Abstract

Light sheet fluorescence microscopy (LSFM) has become an indispensable tool in biomedical studies owing to its depth-sectioning capability and low photo-bleaching. The axial resolution in LSFM is determined mainly by the thickness of the illumination sheet, and a high numerical-aperture lens is thus preferred in the illumination to increase the axial resolution. However, a rapid divergence of the illumination beam limits the effective field-of-view (FoV), that provides high-resolution images. Several strategies have been demonstrated for FoV enhancement, which involve the use of Bessel or Airy beams, for example. However, the generation of these beams requires complicated optical setup or phase filters with continuous phase distributions, which are difficult to manufacture. In contrast, a binary phase filter (BPF) comprising concentric rings with 0 or π phases produces a response similar to its continuous original and is easy to realize. Here, we present a novel form of LSFM that integrates BPFs derived from two representative axi-symmetric aberrations, including phase axicon and spherical aberrations, to improve the imaging performance. We demonstrate that these BPFs significantly increase the FoV, and those derived from axicon generate self-reconstructing beams, which are highly desirable in imaging through scattering specimens. We validate its high-contrast imaging capability over extended FoV by presenting three-dimensional images of microspheres, imaginal disc of Drosophila larva, and Arabidopsis.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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2019 (2)

D. Wang, Y. Jin, R. Feng, Y. Chen, and L. Gao, “Tiling light sheet selective plane illumination microscopy using discontinuous light sheets,” Opt. Express 27(23), 34472–34483 (2019).
[Crossref]

C. Vetter, R. Steinkopf, K. Bergner, M. Ornigotti, S. Nolte, H. Gross, and A. Szameit, “Realization of Free-Space Long-Distance Self-Healing Bessel Beams,” Laser Photonics Rev. 13(10), 1900103 (2019).
[Crossref]

2018 (2)

O. Barlev and M. A. Golub, “Multifunctional binary diffractive optical elements for structured light projectors,” Opt. Express 26(16), 21092–21107 (2018).
[Crossref]

M. Pende, K. Becker, M. Wanis, S. Saghafi, R. Kaur, C. Hahn, N. Pende, M. Foroughipour, T. Hummel, and H.-U. Dodt, “High-resolution ultramicroscopy of the developing and adult nervous system in optically cleared Drosophila melanogaster,” Nat. Commun. 9(1), 4731 (2018).
[Crossref]

2017 (2)

2016 (3)

D. Wilding, P. Pozzi, O. Soloviev, G. Vdovin, C. J. Sheppard, and M. Verhaegen, “Pupil filters for extending the field-of-view in light-sheet microscopy,” Opt. Lett. 41(6), 1205–1208 (2016).
[Crossref]

J. V. Beira and R. Paro, “The legacy of Drosophila imaginal discs,” Chromosoma 125(4), 573–592(2016).
[Crossref]

X. Li, L. Wei, R. H. Poelma, S. Vollebregt, J. Wei, H. P. Urbach, P. M. Sarro, and G. Q. Zhang, “Stretchable Binary Fresnel Lens for Focus Tuning,” Sci. Rep. 6(1), 25348 (2016).
[Crossref]

2015 (2)

W. Wen and X. Chu, “Quantitative comparison of self-healing ability between Bessel–Gaussian beam and Airy beam,” Ann. Phys. 360, 549–555 (2015).
[Crossref]

L. Gao, “Extend the field of view of selective plan illumination microscopy by tiling the excitation light sheet,” Opt. Express 23(5), 6102–6111 (2015).
[Crossref]

2014 (2)

T. Vettenburg, H. I. C. Dalgarno, J. Nylk, C. Coll-Lladó, D. E. K. Ferrier, T. Čižmár, F. J. Gunn-Moore, and K. Dholakia, “Light-sheet microscopy using an Airy beam,” Nat. Methods 11(5), 541–544 (2014).
[Crossref]

B.-C. Chen, W. R. Legant, K. Wang, L. Shao, D. E. Milkie, M. W. Davidson, C. Janetopoulos, X. S. Wu, J. A. Hammer, Z. Liu, B. P. English, Y. Mimori-Kiyosue, D. P. Romero, A. T. Ritter, J. Lippincott-Schwartz, L. Fritz-Laylin, R. D. Mullins, D. M. Mitchell, J. N. Bembenek, A.-C. Reymann, R. Böhme, S. W. Grill, J. T. Wang, G. Seydoux, U. S. Tulu, D. P. Kiehart, and E. Betzig, “Lattice light-sheet microscopy: Imaging molecules to embryos at high spatiotemporal resolution,” Science 346(6208), 1257998 (2014).
[Crossref]

