Abstract

An in-line holographic system for in situ detection of atmospheric cloud particles [Holographic Detector for Clouds (HOLODEC)] has been developed and flown on the National Center for Atmospheric Research C-130 research aircraft. Clear holograms are obtained in daylight conditions at typical aircraft speeds of 100 m s-1. The instrument is fully digital and is interfaced to a control and data-acquisition system in the aircraft via optical fiber. It is operable at temperatures of less than -30 °C and at typical cloud humidities. Preliminary data from the experiment show its utility for studies of the three-dimensional spatial distribution of cloud particles and ice crystal shapes.

© 2004 Optical Society of America

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References

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  1. R. A. Shaw, “Particle-turbulence interactions in atmospheric clouds,” Ann. Rev. Fluid Mech. 35, 183–227 (2003).
    [CrossRef]
  2. A. B. Kostinski, R. A. Shaw, “Scale-dependent droplet clustering in turbulent clouds,” J. Fluid Mech. 434, 389–398 (2001).
    [CrossRef]
  3. A. Kozikowska, K. Haman, J. Supronowicz, “Preliminary results of an investigation of the spatial distribution of fog droplets by a holographic method,” Q. J. R. Meteorol. Soc. 110, 65–73 (1984).
    [CrossRef]
  4. E. Uhlig, S. Borrmann, R. Jaenicke, “Holographic in situ measurements of the spatial droplet distribution in stratiform clouds,” Tellus Ser. B 50, 377–387 (1998).
    [CrossRef]
  5. L. Cao, G. Pan, H. Meng, “Three-dimensional measurement of aerosol particle clustering in homogeneous isotropic turbulence,” in ASME Summer Heat Transfer Conference, Las Vegas, Nev. (American Society of Mechanical Engineers, New York, 2003), doi: HT2003–47435.
  6. G. L. Holtzer, L. R. Collins, “Relationship between the intrinsic radial distribution function for an isotropic field of particles and lower-dimensional measurements,” J. Fluid Mech. 459, 93–102 (2002).
    [CrossRef]
  7. H. Meng, G. Pan, Y. Pu, S. H. Woodward, “Holographic particle image velocimetry: from film to digital recording,” Meas. Sci. Technol. 15, 673–685 (2004).
    [CrossRef]
  8. K. D. Hinsch, “Holographic particle image velocimetry,” Meas. Sci. Technol. 13, R61–R72 (2002).
    [CrossRef]
  9. B. J. Thompson, “Holographic particle sizing techniques,” J. Phys. E 7, 781–788 (1974).
    [CrossRef]
  10. S. Borrmann, R. Jaenicke, “Application of microholography for ground-based in situ measurements in stratus cloud layers: a case study,” J. Atmos. Ocean. Technol. 10, 277–293 (1993).
    [CrossRef]
  11. R. P. Lawson, R. H. Cormack, “Theoretical design and preliminary tests of two new particle spectrometers for cloud microphysics research,” Atmos. Res. 35, 315–348 (1995).
    [CrossRef]
  12. H. Meng, W. L. Anderson, F. Hussain, D. D. Liu, “Intrinsic speckle noise in in-line particle holography,” J. Opt. Soc. Am. A 10, 2046–2058 (1993).
    [CrossRef]
  13. R. B. Owen, A. A. Zozulya, “In-line digital holographic sensor for monitoring and characterizing marine particulates,” Opt. Eng. 39, 2187–2197 (2000).
    [CrossRef]
  14. G. Pan, H. Meng, “Digital holography of particle fields: reconstruction by use of complex amplitude,” Appl. Opt. 42, 827–833 (2003).
    [CrossRef] [PubMed]
  15. W. Xu, M. H. Jericho, I. A. Meinertzhagen, H. J. Kreuzer, “Digital in-line holography of microspheres,” Appl. Opt. 41, 5367–5375 (2002).
    [CrossRef] [PubMed]
  16. H. R. Pruppacher, J. D. Klett, Microphysics of Clouds and Precipitation (Kluwer Academic, Boston, Mass., 1997), pp. 15–24.
  17. C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley-Interscience, New York, 1983), Chap. 4, pp. 107–111.
  18. F. Slimani, G. Grehan, G. Gouesbet, D. Allano, “Near-field Lorenz-Mie theory and its application to microholography,” Appl. Opt. 23, 4140–4148 (1984).
    [CrossRef] [PubMed]
  19. M. Born, E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, Cambridge, UK, 1999), Chap. 8, pp. 412–516.
  20. G. A. Tyler, B. J. Thompson, “Fraunhofer holography applied to particle size analysis: a reassessment,” Opt. Acta 23, 685–700 (1976).
    [CrossRef]
  21. J. I. MacPherson, D. Baumgardner, “Airflow about King Air wingtip-mounted cloud particle measurement probes,” J. Atmos. Ocean. Technol. 5, 259–273 (1988).
    [CrossRef]
  22. P. K. Kundu, I. M. Cohen, Fluid Mechanics (Academic, San Diego, Calif., 2002), Chaps. 6, 10, pp. 148–192, 312–377.
  23. D. Nagel, GKSS-Research Centre, Institute for Coastal Research, Max-Planck-Strasse 1, D-21502 Geesthacht, Germany (personal communication, 2003).
  24. IDEAS 3 was held at the National Center for Atmospheric Research, Research Aviation Facility, sponsored by the National Science Foundation. Details of the IDEAS 3 project are available at http://raf.atd.ucar.edu/Projects/IDEAS-3/ .
  25. J. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, Boston, Mass., 1996), Eq. (3-74).

