[CrossRef]

B. M. Hennelly and J. T. Sheridan, “Generalizing, optimizing, and inventing numerical algorithms for the fractional Fourier, Fresnel, and linear canonical transforms,” J. Opt. Soc. Am. A 22, 917-927 (2005).
[CrossRef]

A. Stern and B. Javidi, “Sampling in the light of Wigner distribution,” J. Opt. Soc. Am. A 21, 360-366 (2004).
[CrossRef]

A. Stern and B. Javidi, “Sampling in the light of Wigner distribution: errata,” J. Opt. Soc. Am. A 21, 2038-2038 (2004).
[CrossRef]

C. J. R. Sheppard, “Generalized Bessel pulse beams,” J. Opt. Soc. Am. A 19, 2218-2222 (2002).
[CrossRef]

R. W. Ziolkowski, “Propagation characteristics of ultrawide-bandwidth pulsed Gaussian beams,” J. Opt. Soc. Am. A 9, 2021-2030 (1992).
[CrossRef]

M. Gu and X. S. Gan, “Fresnel diffraction by circular and serrated apertures illuminated with an ultrashort pulsed-laser beam,” J. Opt. Soc. Am. A 13, 771-778 (1996).
[CrossRef]

A. Gürtler, C. Winnewisser, H. Helm, and P. Uhd Jepsen, “Terahertz pulse propagation in the near field and the far field,” J. Opt. Soc. Am. A 17, 74-83 (2000).
[CrossRef]

C. Rydberg and J. Bengtsson, “Efficient numerical representation of the optical field for the propagation of partially coherent radiation with a specified spatial and temporal coherence function,” J. Opt. Soc. Am. A 23, 1616-1625 (2006).
[CrossRef]

J. Opt. Soc. Am. B

Opt. Commun.

G. P. Agrawal, “Spectrum-induced changes in diffraction of pulsed optical beams,” Opt. Commun. 157, 52-56 (1998).
[CrossRef]

C. J. R. Sheppard and X. Gan, “Free-space propagation of femto-second light pulses,” Opt. Commun. 133, 1-6 (1997).
[CrossRef]

D. P. Kelly, J. T. Sheridan, and W. T. Rhodes, “Finite-aperture effects for Fourier transform systems with convergent illumination. Part I: 2-D system analysis,” Opt. Commun. 263, 171-179 (2006).
[CrossRef]

B. M. Hennelly and J. T. Sheridan, “Optical encryption and the space bandwidth product,” Opt. Commun. 247, 291-305 (2005).
[CrossRef]

Opt. Eng. (Bellingham)

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

Opt. Express

Opt. Lett.

Phys. Rev. E

Z. L. Horvath and Zs. Bor, “Diffraction of short pulses with boundary diffraction wave theory,” Phys. Rev. E 63, 026601 (2001).
[CrossRef]

M. A. Porras, “Ultrashort pulsed Gaussian light beams,” Phys. Rev. E 58, 1086-1093 (1998).
[CrossRef]

Proc. IEEE

M. Unser, “Sampling--50 years after Shannon,” Proc. IEEE 88, 569-587 (2000).
[CrossRef]

Proc. SPIE

D. P. Kelly, B. M. Hennelly, A. Grün, J. Darmo, and K. Unterrainer, “Fast numerical algorithm for ultrashort THz pulse diffraction,” Proc. SPIE 6697, 66970Q (2007).
[CrossRef]

Other

R. N. Bracewell, The Fourier Transform and Its Applications (McGraw-Hill, 1965).

G. B. Thomas, Jr., and R. L. Finney, Calculus (Addision-Wesley, 1996).

S. Wolf, The Mathematica Book, 4th ed. (Cambridge U. Press, 1999).

A. W. Lohmann, Optical Information Processing, S.Sinzinger, ed. (Universitätsverlag Ilmenau, 2006).

J. W. Goodman, Introduction to Fourier Optics, 3rd ed. (Roberts & Co., 2005).

E. Hecht, Optics, 2nd edition (Addison-Wesley, 1989).

M. Gu, Advanced Optical Imaging Theory, 1st ed. (Springer-Verlag, 2000).

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Figures (7)

Fig. 1
Fig. 1

Contour plot of U S ( x , t R , z = 1 ) for 0.075 m < x < 0.075 m and 5 < t R T < 25 . There are ten contour levels spanning the range 0 < U S ( x , t R , z = 1 ) < 3.56 × 10 6 . T1, T2, T3, T4, pulse tails 1–4; AR, aliasing free region delimited by ( x SL , t R ) ; SR, sampled replica; C, curve plotted using Eq. (9b).

