Abstract

A dual-wavelength femtosecond laser pulse source and its application for digital holographic single-shot contouring are presented. The synthesized laser source combines sub-picosecond time scales with a wide reconstruction range. A center wavelength distance of the two separated pulses of only 15 nm with a high contrast was demonstrated by spectral shaping of the 50 nm broad seed spectrum centered at 800 nm. Owing to the resulting synthetic wavelength, the scan depth range without phase ambiguity is extended to the 100-μm-range. Single-shot dual-wavelength imaging is achieved by using two CMOS cameras in a Twyman-Green interferometer, which is extended by a polarization encoding sequence to separate the holograms. The principle of the method is revealed, and experimental results concerning a single axis scanner mirror operating at a resonance frequency of 0.5 kHz are presented. Within the synthetic wavelength, the phase difference information of the object was unambiguously retrieved and the 3D-shape calculated. To the best of our knowledge, this is the first time that single-shot two-wavelength contouring on a sub-ps time scale is reported.

©2009 Optical Society of America

Full Article  |  PDF Article
OSA Recommended Articles
Real-time dual-wavelength digital holographic microscopy with a single hologram acquisition

Jonas Kühn, Tristan Colomb, Frédéric Montfort, Florian Charrière, Yves Emery, Etienne Cuche, Pierre Marquet, and Christian Depeursinge
Opt. Express 15(12) 7231-7242 (2007)

Pulsed digital holography for high-speed contouring that uses a two-wavelength method

Giancarlo Pedrini, Philipp Fröning, Hans J. Tiziani, and Mikhail E. Gusev
Appl. Opt. 38(16) 3460-3467 (1999)

Simultaneous three-dimensional dynamic deformation measurements with pulsed digital holography

Staffan Schedin, Giancarlo Pedrini, Hans J. Tiziani, and Fernando Mendoza Santoyo
Appl. Opt. 38(34) 7056-7062 (1999)

