Abstract

In this paper, we address the problem of calculating Fresnel diffraction integrals using a finite number of uniformly spaced samples. General and simple sampling rules of thumb are derived that allow the user to calculate the distribution for any propagation distance. It is shown how these rules can be extended to fast-Fourier-transform-based algorithms to increase calculation efficiency. A comparison with other theoretical approaches is made.

© 2014 Optical Society of America

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  6. G. Pedrini, P. Fröning, H. J. Tiziani, M. E. Gusev, “Pulsed digital holography for high-speed contouring that uses a two-wavelength method,” Appl. Opt. 38, 3460–3467 (1999).
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  7. P. Picart, J. Leval, D. Mounier, S. Gougeon, “Some opportunities for vibration analysis with time averaging in digital Fresnel holography,” Appl. Opt. 44, 337–343 (2005).
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  8. P. Picart, J. Leval, “General theoretical formulation of image formation in digital Fresnel holography,” J. Opt. Soc. Am. A 25, 1744–1761 (2008).
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  9. D. P. Kelly, B. M. Hennelly, N. Pandey, T. J. Naughton, W. T. Rhodes, “Resolution limits in practical digital holographic systems,” Opt. Eng. 48, 095801 (2009).
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  18. V. Arrizón, M. Testorf, S. Sinzinger, J. Jahns, “Iterative optimization of phase-only diffractive optical elements based on a lenslet array,” J. Opt. Soc. Am. A 17, 2157–2164 (2000).
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  19. R. Kleindienst, L. Moeller, S. Sinzinger, “Highly efficient refractive Gaussian-to-tophat beam shaper for compact terahertz imager,” Appl. Opt. 49, 1757–1763 (2010).
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  22. J. W. Goodman, Speckle Phenomena in Optics (Roberts and Company, 2007).
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  24. D. P. Kelly, J. E. Ward, U. Gopinathan, B. M. Hennelly, F. T. O’Neill, J. T. Sheridan, “Generalized Yamaguchi correlation factor for coherent quadratic phase speckle metrology systems with an aperture,” Opt. Lett. 31, 3444–3446 (2006).
    [CrossRef]
  25. D. Mas, J. Garcia, C. Ferreira, L. M. Bernardo, F. Marinho, “Fast algorithms for free-space diffraction patterns calculation,” Opt. Commun. 164, 233–245 (1999).
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  26. D. Mas, J. Garcia, C. Ferreira, L. M. Bernardo, F. Marinho, “Fast numerical calculation of Fresnel patterns in convergent systems,” Opt. Commun. 227, 245–258 (2003).
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  27. B. M. Hennelly, 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]
  28. D. P. Kelly, B. M. Hennelly, W. T. Rhodes, J. T. Sheridan, “Analytical and numerical analysis of linear optical systems,” Opt. Eng. 45, 088201 (2006).
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  29. F. Shen, A. Wang, “Fast-Fourier-transform based numerical integration method for the Rayleigh–Sommerfeld diffraction formula,” Appl. Opt. 45, 1102–1110 (2006).
    [CrossRef]
  30. D. G. Voelz, M. C. Roggemann, “Digital simulation of scalar optical diffraction: revisiting chirp function sampling criteria and consequences,” Appl. Opt. 48, 6132–6142 (2009).
    [CrossRef]
  31. T. Kozacki, K. Falaggis, M. Kujawinska, “Computation of diffracted fields for the case of high numerical aperture using the angular spectrum method,” Appl. Opt. 51, 7080–7088 (2012).
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  32. A. W. Lohmann, “Image rotation, Wigner rotation, and the fractional Fourier transform,” J. Opt. Soc. Am. A 10, 2181–2186 (1993).
