Abstract

The main obstacle in retrieving quantitative phase with high sensitivity is posed by the phase noise due to mechanical vibrations and air fluctuations that typically affect any interferometric system. In this paper, we review diffraction phase microscopy (DPM), which is a common-path quantitative phase imaging (QPI) method that significantly alleviates the noise problem. DPM utilizes a compact Mach–Zehnder interferometer to combine several attributes of current QPI methods. This compact configuration inherently cancels out most mechanisms responsible for noise and is single-shot, meaning that the acquisition speed is limited only by the speed of the camera employed. This technique is also nondestructive and does not require staining or coating of the specimen. This unique collection of features enables the DPM system to accurately monitor the dynamics of various nanoscale phenomena in a wide variety of environments. The DPM system can operate in both transmission and reflection modes in order to accommodate both transparent and opaque samples, respectively. Thus, current applications of DPM include measuring the dynamics of biological samples, semiconductor wet etching and photochemical etching processes, surface wetting and evaporation of water droplets, self-assembly of nanotubes, expansion and deformation of materials, and semiconductor wafer defect detection. Finally, DPM with white light averages out much of the speckle background and also offers potential for spectroscopic measurements.

© 2014 Optical Society of America

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T. Kim, R. Zhou, M. Mir, D. S. Babacan, S. P. Carney, L. L. Goddard, and G. Popescu, “White light diffraction tomography of unlabeled live cells,” Nat. Photonics 8, 256–263 (2014).
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C. Edwards, B. Bhaduri, T. Nguyen, B. Griffin, H. Pham, T. Kim, G. Popescu, and L. L. Goddard, “Effects of spatial coherence in diffraction phase microscopy,” Opt. Express 22, 5133–5146 (2014).

2013 (7)

C. Edwards, K. Wang, R. Zhou, B. Bhaduri, G. Popescu, and L. L. Goddard, “Digital projection photochemical etching defines gray-scale features,” Opt. Express 21, 13547–13554 (2013).
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R. Zhou, C. Edwards, A. Arbabi, G. Popescu, and L. L. Goddard, “Detecting 20 nm wide defects in large area nanopatterns using optical interferometric microscopy,” Nano Lett. 13, 3716–3721 (2013).
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R. Zhou, G. Popescu, and L. L. Goddard, “22 nm node wafer inspection using diffraction phase microscopy and image post-processing,” Proc. SPIE 8681, 86810G (2013).
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H. V. Pham, B. Bhaduri, K. Tangella, C. Best-Popescu, and G. Popescu, “Real time blood testing using quantitative phase imaging,” PLOS One 8, e55676 (2013).
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H. V. Pham, C. Edwards, L. L. Goddard, and G. Popescu, “Fast phase reconstruction in white light diffraction phase microscopy,” Appl. Opt. 52, A97–A101 (2013).
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B. Bhaduri, K. Tangella, and G. Popescu, “Fourier phase microscopy with white light,” Biomed. Opt. Express 4, 1434–1441 (2013).
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K. Lee, K. Kim, J. Jung, J. Heo, S. Cho, S. Lee, G. Chang, Y. Jo, H. Park, and Y. Park, “Quantitative phase imaging techniques for the study of cell pathophysiology: from principles to applications,” Sensors 13, 4170–4191 (2013).
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2012 (10)

C. Edwards, A. Arbabi, G. Popescu, and L. L. Goddard, “Optically monitoring and controlling nanoscale topography during semiconductor etching,” Light Sci. Appl. 1, e30 (2012).
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B. Bhaduri, H. Pham, M. Mir, and G. Popescu, “Diffraction phase microscopy with white light,” Opt. Lett. 37, 1094–1096 (2012).
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M. Rinehart, Y. Zhu, and A. Wax, “Quantitative phase spectroscopy,” Biomed. Opt. Express 3, 958–965 (2012).
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H. Pham, B. Bhaduri, H. F. Ding, and G. Popescu, “Spectroscopic diffraction phase microscopy,” Opt. Lett. 37, 3438–3440 (2012).
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R. S. Weinstein, A. R. Graham, F. Lian, B. L. Braunhut, G. R. Barker, E. A. Krupinski, and A. K. Bhattacharyya, “Reconciliation of diverse telepathology system designs. Historic issues and implications for emerging markets and new applications,” APMIS 120, 256–275 (2012).
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M. D. Seaberg, D. E. Adams, B. Zhang, D. F. Gardner, M. M. Murnane, and H. C. Kapteyn, “Ultrahigh 22 nm resolution EUV coherent diffraction imaging using a tabletop 13 nm high harmonic source,” Proc. SPIE 8324, 83240D (2012).
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S. J. Lim, W. Kim, and S. K. Shin, “Surface-dependent, ligand-mediated photochemical etching of CdSe nanoplatelets,” J. Am. Chem. Soc. 134, 7576–7579 (2012).
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2011 (6)

