Abstract

We present a high-speed holographic microscopic technique for quantitative measurement of polarization light-field, referred to as polarization holographic microscopy (PHM). Employing the principle of common-path interferometry, PHM quantitatively measures the spatially resolved Jones matrix components of anisotropic samples with only two consecutive measurements of spatially modulated holograms. We demonstrate the features of PHM with imaging the dynamics of liquid crystal droplets at a video-rate.

©2012 Optical Society of America

1. Introduction

Polarization is a fundamental property of light that describes the orientation of an oscillating electric field (E-field). Since the polarization of light is sensitive to birefringence, polarization light microscopy has been widely used for probing molecular structures and orientations in anisotropic materials or induced birefringence due to an externally applied field such as magnetic, electric and/or mechanical. The applications of polarization light microscopy span the fields of mineralogy, chemistry, soft matter physics, and cell biology.

Qualitative polarization light microscopy is widely used in practice with numerous applications. However, quantitative polarization light microscopy has been primarily employed to probe relatively thick and non-transparent samples such as minerals. This is because quantitative polarization light microscopy has been largely limited to intensity-only measurements, and the Stokes–Muller formalism, an intensity-based framework describing the polarization properties of material, has been extensively used [1,2]. However, Stokes vectors and Mueller matrices do not describe the field nature of light polarization; for example, small retardation phases or interference effect in thin and transparent samples such as biological samples are suitable to be effectively addressed. Generally polarization of light can be described in field-based formalism with Jones matrices with complex vectors [3]. In general, coherent light must be treated with Jones formalism rather than Stokes formalism because the former works with complex the amplitude of the E-field rather than intensity.

Recently, quantitative phase imaging or digital holographic microscopy techniques have been reported to quantitatively measure polarization sensitive phase information from birefringent samples [410]. However, the Jones matrix information could not be extracted; the previous polarization sensitive holographic measurements have been thus used to retrieve birefringent phase [4,810] and Strokes parameters [57]. More recently, Wang et al. reported Jones phase microscopy that can extract the Jones matrix information from a thin sample using an interferometric microscopy technique [11]. However, Jones phase microscopy is not suitable for studying dynamics in birefringent samples since four measurements with different polarization states are required to extract one spatially resolved Jones matrix map.

Here, we present a novel field-based polarization microscopy technique, referred to as polarization holographic microscopy (PHM), capable of extracting the spatially resolved Jones matrix associated with anisotropic samples. PHM employs the principle of common-path interferometry to quantitatively retrieve both the amplitude and phase information of a polarized light field. More importantly, the spatially resolved Jones matrix information can be measured at high speed. We demonstrate the features of PHM with quantitatively imaging the dynamics of an individual LC droplet at a video rate.

2. Methods

2.1 Polarization holographic microscopy

The setup of PHM is shown in Fig. 1 . Incident light on a sample can be repeatedly changed between two orthogonal states of polarization and the transmitted light is split in two orthogonal polarization states and then holographically recorded in common-path geometry.

 figure: Fig. 1

Fig. 1 Illustration of the experimental configuration for PHM. PBS, polarized beam splitter; OC, optical chopper; HWP, half waveplate; P, polarizer; SF, spatial filter (extended in the inset as a front and a side views). The arrows indicate to the polarization direction of the beam.

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An unpolarized He-Ne laser (5mW, Thorlabs Inc.) was employed as an illumination source. The laser beam is first spatially filtered (not shown in Fig. 1). In order to repeatedly alternate between orthogonal polarization states from the unpolarized laser, we used a pair of polarized beam splitters (PBSs) and a custom-made optical chopper (OC,) with a servo motor system. At PBS1, the collimated input beam is first split into two orthogonal polarization states (linearly polarized beam along the vertical and horizontal axis) and then combined into the same optical axis by PBS2. The OC located between PBS1 and PBS2, selects only one polarization state at a time; half of the rotating wheel at the OC is an open slot allowing the beam to pass and the other half is a close slot blocking the beam, the combined beam then passes a half wave plate (HWP) and the polarization directions are rotated to −45° and + 45°.

The gated polarized beam then impinges into the target sample. We employed an inverted microscope (IX71, Olympus Inc.) to place the sample. The transmitted light field is first projected to the image plane 1 (IP1) and then the beam is guided to the IP2 with magnification via a 4-f imaging system. At IP2, a two-dimensional transmission grating is located to duplicate the light-field information in two independent spatial frequencies. To simultaneously obtain both vertical and horizontal polarization components of the E-field induced from the sample, the 1st order diffracted beams in both horizontal and vertical directions are used. Two linear polarizers (P1, P2) with the transmission axes of the vertical and horizontal are set at the Fourier plane; the 1st order vertically diffracted beam passes the P1 and the 1st order horizontally diffracted beam passes the P2. The 0th order diffracted beam acts as a reference beam; the beam is spatially filtered by a 25-μm-diameter pinhole at the Fourier plane and converted into a clean plane wave at the detector plane. The 0th order reference beam independently interferes with the 1st order diffracted beams. The interference between the reference beam (the 0th order diffraction) and two sample beams (the 1st order horizontal diffraction [horizontally aligned analyzer] and the 1st order vertical diffraction [vertically aligned analyzer]) results in an interferogram with two-dimensional modulating fringe patterns.

The interferogram is measured by a high-speed CMOS camera (FASTCAM 1024 PCI, Photron Inc.) which has 1024 × 1024 pixels with 17 µm pixel pitch that corresponds to 32 nm in the sample plane considering the total magnification of the imaging system. The common-path interferometric geometry is used; both sample and reference arms share identical optical paths before the IP2; two arms pass the same optical components with minimal lateral displacement after the IP2 to the detector. Thus, the phase noise resulting from mechanical vibration, temperature gradient, and air fluctuation can be mutually canceled upon interference. The high stability of the common-path interferometry has previously been validated in the quantitative phase imaging [12,13].

2.2 Preparation of nematic liquid crystal droplet

Dispersed LC droplets, 4'-pentyl-4-biphenylcarbonitrile (5CB, Sigma-Aldrich), were prepared in water with a small amount of sodium dodecyl sulfate (SDS, Fluka) added to induce homeotropic anchoring of LC molecules and to stabilize the interface [14,15]. To make small droplets of several microns in diameter, we thoroughly agitated the sample by shaking and pipetting. The 5CB droplets are in the nematic phase at room temperature and have a radial configuration due to the homeotropic anchoring and spherical confinement [16]. All the measurements are performed at room temperature and ambient pressure.

3. Results

Successively measured interferograms of a single LC droplet with the nematic phase are shown in Fig. 2(a) with elapsed time and the polarization direction of the incident beam. The field nature of light polarization can be fully formalized by the Jones matrix – Jones vector. The Jones matrix of an arbitrary sample is described as:

J_=[J11J12J21J22],
With complex-valued components and the subscripts represent the polarization states of incoming and outgoing beams, respectively. To extract the Jones matrix from the sample, at least four independent sets on different polarization states with respect to the input and output beams should be measured.

 figure: Fig. 2

Fig. 2 (a) Successively measured interferograms of a single liquid crystal droplet. Elapsed time and polarization direction of the incident beam are noted. Scale bar, 5 µm. Illustrations of polarization states right after (b) the He-Ne laser and (c) PBS2 in Fig. 1.

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Using the polarized beam splitters and optical chopper in the experimental setup, successively alternating orthogonally polarized beams can be obtained from an unpolarized continuous laser [Figs. 2(b)-2(c)]. The rotation speed of the optical chopper is synchronized with the acquisition speed of the camera. Two consecutive interferograms with different orthogonal polarization states can be used for reconstructing one spatially resolved Jones matrix map from the sample. Thus, the shown interferogram can provide the dynamics Jones matrix information every 32 ms. We note that the acquisition speed of the present technique is only limited to the camera speed as long as the optical chopper rotation is synchronized.

In the PHM configuration, the polarization state of the beam after the PBS2 can be mathematically described using Jones vectors as follows,

[Ex(t)Ey(t)]=n=0[δ(t2nTT)[10]+δ(t2nT)[01]]
where δ() is a delta function and T is a time interval between two orthogonal polarization states. After passing the HWP, the directions of the linearly polarized beams are rotated such that the two linearly polarized beams with + 45° and −45° polarization angles are used as input beams, whose Jones vectors can be expressed as:
E+45=c1(11), E45=c2(11),
where c1, c2 are real-values constants that can be obtained from measurement without the sample.

