Abstract

Point-source digital in-line holographic microscopy with numerical reconstruction is ideally suited for quantitative phase measurements to determine optical path lengths and to extract changes in refractive index within accuracy close to 0.001 on the submicrometer length scale. This is demonstrated with simulated holograms and with detailed measurements on a number of different micrometer-sized samples such as suspended drops, optical fibers, as well as organisms of biological interest such as E. coli bacteria, HeLa cells, and fibroblast cells.

© 2012 Optical Society of America

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References

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  1. W. Xu, M. H. Jericho, I. A. Meinertzhagen, and H. J. Kreuzer, “Digital in-line holography for biological applications,” Proc. Natl. Acad. Sci. USA 98, 11301–11305 (2001).
    [CrossRef]
  2. J. Garcia-Sucerquia, W. Xu, S. K. Jericho, M. H. Jericho, P. Klages, and H. J. Kreuzer, “Digital in-line holographic microscopy,” Appl. Opt. 45, 836–850 (2006).
    [CrossRef]
  3. H. J. Kreuzer, M. H. Jericho, I. A. Meinertzhagen, and W. B. Xu, “Digital in-line holography with photons and electrons,” J. Phys. Condens. Matter 13, 10729–10741 (2001).
    [CrossRef]
  4. M. Kanka, R. Riesenberg, and H. J. Kreuzer, “Reconstruction of high-resolution holographic microscopic images,” Opt. Lett. 34, 1162–1164 (2009).
    [CrossRef]
  5. H. J. Kreuzer, K. Nakamura, A. Wierzbicki, H.-W. Fink, and H. Schmid, “Theory of the point source electron microscope,” Ultramicroscopy 45, 381–403 (1992).
    [CrossRef]
  6. M. H. Jericho and H. J. Kreuzer, “Point source digital in-line holographic microscopy,” in Coherent Light Microscopy, Vol. 46 of Springer Series in Surface Sciences, P. Ferraro, A. Wax, and Z. Zalevsky, eds. (Springer-Verlag, 2011), pp. 3–30.
    [CrossRef]
  7. W. Qu, O. C. Chee, Y. Yu, and A. Asundi, “Recording and reconstruction of digital Gabor holograms,” Optik 121, 2179–2184 (2010).
    [CrossRef]
  8. D. Gabor, “Microscopy by reconstructed wave fronts,” Proc. R. Soc. A 197, 454–487 (1949).
    [CrossRef]
  9. S. K. Jericho, J. Garcia-Sucerquia, W. Xu, M. H. Jericho, and H. J. Kreuzer, “Submersible digital in-line holographic microscope,” Rev. Sci. Instrum. 77, 043706 (2006).
    [CrossRef]
  10. www.resolutionoptics.com .
  11. A. Wuttig, M. Kanka, H. J. Kreuzer, and R. Ranier, “Packed domain Rayleigh–Sommerfeld wavefield propagation for large targets,” Opt. Express 18, 27036–27047 (2010).
    [CrossRef]
  12. J. Y. Lee, C. W. Lee, E. H. Lin, and P. K. Wei, “Single live cell refractometer using nanoparticle coated fiber tip,” Appl. Phys. Lett. 93, 173110 (2008).
    [CrossRef]
  13. F. Lanni, A. S. Waggoner, and D. L. Taylor, “Structural organization of interphase 3T3 fibroblasts studied by total internal reflection fluorescence microscopy,” J. Cell Biol. 100, 1091–1102 (1985).
    [CrossRef]
  14. N. Lue, G. Popescu, T. Ikeda, R. R. Dasari, K. Badizadegan, and M. S. Feld, ”Live cell refractometry using microfluidic devices,” Opt. Lett. 31, 2759–2761 (2006).
    [CrossRef]
  15. A. P. Kononenko, K. I. Kononenko, and D. M. Mikhov, “Dependence of refractive index on physiological state of microbial population,” Zh. Prikl. Spektrosk. 11, 114–117 (1969), translation.
  16. S. P. Anokhov, “Plane wave diffraction by a perfectly transparent half-plane,” J. Opt. Soc. Am. 24, 2493–2498 (2007).
    [CrossRef]

2010 (2)

W. Qu, O. C. Chee, Y. Yu, and A. Asundi, “Recording and reconstruction of digital Gabor holograms,” Optik 121, 2179–2184 (2010).
[CrossRef]

