Abstract

We describe various methods to process the data collected with a digital confocal microscope (DCM) in order to get more information than what we could get from a conventional confocal system. Different metrics can be extracted from the data collected with the DCM in order to produce images that reveal different features of the sample. The integrated phase of the scattered field allows for the three-dimensional reconstruction of the refractive index distribution. In a similar way, the integration of the field intensity yields the absorption coefficient distribution. The deflection of the digitally reconstructed focus reveals the sample-induced aberrations and the RMS width of the focus gives an indication on the local scattering coefficient. Finally, in addition to the conventional confocal metric, which consists in integrating the intensity within the pinhole, the DCM allows for the measurement of the phase within the pinhole. This metrics is close to the whole-field integrated phase and thus gives a qualitative image of the refractive index distribution.

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References

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  1. M. Minsky, “Microscopy apparatus,” U.S. patent 3,013,467 (1961).
  2. C. J. R. Sheppard and A. Coudhury, “Image formation in the scanning microscope,” Opt. Acta24(10), 1051–1073 (1977).
    [CrossRef]
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    [CrossRef]
  5. G. Barbastathis, M. Balberg, and D. J. Brady, “Confocal microscopy with a volume holographic filter,” Opt. Lett.24, 811–813 (1999).
    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  22. N. Lue, W. Choi, K. Badizadegan, R. R. Dasari, M. S. Feld, and G. Popescu, “Confocal diffraction phase microscopy of live cells,” Opt. Lett.33, 2074–2076 (2008).
    [CrossRef] [PubMed]
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    [CrossRef]

2012

2010

K. Dillon and Y. Fainman, “Depth sectioning of attenuation,” J. Opt. Soc. Am. A27, 1347–1354 (2010).
[CrossRef]

K. Dillon and Y. Fainman, “Computational confocal tomography for simultaneous reconstruction of objects, occlusions and aberrations,” App. Opt.49(13), 2529–2538 (2010).
[CrossRef]

2008

2007

W. Choi, C. Fang-Yen, K. Badizadegan, S. Oh, N. Lue, R. R. Dasari, and M. S. Feld, “Tomographic phase microscopy,” Nat. Methods4, 717–719 (2007).
[CrossRef] [PubMed]

S. Lai, R. A. McLeod, P. Jacquemin, S. Atalick, and R. Herring, “An algorithm for 3-D refractive index measurement in holographic confocal microscopy,” Ultramicroscopy107, 196–201 (2007).
[CrossRef]

2006

2004

G. N. Vishnyakov, G. G. Levin, V. L. Minaev, V. V. Pickalov, and A. V. Likhachev, “Tomographic interference microscopy of living cells,” Microscopy and Analysis18, 15–17 (2004).

2003

2000

E. Cuche, P. Marquet, and C. Depeursinge, “Spatial filtering for zero-order and twin-image elimination in digital off-axis holography,” App. Opt.39, 4070–4075 (2000).
[CrossRef]

1999

1991

A. E. Dixon, S. Damaskinos, and M. R. Atkinson, “A scanning confocal microscope for transmission and reflection imaging,” Nature351, 551–553 (1991).
[CrossRef]

1988

M. D. Feit and J. A. Fleck, “Beam nonparaxiality, filament formation, and beam breakup in the self-focusing of optical beams,” J. Opt. Soc. Am. B5(3), 633–640 (1988).
[CrossRef]

R. M. Goldstein, H. A. Zebken, and C. L. Werner, “Satellite radar interferometry: two-dimensional phase unwrapping,” Radio Sci.23(4), 713–720 (1988).
[CrossRef]

1977

C. J. R. Sheppard and A. Coudhury, “Image formation in the scanning microscope,” Opt. Acta24(10), 1051–1073 (1977).
[CrossRef]

Arnison, M. R.

J. W. OByrne, P. W. Fekete, M. R. Arnison, H. Zhao, M. Serrano, D. Philp, W. Sudiarta, and C. J. Cogswell, “Adaptive optics in confocal microscopy,” in Proceedings of the 2nd International Workshop on Adaptive Optics for Industry and Medicine, G. D. Love, ed. (World Scientific, 1999).

Atalick, S.

S. Lai, R. A. McLeod, P. Jacquemin, S. Atalick, and R. Herring, “An algorithm for 3-D refractive index measurement in holographic confocal microscopy,” Ultramicroscopy107, 196–201 (2007).
[CrossRef]

Atkinson, M. R.

