Abstract

Measuring the morphology of red blood cells is important for clinical diagnosis, providing valuable indications on a patient’s health. In this work, we have simulated the appearance of normal red blood cells under a reflectance confocal microscope and discovered unique relations between the morphological parameters and the resulting characteristic interference patterns of the cell. The simulation results showed good agreement with in vitro reflectance confocal images of red blood cells, acquired using spectrally encoded flow cytometry that imaged the cells in a linear flow without artificial staining. By matching the simulated patterns to confocal images of the cells, this method could be used for measuring cell morphology in three dimensions and for studying their physiology.

© 2015 Optical Society of America

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References

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2013 (3)

E. M. Strohm, E. S. Berndl, and M. C. Kolios, “Probing red blood cell morphology using high-frequency photoacoustics,” Biophys. J. 105(1), 59–67 (2013).
[Crossref] [PubMed]

E. M. Strohm, E. S. L. Berndl, and M. C. Kolios, “High frequency label-free photoacoustic microscopy of single cells,” Photoacoustics 1(3-4), 49–53 (2013).
[Crossref] [PubMed]

J. Ford, “Red blood cell morphology,” Int. J. Lab. Hematol. 35(3), 351–357 (2013).
[Crossref] [PubMed]

2012 (2)

2011 (2)

2010 (4)

W.-H. Kim, C.-I. Kim, S.-W. Lee, S.-H. Lim, C.-W. Park, H. Lee, and M.-K. Park, “Particle image velocimetry of the blood flow in a micro-channel using the confocal laser scanning microscope,” J. Opt. Soc. Korea 14(1), 42–48 (2010).
[Crossref]

L. Golan and D. Yelin, “Flow cytometry using spectrally encoded confocal microscopy,” Opt. Lett. 35(13), 2218–2220 (2010).
[Crossref] [PubMed]

M. Diez-Silva, M. Dao, J. Han, C.-T. Lim, and S. Suresh, “Shape and biomechanical characteristics of human red blood cells in health and disease,” MRS Bull. 35(5), 382–388 (2010).
[Crossref] [PubMed]

D. Kang, M. J. Suter, C. Boudoux, H. Yoo, P. S. Yachimski, W. P. Puricelli, N. S. Nishioka, M. Mino-Kenudson, G. Y. Lauwers, B. E. Bouma, and G. J. Tearney, “Comprehensive imaging of gastroesophageal biopsy samples by spectrally encoded confocal microscopy,” Gastrointest. Endosc. 71(1), 35–43 (2010).
[Crossref] [PubMed]

2009 (1)

2007 (1)

2005 (1)

2001 (1)

S. González, R. Sackstein, R. R. Anderson, and M. Rajadhyaksha, “Real-time evidence of in vivo leukocyte trafficking in human skin by reflectance confocal microscopy,” J. Invest. Dermatol. 117(2), 384–386 (2001).
[Crossref] [PubMed]

2000 (1)

R. A. Drezek, T. Collier, C. K. Brookner, A. Malpica, R. Lotan, R. R. Richards-Kortum, and M. Follen, “Laser scanning confocal microscopy of cervical tissue before and after application of acetic acid,” Am. J. Obstet. Gynecol. 182(5), 1135–1139 (2000).
[Crossref] [PubMed]

1999 (3)

M. Rajadhyaksha, S. González, J. M. Zavislan, R. R. Anderson, and R. H. Webb, “In vivo confocal scanning laser microscopy of human skin II: advances in instrumentation and comparison with histology,” J. Invest. Dermatol. 113(3), 293–303 (1999).
[Crossref] [PubMed]

W. M. White, M. Rajadhyaksha, S. González, R. L. Fabian, and R. R. Anderson, “Noninvasive imaging of human oral mucosa in vivo by confocal reflectance microscopy,” Laryngoscope 109(10), 1709–1717 (1999).
[Crossref] [PubMed]

M. Rajadhyaksha, R. R. Anderson, and R. H. Webb, “Video-rate confocal scanning laser microscope for imaging human tissues in vivo,” Appl. Opt. 38(10), 2105–2115 (1999).
[Crossref] [PubMed]

1998 (1)

1995 (1)

M. Rajadhyaksha, M. Grossman, D. Esterowitz, R. H. Webb, and R. R. Anderson, “In vivo confocal scanning laser microscopy of human skin: melanin provides strong contrast,” J. Invest. Dermatol. 104(6), 946–952 (1995).
[Crossref] [PubMed]

1993 (1)

1987 (1)

1972 (1)

E. Evans and Y.-C. Fung, “Improved measurements of the erythrocyte geometry,” Microvasc. Res. 4(4), 335–347 (1972).
[Crossref] [PubMed]

Anderson, R. R.