2013 (2)

G. M. De Luca, R. M. Breedijk, R. A. Brandt, C. H. Zeelenberg, B. E. de Jong, W. Timmermans, L. N. Azar, R. A. Hoebe, S. Stallinga, and E. M. Manders, “Re-scan confocal microscopy: scanning twice for better resolution,” Biomed. Opt. Express 4(11), 2644–2656 (2013).
[Crossref]

P. G. Pitrone, J. Schindelin, L. Stuyvenberg, S. Preibisch, M. Weber, K. W. Eliceiri, J. Huisken, and P. Tomancak, “OpenSPIM: an open-access light-sheet microscopy platform,” Nat. Methods 10(7), 598–599 (2013).
[Crossref]

2012 (2)

F. O. Fahrbach and A. Rohrbach, “Propagation stability of self-reconstructing Bessel beams enables contrast-enhanced imaging in thick media,” Nat. Commun. 3(1), 632 (2012).
[Crossref]

M. Persson, D. Engström, and M. Goksör, “Reducing the effect of pixel crosstalk in phase only spatial light modulators,” Opt. Express 20(20), 22334–22343 (2012).
[Crossref]

2011 (2)

T. A. Planchon, L. Gao, D. E. Milkie, M. W. Davidson, J. A. Galbraith, C. G. Galbraith, and E. Betzig, “Rapid three-dimensional isotropic imaging of living cells using Bessel beam plane illumination,” Nat. Methods 8(5), 417–423 (2011).
[Crossref]

C. J. R. Sheppard, “Binary phase filters with a maximally-flat response,” Opt. Lett. 36(8), 1386–1388 (2011).
[Crossref]

2010 (3)

M. Mazilu, D. J. Stevenson, F. Gunn-Moore, and K. Dholakia, “Light beats the spread: “non-diffracting” beams,” Laser Photonics Rev. 4(4), 529–547 (2010).
[Crossref]

F. O. Fahrbach, P. Simon, and A. Rohrbach, “Microscopy with self-reconstructing beams,” Nat. Photonics 4(11), 780–785 (2010).
[Crossref]

C. B. Müller and J. Enderlein, “Image Scanning Microscopy,” Phys. Rev. Lett. 104(19), 198101 (2010).
[Crossref]

2009 (4)

A. Rohrbach, “Artifacts resulting from imaging in scattering media: a theoretical prediction,” Opt. Lett. 34(19), 3041–3043 (2009).
[Crossref]

A. Zlotnik, S. B. Yaish, O. Yehezkel, K. Lahav-Yacouel, M. Belkin, and Z. Zalevsky, “Extended depth of focus contact lenses for presbyopia,” Opt. Lett. 34(14), 2219–2221 (2009).
[Crossref]

Z. Zhai, S. Ding, Q. Lv, X. Wang, and Y. Zhong, “Extended depth of field through an axicon,” J. Mod. Opt. 56(11), 1304–1308 (2009).
[Crossref]

J. Huisken and D. Y. R. Stainier, “Selective plane illumination microscopy techniques in developmental biology,” Development 136(12), 1963–1975 (2009).
[Crossref]

2005 (2)

2004 (1)

J. Huisken, J. Swoger, F. D. Bene, J. Wittbrodt, and E. H. K. Stelzer, “Optical sectioning deep inside live embryos by selective plane illumination microscopy,” Science 305(5686), 1007–1009 (2004).
[Crossref]

2000 (1)

M. G. L. Gustafsson, “Surpassing the lateral resolution limit by a factor of two using structured illumination microscopy,” J. Microsc. 198(2), 82–87 (2000).
[Crossref]

1993 (1)

1982 (1)

A. C. Spradling and G. M. Rubin, “Transposition of cloned P elements into Drosophila germ line chromosomes,” Science 218(4570), 341–347 (1982).
[Crossref]

1974 (2)

A. Shearn and A. Garen, “Genetic control of imaginal disc development in Drosophila,” Proc. Natl. Acad. Sci. 71(4), 1393–1397 (1974).
[Crossref]