2010 (8)

S. S. Kou, L. Waller, G. Barbastathis, and C. J. R. Sheppard, “Transport-of-intensity approach to differential interference contrast (TI-DIC) microscopy for quantitative phase imaging,” Opt. Lett. 35, 447–449 (2010).
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A. Anand, V. K. Chhaniwal, and B. Javidi, “Real-time digital holographic microscopy for phase contrast 3D imaging of dynamic phenomena,” J. Disp. Technol. 6, 500–505 (2010).
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K. A. Hawick, A. Leist, and D. P. Playne, “Parallel graph component labelling with GPUs and CUDA,” Parallel Comput. 36, 655–678 (2010).
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D. Fu, W. Choi, Y. J. Sung, Z. Yaqoob, R. R. Dasari, and M. Feld, “Quantitative dispersion microscopy,” Biomed. Opt. Express 1, 347–353 (2010).
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D. Tseng, O. Mudanyali, C. Oztoprak, S. O. Isikman, I. Sencan, O. Yaglidere, and A. Ozcan, “Lensfree microscopy on a cellphone,” Lab Chip 10, 1787–1792 (2010).
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Y. K. Park, C. A. Best, T. Auth, N. S. Gov, S. A. Safran, G. Popescu, S. Suresh, and M. S. Feld, “Metabolic remodeling of the human red blood cell membrane,” Proc. Natl. Acad. Sci. USA 107, 1289–1294 (2010).
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E. Spyratou, I. Asproudis, D. Tsoutsi, C. Bacharis, K. Moutsouris, M. Makropoulou, and A. A. Serafetinides, “UV laser ablation of intraocular lenses: SEM and AFM microscopy examination of the biomaterial surface,” Appl. Surf. Sci. 256, 2539–2545 (2010).
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Y. K. Park, C. A. Best, K. Badizadegan, R. R. Dasari, M. S. Feld, T. Kuriabova, M. L. Henle, A. J. Levine, and G. Popescu, “Measurement of red blood cell mechanics during morphological changes,” Proc. Natl. Acad. Sci. USA 107, 6731–6736 (2010).
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2009 (4)

2008 (8)

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N. Mohandas and P. G. Gallagher, “Red cell membrane: past, present, and future,” Blood 112, 3939–3948 (2008).
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J. D. Wan, W. D. Ristenpart, and H. A. Stone, “Dynamics of shear-induced ATP release from red blood cells,” Proc. Natl. Acad. Sci. USA 105, 16432–16437 (2008).
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G. Popescu, Y. Park, N. Lue, C. Best-Popescu, L. Deflores, R. R. Dasari, M. S. Feld, and K. Badizadegan, “Optical imaging of cell mass and growth dynamics,” Am. J. Physiol. Cell Physiol. 295, C538–C544 (2008).
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B. Rappaz, F. Charrière, C. Depeursinge, P. J. Magistretti, and P. Marquet, “Simultaneous cell morphometry and refractive index measurement with dual-wavelength digital holographic microscopy and dye-enhanced dispersion of perfusion medium,” Opt. Lett. 33, 744–746 (2008).
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Y. K. Park, M. Diez-Silva, G. Popescu, G. Lykotrafitis, W. Choi, M. S. Feld, and S. Suresh, “Refractive index maps and membrane dynamics of human red blood cells parasitized by Plasmodium falciparum,” Proc. Natl. Acad. Sci. USA 105, 13730 (2008).
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2007 (5)

G. Popescu, Y. Park, R. R. Dasari, K. Badizadegan, and M. S. Feld, “Coherence properties of red blood cell membrane motions,” Phys. Rev. E 76, 031902 (2007).
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K. J. Chalut, W. J. Brown, and A. Wax, “Quantitative phase microscopy with asynchronous digital holography,” Opt. Express 15, 3047–3052 (2007).
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S. S. Kou and C. J. R. Sheppard, “Imaging in digital holographic microscopy,” Opt. Express 15, 13640–13648 (2007).
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2006 (8)

