Abstract

A multi-angle lensfree holographic imaging platform that can accurately characterize both the axial and lateral positions of cells located within multi-layered micro-channels is introduced. In this platform, lensfree digital holograms of the micro-objects on the chip are recorded at different illumination angles using partially coherent illumination. These digital holograms start to shift laterally on the sensor plane as the illumination angle of the source is tilted. Since the exact amount of this lateral shift of each object hologram can be calculated with an accuracy that beats the diffraction limit of light, the height of each cell from the substrate can be determined over a large field of view without the use of any lenses. We demonstrate the proof of concept of this multi-angle lensless imaging platform by using light emitting diodes to characterize various sized microparticles located on a chip with sub-micron axial and lateral localization over ~60 mm2 field of view. Furthermore, we successfully apply this lensless imaging approach to simultaneously characterize blood samples located at multi-layered micro-channels in terms of the counts, individual thicknesses and the volumes of the cells at each layer. Because this platform does not require any lenses, lasers or other bulky optical/mechanical components, it provides a compact and high-throughput alternative to conventional approaches for cytometry and diagnostics applications involving lab on a chip systems.

© 2010 OSA

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2010 (2)

S. O. Isikman, I. Sencan, O. Mudanyali, W. Bishara, C. Oztoprak, and A. Ozcan, “Color and monochrome lensless on-chip imaging of Caenorhabditis elegans over a wide field-of-view,” Lab Chip 10(9), 1109–1112 (2010).
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[CrossRef] [PubMed]

2009 (2)

M. Mir, Z. Wang, K. Tangella, and G. Popescu, “Diffraction Phase Cytometry: Blood on a CD-Rom,” Opt. Express 17(4), 2579–2585 (2009).
[CrossRef] [PubMed]

S. Seo, T. W. Su, D. K. Tseng, A. Erlinger, and A. Ozcan, “Lensfree holographic imaging for on-chip cytometry and diagnostics,” Lab Chip 9(6), 777–787 (2009).
[CrossRef] [PubMed]

2008 (2)

N. G. Clack, K. Salaita, and J. T. Groves, “Electrostatic readout of DNA microarrays with charged microspheres,” Nat. Biotechnol. 26(7), 825–830 (2008).
[CrossRef] [PubMed]

J. Rosen and G. Brooker, “Non-scanning motionless fluorescence three-dimensional holographic microscopy,” Nat. Photonics 2(3), 190–195 (2008).
[CrossRef]

2007 (6)

2006 (8)

2005 (3)

2004 (2)

2002 (2)

G. Pedrini and H. J. Tiziani, “Short-coherence digital microscopy by use of a lensless holographic imaging system,” Appl. Opt. 41(22), 4489–4496 (2002).
[CrossRef] [PubMed]

D. R. Meldrum and M. R. Holl, “Tech.Sight. Microfluidics. Microscale bioanalytical systems,” Science 297(5584), 1197–1198 (2002).
[CrossRef] [PubMed]

2001 (1)

W. Xu, M. H. Jericho, I. A. Meinertzhagen, and H. J. Kreuzer, “Digital in-line holography for biological applications,” Proc. Natl. Acad. Sci. U.S.A. 98(20), 11301–11305 (2001).
[CrossRef] [PubMed]

1999 (1)

1992 (1)

1991 (1)

B. F. Alexander and K. C. Ng, “Elimination of systematic error in subpixel accuracy centroid estimation,” Opt. Eng. 30(9), 1320–1331 (1991).
[CrossRef]

1989 (1)

1978 (1)

1968 (1)

P. B. Canham and A. C. Burton, “Distribution of size and shape in populations of normal human red cells,” Circ. Res. 22(3), 405–422 (1968).
[PubMed]

Alexander, B. F.

B. F. Alexander and K. C. Ng, “Elimination of systematic error in subpixel accuracy centroid estimation,” Opt. Eng. 30(9), 1320–1331 (1991).
[CrossRef]

Alferi, D.

Ares, J.

Arines, J.

Badizadegan, K.