S. González, R. Sackstein, R. R. Anderson, and M. Rajadhyaksha, “Real-time evidence of in vivo leukocyte trafficking in human skin by reflectance confocal microscopy,” J. Invest. Dermatol. 117(2), 384–386 (2001).
[Crossref] [PubMed]

M. Rajadhyaksha, R. R. Anderson, and R. H. Webb, “Video-rate confocal scanning laser microscope for imaging human tissues in vivo,” Appl. Opt. 38(10), 2105–2115 (1999).
[Crossref] [PubMed]

M. Rajadhyaksha, S. González, J. M. Zavislan, R. R. Anderson, and R. H. Webb, “In vivo confocal scanning laser microscopy of human skin II: advances in instrumentation and comparison with histology,” J. Invest. Dermatol. 113(3), 293–303 (1999).
[Crossref] [PubMed]

W. M. White, M. Rajadhyaksha, S. González, R. L. Fabian, and R. R. Anderson, “Noninvasive imaging of human oral mucosa in vivo by confocal reflectance microscopy,” Laryngoscope 109(10), 1709–1717 (1999).
[Crossref] [PubMed]

M. Rajadhyaksha, M. Grossman, D. Esterowitz, R. H. Webb, and R. R. Anderson, “In vivo confocal scanning laser microscopy of human skin: melanin provides strong contrast,” J. Invest. Dermatol. 104(6), 946–952 (1995).
[Crossref] [PubMed]

Auran, J. D.

Berndl, E. S.

E. M. Strohm, E. S. Berndl, and M. C. Kolios, “Probing red blood cell morphology using high-frequency photoacoustics,” Biophys. J. 105(1), 59–67 (2013).
[Crossref] [PubMed]

Berndl, E. S. L.

E. M. Strohm, E. S. L. Berndl, and M. C. Kolios, “High frequency label-free photoacoustic microscopy of single cells,” Photoacoustics 1(3-4), 49–53 (2013).
[Crossref] [PubMed]

Boss, D.

Boudoux, C.

D. Kang, M. J. Suter, C. Boudoux, H. Yoo, P. S. Yachimski, W. P. Puricelli, N. S. Nishioka, M. Mino-Kenudson, G. Y. Lauwers, B. E. Bouma, and G. J. Tearney, “Comprehensive imaging of gastroesophageal biopsy samples by spectrally encoded confocal microscopy,” Gastrointest. Endosc. 71(1), 35–43 (2010).
[Crossref] [PubMed]

D. Yelin, C. Boudoux, B. E. Bouma, and G. J. Tearney, “Large area confocal microscopy,” Opt. Lett. 32(9), 1102–1104 (2007).
[Crossref] [PubMed]

C. Boudoux, S. Yun, W. Oh, W. White, N. Iftimia, M. Shishkov, B. Bouma, and G. Tearney, “Rapid wavelength-swept spectrally encoded confocal microscopy,” Opt. Express 13(20), 8214–8221 (2005).
[Crossref] [PubMed]

Bouma, B.

Bouma, B. E.

D. Kang, M. J. Suter, C. Boudoux, H. Yoo, P. S. Yachimski, W. P. Puricelli, N. S. Nishioka, M. Mino-Kenudson, G. Y. Lauwers, B. E. Bouma, and G. J. Tearney, “Comprehensive imaging of gastroesophageal biopsy samples by spectrally encoded confocal microscopy,” Gastrointest. Endosc. 71(1), 35–43 (2010).
[Crossref] [PubMed]

D. Yelin, C. Boudoux, B. E. Bouma, and G. J. Tearney, “Large area confocal microscopy,” Opt. Lett. 32(9), 1102–1104 (2007).
[Crossref] [PubMed]

G. J. Tearney, R. H. Webb, and B. E. Bouma, “Spectrally encoded confocal microscopy,” Opt. Lett. 23(15), 1152–1154 (1998).
[Crossref] [PubMed]

Brookner, C. K.

