Abstract

Scanning holographic microscopy is a two-pupil synthesis method allowing the capture of single-sideband in-line holograms of noncoherent (e.g., fluorescent) three-dimensional specimens in a single two-dimensional scan. The flexibility offered by the two-pupil method in synthesizing unusual point-spread functions is discussed. We illustrate and compare three examples of holographic recording, using computer simulations. The first example is the classical hologram in which each object point is encoded as a spherical wave. The second example uses pupils with spherical phase distributions having opposite curvatures, leading to reconstructed images with a resolution limit that is half that of the objective. In the third example, axicon pupils are used to obtain axially sectioned images.

© 2006 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. D. Gabor, 'A new microscopic principle,' Nature 161, 777-778 (1948).
    [CrossRef] [PubMed]
  2. E. N. Leith and J. Upatnieks, 'Reconstructed wavefronts and communication theory,' J. Opt. Soc. Am. 52, 1123-1130 (1962).
    [CrossRef]
  3. J. W. Goodman and R. W. Lawrence, 'Digital image information from electronically detected holograms,' Appl. Phys. Lett. 11, 77-79 (1967).
    [CrossRef]
  4. U. Schnars and W. P. O. Juptner, 'Digital recording and numerical reconstruction of holograms,' Meas. Sci. Technol. 13, R85-R101 (2002).
    [CrossRef]
  5. E. Cuche, P. Poscio, and C. Depeursinge, 'Optical tomography at the microscopic scale by means of a numerical low-coherence holographic technique,' in Optical and Imaging Techniques for Biomonitoring II, H.J.Foth, R.Marchesini, and H.Podbielska, eds., Proc. SPIE 2927, 61-66 (1996).
  6. W. Xu, M. H. Jerico, I. A. Meinertzhagen, and H. J. Kreuzer, 'Digital in-line holography for biological applications,' Proc. Natl. Acad. Sci. USA 98, 11301-11305 (2002).
    [CrossRef]
  7. E. Cuche, F. Bevilacqua, and C. Depeursinge, 'Digital holography for quantitative phase-contrast imaging,' Opt. Lett. 24, 291-293 (1999).
    [CrossRef]
  8. S. Grilli, P. Ferraro, S. De Nicola, G. Pierattini, and R. Meucci, 'Whole optical wavefields reconstruction by digital holography,' Opt. Express 9, 294-302 (2001).
    [CrossRef] [PubMed]
  9. G. Indebetouw and W. Zhong, 'Scanning holographic microscopy of three-dimensional fluorescent specimens,' J. Opt. Soc. Am. A 23, 1699-1707 (2006).
    [CrossRef]
  10. W. Lukosz, 'Properties of linear low-pass filters for nonnegative signals,' J. Opt. Soc. Am. 52, 827-829 (1962).
    [CrossRef]
  11. A. W. Lohmann and W. T. Rhodes, 'Two-pupil synthesis of optical transfer functions,' Appl. Opt. 17, 1141-1151 (1978).
    [CrossRef] [PubMed]
  12. W. Stoner, 'Incoherent optical processing via spatially offset pupil masks,' Appl. Opt. 17, 2454-2467 (1978).
    [CrossRef] [PubMed]
  13. T.-C. Poon and A. Korpel, 'Optical transfer function of an acousto-optic heterodyning image processor,' Opt. Lett. 4, 317-319 (1979).
    [CrossRef] [PubMed]
  14. T.-C. Poon, 'Scanning holography and two-dimensional image processing by acousto-optic two-pupil synthesis,' J. Opt. Soc. Am. A 2, 521-527 (1985).
    [CrossRef]
  15. G. Indebetouw, P. Klysubun, T. Kim, and T.-C. Poon, 'Imaging properties of scanning holographic microscopy,' J. Opt. Soc. Am. A 17, 380-390 (2000).
    [CrossRef]
  16. G. Indebetouw, 'Properties of a scanning holographic microscope: improved resolution, extended depth of focus, and/or optical sectioning,' J. Mod. Opt. 49, 1479-1500 (2002).
    [CrossRef]
  17. B. Schilling, T.-C. Poon, G. Indebetouw, B. Storie, K. Shinoda, and M. Wu, 'Three-dimensional holographic fluorescence microscopy,' Opt. Lett. 22, 1506-1508 (1997).
    [CrossRef]
  18. G. Indebetouw, A. El Maghnouji, and R. Foster, 'Scanning holographic microscopy with transverse resolution exceeding the Rayleigh limit and extended depth of focus,' J. Opt. Soc. Am. A 22, 829-898 (2005).
    [CrossRef]
  19. E. R. Dowski and W. T. Cathey, 'Extended depth of field through wave-front coding,' Appl. Opt. 34, 1859-1866 (1995).
    [CrossRef] [PubMed]
  20. D. A. Agard, 'Optical sectioning microscopy: cellular architecture in three dimensions,' Annu. Rev. Biophys. Bioeng. 13, 191-219 (1984).
    [CrossRef] [PubMed]
  21. A.Diaspro, ed., Confocal and Two-Photon Microscopy: Foundations, Applications, and Advances (Wiley-Liss, 2002).
  22. T.Wilson, ed., Confocal Microscopy (Academic, 1990).
  23. M. A. A. Neil, R. Juskaitis, and T. Wilson, 'Real time 3D fluorescence microscopy by two beam interference illumination,' Opt. Commun. 153, 1-4 (1998).
    [CrossRef]
  24. C. W. Mc Cutchen, 'Generalized aperture and the three-dimensional diffraction image,' J. Opt. Soc. Am. 54, 240-244 (1964).
    [CrossRef]
  25. W. Wang, A. T. Friberg, and E. Wolf, 'Structure of focused fields in systems with large Fresnel numbers,' J. Opt. Soc. Am. A 12, 1947-1953 (1995).
    [CrossRef]
  26. D. S. C. Biggs, 'Clearing up deconvolution,' Biophotonics Int. 11, 32-36 (2004).
  27. P. J. Shaw, 'Comparison of wide-field/deconvolution and confocal microscopy,' in Handbook of Biological Confocal Microscopy, 2nd ed., J.B.Pawley, ed. (Plenum, 1995), pp. 373-387.
  28. J. H. McLeod, 'The axicon: a new type of optical element,' J. Opt. Soc. Am. 44, 592-597 (1955).
    [CrossRef]
  29. J. H. McLeod, 'Axicons and their uses,' J. Opt. Soc. Am. 50, 166-169 (1960).
    [CrossRef]