A. E. Dixon, S. Damaskinos, and M. R. Atkinson, “A scanning confocal microscope for transmission and reflection imaging,” Nature351, 551–553 (1991).
[CrossRef]

Badizadegan, K.

Balberg, M.

Barbastathis, G.

Brady, D. J.

Charrière, F.

Choi, W.

Cogswell, C. J.

J. W. OByrne, P. W. Fekete, M. R. Arnison, H. Zhao, M. Serrano, D. Philp, W. Sudiarta, and C. J. Cogswell, “Adaptive optics in confocal microscopy,” in Proceedings of the 2nd International Workshop on Adaptive Optics for Industry and Medicine, G. D. Love, ed. (World Scientific, 1999).

Colomb, T.

Coudhury, A.

C. J. R. Sheppard and A. Coudhury, “Image formation in the scanning microscope,” Opt. Acta24(10), 1051–1073 (1977).
[CrossRef]

Cuche, E.

F. Charrière, A. Marian, F. Montfort, J. Kuehn, T. Colomb, E. Cuche, P. Marquet, and C. Depeursinge, “Cell refractive index tomography by digital holographic microscopy,” Opt. Lett.31, 178–180 (2006).
[CrossRef] [PubMed]

E. Cuche, P. Marquet, and C. Depeursinge, “Spatial filtering for zero-order and twin-image elimination in digital off-axis holography,” App. Opt.39, 4070–4075 (2000).
[CrossRef]

Damaskinos, S.

A. E. Dixon, S. Damaskinos, and M. R. Atkinson, “A scanning confocal microscope for transmission and reflection imaging,” Nature351, 551–553 (1991).
[CrossRef]

Dasari, R. R.

Debailleul, M.

M. Debailleul, B. Simon, V. Georges, O. Haeberle, and V. Lauer, “Holographic microscopy and diffractive microtomography of transparent samples,” Meas. Sci. Technol.19, 074009 (2008).
[CrossRef]

Depeursinge, C.

Dillon, K.

K. Dillon and Y. Fainman, “Depth sectioning of attenuation,” J. Opt. Soc. Am. A27, 1347–1354 (2010).
[CrossRef]

K. Dillon and Y. Fainman, “Computational confocal tomography for simultaneous reconstruction of objects, occlusions and aberrations,” App. Opt.49(13), 2529–2538 (2010).
[CrossRef]

Dixon, A. E.

A. E. Dixon, S. Damaskinos, and M. R. Atkinson, “A scanning confocal microscope for transmission and reflection imaging,” Nature351, 551–553 (1991).
[CrossRef]

Fainman, Y.

K. Dillon and Y. Fainman, “Computational confocal tomography for simultaneous reconstruction of objects, occlusions and aberrations,” App. Opt.49(13), 2529–2538 (2010).
[CrossRef]

K. Dillon and Y. Fainman, “Depth sectioning of attenuation,” J. Opt. Soc. Am. A27, 1347–1354 (2010).
[CrossRef]

Fang-Yen, C.

Feit, M. D.

Fekete, P. W.

J. W. OByrne, P. W. Fekete, M. R. Arnison, H. Zhao, M. Serrano, D. Philp, W. Sudiarta, and C. J. Cogswell, “Adaptive optics in confocal microscopy,” in Proceedings of the 2nd International Workshop on Adaptive Optics for Industry and Medicine, G. D. Love, ed. (World Scientific, 1999).

Feld, M. S.

Fleck, J. A.

Georges, V.

M. Debailleul, B. Simon, V. Georges, O. Haeberle, and V. Lauer, “Holographic microscopy and diffractive microtomography of transparent samples,” Meas. Sci. Technol.19, 074009 (2008).
[CrossRef]

Goldstein, R. M.

R. M. Goldstein, H. A. Zebken, and C. L. Werner, “Satellite radar interferometry: two-dimensional phase unwrapping,” Radio Sci.23(4), 713–720 (1988).
[CrossRef]

Goy, A.

Haeberle, O.

M. Debailleul, B. Simon, V. Georges, O. Haeberle, and V. Lauer, “Holographic microscopy and diffractive microtomography of transparent samples,” Meas. Sci. Technol.19, 074009 (2008).
[CrossRef]

Helgason, S.