W. Choi, C. Fang-Yen, K. Badizadegan, S. Oh, N. Lue, R. R. Dasari, and M. S. Feld, “Tomographic phase microscopy,” Nat. Methods 4(9), 717–719 (2007).
[CrossRef] [PubMed]

Y. Park, G. Popescu, K. Badizadegan, R. R. Dasari, and M. S. Feld, “Diffraction phase and fluorescence microscopy,” Opt. Express 14(18), 8263–8268 (2006).
[CrossRef] [PubMed]

Balis, U. J.

S. Nagrath, L. V. Sequist, S. Maheswaran, D. W. Bell, D. Irimia, L. Ulkus, M. R. Smith, E. L. Kwak, S. Digumarthy, A. Muzikansky, P. Ryan, U. J. Balis, R. G. Tompkins, D. A. Haber, and M. Toner, “Isolation of rare circulating tumour cells in cancer patients by microchip technology,” Nature 450(7173), 1235–1239 (2007).
[CrossRef] [PubMed]

Bell, D. W.

S. Nagrath, L. V. Sequist, S. Maheswaran, D. W. Bell, D. Irimia, L. Ulkus, M. R. Smith, E. L. Kwak, S. Digumarthy, A. Muzikansky, P. Ryan, U. J. Balis, R. G. Tompkins, D. A. Haber, and M. Toner, “Isolation of rare circulating tumour cells in cancer patients by microchip technology,” Nature 450(7173), 1235–1239 (2007).
[CrossRef] [PubMed]

Bishara, W.

S. O. Isikman, I. Sencan, O. Mudanyali, W. Bishara, C. Oztoprak, and A. Ozcan, “Color and monochrome lensless on-chip imaging of Caenorhabditis elegans over a wide field-of-view,” Lab Chip 10(9), 1109–1112 (2010).
[CrossRef] [PubMed]

Boyer, K.

Brooker, G.

J. Rosen and G. Brooker, “Non-scanning motionless fluorescence three-dimensional holographic microscopy,” Nat. Photonics 2(3), 190–195 (2008).
[CrossRef]

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P. B. Canham and A. C. Burton, “Distribution of size and shape in populations of normal human red cells,” Circ. Res. 22(3), 405–422 (1968).
[PubMed]

Canham, P. B.

P. B. Canham and A. C. Burton, “Distribution of size and shape in populations of normal human red cells,” Circ. Res. 22(3), 405–422 (1968).
[PubMed]

Carapezza, E.

Chernyshev, A. V.

Choi, W.

W. Choi, C. Fang-Yen, K. Badizadegan, S. Oh, N. Lue, R. R. Dasari, and M. S. Feld, “Tomographic phase microscopy,” Nat. Methods 4(9), 717–719 (2007).
[CrossRef] [PubMed]

Clack, N. G.

N. G. Clack, K. Salaita, and J. T. Groves, “Electrostatic readout of DNA microarrays with charged microspheres,” Nat. Biotechnol. 26(7), 825–830 (2008).
[CrossRef] [PubMed]

Cullen, D.

DaneshPanah, M.

Dasari, R. R.

W. Choi, C. Fang-Yen, K. Badizadegan, S. Oh, N. Lue, R. R. Dasari, and M. S. Feld, “Tomographic phase microscopy,” Nat. Methods 4(9), 717–719 (2007).
[CrossRef] [PubMed]

Y. Park, G. Popescu, K. Badizadegan, R. R. Dasari, and M. S. Feld, “Diffraction phase and fluorescence microscopy,” Opt. Express 14(18), 8263–8268 (2006).
[CrossRef] [PubMed]

De Nicola, S.

De Petrocellis, L.

Denis, L.

Digumarthy, S.

S. Nagrath, L. V. Sequist, S. Maheswaran, D. W. Bell, D. Irimia, L. Ulkus, M. R. Smith, E. L. Kwak, S. Digumarthy, A. Muzikansky, P. Ryan, U. J. Balis, R. G. Tompkins, D. A. Haber, and M. Toner, “Isolation of rare circulating tumour cells in cancer patients by microchip technology,” Nature 450(7173), 1235–1239 (2007).
[CrossRef] [PubMed]

Edwards, T.

P. Yager, T. Edwards, E. Fu, K. Helton, K. Nelson, M. R. Tam, and B. H. Weigl, “Microfluidic diagnostic technologies for global public health,” Nature 442(7101), 412–418 (2006).
[CrossRef] [PubMed]

El-Ali, J.