2006 (1)

2005 (1)

G. Indebetouw, A. El Maghnouji, and R. Foster, 'Scanning holographic microscopy with transverse resolution exceeding the Rayleigh limit and extended depth of focus,' J. Opt. Soc. Am. A 22, 829-898 (2005).
[CrossRef]

2004 (1)

D. S. C. Biggs, 'Clearing up deconvolution,' Biophotonics Int. 11, 32-36 (2004).

2002 (3)

G. Indebetouw, 'Properties of a scanning holographic microscope: improved resolution, extended depth of focus, and/or optical sectioning,' J. Mod. Opt. 49, 1479-1500 (2002).
[CrossRef]

U. Schnars and W. P. O. Juptner, 'Digital recording and numerical reconstruction of holograms,' Meas. Sci. Technol. 13, R85-R101 (2002).
[CrossRef]

W. Xu, M. H. Jerico, I. A. Meinertzhagen, and H. J. Kreuzer, 'Digital in-line holography for biological applications,' Proc. Natl. Acad. Sci. USA 98, 11301-11305 (2002).
[CrossRef]

2001 (1)

2000 (1)

1999 (1)

1998 (1)

M. A. A. Neil, R. Juskaitis, and T. Wilson, 'Real time 3D fluorescence microscopy by two beam interference illumination,' Opt. Commun. 153, 1-4 (1998).
[CrossRef]

1997 (1)

1995 (2)

1985 (1)

1984 (1)

D. A. Agard, 'Optical sectioning microscopy: cellular architecture in three dimensions,' Annu. Rev. Biophys. Bioeng. 13, 191-219 (1984).
[CrossRef] [PubMed]

1979 (1)

1978 (2)

1967 (1)

J. W. Goodman and R. W. Lawrence, 'Digital image information from electronically detected holograms,' Appl. Phys. Lett. 11, 77-79 (1967).
[CrossRef]

1964 (1)

1962 (2)

1960 (1)

1955 (1)

1948 (1)

D. Gabor, 'A new microscopic principle,' Nature 161, 777-778 (1948).
[CrossRef] [PubMed]

Agard, D. A.