A. Wuttig, M. Kanka, H. J. Kreuzer, and R. Ranier, “Packed domain Rayleigh–Sommerfeld wavefield propagation for large targets,” Opt. Express 18, 27036–27047 (2010).
[CrossRef]

2009 (1)

2008 (1)

J. Y. Lee, C. W. Lee, E. H. Lin, and P. K. Wei, “Single live cell refractometer using nanoparticle coated fiber tip,” Appl. Phys. Lett. 93, 173110 (2008).
[CrossRef]

2007 (1)

S. P. Anokhov, “Plane wave diffraction by a perfectly transparent half-plane,” J. Opt. Soc. Am. 24, 2493–2498 (2007).
[CrossRef]

2006 (3)

2001 (2)

W. Xu, M. H. Jericho, I. A. Meinertzhagen, and H. J. Kreuzer, “Digital in-line holography for biological applications,” Proc. Natl. Acad. Sci. USA 98, 11301–11305 (2001).
[CrossRef]

H. J. Kreuzer, M. H. Jericho, I. A. Meinertzhagen, and W. B. Xu, “Digital in-line holography with photons and electrons,” J. Phys. Condens. Matter 13, 10729–10741 (2001).
[CrossRef]

1992 (1)

H. J. Kreuzer, K. Nakamura, A. Wierzbicki, H.-W. Fink, and H. Schmid, “Theory of the point source electron microscope,” Ultramicroscopy 45, 381–403 (1992).
[CrossRef]

1985 (1)

F. Lanni, A. S. Waggoner, and D. L. Taylor, “Structural organization of interphase 3T3 fibroblasts studied by total internal reflection fluorescence microscopy,” J. Cell Biol. 100, 1091–1102 (1985).
[CrossRef]

1969 (1)

A. P. Kononenko, K. I. Kononenko, and D. M. Mikhov, “Dependence of refractive index on physiological state of microbial population,” Zh. Prikl. Spektrosk. 11, 114–117 (1969), translation.

1949 (1)

D. Gabor, “Microscopy by reconstructed wave fronts,” Proc. R. Soc. A 197, 454–487 (1949).
[CrossRef]

Anokhov, S. P.

S. P. Anokhov, “Plane wave diffraction by a perfectly transparent half-plane,” J. Opt. Soc. Am. 24, 2493–2498 (2007).
[CrossRef]

Asundi, A.

W. Qu, O. C. Chee, Y. Yu, and A. Asundi, “Recording and reconstruction of digital Gabor holograms,” Optik 121, 2179–2184 (2010).
[CrossRef]

Badizadegan, K.

Chee, O. C.

W. Qu, O. C. Chee, Y. Yu, and A. Asundi, “Recording and reconstruction of digital Gabor holograms,” Optik 121, 2179–2184 (2010).
[CrossRef]

Dasari, R. R.

Feld, M. S.

Fink, H.-W.

H. J. Kreuzer, K. Nakamura, A. Wierzbicki, H.-W. Fink, and H. Schmid, “Theory of the point source electron microscope,” Ultramicroscopy 45, 381–403 (1992).
[CrossRef]

Gabor, D.

D. Gabor, “Microscopy by reconstructed wave fronts,” Proc. R. Soc. A 197, 454–487 (1949).
[CrossRef]

Garcia-Sucerquia, J.

S. K. Jericho, J. Garcia-Sucerquia, W. Xu, M. H. Jericho, and H. J. Kreuzer, “Submersible digital in-line holographic microscope,” Rev. Sci. Instrum. 77, 043706 (2006).
[CrossRef]

J. Garcia-Sucerquia, W. Xu, S. K. Jericho, M. H. Jericho, P. Klages, and H. J. Kreuzer, “Digital in-line holographic microscopy,” Appl. Opt. 45, 836–850 (2006).
[CrossRef]

Ikeda, T.

Jericho, M. H.

