Abstract

We show that phase objects may be computed accurately from a single color image in a brightfield microscope, with no hardware modification. Our technique uses the chromatic aberration that is inherent to every lens-based imaging system as a phase contrast mechanism. This leads to a simple and inexpensive way of achieving single-shot quantitative phase recovery by a modified Transport of Intensity Equation (TIE) solution, allowing real-time phase imaging in a traditional microscope.

©2010 Optical Society of America

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References

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  1. F. Zernike, “How I discovered phase contrast,” Science 121(3141), 345–349 (1955).
    [Crossref] [PubMed]
  2. C. J. R. Sheppard, “Defocused transfer function for a partially coherent microscope and application to phase retrieval,” J. Opt. Soc. Am. A 21(5), 828–831 (2004).
    [Crossref]
  3. H. Hopkins, “The frequency response of a defocused optical system,” Proc. Royal Soc. London, Ser. A 231(1184), 91–103 (1955).
    [Crossref]
  4. F. Zernike, “Phase contrast, a new method for the microscopic observation of transparent objects,” Physica 9(7), 686–698 (1942).
    [Crossref]
  5. P. Marquet, B. Rappaz, P. J. Magistretti, E. Cuche, Y. Emery, T. Colomb, and C. Depeursinge, “Digital holographic microscopy: a noninvasive contrast imaging technique allowing quantitative visualization of living cells with subwavelength axial accuracy,” Opt. Lett. 30(5), 468–470 (2005).
    [Crossref] [PubMed]
  6. G. Popescu, L. P. Deflores, J. C. Vaughan, K. Badizadegan, H. Iwai, R. R. Dasari, and M. S. Feld, “Fourier phase microscopy for investigation of biological structures and dynamics,” Opt. Lett. 29(21), 2503–2505 (2004).
    [Crossref] [PubMed]
  7. J. Millerd, N. Brock, J. Hayes, M. North-Morris, B. Kimbrough, and J. Wyant, Fringe 2005: Pixelated Phase-mask Dynamic Interferometers (Springer, 2003).
  8. B. C. Platt and R. Shack, “History and principles of Shack–Hartmann wavefront sensing,” J. Refract. Surg. 17(5), S573–S577 (2001).
    [PubMed]
  9. X. Cui, J. Ren, G. J. Tearney, and C. Yang, “Wavefront image sensor chip,” Opt. Express 18(16), 16685–16701 (2010).
    [Crossref] [PubMed]
  10. L. Waller, and G. Barbastathis, “Phase from Defocused Color Images,” in Frontiers in Optics, OSA Technical Digest (CD) (Optical Society of America, 2009), paper FThR3.
  11. M. Born, and E. Wolf, Principles of Optics (Cambridge Univ. Press, 1999).
  12. H. King, The History of the Telescope (Dover, 2003).
  13. M. Teague, “Deterministic phase retrieval: a Green’s function solution,” J. Opt. Soc. Am. A 73, 1434–1441(1983).
    [Crossref]
  14. D. Paganin and K. Nugent, “Noninterferometric phase imaging with partially coherent light,” Phys. Rev. Lett. 80(12), 2586–2589 (1998).
    [Crossref]
  15. E. D. Barone-Nugent, A. Barty, and K. A. Nugent, “Quantitative phase-amplitude microscopy I: optical microscopy,” J. Microsc. 206(Pt 3), 194–203 (2002).
    [Crossref] [PubMed]
  16. N. Streibl, “Phase imaging by the transport of equation of intensity,” Opt. Commun. 49(1), 6–10 (1984).
    [Crossref]
  17. L. Waller, Y. Luo, S.-Y. Yang, and G. Barbastathis, “Transport of intensity phase imaging in a volume holographic microscope,” Opt. Lett. 35(17), 2961–2963 (2010).
    [Crossref] [PubMed]
  18. S. S. Kou, L. Waller, G. Barbastathis, and C. J. Sheppard, “Transport-of-intensity approach to differential interference contrast (TI-DIC) microscopy for quantitative phase imaging,” Opt. Lett. 35(3), 447–449 (2010).
    [Crossref] [PubMed]
  19. L. Waller, L. Tian, and G. Barbastathis, “Transport of Intensity phase-amplitude imaging with higher order intensity derivatives,” Opt. Express 18(12), 12552–12561 (2010).
    [Crossref] [PubMed]
  20. M. Beleggia, M. Schofield, V. Volkov, and Y. Zhu, “On the transport of intensity technique for phase retrieval,” Ultramicroscopy 102, 37–49 (2004).
    [Crossref] [PubMed]
  21. J. Goodman, Introduction to Fourier Optics, (McGraw-Hill, 1996).
  22. B. Saleh, and M. Teich, Fundamentals of Photonics (John Wiley & Sons, 2010).
  23. T. Gureyev and S. Wilkins, “On X-ray phase retrieval from polychromatic images," Opt. Commun. 147, 229–232 (1998) (Erratum: Opt. Commun. 154, 391).
    [Crossref]
  24. T. E. Gureyev, S. Mayo, S. W. Wilkins, D. Paganin, and A. W. Stevenson, “Quantitative in-line phase-contrast imaging with multienergy X rays,” Phys. Rev. Lett. 86(25), 5827–5830 (2001).
    [Crossref] [PubMed]
  25. M. A. Anastasio, Q. Xu, and D. Shi, “Multispectral intensity diffraction tomography: single material objects with variable densities,” J. Opt. Soc. Am. A 26(2), 403–412 (2009).
    [Crossref]
  26. G. Strang, Computational Science and Engineering (Wellesley-Cambridge Press, 2010).
  27. L. Allen and M. Oxley, “Phase retrieval from series of images obtained by defocus variation,” Opt. Commun. 199(1-4), 65–75 (2001).
    [Crossref]
  28. N. Loomis, L. Waller, G. Barbastathis, “High-speed phase recovery using chromatic transport of intensity computation in graphics processing units,” Proc. Digital Holography meeting of the OSA: JMA7 (2010).
  29. G. Molesini and F. Quercioli, “Pseudocolor effects of longitudinal chromatic aberration,” J. Opt. (Paris) 17, 279–282 (1986).
  30. J. S. Courtney-Pratt and R. L. Gregory, “Microscope with enhanced depth of field and 3-D capability,” Appl. Opt. 12(10), 2509–2519 (1973).
    [Crossref] [PubMed]
  31. T. Bifano, R. K. Mali, J. K. Dorton, J. A. Perreault, N. Vandelli, M. N. Horenstein, and D. A. Castanon, “Continuous-membrane silicon deformable mirror,” Opt. Eng. 36, 1354–1360 (1997).
    [Crossref]
  32. A. M. Zysk, R. W. Schoonover, P. S. Carney, and M. A. Anastasio, “Transport of intensity and spectrum for partially coherent fields,” Opt. Lett. 35(13), 2239–2241 (2010).
    [Crossref] [PubMed]