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D. C. B. Kemper, J. Schnekenburger, M. Schäfer, W. Domschke, G. von Bally, I. Bredebusch, and D. Carl, “Investigation of living pancreas tumor cells by digital holographic microscopy,” J. Biomed. Opt. 11, 034005 (2006).
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G. Popescu, T. Ikeda, K. Goda, C. A. Best-Popescu, M. Laposata, S. Manley, R. R. Dasari, K. Badizadegan, and M. S. Feld, “Optical measurement of cell membrane tension,” Phys. Rev. Lett. 97, 218101 (2006).
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G. Popescu, T. Ikeda, R. R. Dasari, and M. S. Feld, “Diffraction phase microscopy for quantifying cell structure and dynamics,” Opt. Lett. 31, 775–777 (2006).
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C. L. L. Lawrence, N. Gov, and F. L. H. Brown, “Nonequilibrium membrane fluctuations driven by active proteins,” J. Chem. Phys. 124, 074903 (2006).
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M. Friebel and M. Meinke, “Model function to calculate the refractive index of native hemoglobin in the wavelength range of 250–1100 nm dependent on concentration,” Appl. Opt. 45, 2838–2842 (2006).
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D. Psaltis, S. R. Quake, and C. H. Yang, “Developing optofluidic technology through the fusion of microfluidics and optics,” Nature 442, 381–386 (2006).
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J. Li, M. Dao, C. T. Lim, and S. Suresh, “Spectrin-level modeling of the cytoskeleton and optical tweezers stretching of the erythrocyte,” Biophys. J. 88, 3707–3719 (2005).
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S. Suresh, J. Spatz, J. P. Mills, A. Micoulet, M. Dao, C. T. Lim, M. Beil, and T. Seufferlein, “Connections between single-cell biomechanics and human disease states: gastrointestinal cancer and malaria,” Acta Biomater. 1, 15–30 (2005).
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N. S. Gov and S. A. Safran, “Red blood cell membrane fluctuations and shape controlled by ATP-induced cytoskeletal defects,” Biophys. J. 88, 1859–1874 (2005).
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B. Rappaz, P. Marquet, E. Cuche, Y. Emery, C. Depeursinge, and P. J. Magistretti, “Measurement of the integral refractive index and dynamic cell morphometry of living cells with digital holographic microscopy,” Opt. Express 13, 9361–9373 (2005).
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T. Ikeda, G. Popescu, R. R. Dasari, and M. S. Feld, “Hilbert phase microscopy for investigating fast dynamics in transparent systems,” Opt. Lett. 30, 1165–1168 (2005).
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C. J. Mann, L. F. Yu, C. M. Lo, and M. K. Kim, “High-resolution quantitative phase-contrast microscopy by digital holography,” Opt. Express 13, 8693–8698 (2005).
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P. Marquet, B. Rappaz, P. J. Magistretti, E. Cuche, Y. Emery, T. Colomb, and C. Depeursinge, “Digital holographic microscopy: a noninvasive contrast imaging technique allowing quantitative visualization of living cells with subwavelength axial accuracy,” Opt. Lett. 30, 468–470 (2005).
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L. Xu, X. Y. Peng, Z. X. Guo, J. M. Miao, and A. Asundi, “Imaging analysis of digital holography,” Opt. Express 13, 2444–2452 (2005).
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2004 (4)

C. L. Curl, C. J. Bellair, P. J. Harris, B. E. Allman, A. Roberts, K. A. Nugent, and L. M. D. Delbridge, “Quantitative phase microscopy: a new tool for investigating the structure and function of unstained live cells,” Clin. Exp. Pharmacol. Physiol. 31, 896–901 (2004).
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D. Carl, B. Kemper, G. Wernicke, and G. von Bally, “Parameter-optimized digital holographic microscope for high-resolution living-cell analysis,” Appl. Opt. 43, 6536–6544 (2004).
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G. Popescu, L. P. Deflores, J. C. Vaughan, K. Badizadegan, H. Iwai, R. R. Dasari, and M. S. Feld, “Fourier phase microscopy for investigation of biological structures and dynamics,” Opt. Lett. 29, 2503–2505 (2004).
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Figures (34)