J. Garcia-Sucerquia, W. Xu, S. K. Jericho, M. H. Jericho, P. Klages, and H. J. Kreuzer, “Digital in-line holographic microscopy,” Appl. Opt. 45, 836–850 (2006).
[CrossRef]

S. K. Jericho, J. Garcia-Sucerquia, W. Xu, M. H. Jericho, and H. J. Kreuzer, “Submersible digital in-line holographic microscope,” Rev. Sci. Instrum. 77, 043706 (2006).
[CrossRef]

W. Xu, M. H. Jericho, I. A. Meinertzhagen, and H. J. Kreuzer, “Digital in-line holography for biological applications,” Proc. Natl. Acad. Sci. USA 98, 11301–11305 (2001).
[CrossRef]

H. J. Kreuzer, M. H. Jericho, I. A. Meinertzhagen, and W. B. Xu, “Digital in-line holography with photons and electrons,” J. Phys. Condens. Matter 13, 10729–10741 (2001).
[CrossRef]

M. H. Jericho and H. J. Kreuzer, “Point source digital in-line holographic microscopy,” in Coherent Light Microscopy, Vol. 46 of Springer Series in Surface Sciences, P. Ferraro, A. Wax, and Z. Zalevsky, eds. (Springer-Verlag, 2011), pp. 3–30.
[CrossRef]

Jericho, S. K.

S. K. Jericho, J. Garcia-Sucerquia, W. Xu, M. H. Jericho, and H. J. Kreuzer, “Submersible digital in-line holographic microscope,” Rev. Sci. Instrum. 77, 043706 (2006).
[CrossRef]

J. Garcia-Sucerquia, W. Xu, S. K. Jericho, M. H. Jericho, P. Klages, and H. J. Kreuzer, “Digital in-line holographic microscopy,” Appl. Opt. 45, 836–850 (2006).
[CrossRef]

Kanka, M.

Klages, P.

Kononenko, A. P.

A. P. Kononenko, K. I. Kononenko, and D. M. Mikhov, “Dependence of refractive index on physiological state of microbial population,” Zh. Prikl. Spektrosk. 11, 114–117 (1969), translation.

Kononenko, K. I.

A. P. Kononenko, K. I. Kononenko, and D. M. Mikhov, “Dependence of refractive index on physiological state of microbial population,” Zh. Prikl. Spektrosk. 11, 114–117 (1969), translation.

Kreuzer, H. J.

A. Wuttig, M. Kanka, H. J. Kreuzer, and R. Ranier, “Packed domain Rayleigh–Sommerfeld wavefield propagation for large targets,” Opt. Express 18, 27036–27047 (2010).
[CrossRef]

M. Kanka, R. Riesenberg, and H. J. Kreuzer, “Reconstruction of high-resolution holographic microscopic images,” Opt. Lett. 34, 1162–1164 (2009).
[CrossRef]

J. Garcia-Sucerquia, W. Xu, S. K. Jericho, M. H. Jericho, P. Klages, and H. J. Kreuzer, “Digital in-line holographic microscopy,” Appl. Opt. 45, 836–850 (2006).
[CrossRef]

S. K. Jericho, J. Garcia-Sucerquia, W. Xu, M. H. Jericho, and H. J. Kreuzer, “Submersible digital in-line holographic microscope,” Rev. Sci. Instrum. 77, 043706 (2006).
[CrossRef]

H. J. Kreuzer, M. H. Jericho, I. A. Meinertzhagen, and W. B. Xu, “Digital in-line holography with photons and electrons,” J. Phys. Condens. Matter 13, 10729–10741 (2001).
[CrossRef]

W. Xu, M. H. Jericho, I. A. Meinertzhagen, and H. J. Kreuzer, “Digital in-line holography for biological applications,” Proc. Natl. Acad. Sci. USA 98, 11301–11305 (2001).
[CrossRef]

H. J. Kreuzer, K. Nakamura, A. Wierzbicki, H.-W. Fink, and H. Schmid, “Theory of the point source electron microscope,” Ultramicroscopy 45, 381–403 (1992).
[CrossRef]

M. H. Jericho and H. J. Kreuzer, “Point source digital in-line holographic microscopy,” in Coherent Light Microscopy, Vol. 46 of Springer Series in Surface Sciences, P. Ferraro, A. Wax, and Z. Zalevsky, eds. (Springer-Verlag, 2011), pp. 3–30.
[CrossRef]

Lanni, F.

F. Lanni, A. S. Waggoner, and D. L. Taylor, “Structural organization of interphase 3T3 fibroblasts studied by total internal reflection fluorescence microscopy,” J. Cell Biol. 100, 1091–1102 (1985).
[CrossRef]

Lee, C. W.