2004

H. Meng, G. Pan, Y. Pu, S. H. Woodward, “Holographic particle image velocimetry: from film to digital recording,” Meas. Sci. Technol. 15, 673–685 (2004).
[CrossRef]

2003

R. A. Shaw, “Particle-turbulence interactions in atmospheric clouds,” Ann. Rev. Fluid Mech. 35, 183–227 (2003).
[CrossRef]

G. Pan, H. Meng, “Digital holography of particle fields: reconstruction by use of complex amplitude,” Appl. Opt. 42, 827–833 (2003).
[CrossRef] [PubMed]

2002

W. Xu, M. H. Jericho, I. A. Meinertzhagen, H. J. Kreuzer, “Digital in-line holography of microspheres,” Appl. Opt. 41, 5367–5375 (2002).
[CrossRef] [PubMed]

G. L. Holtzer, L. R. Collins, “Relationship between the intrinsic radial distribution function for an isotropic field of particles and lower-dimensional measurements,” J. Fluid Mech. 459, 93–102 (2002).
[CrossRef]

K. D. Hinsch, “Holographic particle image velocimetry,” Meas. Sci. Technol. 13, R61–R72 (2002).
[CrossRef]

2001

A. B. Kostinski, R. A. Shaw, “Scale-dependent droplet clustering in turbulent clouds,” J. Fluid Mech. 434, 389–398 (2001).
[CrossRef]

2000

R. B. Owen, A. A. Zozulya, “In-line digital holographic sensor for monitoring and characterizing marine particulates,” Opt. Eng. 39, 2187–2197 (2000).
[CrossRef]

1998

E. Uhlig, S. Borrmann, R. Jaenicke, “Holographic in situ measurements of the spatial droplet distribution in stratiform clouds,” Tellus Ser. B 50, 377–387 (1998).
[CrossRef]

1995

R. P. Lawson, R. H. Cormack, “Theoretical design and preliminary tests of two new particle spectrometers for cloud microphysics research,” Atmos. Res. 35, 315–348 (1995).
[CrossRef]

1993

S. Borrmann, R. Jaenicke, “Application of microholography for ground-based in situ measurements in stratus cloud layers: a case study,” J. Atmos. Ocean. Technol. 10, 277–293 (1993).
[CrossRef]

H. Meng, W. L. Anderson, F. Hussain, D. D. Liu, “Intrinsic speckle noise in in-line particle holography,” J. Opt. Soc. Am. A 10, 2046–2058 (1993).
[CrossRef]

1988

J. I. MacPherson, D. Baumgardner, “Airflow about King Air wingtip-mounted cloud particle measurement probes,” J. Atmos. Ocean. Technol. 5, 259–273 (1988).
[CrossRef]