L. B. Lucy, “An iterative technique for the rectification of observed distributions,” Astron J. 79, 745 (1974).
[Crossref]

1972 (1)

1971 (1)

P. J. Bryant, “Regeneration and duplication following operations in situ on the imaginal discs of Drosophila melanogaster,” Dev. Biol. 26 (4), 637–651(1971).
[Crossref]

1902 (1)

H. Siedentopf and R. Zsigmondy, “Uber Sichtbarmachung und Größenbestimmung ultramikoskopischer† Teilchen, mit besonderer Anwendung auf Goldrubingläser,” Ann. Phys. 315(1), 1–39 (1902).
[Crossref]

Agarwal, G. S.

Aiello, A.

Azar, L. N.

Barlev, O.

Becker, K.

M. Pende, K. Becker, M. Wanis, S. Saghafi, R. Kaur, C. Hahn, N. Pende, M. Foroughipour, T. Hummel, and H.-U. Dodt, “High-resolution ultramicroscopy of the developing and adult nervous system in optically cleared Drosophila melanogaster,” Nat. Commun. 9(1), 4731 (2018).
[Crossref]

Beira, J. V.

J. V. Beira and R. Paro, “The legacy of Drosophila imaginal discs,” Chromosoma 125(4), 573–592(2016).
[Crossref]

Belkin, M.

Bembenek, J. N.

B.-C. Chen, W. R. Legant, K. Wang, L. Shao, D. E. Milkie, M. W. Davidson, C. Janetopoulos, X. S. Wu, J. A. Hammer, Z. Liu, B. P. English, Y. Mimori-Kiyosue, D. P. Romero, A. T. Ritter, J. Lippincott-Schwartz, L. Fritz-Laylin, R. D. Mullins, D. M. Mitchell, J. N. Bembenek, A.-C. Reymann, R. Böhme, S. W. Grill, J. T. Wang, G. Seydoux, U. S. Tulu, D. P. Kiehart, and E. Betzig, “Lattice light-sheet microscopy: Imaging molecules to embryos at high spatiotemporal resolution,” Science 346(6208), 1257998 (2014).
[Crossref]

Bene, F. D.

J. Huisken, J. Swoger, F. D. Bene, J. Wittbrodt, and E. H. K. Stelzer, “Optical sectioning deep inside live embryos by selective plane illumination microscopy,” Science 305(5686), 1007–1009 (2004).
[Crossref]

Bergner, K.

C. Vetter, R. Steinkopf, K. Bergner, M. Ornigotti, S. Nolte, H. Gross, and A. Szameit, “Realization of Free-Space Long-Distance Self-Healing Bessel Beams,” Laser Photonics Rev. 13(10), 1900103 (2019).
[Crossref]

Betzig, E.

B.-C. Chen, W. R. Legant, K. Wang, L. Shao, D. E. Milkie, M. W. Davidson, C. Janetopoulos, X. S. Wu, J. A. Hammer, Z. Liu, B. P. English, Y. Mimori-Kiyosue, D. P. Romero, A. T. Ritter, J. Lippincott-Schwartz, L. Fritz-Laylin, R. D. Mullins, D. M. Mitchell, J. N. Bembenek, A.-C. Reymann, R. Böhme, S. W. Grill, J. T. Wang, G. Seydoux, U. S. Tulu, D. P. Kiehart, and E. Betzig, “Lattice light-sheet microscopy: Imaging molecules to embryos at high spatiotemporal resolution,” Science 346(6208), 1257998 (2014).
[Crossref]

T. A. Planchon, L. Gao, D. E. Milkie, M. W. Davidson, J. A. Galbraith, C. G. Galbraith, and E. Betzig, “Rapid three-dimensional isotropic imaging of living cells using Bessel beam plane illumination,” Nat. Methods 8(5), 417–423 (2011).
[Crossref]

Böhme, R.

B.-C. Chen, W. R. Legant, K. Wang, L. Shao, D. E. Milkie, M. W. Davidson, C. Janetopoulos, X. S. Wu, J. A. Hammer, Z. Liu, B. P. English, Y. Mimori-Kiyosue, D. P. Romero, A. T. Ritter, J. Lippincott-Schwartz, L. Fritz-Laylin, R. D. Mullins, D. M. Mitchell, J. N. Bembenek, A.-C. Reymann, R. Böhme, S. W. Grill, J. T. Wang, G. Seydoux, U. S. Tulu, D. P. Kiehart, and E. Betzig, “Lattice light-sheet microscopy: Imaging molecules to embryos at high spatiotemporal resolution,” Science 346(6208), 1257998 (2014).
[Crossref]

Brandt, R. A.