2010 (5)

2009 (1)

2005 (1)

2004 (3)

2002 (1)

E. D. Barone-Nugent, A. Barty, and K. A. Nugent, “Quantitative phase-amplitude microscopy I: optical microscopy,” J. Microsc. 206(Pt 3), 194–203 (2002).
[Crossref] [PubMed]

2001 (3)

T. E. Gureyev, S. Mayo, S. W. Wilkins, D. Paganin, and A. W. Stevenson, “Quantitative in-line phase-contrast imaging with multienergy X rays,” Phys. Rev. Lett. 86(25), 5827–5830 (2001).
[Crossref] [PubMed]

L. Allen and M. Oxley, “Phase retrieval from series of images obtained by defocus variation,” Opt. Commun. 199(1-4), 65–75 (2001).
[Crossref]

B. C. Platt and R. Shack, “History and principles of Shack–Hartmann wavefront sensing,” J. Refract. Surg. 17(5), S573–S577 (2001).
[PubMed]

1998 (2)

D. Paganin and K. Nugent, “Noninterferometric phase imaging with partially coherent light,” Phys. Rev. Lett. 80(12), 2586–2589 (1998).
[Crossref]

T. Gureyev and S. Wilkins, “On X-ray phase retrieval from polychromatic images," Opt. Commun. 147, 229–232 (1998) (Erratum: Opt. Commun. 154, 391).
[Crossref]

1997 (1)

T. Bifano, R. K. Mali, J. K. Dorton, J. A. Perreault, N. Vandelli, M. N. Horenstein, and D. A. Castanon, “Continuous-membrane silicon deformable mirror,” Opt. Eng. 36, 1354–1360 (1997).
[Crossref]

1986 (1)

G. Molesini and F. Quercioli, “Pseudocolor effects of longitudinal chromatic aberration,” J. Opt. (Paris) 17, 279–282 (1986).