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  33. A. W. Lohmann, R. G. Dorsch, D. Mendlovic, Z. Zalevsky, C. Ferreira, “Space–bandwidth product of optical signals and systems,” J. Opt. Soc. Am. A 13, 470–473 (1996).
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  34. D. Mendlovic, A. W. Lohmann, “Space–bandwidth product adaptation and its application to superresolution: fundamentals,” J. Opt. Soc. Am. A 14, 558–562 (1997).
    [CrossRef]
  35. D. Mendlovic, A. W. Lohmann, Z. Zalevsky, “Space–bandwidth product adaptation and its application to superresolution: examples,” J. Opt. Soc. Am. A 14, 563–567 (1997).
    [CrossRef]
  36. F. Gori, “Fresnel transform and sampling theorem,” Opt. Commun. 39, 293–297 (1981).
    [CrossRef]
  37. L. Onural, “Sampling of the diffraction field,” Appl. Opt. 39, 5929–5935 (2000).
    [CrossRef]
  38. A. Stern, B. Javidi, “Analysis of practical sampling and reconstruction from Fresnel fields,” Opt. Eng. 43, 239–250 (2004).
    [CrossRef]
  39. A. Sokołowski, “Consequences of sampling an image and Fourier planes in a numerical light propagation model based on the Helmholtz–Kirchhoff approximation,” J. Opt. Soc. Am. A 23, 2764–2767 (2006).
    [CrossRef]
  40. L. Onural, “Exact analysis of the effects of sampling of the scalar diffraction field,” J. Opt. Soc. Am. A 24, 359–367 (2007).
    [CrossRef]
  41. A. Sokołowski, T. Więcek, “Consequences of sampling an image and Fourier planes in a numerical light propagation model based on the Helmholtz–Kirchhoff approximation. ii. comparison between two numerical algorithms,” J. Opt. Soc. Am. A 27, 1688–1693 (2010).
    [CrossRef]
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    [CrossRef]
  44. A. W. Lohmann, R. G. Dorsch, D. Mendolovic, Z. Zalevsky, C. Ferreira, “Space–bandwidth product of optical signals and systems,” J. Opt. Soc. Am. A 13, 470–473 (1996).
    [CrossRef]
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    [CrossRef]
  47. A. Stern, B. Javidi, “Sampling in the light of wigner distribution,” J. Opt. Soc. Am. A 21, 360–366 (2004).
    [CrossRef]
  48. A. Stern, B. Javidi, “Sampling in the light of Wigner distribution: errata,” J. Opt. Soc. Am. A 21, 2038 (2004).
    [CrossRef]
  49. D. P. Kelly, D. Claus, “Filtering role of the sensor pixel in Fourier and Fresnel digital holography,” Appl. Opt. 52, A336–A345 (2013).
    [CrossRef]
  50. L. Onural, “Diffraction from a wavelet point of view,” Opt. Lett. 18, 846–848 (1993).
    [CrossRef]
  51. M. Liebling, T. Blu, M. Unser, “Fresnelets: new multiresolution wavelet bases for digital holography,” IEEE Trans. Image Process. 12, 29–43 (2003).
    [CrossRef]
  52. M. Liebling, “On Fresnelets, interference fringes, and digital holography,” Ph.D. thesis (Swiss Federal Institute of Technology Lausanne, 2004).
  53. N. Chacko, M. Liebling, T. Blu, “Discretization of continuous convolution operators for accurate modeling of wave propagation in digital holography,” J. Opt. Soc. Am. A 30, 2012–2020 (2013).
    [CrossRef]
  54. M. Gu, 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]
  55. D. P. Kelly, B. M. Hennelly, A. Grün, K. Unterrainer, “Numerical sampling rules for paraxial regime pulse diffraction calculations,” J. Opt. Soc. Am. A 25, 2299–2308 (2008).
    [CrossRef]
  56. R. J. Mahon, J. A. Murphy, “Diffraction of an optical pulse as an expansion in ultrashort orthogonal Gaussian beam modes,” J. Opt. Soc. Am. A 30, 215–226 (2013).
    [CrossRef]
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  58. A. Stern, “Sampling of linear canonical transformed signals,” Signal Process. 86, 1421–1425 (2006).
    [CrossRef]

