A novel application of quantitative phase imaging under linearly polarized light is introduced for studying unstained anisotropic live cells. The method is first validated as a technique for mapping the twodimensional retardation distribution of a well-characterized optical fiber and is then applied to the characterization of unstained isolated cardiac cells. The experimental retardation measurements are in very good agreement with the established Brace-Köhler method, and additionally provide spatially resolved cell birefringence and phase data.
©2007 Optical Society of America
Various imaging-based measurement techniques are capable of delivering information in the form of phase [1, 2]. Historically, study of translucent specimens has been achieved using phase imaging methods and polarized microscopy to extract anisotropy information [1, 3]. Anisotropy is generated by local structural order existent at very small scale including at the level of atomic bonds and molecular shape in the specimen’s structure. Specimen birefringence characterizes the differential speed of propagation between two orthogonal polarized states of light, which is also a direct result of the existence of more than one index of refraction. There is considerable interest in spatially quantifying the parameters describing optical anisotropy [4–8]. Many living biological specimens [5–9] as well as photonic devices [9–11] exhibit low-level birefringence In such specimens the relative retardation is less than about λ/30, where λ is the wavelength of the light source. The interaction between the polarized light with the molecular organization enables the interpretation of this organization even though the molecular arrangements are below the spatial limit of the light microscope .
Several techniques for determining the optical birefringence have been used including compensator-based [3, 8, 13], polarimetric [5, 6, 14] and interferometric [4, 7, 9, 15] methods. In the case of compensator-based methods, the Brace-Köhler method is recognized as the standard methodology for measuring low-level birefringence . The measurements involve finding a minimum of intensity by rotating a calibrated compensator plate when the sample is observed through crossed-polarizers. Based on a similar principle, a two-wave-plate compensator technique has been introduced for single-point  and full-field  retardation measurements. This method has been introduced to complement the limitations of the Brace-Köhler technique allowing for a null of intensity to be measured. However, the method relies on determining the extinction by fitting the variations of the monochromatic light component in the images determined by fitting variations in the intensity of the transmitted light obtained for different compensator and analyzer settings.
Generally, in the case of conventional polarimetric methods, [5, 6], several settings are required to record a number of images using polarization-analyzing optics, consisting of polarizers, retarders and rotators. In addition due to the complexities involved in the acquisition of several images from which the states of polarization parameters are computed, considerable effort is required for precise control of the polarization-analyzing optics. The microscope system introduced by Oldenbourg and Mei  has been additionally developed  to improve the polarization control by using electro-optic modulators. The technique relies on measuring four intensity images recorded for various birefringence settings of a compensator. Oka and Kaneko  introduced a system of four birefringent wedge prisms to generate multiple fringe patterns having different spatial frequencies to eliminate the need to control the polarization-analyzing optics while recording the information about the state of polarization of interest.
A number of interferometric techniques exist for recording an interferogram resulting from the interference between an object wave and two orthogonal linearly polarized reference waves [9, 15]. Stokes parameters describing the state of polarization are then computed via Fourier transform algorithms. Furthermore, the digital holographic imaging system  involves a Mach-Zender type interferometer similar to that introduced by Ohtsuka and Oka  without using an imaging lens but by recording a hologram in off-axis geometry. The hologram is reconstructed in amplitude and phase for each polarization component by digital reference waves. The technique has been used to demonstrate birefringence measurement of only optical fibers.
Cardiac, skeletal and smooth muscle specimens, which constitute orderly arrays of tethered contractile proteins, have been previously interrogated as biological samples of birefringence interest [16–18]. Muscle histological sections exhibit spatial disturbance of birefringence signal when infective or fibrotic lesions are present [16, 17, 19] and when tissue integrity is modified by grafting or by necrotic damage [18, 20]. In vitro investigations of functional muscle preparations have demonstrated that the birefringence signal is modulated during the course of a single contractile cycle [21, 22]. In cryopreserved muscle sections, increased enzymatic activity has been observed in association with birefringence modulation . These effects are interpreted to reflect a variable degree of Ca-dependent interaction occurring between sarcomeric proteins during excitation-contraction coupling . Information relating to sub-cellular spatial variation in birefringence of single, intact muscle cell specimens has not been previously reported.