More Recommended Articles

References

  • View by:
  • Article Order
  • |
  • Year
  • |
  • Author
  • |
  • Publication
  1. Z. Liu, M. Centurion, G. Panotopoulus, J. Hong, and D. Psaltis, “Holographic recording of fast events on a CCD camera,” Opt. Lett. 27, 22–24 (2002).
    [Crossref]
  2. L. O. Heflinger, R. F. Wuerker, and R. E. Brooks, “Holographic Interferometry,” J. Appl. Phys. 37, 642–649 (1966).
    [Crossref]
  3. T. Tschudi, C. Yamanaka, T. Sasaki, K. Yoshida, and K. Tanaka, “A study of high-power laser effects in dielectrics using multiframe picosecond holography,” J. Phys. D11, 177–180 (1978).
  4. W. Hentschel and W. Lauterborn, “High-Speed Holographic Movie Camera,” Opt. Eng. 24, 687–691 (1985).
  5. P. Picart, J. Leval, D. Mounier, and S. Gougeon, “Time-averaged digital holography,” Opt. Lett. 28, 1900–1902 (2003).
    [Crossref] [PubMed]
  6. U. Schnars and W. Jueptner, Digital Holography: Digital Hologram Recording, Numerical Reconstruction, and Related Techniques, (Springer Verlag, 2005).
  7. B. P. Hildebrand and K. A. Haines,”Multiple-wavelength and multiple-source holography applied to contour generation,” J. Opt. Soc. Am. 57, 155–162 (1967).
    [Crossref]
  8. A. A. Friesem and U. Levy, “Fringe formation in two-wavelength contour holography,” Appl. Opt. 15, 3009–3020 (1976).
    [Crossref] [PubMed]
  9. T. Gessner, J. Bonitz, C. Kaufmann, S. Kurth, and H. Specht, “MEMS based micro scanners: Components, Technologies and Applications,” Actuator 2006, 10th Intern. Conf. on New Actuators, 193–198 (2006).
  10. U. Schnars and W. Jüptner, “Direct recording of holograms by a CCD target and numerical reconstruction,” Appl. Opt. 33, 179–181 (1994).
    [Crossref] [PubMed]
  11. C. Wagner, W. Osten, and S. Seebacher, “Direct shape measurement by digital wavefront reconstruction and multiwavelength contouring,” Opt. Eng. 39, 79–85 (2000).
    [Crossref]
  12. J. Kühn, T. Colomb, F. Montfort, F. Charrière, Y. Emery, E. Cuche, P. Marquet, and C. Depeursinge, “Realtime dual-wavelength digital holographic microscopy with a single hologram acquisition,” Opt. Express 15, 7231–7242 (2007).
    [Crossref] [PubMed]
  13. X. Wang, H. Zhai, and G. Mu, “Pulsed digital holography system recording ultrafast process of the femtosecond order,” Opt. Lett. 31, 1636–1638 (2006).
    [Crossref] [PubMed]
  14. G. Pedrini, P. Froening, H. J. Tiziani, and M. E. Gusev, “Pulsed digital holography for high-speed contouring that uses a two-wavelength method, ” Appl. Opt. 38, 3460–3467 (1999).
    [Crossref]
  15. J. F. Xia, J. Song, and D. Strickland, “Development of a dual-wavelength Ti:sapphire multi-pass amplifier and its application to intense mid-infrared generation,” Opt. Commun. 206, 149–157 (2002).
    [Crossref]
  16. Z. Zhang, A. M. Deslauriers, and D. Strickland, “Dual-wavelength chirped-pulse amplification system,” Opt. Lett. 25, 581–583 (2000).
    [Crossref]
  17. C. P. J. Barty, G. Korn, F. Raksi, C. Rose-Petruck, J. Squier, A.-C. Tien, K. R. Wilson, V. V. Yakovlev, and K. Yamakawa, “Regenerative pulse shaping and amplification of ultrabroadband optical pulses,” Opt. Lett. 21, 219–221 (1996).
    [Crossref] [PubMed]
  18. A. Leitenstorfer, C. Fürst, and A. Laubereau, “Widely tunable two-color mode-locked Ti:sapphire laser with pulse jitter of less than 2 fs,” Opt. Lett. 20, 916–918 (1995)
    [Crossref] [PubMed]
  19. P. de Groot and S. Kishner, “Synthetic wavelength stabilization for two-color laser-diode interferometry,” Appl. Opt. 30, 4026–4033 (1991).
    [Crossref] [PubMed]
  20. E. N. Leith and J. Upatnieks, “Reconstructed wavefronts and communication theory,” J. Opt. Soc. Am. 52, 1123–1130 (1962).
    [Crossref]
  21. W. Lauterbach and T. Kurz, Coherent Optics: fundamentals and applications, (Springer Verlag, 2003).
  22. B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics, (Wiley & Sons Inc., 1991).
  23. M. Hentschel, Z. Cheng, F. Krausz, and C. Spielmann, “Generation of 0:1-TWoptical pulses with a single stage Ti:sapphire amplifier at a 1-kHz repetition rate,” Appl. Phys. B 70, 161–164 (2000).
  24. M. Takeda, I. Ina, and S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. 72, 156–160 (1982).
    [Crossref]
  25. U. Schnars, T. M. Kreis, and W. Jüptner, “Digital recording and numerical reconstruction of holograms: reduction of the spatial frequency spectrum,” Opt. Eng. 35, 977–982 (1996).
    [Crossref]
  26. T. Kreis, Holographic Interferometry: Principles and Methods (Akademie Verlag, 1996).
  27. J. W. Goodman, Introduction to Fourier Optics, 3rd ed (Roberts & Company Publishers, 2005).

2007 (1)

2006 (1)

2005 (1)

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

2003 (1)

2002 (2)

Z. Liu, M. Centurion, G. Panotopoulus, J. Hong, and D. Psaltis, “Holographic recording of fast events on a CCD camera,” Opt. Lett. 27, 22–24 (2002).
[Crossref]

J. F. Xia, J. Song, and D. Strickland, “Development of a dual-wavelength Ti:sapphire multi-pass amplifier and its application to intense mid-infrared generation,” Opt. Commun. 206, 149–157 (2002).
[Crossref]

2000 (3)

M. Hentschel, Z. Cheng, F. Krausz, and C. Spielmann, “Generation of 0:1-TWoptical pulses with a single stage Ti:sapphire amplifier at a 1-kHz repetition rate,” Appl. Phys. B 70, 161–164 (2000).