2013

D. P. Kelly, D. Claus, “Filtering role of the sensor pixel in Fourier and Fresnel digital holography,” Appl. Opt. 52, A336–A345 (2013).
[CrossRef]

N. Chacko, M. Liebling, T. Blu, “Discretization of continuous convolution operators for accurate modeling of wave propagation in digital holography,” J. Opt. Soc. Am. A 30, 2012–2020 (2013).
[CrossRef]

R. J. Mahon, J. A. Murphy, “Diffraction of an optical pulse as an expansion in ultrashort orthogonal Gaussian beam modes,” J. Opt. Soc. Am. A 30, 215–226 (2013).
[CrossRef]

2012

M. Guizar-Sicairos, J. R. Fienup, “Understanding the twin-image problem in phase retrieval,” J. Opt. Soc. Am. A 29, 2367–2375 (2012).
[CrossRef]

I. Eriksson, P. Haglund, J. Powell, M. Sjodahl, A. F. H. Kaplan, “Holographic measurement of thermal distortion during laser spot welding,” Opt. Eng. 51, 030501 (2012).
[CrossRef]

T. Kozacki, K. Falaggis, M. Kujawinska, “Computation of diffracted fields for the case of high numerical aperture using the angular spectrum method,” Appl. Opt. 51, 7080–7088 (2012).
[CrossRef]

2011

D. P. Kelly, J. J. Healy, J. T. S. B. M. Hennelly, “Quantifying the 2.5d imaging performance of digital holographic systems,” J. Eur. Opt. Soc. Rapid Pub. 6, 11034 (2011).
[CrossRef]

2010

A. Sokołowski, T. Więcek, “Consequences of sampling an image and Fourier planes in a numerical light propagation model based on the Helmholtz–Kirchhoff approximation. ii. comparison between two numerical algorithms,” J. Opt. Soc. Am. A 27, 1688–1693 (2010).
[CrossRef]

R. Kleindienst, L. Moeller, S. Sinzinger, “Highly efficient refractive Gaussian-to-tophat beam shaper for compact terahertz imager,” Appl. Opt. 49, 1757–1763 (2010).
[CrossRef]

T. Meinecke, N. Sabitov, S. Sinzinger, “Information extraction from digital holograms for particle flow analysis,” Appl. Opt. 49, 2446–2455 (2010).
[CrossRef]

K. M. Molony, B. M. Hennelly, D. P. Kelly, T. J. Naughton, “Reconstruction algorithms applied to in-line Gabor digital holographic microscopy,” Opt. Commun. 283, 903–909 (2010).
[CrossRef]

2009

D. S. Monaghan, D. P. Kelly, N. Pandey, B. M. Hennelly, “Twin removal in digital holography using diffuse illumination,” Opt. Lett. 34, 3610–3612 (2009).
[CrossRef]

D. P. Kelly, B. M. Hennelly, N. Pandey, T. J. Naughton, W. T. Rhodes, “Resolution limits in practical digital holographic systems,” Opt. Eng. 48, 095801 (2009).
[CrossRef]

P. F. Almoro, P. N. Gundu, S. G. Hanson, “Numerical correction of aberrations via phase retrieval with speckle illumination,” Opt. Lett. 34, 521–523 (2009).
[CrossRef]

D. G. Voelz, M. C. Roggemann, “Digital simulation of scalar optical diffraction: revisiting chirp function sampling criteria and consequences,” Appl. Opt. 48, 6132–6142 (2009).
[CrossRef]

2008

P. Picart, J. Leval, “General theoretical formulation of image formation in digital Fresnel holography,” J. Opt. Soc. Am. A 25, 1744–1761 (2008).
[CrossRef]