S. Helgason, The Radon Transform, 2nd ed. (Birkhauser, 1999).

Herring, R.

S. Lai, R. A. McLeod, P. Jacquemin, S. Atalick, and R. Herring, “An algorithm for 3-D refractive index measurement in holographic confocal microscopy,” Ultramicroscopy107, 196–201 (2007).
[CrossRef]

Jacquemin, P.

S. Lai, R. A. McLeod, P. Jacquemin, S. Atalick, and R. Herring, “An algorithm for 3-D refractive index measurement in holographic confocal microscopy,” Ultramicroscopy107, 196–201 (2007).
[CrossRef]

Kuehn, J.

Lai, S.

S. Lai, R. A. McLeod, P. Jacquemin, S. Atalick, and R. Herring, “An algorithm for 3-D refractive index measurement in holographic confocal microscopy,” Ultramicroscopy107, 196–201 (2007).
[CrossRef]

Landkof, N. S.

N. S. Landkof, Foundation of Modern Potential Theory (Springer Verlag, 1972).
[CrossRef]

Lauer, V.

M. Debailleul, B. Simon, V. Georges, O. Haeberle, and V. Lauer, “Holographic microscopy and diffractive microtomography of transparent samples,” Meas. Sci. Technol.19, 074009 (2008).
[CrossRef]

Levin, G. G.

G. N. Vishnyakov, G. G. Levin, V. L. Minaev, V. V. Pickalov, and A. V. Likhachev, “Tomographic interference microscopy of living cells,” Microscopy and Analysis18, 15–17 (2004).

Likhachev, A. V.

G. N. Vishnyakov, G. G. Levin, V. L. Minaev, V. V. Pickalov, and A. V. Likhachev, “Tomographic interference microscopy of living cells,” Microscopy and Analysis18, 15–17 (2004).

Lue, N.

N. Lue, W. Choi, K. Badizadegan, R. R. Dasari, M. S. Feld, and G. Popescu, “Confocal diffraction phase microscopy of live cells,” Opt. Lett.33, 2074–2076 (2008).
[CrossRef] [PubMed]

W. Choi, C. Fang-Yen, K. Badizadegan, S. Oh, N. Lue, R. R. Dasari, and M. S. Feld, “Tomographic phase microscopy,” Nat. Methods4, 717–719 (2007).
[CrossRef] [PubMed]

Marian, A.

Marquet, P.

F. Charrière, A. Marian, F. Montfort, J. Kuehn, T. Colomb, E. Cuche, P. Marquet, and C. Depeursinge, “Cell refractive index tomography by digital holographic microscopy,” Opt. Lett.31, 178–180 (2006).
[CrossRef] [PubMed]

E. Cuche, P. Marquet, and C. Depeursinge, “Spatial filtering for zero-order and twin-image elimination in digital off-axis holography,” App. Opt.39, 4070–4075 (2000).
[CrossRef]

McLeod, R. A.

S. Lai, R. A. McLeod, P. Jacquemin, S. Atalick, and R. Herring, “An algorithm for 3-D refractive index measurement in holographic confocal microscopy,” Ultramicroscopy107, 196–201 (2007).
[CrossRef]

Mertz, J.

Minaev, V. L.

G. N. Vishnyakov, G. G. Levin, V. L. Minaev, V. V. Pickalov, and A. V. Likhachev, “Tomographic interference microscopy of living cells,” Microscopy and Analysis18, 15–17 (2004).

Minsky, M.

M. Minsky, “Microscopy apparatus,” U.S. patent 3,013,467 (1961).

Montfort, F.

OByrne, J. W.

J. W. OByrne, P. W. Fekete, M. R. Arnison, H. Zhao, M. Serrano, D. Philp, W. Sudiarta, and C. J. Cogswell, “Adaptive optics in confocal microscopy,” in Proceedings of the 2nd International Workshop on Adaptive Optics for Industry and Medicine, G. D. Love, ed. (World Scientific, 1999).

Oh, S.

W. Choi, C. Fang-Yen, K. Badizadegan, S. Oh, N. Lue, R. R. Dasari, and M. S. Feld, “Tomographic phase microscopy,” Nat. Methods4, 717–719 (2007).
[CrossRef] [PubMed]

Pavillon, N.

Philp, D.