J. Y. Lee, C. W. Lee, E. H. Lin, and P. K. Wei, “Single live cell refractometer using nanoparticle coated fiber tip,” Appl. Phys. Lett. 93, 173110 (2008).
[CrossRef]

Lee, J. Y.

J. Y. Lee, C. W. Lee, E. H. Lin, and P. K. Wei, “Single live cell refractometer using nanoparticle coated fiber tip,” Appl. Phys. Lett. 93, 173110 (2008).
[CrossRef]

Lin, E. H.

J. Y. Lee, C. W. Lee, E. H. Lin, and P. K. Wei, “Single live cell refractometer using nanoparticle coated fiber tip,” Appl. Phys. Lett. 93, 173110 (2008).
[CrossRef]

Lue, N.

Meinertzhagen, I. A.

H. J. Kreuzer, M. H. Jericho, I. A. Meinertzhagen, and W. B. Xu, “Digital in-line holography with photons and electrons,” J. Phys. Condens. Matter 13, 10729–10741 (2001).
[CrossRef]

W. Xu, M. H. Jericho, I. A. Meinertzhagen, and H. J. Kreuzer, “Digital in-line holography for biological applications,” Proc. Natl. Acad. Sci. USA 98, 11301–11305 (2001).
[CrossRef]

Mikhov, D. M.

A. P. Kononenko, K. I. Kononenko, and D. M. Mikhov, “Dependence of refractive index on physiological state of microbial population,” Zh. Prikl. Spektrosk. 11, 114–117 (1969), translation.

Nakamura, K.

H. J. Kreuzer, K. Nakamura, A. Wierzbicki, H.-W. Fink, and H. Schmid, “Theory of the point source electron microscope,” Ultramicroscopy 45, 381–403 (1992).
[CrossRef]

Popescu, G.

Qu, W.

W. Qu, O. C. Chee, Y. Yu, and A. Asundi, “Recording and reconstruction of digital Gabor holograms,” Optik 121, 2179–2184 (2010).
[CrossRef]

Ranier, R.

Riesenberg, R.

Schmid, H.

H. J. Kreuzer, K. Nakamura, A. Wierzbicki, H.-W. Fink, and H. Schmid, “Theory of the point source electron microscope,” Ultramicroscopy 45, 381–403 (1992).
[CrossRef]

Taylor, D. L.

F. Lanni, A. S. Waggoner, and D. L. Taylor, “Structural organization of interphase 3T3 fibroblasts studied by total internal reflection fluorescence microscopy,” J. Cell Biol. 100, 1091–1102 (1985).
[CrossRef]

Waggoner, A. S.

F. Lanni, A. S. Waggoner, and D. L. Taylor, “Structural organization of interphase 3T3 fibroblasts studied by total internal reflection fluorescence microscopy,” J. Cell Biol. 100, 1091–1102 (1985).
[CrossRef]

Wei, P. K.

J. Y. Lee, C. W. Lee, E. H. Lin, and P. K. Wei, “Single live cell refractometer using nanoparticle coated fiber tip,” Appl. Phys. Lett. 93, 173110 (2008).
[CrossRef]

Wierzbicki, A.

H. J. Kreuzer, K. Nakamura, A. Wierzbicki, H.-W. Fink, and H. Schmid, “Theory of the point source electron microscope,” Ultramicroscopy 45, 381–403 (1992).
[CrossRef]

Wuttig, A.

Xu, W.

S. K. Jericho, J. Garcia-Sucerquia, W. Xu, M. H. Jericho, and H. J. Kreuzer, “Submersible digital in-line holographic microscope,” Rev. Sci. Instrum. 77, 043706 (2006).
[CrossRef]

J. Garcia-Sucerquia, W. Xu, S. K. Jericho, M. H. Jericho, P. Klages, and H. J. Kreuzer, “Digital in-line holographic microscopy,” Appl. Opt. 45, 836–850 (2006).
[CrossRef]

W. Xu, M. H. Jericho, I. A. Meinertzhagen, and H. J. Kreuzer, “Digital in-line holography for biological applications,” Proc. Natl. Acad. Sci. USA 98, 11301–11305 (2001).
[CrossRef]

Xu, W. B.

H. J. Kreuzer, M. H. Jericho, I. A. Meinertzhagen, and W. B. Xu, “Digital in-line holography with photons and electrons,” J. Phys. Condens. Matter 13, 10729–10741 (2001).
[CrossRef]

Yu, Y.