D. P. Kelly, B. M. Hennelly, A. Grün, K. Unterrainer, “Numerical sampling rules for paraxial regime pulse diffraction calculations,” J. Opt. Soc. Am. A 25, 2299–2308 (2008).
[CrossRef]

2007

L. Onural, “Exact analysis of the effects of sampling of the scalar diffraction field,” J. Opt. Soc. Am. A 24, 359–367 (2007).
[CrossRef]

2006

A. Stern, B. Javidi, “Improved-resolution digital holography using the generalized sampling theorem for locally band-limited fields,” J. Opt. Soc. Am. A 23, 1227–1235 (2006).
[CrossRef]

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

O. V. Angelsky, A. P. Maksimyak, P. P. Maksimyak, S. G. Hanson, “Optical correlation diagnostics of rough surfaces with large surface inhomogeneities,” Opt. Express 14, 7299–7311 (2006).
[CrossRef]

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

F. Shen, A. Wang, “Fast-Fourier-transform based numerical integration method for the Rayleigh–Sommerfeld diffraction formula,” Appl. Opt. 45, 1102–1110 (2006).
[CrossRef]

D. P. Kelly, J. E. Ward, U. Gopinathan, B. M. Hennelly, F. T. O’Neill, J. T. Sheridan, “Generalized Yamaguchi correlation factor for coherent quadratic phase speckle metrology systems with an aperture,” Opt. Lett. 31, 3444–3446 (2006).
[CrossRef]

A. Sokołowski, “Consequences of sampling an image and Fourier planes in a numerical light propagation model based on the Helmholtz–Kirchhoff approximation,” J. Opt. Soc. Am. A 23, 2764–2767 (2006).
[CrossRef]

2005

B. M. Hennelly, 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]

D. P. Kelly, B. M. Hennelly, J. T. Sheridan, “Magnitude and direction of motion with speckle correlation and the optical fractional fourier transform,” Appl. Opt. 44, 2720–2727 (2005).
[CrossRef]

P. Picart, J. Leval, D. Mounier, S. Gougeon, “Some opportunities for vibration analysis with time averaging in digital Fresnel holography,” Appl. Opt. 44, 337–343 (2005).
[CrossRef]

2004

A. Stern, B. Javidi, “Analysis of practical sampling and reconstruction from Fresnel fields,” Opt. Eng. 43, 239–250 (2004).
[CrossRef]

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

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

2003

M. Liebling, T. Blu, M. Unser, “Fresnelets: new multiresolution wavelet bases for digital holography,” IEEE Trans. Image Process. 12, 29–43 (2003).
[CrossRef]

D. Mas, J. Garcia, C. Ferreira, L. M. Bernardo, F. Marinho, “Fast numerical calculation of Fresnel patterns in convergent systems,” Opt. Commun. 227, 245–258 (2003).
[CrossRef]

2002

C. J. R. Sheppard, “Three-dimensional phase imaging with the intensity transport equation,” Appl. Opt. 41, 5951–5955 (2002).
[CrossRef]

2000

V. Arrizón, M. Testorf, S. Sinzinger, J. Jahns, “Iterative optimization of phase-only diffractive optical elements based on a lenslet array,” J. Opt. Soc. Am. A 17, 2157–2164 (2000).
[CrossRef]

L. Onural, “Sampling of the diffraction field,” Appl. Opt. 39, 5929–5935 (2000).
[CrossRef]

1999

D. Mas, J. Garcia, C. Ferreira, L. M. Bernardo, F. Marinho, “Fast algorithms for free-space diffraction patterns calculation,” Opt. Commun. 164, 233–245 (1999).
[CrossRef]

G. Pedrini, P. Fröning, H. J. Tiziani, M. E. Gusev, “Pulsed digital holography for high-speed contouring that uses a two-wavelength method,” Appl. Opt. 38, 3460–3467 (1999).
[CrossRef]

1997

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

1996

A. W. Lohmann, R. G. Dorsch, D. Mendlovic, Z. Zalevsky, C. Ferreira, “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]