R. A. Drezek, T. Collier, C. K. Brookner, A. Malpica, R. Lotan, R. R. Richards-Kortum, and M. Follen, “Laser scanning confocal microscopy of cervical tissue before and after application of acetic acid,” Am. J. Obstet. Gynecol. 182(5), 1135–1139 (2000).
[Crossref] [PubMed]

Carlini, A. R.

Choi, Y.-S.

Collier, T.

R. A. Drezek, T. Collier, C. K. Brookner, A. Malpica, R. Lotan, R. R. Richards-Kortum, and M. Follen, “Laser scanning confocal microscopy of cervical tissue before and after application of acetic acid,” Am. J. Obstet. Gynecol. 182(5), 1135–1139 (2000).
[Crossref] [PubMed]

Dann, E. J.

Dao, M.

M. Diez-Silva, M. Dao, J. Han, C.-T. Lim, and S. Suresh, “Shape and biomechanical characteristics of human red blood cells in health and disease,” MRS Bull. 35(5), 382–388 (2010).
[Crossref] [PubMed]

Dey, D. K.

Diez-Silva, M.

M. Diez-Silva, M. Dao, J. Han, C.-T. Lim, and S. Suresh, “Shape and biomechanical characteristics of human red blood cells in health and disease,” MRS Bull. 35(5), 382–388 (2010).
[Crossref] [PubMed]

Ding, H.

Drezek, R. A.

R. A. Drezek, T. Collier, C. K. Brookner, A. Malpica, R. Lotan, R. R. Richards-Kortum, and M. Follen, “Laser scanning confocal microscopy of cervical tissue before and after application of acetic acid,” Am. J. Obstet. Gynecol. 182(5), 1135–1139 (2000).
[Crossref] [PubMed]

Esterowitz, D.

M. Rajadhyaksha, M. Grossman, D. Esterowitz, R. H. Webb, and R. R. Anderson, “In vivo confocal scanning laser microscopy of human skin: melanin provides strong contrast,” J. Invest. Dermatol. 104(6), 946–952 (1995).
[Crossref] [PubMed]

Evans, E.

E. Evans and Y.-C. Fung, “Improved measurements of the erythrocyte geometry,” Microvasc. Res. 4(4), 335–347 (1972).
[Crossref] [PubMed]

Fabian, R. L.

W. M. White, M. Rajadhyaksha, S. González, R. L. Fabian, and R. R. Anderson, “Noninvasive imaging of human oral mucosa in vivo by confocal reflectance microscopy,” Laryngoscope 109(10), 1709–1717 (1999).
[Crossref] [PubMed]

Florakis, G. J.

Follen, M.

R. A. Drezek, T. Collier, C. K. Brookner, A. Malpica, R. Lotan, R. R. Richards-Kortum, and M. Follen, “Laser scanning confocal microscopy of cervical tissue before and after application of acetic acid,” Am. J. Obstet. Gynecol. 182(5), 1135–1139 (2000).
[Crossref] [PubMed]

Ford, J.

J. Ford, “Red blood cell morphology,” Int. J. Lab. Hematol. 35(3), 351–357 (2013).
[Crossref] [PubMed]

Fung, Y.-C.

E. Evans and Y.-C. Fung, “Improved measurements of the erythrocyte geometry,” Microvasc. Res. 4(4), 335–347 (1972).
[Crossref] [PubMed]

Golan, L.

González, S.

S. González, R. Sackstein, R. R. Anderson, and M. Rajadhyaksha, “Real-time evidence of in vivo leukocyte trafficking in human skin by reflectance confocal microscopy,” J. Invest. Dermatol. 117(2), 384–386 (2001).
[Crossref] [PubMed]

W. M. White, M. Rajadhyaksha, S. González, R. L. Fabian, and R. R. Anderson, “Noninvasive imaging of human oral mucosa in vivo by confocal reflectance microscopy,” Laryngoscope 109(10), 1709–1717 (1999).
[Crossref] [PubMed]

M. Rajadhyaksha, S. González, J. M. Zavislan, R. R. Anderson, and R. H. Webb, “In vivo confocal scanning laser microscopy of human skin II: advances in instrumentation and comparison with histology,” J. Invest. Dermatol. 113(3), 293–303 (1999).
[Crossref] [PubMed]

Grossman, M.

M. Rajadhyaksha, M. Grossman, D. Esterowitz, R. H. Webb, and R. R. Anderson, “In vivo confocal scanning laser microscopy of human skin: melanin provides strong contrast,” J. Invest. Dermatol. 104(6), 946–952 (1995).
[Crossref] [PubMed]

Han, J.