Breedijk, R. M.

Bryant, P. J.

P. J. Bryant, “Regeneration and duplication following operations in situ on the imaginal discs of Drosophila melanogaster,” Dev. Biol. 26 (4), 637–651(1971).
[Crossref]

Chen, B.-C.

B.-C. Chen, W. R. Legant, K. Wang, L. Shao, D. E. Milkie, M. W. Davidson, C. Janetopoulos, X. S. Wu, J. A. Hammer, Z. Liu, B. P. English, Y. Mimori-Kiyosue, D. P. Romero, A. T. Ritter, J. Lippincott-Schwartz, L. Fritz-Laylin, R. D. Mullins, D. M. Mitchell, J. N. Bembenek, A.-C. Reymann, R. Böhme, S. W. Grill, J. T. Wang, G. Seydoux, U. S. Tulu, D. P. Kiehart, and E. Betzig, “Lattice light-sheet microscopy: Imaging molecules to embryos at high spatiotemporal resolution,” Science 346(6208), 1257998 (2014).
[Crossref]

Chen, Y.

Chu, X.

W. Wen and X. Chu, “Quantitative comparison of self-healing ability between Bessel–Gaussian beam and Airy beam,” Ann. Phys. 360, 549–555 (2015).
[Crossref]

Cižmár, T.

T. Vettenburg, H. I. C. Dalgarno, J. Nylk, C. Coll-Lladó, D. E. K. Ferrier, T. Čižmár, F. J. Gunn-Moore, and K. Dholakia, “Light-sheet microscopy using an Airy beam,” Nat. Methods 11(5), 541–544 (2014).
[Crossref]

Cloonan, T. J.

Coll-Lladó, C.

T. Vettenburg, H. I. C. Dalgarno, J. Nylk, C. Coll-Lladó, D. E. K. Ferrier, T. Čižmár, F. J. Gunn-Moore, and K. Dholakia, “Light-sheet microscopy using an Airy beam,” Nat. Methods 11(5), 541–544 (2014).
[Crossref]

Dalgarno, H. I. C.

T. Vettenburg, H. I. C. Dalgarno, J. Nylk, C. Coll-Lladó, D. E. K. Ferrier, T. Čižmár, F. J. Gunn-Moore, and K. Dholakia, “Light-sheet microscopy using an Airy beam,” Nat. Methods 11(5), 541–544 (2014).
[Crossref]

Davidson, M. W.

B.-C. Chen, W. R. Legant, K. Wang, L. Shao, D. E. Milkie, M. W. Davidson, C. Janetopoulos, X. S. Wu, J. A. Hammer, Z. Liu, B. P. English, Y. Mimori-Kiyosue, D. P. Romero, A. T. Ritter, J. Lippincott-Schwartz, L. Fritz-Laylin, R. D. Mullins, D. M. Mitchell, J. N. Bembenek, A.-C. Reymann, R. Böhme, S. W. Grill, J. T. Wang, G. Seydoux, U. S. Tulu, D. P. Kiehart, and E. Betzig, “Lattice light-sheet microscopy: Imaging molecules to embryos at high spatiotemporal resolution,” Science 346(6208), 1257998 (2014).
[Crossref]

T. A. Planchon, L. Gao, D. E. Milkie, M. W. Davidson, J. A. Galbraith, C. G. Galbraith, and E. Betzig, “Rapid three-dimensional isotropic imaging of living cells using Bessel beam plane illumination,” Nat. Methods 8(5), 417–423 (2011).
[Crossref]

de Jong, B. E.

De Luca, G. M.

Dholakia, K.

T. Vettenburg, H. I. C. Dalgarno, J. Nylk, C. Coll-Lladó, D. E. K. Ferrier, T. Čižmár, F. J. Gunn-Moore, and K. Dholakia, “Light-sheet microscopy using an Airy beam,” Nat. Methods 11(5), 541–544 (2014).
[Crossref]

M. Mazilu, D. J. Stevenson, F. Gunn-Moore, and K. Dholakia, “Light beats the spread: “non-diffracting” beams,” Laser Photonics Rev. 4(4), 529–547 (2010).
[Crossref]

D. McGloin and K. Dholakia, “Bessel beams: diffraction in a new light,” Contemp. Phys. 46(1), 15–28 (2005).
[Crossref]

Ding, S.