1984

F. Slimani, G. Grehan, G. Gouesbet, D. Allano, “Near-field Lorenz-Mie theory and its application to microholography,” Appl. Opt. 23, 4140–4148 (1984).
[CrossRef] [PubMed]

A. Kozikowska, K. Haman, J. Supronowicz, “Preliminary results of an investigation of the spatial distribution of fog droplets by a holographic method,” Q. J. R. Meteorol. Soc. 110, 65–73 (1984).
[CrossRef]

1976

G. A. Tyler, B. J. Thompson, “Fraunhofer holography applied to particle size analysis: a reassessment,” Opt. Acta 23, 685–700 (1976).
[CrossRef]

1974

B. J. Thompson, “Holographic particle sizing techniques,” J. Phys. E 7, 781–788 (1974).
[CrossRef]

Allano, D.

Anderson, W. L.

Baumgardner, D.

J. I. MacPherson, D. Baumgardner, “Airflow about King Air wingtip-mounted cloud particle measurement probes,” J. Atmos. Ocean. Technol. 5, 259–273 (1988).
[CrossRef]

Bohren, C. F.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley-Interscience, New York, 1983), Chap. 4, pp. 107–111.

Born, M.

M. Born, E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, Cambridge, UK, 1999), Chap. 8, pp. 412–516.

Borrmann, S.

E. Uhlig, S. Borrmann, R. Jaenicke, “Holographic in situ measurements of the spatial droplet distribution in stratiform clouds,” Tellus Ser. B 50, 377–387 (1998).
[CrossRef]

S. Borrmann, R. Jaenicke, “Application of microholography for ground-based in situ measurements in stratus cloud layers: a case study,” J. Atmos. Ocean. Technol. 10, 277–293 (1993).
[CrossRef]

Cao, L.

L. Cao, G. Pan, H. Meng, “Three-dimensional measurement of aerosol particle clustering in homogeneous isotropic turbulence,” in ASME Summer Heat Transfer Conference, Las Vegas, Nev. (American Society of Mechanical Engineers, New York, 2003), doi: HT2003–47435.

Cohen, I. M.

P. K. Kundu, I. M. Cohen, Fluid Mechanics (Academic, San Diego, Calif., 2002), Chaps. 6, 10, pp. 148–192, 312–377.

Collins, L. R.

G. L. Holtzer, L. R. Collins, “Relationship between the intrinsic radial distribution function for an isotropic field of particles and lower-dimensional measurements,” J. Fluid Mech. 459, 93–102 (2002).
[CrossRef]

Cormack, R. H.

R. P. Lawson, R. H. Cormack, “Theoretical design and preliminary tests of two new particle spectrometers for cloud microphysics research,” Atmos. Res. 35, 315–348 (1995).
[CrossRef]

Goodman, J.

J. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, Boston, Mass., 1996), Eq. (3-74).

Gouesbet, G.

Grehan, G.

Haman, K.

A. Kozikowska, K. Haman, J. Supronowicz, “Preliminary results of an investigation of the spatial distribution of fog droplets by a holographic method,” Q. J. R. Meteorol. Soc. 110, 65–73 (1984).
[CrossRef]

Hinsch, K. D.

K. D. Hinsch, “Holographic particle image velocimetry,” Meas. Sci. Technol. 13, R61–R72 (2002).
[CrossRef]

Holtzer, G. L.

G. L. Holtzer, L. R. Collins, “Relationship between the intrinsic radial distribution function for an isotropic field of particles and lower-dimensional measurements,” J. Fluid Mech. 459, 93–102 (2002).
[CrossRef]

Huffman, D. R.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley-Interscience, New York, 1983), Chap. 4, pp. 107–111.

Hussain, F.

Jaenicke, R.

E. Uhlig, S. Borrmann, R. Jaenicke, “Holographic in situ measurements of the spatial droplet distribution in stratiform clouds,” Tellus Ser. B 50, 377–387 (1998).
[CrossRef]

S. Borrmann, R. Jaenicke, “Application of microholography for ground-based in situ measurements in stratus cloud layers: a case study,” J. Atmos. Ocean. Technol. 10, 277–293 (1993).
[CrossRef]

Jericho, M. H.

Klett, J. D.