1984 (1)

N. Streibl, “Phase imaging by the transport of equation of intensity,” Opt. Commun. 49(1), 6–10 (1984).
[Crossref]

1983 (1)

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

1973 (1)

1955 (2)

F. Zernike, “How I discovered phase contrast,” Science 121(3141), 345–349 (1955).
[Crossref] [PubMed]

H. Hopkins, “The frequency response of a defocused optical system,” Proc. Royal Soc. London, Ser. A 231(1184), 91–103 (1955).
[Crossref]

1942 (1)

F. Zernike, “Phase contrast, a new method for the microscopic observation of transparent objects,” Physica 9(7), 686–698 (1942).
[Crossref]

Allen, L.

L. Allen and M. Oxley, “Phase retrieval from series of images obtained by defocus variation,” Opt. Commun. 199(1-4), 65–75 (2001).
[Crossref]

Anastasio, M. A.

Badizadegan, K.

Barbastathis, G.

Barone-Nugent, E. D.

E. D. Barone-Nugent, A. Barty, and K. A. Nugent, “Quantitative phase-amplitude microscopy I: optical microscopy,” J. Microsc. 206(Pt 3), 194–203 (2002).
[Crossref] [PubMed]

Barty, A.

E. D. Barone-Nugent, A. Barty, and K. A. Nugent, “Quantitative phase-amplitude microscopy I: optical microscopy,” J. Microsc. 206(Pt 3), 194–203 (2002).
[Crossref] [PubMed]

Beleggia, M.

M. Beleggia, M. Schofield, V. Volkov, and Y. Zhu, “On the transport of intensity technique for phase retrieval,” Ultramicroscopy 102, 37–49 (2004).
[Crossref] [PubMed]

Bifano, T.

T. Bifano, R. K. Mali, J. K. Dorton, J. A. Perreault, N. Vandelli, M. N. Horenstein, and D. A. Castanon, “Continuous-membrane silicon deformable mirror,” Opt. Eng. 36, 1354–1360 (1997).
[Crossref]

Carney, P. S.

Castanon, D. A.

T. Bifano, R. K. Mali, J. K. Dorton, J. A. Perreault, N. Vandelli, M. N. Horenstein, and D. A. Castanon, “Continuous-membrane silicon deformable mirror,” Opt. Eng. 36, 1354–1360 (1997).
[Crossref]

Colomb, T.

Courtney-Pratt, J. S.

Cuche, E.

Cui, X.

Dasari, R. R.

Deflores, L. P.

Depeursinge, C.

Dorton, J. K.

T. Bifano, R. K. Mali, J. K. Dorton, J. A. Perreault, N. Vandelli, M. N. Horenstein, and D. A. Castanon, “Continuous-membrane silicon deformable mirror,” Opt. Eng. 36, 1354–1360 (1997).
[Crossref]

Emery, Y.

Feld, M. S.

Gregory, R. L.

Gureyev, T.

T. Gureyev and S. Wilkins, “On X-ray phase retrieval from polychromatic images," Opt. Commun. 147, 229–232 (1998) (Erratum: Opt. Commun. 154, 391).
[Crossref]

Gureyev, T. E.

T. E. Gureyev, S. Mayo, S. W. Wilkins, D. Paganin, and A. W. Stevenson, “Quantitative in-line phase-contrast imaging with multienergy X rays,” Phys. Rev. Lett. 86(25), 5827–5830 (2001).
[Crossref] [PubMed]

Hopkins, H.

H. Hopkins, “The frequency response of a defocused optical system,” Proc. Royal Soc. London, Ser. A 231(1184), 91–103 (1955).
[Crossref]

Horenstein, M. N.

T. Bifano, R. K. Mali, J. K. Dorton, J. A. Perreault, N. Vandelli, M. N. Horenstein, and D. A. Castanon, “Continuous-membrane silicon deformable mirror,” Opt. Eng. 36, 1354–1360 (1997).
[Crossref]

Iwai, H.

Kou, S. S.

Luo, Y.

Magistretti, P. J.

Mali, R. K.

T. Bifano, R. K. Mali, J. K. Dorton, J. A. Perreault, N. Vandelli, M. N. Horenstein, and D. A. Castanon, “Continuous-membrane silicon deformable mirror,” Opt. Eng. 36, 1354–1360 (1997).
[Crossref]

Marquet, P.

Mayo, S.