Figure 1
Figure 1

DPM. (a) A diffraction grating is used in conjunction with a 4f system in order to produce interference and obtain the phase information from the scattered fields. The grating is placed at the output image plane of the microscope, which creates copies of the image at different angles, some of which are captured by the first lens. Under a 4f configuration, the first lens takes a Fourier transform, creating a Fourier plane. (b) A spatial filter is placed in the Fourier plane, which allows the full 0th order to pass, and the 1st order is filtered down using a small pinhole such that after the second lens takes a Fourier transform, the field becomes a uniform plane wave and serves as the reference to the interferometer. The two fields interfere at the final image plane on the CCD sensor to create the interferogram.

Figure 2
Figure 2

Fourier transform of the intensity at the CCD as represented in the sample plane. Because of interference, the radius of the central lobe is 2k0NAobj and the radius of the sidelobes is k0NAobj. To avoid aliasing effects and allow for proper reconstruction of the desired signal, the modulation frequency β must be 3k0NAobj and the sampling (pixel) frequency ks2(β+k0NAobj). (a) Aliasing as a result of the grating period being too large. (b) Grating period is small enough to avoid aliasing. (c) Sampling by the CCD pixels does not meet the Nyquist criterion, resulting in aliasing even if the grating period is chosen correctly. (d) The grating period is small enough to push the modulation outside the central lobe and sampling by the CCD pixels satisfies the Nyquist criterion; no aliasing occurs.

Figure 3
Figure 3

Raw images. (a) Calibration/background image taken from a flat featureless portion of the sample. (b) Image of micropillar (control sample). (c) Zoomed-in portion of the calibration image in (a), showing no shift in the fringes. (d) Zoomed-in portion of the micropillar’s edge, showing a horizontal shift in the fringes due to a change in height. The shift of the fringes is proportional to the change in the optical path length.

Figure 4
Figure 4

Phase reconstruction. (a) Fourier Transform of raw interferogram. (b) A simple bandpass filter is used to pick out the spatially modulated signal. (c) The signal is brought back to baseband. From here, the phase can be extracted and used to reconstruct the surface profile.

Figure 5
Figure 5

Quantitative phase images obtained via DPM. (a) Height map obtained via DPM. (b) Cross section of the micropillar showing height and diameter. (c) Topographic reconstruction of micropillar in (a).

Figure 6
Figure 6

Comparison between standard digital BPF and apodized BPF. (a) Phase image processed using standard digital BPF. (b) Phase image processed using optimized hybrid filter. For both sets of images, region 1 is a cross-sectional profile of the micropillar taken along the vertical line indicated in the DPM image. Region 2 indicates the noise and roughness on top of the pillar, and region 3 indicates the background region near the edge of the micropillar where ringing is most prominent. Notice a clear reduction in the ringing and noise while maintaining the quantitative values. This is not possible using an abrupt BPF or a Gaussian alone; a combination must be employed.

Figure 7
Figure 7

Noise characterization and system verification. (a) Height image of plain unprocessed n+GaAs wafer. The heights should ideally be zero in all locations. Thus, the standard deviation of the height is a measure of the spatial noise. (b) Images of the standard deviation of each pixel projected over the entire 256 frame sequence. This gives the standard deviation of each pixel over time, which is a measure of the temporal noise in the system. (c) Height map of our micropillar control sample. (d) Histogram of (a) used to extract pillar height. All measured dimensions were verified using the Tencor Alpha-Step 200 surface profiler and the Hitachi S-4800 scanning electron microscope. All measured heights were accurate to within the spatial noise floor, and all lateral dimensions were accurate to within the diffraction spot. Adapted from [50].

Figure 8
Figure 8

Derivative method for phase calculation. (a) Original DPM image. (b) First-order derivative of (a) with respect to x. (c) Second-order derivative of (a) with respect to x. (d) The reconstructed unwrapped phase. (e) Phase obtained by Hilbert transform; the color bars show the phase in radians.

Figure 9
Figure 9

Laser DPM setup operating in transmission.

Figure 10
Figure 10

Live neuron cell imaging. (a) Transmission DPM interferogram of a live neuron cell. (b) Reconstructed phase with Hilbert transform. Color bar shows the phase in radians.