M. Diez-Silva, M. Dao, J. Han, C.-T. Lim, and S. Suresh, “Shape and biomechanical characteristics of human red blood cells in health and disease,” MRS Bull. 35(5), 382–388 (2010).
[Crossref] [PubMed]

Iftimia, N.

Javidi, B.

Kang, D.

D. Kang, M. J. Suter, C. Boudoux, H. Yoo, P. S. Yachimski, W. P. Puricelli, N. S. Nishioka, M. Mino-Kenudson, G. Y. Lauwers, B. E. Bouma, and G. J. Tearney, “Comprehensive imaging of gastroesophageal biopsy samples by spectrally encoded confocal microscopy,” Gastrointest. Endosc. 71(1), 35–43 (2010).
[Crossref] [PubMed]

Kim, C.-I.

Kim, W.-H.

Koester, C. J.

Kolios, M. C.

E. M. Strohm, E. S. Berndl, and M. C. Kolios, “Probing red blood cell morphology using high-frequency photoacoustics,” Biophys. J. 105(1), 59–67 (2013).
[Crossref] [PubMed]

E. M. Strohm, E. S. L. Berndl, and M. C. Kolios, “High frequency label-free photoacoustic microscopy of single cells,” Photoacoustics 1(3-4), 49–53 (2013).
[Crossref] [PubMed]

Lauwers, G. Y.

D. Kang, M. J. Suter, C. Boudoux, H. Yoo, P. S. Yachimski, W. P. Puricelli, N. S. Nishioka, M. Mino-Kenudson, G. Y. Lauwers, B. E. Bouma, and G. J. Tearney, “Comprehensive imaging of gastroesophageal biopsy samples by spectrally encoded confocal microscopy,” Gastrointest. Endosc. 71(1), 35–43 (2010).
[Crossref] [PubMed]

Lee, H.

Lee, S.-J.

Lee, S.-W.

Lim, C.-T.

M. Diez-Silva, M. Dao, J. Han, C.-T. Lim, and S. Suresh, “Shape and biomechanical characteristics of human red blood cells in health and disease,” MRS Bull. 35(5), 382–388 (2010).
[Crossref] [PubMed]

Lim, J.

Lim, S.-H.

Liu, R.

Lotan, R.

R. A. Drezek, T. Collier, C. K. Brookner, A. Malpica, R. Lotan, R. R. Richards-Kortum, and M. Follen, “Laser scanning confocal microscopy of cervical tissue before and after application of acetic acid,” Am. J. Obstet. Gynecol. 182(5), 1135–1139 (2000).
[Crossref] [PubMed]

Malpica, A.

R. A. Drezek, T. Collier, C. K. Brookner, A. Malpica, R. Lotan, R. R. Richards-Kortum, and M. Follen, “Laser scanning confocal microscopy of cervical tissue before and after application of acetic acid,” Am. J. Obstet. Gynecol. 182(5), 1135–1139 (2000).
[Crossref] [PubMed]

Marquet, P.

Minai, L.

Mino-Kenudson, M.

D. Kang, M. J. Suter, C. Boudoux, H. Yoo, P. S. Yachimski, W. P. Puricelli, N. S. Nishioka, M. Mino-Kenudson, G. Y. Lauwers, B. E. Bouma, and G. J. Tearney, “Comprehensive imaging of gastroesophageal biopsy samples by spectrally encoded confocal microscopy,” Gastrointest. Endosc. 71(1), 35–43 (2010).
[Crossref] [PubMed]

Mir, M.

Nishioka, N. S.

D. Kang, M. J. Suter, C. Boudoux, H. Yoo, P. S. Yachimski, W. P. Puricelli, N. S. Nishioka, M. Mino-Kenudson, G. Y. Lauwers, B. E. Bouma, and G. J. Tearney, “Comprehensive imaging of gastroesophageal biopsy samples by spectrally encoded confocal microscopy,” Gastrointest. Endosc. 71(1), 35–43 (2010).
[Crossref] [PubMed]

Oh, W.

Park, C.-W.

Park, M.-K.

Popescu, G.

Puricelli, W. P.