Each input beam with different polarization states (E+45E45) passes through the sample. For each polarization input beam, the field information of the sample is decomposed into two independent polarization states as the beam passes through the two spatially-decoupled orthogonal analyzers (P1, P2), which are measured at the CCD plane simultaneously. The representative data are shown in Fig. 3 . Two holograms of a single LC droplet that were successively measured with different input polarization states are shown in Figs. 2(a)-2(b). The zoom-in image of the hologram [inset, Fig. 3(a)] clearly shows the two-dimensional fringe patterns in the hologram. Figures 3(c)-3(d) show the Fourier spectra of the measured interferograms [Figs. 3(a)-3(b)] obtained via 2D Fourier transformation. Each Fourier spectrum shows that the polarization dependent field information, passing different orientated analyzers, is spatially decoupled. From the two spatially modulated two-dimensional holograms [Figs. 3(a)-3(b)], four independent polarization sensitive field information maps, Yij can be retrieved with the appropriate field-retrieval algorithm.

 figure: Fig. 3

Fig. 3 Experimental results for a nematic LC droplet. (a,b) Interferograms of a nematic LC droplet measured with linearly polarized beams oscillating in + 45° and −45°, respectively. (c,d) Logarithmic amplitudes of Fourier spectra from the measured interferograms (a,b) via 2D Fourier transformation. The indicated areas of Y11, Y21 and Y12, Y21 correspond to Fourier spectra that pass the horizontally and vertically aligned linear polarizers, respectively. (e) Amplitude and phase maps of the sample associated with Y11, Y12, Y21, and Y22, respectively. The arrows in (a,b) represent the polarization direction of the incident beam.

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The details on the field retrieval process with different algorithms can be found elsewhere [17,18]. Yij (i,j = 1,2) indicates the complex field of the sample with the ith polarization states of the incident beam and jth polarization states of the analyzer orientation. For example, Y11 represents the sample field information that was measured with a linear polarized incident beam with a + 45° oscillating angle, and measured with the horizontally aligned polarizer. Figure 3(e) shows the polarization sensitive E-field maps to be used to reconstruct one spatially resolved Jones matrix map. Note that two polarization-dependent images of the LC droplet with lateral displacement can be seen in Figs. 3(a)-3(b); they result from slight angle misalignment of P1,2, which can be conveniently corrected by numerical shifting of the images in the final E-field images.

The relation between the Jones matrix component and the measured polarization sensitive field information can be represented as:

[Y11Y12Y21Y22]=[c1c100c2c20000c1c100c2c2][J11J12J21J22].
To normalize the intensity of the input beam by finding the real-values constants c1, c2, the E-fields of the sample are divided by the E-fields measured without the sample. The complex-values components of the Jones matrix map can then be retrieved from the measured polarization dependent holograms Yij (i,j = 1,2) using Eq. (4). The results are shown in Fig. 4 (see also Media 1). The amplitude and phase maps of the Jones matrix components are shown in Fig. 4(a) and Fig. 4(b), respectively. The background area, corresponding to isotropic fluid, shows values of 1 for the |J11| and |J22| components, but values of 0 for the |J21| and |J21| components. The sample area shows complex but symmetric patterns for the Jones matrix components. The amplitude images of cross-polarization configurations, |J11| or |J22|, are consistent with the images that can be obtained via intensity-based cross-polarization imaging. Note that cross-polarization imaging which is extensively used for imaging LC droplets only address one components of the Jones matrix, |J11| or |J22|, and cannot investigate the other seven components of the Jones matrix information. It takes less than 100 ms to compute for extracting the complex Jones matrix with the size of 512 × 512 pixels on a typical desktop computer (Intel® Core i7-870 CPU, 2.93GHz) with a custom MatLab® script.

 figure: Fig. 4

Fig. 4 (Media 1) Amplitude (a) and phase (b) maps of Jones matrix components, J11, J12, J21, J22, measured from a nematic LC droplet.

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4. Discussions and conclusions

A liquid crystal (LC) is a classic example of soft matter, exhibiting diverse degrees of orientational and positional ordering of anisotropic molecules, and thereby shows strong birefringence [19]. LCs are found ubiquitously and related to both fundamental and technological problems ranging from biological membranes to liquid crystal displays. Due to this importance, many optical imaging techniques including cross-polarization microscopy and fluorescence confocal polarizing microscopy have been used to study the director field of LCs [20,21]. Recently, nonlinear optical imaging techniques have enabled three-dimensional (3D) label-free imaging of the director field [2229]. Despite substantial progress, the development of imaging techniques for the study of LCs has been stymied by fundamental limitations; in particular, most of the conventional imaging techniques are not suitable to directly retrieve the Jones matrix components of dynamically changing LCs.

The present technique, PHM makes it possible, for the first time to our knowledge, to quantitatively investigate the spatio-temporal information of Jones matrix components of an anisotropic sample. Four independent polarization sensitive E-field maps can be acquired in two consecutively measured interferograms at high speed, from which the spatially resolved Jones matrix map can be retrieved. We demonstrate the retrieval of the Jones matrix information at a speed of 31 Hz, but higher speed can easily be achieved by increasing the acquisition speed of the detector. The Jones matrix – Jones vector formalism is a general formalism for light polarization based on the light field. The spatially resolved Jones matrix measured here can be readily converted into Muller matrices, but the converse is not possible. Also noteworthy is that the present PHM setup is based on common-path interferometry and the instrument should exhibit high stability against phase noise. The high temporal stability of the common-path interferometry will be advantageous especially when highly sensitive E-field measurements are required over a long period of time.

In this manuscript we demonstrate the proof of this principle of PHM through simple applications to the imaging of dynamics of individual nematic LC droplets in vitro. This new technique is sufficiently broad and general to allow for potential applications in soft matter, cell biology, biophysics, biochemistry, and geology, wherever imaging birefringence from anisotropic material can have an impact. For example, dynamics of structural and phase changes of liquid crystal molecules including nematic, smectic A, and chiral phases can be directly addressed with PHM. Dynamics of the molecular architectures of several birefringent biological molecules are also direct applications of PHM: structures and dynamics of cytoskeletal filaments and collagen fiber networks [3032]; birefringent hemoglobin S proteins and its implications in sickle cell diseases and malaria infection [3337]; the local molecular orientations in lipid and cell membranes and its biomechanical properties [3841].

Acknowledgments

This work was supported by KAIST (N10110038, N10110048, G04100075), KAIST Institute for Optical Science and Technology, the Korean Ministry of Education, Science and Technology (MEST) grant No. 2009-0087691 (BRL) and grant No. R33-2008-000-10163-0 (WCU), the Korean Ministry for Health, Welfare & Family Affairs grant No. A040041 (the Korea Healthcare technology R&D Project), National Research Foundation (NRF - 2011 - 355 - c00039). YKP acknowledges support from POSCO TJ Park Fellowship. The authors thank to Prof. Yong-Hee Lee and Prof. Byung Yoon Kim for stimulating discussions.

References and links

1. F. Massoumian, R. Juškaitis, M. A. Neil, and T. Wilson, “Quantitative polarized light microscopy,” J. Microsc. 209(1), 13–22 (2003). [CrossRef]   [PubMed]  

2. S. Ross, R. Newton, Y. M. Zhou, J. Haffegee, M. W. Ho, J. Bolton, and D. Knight, “Quantitative image analysis of birefringent biological material,” J. Microsc. 187(1), 62–67 (1997). [CrossRef]  

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5. T. Nomura, B. Javidi, S. Murata, E. Nitanai, and T. Numata, “Polarization imaging of a 3D object by use of on-axis phase-shifting digital holography,” Opt. Lett. 32(5), 481–483 (2007). [CrossRef]   [PubMed]  

6. T. Tahara, Y. Awatsuji, Y. Shimozato, T. Kakue, K. Nishio, S. Ura, T. Kubota, and O. Matoba, “Single-shot polarization-imaging digital holography based on simultaneous phase-shifting interferometry,” Opt. Lett. 36(16), 3254–3256 (2011). [CrossRef]   [PubMed]  

7. R. K. Singh, D. N. Naik, H. Itou, Y. Miyamoto, and M. Takeda, “Stokes holography,” Opt. Lett. 37(5), 966–968 (2012). [CrossRef]   [PubMed]  

8. J. R. Kuhn, Z. Wu, and M. Poenie, “Modulated polarization microscopy: a promising new approach to visualizing cytoskeletal dynamics in living cells,” Biophys. J. 80(2), 972–985 (2001). [CrossRef]   [PubMed]  

9. I. H. Shin, S. M. Shin, and D. Y. Kim, “New, simple theory-based, accurate polarization microscope for birefringence imaging of biological cells,” J. Biomed. Opt. 15(1), 016028 (2010). [CrossRef]   [PubMed]  