H. R. Pruppacher, J. D. Klett, Microphysics of Clouds and Precipitation (Kluwer Academic, Boston, Mass., 1997), pp. 15–24.

Kostinski, A. B.

A. B. Kostinski, R. A. Shaw, “Scale-dependent droplet clustering in turbulent clouds,” J. Fluid Mech. 434, 389–398 (2001).
[CrossRef]

Kozikowska, A.

A. Kozikowska, K. Haman, J. Supronowicz, “Preliminary results of an investigation of the spatial distribution of fog droplets by a holographic method,” Q. J. R. Meteorol. Soc. 110, 65–73 (1984).
[CrossRef]

Kreuzer, H. J.

Kundu, P. K.

P. K. Kundu, I. M. Cohen, Fluid Mechanics (Academic, San Diego, Calif., 2002), Chaps. 6, 10, pp. 148–192, 312–377.

Lawson, R. P.

R. P. Lawson, R. H. Cormack, “Theoretical design and preliminary tests of two new particle spectrometers for cloud microphysics research,” Atmos. Res. 35, 315–348 (1995).
[CrossRef]

Liu, D. D.

MacPherson, J. I.

J. I. MacPherson, D. Baumgardner, “Airflow about King Air wingtip-mounted cloud particle measurement probes,” J. Atmos. Ocean. Technol. 5, 259–273 (1988).
[CrossRef]

Meinertzhagen, I. A.

Meng, H.

H. Meng, G. Pan, Y. Pu, S. H. Woodward, “Holographic particle image velocimetry: from film to digital recording,” Meas. Sci. Technol. 15, 673–685 (2004).
[CrossRef]

G. Pan, H. Meng, “Digital holography of particle fields: reconstruction by use of complex amplitude,” Appl. Opt. 42, 827–833 (2003).
[CrossRef] [PubMed]

H. Meng, W. L. Anderson, F. Hussain, D. D. Liu, “Intrinsic speckle noise in in-line particle holography,” J. Opt. Soc. Am. A 10, 2046–2058 (1993).
[CrossRef]

L. Cao, G. Pan, H. Meng, “Three-dimensional measurement of aerosol particle clustering in homogeneous isotropic turbulence,” in ASME Summer Heat Transfer Conference, Las Vegas, Nev. (American Society of Mechanical Engineers, New York, 2003), doi: HT2003–47435.

Nagel, D.

D. Nagel, GKSS-Research Centre, Institute for Coastal Research, Max-Planck-Strasse 1, D-21502 Geesthacht, Germany (personal communication, 2003).

Owen, R. B.

R. B. Owen, A. A. Zozulya, “In-line digital holographic sensor for monitoring and characterizing marine particulates,” Opt. Eng. 39, 2187–2197 (2000).
[CrossRef]

Pan, G.

H. Meng, G. Pan, Y. Pu, S. H. Woodward, “Holographic particle image velocimetry: from film to digital recording,” Meas. Sci. Technol. 15, 673–685 (2004).
[CrossRef]

G. Pan, H. Meng, “Digital holography of particle fields: reconstruction by use of complex amplitude,” Appl. Opt. 42, 827–833 (2003).
[CrossRef] [PubMed]

L. Cao, G. Pan, H. Meng, “Three-dimensional measurement of aerosol particle clustering in homogeneous isotropic turbulence,” in ASME Summer Heat Transfer Conference, Las Vegas, Nev. (American Society of Mechanical Engineers, New York, 2003), doi: HT2003–47435.

Pruppacher, H. R.

H. R. Pruppacher, J. D. Klett, Microphysics of Clouds and Precipitation (Kluwer Academic, Boston, Mass., 1997), pp. 15–24.

Pu, Y.

H. Meng, G. Pan, Y. Pu, S. H. Woodward, “Holographic particle image velocimetry: from film to digital recording,” Meas. Sci. Technol. 15, 673–685 (2004).
[CrossRef]

Shaw, R. A.

R. A. Shaw, “Particle-turbulence interactions in atmospheric clouds,” Ann. Rev. Fluid Mech. 35, 183–227 (2003).
[CrossRef]

A. B. Kostinski, R. A. Shaw, “Scale-dependent droplet clustering in turbulent clouds,” J. Fluid Mech. 434, 389–398 (2001).
[CrossRef]

Slimani, F.