Z. Zhai, S. Ding, Q. Lv, X. Wang, and Y. Zhong, “Extended depth of field through an axicon,” J. Mod. Opt. 56(11), 1304–1308 (2009).
[Crossref]

Dodt, H.-U.

M. Pende, K. Becker, M. Wanis, S. Saghafi, R. Kaur, C. Hahn, N. Pende, M. Foroughipour, T. Hummel, and H.-U. Dodt, “High-resolution ultramicroscopy of the developing and adult nervous system in optically cleared Drosophila melanogaster,” Nat. Commun. 9(1), 4731 (2018).
[Crossref]

Eliceiri, K. W.

P. G. Pitrone, J. Schindelin, L. Stuyvenberg, S. Preibisch, M. Weber, K. W. Eliceiri, J. Huisken, and P. Tomancak, “OpenSPIM: an open-access light-sheet microscopy platform,” Nat. Methods 10(7), 598–599 (2013).
[Crossref]

Enderlein, J.

C. B. Müller and J. Enderlein, “Image Scanning Microscopy,” Phys. Rev. Lett. 104(19), 198101 (2010).
[Crossref]

English, B. P.

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[Crossref]

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Supplementary Material (4)

NameDescription
» Visualization 1       BPF-LSFM 3D-rendered image of Drosophila imaginal disc
» Visualization 2       BPF-LSFM raw z-stack images of Drosophila imaginal disc
» Visualization 3       BPF-LSFM 3D-rendered image of Arabidopsis
» Visualization 4       BPF-LSFM raw z-stack images of Arabidopsis

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Figures (8)