10. N. M. Dragomir, X. M. Goh, C. L. Curl, L. M. Delbridge, and A. Roberts, “Quantitative polarized phase microscopy for birefringence imaging,” Opt. Express 15(26), 17690–17698 (2007). [CrossRef]   [PubMed]  

11. Z. Wang, L. J. Millet, M. U. Gillette, and G. Popescu, “Jones phase microscopy of transparent and anisotropic samples,” Opt. Lett. 33(11), 1270–1272 (2008). [CrossRef]   [PubMed]  

12. G. Popescu, T. Ikeda, R. R. Dasari, and M. S. Feld, “Diffraction phase microscopy for quantifying cell structure and dynamics,” Opt. Lett. 31(6), 775–777 (2006). [CrossRef]   [PubMed]  

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

14. P. Poulin and D. A. Weitz, “Inverted and multiple nematic emulsions,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 57(1), 626–637 (1998). [CrossRef]  

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36. M. Aidoo, D. J. Terlouw, M. S. Kolczak, P. D. McElroy, F. O. ter Kuile, S. Kariuki, B. L. Nahlen, A. A. Lal, and V. Udhayakumar, “Protective effects of the sickle cell gene against malaria morbidity and mortality,” Lancet 359(9314), 1311–1312 (2002). [CrossRef]   [PubMed]  

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39. Y. 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. U.S.A. 107(15), 6731–6736 (2010). [CrossRef]   [PubMed]  

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References

  • View by:

  1. F. Massoumian, R. Juškaitis, M. A. Neil, and T. Wilson, “Quantitative polarized light microscopy,” J. Microsc. 209(1), 13–22 (2003).
    [Crossref] [PubMed]
  2. S. Ross, R. Newton, Y. M. Zhou, J. Haffegee, M. W. Ho, J. Bolton, and D. Knight, “Quantitative image analysis of birefringent biological material,” J. Microsc. 187(1), 62–67 (1997).
    [Crossref]
  3. R. C. Jones, “A new calculus for the treatment of optical systems. IV,” J. Opt. Soc. Am. 32(8), 486–493 (1942).
    [Crossref]
  4. T. Colomb, F. Dürr, E. Cuche, P. Marquet, H. G. Limberger, R. P. Salathé, and C. Depeursinge, “Polarization microscopy by use of digital holography: application to optical-fiber birefringence measurements,” Appl. Opt. 44(21), 4461–4469 (2005).
    [Crossref] [PubMed]
  5. T. Nomura, B. Javidi, S. Murata, E. Nitanai, and T. Numata, “Polarization imaging of a 3D object by use of on-axis phase-shifting digital holography,” Opt. Lett. 32(5), 481–483 (2007).
    [Crossref] [PubMed]
  6. T. Tahara, Y. Awatsuji, Y. Shimozato, T. Kakue, K. Nishio, S. Ura, T. Kubota, and O. Matoba, “Single-shot polarization-imaging digital holography based on simultaneous phase-shifting interferometry,” Opt. Lett. 36(16), 3254–3256 (2011).
    [Crossref] [PubMed]
  7. R. K. Singh, D. N. Naik, H. Itou, Y. Miyamoto, and M. Takeda, “Stokes holography,” Opt. Lett. 37(5), 966–968 (2012).
    [Crossref] [PubMed]
  8. J. R. Kuhn, Z. Wu, and M. Poenie, “Modulated polarization microscopy: a promising new approach to visualizing cytoskeletal dynamics in living cells,” Biophys. J. 80(2), 972–985 (2001).
    [Crossref] [PubMed]
  9. I. H. Shin, S. M. Shin, and D. Y. Kim, “New, simple theory-based, accurate polarization microscope for birefringence imaging of biological cells,” J. Biomed. Opt. 15(1), 016028 (2010).
    [Crossref] [PubMed]
  10. N. M. Dragomir, X. M. Goh, C. L. Curl, L. M. Delbridge, and A. Roberts, “Quantitative polarized phase microscopy for birefringence imaging,” Opt. Express 15(26), 17690–17698 (2007).
    [Crossref] [PubMed]
  11. Z. Wang, L. J. Millet, M. U. Gillette, and G. Popescu, “Jones phase microscopy of transparent and anisotropic samples,” Opt. Lett. 33(11), 1270–1272 (2008).
    [Crossref] [PubMed]
  12. G. Popescu, T. Ikeda, R. R. Dasari, and M. S. Feld, “Diffraction phase microscopy for quantifying cell structure and dynamics,” Opt. Lett. 31(6), 775–777 (2006).
    [Crossref] [PubMed]
  13. G. Popescu, Y. K. Park, K. Badizadegan, R. R. Dasari, and M. S. Feld, “Diffraction phase and fluorescence microscopy,” Opt. Express 14(18), 8263–8268 (2006).
    [Crossref] [PubMed]
  14. P. Poulin and D. A. Weitz, “Inverted and multiple nematic emulsions,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 57(1), 626–637 (1998).
    [Crossref]
  15. T. Tixier, M. Heppenstall-Butler, and E. M. Terentjev, “Spontaneous size selection in cholesteric and nematic emulsions,” Langmuir 22(5), 2365–2370 (2006).
    [Crossref] [PubMed]
  16. P. Drzaic, Liquid Crystal Dispersions (World Scientific, Singapore, 1995).
  17. G. Popescu, Quantitative Phase Imaging of Cells and Tissues (McGraw-Hill Professional, 2011).
  18. S. K. Debnath and Y. Park, “Real-time quantitative phase imaging with a spatial phase-shifting algorithm,” Opt. Lett. 36(23), 4677–4679 (2011).
    [Crossref] [PubMed]
  19. P. G. De Gennes and J. Prost, The Physics of Liquid Crystals (Clarendon Press, Oxford, 1993).
  20. I. I. Smalyukh, S. V. Shiyanovskii, and O. D. Lavrentovich, “Three-dimensional imaging of orientational order by Fluorescence confocal polarizing microscopy,” Chem. Phys. Lett. 336(1-2), 88–96 (2001).
    [Crossref]
  21. T. Lee, B. Senyuk, R. P. Trivedi, and I. I. Smalyukh, “Optical microscopy of soft matter systems,” arXiv:1108.3287 (2011).
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  23. R. S. Pillai, M. Oh-E, H. Yokoyama, G. J. Brakenhoff, and M. Müller, “Imaging colloidal particle induced topological defects in a nematic liquid crystal using third harmonic generation microscopy,” Opt. Express 14(26), 12976–12983 (2006).
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  24. A. V. Kachynski, A. N. Kuzmin, P. N. Prasad, and I. I. Smalyukh, “Coherent anti-Stokes Raman scattering polarized microscopy of three-dimensional director structures in liquid crystals,” Appl. Phys. Lett. 91(15), 151905 (2007).
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  25. D. A. Higgins and B. J. Luther, “Watching molecules reorient in liquid crystal droplets with multiphoton-excited fluorescence microscopy,” J. Chem. Phys. 119(7), 3935–3942 (2003).
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  26. B. G. Saar, H.-S. Park, X. S. Xie, and O. D. Lavrentovich, “Three-dimensional imaging of chemical bond orientation in liquid crystals by coherent anti- Stokes Raman scattering microscopy,” Opt. Express 15(21), 13585–13596 (2007).
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  27. B.-C. Chen and S.-H. Lim, “Three-dimensional imaging of director field orientations in liquid crystals by polarized four-wave mixing microscopy,” Appl. Phys. Lett. 94(17), 171911 (2009).
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  28. T. Lee, R. P. Trivedi, and I. I. Smalyukh, “Multimodal nonlinear optical polarizing microscopy of long-range molecular order in liquid crystals,” Opt. Lett. 35(20), 3447–3449 (2010).
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  29. R. P. Trivedi, T. Lee, K. A. Bertness, and I. I. Smalyukh, “Three dimensional optical manipulation and structural imaging of soft materials by use of laser tweezers and multimodal nonlinear microscopy,” Opt. Express 18(26), 27658–27669 (2010).
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  30. K. Katoh, K. Hammar, P. J. S. Smith, and R. Oldenbourg, “Birefringence imaging directly reveals architectural dynamics of filamentous actin in living growth cones,” Mol. Biol. Cell 10(1), 197–210 (1999).
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  31. R. Oldenbourg, “A new view on polarization microscopy,” Nature 381(6585), 811–812 (1996).
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  32. M. D. Shoulders and R. T. Raines, “Collagen structure and stability,” Annu. Rev. Biochem. 78(1), 929–958 (2009).
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  34. G. A. Barabino, M. O. Platt, and D. K. Kaul, “Sickle cell biomechanics,” Annu. Rev. Biomed. Eng. 12(1), 345–367 (2010).
    [Crossref] [PubMed]
  35. Y. Kim, J. M. Higgins, R. R. Dasari, S. Suresh, and Y. K. Park, “Anisotropic light scattering of individual sickle red blood cells,” J. Biomed. Opt. 17(4), 040501 (2012).
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  36. M. Aidoo, D. J. Terlouw, M. S. Kolczak, P. D. McElroy, F. O. ter Kuile, S. Kariuki, B. L. Nahlen, A. A. Lal, and V. Udhayakumar, “Protective effects of the sickle cell gene against malaria morbidity and mortality,” Lancet 359(9314), 1311–1312 (2002).
    [Crossref] [PubMed]
  37. S. Cho, S. Kim, Y. Kim, and Y. K. Park, “Optical imaging techniques for the study of malaria,” Trends Biotechnol. 30(2), 71–79 (2012).
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  40. Y. Park, M. Diez-Silva, D. Fu, G. Popescu, W. Choi, I. Barman, S. Suresh, and M. S. Feld, “Static and dynamic light scattering of healthy and malaria-parasite invaded red blood cells,” J. Biomed. Opt. 15(2), 020506 (2010).
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2012 (3)