D. Kang, M. J. Suter, C. Boudoux, H. Yoo, P. S. Yachimski, W. P. Puricelli, N. S. Nishioka, M. Mino-Kenudson, G. Y. Lauwers, B. E. Bouma, and G. J. Tearney, “Comprehensive imaging of gastroesophageal biopsy samples by spectrally encoded confocal microscopy,” Gastrointest. Endosc. 71(1), 35–43 (2010).
[Crossref] [PubMed]

Rajadhyaksha, M.

S. González, R. Sackstein, R. R. Anderson, and M. Rajadhyaksha, “Real-time evidence of in vivo leukocyte trafficking in human skin by reflectance confocal microscopy,” J. Invest. Dermatol. 117(2), 384–386 (2001).
[Crossref] [PubMed]

M. Rajadhyaksha, R. R. Anderson, and R. H. Webb, “Video-rate confocal scanning laser microscope for imaging human tissues in vivo,” Appl. Opt. 38(10), 2105–2115 (1999).
[Crossref] [PubMed]

M. Rajadhyaksha, S. González, J. M. Zavislan, R. R. Anderson, and R. H. Webb, “In vivo confocal scanning laser microscopy of human skin II: advances in instrumentation and comparison with histology,” J. Invest. Dermatol. 113(3), 293–303 (1999).
[Crossref] [PubMed]

W. M. White, M. Rajadhyaksha, S. González, R. L. Fabian, and R. R. Anderson, “Noninvasive imaging of human oral mucosa in vivo by confocal reflectance microscopy,” Laryngoscope 109(10), 1709–1717 (1999).
[Crossref] [PubMed]

M. Rajadhyaksha, M. Grossman, D. Esterowitz, R. H. Webb, and R. R. Anderson, “In vivo confocal scanning laser microscopy of human skin: melanin provides strong contrast,” J. Invest. Dermatol. 104(6), 946–952 (1995).
[Crossref] [PubMed]

Richards-Kortum, R. R.

R. A. Drezek, T. Collier, C. K. Brookner, A. Malpica, R. Lotan, R. R. Richards-Kortum, and M. Follen, “Laser scanning confocal microscopy of cervical tissue before and after application of acetic acid,” Am. J. Obstet. Gynecol. 182(5), 1135–1139 (2000).
[Crossref] [PubMed]

Rosskothen, H. D.

Sackstein, R.

S. González, R. Sackstein, R. R. Anderson, and M. Rajadhyaksha, “Real-time evidence of in vivo leukocyte trafficking in human skin by reflectance confocal microscopy,” J. Invest. Dermatol. 117(2), 384–386 (2001).
[Crossref] [PubMed]

Shishkov, M.

Strohm, E. M.

E. M. Strohm, E. S. Berndl, and M. C. Kolios, “Probing red blood cell morphology using high-frequency photoacoustics,” Biophys. J. 105(1), 59–67 (2013).
[Crossref] [PubMed]

E. M. Strohm, E. S. L. Berndl, and M. C. Kolios, “High frequency label-free photoacoustic microscopy of single cells,” Photoacoustics 1(3-4), 49–53 (2013).
[Crossref] [PubMed]

Suresh, S.

M. Diez-Silva, M. Dao, J. Han, C.-T. Lim, and S. Suresh, “Shape and biomechanical characteristics of human red blood cells in health and disease,” MRS Bull. 35(5), 382–388 (2010).
[Crossref] [PubMed]

Suter, M. J.

D. Kang, M. J. Suter, C. Boudoux, H. Yoo, P. S. Yachimski, W. P. Puricelli, N. S. Nishioka, M. Mino-Kenudson, G. Y. Lauwers, B. E. Bouma, and G. J. Tearney, “Comprehensive imaging of gastroesophageal biopsy samples by spectrally encoded confocal microscopy,” Gastrointest. Endosc. 71(1), 35–43 (2010).
[Crossref] [PubMed]

Tackaberry, R. B.

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E. M. Strohm, E. S. Berndl, and M. C. Kolios, “Probing red blood cell morphology using high-frequency photoacoustics,” Biophys. J. 105(1), 59–67 (2013).
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D. Kang, M. J. Suter, C. Boudoux, H. Yoo, P. S. Yachimski, W. P. Puricelli, N. S. Nishioka, M. Mino-Kenudson, G. Y. Lauwers, B. E. Bouma, and G. J. Tearney, “Comprehensive imaging of gastroesophageal biopsy samples by spectrally encoded confocal microscopy,” Gastrointest. Endosc. 71(1), 35–43 (2010).
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J. Ford, “Red blood cell morphology,” Int. J. Lab. Hematol. 35(3), 351–357 (2013).
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W. M. White, M. Rajadhyaksha, S. González, R. L. Fabian, and R. R. Anderson, “Noninvasive imaging of human oral mucosa in vivo by confocal reflectance microscopy,” Laryngoscope 109(10), 1709–1717 (1999).
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Opt. Express (1)