J. W. OByrne, P. W. Fekete, M. R. Arnison, H. Zhao, M. Serrano, D. Philp, W. Sudiarta, and C. J. Cogswell, “Adaptive optics in confocal microscopy,” in Proceedings of the 2nd International Workshop on Adaptive Optics for Industry and Medicine, G. D. Love, ed. (World Scientific, 1999).

Pickalov, V. V.

G. N. Vishnyakov, G. G. Levin, V. L. Minaev, V. V. Pickalov, and A. V. Likhachev, “Tomographic interference microscopy of living cells,” Microscopy and Analysis18, 15–17 (2004).

Popescu, G.

Psaltis, D.

Serrano, M.

J. W. OByrne, P. W. Fekete, M. R. Arnison, H. Zhao, M. Serrano, D. Philp, W. Sudiarta, and C. J. Cogswell, “Adaptive optics in confocal microscopy,” in Proceedings of the 2nd International Workshop on Adaptive Optics for Industry and Medicine, G. D. Love, ed. (World Scientific, 1999).

Sheppard, C.

C. Sheppard and D. Shotton, Confocal Laser Scanning Microscopy (BIOS Scientific Publishers, 1997).

Sheppard, C. J. R.

C. J. R. Sheppard and A. Coudhury, “Image formation in the scanning microscope,” Opt. Acta24(10), 1051–1073 (1977).
[CrossRef]

Shotton, D.

C. Sheppard and D. Shotton, Confocal Laser Scanning Microscopy (BIOS Scientific Publishers, 1997).

Simon, B.

M. Debailleul, B. Simon, V. Georges, O. Haeberle, and V. Lauer, “Holographic microscopy and diffractive microtomography of transparent samples,” Meas. Sci. Technol.19, 074009 (2008).
[CrossRef]

Sudiarta, W.

J. W. OByrne, P. W. Fekete, M. R. Arnison, H. Zhao, M. Serrano, D. Philp, W. Sudiarta, and C. J. Cogswell, “Adaptive optics in confocal microscopy,” in Proceedings of the 2nd International Workshop on Adaptive Optics for Industry and Medicine, G. D. Love, ed. (World Scientific, 1999).

Vishnyakov, G. N.

G. N. Vishnyakov, G. G. Levin, V. L. Minaev, V. V. Pickalov, and A. V. Likhachev, “Tomographic interference microscopy of living cells,” Microscopy and Analysis18, 15–17 (2004).

Werner, C. L.

R. M. Goldstein, H. A. Zebken, and C. L. Werner, “Satellite radar interferometry: two-dimensional phase unwrapping,” Radio Sci.23(4), 713–720 (1988).
[CrossRef]

Yang, C.

Zebken, H. A.

R. M. Goldstein, H. A. Zebken, and C. L. Werner, “Satellite radar interferometry: two-dimensional phase unwrapping,” Radio Sci.23(4), 713–720 (1988).
[CrossRef]

Zhao, H.

J. W. OByrne, P. W. Fekete, M. R. Arnison, H. Zhao, M. Serrano, D. Philp, W. Sudiarta, and C. J. Cogswell, “Adaptive optics in confocal microscopy,” in Proceedings of the 2nd International Workshop on Adaptive Optics for Industry and Medicine, G. D. Love, ed. (World Scientific, 1999).

App. Opt.

E. Cuche, P. Marquet, and C. Depeursinge, “Spatial filtering for zero-order and twin-image elimination in digital off-axis holography,” App. Opt.39, 4070–4075 (2000).
[CrossRef]

K. Dillon and Y. Fainman, “Computational confocal tomography for simultaneous reconstruction of objects, occlusions and aberrations,” App. Opt.49(13), 2529–2538 (2010).
[CrossRef]

J. Opt. Soc. Am. A

J. Opt. Soc. Am. B

Meas. Sci. Technol.

M. Debailleul, B. Simon, V. Georges, O. Haeberle, and V. Lauer, “Holographic microscopy and diffractive microtomography of transparent samples,” Meas. Sci. Technol.19, 074009 (2008).
[CrossRef]

Microscopy and Analysis

G. N. Vishnyakov, G. G. Levin, V. L. Minaev, V. V. Pickalov, and A. V. Likhachev, “Tomographic interference microscopy of living cells,” Microscopy and Analysis18, 15–17 (2004).