R. K. Singh, D. N. Naik, H. Itou, Y. Miyamoto, and M. Takeda, “Stokes holography,” Opt. Lett. 37(5), 966–968 (2012).
[Crossref] [PubMed]

Y. Kim, J. M. Higgins, R. R. Dasari, S. Suresh, and Y. K. Park, “Anisotropic light scattering of individual sickle red blood cells,” J. Biomed. Opt. 17(4), 040501 (2012).
[Crossref]

S. Cho, S. Kim, Y. Kim, and Y. K. Park, “Optical imaging techniques for the study of malaria,” Trends Biotechnol. 30(2), 71–79 (2012).
[Crossref] [PubMed]

2011 (2)

2010 (7)

I. H. Shin, S. M. Shin, and D. Y. Kim, “New, simple theory-based, accurate polarization microscope for birefringence imaging of biological cells,” J. Biomed. Opt. 15(1), 016028 (2010).
[Crossref] [PubMed]

T. Lee, R. P. Trivedi, and I. I. Smalyukh, “Multimodal nonlinear optical polarizing microscopy of long-range molecular order in liquid crystals,” Opt. Lett. 35(20), 3447–3449 (2010).
[Crossref] [PubMed]

R. P. Trivedi, T. Lee, K. A. Bertness, and I. I. Smalyukh, “Three dimensional optical manipulation and structural imaging of soft materials by use of laser tweezers and multimodal nonlinear microscopy,” Opt. Express 18(26), 27658–27669 (2010).
[Crossref] [PubMed]

Y. 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. U.S.A. 107(15), 6731–6736 (2010).
[Crossref] [PubMed]

Y. Park, M. Diez-Silva, D. Fu, G. Popescu, W. Choi, I. Barman, S. Suresh, and M. S. Feld, “Static and dynamic light scattering of healthy and malaria-parasite invaded red blood cells,” J. Biomed. Opt. 15(2), 020506 (2010).
[Crossref] [PubMed]

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. U.S.A. 107(4), 1289–1294 (2010).
[Crossref] [PubMed]

G. A. Barabino, M. O. Platt, and D. K. Kaul, “Sickle cell biomechanics,” Annu. Rev. Biomed. Eng. 12(1), 345–367 (2010).
[Crossref] [PubMed]

2009 (3)

A. Gasecka, T. J. Han, C. Favard, B. R. Cho, and S. Brasselet, “Quantitative imaging of molecular order in lipid membranes using two-photon fluorescence polarimetry,” Biophys. J. 97(10), 2854–2862 (2009).
[Crossref] [PubMed]

B.-C. Chen and S.-H. Lim, “Three-dimensional imaging of director field orientations in liquid crystals by polarized four-wave mixing microscopy,” Appl. Phys. Lett. 94(17), 171911 (2009).
[Crossref]

M. D. Shoulders and R. T. Raines, “Collagen structure and stability,” Annu. Rev. Biochem. 78(1), 929–958 (2009).
[Crossref] [PubMed]

2008 (1)

2007 (5)

2006 (4)

2005 (2)

T. Colomb, F. Dürr, E. Cuche, P. Marquet, H. G. Limberger, R. P. Salathé, and C. Depeursinge, “Polarization microscopy by use of digital holography: application to optical-fiber birefringence measurements,” Appl. Opt. 44(21), 4461–4469 (2005).
[Crossref] [PubMed]

K. Yoshiki, M. Hashimoto, and T. Araki, “Second-harmonic-generation microscopy using excitation beam with controlled polarization pattern to determine three-dimensional molecular orientation,” Jpn. J. Appl. Phys. 44(34), L1066–L1068 (2005).
[Crossref]

2003 (2)

D. A. Higgins and B. J. Luther, “Watching molecules reorient in liquid crystal droplets with multiphoton-excited fluorescence microscopy,” J. Chem. Phys. 119(7), 3935–3942 (2003).
[Crossref]

F. Massoumian, R. Juškaitis, M. A. Neil, and T. Wilson, “Quantitative polarized light microscopy,” J. Microsc. 209(1), 13–22 (2003).
[Crossref] [PubMed]

2002 (1)

M. Aidoo, D. J. Terlouw, M. S. Kolczak, P. D. McElroy, F. O. ter Kuile, S. Kariuki, B. L. Nahlen, A. A. Lal, and V. Udhayakumar, “Protective effects of the sickle cell gene against malaria morbidity and mortality,” Lancet 359(9314), 1311–1312 (2002).
[Crossref] [PubMed]

2001 (2)

J. R. Kuhn, Z. Wu, and M. Poenie, “Modulated polarization microscopy: a promising new approach to visualizing cytoskeletal dynamics in living cells,” Biophys. J. 80(2), 972–985 (2001).
[Crossref] [PubMed]

I. I. Smalyukh, S. V. Shiyanovskii, and O. D. Lavrentovich, “Three-dimensional imaging of orientational order by Fluorescence confocal polarizing microscopy,” Chem. Phys. Lett. 336(1-2), 88–96 (2001).
[Crossref]

1999 (1)

K. Katoh, K. Hammar, P. J. S. Smith, and R. Oldenbourg, “Birefringence imaging directly reveals architectural dynamics of filamentous actin in living growth cones,” Mol. Biol. Cell 10(1), 197–210 (1999).
[PubMed]

1998 (1)

P. Poulin and D. A. Weitz, “Inverted and multiple nematic emulsions,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 57(1), 626–637 (1998).
[Crossref]

1997 (1)

S. Ross, R. Newton, Y. M. Zhou, J. Haffegee, M. W. Ho, J. Bolton, and D. Knight, “Quantitative image analysis of birefringent biological material,” J. Microsc. 187(1), 62–67 (1997).
[Crossref]

1996 (1)

R. Oldenbourg, “A new view on polarization microscopy,” Nature 381(6585), 811–812 (1996).
[Crossref] [PubMed]

1942 (1)

Aidoo, M.

M. Aidoo, D. J. Terlouw, M. S. Kolczak, P. D. McElroy, F. O. ter Kuile, S. Kariuki, B. L. Nahlen, A. A. Lal, and V. Udhayakumar, “Protective effects of the sickle cell gene against malaria morbidity and mortality,” Lancet 359(9314), 1311–1312 (2002).
[Crossref] [PubMed]

Araki, T.

K. Yoshiki, M. Hashimoto, and T. Araki, “Second-harmonic-generation microscopy using excitation beam with controlled polarization pattern to determine three-dimensional molecular orientation,” Jpn. J. Appl. Phys. 44(34), L1066–L1068 (2005).
[Crossref]

Atweh, G. F.

P. S. Frenette and G. F. Atweh, “Sickle cell disease: old discoveries, new concepts, and future promise,” J. Clin. Invest. 117(4), 850–858 (2007).
[Crossref] [PubMed]

Auth, T.

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. U.S.A. 107(4), 1289–1294 (2010).
[Crossref] [PubMed]

Awatsuji, Y.

Badizadegan, K.

Y. 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. U.S.A. 107(15), 6731–6736 (2010).
[Crossref] [PubMed]

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

Barabino, G. A.

G. A. Barabino, M. O. Platt, and D. K. Kaul, “Sickle cell biomechanics,” Annu. Rev. Biomed. Eng. 12(1), 345–367 (2010).
[Crossref] [PubMed]

Barman, I.