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E. M. Strohm, E. S. L. Berndl, and M. C. Kolios, “High frequency label-free photoacoustic microscopy of single cells,” Photoacoustics 1(3-4), 49–53 (2013).
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Figures (7)

Fig. 1
Fig. 1 Schematic description of the simulated imaging system. L1, L2 – lenses. An image of the cell is acquired by scanning in the lateral x-y plane. The small black and red triangles represent the origin of the optical system and the cell coordinates, respectively.
Fig. 2
Fig. 2 (a) Lateral (left) and axial (right) field amplitude distributions around the focal point for NA = 0.6 lens and 840 nm wavelength. The simulated fields (blue circles) agree well with theoretical prediction (dashed lines) of Eq. (2). (b) An exemplary x-z cross-section of a simulated cell with a 20° tilt angle and 1 μm below the focal plane, drawn to scale with the cross-sectional field amplitude distribution (red colormap, arbitrary units).
Fig. 3
Fig. 3 Simulated RCM images of a red blood cell for different axial positions and tilt angles. Size of each panel is 8 μm × 8 μm. Cell shape is calculated according to Eq. (3) with the parameters reported in Ref [24]: R0 = 3.91 μm, C0 = 0.81 μm, C2 = 7.83 μm, and C4 = -4.39 μm.
Fig. 4
Fig. 4 Schematic illustration of the bench-top SEFC system for confocal imaging of flowing blood cells. SLD: super-luminescence diode array. L1-5 – lenses. BS – beam splitter. G – transmission grating. CM – cold mirror.
Fig. 5
Fig. 5 SEFC image of flowing RBCs. 5 × magnified views are shown next to selected cells.
Fig. 6
Fig. 6 Top panels: Comparison between six SEFC images of RBCs and simulated confocal patterns that approximate each image. Table: The parameters used to simulate the patterns that correspond to each cell image. Cells’ profiles (red curves) are derived using Eq. (3). Dashed curves represent the averaged cell shape in Ref [24]. CV – corpuscular volume. Scale bars denote 1 µm.
Fig. 7
Fig. 7 Comparison between transmission-mode widefield image and co-registered confocal image of the same RBCs. Loss of the characteristic interference rings patterns was most likely due to aberrations induced by nearby out-of-focus cells. Scale bars represent 2 μm.

Equations (8)

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U 2 ( x,y,z )= i e ik( f 1 z ) λ( f 1 z ) U 1 P L1 e i k 2 f 1 ( x 2 + y 2 ) e iπ ( xx' ) 2 + ( yy' ) 2 λ( f 1 z ) dx'dy' ,
U( u,v )= 0 1 J 0 ( vρ ) e i 2 u ρ 2 ρdρ ,
Z c ± ( x c , y c )=[ C 0 + C 2 ( x c 2 + y c 2 R 0 ) 2 + C 4 ( x c 2 + y c 2 R 0 ) 4 ]( ± ) 1 ( x c 2 + y c 2 R 0 ) 2 ,
U ± ( x,y, z ± ; x c , y c )= U 2 ( x,y, z ± ) e i2k[ Z c ± ( x+ x c ,y+ y c ) Z c ± ( x c , y c ) ] P cell ,
U 3 ( x,y )= i e ik( f 1 z + ) λ( f 1 z + ) U + e iπ ( xx' ) 2 + ( yy' ) 2 λ( f 1 z + ) d x d y + i e ik( f 1 z ) λ( f 1 z ) U e iπ ( xx' ) 2 + ( yy' ) 2 λ( f 1 z ) d x d y .
U 4 ( x,y )= i e ik f 2 λ f 2 U 3 P L1 e i k 2 f 1 ( x 2 + y 2 ) e iπ ( x x ) 2 + ( y y ) 2 λ f 2 d x d y .
U 5 ( x,y )= i e ik f 2 λ f 2 U 4 P L2 e i k 2 f 2 ( x 2 + y 2 ) e iπ ( x x ) 2 + ( y y ) 2 λ f 2 d x d y .
CV=2π 0 R 0 2 Z c + ( r )rdr .

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