Supronowicz, J.

A. Kozikowska, K. Haman, J. Supronowicz, “Preliminary results of an investigation of the spatial distribution of fog droplets by a holographic method,” Q. J. R. Meteorol. Soc. 110, 65–73 (1984).
[CrossRef]

Thompson, B. J.

G. A. Tyler, B. J. Thompson, “Fraunhofer holography applied to particle size analysis: a reassessment,” Opt. Acta 23, 685–700 (1976).
[CrossRef]

B. J. Thompson, “Holographic particle sizing techniques,” J. Phys. E 7, 781–788 (1974).
[CrossRef]

Tyler, G. A.

G. A. Tyler, B. J. Thompson, “Fraunhofer holography applied to particle size analysis: a reassessment,” Opt. Acta 23, 685–700 (1976).
[CrossRef]

Uhlig, E.

E. Uhlig, S. Borrmann, R. Jaenicke, “Holographic in situ measurements of the spatial droplet distribution in stratiform clouds,” Tellus Ser. B 50, 377–387 (1998).
[CrossRef]

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, Cambridge, UK, 1999), Chap. 8, pp. 412–516.

Woodward, S. H.

H. Meng, G. Pan, Y. Pu, S. H. Woodward, “Holographic particle image velocimetry: from film to digital recording,” Meas. Sci. Technol. 15, 673–685 (2004).
[CrossRef]

Xu, W.

Zozulya, A. A.

R. B. Owen, A. A. Zozulya, “In-line digital holographic sensor for monitoring and characterizing marine particulates,” Opt. Eng. 39, 2187–2197 (2000).
[CrossRef]

Ann. Rev. Fluid Mech.

R. A. Shaw, “Particle-turbulence interactions in atmospheric clouds,” Ann. Rev. Fluid Mech. 35, 183–227 (2003).
[CrossRef]

Appl. Opt.

Atmos. Res.

R. P. Lawson, R. H. Cormack, “Theoretical design and preliminary tests of two new particle spectrometers for cloud microphysics research,” Atmos. Res. 35, 315–348 (1995).
[CrossRef]

J. Atmos. Ocean. Technol.

J. I. MacPherson, D. Baumgardner, “Airflow about King Air wingtip-mounted cloud particle measurement probes,” J. Atmos. Ocean. Technol. 5, 259–273 (1988).
[CrossRef]

S. Borrmann, R. Jaenicke, “Application of microholography for ground-based in situ measurements in stratus cloud layers: a case study,” J. Atmos. Ocean. Technol. 10, 277–293 (1993).
[CrossRef]

J. Fluid Mech.

G. L. Holtzer, L. R. Collins, “Relationship between the intrinsic radial distribution function for an isotropic field of particles and lower-dimensional measurements,” J. Fluid Mech. 459, 93–102 (2002).
[CrossRef]

A. B. Kostinski, R. A. Shaw, “Scale-dependent droplet clustering in turbulent clouds,” J. Fluid Mech. 434, 389–398 (2001).
[CrossRef]

J. Opt. Soc. Am. A

J. Phys. E

B. J. Thompson, “Holographic particle sizing techniques,” J. Phys. E 7, 781–788 (1974).
[CrossRef]

Meas. Sci. Technol.

H. Meng, G. Pan, Y. Pu, S. H. Woodward, “Holographic particle image velocimetry: from film to digital recording,” Meas. Sci. Technol. 15, 673–685 (2004).
[CrossRef]

K. D. Hinsch, “Holographic particle image velocimetry,” Meas. Sci. Technol. 13, R61–R72 (2002).
[CrossRef]

Opt. Acta

G. A. Tyler, B. J. Thompson, “Fraunhofer holography applied to particle size analysis: a reassessment,” Opt. Acta 23, 685–700 (1976).
[CrossRef]

Opt. Eng.