Y. Park, M. Diez-Silva, D. Fu, G. Popescu, W. Choi, I. Barman, S. Suresh, and M. S. Feld, “Static and dynamic light scattering of healthy and malaria-parasite invaded red blood cells,” J. Biomed. Opt. 15(2), 020506 (2010).
[Crossref] [PubMed]

Bertness, K. A.

Best, C. A.

Y. 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. U.S.A. 107(15), 6731–6736 (2010).
[Crossref] [PubMed]

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. U.S.A. 107(4), 1289–1294 (2010).
[Crossref] [PubMed]

Bolton, J.

S. Ross, R. Newton, Y. M. Zhou, J. Haffegee, M. W. Ho, J. Bolton, and D. Knight, “Quantitative image analysis of birefringent biological material,” J. Microsc. 187(1), 62–67 (1997).
[Crossref]

Brakenhoff, G. J.

Brasselet, S.

A. Gasecka, T. J. Han, C. Favard, B. R. Cho, and S. Brasselet, “Quantitative imaging of molecular order in lipid membranes using two-photon fluorescence polarimetry,” Biophys. J. 97(10), 2854–2862 (2009).
[Crossref] [PubMed]

Chen, B.-C.

B.-C. Chen and S.-H. Lim, “Three-dimensional imaging of director field orientations in liquid crystals by polarized four-wave mixing microscopy,” Appl. Phys. Lett. 94(17), 171911 (2009).
[Crossref]

Cho, B. R.

A. Gasecka, T. J. Han, C. Favard, B. R. Cho, and S. Brasselet, “Quantitative imaging of molecular order in lipid membranes using two-photon fluorescence polarimetry,” Biophys. J. 97(10), 2854–2862 (2009).
[Crossref] [PubMed]

Cho, S.

S. Cho, S. Kim, Y. Kim, and Y. K. Park, “Optical imaging techniques for the study of malaria,” Trends Biotechnol. 30(2), 71–79 (2012).
[Crossref] [PubMed]

Choi, W.

Y. Park, M. Diez-Silva, D. Fu, G. Popescu, W. Choi, I. Barman, S. Suresh, and M. S. Feld, “Static and dynamic light scattering of healthy and malaria-parasite invaded red blood cells,” J. Biomed. Opt. 15(2), 020506 (2010).
[Crossref] [PubMed]

Colomb, T.

Cuche, E.

Curl, C. L.

Dasari, R. R.

Y. Kim, J. M. Higgins, R. R. Dasari, S. Suresh, and Y. K. Park, “Anisotropic light scattering of individual sickle red blood cells,” J. Biomed. Opt. 17(4), 040501 (2012).
[Crossref]

Y. 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. U.S.A. 107(15), 6731–6736 (2010).
[Crossref] [PubMed]

G. Popescu, T. Ikeda, R. R. Dasari, and M. S. Feld, “Diffraction phase microscopy for quantifying cell structure and dynamics,” Opt. Lett. 31(6), 775–777 (2006).
[Crossref] [PubMed]

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

Debnath, S. K.

Delbridge, L. M.

Depeursinge, C.

Diez-Silva, M.

Y. Park, M. Diez-Silva, D. Fu, G. Popescu, W. Choi, I. Barman, S. Suresh, and M. S. Feld, “Static and dynamic light scattering of healthy and malaria-parasite invaded red blood cells,” J. Biomed. Opt. 15(2), 020506 (2010).
[Crossref] [PubMed]

Dragomir, N. M.

Dürr, F.

Favard, C.

A. Gasecka, T. J. Han, C. Favard, B. R. Cho, and S. Brasselet, “Quantitative imaging of molecular order in lipid membranes using two-photon fluorescence polarimetry,” Biophys. J. 97(10), 2854–2862 (2009).
[Crossref] [PubMed]

Feld, M. S.

Y. 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. U.S.A. 107(15), 6731–6736 (2010).
[Crossref] [PubMed]

Y. Park, M. Diez-Silva, D. Fu, G. Popescu, W. Choi, I. Barman, S. Suresh, and M. S. Feld, “Static and dynamic light scattering of healthy and malaria-parasite invaded red blood cells,” J. Biomed. Opt. 15(2), 020506 (2010).
[Crossref] [PubMed]

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. U.S.A. 107(4), 1289–1294 (2010).
[Crossref] [PubMed]

G. Popescu, T. Ikeda, R. R. Dasari, and M. S. Feld, “Diffraction phase microscopy for quantifying cell structure and dynamics,” Opt. Lett. 31(6), 775–777 (2006).
[Crossref] [PubMed]

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

Frenette, P. S.

P. S. Frenette and G. F. Atweh, “Sickle cell disease: old discoveries, new concepts, and future promise,” J. Clin. Invest. 117(4), 850–858 (2007).
[Crossref] [PubMed]

Fu, D.

Y. Park, M. Diez-Silva, D. Fu, G. Popescu, W. Choi, I. Barman, S. Suresh, and M. S. Feld, “Static and dynamic light scattering of healthy and malaria-parasite invaded red blood cells,” J. Biomed. Opt. 15(2), 020506 (2010).
[Crossref] [PubMed]

Gasecka, A.

A. Gasecka, T. J. Han, C. Favard, B. R. Cho, and S. Brasselet, “Quantitative imaging of molecular order in lipid membranes using two-photon fluorescence polarimetry,” Biophys. J. 97(10), 2854–2862 (2009).
[Crossref] [PubMed]

Gillette, M. U.

Goh, X. M.

Gov, N. S.

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. U.S.A. 107(4), 1289–1294 (2010).
[Crossref] [PubMed]

Haffegee, J.

S. Ross, R. Newton, Y. M. Zhou, J. Haffegee, M. W. Ho, J. Bolton, and D. Knight, “Quantitative image analysis of birefringent biological material,” J. Microsc. 187(1), 62–67 (1997).
[Crossref]

Hammar, K.

K. Katoh, K. Hammar, P. J. S. Smith, and R. Oldenbourg, “Birefringence imaging directly reveals architectural dynamics of filamentous actin in living growth cones,” Mol. Biol. Cell 10(1), 197–210 (1999).
[PubMed]

Han, T. J.

A. Gasecka, T. J. Han, C. Favard, B. R. Cho, and S. Brasselet, “Quantitative imaging of molecular order in lipid membranes using two-photon fluorescence polarimetry,” Biophys. J. 97(10), 2854–2862 (2009).
[Crossref] [PubMed]

Hashimoto, M.

K. Yoshiki, M. Hashimoto, and T. Araki, “Second-harmonic-generation microscopy using excitation beam with controlled polarization pattern to determine three-dimensional molecular orientation,” Jpn. J. Appl. Phys. 44(34), L1066–L1068 (2005).
[Crossref]

Henle, M. L.

Y. 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. U.S.A. 107(15), 6731–6736 (2010).
[Crossref] [PubMed]

Heppenstall-Butler, M.

T. Tixier, M. Heppenstall-Butler, and E. M. Terentjev, “Spontaneous size selection in cholesteric and nematic emulsions,” Langmuir 22(5), 2365–2370 (2006).
[Crossref] [PubMed]

Higgins, D. A.

D. A. Higgins and B. J. Luther, “Watching molecules reorient in liquid crystal droplets with multiphoton-excited fluorescence microscopy,” J. Chem. Phys. 119(7), 3935–3942 (2003).
[Crossref]

Higgins, J. M.

Y. Kim, J. M. Higgins, R. R. Dasari, S. Suresh, and Y. K. Park, “Anisotropic light scattering of individual sickle red blood cells,” J. Biomed. Opt. 17(4), 040501 (2012).
[Crossref]

Ho, M. W.

S. Ross, R. Newton, Y. M. Zhou, J. Haffegee, M. W. Ho, J. Bolton, and D. Knight, “Quantitative image analysis of birefringent biological material,” J. Microsc. 187(1), 62–67 (1997).
[Crossref]

Ikeda, T.

Itou, H.

Javidi, B.

Jones, R. C.

Juškaitis, R.

F. Massoumian, R. Juškaitis, M. A. Neil, and T. Wilson, “Quantitative polarized light microscopy,” J. Microsc. 209(1), 13–22 (2003).
[Crossref] [PubMed]

Kachynski, A. V.

A. V. Kachynski, A. N. Kuzmin, P. N. Prasad, and I. I. Smalyukh, “Coherent anti-Stokes Raman scattering polarized microscopy of three-dimensional director structures in liquid crystals,” Appl. Phys. Lett. 91(15), 151905 (2007).
[Crossref]

Kakue, T.

Kariuki, S.

