Field-based polarization measurements are essential for the completeness of information when exploiting the complex nature of optical responses of target objects. Here, we demonstrate digital holographic microscopy for quantifying a polarization-sensitive map of an object with a single-shot measurement. Using the image-splitting device generating four different copies of an object image and a separate reference beam of an off-axis configuration enables single-shot and multi-imaging capability. With the use of two polarization filters, four complex field images containing an object’s polarization response are obtained simultaneously. With this method, we can construct a complete set of 2-by-2 Jones matrix at every single point of the object’s images, and thus clearly visualize the anisotropic structures of biological tissues with low level of birefringence. This method will facilitate the high-precision measurements for fast dynamics of the polarization properties of biological specimens.
© 2016 Optical Society of America
Polarization is an intrinsic property of light that represents the direction of oscillating electric field of light. The polarization of light is inherently susceptible to optical anisotropy, which reflects the local orderness of small structures such as the collective rotation of atomic bonding and the overall alignment of molecular axes within an object. Some materials including biological samples exhibit distinctive optical characteristics, so called birefringence, that give the materials two different refractive indices along two characteristic directions of polarization. Thus, polarization of light is exceptionally sensitive to birefringence and can effectively visualize the structural details of target samples that have optical anisotropy.
There have been great efforts to quantify the degree of anisotropy of an object using polarization-based measurements. To measure optical anisotropy, polarized light microscopy has been intensively used. In the compensator-based methods, an object was placed together with a compensator between two orthogonally aligned polarization filters. The retardation of the object was then measured by rotating the compensator until the retardation introduced by the object was eliminated [1–3]. In the polarimetric approaches [4–8], multiple images were acquired under various settings of polarization filters, retarders, and rotators, and from the taken images, the polarization properties of a sample were computed. Oldenbourg et al. circumvented the need for intensive work to precisely control the polarization optics by employing electro-optic modulators [4, 5], and a lock-in technique in conjunction with rotating analyzers was also used to measure the retardation and orientation of polarization induced by specimen’s birefringence . These intensity based measurements for the degree of polarization, however, can only describe the incoherent statistical properties of light rather than the full information for the deterministic polarization states of light.
Due to the complex nature of polarization of light, a field-based description is required to work with polarized light for more comprehensiveness. For this reason, digital holographic microscopy alternatively has been widely used for measuring the polarization responses of target specimens. Examples include investigating weak birefringence of optical fibers introduced by bending strain [10–12], characterizing the polarization-dependent scattering properties of random media [13, 14], and measuring retardation in biological samples [15, 16]. With the capability of measuring complex fields of light, digital holography can retrieve the complete information regarding all possible combinations of input and output polarization states of an object, which are represented with a 2-by-2 matrix called a Jones matrix [12, 16–21]. Due to the multi-order formalism associated with Jones matrices , however, multiple images taken with appropriately prepared combinations of polarization filters were required for determining all elements in a single Jones matrix , and this significantly limited the acquisition speed, making these techniques unsuitable for fast measurements. To overcome this limitation, a method has been introduced that can measure a sample’s Jones matrix with two consecutive acquisitions . In the method, two Jones matrix elements were taken simultaneously by multiplexing two different interference patterns in a single shot; thus, the full set of a Jones matrix was obtained by acquiring two images while synchronously altering the polarization state of the input beam. Recently, Liu et al. advanced the method so that a complete Jones matrix of an object could be obtained with a single measurement . They acquired four different images at a time by multiplexing four different interference patterns generated by two independent laser sources in a single measurement. These two techniques, however, were subject to considerable efforts for precise alignment of the optics due to the use of a spatial filter in the Fourier plane to generate a clean reference beam. Furthermore, multiplexing by overlapping multiple images in a single or two images inevitably shares the dynamic range of detectors among the images and thus can reduce measurement sensitivity.
In this paper, we demonstrate digital holographic microscopy that can measure a complete information space spanned by the complex 2x2 Jones matrix at each point in a 2-D object space. This method features single-shot operation by acquiring four copies of an object image using the image-splitting device (ISD) that we previously developed . With two linear polarization filters placed in the detection arm configured as off-axis interferometry, all the complex elements for the Jones matrix of an object can be retrieved at once. With the simple combination of the ISD and the polarization control, we successfully obtained the complete map of the polarization responses of biological samples with relatively low level of optical anisotropy. In addition, by analyzing the measured information in systematic ways, we could extract the characteristic quantities related to the optical anisotropy of the tissue. We call this method polarization-sensitive quantitative phase microscopy (PSQPM).
2. Methods and experimental setup
2.1 Experimental setup
The optical configuration of PSQPM is depicted in Fig. 1(a); the overall shape is based on a Mach-Zehnder-type quantitative phase microscope employing an off-axis detection configuration. Two He-Ne lasers (Thorlabs: HNL210L) with a wavelength of 632.8 nm were used to generate two independent light waves. The light from laser 1 was split into two for an interferometric measurement by a polarizing beam splitter 1 (PBS1). The fraction of the light with vertical polarization (v-pol.) was reflected off by PBS1 and then redirected to the sample beam path while the other fraction with horizontal polarization (h-pol.) was transmitted through PBS1 and then steered to the reference beam path. The second light from laser 2 had the opposite configuration at PBS2. The fraction of light with h-pol. transmitting through PBS2 took another transmission through PBS1 and then co-propagated along the sample path while sharing the spatial mode with no mutual coherence with the light from laser 1. The other fraction of light with v-pol. from laser 2 was reflected off by PBS2 and then sent to a separate reference path that differed from that of laser 1. In both the sample and reference paths, the polarizations of two light waves from the two light sources were perpendicular; one was vertically and the other was horizontally polarized.