T. E. Gureyev, S. Mayo, S. W. Wilkins, D. Paganin, and A. W. Stevenson, “Quantitative in-line phase-contrast imaging with multienergy X rays,” Phys. Rev. Lett. 86(25), 5827–5830 (2001).
[Crossref] [PubMed]

Molesini, G.

G. Molesini and F. Quercioli, “Pseudocolor effects of longitudinal chromatic aberration,” J. Opt. (Paris) 17, 279–282 (1986).

Nugent, K.

D. Paganin and K. Nugent, “Noninterferometric phase imaging with partially coherent light,” Phys. Rev. Lett. 80(12), 2586–2589 (1998).
[Crossref]

Nugent, K. A.

E. D. Barone-Nugent, A. Barty, and K. A. Nugent, “Quantitative phase-amplitude microscopy I: optical microscopy,” J. Microsc. 206(Pt 3), 194–203 (2002).
[Crossref] [PubMed]

Oxley, M.

L. Allen and M. Oxley, “Phase retrieval from series of images obtained by defocus variation,” Opt. Commun. 199(1-4), 65–75 (2001).
[Crossref]

Paganin, D.

T. E. Gureyev, S. Mayo, S. W. Wilkins, D. Paganin, and A. W. Stevenson, “Quantitative in-line phase-contrast imaging with multienergy X rays,” Phys. Rev. Lett. 86(25), 5827–5830 (2001).
[Crossref] [PubMed]

D. Paganin and K. Nugent, “Noninterferometric phase imaging with partially coherent light,” Phys. Rev. Lett. 80(12), 2586–2589 (1998).
[Crossref]

Perreault, J. A.

T. Bifano, R. K. Mali, J. K. Dorton, J. A. Perreault, N. Vandelli, M. N. Horenstein, and D. A. Castanon, “Continuous-membrane silicon deformable mirror,” Opt. Eng. 36, 1354–1360 (1997).
[Crossref]

Platt, B. C.

B. C. Platt and R. Shack, “History and principles of Shack–Hartmann wavefront sensing,” J. Refract. Surg. 17(5), S573–S577 (2001).
[PubMed]

Popescu, G.

Quercioli, F.

G. Molesini and F. Quercioli, “Pseudocolor effects of longitudinal chromatic aberration,” J. Opt. (Paris) 17, 279–282 (1986).

Rappaz, B.

Ren, J.

Schofield, M.

M. Beleggia, M. Schofield, V. Volkov, and Y. Zhu, “On the transport of intensity technique for phase retrieval,” Ultramicroscopy 102, 37–49 (2004).
[Crossref] [PubMed]

Schoonover, R. W.

Shack, R.

B. C. Platt and R. Shack, “History and principles of Shack–Hartmann wavefront sensing,” J. Refract. Surg. 17(5), S573–S577 (2001).
[PubMed]

Sheppard, C. J.

Sheppard, C. J. R.

Shi, D.

Stevenson, A. W.

T. E. Gureyev, S. Mayo, S. W. Wilkins, D. Paganin, and A. W. Stevenson, “Quantitative in-line phase-contrast imaging with multienergy X rays,” Phys. Rev. Lett. 86(25), 5827–5830 (2001).
[Crossref] [PubMed]

Streibl, N.

N. Streibl, “Phase imaging by the transport of equation of intensity,” Opt. Commun. 49(1), 6–10 (1984).
[Crossref]

Teague, M.

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

Tearney, G. J.

Tian, L.

Vandelli, N.

T. Bifano, R. K. Mali, J. K. Dorton, J. A. Perreault, N. Vandelli, M. N. Horenstein, and D. A. Castanon, “Continuous-membrane silicon deformable mirror,” Opt. Eng. 36, 1354–1360 (1997).
[Crossref]

Vaughan, J. C.

Volkov, V.

M. Beleggia, M. Schofield, V. Volkov, and Y. Zhu, “On the transport of intensity technique for phase retrieval,” Ultramicroscopy 102, 37–49 (2004).
[Crossref] [PubMed]

Waller, L.

Wilkins, S.

T. Gureyev and S. Wilkins, “On X-ray phase retrieval from polychromatic images," Opt. Commun. 147, 229–232 (1998) (Erratum: Opt. Commun. 154, 391).
[Crossref]

Wilkins, S. W.