M. Aidoo, D. J. Terlouw, M. S. Kolczak, P. D. McElroy, F. O. ter Kuile, S. Kariuki, B. L. Nahlen, A. A. Lal, and V. Udhayakumar, “Protective effects of the sickle cell gene against malaria morbidity and mortality,” Lancet 359(9314), 1311–1312 (2002).
[Crossref] [PubMed]

Katoh, K.

K. Katoh, K. Hammar, P. J. S. Smith, and R. Oldenbourg, “Birefringence imaging directly reveals architectural dynamics of filamentous actin in living growth cones,” Mol. Biol. Cell 10(1), 197–210 (1999).
[PubMed]

Kaul, D. K.

G. A. Barabino, M. O. Platt, and D. K. Kaul, “Sickle cell biomechanics,” Annu. Rev. Biomed. Eng. 12(1), 345–367 (2010).
[Crossref] [PubMed]

Kim, D. Y.

I. H. Shin, S. M. Shin, and D. Y. Kim, “New, simple theory-based, accurate polarization microscope for birefringence imaging of biological cells,” J. Biomed. Opt. 15(1), 016028 (2010).
[Crossref] [PubMed]

Kim, S.

S. Cho, S. Kim, Y. Kim, and Y. K. Park, “Optical imaging techniques for the study of malaria,” Trends Biotechnol. 30(2), 71–79 (2012).
[Crossref] [PubMed]

Kim, Y.

S. Cho, S. Kim, Y. Kim, and Y. K. Park, “Optical imaging techniques for the study of malaria,” Trends Biotechnol. 30(2), 71–79 (2012).
[Crossref] [PubMed]

Y. Kim, J. M. Higgins, R. R. Dasari, S. Suresh, and Y. K. Park, “Anisotropic light scattering of individual sickle red blood cells,” J. Biomed. Opt. 17(4), 040501 (2012).
[Crossref]

Knight, D.

S. Ross, R. Newton, Y. M. Zhou, J. Haffegee, M. W. Ho, J. Bolton, and D. Knight, “Quantitative image analysis of birefringent biological material,” J. Microsc. 187(1), 62–67 (1997).
[Crossref]

Kolczak, M. S.

M. Aidoo, D. J. Terlouw, M. S. Kolczak, P. D. McElroy, F. O. ter Kuile, S. Kariuki, B. L. Nahlen, A. A. Lal, and V. Udhayakumar, “Protective effects of the sickle cell gene against malaria morbidity and mortality,” Lancet 359(9314), 1311–1312 (2002).
[Crossref] [PubMed]

Kubota, T.

Kuhn, J. R.

J. R. Kuhn, Z. Wu, and M. Poenie, “Modulated polarization microscopy: a promising new approach to visualizing cytoskeletal dynamics in living cells,” Biophys. J. 80(2), 972–985 (2001).
[Crossref] [PubMed]

Kuriabova, T.

Y. 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. U.S.A. 107(15), 6731–6736 (2010).
[Crossref] [PubMed]

Kuzmin, A. N.

A. V. Kachynski, A. N. Kuzmin, P. N. Prasad, and I. I. Smalyukh, “Coherent anti-Stokes Raman scattering polarized microscopy of three-dimensional director structures in liquid crystals,” Appl. Phys. Lett. 91(15), 151905 (2007).
[Crossref]

Lal, A. A.

M. Aidoo, D. J. Terlouw, M. S. Kolczak, P. D. McElroy, F. O. ter Kuile, S. Kariuki, B. L. Nahlen, A. A. Lal, and V. Udhayakumar, “Protective effects of the sickle cell gene against malaria morbidity and mortality,” Lancet 359(9314), 1311–1312 (2002).
[Crossref] [PubMed]

Lavrentovich, O. D.

B. G. Saar, H.-S. Park, X. S. Xie, and O. D. Lavrentovich, “Three-dimensional imaging of chemical bond orientation in liquid crystals by coherent anti- Stokes Raman scattering microscopy,” Opt. Express 15(21), 13585–13596 (2007).
[Crossref] [PubMed]

I. I. Smalyukh, S. V. Shiyanovskii, and O. D. Lavrentovich, “Three-dimensional imaging of orientational order by Fluorescence confocal polarizing microscopy,” Chem. Phys. Lett. 336(1-2), 88–96 (2001).
[Crossref]

Lee, T.

Levine, A. J.

Y. 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. U.S.A. 107(15), 6731–6736 (2010).
[Crossref] [PubMed]

Lim, S.-H.

B.-C. Chen and S.-H. Lim, “Three-dimensional imaging of director field orientations in liquid crystals by polarized four-wave mixing microscopy,” Appl. Phys. Lett. 94(17), 171911 (2009).
[Crossref]

Limberger, H. G.

Luther, B. J.

D. A. Higgins and B. J. Luther, “Watching molecules reorient in liquid crystal droplets with multiphoton-excited fluorescence microscopy,” J. Chem. Phys. 119(7), 3935–3942 (2003).
[Crossref]

Marquet, P.

Massoumian, F.

F. Massoumian, R. Juškaitis, M. A. Neil, and T. Wilson, “Quantitative polarized light microscopy,” J. Microsc. 209(1), 13–22 (2003).
[Crossref] [PubMed]

Matoba, O.

McElroy, P. D.

M. Aidoo, D. J. Terlouw, M. S. Kolczak, P. D. McElroy, F. O. ter Kuile, S. Kariuki, B. L. Nahlen, A. A. Lal, and V. Udhayakumar, “Protective effects of the sickle cell gene against malaria morbidity and mortality,” Lancet 359(9314), 1311–1312 (2002).
[Crossref] [PubMed]

Millet, L. J.

Miyamoto, Y.

Müller, M.

Murata, S.

Nahlen, B. L.

M. Aidoo, D. J. Terlouw, M. S. Kolczak, P. D. McElroy, F. O. ter Kuile, S. Kariuki, B. L. Nahlen, A. A. Lal, and V. Udhayakumar, “Protective effects of the sickle cell gene against malaria morbidity and mortality,” Lancet 359(9314), 1311–1312 (2002).
[Crossref] [PubMed]

Naik, D. N.

Neil, M. A.

F. Massoumian, R. Juškaitis, M. A. Neil, and T. Wilson, “Quantitative polarized light microscopy,” J. Microsc. 209(1), 13–22 (2003).
[Crossref] [PubMed]

Newton, R.

S. Ross, R. Newton, Y. M. Zhou, J. Haffegee, M. W. Ho, J. Bolton, and D. Knight, “Quantitative image analysis of birefringent biological material,” J. Microsc. 187(1), 62–67 (1997).
[Crossref]

Nishio, K.

Nitanai, E.

Nomura, T.

Numata, T.

Oh-E, M.

Oldenbourg, R.

K. Katoh, K. Hammar, P. J. S. Smith, and R. Oldenbourg, “Birefringence imaging directly reveals architectural dynamics of filamentous actin in living growth cones,” Mol. Biol. Cell 10(1), 197–210 (1999).
[PubMed]

R. Oldenbourg, “A new view on polarization microscopy,” Nature 381(6585), 811–812 (1996).
[Crossref] [PubMed]

Park, H.-S.

Park, Y.

S. K. Debnath and Y. Park, “Real-time quantitative phase imaging with a spatial phase-shifting algorithm,” Opt. Lett. 36(23), 4677–4679 (2011).
[Crossref] [PubMed]

Y. Park, M. Diez-Silva, D. Fu, G. Popescu, W. Choi, I. Barman, S. Suresh, and M. S. Feld, “Static and dynamic light scattering of healthy and malaria-parasite invaded red blood cells,” J. Biomed. Opt. 15(2), 020506 (2010).
[Crossref] [PubMed]

Y. 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. U.S.A. 107(15), 6731–6736 (2010).
[Crossref] [PubMed]

Park, Y. K.

S. Cho, S. Kim, Y. Kim, and Y. K. Park, “Optical imaging techniques for the study of malaria,” Trends Biotechnol. 30(2), 71–79 (2012).
[Crossref] [PubMed]

Y. Kim, J. M. Higgins, R. R. Dasari, S. Suresh, and Y. K. Park, “Anisotropic light scattering of individual sickle red blood cells,” J. Biomed. Opt. 17(4), 040501 (2012).
[Crossref]

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. U.S.A. 107(4), 1289–1294 (2010).
[Crossref] [PubMed]

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

Pillai, R. S.

Platt, M. O.

G. A. Barabino, M. O. Platt, and D. K. Kaul, “Sickle cell biomechanics,” Annu. Rev. Biomed. Eng. 12(1), 345–367 (2010).
[Crossref] [PubMed]

Poenie, M.