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

1993

L. Onural, “Diffraction from a wavelet point of view,” Opt. Lett. 18, 846–848 (1993).
[CrossRef]

A. W. Lohmann, “Image rotation, Wigner rotation, and the fractional Fourier transform,” J. Opt. Soc. Am. A 10, 2181–2186 (1993).
[CrossRef]

1983

M. R. Teague, “Deterministic phase retrieval: a Green’s function solution,” J. Opt. Soc. Am. 73, 1434–1441 (1983).
[CrossRef]

1982

J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21, 2758–2769 (1982).
[CrossRef]

1981

W. H. Southwell, “Validity of the Fresnel approximation in the near field,” J. Opt. Soc. Am. 71, 7–14 (1981).
[CrossRef]

F. Gori, “Fresnel transform and sampling theorem,” Opt. Commun. 39, 293–297 (1981).
[CrossRef]

Almoro, P. F.

P. F. Almoro, P. N. Gundu, S. G. Hanson, “Numerical correction of aberrations via phase retrieval with speckle illumination,” Opt. Lett. 34, 521–523 (2009).
[CrossRef]

Angelsky, O. V.

O. V. Angelsky, A. P. Maksimyak, P. P. Maksimyak, S. G. Hanson, “Optical correlation diagnostics of rough surfaces with large surface inhomogeneities,” Opt. Express 14, 7299–7311 (2006).
[CrossRef]

Arrizón, V.

V. Arrizón, M. Testorf, S. Sinzinger, J. Jahns, “Iterative optimization of phase-only diffractive optical elements based on a lenslet array,” J. Opt. Soc. Am. A 17, 2157–2164 (2000).
[CrossRef]

Bernardo, L. M.

D. Mas, J. Garcia, C. Ferreira, L. M. Bernardo, F. Marinho, “Fast numerical calculation of Fresnel patterns in convergent systems,” Opt. Commun. 227, 245–258 (2003).
[CrossRef]

D. Mas, J. Garcia, C. Ferreira, L. M. Bernardo, F. Marinho, “Fast algorithms for free-space diffraction patterns calculation,” Opt. Commun. 164, 233–245 (1999).
[CrossRef]

Blu, T.

N. Chacko, M. Liebling, T. Blu, “Discretization of continuous convolution operators for accurate modeling of wave propagation in digital holography,” J. Opt. Soc. Am. A 30, 2012–2020 (2013).
[CrossRef]

M. Liebling, T. Blu, M. Unser, “Fresnelets: new multiresolution wavelet bases for digital holography,” IEEE Trans. Image Process. 12, 29–43 (2003).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, 7th ed. (Cambridge University, 1999).

Bracewell, R.

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

Chacko, N.

N. Chacko, M. Liebling, T. Blu, “Discretization of continuous convolution operators for accurate modeling of wave propagation in digital holography,” J. Opt. Soc. Am. A 30, 2012–2020 (2013).
[CrossRef]

Claus, D.

D. P. Kelly, D. Claus, “Filtering role of the sensor pixel in Fourier and Fresnel digital holography,” Appl. Opt. 52, A336–A345 (2013).
[CrossRef]

Dorsch, R. G.

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

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

Eriksson, I.

I. Eriksson, P. Haglund, J. Powell, M. Sjodahl, A. F. H. Kaplan, “Holographic measurement of thermal distortion during laser spot welding,” Opt. Eng. 51, 030501 (2012).
[CrossRef]

Falaggis, K.

T. Kozacki, K. Falaggis, M. Kujawinska, “Computation of diffracted fields for the case of high numerical aperture using the angular spectrum method,” Appl. Opt. 51, 7080–7088 (2012).
[CrossRef]

Ferreira, C.