D. A. Agard, 'Optical sectioning microscopy: cellular architecture in three dimensions,' Annu. Rev. Biophys. Bioeng. 13, 191-219 (1984).
[CrossRef] [PubMed]

Bevilacqua, F.

Biggs, D. S. C.

D. S. C. Biggs, 'Clearing up deconvolution,' Biophotonics Int. 11, 32-36 (2004).

Cathey, W. T.

Cuche, E.

E. Cuche, F. Bevilacqua, and C. Depeursinge, 'Digital holography for quantitative phase-contrast imaging,' Opt. Lett. 24, 291-293 (1999).
[CrossRef]

E. Cuche, P. Poscio, and C. Depeursinge, 'Optical tomography at the microscopic scale by means of a numerical low-coherence holographic technique,' in Optical and Imaging Techniques for Biomonitoring II, H.J.Foth, R.Marchesini, and H.Podbielska, eds., Proc. SPIE 2927, 61-66 (1996).

De Nicola, S.

Depeursinge, C.

E. Cuche, F. Bevilacqua, and C. Depeursinge, 'Digital holography for quantitative phase-contrast imaging,' Opt. Lett. 24, 291-293 (1999).
[CrossRef]

E. Cuche, P. Poscio, and C. Depeursinge, 'Optical tomography at the microscopic scale by means of a numerical low-coherence holographic technique,' in Optical and Imaging Techniques for Biomonitoring II, H.J.Foth, R.Marchesini, and H.Podbielska, eds., Proc. SPIE 2927, 61-66 (1996).

Dowski, E. R.

El Maghnouji, A.

G. Indebetouw, A. El Maghnouji, and R. Foster, 'Scanning holographic microscopy with transverse resolution exceeding the Rayleigh limit and extended depth of focus,' J. Opt. Soc. Am. A 22, 829-898 (2005).
[CrossRef]

Ferraro, P.

Foster, R.

G. Indebetouw, A. El Maghnouji, and R. Foster, 'Scanning holographic microscopy with transverse resolution exceeding the Rayleigh limit and extended depth of focus,' J. Opt. Soc. Am. A 22, 829-898 (2005).
[CrossRef]

Friberg, A. T.

Gabor, D.

D. Gabor, 'A new microscopic principle,' Nature 161, 777-778 (1948).
[CrossRef] [PubMed]

Goodman, J. W.

J. W. Goodman and R. W. Lawrence, 'Digital image information from electronically detected holograms,' Appl. Phys. Lett. 11, 77-79 (1967).
[CrossRef]

Grilli, S.

Indebetouw, G.

G. Indebetouw and W. Zhong, 'Scanning holographic microscopy of three-dimensional fluorescent specimens,' J. Opt. Soc. Am. A 23, 1699-1707 (2006).
[CrossRef]

G. Indebetouw, A. El Maghnouji, and R. Foster, 'Scanning holographic microscopy with transverse resolution exceeding the Rayleigh limit and extended depth of focus,' J. Opt. Soc. Am. A 22, 829-898 (2005).
[CrossRef]

G. Indebetouw, 'Properties of a scanning holographic microscope: improved resolution, extended depth of focus, and/or optical sectioning,' J. Mod. Opt. 49, 1479-1500 (2002).
[CrossRef]

G. Indebetouw, P. Klysubun, T. Kim, and T.-C. Poon, 'Imaging properties of scanning holographic microscopy,' J. Opt. Soc. Am. A 17, 380-390 (2000).
[CrossRef]

B. Schilling, T.-C. Poon, G. Indebetouw, B. Storie, K. Shinoda, and M. Wu, 'Three-dimensional holographic fluorescence microscopy,' Opt. Lett. 22, 1506-1508 (1997).
[CrossRef]

Jerico, M. H.

W. Xu, M. H. Jerico, I. A. Meinertzhagen, and H. J. Kreuzer, 'Digital in-line holography for biological applications,' Proc. Natl. Acad. Sci. USA 98, 11301-11305 (2002).
[CrossRef]

Juptner, W. P. O.

U. Schnars and W. P. O. Juptner, 'Digital recording and numerical reconstruction of holograms,' Meas. Sci. Technol. 13, R85-R101 (2002).
[CrossRef]

Juskaitis, R.

M. A. A. Neil, R. Juskaitis, and T. Wilson, 'Real time 3D fluorescence microscopy by two beam interference illumination,' Opt. Commun. 153, 1-4 (1998).
[CrossRef]

Kim, T.