Fig. 1.
Fig. 1. (a, b) Computer-generated transmission functions and corresponding focal responses of clear aperture and representative binary phase filters (BPFs). The pupil coordinates $\zeta $ and $\eta $ are the spatial frequencies normalized with NA/λ. The design parameters of the BPFs are provided in Table 1 and Table 2. It can be noted that the focus is elongated significantly as the number of rings in the BPFs increases. u and v are the axial and lateral coordinates in optical units, respectively [10]. (c, d) Comparisons of axial and lateral focal responses of the BPFs against those of a clear aperture. The BPFs with a large number of rings provide elongated focal responses with broader widths. The lateral responses of BPF-AXI filters exhibit narrower mainlobes than those of clear aperture, but with pronounced sidelobes. The lateral responses of BPF-SA filters, however, are characterized by broader mainlobes with reduced sidelobes.
Fig. 2.
Fig. 2. (a1) Schematic of BPF-LSFM. A 473-nm light illuminated a spatial light modulator (SLM) that was employed to implement various BPF designs. The light was then imaged into the back focal plane of illumination objective (OBJ1) to generate elongated focus, which scanned through a specimen via a two-dimensional galvanometric scanner (2D-GV). The fluorescence emission was imaged by an imaging setup comprising the detection objective (OBJ2), emission filter (EM), and relay lenses. ETL: electrical tunable lens, sCMOS: scientific CMOS camera, M: mirror, HWP: half-wave plate, BD: beam dump, PBS: polarizing beam splitter, PMF: polarization-maintaining fiber, AS: aperture stop. (a2) Axial focal plane scanning was performed with ETL, which operated synchronously with the 2D-GV. The rolling shutter of sCMOS scanned synchronously with Y-directional scanned focus. (b) Raw and post-processed Y-Z images of 100-nm fluorescent beads. The scale bar denotes 1 µm. (c-d) Raw and deconvolved point-spread functions along the Y- and Z-directions.
Fig. 3.
Fig. 3. XZ maximum intensity projection (MIP) images generated from raw 3D datasets of 100-nm fluorescent beads obtained with clear aperture (a), BPF-AXIs (b), and BPF-SAs (c). (d–e) The measured FWHMs of the axial PSFs (relative to that at X=0) for clear aperture, BPF-AXIs and BPF-SAs. It is clearly observed that BPF-AXI and BPF-SA filters provide extended FoV along the X direction.
Fig. 4.
Fig. 4. (a1, b1, c1) XY slice images of 5-µm silica microspheres embedded in fluorescent gels, obtained with the clear aperture, BPF-AXI10, and BPF-SA10. The illumination beams propagated from left to right. (a2, b2, c2) Corresponding intensity profiles along the black and white dashed lines in the X direction in (a1, b1, c1). The black and white dashed lines represent the cases where the illumination beams do and do not cross the microspheres, respectively. (a3, b3, c3) Intensity distributions along the black and white dashed lines in the Y direction in (a1, b1, c1), which correspond to the intensity profiles before and after the microspheres. The beam from BPF-AXI10 recovers rapidly after the microspheres, while other beams spread after the scatterers.
Fig. 5.
Fig. 5. (a) XY slice images of 5-µm silica beads embedded in fluorescing gel obtained with clear aperture, BPF-AXI10, and BPF-SA10. (b) Evaluation results of Eq. (4) for the three pupil filters. It is evident that the scattered light from the bead rapidly recovers and returns to its original unperturbed light for BPF-AXI10, while it deviates significantly from the unperturbed light for clear aperture and BPF-SA10.
Fig. 6.
Fig. 6. (a–c) 3D-rendered and (d–f) maximum intensity projection (MIP) images along the Z-direction of a Drosophila imaginal disc acquired with the clear aperture, BPF-AXI10, and BPF-SA10. The illumination beams propagate from left to right in the X-direction. The insets in (d–f) show the averaged intensity profiles as a function of propagation distance. High-contrast imaging over extended volumes is achieved with BPF-AXI10 and BPF-SA10. (g–i) Magnified XY slice images of the regions marked with dashed white rectangles in (d-f). The images correspond to the XY plane at the position of 72.9 µm from the bottom of imaging volumes. It is evident that the use of BPF-AXI10 enables imaging with significant reduction of the shadow artifacts. Visualization 1 and Visualization 2 demonstrate the imaging capability of BPF-LSFMs over the extended volume.
Fig. 7.
Fig. 7. (a–c) 3D-rendered and (d–f) MIP images of the Arabidopsis root meristem along the Z-direction, acquired with the clear aperture, BPF-AXI10, and BPF-SA10. It is evident that imaging over extended volumes can be achieved through BPF-AXI and BPF-SA. The insets in (d–f) show the averaged intensity profiles as a function of propagation distance. Figures (g–i) present the corresponding XY slice images acquired at 93.6 µm from the bottom of the imaging volumes, which clearly demonstrate the dramatic reduction of shadow artifact in the image with BPF-AXI10. Visualization 3 and Visualization 4 manifest high-contrast imaging capability of BPF-LSFMs over extended volume.
Fig. 8.
Fig. 8. (a) Pupil transmission functions of BPF-AXI10, BPF-SA10, and their respective continuous counterparts (i.e., AXI10 and SA10). The BPFs are generated from the continuous pupil functions with the same axicon (α=10) and spherical aberration (γ=10) parameters, with the proper incorporation of defocus parameters to combine two foci in the optical axis. (b) 3D focal responses from the BPFs and their respective originals. Note that quadratic phase terms have been multiplied to the continuous pupil functions to locate their focal responses proximal to the focal plane of a lens. (c, d) Comparisons of the axial and lateral focal responses of the BPF-AXI10 and BPF-SA10 against those of their continuous originals.

Tables (2)

Tables Icon

Table 1. Design parameters for axicon-derived BPFs (BPF-AXI).

Tables Icon

Table 2. Design parameters for spherical aberration derived BPFs (BPF-SA).

Equations (5)

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P B P F A X I ( ρ ) = Bin [ exp ( i 2 π ( ψ ρ 2 + α ρ ) ) ] , P B P F S A ( ρ ) = Bin [ exp ( i 2 π ( ψ ρ 2 + γ ρ 4 ) ) ] ,
Bin [ P ( ρ ) ] = { 1 ( Re [ P ( ρ ) ] 0 ) 1 ( Re [ P ( ρ ) ] < 0 ) ,
ψ = Δ x λ N A 2 2 n ,
σ ( x ) 2 = 1 I ¯ u ( x ) 2 ( I ¯ u ( x ) I ¯ p ( x , b ) ) 2 d b ,
I n + 1 = I n ( I 0 I n h h ) .