D. Mas, J. Garcia, C. Ferreira, L. M. Bernardo, F. Marinho, “Fast numerical calculation of Fresnel patterns in convergent systems,” Opt. Commun. 227, 245–258 (2003).
[CrossRef]

D. Mas, J. Garcia, C. Ferreira, L. M. Bernardo, F. Marinho, “Fast algorithms for free-space diffraction patterns calculation,” Opt. Commun. 164, 233–245 (1999).
[CrossRef]

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

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

Fienup, J. R.

M. Guizar-Sicairos, J. R. Fienup, “Understanding the twin-image problem in phase retrieval,” J. Opt. Soc. Am. A 29, 2367–2375 (2012).
[CrossRef]

J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21, 2758–2769 (1982).
[CrossRef]

Forbes, G. W.

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

Fröning, P.

G. Pedrini, P. Fröning, H. J. Tiziani, M. E. Gusev, “Pulsed digital holography for high-speed contouring that uses a two-wavelength method,” Appl. Opt. 38, 3460–3467 (1999).
[CrossRef]

Gan, X. S.

M. Gu, 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]

Garcia, J.

D. Mas, J. Garcia, C. Ferreira, L. M. Bernardo, F. Marinho, “Fast numerical calculation of Fresnel patterns in convergent systems,” Opt. Commun. 227, 245–258 (2003).
[CrossRef]

D. Mas, J. Garcia, C. Ferreira, L. M. Bernardo, F. Marinho, “Fast algorithms for free-space diffraction patterns calculation,” Opt. Commun. 164, 233–245 (1999).
[CrossRef]

Goodman, J.

J. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1966).

Goodman, J. W.

J. W. Goodman, Speckle Phenomena in Optics (Roberts and Company, 2007).

Gopinathan, U.

D. P. Kelly, J. E. Ward, U. Gopinathan, B. M. Hennelly, F. T. O’Neill, J. T. Sheridan, “Generalized Yamaguchi correlation factor for coherent quadratic phase speckle metrology systems with an aperture,” Opt. Lett. 31, 3444–3446 (2006).
[CrossRef]

Gori, F.

F. Gori, “Fresnel transform and sampling theorem,” Opt. Commun. 39, 293–297 (1981).
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Figures (4)

Fig. 1.
Fig. 1.

Illustration of the diffraction process. An infinite plane wave is incident on the aperture in Plane 1, denoted by the spatial variable X. Consider how to calculate the complex amplitude at P ( x , z ) . Only the light that lies within the aperture opening can contribute to the complex amplitude at P ( x , z ) . We envisage the light within the aperture opening as consisting of an infinite number of secondary point sources (PS), each with an intensity and phase value (only a few PS are shown here). Since the aperture is illuminated with an ideal plane wave, each secondary PS will have an identical intensity and phase value in Plane 1. Nevertheless, each PS will accumulate a different phase value P ( x , z ) due to the different path lengths, compared to the distance l 1 and l 4 .

Fig. 2.
Fig. 2.

Result of the diffraction calculation. The presence of a replica is clearly visible at x = 2.5 mm . The red dotted line indicates the location of S E / 2 , illustrating the extent of the diffracted field. The distribution plotted in black has been spatially filtered and contains no spatial frequencies higher than 35 lines / mm . Note the finite extent of the signal in comparison to the blue plot. (For presentation purposes, the central order replica in blue is not plotted over the range S E / 2 x S E / 2 .)

Fig. 3.
Fig. 3.

Result of the diffraction calculation for the phase distributions. Both plots are over a range of δ X . The plot on the left is the phase of the zero-order replica (offset to zero). The plot on the right is for the first replica, where there is a constant phase offset of π / 4 and a linear phase with slope 1 / δ X .

Fig. 4.
Fig. 4.

Reproducing the numerical results from Voelz and Roggemann [30].