Fig. 2
Fig. 2

Contour plot of ψ S for 0.075 m < x < 0.075 m and 5 < t R T < 25 . There are ten contour levels spanning the range 0 < ψ S < 323 × 10 3 . T1, T2, T3, T4, pulse tails 1–4; AR, aliasing free region delimited by ( x SL , t R ) ; SR, sampled replica.

Fig. 3
Fig. 3

Plot of Eq. (9b) with L = 25 mm . SR, sampled replica; NSV, numerical search values (these values were found by performing a maximum value numerical search of U S ( x , t R , z = 1 ) over the range 0 < x < 0.07 m and 0 < t R T < 25 ).

Fig. 4
Fig. 4

Comparison of V ( ω ) and V 2 ( ω ) , where ω × 10 12 rad s 1 . SV, sampled values; ESV, extra sampled values. For Section 4, gray triangles indicate the SV for plane z 1 , black boxes indicate the SV for plane z 2 , and ESV represent the two extra sampled points necessary for diffraction pattern calculation at z 2 .

Fig. 5
Fig. 5

(a) Numerical results of U S for a square aperture, with t v = 0 s , z = 0.1 m , and L = 0.1 mm over the range 23 mm < x < 23 mm , 23 mm < y < 23 mm . (b) Contour plot of U S for a square aperture, with t v = 0 s , z = 0.1 m , and L = 0.1 mm over the range 23 mm < x < 23 mm , 23 mm < y < 23 mm . Coordinates of the gray box (red online) defined in text, with x SL = 20.3126 mm . AR, aliasing region.

Fig. 6
Fig. 6

(a) Numerical results of U S for a circular aperture, with t v = 0 , z = 0.1 m , L = 0.1 mm over the range 23 mm < x < 23 mm , 23 mm < y < 23 mm . (b) Contour plot of U S for a circular aperture, with t v = 0 s , z = 0.1 m , and L = 0.1 mm over the range 23 mm < x < 23 mm , 23 mm < y < 23 mm . Coordinates of the gray box (red online) defined in text, with x SL = 20.3126 mm . AR, aliasing region.

Fig. 7
Fig. 7

(a) Numerical results of U S for a square aperture, with t v = 0.5 × 10 12 s , z = 0.1 m , and L = 0.1 mm over the range 23 mm < x < 23 mm , 23 mm < y < 23 mm . Ten contour levels spanning 0 to 1.13 × 10 3 . Coordinates of the gray box (red online) given by x SL = 19.564 mm . AR, aliasing region. (b) Contour plot of U S for a square aperture, with t v = 1 × 10 12 s , z = 0.1 m , and L = 0.1 mm over the range 23 mm < x < 23 mm , 23 mm < y < 23 mm . Ten contour levels spanning 0 to 3.78 × 10 4 . Coordinates of the gray box (red online) given by x SL = 18.785 mm . AR, aliasing region.

Equations (37)