Here, the use of a non-interferometric polarized phase-based imaging technique, termed quantitative polarized phase microscopy (QPPM), to examine the low-level birefringence of anisotropic specimens is demonstrated. The method uses polarized, partially coherent illumination and is applied to the two-dimensional (2D) mapping of the retardation of an anisotropic transparent specimen without the use of additional compensators. It involves only the use of a polarizer and a priori knowledge of the optical axis orientation of the specimen.In addition, this birefringence quantification technique allows simultaneous phase and absorption determination. QPPM has previously been utilized to determine the stress profile of optical fibers and fiber devices [10, 11]. In this report we extend our earlier findings in relation to optical fiber birefringence analysis and apply the validated methodology to the examination of single, isolated cardiac muscle cells. The two-dimensional retardation of both the cardiac muscle cell and optical fiber specimens is computed and analyzed. The results are then compared with images obtained using the conventional Brace-Köhler technique of birefringence measurement.
2. Materials and methods
Quantitative Phase Microscopy (QPM)  is a non-interferometric method that can be used with partially coherent light in a conventional bright field microscope to determine the phase shift introduced into an optical wavefield by a transparent specimen. The use of partially coherent illumination allows for high spatial resolution to be achieved . QPM processes a series of through-focal bright field images of a specimen to produce simultaneous but separate information about the absorption and phase . The phase of the specimen is computed using commercial software QPmTm (QPm V2.1 IATIA, Ltd.). In addition to its application to biological specimens [2, 28] the application to reconstructing the three-dimensional refractive index distribution for a series of optical devices [29–31] has been demonstrated.
Quantitative polarized phase microscopy makes use of linearly polarized light that is obtained by the insertion of a single linear polarizer into the optical pathway. The determination of the retardation of a uniaxial specimen implies a priori knowledge about the optical axis orientation and involves taking two phase measurements with incident polarized light at different orientations: parallel, φ||, and perpendicular, φ⊥, to the optic axis respectively. The phase retardation, expressed in radian units, is then computed pixel by pixel according to:
In the case when an homogeneous anisotropic specimen is placed between crossedpolarizers, with its optical axis in the focal plane and aligned at 45° relative to the polarizers’ axes, the transmitted intensity, I(x,y) is proportional to the phase retardation, Γ(x, y) according to :
where I 0(x,y) is the intensity of light in the absence of any polarizing elements.
The Brace-Köhler compensator retardation method , also known as the elliptic compensator method is sufficiently sensitive that retardation measurements down to 5Å have been reported . Using a polarizing microscope the method involves finding a minimum in intensity by rotating a compensator wave-plate when the specimen, oriented at 45° from extinction, is observed through crossed-polarizers. The compensator introduces a retardation equal to that of the studied specimen, but of opposite sign. With the optic axis of the sample oriented at 45° to the polarizer-analyzer axes from extinction and the compensator angle, which is the minimum angle measured from the background extinction position, one can compute the average retardation of the sample, Γs :
where Γc is the maximum retardation of the compensator and θc is the rotation compensator angle. It is important to note that this equation is valid only for the case when the retardation of the sample is smaller than the compensator retardation.
The optical axis orientation of the specimen was determined experimentally following a procedure outlined elsewhere [1, 3, 34]. The sample is placed on the rotatable stage of a polarizing microscope (Olympus BX51-P) and observed under crossed polarisers. The stage is rotated until full extinction is observed; at the position where the sample is aligned with the vibrational axis of the first polariser. From this position the stage is rotated 45° and is thus presented for Brace-Köhler observation. This is achieved by additionally inserting a compensator of known retardation.
Optical fibers provide robust test specimens for studying birefringence measurements techniques , therefore we chose a commercially available optical fiber (Corning Multimode (MM) graded-index 62.5/125 µm core/cladding diameter) for testing the accuracy of the method illustrated here.
Cardiac myocytes were used to demonstrate the application of QPPM to live unstained specimens. The birefringence of a cardiac muscle cell can be most simply understood by considering that it has the same approximate characteristics as a uniaxial positive anisotropic material whose optic axis is considered to be parallel to the long axis of the cell .