Z. Zhang, A. M. Deslauriers, and D. Strickland, “Dual-wavelength chirped-pulse amplification system,” Opt. Lett. 25, 581–583 (2000).
[Crossref]

C. Wagner, W. Osten, and S. Seebacher, “Direct shape measurement by digital wavefront reconstruction and multiwavelength contouring,” Opt. Eng. 39, 79–85 (2000).
[Crossref]

1999 (1)

1996 (2)

C. P. J. Barty, G. Korn, F. Raksi, C. Rose-Petruck, J. Squier, A.-C. Tien, K. R. Wilson, V. V. Yakovlev, and K. Yamakawa, “Regenerative pulse shaping and amplification of ultrabroadband optical pulses,” Opt. Lett. 21, 219–221 (1996).
[Crossref] [PubMed]

U. Schnars, T. M. Kreis, and W. Jüptner, “Digital recording and numerical reconstruction of holograms: reduction of the spatial frequency spectrum,” Opt. Eng. 35, 977–982 (1996).
[Crossref]

1995 (1)

1994 (1)

1991 (1)

1985 (1)

W. Hentschel and W. Lauterborn, “High-Speed Holographic Movie Camera,” Opt. Eng. 24, 687–691 (1985).

1982 (1)

1978 (1)

T. Tschudi, C. Yamanaka, T. Sasaki, K. Yoshida, and K. Tanaka, “A study of high-power laser effects in dielectrics using multiframe picosecond holography,” J. Phys. D11, 177–180 (1978).

1976 (1)

1967 (1)

1966 (1)

L. O. Heflinger, R. F. Wuerker, and R. E. Brooks, “Holographic Interferometry,” J. Appl. Phys. 37, 642–649 (1966).
[Crossref]

1962 (1)

Barty, C. P. J.

Bonitz, J.

T. Gessner, J. Bonitz, C. Kaufmann, S. Kurth, and H. Specht, “MEMS based micro scanners: Components, Technologies and Applications,” Actuator 2006, 10th Intern. Conf. on New Actuators, 193–198 (2006).

Brooks, R. E.

L. O. Heflinger, R. F. Wuerker, and R. E. Brooks, “Holographic Interferometry,” J. Appl. Phys. 37, 642–649 (1966).
[Crossref]

Centurion, M.

Charrière, F.

Cheng, Z.

M. Hentschel, Z. Cheng, F. Krausz, and C. Spielmann, “Generation of 0:1-TWoptical pulses with a single stage Ti:sapphire amplifier at a 1-kHz repetition rate,” Appl. Phys. B 70, 161–164 (2000).

Colomb, T.

Cuche, E.

de Groot, P.

Depeursinge, C.

Deslauriers, A. M.

Emery, Y.

Friesem, A. A.

Froening, P.

Fürst, C.

Gessner, T.

T. Gessner, J. Bonitz, C. Kaufmann, S. Kurth, and H. Specht, “MEMS based micro scanners: Components, Technologies and Applications,” Actuator 2006, 10th Intern. Conf. on New Actuators, 193–198 (2006).

Goodman, J. W.

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

Gougeon, S.

Gusev, M. E.

Haines, K. A.

Heflinger, L. O.

L. O. Heflinger, R. F. Wuerker, and R. E. Brooks, “Holographic Interferometry,” J. Appl. Phys. 37, 642–649 (1966).
[Crossref]

Hentschel, M.

M. Hentschel, Z. Cheng, F. Krausz, and C. Spielmann, “Generation of 0:1-TWoptical pulses with a single stage Ti:sapphire amplifier at a 1-kHz repetition rate,” Appl. Phys. B 70, 161–164 (2000).

Hentschel, W.

W. Hentschel and W. Lauterborn, “High-Speed Holographic Movie Camera,” Opt. Eng. 24, 687–691 (1985).

Hildebrand, B. P.

Hong, J.

Ina, I.

Jueptner, W.

U. Schnars and W. Jueptner, Digital Holography: Digital Hologram Recording, Numerical Reconstruction, and Related Techniques, (Springer Verlag, 2005).

Jüptner, W.

U. Schnars, T. M. Kreis, and W. Jüptner, “Digital recording and numerical reconstruction of holograms: reduction of the spatial frequency spectrum,” Opt. Eng. 35, 977–982 (1996).
[Crossref]

U. Schnars and W. Jüptner, “Direct recording of holograms by a CCD target and numerical reconstruction,” Appl. Opt. 33, 179–181 (1994).
[Crossref] [PubMed]

Kaufmann, C.