J. El-Ali, P. K. Sorger, and K. F. Jensen, “Cells on chips,” Nature 442(7101), 403–411 (2006).
[CrossRef] [PubMed]

Erlinger, A.

S. Seo, T. W. Su, D. K. Tseng, A. Erlinger, and A. Ozcan, “Lensfree holographic imaging for on-chip cytometry and diagnostics,” Lab Chip 9(6), 777–787 (2009).
[CrossRef] [PubMed]

Fang-Yen, C.

W. Choi, C. Fang-Yen, K. Badizadegan, S. Oh, N. Lue, R. R. Dasari, and M. S. Feld, “Tomographic phase microscopy,” Nat. Methods 4(9), 717–719 (2007).
[CrossRef] [PubMed]

Feld, M. S.

W. Choi, C. Fang-Yen, K. Badizadegan, S. Oh, N. Lue, R. R. Dasari, and M. S. Feld, “Tomographic phase microscopy,” Nat. Methods 4(9), 717–719 (2007).
[CrossRef] [PubMed]

Y. Park, G. Popescu, K. Badizadegan, R. R. Dasari, and M. S. Feld, “Diffraction phase and fluorescence microscopy,” Opt. Express 14(18), 8263–8268 (2006).
[CrossRef] [PubMed]

Ferraro, P.

Fienup, J. R.

Finizio, A.

Fournier, C.

Fu, E.

P. Yager, T. Edwards, E. Fu, K. Helton, K. Nelson, M. R. Tam, and B. H. Weigl, “Microfluidic diagnostic technologies for global public health,” Nature 442(7101), 412–418 (2006).
[CrossRef] [PubMed]

Garcia-Sucerquia, J.

Goepfert, C.

Grier, D. G.

Groves, J. T.

N. G. Clack, K. Salaita, and J. T. Groves, “Electrostatic readout of DNA microarrays with charged microspheres,” Nat. Biotechnol. 26(7), 825–830 (2008).
[CrossRef] [PubMed]

Haber, D. A.

S. Nagrath, L. V. Sequist, S. Maheswaran, D. W. Bell, D. Irimia, L. Ulkus, M. R. Smith, E. L. Kwak, S. Digumarthy, A. Muzikansky, P. Ryan, U. J. Balis, R. G. Tompkins, D. A. Haber, and M. Toner, “Isolation of rare circulating tumour cells in cancer patients by microchip technology,” Nature 450(7173), 1235–1239 (2007).
[CrossRef] [PubMed]

Haddad, W. S.

Helton, K.

P. Yager, T. Edwards, E. Fu, K. Helton, K. Nelson, M. R. Tam, and B. H. Weigl, “Microfluidic diagnostic technologies for global public health,” Nature 442(7101), 412–418 (2006).
[CrossRef] [PubMed]

Holl, M. R.

D. R. Meldrum and M. R. Holl, “Tech.Sight. Microfluidics. Microscale bioanalytical systems,” Science 297(5584), 1197–1198 (2002).
[CrossRef] [PubMed]

Irimia, D.

S. Nagrath, L. V. Sequist, S. Maheswaran, D. W. Bell, D. Irimia, L. Ulkus, M. R. Smith, E. L. Kwak, S. Digumarthy, A. Muzikansky, P. Ryan, U. J. Balis, R. G. Tompkins, D. A. Haber, and M. Toner, “Isolation of rare circulating tumour cells in cancer patients by microchip technology,” Nature 450(7173), 1235–1239 (2007).
[CrossRef] [PubMed]

Isikman, S. O.

C. Oh, S. O. Isikman, B. Khademhosseinieh, and A. Ozcan, “On-chip differential interference contrast microscopy using lensless digital holography,” Opt. Express 18(5Issue 5), 4717–4726 (2010).
[CrossRef] [PubMed]

S. O. Isikman, I. Sencan, O. Mudanyali, W. Bishara, C. Oztoprak, and A. Ozcan, “Color and monochrome lensless on-chip imaging of Caenorhabditis elegans over a wide field-of-view,” Lab Chip 10(9), 1109–1112 (2010).
[CrossRef] [PubMed]

Javidi, B.

Jenkins, E. B.