Nat. Methods

W. Choi, C. Fang-Yen, K. Badizadegan, S. Oh, N. Lue, R. R. Dasari, and M. S. Feld, “Tomographic phase microscopy,” Nat. Methods4, 717–719 (2007).
[CrossRef] [PubMed]

Nature

A. E. Dixon, S. Damaskinos, and M. R. Atkinson, “A scanning confocal microscope for transmission and reflection imaging,” Nature351, 551–553 (1991).
[CrossRef]

Opt. Acta

C. J. R. Sheppard and A. Coudhury, “Image formation in the scanning microscope,” Opt. Acta24(10), 1051–1073 (1977).
[CrossRef]

Opt. Express

Opt. Lett.

Radio Sci.

R. M. Goldstein, H. A. Zebken, and C. L. Werner, “Satellite radar interferometry: two-dimensional phase unwrapping,” Radio Sci.23(4), 713–720 (1988).
[CrossRef]

Ultramicroscopy

S. Lai, R. A. McLeod, P. Jacquemin, S. Atalick, and R. Herring, “An algorithm for 3-D refractive index measurement in holographic confocal microscopy,” Ultramicroscopy107, 196–201 (2007).
[CrossRef]

Other

S. Helgason, The Radon Transform, 2nd ed. (Birkhauser, 1999).

N. S. Landkof, Foundation of Modern Potential Theory (Springer Verlag, 1972).
[CrossRef]

J. W. OByrne, P. W. Fekete, M. R. Arnison, H. Zhao, M. Serrano, D. Philp, W. Sudiarta, and C. J. Cogswell, “Adaptive optics in confocal microscopy,” in Proceedings of the 2nd International Workshop on Adaptive Optics for Industry and Medicine, G. D. Love, ed. (World Scientific, 1999).

M. Minsky, “Microscopy apparatus,” U.S. patent 3,013,467 (1961).

C. Sheppard and D. Shotton, Confocal Laser Scanning Microscopy (BIOS Scientific Publishers, 1997).

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Figures (7)

Fig. 1
Fig. 1

(a) Experimental apparatus for digital confocal imaging in transmission. The sample is illuminated with a focused beam through lens MO1 (Olympus 100x oil immersion N.A. = 1.4). The transmitted light is collimated by lens MO2 (same lens than MO1) the back focal plane of which is imaged by a 4f system (lenses L3 and L4) onto a CCD camera. A reference beam is combined before the camera in order to record an off-axis digital hologram. The optical path of the reference can be adjusted with the delay line (DL). It is possible to reject parasitic reflections from the objective by using a short coherence length laser and adjust the delay line so that only the reflection from the focus interferes with the reference. (b) Modified apparatus to improve the stability for phase imaging. The signal and reference beams pass through the same optical components in order to reduce mechanical variations that would change the relative optical paths. (c) Closer look at the sample in the optical system described in (b). The reference beam is shown with a dashed line and the signal beam probing the sample with a solid line.

Fig. 2
Fig. 2

(a) Diagram showing the reconstruction of the field in the 3D space around the focus. The thick 2D frame in the center is the Fourier transform of the field extracted from the digital hologram. It corresponds to the ideal imaging plane where the light would focus in the absence of scattering. A Fourier beam propagation method is used to propagate the field in the +z and −z directions. (b) xz cross-section of the ideal focus amplitude (calculated). (c) xz cross-section of the focus for a real measurement. (d) Graph of the maximum of the amplitude of the 2D (xy) correlation function between the ideal focus spot at its waist and the probe beam. The dashed curve shows the correlation for the ideal beam shown in (b) and the solid curve shows the correlation for the measured beam shown in (c).

Fig. 3
Fig. 3

(a) Image of an epithelial cell obtained with the digital confocal with a dynamic pinhole. (b) Beam deflection along the x axis (vertical axis in this picture). (c) Beam deflection along the y axis (horizontal axis in this picture). (d) Beam defocus, i.e. distance along the z direction between the ideal and measured foci. (e) Image produced by plotting the distance in the xy plane between the ideal focus and the measured focus. Bright pixels correspond to large distances and thus to a deflection of the probe beam. (f) Same as in (e) but with the distance measured in three dimensions.