T. E. Gureyev, S. Mayo, S. W. Wilkins, D. Paganin, and A. W. Stevenson, “Quantitative in-line phase-contrast imaging with multienergy X rays,” Phys. Rev. Lett. 86(25), 5827–5830 (2001).
[Crossref] [PubMed]

Xu, Q.

Yang, C.

Yang, S.-Y.

Zernike, F.

F. Zernike, “How I discovered phase contrast,” Science 121(3141), 345–349 (1955).
[Crossref] [PubMed]

F. Zernike, “Phase contrast, a new method for the microscopic observation of transparent objects,” Physica 9(7), 686–698 (1942).
[Crossref]

Zhu, Y.

M. Beleggia, M. Schofield, V. Volkov, and Y. Zhu, “On the transport of intensity technique for phase retrieval,” Ultramicroscopy 102, 37–49 (2004).
[Crossref] [PubMed]

Zysk, A. M.

Appl. Opt. (1)

J. Microsc. (1)

E. D. Barone-Nugent, A. Barty, and K. A. Nugent, “Quantitative phase-amplitude microscopy I: optical microscopy,” J. Microsc. 206(Pt 3), 194–203 (2002).
[Crossref] [PubMed]

J. Opt. (Paris) (1)

G. Molesini and F. Quercioli, “Pseudocolor effects of longitudinal chromatic aberration,” J. Opt. (Paris) 17, 279–282 (1986).

J. Opt. Soc. Am. A (3)

J. Refract. Surg. (1)

B. C. Platt and R. Shack, “History and principles of Shack–Hartmann wavefront sensing,” J. Refract. Surg. 17(5), S573–S577 (2001).
[PubMed]

Opt. Commun. (3)

N. Streibl, “Phase imaging by the transport of equation of intensity,” Opt. Commun. 49(1), 6–10 (1984).
[Crossref]

T. Gureyev and S. Wilkins, “On X-ray phase retrieval from polychromatic images," Opt. Commun. 147, 229–232 (1998) (Erratum: Opt. Commun. 154, 391).
[Crossref]

L. Allen and M. Oxley, “Phase retrieval from series of images obtained by defocus variation,” Opt. Commun. 199(1-4), 65–75 (2001).
[Crossref]

Opt. Eng. (1)

T. Bifano, R. K. Mali, J. K. Dorton, J. A. Perreault, N. Vandelli, M. N. Horenstein, and D. A. Castanon, “Continuous-membrane silicon deformable mirror,” Opt. Eng. 36, 1354–1360 (1997).
[Crossref]

Opt. Express (2)

Opt. Lett. (5)

Phys. Rev. Lett. (2)

T. E. Gureyev, S. Mayo, S. W. Wilkins, D. Paganin, and A. W. Stevenson, “Quantitative in-line phase-contrast imaging with multienergy X rays,” Phys. Rev. Lett. 86(25), 5827–5830 (2001).
[Crossref] [PubMed]

D. Paganin and K. Nugent, “Noninterferometric phase imaging with partially coherent light,” Phys. Rev. Lett. 80(12), 2586–2589 (1998).
[Crossref]

Physica (1)

F. Zernike, “Phase contrast, a new method for the microscopic observation of transparent objects,” Physica 9(7), 686–698 (1942).
[Crossref]

Proc. Royal Soc. London, Ser. A (1)

H. Hopkins, “The frequency response of a defocused optical system,” Proc. Royal Soc. London, Ser. A 231(1184), 91–103 (1955).
[Crossref]

Science (1)

F. Zernike, “How I discovered phase contrast,” Science 121(3141), 345–349 (1955).
[Crossref] [PubMed]

Ultramicroscopy (1)

M. Beleggia, M. Schofield, V. Volkov, and Y. Zhu, “On the transport of intensity technique for phase retrieval,” Ultramicroscopy 102, 37–49 (2004).
[Crossref] [PubMed]

Other (8)

J. Goodman, Introduction to Fourier Optics, (McGraw-Hill, 1996).

B. Saleh, and M. Teich, Fundamentals of Photonics (John Wiley & Sons, 2010).

G. Strang, Computational Science and Engineering (Wellesley-Cambridge Press, 2010).

N. Loomis, L. Waller, G. Barbastathis, “High-speed phase recovery using chromatic transport of intensity computation in graphics processing units,” Proc. Digital Holography meeting of the OSA: JMA7 (2010).

L. Waller, and G. Barbastathis, “Phase from Defocused Color Images,” in Frontiers in Optics, OSA Technical Digest (CD) (Optical Society of America, 2009), paper FThR3.