Jensen, K. F.

J. El-Ali, P. K. Sorger, and K. F. Jensen, “Cells on chips,” Nature 442(7101), 403–411 (2006).
[CrossRef] [PubMed]

Jericho, M. H.

J. Garcia-Sucerquia, W. Xu, M. H. Jericho, and H. J. Kreuzer, “Immersion digital in-line holographic microscopy,” Opt. Lett. 31(9), 1211–1213 (2006).
[CrossRef] [PubMed]

W. Xu, M. H. Jericho, I. A. Meinertzhagen, and H. J. Kreuzer, “Digital in-line holography for biological applications,” Proc. Natl. Acad. Sci. U.S.A. 98(20), 11301–11305 (2001).
[CrossRef] [PubMed]

Katz, J.

Khademhosseinieh, B.

Kim, M.

Kim, S.

Kreuzer, H. J.

J. Garcia-Sucerquia, W. Xu, M. H. Jericho, and H. J. Kreuzer, “Immersion digital in-line holographic microscopy,” Opt. Lett. 31(9), 1211–1213 (2006).
[CrossRef] [PubMed]

W. Xu, M. H. Jericho, I. A. Meinertzhagen, and H. J. Kreuzer, “Digital in-line holography for biological applications,” Proc. Natl. Acad. Sci. U.S.A. 98(20), 11301–11305 (2001).
[CrossRef] [PubMed]

Kwak, E. L.

S. Nagrath, L. V. Sequist, S. Maheswaran, D. W. Bell, D. Irimia, L. Ulkus, M. R. Smith, E. L. Kwak, S. Digumarthy, A. Muzikansky, P. Ryan, U. J. Balis, R. G. Tompkins, D. A. Haber, and M. Toner, “Isolation of rare circulating tumour cells in cancer patients by microchip technology,” Nature 450(7173), 1235–1239 (2007).
[CrossRef] [PubMed]

Lee, S.

Lo, C. M.

Longworth, J. W.

Lue, N.

W. Choi, C. Fang-Yen, K. Badizadegan, S. Oh, N. Lue, R. R. Dasari, and M. S. Feld, “Tomographic phase microscopy,” Nat. Methods 4(9), 717–719 (2007).
[CrossRef] [PubMed]

Maheswaran, S.

S. Nagrath, L. V. Sequist, S. Maheswaran, D. W. Bell, D. Irimia, L. Ulkus, M. R. Smith, E. L. Kwak, S. Digumarthy, A. Muzikansky, P. Ryan, U. J. Balis, R. G. Tompkins, D. A. Haber, and M. Toner, “Isolation of rare circulating tumour cells in cancer patients by microchip technology,” Nature 450(7173), 1235–1239 (2007).
[CrossRef] [PubMed]

Malkiel, E.

Maltsev, V. P.

Mann, C.

McPherson, A.

Meinertzhagen, I. A.

W. Xu, M. H. Jericho, I. A. Meinertzhagen, and H. J. Kreuzer, “Digital in-line holography for biological applications,” Proc. Natl. Acad. Sci. U.S.A. 98(20), 11301–11305 (2001).
[CrossRef] [PubMed]

Meldrum, D. R.

D. R. Meldrum and M. R. Holl, “Tech.Sight. Microfluidics. Microscale bioanalytical systems,” Science 297(5584), 1197–1198 (2002).
[CrossRef] [PubMed]

Mir, M.

Moon, I.

Morgan, J. S.

Mudanyali, O.

S. O. Isikman, I. Sencan, O. Mudanyali, W. Bishara, C. Oztoprak, and A. Ozcan, “Color and monochrome lensless on-chip imaging of Caenorhabditis elegans over a wide field-of-view,” Lab Chip 10(9), 1109–1112 (2010).
[CrossRef] [PubMed]

Muzikansky, A.

S. Nagrath, L. V. Sequist, S. Maheswaran, D. W. Bell, D. Irimia, L. Ulkus, M. R. Smith, E. L. Kwak, S. Digumarthy, A. Muzikansky, P. Ryan, U. J. Balis, R. G. Tompkins, D. A. Haber, and M. Toner, “Isolation of rare circulating tumour cells in cancer patients by microchip technology,” Nature 450(7173), 1235–1239 (2007).
[CrossRef] [PubMed]

Nagrath, S.