Figure 11
Figure 11

RBC topography (left column) and instantaneous displacement maps (right column) for (a) a discocyte, (b) an echinocyte, and (c) a spherocyte, as indicated by DC, EC, and SC, respectively. Adapted from Park et al., “Measurement of red blood cell mechanics during morphological changes,” Proc. Natl. Acad. Sci. U.S.A. 107, 6731–6736 (2010) [80] with permission.

Figure 12
Figure 12

Topographic images and effective elastic constant maps of Pf-RBCs. (a) and (e) Healthy RBC. (b) and (f) Ring stage. (c) and (g) Trophozoite stage. (d) and (h) Schizont stage. The topographic images in (a)–(d) are the instant thickness map of Pf-RBCs. The effective elastic constant maps were calculated from the RMS displacement of the thermal membrane fluctuations in the Pf-RBC membranes. Black arrows indicate the location of P. falciparum, and the gray arrows indicate the location of hemozoin. (Scale bar, 1.5 μm.) Adapted with permission from Park et al., “Refractive index maps and membrane dynamics of human red blood cells parasitized by Plasmodium falciparum,” Proc. Natl. Acad. Sci. U.S.A. 105, 13730 (2008) [94]. Copyright 2008 National Academy of Sciences, U.S.A.

Figure 13
Figure 13

Laser DPM setup operating in reflection mode.

Figure 14
Figure 14

Real-time imaging during wet etching of the University of Illinois logo. A 5× objective was used for monitoring the etch, resulting in a FOV of 320μm×240μm for each of the images shown. The video was acquired at 8.93frames/s. It took roughly 10 s for the etchant to diffuse into the FOV and begin etching. Still frames over a 30 s interval from 14–44 s are displayed, showing the dynamics of the etching process. Diffusion of the etchant from the top-left corner to the bottom-right corner is observed. Adapted from [50].

Figure 15
Figure 15

Analysis of real-time etching dynamics. (a) Final depth profile after etching was completed. (b) Plot of etching depths over time to show variation of depths and etch rates at points indicated in (f). (c)–(e) Etch rates at each pixel across the FOV at frame 150 (16.8 s), frame 250 (28.0 s), and frame 350 (39.2 s). (f) Overall etch rate averaged over the entire etch process. The etch rate may vary by several nanometers/second over the FOV at any given time, but average out over time. Adapted from [50].

Figure 16
Figure 16

Digital projection PC etching system. Illustration showing the integration of our newly developed digital projection PC etching and epi-DPM methods. The bold solid lines represent the light path from the projector, which was focused on the sample and aligned using CCD1. The epi-DPM system was used to measure the height of features on the sample using CCD2. The light path for epi-DPM is indicated by the dotted lines. Adapted from [96].

Figure 17
Figure 17

System characterization. (a) epi-DPM image of horizontal and vertical lines etched using the USAF-1951 resolution target. Image was captured using a 20×, NA=0.5 objective (650 nm lateral resolution). (b) Cross section of horizontal lines taken along the dotted line in (c). Based on this resolution test, the etch feature resolution using our PC etching setup is approximately 2 μm. (c) DPM height map of a PC etched micropillar. Mask patterns with gray levels of 0 and 136 were used for the pillar and background, respectively. PC etching was performed for 60 s, resulting in a mean height of 304.9 nm. It was calibrated to have a 100 μm diameter and a height of 300 nm. (d) Cross section of the pillar in (c) showing the dimensions and edge resolution. Adapted from [96].

Figure 18
Figure 18

Etched structures. (a) DPM height image of stacked cubes. A projected image with gray levels of 0, 60, and 78 were used for the mask pattern. PC etching was performed for 33 s, resulting in mean heights of 0, 50.7, and 101.6 nm. It was calibrated to have widths of 150 and 75 μm with heights of 0, 50, and 100 nm. (b) Histogram of image in (c) showing the heights of the three levels. (c) Topographical reconstruction of a microlens array fabricated using PC etching. (d) Comparison between s cross section of the projected mask pattern and horizontal and vertical cross sections from an etched microlens. (e) Topographical profile of PC etched Archimedean spiral. Adapted from [50,96].

Figure 19
Figure 19

Illustration of different defect types. (a) An isolated defect, (b) a perpendicular bridge defect, and (c) a parallel bridge defect. Adapted from Zhou et al., “22 nm node wafer inspection using diffraction phase microscopy and image post-processing,” Proc. SPIE 8681, 8610G (2013) [103] with permission from SPIE.