Ventricular cardiac myocytes were obtained from hearts of young male Sprague-Dawley rats by an enzymatic digestion procedure (Worthington Type II collagenase; 0.5 mg/ml, 20 min retrograde perfusion). Dissociated myocytes were filtered, washed and re-suspended in HEPES buffered solution (in mM: 118 NaCl, 4.8 KCl, 1.2 MgSO4, 1.2 KH2PO4, 25.0 Na-HEPES, 11.0 glucose with the final pH of 7.4, no added Ca2+). To ensure myocyte quiescence, the myofilament immobilizing agent 2,3-butanedione monoxime (BDM) was included in the incubation solution (25mM). An aliquot of myocyte suspension was placed in a chamber formed between glass slides using two glass fibers as spacers to minimize the introduction of specious tilts into the phase images .
To acquire a phase image using polarized light, a linear polarizer mounted on a calibrated rotation mount was used. Intensity images were captured in transmission with a 12-bit black-and-white 1392×1040 pixel charged coupled device (CCD) (Roper Scientific CoolSNAP HQ, 6.45 µm pixel size) camera mounted on a polarizing microscope (Olympus BX51) using a low numerical aperture (NA) strain free objective lens (Olympus Uplan Fl 40×/0.70 P, ∞/0.17). The extinction of the system was 1×104, with the condenser set at NA=0.2 and the field stop closed to its minimal diameter. The measurements were performed in monochromatic light by insertion of a narrow-band interference filter (λ=546±10 nm) in front of the light source. An effective pixel size of 0.322 µm in the recorded images was obtained by the insertion of a 0.5×relay lens onto the microscope imaging tube. To capture the through-focal series the defocus distance was achieved using a piezoelectric nanopositioning device (PIFOC, Physik Instrumente). Phase images were processed using commercial software QPmTm. Retardation images were computed from phase images recorded at two orthogonal polarization inputs using Eq.(1).
The Brace-Köhler retardation compensator method  was implemented in its traditional form using monochromatic light. A Brace-Köhler rotating compensator (Olympus UCBR-2), having the retardation specified by the manufacturer, (Γc=21.84 nm @ 546 nm), was mounted at the back of the focal plane of the objective. The average point retardation was estimated on the axis of the specimen from four measurements of the compensator angle, in unit degrees and knowledge of the compensator retardation, Γc. The retardation was measured for two types of specimens, having relatively close retardation values: an optical fiber and a number of unstained cardiac muscle cells. Retardation measurements for both samples were performed using Eq. (3), since they were all less than 20 nm and as such, satisfied the lowlevel birefringence condition. Although the measurements using this method are affected by the various prisms in the binocular head, recent analysis of error measurements using a UCBR-2 compensator have been shown to be within reasonable values 0.74 % .
The transverse phase images of the optical fiber shown in Fig. 1(a) and (b) were computed from intensity images recorded at a defocus of ±3 µm and obtained with the polarizer oriented parallel and perpendicular to the axis of the fiber respectively. Because the fiber was immersed in index matching fluid (Cargille oil n D=1.456±0.002), that matched closely to that of the cladding, there were no phase jumps at the interface between the fluid and the cladding of the fiber, i.e. the black background and only the phase due to the core of the fiber is visible at the centre of the fiber.
The two phase images are sufficient to compute the 2D retardation represented in Fig. 1(c). The retardation was computed using simple image arithmetic involving the two phase images according to Eq. (1). A Roberts edge enhancement operator was employed to ensure proper alignment of the images before computing the retardation . The phase difference standard deviation in the designated rectangular area in Fig. 1(c), 71×101 pixels, of ~0.0068 radians (~0.4 degrees) is indicative of the accuracy of the method. Another way to evaluate the accuracy of the method is to use the retardation measurement illustrated in Fig. 1(c) to calculate the transmittance of the light transversely through the fiber and compare it with the transmitted intensity measured through crossed-polarizers. The normalized calculated transmittance, dashed line, was computed from the retardation image illustrated in Fig. 1(c) using Eq. (2). This compares well with the measured transmitted intensity through crossedpolarizers along the same section of the optical fiber, the solid curve, Fig. 2. Error bars shown on the calculated transmittance are representative of standard deviation. It can be clearly seen that the limited use of the dynamic range of the CCD array has a negative impact on the measured transmittance. The average retardation measured using the Brace-Köhler method, 19.75±0.04 nm was in very good agreement with the averaged value of 21.8±0.6 nm, over the core region using the QPPM technique.