S. Nagrath, L. V. Sequist, S. Maheswaran, D. W. Bell, D. Irimia, L. Ulkus, M. R. Smith, E. L. Kwak, S. Digumarthy, A. Muzikansky, P. Ryan, U. J. Balis, R. G. Tompkins, D. A. Haber, and M. Toner, “Isolation of rare circulating tumour cells in cancer patients by microchip technology,” Nature 450(7173), 1235–1239 (2007).
[CrossRef] [PubMed]

Nelson, K.

P. Yager, T. Edwards, E. Fu, K. Helton, K. Nelson, M. R. Tam, and B. H. Weigl, “Microfluidic diagnostic technologies for global public health,” Nature 442(7101), 412–418 (2006).
[CrossRef] [PubMed]

Ng, K. C.

B. F. Alexander and K. C. Ng, “Elimination of systematic error in subpixel accuracy centroid estimation,” Opt. Eng. 30(9), 1320–1331 (1991).
[CrossRef]

Oh, C.

Oh, S.

W. Choi, C. Fang-Yen, K. Badizadegan, S. Oh, N. Lue, R. R. Dasari, and M. S. Feld, “Tomographic phase microscopy,” Nat. Methods 4(9), 717–719 (2007).
[CrossRef] [PubMed]

Ozcan, A.

S. O. Isikman, I. Sencan, O. Mudanyali, W. Bishara, C. Oztoprak, and A. Ozcan, “Color and monochrome lensless on-chip imaging of Caenorhabditis elegans over a wide field-of-view,” Lab Chip 10(9), 1109–1112 (2010).
[CrossRef] [PubMed]

C. Oh, S. O. Isikman, B. Khademhosseinieh, and A. Ozcan, “On-chip differential interference contrast microscopy using lensless digital holography,” Opt. Express 18(5Issue 5), 4717–4726 (2010).
[CrossRef] [PubMed]

S. Seo, T. W. Su, D. K. Tseng, A. Erlinger, and A. Ozcan, “Lensfree holographic imaging for on-chip cytometry and diagnostics,” Lab Chip 9(6), 777–787 (2009).
[CrossRef] [PubMed]

Oztoprak, C.

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[CrossRef] [PubMed]

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Figures (12)

Fig. 1
Fig. 1

The schematic diagram illustrating the principles of multi-angle lensfree holographic imaging. For each illumination angle, a spatially incoherent source such as a light emitting diode is filtered by a large aperture (~0.05mm - 0.1mm diameter), which is placed ~6 cm away from the object plane. Note that unlike conventional in-line holography approaches, the sample plane is much closer to the detector plane with a vertical distance of ~1mm, such that the entire active area of the sensor becomes the imaging FOV. (a) The shadow of each cell shifts laterally on the sensor plane as a function of the illumination angle of the incoherent source, encoding its axial position. (b) Matching of the cells’ shadows acquired at different illumination angles can be achieved by forming imaginary rays between each cell shadow and the corresponding source.

Fig. 2
Fig. 2

The details of the depth resolved imaging process using multi-angle lensfree holography is summarized.

Fig. 3
Fig. 3

The validation of sub-micron localization performance over a large field of view of ~60 mm2. (a-b-c) Lensfree holograms captured with three different illumination angles are illustrated. (d-e-f) Raw hologram signatures (digitally cropped from the vertical illumination hologram shown in (a)) and their corresponding reconstructed amplitude images for (d) 5 µm, (e) 10 µm, and (f) 20 µm microbeads are illustrated. (g-h-i) Raw hologram signatures (digitally cropped from the oblique illumination hologram shown in (b)) and their corresponding reconstructed amplitude images for (g) 5 µm, (h) 10 µm, and (i) 20 µm microbeads are illustrated. (j-k-l) Raw hologram signatures (digitally cropped from the oblique illumination hologram shown in (c)) and their corresponding reconstructed amplitude images for (j) 5 µm, (k) 10 µm, and (l) 20 µm microbeads are also illustrated. The scale bars in (d-l) are 40 µm long. (m) The cross-sectional structure of the imaged sample is shown. (n) The 2-D distribution of the characterized microbeads is illustrated, with their physical size and the relative height coded by the spot size and the colormap, respectively. (o) The height histogram is calculated from (n) showing three distinct peaks for the 5 µm, 10 µm, 20 µm beads in the mixture. In (n) and (o), the relative height of the substrate surface is arbitrarily assumed to be 0 µm.