Figure 20
Figure 20

Stretched image using moving averaging. (a) Stretched amplitude image and (b) stretched phase image. In each stretched image, the location of the defect is marked by a red box. Adapted from Zhou et al., “22 nm node wafer inspection using diffraction phase microscopy and image post-processing,” Proc. SPIE 8681, 8610G (2013) [103] with permission from SPIE.

Figure 21
Figure 21

wDPM experimental setup.

Figure 22
Figure 22

Noise stability of wDPM. (a) Spatial noise of wDPM. (b) Spatial noise of laser DPM. (c) and (d) Spatiotemporal power spectral density of wDPM in log scale at ky=0 and ky=2π, respectively; color bars in (c) and (d) are nm2/[(rad/s)·(rad/μm)2]. Adapted from [54].

Figure 23
Figure 23

Temporal coherence of the wDPM imaging system. (a) Spectrum of the HAL 100 halogen lamp showing the power-equivalent bandwidth approximation. The center wavelength at a color temperature of 3200 K is 574 nm. This wavelength is used to convert the measured phase to height. (b) Temporal autocorrelation function and its envelope obtained via a Hilbert transform. The temporal coherence length obtained from the envelope is 2.1 μm. Adapted from [104].

Figure 24
Figure 24

Measured phase value with respect to different sample size and various values of degree of spatial correlation. T. Nguyen, C. Edwards, L. L. Goddard, and G. Popescu are preparing a manuscript to be called “Quantitative phase imaging with partially coherent illumination,” available from G. Popescu, gpopescu@illinois.edu.

Figure 25
Figure 25

RBC membrane fluctuation measurement with wDPM. (a) RBC thickness map. (b) Generated binary mask. (c) Standard deviation map. Color bar in (a) shows thickness in micrometers and that in (c) shows standard deviation in nanometers. The mean of the standard deviation in the masked area is 30.9 nm.

Figure 26
Figure 26

Beating cardiomyocyte imaging with wDPM. (a) Phase image. (b) Phase profile along line AB. (c) Phase profile along line CD. Color bar shows phase in radians.

Figure 27
Figure 27

Cell growth measurement with wDPM. (a) and (b) Time-lapse quantitative phase images of an isolated HeLa cell at T=2h and T=16h. (c) Variation of dry mass with time for the HeLa cell during its growth. Color bars represent phase in radians. Adapted from [54].

Figure 28
Figure 28

Experimental setup of sDPM. Inset a: scanning filters on the SLM. Inset b: first-order diffraction spectrum (outer envelope) and spectra at different wavelengths.

Figure 29
Figure 29

Reconstructed phase maps in radians at different wavelengths and dispersion curves of microbeads and red blood cells. (a) 3 μm beads in Zeiss immersion oil. (b) Red blood cell. Measured (dashed line) and expected (solid line) dispersion curves. The error bars indicate standard deviation (N=10). Adapted from [117].

Figure 30
Figure 30

Phase reconstruction procedure of an off-axis quantitative phase imaging system.

Figure 31
Figure 31

Flow chart of the unwrapping step.

Figure 32
Figure 32

Overview of the blood-testing instrument. The white-light diffraction phase microscope is used as the imaging system. Different CUDA modules perform phase reconstruction and image analysis to calculate several morphological parameters of individual RBCs. The output information is in the form of arrays of numbers and can be easily transmitted for remote diagnosis.

Figure 33
Figure 33

Morphological parameter distribution of RBCs of a healthy patient (N=6181 cells) and patients with microcytic (N=8442 cells) and macrocytic (N=4535 cells) anemias. (a) Cell volume and comparison of measured data versus hospital data. (b) RBC surface area distribution of the three patients. Adapted from Pham et al., “Real time blood testing using quantitative phase imaging,” PLOS One 8, e55676 (2013) [128].

Figure 34
Figure 34

Examples of output distributions of other morphological parameters. (a) Sphericity index. (b) MCD. Adapted from Pham et al., “Real time blood testing using quantitative phase imaging,” PLOS One 8, e55676 (2013) [128].