R. B. Owen, A. A. Zozulya, “In-line digital holographic sensor for monitoring and characterizing marine particulates,” Opt. Eng. 39, 2187–2197 (2000).
[CrossRef]

Q. J. R. Meteorol. Soc.

A. Kozikowska, K. Haman, J. Supronowicz, “Preliminary results of an investigation of the spatial distribution of fog droplets by a holographic method,” Q. J. R. Meteorol. Soc. 110, 65–73 (1984).
[CrossRef]

Tellus Ser. B

E. Uhlig, S. Borrmann, R. Jaenicke, “Holographic in situ measurements of the spatial droplet distribution in stratiform clouds,” Tellus Ser. B 50, 377–387 (1998).
[CrossRef]

Other

L. Cao, G. Pan, H. Meng, “Three-dimensional measurement of aerosol particle clustering in homogeneous isotropic turbulence,” in ASME Summer Heat Transfer Conference, Las Vegas, Nev. (American Society of Mechanical Engineers, New York, 2003), doi: HT2003–47435.

H. R. Pruppacher, J. D. Klett, Microphysics of Clouds and Precipitation (Kluwer Academic, Boston, Mass., 1997), pp. 15–24.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley-Interscience, New York, 1983), Chap. 4, pp. 107–111.

M. Born, E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, Cambridge, UK, 1999), Chap. 8, pp. 412–516.

P. K. Kundu, I. M. Cohen, Fluid Mechanics (Academic, San Diego, Calif., 2002), Chaps. 6, 10, pp. 148–192, 312–377.

D. Nagel, GKSS-Research Centre, Institute for Coastal Research, Max-Planck-Strasse 1, D-21502 Geesthacht, Germany (personal communication, 2003).

IDEAS 3 was held at the National Center for Atmospheric Research, Research Aviation Facility, sponsored by the National Science Foundation. Details of the IDEAS 3 project are available at http://raf.atd.ucar.edu/Projects/IDEAS-3/ .

J. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, Boston, Mass., 1996), Eq. (3-74).

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Figures (5)

Fig. 1
Fig. 1

(a) HOLODEC instrument mounted on the right wing pod of the National Center for Atmospheric Research/National Science Foundation (NCAR/NSF) C-130. An idea is given of the measurement environment and geometry, and the resulting design constraints are shown. (b) Schematic of the electro-optical system of HOLODEC. Cloud particles flow to the left relative to the instrument at a true air speed of approximately 100 m s-1.

Fig. 2
Fig. 2

(a) Whole reconstruction slice at 58 mm from the CCD. The droplet generator tip is shown at the top. A 50-μm-diameter droplet is visible about two-thirds down the frame below the tip. (b) Closeup of the 50-μm-diameter drop. (c) A 25-μm-diameter drop. Note that the reconstructed images of the droplets are within a pixel (4.65 μm) of the known diameter. Help with the calibration droplets was graciously provided by U. Maixner and D. Nagel from GKSS Forschungszentrum.

Fig. 3
Fig. 3

Images from a small section of a hologram obtained during flight traverse of a cloud: (a)–(d) Digitally reconstructed images in 500-μm steps along the optical axis, as indicated above each image. Each frame shows cloud droplets that come in and out of focus as the frames advance along the optical axis. Units on each axis are in millimeters. Other instruments on the aircraft indicated that the largest drops in this cloud are approximately 10–20 μm in diameter.

Fig. 4
Fig. 4

Various reconstructions of small ice crystals and one of the original holograms. The distance from the CCD is indicated above each frame, and the units on each axis are in millimeters. (a) Section of the hologram from which the ice crystal in (b) was reconstructed; (c), (d) other reconstructions of small ice crystals.

Fig. 5
Fig. 5

Various reconstructions of large ice crystals and one of the original holograms. The distance from the CCD is indicated above each frame. The units on each axis are millimeters. (a) Section of the hologram from which the ice crystal in frame (b) was reconstructed; (c), (d), (e) other reconstructions of large ice crystals.

Equations (7)

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

pN=N¯N exp-N¯N!,
ηr=Nr0Nr0+r¯N¯2-1,
IHx, y=EHEH*=ERER*+ERES*+ER*ES+ESES*.
IH=1-2CQ sinΦ+C2Q2.
V=CQ=d2r J1πrdλz.
Uνx, νy; z=z0=Uνx, νy; z=0Hνx, νy; z0,
Hνx, νy; z0=expjkz01-λ2νx2+νy21/2,

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