T. Gessner, J. Bonitz, C. Kaufmann, S. Kurth, and H. Specht, “MEMS based micro scanners: Components, Technologies and Applications,” Actuator 2006, 10th Intern. Conf. on New Actuators, 193–198 (2006).

Kishner, S.

Kobayashi, S.

Korn, G.

Krausz, F.

M. Hentschel, Z. Cheng, F. Krausz, and C. Spielmann, “Generation of 0:1-TWoptical pulses with a single stage Ti:sapphire amplifier at a 1-kHz repetition rate,” Appl. Phys. B 70, 161–164 (2000).

Kreis, T.

T. Kreis, Holographic Interferometry: Principles and Methods (Akademie Verlag, 1996).

Kreis, T. M.

U. Schnars, T. M. Kreis, and W. Jüptner, “Digital recording and numerical reconstruction of holograms: reduction of the spatial frequency spectrum,” Opt. Eng. 35, 977–982 (1996).
[Crossref]

Kühn, J.

Kurth, S.

T. Gessner, J. Bonitz, C. Kaufmann, S. Kurth, and H. Specht, “MEMS based micro scanners: Components, Technologies and Applications,” Actuator 2006, 10th Intern. Conf. on New Actuators, 193–198 (2006).

Kurz, T.

W. Lauterbach and T. Kurz, Coherent Optics: fundamentals and applications, (Springer Verlag, 2003).

Laubereau, A.

Lauterbach, W.

W. Lauterbach and T. Kurz, Coherent Optics: fundamentals and applications, (Springer Verlag, 2003).

Lauterborn, W.

W. Hentschel and W. Lauterborn, “High-Speed Holographic Movie Camera,” Opt. Eng. 24, 687–691 (1985).

Leitenstorfer, A.

Leith, E. N.

Leval, J.

Levy, U.

Liu, Z.

Marquet, P.

Montfort, F.

Mounier, D.

Mu, G.

Osten, W.

C. Wagner, W. Osten, and S. Seebacher, “Direct shape measurement by digital wavefront reconstruction and multiwavelength contouring,” Opt. Eng. 39, 79–85 (2000).
[Crossref]

Panotopoulus, G.

Pedrini, G.

Picart, P.

Psaltis, D.

Raksi, F.

Rose-Petruck, C.

Saleh, B. E. A.

B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics, (Wiley & Sons Inc., 1991).

Sasaki, T.

T. Tschudi, C. Yamanaka, T. Sasaki, K. Yoshida, and K. Tanaka, “A study of high-power laser effects in dielectrics using multiframe picosecond holography,” J. Phys. D11, 177–180 (1978).

Schnars, U.

U. Schnars, T. M. Kreis, and W. Jüptner, “Digital recording and numerical reconstruction of holograms: reduction of the spatial frequency spectrum,” Opt. Eng. 35, 977–982 (1996).
[Crossref]

U. Schnars and W. Jüptner, “Direct recording of holograms by a CCD target and numerical reconstruction,” Appl. Opt. 33, 179–181 (1994).
[Crossref] [PubMed]

U. Schnars and W. Jueptner, Digital Holography: Digital Hologram Recording, Numerical Reconstruction, and Related Techniques, (Springer Verlag, 2005).

Seebacher, S.

C. Wagner, W. Osten, and S. Seebacher, “Direct shape measurement by digital wavefront reconstruction and multiwavelength contouring,” Opt. Eng. 39, 79–85 (2000).
[Crossref]

Song, J.

J. F. Xia, J. Song, and D. Strickland, “Development of a dual-wavelength Ti:sapphire multi-pass amplifier and its application to intense mid-infrared generation,” Opt. Commun. 206, 149–157 (2002).
[Crossref]

Specht, H.

T. Gessner, J. Bonitz, C. Kaufmann, S. Kurth, and H. Specht, “MEMS based micro scanners: Components, Technologies and Applications,” Actuator 2006, 10th Intern. Conf. on New Actuators, 193–198 (2006).

Spielmann, C.

M. Hentschel, Z. Cheng, F. Krausz, and C. Spielmann, “Generation of 0:1-TWoptical pulses with a single stage Ti:sapphire amplifier at a 1-kHz repetition rate,” Appl. Phys. B 70, 161–164 (2000).