Klysubun, P.

Korpel, A.

Kreuzer, H. J.

W. Xu, M. H. Jerico, I. A. Meinertzhagen, and H. J. Kreuzer, 'Digital in-line holography for biological applications,' Proc. Natl. Acad. Sci. USA 98, 11301-11305 (2002).
[CrossRef]

Lawrence, R. W.

J. W. Goodman and R. W. Lawrence, 'Digital image information from electronically detected holograms,' Appl. Phys. Lett. 11, 77-79 (1967).
[CrossRef]

Leith, E. N.

Lohmann, A. W.

Lukosz, W.

Mc Cutchen, C. W.

McLeod, J. H.

Meinertzhagen, I. A.

W. Xu, M. H. Jerico, I. A. Meinertzhagen, and H. J. Kreuzer, 'Digital in-line holography for biological applications,' Proc. Natl. Acad. Sci. USA 98, 11301-11305 (2002).
[CrossRef]

Meucci, R.

Neil, M. A. A.

M. A. A. Neil, R. Juskaitis, and T. Wilson, 'Real time 3D fluorescence microscopy by two beam interference illumination,' Opt. Commun. 153, 1-4 (1998).
[CrossRef]

Pierattini, G.

Poon, T.-C.

Poscio, P.

E. Cuche, P. Poscio, and C. Depeursinge, 'Optical tomography at the microscopic scale by means of a numerical low-coherence holographic technique,' in Optical and Imaging Techniques for Biomonitoring II, H.J.Foth, R.Marchesini, and H.Podbielska, eds., Proc. SPIE 2927, 61-66 (1996).

Rhodes, W. T.

Schilling, B.

Schnars, U.

U. Schnars and W. P. O. Juptner, 'Digital recording and numerical reconstruction of holograms,' Meas. Sci. Technol. 13, R85-R101 (2002).
[CrossRef]

Shaw, P. J.

P. J. Shaw, 'Comparison of wide-field/deconvolution and confocal microscopy,' in Handbook of Biological Confocal Microscopy, 2nd ed., J.B.Pawley, ed. (Plenum, 1995), pp. 373-387.

Shinoda, K.

Stoner, W.

Storie, B.

Upatnieks, J.

Wang, W.

Wilson, T.

M. A. A. Neil, R. Juskaitis, and T. Wilson, 'Real time 3D fluorescence microscopy by two beam interference illumination,' Opt. Commun. 153, 1-4 (1998).
[CrossRef]

Wolf, E.

Wu, M.

Xu, W.

W. Xu, M. H. Jerico, I. A. Meinertzhagen, and H. J. Kreuzer, 'Digital in-line holography for biological applications,' Proc. Natl. Acad. Sci. USA 98, 11301-11305 (2002).
[CrossRef]

Zhong, W.

Annu. Rev. Biophys. Bioeng. (1)

D. A. Agard, 'Optical sectioning microscopy: cellular architecture in three dimensions,' Annu. Rev. Biophys. Bioeng. 13, 191-219 (1984).
[CrossRef] [PubMed]

Appl. Opt. (3)

Appl. Phys. Lett. (1)

J. W. Goodman and R. W. Lawrence, 'Digital image information from electronically detected holograms,' Appl. Phys. Lett. 11, 77-79 (1967).
[CrossRef]

Biophotonics Int. (1)

D. S. C. Biggs, 'Clearing up deconvolution,' Biophotonics Int. 11, 32-36 (2004).

J. Mod. Opt. (1)

G. Indebetouw, 'Properties of a scanning holographic microscope: improved resolution, extended depth of focus, and/or optical sectioning,' J. Mod. Opt. 49, 1479-1500 (2002).
[CrossRef]

J. Opt. Soc. Am. (5)

J. Opt. Soc. Am. A (5)

Meas. Sci. Technol. (1)

U. Schnars and W. P. O. Juptner, 'Digital recording and numerical reconstruction of holograms,' Meas. Sci. Technol. 13, R85-R101 (2002).
[CrossRef]

Nature (1)

D. Gabor, 'A new microscopic principle,' Nature 161, 777-778 (1948).
[CrossRef] [PubMed]

Opt. Commun. (1)