W. Qu, O. C. Chee, Y. Yu, and A. Asundi, “Recording and reconstruction of digital Gabor holograms,” Optik 121, 2179–2184 (2010).
[CrossRef]

Appl. Opt. (1)

Appl. Phys. Lett. (1)

J. Y. Lee, C. W. Lee, E. H. Lin, and P. K. Wei, “Single live cell refractometer using nanoparticle coated fiber tip,” Appl. Phys. Lett. 93, 173110 (2008).
[CrossRef]

J. Cell Biol. (1)

F. Lanni, A. S. Waggoner, and D. L. Taylor, “Structural organization of interphase 3T3 fibroblasts studied by total internal reflection fluorescence microscopy,” J. Cell Biol. 100, 1091–1102 (1985).
[CrossRef]

J. Opt. Soc. Am. (1)

S. P. Anokhov, “Plane wave diffraction by a perfectly transparent half-plane,” J. Opt. Soc. Am. 24, 2493–2498 (2007).
[CrossRef]

J. Phys. Condens. Matter (1)

H. J. Kreuzer, M. H. Jericho, I. A. Meinertzhagen, and W. B. Xu, “Digital in-line holography with photons and electrons,” J. Phys. Condens. Matter 13, 10729–10741 (2001).
[CrossRef]

Opt. Express (1)

Opt. Lett. (2)

Optik (1)

W. Qu, O. C. Chee, Y. Yu, and A. Asundi, “Recording and reconstruction of digital Gabor holograms,” Optik 121, 2179–2184 (2010).
[CrossRef]

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

W. Xu, M. H. Jericho, I. A. Meinertzhagen, and H. J. Kreuzer, “Digital in-line holography for biological applications,” Proc. Natl. Acad. Sci. USA 98, 11301–11305 (2001).
[CrossRef]

Proc. R. Soc. A (1)

D. Gabor, “Microscopy by reconstructed wave fronts,” Proc. R. Soc. A 197, 454–487 (1949).
[CrossRef]

Rev. Sci. Instrum. (1)

S. K. Jericho, J. Garcia-Sucerquia, W. Xu, M. H. Jericho, and H. J. Kreuzer, “Submersible digital in-line holographic microscope,” Rev. Sci. Instrum. 77, 043706 (2006).
[CrossRef]

Ultramicroscopy (1)

H. J. Kreuzer, K. Nakamura, A. Wierzbicki, H.-W. Fink, and H. Schmid, “Theory of the point source electron microscope,” Ultramicroscopy 45, 381–403 (1992).
[CrossRef]

Zh. Prikl. Spektrosk. (1)

A. P. Kononenko, K. I. Kononenko, and D. M. Mikhov, “Dependence of refractive index on physiological state of microbial population,” Zh. Prikl. Spektrosk. 11, 114–117 (1969), translation.

Other (2)

M. H. Jericho and H. J. Kreuzer, “Point source digital in-line holographic microscopy,” in Coherent Light Microscopy, Vol. 46 of Springer Series in Surface Sciences, P. Ferraro, A. Wax, and Z. Zalevsky, eds. (Springer-Verlag, 2011), pp. 3–30.
[CrossRef]

www.resolutionoptics.com .

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Figures (10)

Fig. 1.
Fig. 1.

A. Schematic drawing of a spherical wave front of radius, R, contacting a reconstruction plane when a phase image of a sample is generated. The phase shift at a distance, w, from the optic axis is determined by the extra distance, S, the wave front has to travel to reach the reconstruction plane. The phase range is limited to 2π so that the phase is reset back to zero every multiple of 2π. This leads to the characteristic concentric ring structure of a phase reconstruction in point-source DIHM. B. Example of such a ring structure for a source-reconstruction plane distance, R, of 840 μm and a wavelength of 307 nm.

Fig. 2.
Fig. 2.

Schematic of the component arrangement for the point-source digital in-line holographic microscope from Resolution Optics, Inc. that was used for the experiments reported in this work.

Fig. 3.
Fig. 3.

Analysis of a simulated hologram for a sphere. A. Setup for the calculation of a hologram of a 15 μm diameter sphere placed on a 170 μm thick coverslip and immersed in water. The point source is an 800 nm diameter hole illuminated by 408 nm wavelength coherent radiation. The detector is a 2048×2048 array of 6.4 μm square pixels. B. Phase reconstruction of the simulated hologram. Image is reconstructed at an effective wavelength of 307 nm. C. Phase variation along a 75 μm long path through the center of the sphere in C (diamonds). Calculated phase variation from the refractive index difference and sphere size as given in the setup diagram A (open circles).The agreement is excellent.