J. R. Kuhn, Z. Wu, and M. Poenie, “Modulated polarization microscopy: a promising new approach to visualizing cytoskeletal dynamics in living cells,” Biophys. J. 80(2), 972–985 (2001).
[Crossref] [PubMed]

Popescu, G.

Y. 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. U.S.A. 107(15), 6731–6736 (2010).
[Crossref] [PubMed]

Y. Park, M. Diez-Silva, D. Fu, G. Popescu, W. Choi, I. Barman, S. Suresh, and M. S. Feld, “Static and dynamic light scattering of healthy and malaria-parasite invaded red blood cells,” J. Biomed. Opt. 15(2), 020506 (2010).
[Crossref] [PubMed]

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. U.S.A. 107(4), 1289–1294 (2010).
[Crossref] [PubMed]

Z. Wang, L. J. Millet, M. U. Gillette, and G. Popescu, “Jones phase microscopy of transparent and anisotropic samples,” Opt. Lett. 33(11), 1270–1272 (2008).
[Crossref] [PubMed]

G. Popescu, T. Ikeda, R. R. Dasari, and M. S. Feld, “Diffraction phase microscopy for quantifying cell structure and dynamics,” Opt. Lett. 31(6), 775–777 (2006).
[Crossref] [PubMed]

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

Poulin, P.

P. Poulin and D. A. Weitz, “Inverted and multiple nematic emulsions,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 57(1), 626–637 (1998).
[Crossref]

Prasad, P. N.

A. V. Kachynski, A. N. Kuzmin, P. N. Prasad, and I. I. Smalyukh, “Coherent anti-Stokes Raman scattering polarized microscopy of three-dimensional director structures in liquid crystals,” Appl. Phys. Lett. 91(15), 151905 (2007).
[Crossref]

Raines, R. T.

M. D. Shoulders and R. T. Raines, “Collagen structure and stability,” Annu. Rev. Biochem. 78(1), 929–958 (2009).
[Crossref] [PubMed]

Roberts, A.

Ross, S.

S. Ross, R. Newton, Y. M. Zhou, J. Haffegee, M. W. Ho, J. Bolton, and D. Knight, “Quantitative image analysis of birefringent biological material,” J. Microsc. 187(1), 62–67 (1997).
[Crossref]

Saar, B. G.

Safran, S. A.

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. U.S.A. 107(4), 1289–1294 (2010).
[Crossref] [PubMed]

Salathé, R. P.

Shimozato, Y.

Shin, I. H.

I. H. Shin, S. M. Shin, and D. Y. Kim, “New, simple theory-based, accurate polarization microscope for birefringence imaging of biological cells,” J. Biomed. Opt. 15(1), 016028 (2010).
[Crossref] [PubMed]

Shin, S. M.

I. H. Shin, S. M. Shin, and D. Y. Kim, “New, simple theory-based, accurate polarization microscope for birefringence imaging of biological cells,” J. Biomed. Opt. 15(1), 016028 (2010).
[Crossref] [PubMed]

Shiyanovskii, S. V.

I. I. Smalyukh, S. V. Shiyanovskii, and O. D. Lavrentovich, “Three-dimensional imaging of orientational order by Fluorescence confocal polarizing microscopy,” Chem. Phys. Lett. 336(1-2), 88–96 (2001).
[Crossref]

Shoulders, M. D.

M. D. Shoulders and R. T. Raines, “Collagen structure and stability,” Annu. Rev. Biochem. 78(1), 929–958 (2009).
[Crossref] [PubMed]

Singh, R. K.

Smalyukh, I. I.

T. Lee, R. P. Trivedi, and I. I. Smalyukh, “Multimodal nonlinear optical polarizing microscopy of long-range molecular order in liquid crystals,” Opt. Lett. 35(20), 3447–3449 (2010).
[Crossref] [PubMed]

R. P. Trivedi, T. Lee, K. A. Bertness, and I. I. Smalyukh, “Three dimensional optical manipulation and structural imaging of soft materials by use of laser tweezers and multimodal nonlinear microscopy,” Opt. Express 18(26), 27658–27669 (2010).
[Crossref] [PubMed]

A. V. Kachynski, A. N. Kuzmin, P. N. Prasad, and I. I. Smalyukh, “Coherent anti-Stokes Raman scattering polarized microscopy of three-dimensional director structures in liquid crystals,” Appl. Phys. Lett. 91(15), 151905 (2007).
[Crossref]

I. I. Smalyukh, S. V. Shiyanovskii, and O. D. Lavrentovich, “Three-dimensional imaging of orientational order by Fluorescence confocal polarizing microscopy,” Chem. Phys. Lett. 336(1-2), 88–96 (2001).
[Crossref]

Smith, P. J. S.

K. Katoh, K. Hammar, P. J. S. Smith, and R. Oldenbourg, “Birefringence imaging directly reveals architectural dynamics of filamentous actin in living growth cones,” Mol. Biol. Cell 10(1), 197–210 (1999).
[PubMed]

Suresh, S.

Y. Kim, J. M. Higgins, R. R. Dasari, S. Suresh, and Y. K. Park, “Anisotropic light scattering of individual sickle red blood cells,” J. Biomed. Opt. 17(4), 040501 (2012).
[Crossref]

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. U.S.A. 107(4), 1289–1294 (2010).
[Crossref] [PubMed]

Y. Park, M. Diez-Silva, D. Fu, G. Popescu, W. Choi, I. Barman, S. Suresh, and M. S. Feld, “Static and dynamic light scattering of healthy and malaria-parasite invaded red blood cells,” J. Biomed. Opt. 15(2), 020506 (2010).
[Crossref] [PubMed]

Tahara, T.

Takeda, M.

ter Kuile, F. O.

M. Aidoo, D. J. Terlouw, M. S. Kolczak, P. D. McElroy, F. O. ter Kuile, S. Kariuki, B. L. Nahlen, A. A. Lal, and V. Udhayakumar, “Protective effects of the sickle cell gene against malaria morbidity and mortality,” Lancet 359(9314), 1311–1312 (2002).
[Crossref] [PubMed]

Terentjev, E. M.

T. Tixier, M. Heppenstall-Butler, and E. M. Terentjev, “Spontaneous size selection in cholesteric and nematic emulsions,” Langmuir 22(5), 2365–2370 (2006).
[Crossref] [PubMed]

Terlouw, D. J.

M. Aidoo, D. J. Terlouw, M. S. Kolczak, P. D. McElroy, F. O. ter Kuile, S. Kariuki, B. L. Nahlen, A. A. Lal, and V. Udhayakumar, “Protective effects of the sickle cell gene against malaria morbidity and mortality,” Lancet 359(9314), 1311–1312 (2002).
[Crossref] [PubMed]

Tixier, T.

T. Tixier, M. Heppenstall-Butler, and E. M. Terentjev, “Spontaneous size selection in cholesteric and nematic emulsions,” Langmuir 22(5), 2365–2370 (2006).
[Crossref] [PubMed]

Trivedi, R. P.

Udhayakumar, V.

M. Aidoo, D. J. Terlouw, M. S. Kolczak, P. D. McElroy, F. O. ter Kuile, S. Kariuki, B. L. Nahlen, A. A. Lal, and V. Udhayakumar, “Protective effects of the sickle cell gene against malaria morbidity and mortality,” Lancet 359(9314), 1311–1312 (2002).
[Crossref] [PubMed]

Ura, S.

Wang, Z.

Weitz, D. A.

P. Poulin and D. A. Weitz, “Inverted and multiple nematic emulsions,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 57(1), 626–637 (1998).
[Crossref]

Wilson, T.

F. Massoumian, R. Juškaitis, M. A. Neil, and T. Wilson, “Quantitative polarized light microscopy,” J. Microsc. 209(1), 13–22 (2003).
[Crossref] [PubMed]

Wu, Z.

J. R. Kuhn, Z. Wu, and M. Poenie, “Modulated polarization microscopy: a promising new approach to visualizing cytoskeletal dynamics in living cells,” Biophys. J. 80(2), 972–985 (2001).
[Crossref] [PubMed]

Xie, X. S.

Yokoyama, H.

Yoshiki, K.

K. Yoshiki, M. Hashimoto, and T. Araki, “Second-harmonic-generation microscopy using excitation beam with controlled polarization pattern to determine three-dimensional molecular orientation,” Jpn. J. Appl. Phys. 44(34), L1066–L1068 (2005).
[Crossref]

Zhou, Y. M.