M. A. A. Neil, R. Juskaitis, and T. Wilson, 'Real time 3D fluorescence microscopy by two beam interference illumination,' Opt. Commun. 153, 1-4 (1998).
[CrossRef]

Opt. Express (1)

Opt. Lett. (3)

Proc. Natl. Acad. Sci. USA (1)

W. Xu, M. H. Jerico, I. A. Meinertzhagen, and H. J. Kreuzer, 'Digital in-line holography for biological applications,' Proc. Natl. Acad. Sci. USA 98, 11301-11305 (2002).
[CrossRef]

Other (4)

E. Cuche, P. Poscio, and C. Depeursinge, 'Optical tomography at the microscopic scale by means of a numerical low-coherence holographic technique,' in Optical and Imaging Techniques for Biomonitoring II, H.J.Foth, R.Marchesini, and H.Podbielska, eds., Proc. SPIE 2927, 61-66 (1996).

A.Diaspro, ed., Confocal and Two-Photon Microscopy: Foundations, Applications, and Advances (Wiley-Liss, 2002).

T.Wilson, ed., Confocal Microscopy (Academic, 1990).

P. J. Shaw, 'Comparison of wide-field/deconvolution and confocal microscopy,' in Handbook of Biological Confocal Microscopy, 2nd ed., J.B.Pawley, ed. (Plenum, 1995), pp. 373-387.

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 (13)

Fig. 1
Fig. 1

Generic arrangement of a two-pupil synthesis setup in scanning mode, implementing an incoherent holographic system with a complex spread function. SLM, spatial light modulator: EO, electro-optic.

Fig. 2
Fig. 2

Amplitude of the hologram of seven incoherent point sources 7.5 μ m apart transversally and 4 μ m ( 1 2 depth of focus) apart axially. Axial positions range from 12 to + 12 μ m from the objective’s focal plane. Hologram obtained with one pointlike pupil and one pupil with spherical phase front. System parameters: numerical aperture sin α = 0.25 , z 0 = 80 μ m , scanning distribution radius a = 20 μ m .

Fig. 3
Fig. 3

(a) Amplitude of the reconstruction of the hologram of Fig. 2 focused on the central point source at z = 0 . (b) Profile of the reconstructed amplitude of the seven point sources.

Fig. 4
Fig. 4

(a) Amplitude of the reconstruction of the hologram of Fig. 2 focused on the first point source at z = 12 μ m . (b) Profile of the reconstructed amplitude.

Fig. 5
Fig. 5

Intensity of the reconstruction of two bars using the same parameters as described in Fig. 2. The bars are 16 μ m (two depths of focus) apart axially. Reconstruction is focused on the vertical bar.

Fig. 6
Fig. 6

Amplitude of the hologram of seven incoherent point sources 7.5 μ m apart transversally and 4 μ m apart axially. Axial positions range from 12 to + 12 μ m from the objective’s focal plane. Hologram obtained with two pupils having spherical phase fronts with opposite curvatures. System parameters: numerical aperture sin α = 0.25 , z 0 = ± 80 μ m , scanning distribution radius a = 20 μ m .

Fig. 7
Fig. 7

(a) Amplitude of the reconstruction of the hologram of Fig. 6 focused on the central point source at z = 0 . (b) Profile of the reconstructed amplitude of the seven point sources, demonstrating a gain of a factor of 2 in transverse resolution and an extended depth of focus.

Fig. 8
Fig. 8

Intensity of the reconstruction of two bars using the same parameters as described in Fig. 5. The bars are 16 μ m apart axially. The reconstruction is focused on the vertical bar, but both are in reasonable focus.

Fig. 9
Fig. 9

Amplitude of the hologram of seven incoherent point sources 7.5 μ m apart transversally and 4 μ m apart axially. Axial positions range from 12 to + 12 μ m from the objective’s focal plane. Hologram obtained with two pupils having linear phase fronts (axicons) with opposite phases. System parameters: numerical aperture sin α = 0.25 , ring scanning distribution radius a = 20 μ m .

Fig. 10
Fig. 10

(a) Amplitude of the reconstruction of the hologram of Fig. 9 focused on the central point source at z = 0 . (b) Profile of the reconstructed amplitude of the seven point sources, demonstrating axial sectioning and suppression of the out-of-focus information.