Computing the intensity transmitted through the crossed-polarizers we evaluated the error of the QPPM for low retardation specimens. For retardation less than λ/10, as is the case for the fiber investigated, the error measurements were less than 1%. Hence, given the fact that the fiber was placed in an index matching fluid that perfectly matched the refractive index of the cladding (as can be appreciated from the phase images in Fig.1 where only the core of the fiber and not the cladding, is visualized) an excellent agreement between the measured transmitted intensity and the computed one is demonstrated in Fig. 2. Additionally, the standard deviation between the measured and the computed transmittance is ~0.02. Thus the accuracy of the QPPM method for measuring low-level retardation, within λ/10, is verified.
Similar measurements were performed for isolated adult rat cardiac muscle cells. Shown in Fig. 3 are the 2D phase image of an isolated cardiomyocyte for light polarized along the cell (Fig 3(a)) and the 2D retardation computed from phase images using the QPPM technique, Fig. 3(b). The average value of the retardation across the cell obtained with QPPM was 17±1 nm. There is a relatively large error in the retardation measurement using QPPM which can be attributed to the phase mismatch due to the refractive index of the immersing fluid and the cell. However, this value compares very well with the measured value of 18.89±0.04 nm obtained with the Brace-Köhler. Further, the computed 2D retardation, in Fig. 3(b), shows all anisotropic structures expected.
The 2D retardation measurements were also used to calculate the transmittance of the light transversely illuminating the cell to compare with the intensity measured between crossedpolarizers (as illustrated in Fig. 4), and to evaluate the accuracy of the retardation measurements. It is interesting to note that the cell investigated here had retardation within λ/30, indicating a very low-level retardation. Furthermore, the complex cellular morphology introducing multiple refractions, as well as the lack of index matching incubation solution, contributed to the larger errors in the retardation determined using QPPM. The discrepancies observed in Fig. 4, between the measured and computed transmittance between crossed polarisers may be attributed to multiple factors. The low-level dynamic range of the CCD camera may be a plausible explanation; however it is unlikely that increasing the light level will improve the results. In dealing with viable cell specimens it is more likely that micromovements occur during the phase acquisition process which involves acquisition of two phase images for two different polarisation inputs. The phase images obtained for the two polariser positions are time delayed (~30s) relative to the measurements of transmittance through crossed polarisers (~1s). Thus differences would be anticipated when comparing the different retardation measures obtained through crossed polarisers and by retardation computation using QPPM. Additional research considering automated control over the polarization illumination input is required to fully investigate this issue.
The quantitative polarized phase microscopy technique has been demonstrated as a quantitative method for two-dimensional retardation mapping of uniaxial anisotropic specimens. It is based on phase imaging using linearly polarized light and is implemented by using only a single linear polarizer without the need for any other polarization-analyzing optics and assuming a knowledge of the optic axis orientation of the specimen.
The results obtained with QPPM were in excellent agreement with the measurements performed using a conventional method. An essential feature of this method is that the retardation magnitude is obtained simultaneously with the phase map, involving a very simple process which also yields information about the specimen amplitude characteristics. Ongoing research aims to extend QPPM for optical axis determination as well as to gain in-depth understanding of other impediments and issues related to QPPM implementation for the study of birefringence in unstained live specimens. With regard to cardiac cells, more extensive measurements of the 2D retardation may be informative in relation to both ‘form’ and ‘intrinsic’ birefringence (as considered here), with potential application in the investigation of excitation-contraction coupling processes.
This research was supported by the Australian Research Council (DP 0663365). Dr C.L. Curl is a current NH&MRC Peter Doherty Fellow. The authors wish to acknowledge useful discussions with Keith Nugent and Peter Harris relating to this work.
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