Fig. 4
Fig. 4

Lensless multi-angle characterization of RBCs located within two-layered micro-channels. (a)-(e) Lensfree holograms are captured with five different illumination angles. The magenta dashed rectangles in (a)-(e) are the regions corresponding to the field-of-view shown in (g); and the yellow rectangles define the regions corresponding to the field of view of the images in Fig. 6. (f) The cross-sectional structure of the 2-layered sample is shown. (g) The 2D distribution of the RBCs located in both vertical channels is calculated with their height coded by the colormap. (h) shows the histogram of the cell heights over the entire field of view, which exhibits a double peaked behavior, as expected, resolving the 2 vertical micro-channels. In (g) and (h), the relative height is arbitrarily assumed to be 0 µm at the surface of the sensor.

Fig. 7
Fig. 7

The shadow overlap probability plotted as a function of both the shadow density (i.e., the throughput) and the multi-angle hologram recording method (parallel vs. sequential). The diameter of the holographic shadows for all the angles and all the vertical layers is assumed to be ~10 μm.

Fig. 8
Fig. 8

Quantified performance comparison of the multi-angle lensfree holographic cell characterization platform as a function of the shadow density and the number of vertical layers on the sensor chip. (a) The true positive rate, (b) the false positive rate, and (c) the total error rate for different cell densities distributed to 1, 2, 3, or 4 vertical layers/channels, where the total error rate includes both the missed cells and the false positives. (d) The maximum permitted shadow density (i.e., the maximum permitted throughput) at the sensor plane is plotted as a function of the number of vertical layers when the total error rate is maintained at a level of 5%. The cells are assumed to be illuminated from 5 different angles as shown in Fig. 1(a), and the ray threshold value is set to 3 for detecting each cell’s 3-D location (refer to Section 2). The shadow width at the sensor plane is assumed to be 10 μm for these numerical simulations.

Fig. 5
Fig. 5

Thickness and volume of each cell within the two-layered micro-channels are calculated over a field of view of >15 mm2. (a) The 3D distribution of the RBCs in both of the vertical channels is illustrated with their cell thickness value coded by the colormap. (b) The thickness histograms of the RBCs in both the upper and lower micro-channels are shown. (c) The 3D distribution of the RBCs in both vertical channels is illustrated with their cell volume coded by the colormap. (d) The volume histograms of the RBCs in both the upper and lower micro-channels are shown.

Fig. 6
Fig. 6

Demonstration of overlapping RBCs from two vertical micro-channels being digitally resolved by the holographic reconstruction process. (a) The raw lensfree hologram of the digitally zoomed region specified with the yellow rectangle in Fig. 2 (a) is illustrated. The amplitude images (b), (c), and (d) were reconstructed from (a) at a height of 1026 µm, 1081 µm, and 1049 µm respectively. In these reconstructed images, “L” and “U” refer to the RBCs located at the lower and upper micro-channels, respectively. The same field of view is also imaged using a 40X objective lens (0.65 NA) by focusing on both the lower (e) and the upper (f) micro-channels for comparison purposes. Note that the field of view that is imaged with Fig. 4 constitutes ~2 orders of magnitude improvement over the 40X microscope images shown in (e-f).

Fig. A1
Fig. A1

Noise characteristics of the CCD image sensor used in Fig. 3 are quantified. (a) reports the variance values of the pixels of the sensor chip measured with different integration times as a function of the illumination intensity. (b) quantifies the decomposition of various noise terms as a function of the illumination intensity. The fitted strengths of individual noise terms in (b) were calculated based on the parameter values estimated in (a). The results indicate that dominant detection noise sources were RIN and SN in our experiments reported in Fig. 3.