Fig. 4
Fig. 4

(a) Scatter plot showing the relationship between the signal through the dynamic pinhole (normalized to the maximum) and the transverse distance d between the ideal focus and the measured focus (vertical axis). The distance is normalized with respect to the ideal focus width w0 = 16μm. A linear regression line representing the global trend is used to define two regions in the graph. The black points correspond to scanning positions for which there is little deviation of the beam and a large confocal signal. The red points are those for which the beam is deflected but only weakly aberrated. Conversely, the blue points correspond to region where the confocal signal is reduced by scattering. (b) Image obtained projecting all the points onto the regression line and then plotting the distance along the line (variable U on the graph) from the point (DC = 1, = 0). The brighter points correspond to larger distance. (c) Image obtained by plotting the distance from the regression line (variable V on the graph). Note that the distance is signed, the blue points being at negative distance from the line. The brighter pixels correspond to points above the line whereas darker pixels correspond to points below the line. (d) Map of the cell with each point painted in the color corresponding to the position in graph (a).

Fig. 5
Fig. 5

Width of the beam at the focus (1/e-radius of the beam at z = zopt) normalized to the minimum width w0 plotted as a function of the dynamic confocal signal normalized to its maximum. There is a clear and expected correlation between the width and the confocal signal. The solid curve represents the expected confocal signal for an ideal Gaussian beam according to Eq. (7). Any deviation from this model is related to scattering in the sample.

Fig. 6
Fig. 6

Principle of optical tomography with a focused beam. The phase delay and the absorption at point (x0, z0) are common to all rays in the probe beam. Conversely, regions away from (x0, z0) will modulate some of the rays but not all of them and the averaged contributions from these regions will be weak. The focused beam naturally performs a back-projection to the point of interest (x0, z0). The data can be filtered later on with a classical tomography high-pass ramp filter.

Fig. 7
Fig. 7

Images of the cell at different depths using the different metrics presented in the text. (a) In the first column is the dynamic confocal signal with the pinhole tracking in x and y. Bright pixels correspond to more light going through the dynamic pinhole. (b) In the second column is the shift in the z direction undergone by the focus. A middle gray tone corresponds to no shift, bright points to positive shifts and dark point to negative shifts. (c) In the third column the images correspond to the transverse distance d. Bright pixels correspond to large deviation of the beam. (d) In the fourth column are images produced by plotting the phase integrated within the digital pinhole. Bright pixels correspond to a larger phase shift and thus to a larger refractive index.

Tables (1)

Tables Icon

Table 1 Summary of the Different Metrics That Can Be Computed from the Data Obtained from a Digital Confocal Microscope*

Equations (17)

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z opt = arg [ max z c ( z ) ] , with c ( z ) = max z [ u i ( x , y , z ) u m ( x , y , z ) ] ,
( Δ x , Δ y ) = ( x , y ) | u m ( x , y , z opt ) | d x d y | u m ( x , y , z opt ) | d x d y .
w = ( x 2 + y 2 ) | u m ( x , y , z opt ) | d x d y | u m ( x , y , z opt ) | d x d y ,
f DC = | u m ( x , y , z opt ) | 2 p ( x Δ x , y Δ y ) d x d y
p ( x , y ) = 1 , if x 2 + y 2 < a 2 / 4 and 0 otherwise ,
Φ DC = ϕ [ u m ( x , y , z opt ) ] p ( x Δ x , y Δ y ) d x d y p ( x , y ) d x d y .
d ^ = β α f ^ DC ,
f ^ DC = 1 exp ( 2 w ^ 2 ) 1 e 2
Φ int = ϕ ( x , y ) d x d y
𝒳 ( f ) ( x , Θ ) = f ( x + t Θ ) d t
g ( x ) = S 2 𝒳 ( f ) ( x , Θ ) d Θ = S 2 f ( x + t Θ ) d t d Θ
= 0 2 π 0 π / 2 f ( x + r Θ ) r 2 r 2 sin θ d r d θ d ϕ
g ( x ) = f ( x x ) | x | 2 d x
f ( x , y , z ) = 1 [ | κ | ( g ) ( κ x , κ y , κ z ) ] , with | κ | = κ x 2 + κ y 2 + κ z 2
n ( x , y , z ) 1 [ | κ | ( Φ int ) ]
u c = exp [ j ϕ ( x , y ) ] exp [ j ( k x x + k y y ) ] d x d y = exp [ j ϕ ( x , y ) ] d x d y
u c = const . + j ϕ ( x , y ) d x d y

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