M. Born, and E. Wolf, Principles of Optics (Cambridge Univ. Press, 1999).

H. King, The History of the Telescope (Dover, 2003).

J. Millerd, N. Brock, J. Hayes, M. North-Morris, B. Kimbrough, and J. Wyant, Fringe 2005: Pixelated Phase-mask Dynamic Interferometers (Springer, 2003).

Supplementary Material (2)

» Media 1: MPEG (1863 KB)     
» Media 2: MPEG (1076 KB)     

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Figures (7)

Fig. 1
Fig. 1 Design of a chromatic 4f system for differential defocus of colour channels. a) Chromatic defocus causes different colors to focus in different planes along the optical axis, b) quantification of chromatic defocus for three values of f 2 given f 1 = 200mm (at 532nm) with BK7 lens dispersion.

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Fig. 2
Fig. 2 a) Experimental setup for deformable mirror experiments. b-d) Phase retrieval from a single colour image. b) Red, green and blue colour channels from c) captured colour image. d) Phase solution giving inverse height profile across the mirror (video in Media 1).

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Fig. 3
Fig. 3 Phase retrieval in a brightfield microscope. (Top row) Color images, (bottom row) recovered phase for (left) PMMA test object height, (middle) live HMVEC cells (also see Media 2), and (right) live HeLa cells (normalized phase).

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Fig. 4
Fig. 4 Nonlinearity error of a test phase object (inset) for varying values of wavelength and distance-dependent defocus. In the absence of noise, the error goes asymptotically to zero with decreasing defocus. Thus, the noise floor will determine the accuracy of the system.

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Fig. 5
Fig. 5 Comparison of results from our technique with profilometer data from a commercial interferometer. (a) Height map from Zygo interferometer compared to (b) height map from our technique (µm). (c) Cross-section along one actuator of the DM array, showing the ‘influence function’ of the DM using both techniques.

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Fig. 6
Fig. 6 Phase imaging with achromats. (a) Typical defocus-wavelength plot for red-blue achromat, (b) phase result using Eq. (1) (very poor contrast) compared to (c) Eq. (5) with achromat.

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Fig. 7
Fig. 7 Comparison of phase results from our technique with traditional TIE. (a) Phase from traditional TIE using two images (radians), (b) phase from our technique using a single color image (radians) and (c) the difference between the two results (radians).

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Equations (10)

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

∂ I ( x , y ) ∂ ξ = − ∇ ⊥ • [ I ( x , y ) ∇ ⊥ φ ( x , y ) ] 2 π ,
Δ f ( λ ) = n ( λ ) − n ( λ 0 ) ( n ( λ ) − 1 ) f ( λ 0 ) ,
Δ f ' ( λ ) = Δ f 2 ( λ ) + f 2 ( λ ) 2 Δ f 1 ( λ ) + Δ f 2 ( λ ) − ( f 1 ( λ ) 2 / Δ f 1 ( λ ) ) ,
∂ I ∂ ξ ≈ I ( ξ R ) − I ( ξ B ) Δ ξ + N R − N B Δ ξ ,
∂ I ∂ ξ ≈ ( I R + I B ) − 2 I G 2 Δ ξ ,
h ( x , y ; ξ ) = e i 2 π z / λ i ξ exp { i π ξ ( x 2 + y 2 ) } .
I ( x , y ; ξ ) = | ψ ( x , y ) ⊗ h ( x , y ; ξ ) | 2 = | ℑ − 1 { Ψ ( u , v ) H ( u , v ; ξ ) } | 2 ,
H ( u , v ; ξ ) = 1 − i π ξ ( u 2 + v 2 ) − ( π ξ ) 2 ( u 2 + v 2 ) 2 2 ! − ... ≈ 1 − i π ξ ( u 2 + v 2 ) .
I ( x , y ; ξ ) = | ψ ( x , y ) + i ξ ∇ ⊥ 2 ψ ( x , y ) 4 π | 2 = I 0 ( x , y ) − ξ 2 π ∇ ⊥ • ( I 0 ( x , y ) ∇ ⊥ φ ( x , y ) ) ,
I ( x , y ; ξ ) − I 0 ( x , y ) ξ ≈ ∂ I ∂ ξ = − ∇ ⊥ • [ I 0 ( x , y ) ∇ ⊥ ϕ ( x , y ) ] 2 π .

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