Fig. 4.
Fig. 4.

A. Phase image of four glycerin drops in type A microscope oil. Bar=10μm. Imaging conditions: SOD=745μm, SSD=20mm, λ=405nm. B. Phase cuts through drops 1 and 2 in A with trace 1 shifted upward by one radian for clarity.

Fig. 5.
Fig. 5.

A. Phase image of four fibroblast cells in growth medium. The cell at a is a floating cell and such cells are known to be spherical in shape. Cell at (b) has a larger diameter and is in the process of attaching to the substrate surface. Imaging parameters: SSD=20mm, SOD=560μm, λ=405nm. B. Phase cut through the center of cell at (a). The cut shows the characteristic signature of a 2π phase wrap. C. Unwrapped phase image of cell (a). D. Phase cut through center of cell at (b). The phase initially decreases as the cell edge is approached and then shows a phase jump indicated by the arrow. The phase changes for this cell are much smaller than the phase changes for the cell at (a) and suggests a flattened cell.

Fig. 6.
Fig. 6.

A. Phase image of an attached HeLa cell. Imaging parameters: SSD=19.4mm, SOD=1287μm, λeff=388. B. Unwrapped phase cut through center of cell at A. C. Phase image of two HeLa cells in close proximity of the substrate surface. Imaging parameters: SSD=19.4mm, SOD=1287μm, λeff=388nm. D. Unwrapped phase cut through the cell on the left. The phase shifts for the mobile cells in C. were larger than the phase shifts for the attached cell in A.

Fig. 7.
Fig. 7.

Analysis of simulated hologram of a transparent rod immersed in water. A. Setup used to generate the hologram. B. Calculated hologram for a 5 μm diameter rod with n=1.38. C. Intensity reconstruction of the rod. D. Phase reconstruction of the rod. E. Phase cut perpendicular to the rod shown in D. The large phase shift from the rod is superimposed on a pattern of phase oscillations on either side of the rod so that the flat background phase is obtained only a few hundred micrometers from the rod. This phase signature is characteristic for thin rods.

Fig. 8.
Fig. 8.

A. Phase reconstruction of a single mode (633 nm) optic fiber. Nominal cladding and core diameters were 125 and 4 μm, respectively. The fiber was immersed in index matching oil to reduce phase shifts from the cladding. The refractive index jump from cladding to core of Δn=0.00493 is clearly visible as the thin white line running down the middle of the fiber. B. Phase cut perpendicular to the fiber. The edges of the cladding show up as phase dips at L and R while the core region at C has the characteristic phase signature of a thin rod. C. An average of ten phase cuts perpendicular to the fiber core taken at equal intervals along the fiber.

Fig. 9.
Fig. 9.

A. Phase reconstruction of a hologram of E. coli bacteria in the initial phases of biofilm formation. Imaging conditions: SSD=15mm, SOD=285μm, λ=397nm. B. Average of three phase cuts taken perpendicular to the bacterium marked by the arrow in A. The phase signature is characteristic for a thin rod and shows background oscillations on either side of the phase shift peak from the bacterium.

Fig. 10.
Fig. 10.

A. Setup for hologram simulation of two 20 μm diameter disks. B. Phase reconstruction of the disks. C. Phase cut through center of one disk along line indicated in B. Besides some background oscillations, the main signature is the phase jump (arrow). This jump signature is very similar to the phase variation through the flattened cell (disklike) of Fig. 5D.This phase jump at a refractive index discontinuity, such as the edge of a plate, may allow estimates of the plate thickness.

Tables (1)

Tables Icon

Table 1. Comparison of Glycerin Drop Size as Determined from Phase Shifts and from Intensity Reconstructionsa

Equations (6)

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

K(r)=Sd3ξI˜(ξ)exp[2πiξ·r/λξ].
ψ(r)=arctanImK(r)ReK(r).
Φ=(2π/λ)S=(2π/λ)[(R2+w2)1/2R].
N=Φ/2π.
Δψ=(nobjnf)2πt/λ0.
OP=(nobjnf)t=(λ0Δψ)/(2π).

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