S. Ross, R. Newton, Y. M. Zhou, J. Haffegee, M. W. Ho, J. Bolton, and D. Knight, “Quantitative image analysis of birefringent biological material,” J. Microsc. 187(1), 62–67 (1997).
[Crossref]

Annu. Rev. Biochem. (1)

M. D. Shoulders and R. T. Raines, “Collagen structure and stability,” Annu. Rev. Biochem. 78(1), 929–958 (2009).
[Crossref] [PubMed]

Annu. Rev. Biomed. Eng. (1)

G. A. Barabino, M. O. Platt, and D. K. Kaul, “Sickle cell biomechanics,” Annu. Rev. Biomed. Eng. 12(1), 345–367 (2010).
[Crossref] [PubMed]

Appl. Opt. (1)

Appl. Phys. Lett. (2)

A. V. Kachynski, A. N. Kuzmin, P. N. Prasad, and I. I. Smalyukh, “Coherent anti-Stokes Raman scattering polarized microscopy of three-dimensional director structures in liquid crystals,” Appl. Phys. Lett. 91(15), 151905 (2007).
[Crossref]

B.-C. Chen and S.-H. Lim, “Three-dimensional imaging of director field orientations in liquid crystals by polarized four-wave mixing microscopy,” Appl. Phys. Lett. 94(17), 171911 (2009).
[Crossref]

Biophys. J. (2)

J. R. Kuhn, Z. Wu, and M. Poenie, “Modulated polarization microscopy: a promising new approach to visualizing cytoskeletal dynamics in living cells,” Biophys. J. 80(2), 972–985 (2001).
[Crossref] [PubMed]

A. Gasecka, T. J. Han, C. Favard, B. R. Cho, and S. Brasselet, “Quantitative imaging of molecular order in lipid membranes using two-photon fluorescence polarimetry,” Biophys. J. 97(10), 2854–2862 (2009).
[Crossref] [PubMed]

Chem. Phys. Lett. (1)

I. I. Smalyukh, S. V. Shiyanovskii, and O. D. Lavrentovich, “Three-dimensional imaging of orientational order by Fluorescence confocal polarizing microscopy,” Chem. Phys. Lett. 336(1-2), 88–96 (2001).
[Crossref]

J. Biomed. Opt. (3)

Y. Kim, J. M. Higgins, R. R. Dasari, S. Suresh, and Y. K. Park, “Anisotropic light scattering of individual sickle red blood cells,” J. Biomed. Opt. 17(4), 040501 (2012).
[Crossref]

I. H. Shin, S. M. Shin, and D. Y. Kim, “New, simple theory-based, accurate polarization microscope for birefringence imaging of biological cells,” J. Biomed. Opt. 15(1), 016028 (2010).
[Crossref] [PubMed]

Y. Park, M. Diez-Silva, D. Fu, G. Popescu, W. Choi, I. Barman, S. Suresh, and M. S. Feld, “Static and dynamic light scattering of healthy and malaria-parasite invaded red blood cells,” J. Biomed. Opt. 15(2), 020506 (2010).
[Crossref] [PubMed]

J. Chem. Phys. (1)

D. A. Higgins and B. J. Luther, “Watching molecules reorient in liquid crystal droplets with multiphoton-excited fluorescence microscopy,” J. Chem. Phys. 119(7), 3935–3942 (2003).
[Crossref]

J. Clin. Invest. (1)

P. S. Frenette and G. F. Atweh, “Sickle cell disease: old discoveries, new concepts, and future promise,” J. Clin. Invest. 117(4), 850–858 (2007).
[Crossref] [PubMed]

J. Microsc. (2)

F. Massoumian, R. Juškaitis, M. A. Neil, and T. Wilson, “Quantitative polarized light microscopy,” J. Microsc. 209(1), 13–22 (2003).
[Crossref] [PubMed]

S. Ross, R. Newton, Y. M. Zhou, J. Haffegee, M. W. Ho, J. Bolton, and D. Knight, “Quantitative image analysis of birefringent biological material,” J. Microsc. 187(1), 62–67 (1997).
[Crossref]

J. Opt. Soc. Am. (1)

Jpn. J. Appl. Phys. (1)

K. Yoshiki, M. Hashimoto, and T. Araki, “Second-harmonic-generation microscopy using excitation beam with controlled polarization pattern to determine three-dimensional molecular orientation,” Jpn. J. Appl. Phys. 44(34), L1066–L1068 (2005).
[Crossref]

Lancet (1)

M. Aidoo, D. J. Terlouw, M. S. Kolczak, P. D. McElroy, F. O. ter Kuile, S. Kariuki, B. L. Nahlen, A. A. Lal, and V. Udhayakumar, “Protective effects of the sickle cell gene against malaria morbidity and mortality,” Lancet 359(9314), 1311–1312 (2002).
[Crossref] [PubMed]

Langmuir (1)

T. Tixier, M. Heppenstall-Butler, and E. M. Terentjev, “Spontaneous size selection in cholesteric and nematic emulsions,” Langmuir 22(5), 2365–2370 (2006).
[Crossref] [PubMed]

Mol. Biol. Cell (1)

K. Katoh, K. Hammar, P. J. S. Smith, and R. Oldenbourg, “Birefringence imaging directly reveals architectural dynamics of filamentous actin in living growth cones,” Mol. Biol. Cell 10(1), 197–210 (1999).
[PubMed]

Nature (1)

R. Oldenbourg, “A new view on polarization microscopy,” Nature 381(6585), 811–812 (1996).
[Crossref] [PubMed]

Opt. Express (5)

Opt. Lett. (7)

Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics (1)

P. Poulin and D. A. Weitz, “Inverted and multiple nematic emulsions,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 57(1), 626–637 (1998).
[Crossref]

Proc. Natl. Acad. Sci. U.S.A. (2)

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. U.S.A. 107(4), 1289–1294 (2010).
[Crossref] [PubMed]

Y. 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. U.S.A. 107(15), 6731–6736 (2010).
[Crossref] [PubMed]

Trends Biotechnol. (1)

S. Cho, S. Kim, Y. Kim, and Y. K. Park, “Optical imaging techniques for the study of malaria,” Trends Biotechnol. 30(2), 71–79 (2012).
[Crossref] [PubMed]

Other (4)

T. Lee, B. Senyuk, R. P. Trivedi, and I. I. Smalyukh, “Optical microscopy of soft matter systems,” arXiv:1108.3287 (2011).

P. G. De Gennes and J. Prost, The Physics of Liquid Crystals (Clarendon Press, Oxford, 1993).

P. Drzaic, Liquid Crystal Dispersions (World Scientific, Singapore, 1995).

G. Popescu, Quantitative Phase Imaging of Cells and Tissues (McGraw-Hill Professional, 2011).

Supplementary Material (1)

Media 1: MOV (597 KB)     

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Figures (4)

Fig. 1
Fig. 1 Illustration of the experimental configuration for PHM. PBS, polarized beam splitter; OC, optical chopper; HWP, half waveplate; P, polarizer; SF, spatial filter (extended in the inset as a front and a side views). The arrows indicate to the polarization direction of the beam.
Fig. 2
Fig. 2 (a) Successively measured interferograms of a single liquid crystal droplet. Elapsed time and polarization direction of the incident beam are noted. Scale bar, 5 µm. Illustrations of polarization states right after (b) the He-Ne laser and (c) PBS2 in Fig. 1.
Fig. 3
Fig. 3 Experimental results for a nematic LC droplet. (a,b) Interferograms of a nematic LC droplet measured with linearly polarized beams oscillating in + 45° and −45°, respectively. (c,d) Logarithmic amplitudes of Fourier spectra from the measured interferograms (a,b) via 2D Fourier transformation. The indicated areas of Y11, Y21 and Y12, Y21 correspond to Fourier spectra that pass the horizontally and vertically aligned linear polarizers, respectively. (e) Amplitude and phase maps of the sample associated with Y11, Y12, Y21, and Y22, respectively. The arrows in (a,b) represent the polarization direction of the incident beam.
Fig. 4
Fig. 4 (Media 1) Amplitude (a) and phase (b) maps of Jones matrix components, J11, J12, J21, J22, measured from a nematic LC droplet.

Equations (4)

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

J _ =[ J 11 J 12 J 21 J 22 ],
[ E x ( t ) E y ( t ) ]= n=0 [ δ( t2nTT )[ 1 0 ]+δ( t2nT )[ 0 1 ] ]
E +45 = c 1 ( 1 1 ),  E 45 = c 2 ( 1 1 ),
[ Y 11 Y 12 Y 21 Y 22 ]=[ c 1 c 1 0 0 c 2 c 2 0 0 0 0 c 1 c 1 0 0 c 2 c 2 ][ J 11 J 12 J 21 J 22 ].

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