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

U ̃ K ( x , K ) = U K ( x , K ) exp ( j z ω c ) ,
U K ( x , K ) = 1 j K u ( X ) p ( X ) exp [ j π K ( x X ) 2 ] d X ,
U ( x , K ) = U ̃ K ( x , K ) V ( ω ) .
U ( x , t R , z ) = 0 U K ( x , K ) V ( ω ) exp ( j ω t R ) d ω ,
t R = t z c .
p ( x ) { 1 , x < L 0 , x > L } ,
U K ( x , K ) = 1 2 { erf [ π K ( 1 ) 3 4 ( L x ) ] + erf [ π K ( 1 ) 3 4 ( L + x ) ] } .
U S ( x , t R , z ) = n = 1 N ω U K ( x , K n ) V ( ω n ) exp ( j ω n t R ) ,
Δ ω 2 + ω 0 Δ ω 2 + ω 0 g ̃ ( ω ) 2 d ω = Δ t 2 Δ t 2 g ( t ) 2 d t = η ω E ω ,
N ω = Δ t Δ ω 4 π .
V ( ω ) = T exp { [ T ( ω ω 0 ) 2 ] 2 } ,
v ( t ) = 1 T exp ( j ω 0 t ) exp [ ( t 2 T ) 2 ] .
N ω = Δ t Δ ω 4 π = 1 4 π ( 40 T ) ( 8 T ) = 80 π ,
Δ t = ( δ ω ) 1 = N ω Δ ω ( 4 π ) .
p T ( x ) δ ( x L ) + δ ( x + L ) ,
ψ K ( x , K ) = 1 j K { exp [ j π K ( x L ) 2 ] + exp [ j π K ( x + L ) 2 ] }
ψ ( x , t R , z ) = I 1 { V ( ω ) [ exp ( j ω R 1 ) + exp ( j ω R 2 ) ] } ( t R ) = I 1 { V ( ω ) exp ( j ω R 1 ) } ( t R ) + I 1 { V ( ω ) exp ( j ω R 2 ) } ( t R ) = v ( t R R 1 ) + v ( t R R 2 ) ,
ψ S ( x , t R , z ) = n = 1 N ω [ exp ( j ω n R 1 ) + exp ( j ω n R 2 ) ] V ( ω n ) exp ( j ω n t R ) .
x = ± 2 z c t R + L for R 1
x = ± 2 z c t R L for R 2 .
x S L 1 ( t v ) = ± 2 z c ( 2 π N ω Δ ω + t v ) + L for R 1
x S L 2 ( t v ) = ± 2 z c ( 2 π N ω Δ ω + t v ) L for R 2 .
V 2 ( ω ) = { ( s + 1 ω b ) s + 1 ( ω ω a ) s Γ ( s + 1 ) exp [ ( s + 1 ) ω ω a ω b ] , for ω > ω a 0 , otherwise } ,
f z ( x , z ) = f 0 ( x ) h ( x ) ,
h ( x ) = exp ( j π x 2 λ z )
f z ( x , z ) = I 1 { F 0 ( f x ) H ( f x ) } ( x ) ,
F 0 ( f x ) = I { f 0 ( x ) } ( f x )
H ( f x ) = I { h ( x ) } ( f x ) = exp ( j 2 π z λ ) exp ( j π λ z f x 2 ) ,
F 0 ( f x ) = L L exp ( j 2 π x f x ) d x = j exp ( j 2 π f x L ) 2 π f x j exp ( j 2 π f x L ) 2 π f x ,
f z ( x , z ) = exp ( j 2 π z λ ) [ j exp ( j 2 π f x L ) 2 π f x j exp ( j 2 π f x L ) 2 π f x ] exp ( j π λ z f x 2 ) exp ( j 2 π x f x ) d f x ,
f z ( x , z ) = exp ( j 2 π z λ ) [ Y ( x L ) Y ( x + L ) ] ,
Y ( x ) = ( j 2 π f x ) exp [ j 2 π f x ( x λ z f x 2 ) ] d f x .
ϕ ( f x ) = x f x λ z f x 2 2 ,
ϕ ( f x ) = d ϕ ( f x ) d f x = x λ z f x .
Y ( x ) ( j λ z 2 π ) λ ϕ ( x ) exp ( j π 4 ) ( 1 x ) exp ( j π x 2 λ z ) .
f z ( x , z ) = exp ( j 2 π z λ ) λ ϕ ( x ) exp ( j π 4 ) ( j λ z 2 π ) × { Θ 1 ( x ) exp [ j π ( x L ) 2 λ z ] + Θ 2 ( x ) exp ( j π ( x + L ) 2 λ z ) } ,
Θ 1 ( x ) = 1 L x , Θ 2 ( x ) = 1 L + x .

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