Equations (34)

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u z ( x ) = FST z { U ( X ) } ( x ) = exp ( j 2 π z / λ ) j λ z U ( X ) exp [ j π λ z ( x X ) 2 ] d X ,
U ( X ) = exp ( X 2 α i 2 ) cos ( 2 π f x X ) ,
U ( X ) = 1 2 exp ( X 2 α i 2 ) [ exp ( j 2 π f x X ) + exp ( j 2 π f x X ) ] ,
u z ( x ) = 1 2 [ u z + ( x ) + u z ( x ) ]
u z + ( x ) = K z exp [ ( x λ z f x ) 2 α z 2 ] exp [ j ( ϕ z l x + ϕ z q x 2 ) ] ,
α z = ( π α i π 2 α i 4 + z 2 λ 2 ) 1 ,
K z = ( 1 + j z λ π α i 2 ) 1 exp ( j π α i 4 π 2 f x 2 z λ π 2 α i 4 + z 2 λ 2 ) ,
ϕ z l = 2 π 3 f x α i 4 π 2 α i 4 + z 2 λ 2 ,
ϕ z q = π z 2 λ 2 π 2 λ z α i 4 + z 3 λ 3 .
S E = 2 ( λ z f x + 2 α z ) .
U = [ U ( X 1 ) , U ( X 2 ) , U ( X N ) ] , = [ U 1 , U 2 , U N ] ,
X = [ X 1 , X 2 , X N ] ,
u z S ( x ) = δ X j λ z n = 1 N U n exp [ j π λ z ( x X n ) 2 ] , = δ X j λ z exp ( j π x 2 λ z ) × { n = 1 N U n exp ( j π X n 2 λ z ) exp ( j 2 π x X n λ z ) } ,
U ( X ) = p L ( X ) cos ( 2 π Γ X ) ,
p L ( x ) = { 1 , when | x | < L , 0 , otherwise .
u z S ( x ) = m = u z ( x m λ z δ X ) × exp ( j π λ z m 2 δ X 2 ) exp ( j 2 π m x δ X ) .
S E < λ z / δ X ,
P i = U ( X ) 2 d X ,
f ˜ ( v ) = FT { f ( x ) } ( v ) = f ( x ) exp ( j 2 π x v ) d x ,
f ( x ) = IFT { f ˜ ( v ) } ( v ) = f ( x ) exp ( j 2 π x v ) d v ,
u z ( x ) = IFT { exp ( j π λ z v 2 ) F T { U ( X ) } ( v ) } ( x ) .
U ˜ S ( v ) = δ X n = 1 N U n exp ( j 2 π v X n ) ,
u z S ( x ) = exp ( j π λ z v 2 ) U ˜ S ( v ) exp ( j 2 π x v ) d v ,
P v = F R / 2 F R / 2 U ˜ ( v ) 2 d v ,
v = [ F R / 2 , F R / 2 + δ v + + F R / 2 δ v ] ,
U ˜ S = [ U ˜ S ( v 1 ) , U ˜ S ( v 2 ) , U ˜ S ( v N ) ] , = [ U ˜ 1 S , U ˜ 2 S , U ˜ N S ] .
u z S S C ( x ) = δ v { n = 1 N U ˜ n S exp ( j π λ z v n 2 ) exp ( + j 2 π x v n ) } ,
S E < 1 / δ v ,
U zp = [ 0 , 0 , 0 , U 0 , 0 , 0 ] ,
U iz = [ 0 , 0 , U 1 , 0 , 0 , U 2 , 0 , 0 , U 3 , 0 , 0 ] ,
u z ( x ) = FST z { U ( X ) } ( x ) .
δ T ( X ) = m = δ ( x m δ X ) ,
δ T ( X ) = 1 δ X m = exp ( j 2 π m δ X X ) .
u z S ( x ) = FST z { U ( X ) δ T ( X ) } ( x ) = m = FST z { U ( X ) exp ( j 2 π m δ X X ) } ( x ) = m = u z ( x m λ z δ X ) × exp ( j π λ z m 2 δ X 2 ) exp ( j 2 π m x δ X ) ,

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