Fig. 11
Fig. 11

Intensity of the reconstruction of two bars using the same parameters as described in Fig. 9. The bars are 16 μ m apart axially. The plane containing the vertical bar was selected for sectioning. The axicon pupils lead to a transfer function suppressing the low spatial frequencies and exhibiting a high-pass filtering behavior.

Fig. 12
Fig. 12

Trace through the reconstruction of the vertical bar in Fig. 11, illustrating the possibility of locating spatial features with an accuracy far exceeding the resolution of the objective. The arrows indicate the precise location of the edges of the bar.

Fig. 13
Fig. 13

Reconstruction of the hologram of Fig. 9 using the algorithm described in the text to recover, to a controllable degree, the low spatial frequencies that have been suppressed in the hologram.

Equations (24)

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

P j ( x , y , z ) = F 2 { P ̃ j ( u , v ) exp [ i π λ z ( u 2 + v 2 ) ] }
d z P 1 ( x , y , z ) + P 2 ( x , y , z ) exp ( i Ω t ) 2 I ( x x s , y y s , z ) ,
H ( x s , y s ) = d x d y d z P 1 ( x , y , z ) P 2 * ( x , y , z ) I ( x x s , y y s , z ) .
S ( x , y , z ) = P 1 ( x , y , z ) P 2 * ( x , y , z )
H ( x , y , z ) = d z S ( x , y , z ) I ( x , y , z ) ,
P ̃ 1 ( u , v ) δ ( u , v ) ,
P ̃ 2 ( u , v ) = exp [ i π λ z 0 ( u 2 + v 2 ) ] P ̃ 0 ( u , v ) .
S ( x , y , z ) = F 2 { exp [ i π λ ( z 0 + z ) ( u 2 + v 2 ) ] P ̃ 0 ( u , v ) } .
I R ( x , y , z R ) = H ( x , y ) S * ( x , y , z R ) = d z I ( x , y , z ) Q ( x , y , z ; z R ) ,
Q ( x , y , z ; z R ) = S ( x , y , z ) S * ( x , y , z R )
I ̃ R ( u , v ; z R ) = H ̃ ( u , v ) S ̃ * ( u , v ; z R ) ,
Q ̃ ( u , v ; z ) = S ̃ ( u , v ; 0 ) S ̃ * ( u , v ; z ) = exp [ i π λ z ( u 2 + v 2 ) ] P ̃ 0 ( u , v ) 2 .
P ̃ 1 ( u , v ) = exp [ i π λ z 0 ( u 2 + v 2 ) ] P ̃ 0 ( u , v ) ,
P ̃ 2 ( u , v ) = exp [ + i π λ z 0 ( u 2 + v 2 ) ] P ̃ 0 ( u , v ) ,
S ( x , y , z ) = ( F 2 { exp [ i π λ ( z 0 + z ) ( u 2 + v 2 ) ] P ̃ 0 ( u , v ) } ) × ( F 2 { exp [ i π λ ( z 0 + z ) ( u 2 + v 2 ) ] P ̃ 0 ( u , v ) } ) .
F 2 { exp [ i π λ z 0 ( u 2 + v 2 ) ] circ ( u 2 + v 2 ρ max ) } exp [ i π ( x 2 + y 2 ) λ z 0 ] circ ( x 2 + y 2 λ z 0 ρ max ) .
S ( x , y , z ) exp [ i π ( x 2 + y 2 ) λ ( z 0 2 ) ( 1 z 2 z 0 2 ) ] circ ( x 2 + y 2 a ) ,
a ( z 0 z ) λ ρ max = ( z 0 z ) sin α ,
Q ̃ ( u , v ; z ) exp [ i π λ ( u 2 + v 2 ) z 2 2 z 0 ] circ ( u 2 + v 2 2 ρ max ) .
DOF = ( Δ z ) 1 = λ 1 ρ max 2 λ sin 2 α .
DOF = ( Δ z ) 2 = ( z 0 2 λ ) 1 2 ρ max 1 = ( λ z 0 2 ) 1 2 sin α .
P ̃ 1 ( u , v ) = exp ( + i 2 π a u 2 + v 2 ) P ̃ 0 ( u , v ) ,
P ̃ 2 ( u , v ) = exp ( i 2 π a u 2 + v 2 ) P ̃ 0 ( u , v ) .
I ̃ g ( u , v ) = H ̃ ( u , v ) S ̃ * ( u , v ) S ̃ ( u , v ) 2 + Γ ,

Metrics