Squier, J.

Strickland, D.

J. F. Xia, J. Song, and D. Strickland, “Development of a dual-wavelength Ti:sapphire multi-pass amplifier and its application to intense mid-infrared generation,” Opt. Commun. 206, 149–157 (2002).
[Crossref]

Z. Zhang, A. M. Deslauriers, and D. Strickland, “Dual-wavelength chirped-pulse amplification system,” Opt. Lett. 25, 581–583 (2000).
[Crossref]

Takeda, M.

Tanaka, K.

T. Tschudi, C. Yamanaka, T. Sasaki, K. Yoshida, and K. Tanaka, “A study of high-power laser effects in dielectrics using multiframe picosecond holography,” J. Phys. D11, 177–180 (1978).

Teich, M. C.

B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics, (Wiley & Sons Inc., 1991).

Tien, A.-C.

Tiziani, H. J.

Tschudi, T.

T. Tschudi, C. Yamanaka, T. Sasaki, K. Yoshida, and K. Tanaka, “A study of high-power laser effects in dielectrics using multiframe picosecond holography,” J. Phys. D11, 177–180 (1978).

Upatnieks, J.

Wagner, C.

C. Wagner, W. Osten, and S. Seebacher, “Direct shape measurement by digital wavefront reconstruction and multiwavelength contouring,” Opt. Eng. 39, 79–85 (2000).
[Crossref]

Wang, X.

Wilson, K. R.

Wuerker, R. F.

L. O. Heflinger, R. F. Wuerker, and R. E. Brooks, “Holographic Interferometry,” J. Appl. Phys. 37, 642–649 (1966).
[Crossref]

Xia, J. F.

J. F. Xia, J. Song, and D. Strickland, “Development of a dual-wavelength Ti:sapphire multi-pass amplifier and its application to intense mid-infrared generation,” Opt. Commun. 206, 149–157 (2002).
[Crossref]

Yakovlev, V. V.

Yamakawa, K.

Yamanaka, C.

T. Tschudi, C. Yamanaka, T. Sasaki, K. Yoshida, and K. Tanaka, “A study of high-power laser effects in dielectrics using multiframe picosecond holography,” J. Phys. D11, 177–180 (1978).

Yoshida, K.

T. Tschudi, C. Yamanaka, T. Sasaki, K. Yoshida, and K. Tanaka, “A study of high-power laser effects in dielectrics using multiframe picosecond holography,” J. Phys. D11, 177–180 (1978).

Zhai, H.

Zhang, Z.

Appl. Opt. (4)

Appl. Phys. B (1)

M. Hentschel, Z. Cheng, F. Krausz, and C. Spielmann, “Generation of 0:1-TWoptical pulses with a single stage Ti:sapphire amplifier at a 1-kHz repetition rate,” Appl. Phys. B 70, 161–164 (2000).

J. Appl. Phys. (1)

L. O. Heflinger, R. F. Wuerker, and R. E. Brooks, “Holographic Interferometry,” J. Appl. Phys. 37, 642–649 (1966).
[Crossref]

J. Opt. Soc. Am. (3)

J. Phys. (1)

T. Tschudi, C. Yamanaka, T. Sasaki, K. Yoshida, and K. Tanaka, “A study of high-power laser effects in dielectrics using multiframe picosecond holography,” J. Phys. D11, 177–180 (1978).

Opt. Commun. (1)

J. F. Xia, J. Song, and D. Strickland, “Development of a dual-wavelength Ti:sapphire multi-pass amplifier and its application to intense mid-infrared generation,” Opt. Commun. 206, 149–157 (2002).
[Crossref]

Opt. Eng. (3)

C. Wagner, W. Osten, and S. Seebacher, “Direct shape measurement by digital wavefront reconstruction and multiwavelength contouring,” Opt. Eng. 39, 79–85 (2000).
[Crossref]

W. Hentschel and W. Lauterborn, “High-Speed Holographic Movie Camera,” Opt. Eng. 24, 687–691 (1985).

U. Schnars, T. M. Kreis, and W. Jüptner, “Digital recording and numerical reconstruction of holograms: reduction of the spatial frequency spectrum,” Opt. Eng. 35, 977–982 (1996).
[Crossref]

Opt. Express (1)

Opt. Lett. (6)

Other (6)

W. Lauterbach and T. Kurz, Coherent Optics: fundamentals and applications, (Springer Verlag, 2003).

B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics, (Wiley & Sons Inc., 1991).