Tables (2)

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Table 1. DPM Design Equations

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Table 2. CUDA Implementation versus C-Based Sequential Implementation

Equations (52)

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UGP(x,y)=U0(x,y)+U1(x,y)eiβx,
U˜FP(kx,ky)=U˜0(kx,ky)+U˜1(kxβ,ky),
kx=2πx1/(λf1)αx1,ky=2πy1/(λf1)αy1,
β=2π/Λ=2πΔx/(λf1)αΔx,Δx=λf1/Λ,
U˜FP+(αx1,αy1)=U˜0(αx1,αy1)+U˜1(αx1β,αy1)*δ(αx1β,αy1)=U˜0(αx1,αy1)+U˜1(0,0)δ(αx1β,αy1).
F.T{U˜FP+(αx1,αy1)}=F.T{U˜0(αx1,αy1)+U˜1(0,0)δ(αx1β,αy1)}=1|α|(U0(ξ/α,η/α)+U˜1(0,0)eiξβ/α),
ξ2πx/λf2,η2πy/λf2.
UCP(x,y)=1|α|[U0(x/M4f,y/M4f)+U1(0,0)eiβx/M4f],
M4ff2/f1,
U0(x,y)A0(x,y)eiϕ0(x,y),U1(x,y)A1(x,y)eiϕ1(x,y).
UCP(x,y)=1|α|(A0(x/M4f,y/M4f)eiϕ0(x/M4f,y/M4f)+A1(0,0)eiβx/M4feiϕ1(0,0)).
UCP(x,y)=A0(x,y)eiϕ0(x,y)+A1eiβxeiϕ1.
ICP(x,y)=UCP(x,y)UCP*(x,y)=|A0(x,y)|2+|A1|2+2|A1||A0(x,y)|cos(βx+Δϕ),
ICP(x,y)=I0(x,y)+I1+2I0(x,y)I1cos(βx+Δϕ),
Δϕ=ϕ0(x,y)ϕ1.
Δρ=1.22λ(NAobj+NAcon)1.22λNAobj,
I˜CP(kx,ky)=FT[ICP(x,y)]=FT[U0(x,y)U0*(x,y)+U1U1*+U0(x,y)U1*eiβx+U0*(x,y)U1eiβx]=U˜0(kx,ky)U˜0*(kx,ky)+|U1|2δ(kx,ky)+U˜0(kx,ky)U1*δ(kxβ,ky)+U˜0*(kx,ky)U1δ(kx+β,ky).
β3k0NAobj,
ΛλMobj3NAobj.
ΛΔρMobj3.66.
ks2kmax=2(β+k0NAobj),
ks=2πaMobj|M4f|2(β+k0NAobj),
|M4f|2a[1Λ+1λNAobjMobj].
|M4f|8a3Λ,
FOV=[m,n]aMobj|M4f|.
sinθm=mλΛ.
tanθm=Δxf1.
tanθmsinθmθm.
Δx=f1λΛ.
NA1λΛ+NAobjMobj.
I(x,y)=I0[2J1(πDρ/λf2)πDρ/λf2]2,
πDρλf2=3.83,
ρ=3.83λf2πD=1.22λf2D.
d=am2+n2,
ρ=1.22λf2Dγd2,
D2.44λf2γd.
NAL2λ|M4f|Λ+1.22λD.
NAL2λ|M4f|Λ+1.22λγD.
η=1.22λMobj|M4f|am2+n2NAobj.
η=ΔρFOVdiagonal.
h(x,y)=φ(x,y)λ02πΔn.
I(x,y)=Ib(x,y)+γ(x,y)cos[ϕ(x,y)+βx],
β=2πsinθ/λ,
I(x,y)x=Ib(x,y)x+cos[ϕ(x,y)+βx]γ(x,y)xγ(x,y)sin[ϕ(x,y)+βx][ϕ(x,y)x+β].
Ibx0,γx0,ϕxβ,
I=I(x,y)x=γβsin[ϕ(x,y)+βx].
I=2I(x,y)x2=γβ2cos[ϕ(x,y)+βx].
ϕ(x,y)=arg(I+ikI)βx.
ke(x,y)=kBT/Δh(x,y)2,
ϕ(k,ω)=α|Aδφ(r,t)ei(ωtk·r)dtd2r|2,
ϕ(x,y)=ϕo(x,y)arg[(exp(iϕo)h)(x,y)],
Δϕ(x,y;λ)=k0[β(λ)C(x,y)+Δnws(λ)]h(x,y),

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