Fig. S1
Fig. S1

Error rate characterization of the reported multi-angle algorithm for analyzing 2 vertical layers of cells with different illumination conditions and ray threshold values. (a) The true positive rate, (b) the false positive rate, and (c) the total error rate as a function of the shadow density at the sensor plane are reported for 5 illumination angles. In each image, the effect of the ray threshold value (denoted by Th - refer to Section 2) on our characterization accuracy is also quantified. (d), (e) and (f) report the same error characterization as a function of the imaging throughput for this time 3 illumination angles, rather than 5. The holographic shadow width of the cells at the sensor plane is assumed to be 10 μm for these calculations.

Fig. S2
Fig. S2

Same as Supplementary Fig. S1, except that the error characterization results as a function of the imaging throughput are now reported for 3 vertical layers of cells with different illumination conditions and ray threshold values (Th).

Fig. S3
Fig. S3

Same as Supplementary Fig. S1, except that the error characterization results as a function of the imaging throughput are now reported for 4 vertical layers of cells with different illumination conditions and ray threshold values (Th).

Tables (2)

Tables Icon

Table A1 Theoretical breakdown of the lateral localization a

Tables Icon

Table A2 Theoretical breakdown of the height localization errors and comparison to measurement results a

Equations (13)

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{ x c = i j x i j n i j i j n i j y c = i j y i j n i j i j n i j ,
{ x c = i j x i j n i j i j n i j = u x v y c = i j y i j n i j i j n i j = u y v ,
{ S x c 2 x c 2 = S u x 2 u x 2 + S v 2 v 2 2 S u v x u x v S y c 2 y c 2 = S u y 2 u y 2 + S v 2 v 2 2 S u v y u y v ,
{ S u x 2 = i j S i j 2 x i j 2 + i k j l S i j k l x i j x k l S u y 2 = i j S i j 2 y i j 2 + i k j l S i j k l y i j y k l S v 2 = i j S i j 2 + i k j l S i j k l S u v x = i j S i j 2 x i j + i k j l S i j k l x i j S u v y = i j S i j 2 y i j + i k j l S i j k l y i j ,
{ S x c 2 = u x 2 v 2 ( S u x 0 2 u x 2 + S v 0 2 v 2 2 S u v x 0 u x v ) S y c 2 = u y 2 v 2 ( S u y 0 2 u y 2 + S v 0 2 v 2 2 S u v y 0 u y v ) ,
S i j 2 = n i j 2 n i j 2 = p i j 4 p i j 2 2 = ( 3 S p i j 4 + 6 p i j 2 S p i j 2 + p i j 4 ) ( S p i j 2 + p i j 2 ) , 2 = 2 S p i j 4 + 4 p i j 2 S p i j 2 ,
S S h i f t 2 = ( S h i f t d x ) 2 S d x 2 + ( S h i f t d y ) 2 S d y 2 + 2 S h i f t d x S h i f t d y S d x d y ( S h i f t d x ) 2 S d x 2 + ( S h i f t d y ) 2 S d y 2 = ( d x S h i f t ) 2 ( S x c 2 + S x c o 2 ) + ( d y S h i f t ) 2 ( S y c 2 + S y c o 2 ) ,
S H e i g h t 2 = S S h i f t 2 cot θ g ,
S H e i g h t , A V G = 1 n a a = 1 n a S H e i g h t , a 2 ,
{ x c = x e ( x , y ) d x d y e ( x , y ) d x d y = x f e ( x , y ) d x d y f e ( x , y ) d x d y = x f e x ( x ) d x f e x ( x ) d x y c = y e ( x , y ) d x d y e ( x , y ) d x d y = y f e ( x , y ) d x d y f e ( x , y ) d x d y = y f e y ( y ) d y f e y ( y ) d y ,
{ δ x = n = 1 F e x ( n / Δ ) sin ( 2 π η x n / Δ ) π [ F e x ( 0 ) + n = 1 F e x ( n / Δ ) cos ( 2 π η x n / Δ ) ] δ y = n = 1 F e y ( n / Δ ) sin ( 2 π η y n / Δ ) π [ F e y ( 0 ) + n = 1 F e y ( n / Δ ) cos ( 2 π η y n / Δ ) ] ,
{ S x c , p x = 1 Δ δ x 2 ( η x ) d η x S y c , p x = 1 Δ δ y 2 ( η y ) d η y .
S I 2 = a 2 I 2 + a 1 I + a t t + a 0 ,

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