T. Kreis, Holographic Interferometry: Principles and Methods (Akademie Verlag, 1996).

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

U. Schnars and W. Jueptner, Digital Holography: Digital Hologram Recording, Numerical Reconstruction, and Related Techniques, (Springer Verlag, 2005).

T. Gessner, J. Bonitz, C. Kaufmann, S. Kurth, and H. Specht, “MEMS based micro scanners: Components, Technologies and Applications,” Actuator 2006, 10th Intern. Conf. on New Actuators, 193–198 (2006).

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. Setup for the generation of two spectrally separated sub-ps pulses via spectral shaping of the seed pulse spectrum.

Download Full Size | PPT Slide | PDF

Fig. 2.
Fig. 2. (a) Dual-wavelength pulse spectrum together with seed spectrum (Dλ - separation of the spectral maxima of 5w; λi; Δλi - spectral width); (b) Autocorrelation traces of the two generated pulses and a single pulse (inset, Δt - temporal pulse separation; τP - single pulse duration).

Download Full Size | PPT Slide | PDF

Fig. 3.
Fig. 3. Setup for digital holographic single shot two wavelength contouring applying two CCD-cameras. The polarization direction is indicated at the position of the cameras. (BS: beam splitter, PBS: polarization beam splitter). Inset: Scheme of the single axis scanner - MEMS (test object).

Download Full Size | PPT Slide | PDF

Fig. 4.
Fig. 4. Optical spectra behind the polarization beam splitter in the two camera arms: (a) Suppression of λ2 for camera 1, (b) Suppression of λ1 for camera 2.

Download Full Size | PPT Slide | PDF

Fig. 5.
Fig. 5. Simultaneously captured holograms of the oscillating mirror, (a) At λ1 = 772 nm by camera 1, (b) At λ2 = 787 nm by camera 2. The recorded mirror elongation is limited by the size of the CCD-camera chip.

Download Full Size | PPT Slide | PDF

Fig. 6.
Fig. 6. Reconstructed single phases with corrected slope, (a) At λ1 = 772 nm for the hologram of camera 1, (b) At λ2 = 787 nm for the hologram of camera 2, (c) Calculated phase difference of the spatially correlated single phases in (a) and (b) without slope correction.

Download Full Size | PPT Slide | PDF

Fig. 7.
Fig. 7. 3D contour map of the unwrapped difference phase image of (a) the static mirror and (b) the oscillating mirror at its turning point. (c) Cut along the surface of the oscillating mirror at its turning point. The corresponding position is marked as white line in (b).

Download Full Size | PPT Slide | PDF

Fig. 8.
Fig. 8. (a) 3D surface shape calculated from the reconstructed single phase at λ2 = 787 nm. (b) Corrected 3D-plot of the mirror at its turning point using the single phase and slope evaluations. (please note the different height-scales of Figs. (a) and (b)).

Download Full Size | PPT Slide | PDF

Equations (10)

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

E h = E ref + E obj .
E ( r , t ) ≕ E ref E ( r , t + τ ) ≕ E obj .
I h = I ref + I obj + 2 Re { G ( τ ) } .
τ c = ( 2 ln ⁡ 2 / π ) 1 / 2 Δ ν ,
l c = c · τ c ,
Δ φ = φ 1 − φ 2 = 2 π l λ 1 − 2 π l λ 2 = 2 πl · ( ∣ λ 1 − λ 2 ∣ λ 1 · λ 2 ) = 2 π l Λ ,
Λ = λ 1 · λ 2 ∣ λ 1 − λ 2 ∣ .
O ( x ′ , y ′ , d i ) = i λ ∫ ∫ − ∞ ∞ h i ( x , y ) r i ( x , y ) exp ⁡ ( − i 2 π λ i R ) R ( 1 2 cos ⁡ ( θ S ) + 1 2 cos ⁡ ( θ B ) ) dxdy ,
R = ( x ′ − x ) 2 + ( y ′ − y ) 2 + d i 2 .
δ = Δ φ 4 π · Λ .

Metrics