In the sample path, the lasers that passed through a sample were collected and delivered to an intermediate image plane (IP1) by a 4-f telescope that consisted of an objective lens (OL: Olympus, 40x, NA 0.65) and a tube lens. The image was then sent to the image-splitting device (ISD), composed of two lenses arranged in a 4-f configuration with a magnification of 1.0x and a 2-D diffraction grating placed in the Fourier plane between the lenses . Each diffracted beam after the grating generated an identical image copy (except for the overall phase) on the image plane (IP2) of the 4-f telescope of the ISD. The images at IP2 were further delivered to the camera plane through another 4-f telescope with a magnification of 2.5x (not shown in Fig. 1). Thus, the total magnification of the imaging system was 100x. Among all the image copies, only four images generated by 0th and + 1st diffraction orders were selected and recorded by a camera (Andor: Neo5.5, sCOMS). The 2-D grating was made by attaching two 1-D gratings face to face in a configuration in which the grid directions were aligned orthogonally. To avoid the uneven power distribution among the diffraction orders, two transmission grating beamsplitters (Edmund Optics, 80 lines/mm) were used for the 2-D grating. At IP2, two linear polarizers (LP1 and LP2) were placed, LP1 with a + 45° polarization axis for the two upper images and LP2 with a −45° axis for the two lower images (see the polarization analyzers shown in the inset on the bottom right in Fig. 1). Consequently, the polarizations of the object images after the analyzers were configured as shown in Fig. 1(b).
The reference beam also had two light waves with orthogonal polarizations, h-pol. from laser 1 and v-pol. from laser 2. Each beam was expanded by a beam expander, which was composed of an objective lens (OL: Olympus, 20x, NA 0.4) and a tube lens, for the spatial match of the interference with the sample beam. After the expansion, one-half of each beam was obstructed by a block; the left side of the light from laser 1 was blocked by BL1, and the right side of the light from laser 2 was blocked by BL2. The two beams were combined at a non-polarizing beam splitter (BS1). At this point, the reference beam had two light waves with orthogonal polarizations, h-pol. on the left side of the camera and v-pol. on the right side, with no spatial overlap, as shown in the inset on the bottom left of Fig. 1(a). The reference and sample beams were combined at BS2 and reached the camera together. The configuration of the resultant combinations for polarizations at the camera plane is depicted in Fig. 1(b). Unlike the previous setup that used the collinear arrangement for interferometry , the off-axis detection configuration enabled this setup to obtain four different complex field images with a single-shot measurement. As a result of imaging by the ISD, the total field of view is reduced to one fourth of a usual system, but the optical resolution is unaffected. Due to the combinations of the polarization filters for the measurement at the camera, the acquired images contained the information for the polarization response of an object, which can be processed into the elements of the Jones matrix for the sample.
2.2 Extraction of Jones matrix
The electric field of an arbitrarily polarized light wave can be expressed using a vector notation asEq. (1). Here C1 and C2 are constants representing the complex amplitude for the static field. In our experimental setup, two light waves with orthogonal polarizations are superimposed in the sample path. Because those waves have no cross coherence, the sample beam is represented as an incoherent mixture of the two polarized fields:Fig. 1(b), is expressed byEq. (3). After some matrix algebra, the four polarization-selective images are represented as
3. Experimental results
3.1 Jones matrix of a linear polarizer
In order to validate the capability of PSQPM to measure an object’s Jones matrix with a single-shot acquisition, we first conducted a test experiment to extract the Jones matrices of a linear polarizer depending on the orientation angles. The Jones matrix for a linear polarizer Jpol is theoretically given byEq. (5). We performed this experiment at the orientation angles of 0°, 45°, and 90°. As shown in Eq. (6), a linear polarizer has contrast only in amplitude, not in the phase of its Jones matrix elements; thus, we considered only the amplitude parts in this experiment. The amplitude images for the polarization-selective measurements are shown in Fig. 2(a) at the different orientation angles, and the amplitudes for the corresponding Jones matrices are presented in Fig. 2(b). As expected from Eq. (6), only |J11| has a strong signal, whereas others have negligible values, near zero at θ = 0°. In contrast, with θ of 90°, |J22| reaches the maximum, but the others remain at approximately zero. When θ was rotated by 45°, unlike the previous cases, all the matrix elements had similar amplitudes, approximately 0.5. Figure 2(c) shows plots for the average amplitudes of the Jones matrices, showing the relative strengths of the matrix elements, which agree well with the values predicted by Eq. (6). Some deviation from the expectation values may be caused by different laser intensity in each image. To minimize this error we normalized the sample images by reference images taken at blank area without the sample.
3.2 Measurement of polarization response of a biological specimen
We also demonstrated the capability of PSQPM for visualizing the optical anisotropy in biological samples with relatively weak birefringence. We performed an experiment with a mouse kidney tissue extracted in the middle of a whole kidney of an adult mouse. The tissue was treated with 4% paraformaldehyde overnight at room temperature for stabilization and fixation. Then, it was prepared as a horizontal cryosection with a thickness of 10 μm and sandwiched between a slide glass and a cover slip for imaging.
We acquired the polarization-selective images for the tissue using PSQPM and then obtained the corresponding Jones matrix elements as shown in Fig. 3. Figure 3 (a) and 3 (b) show the amplitudes |Xij| and phases ∠Xij for the polarization-selective images. Despite the different combinations for the polarization filters, all images look very similar due to the weak optical anisotropy of the sample. In the Jones matrix images, as presented in Figs. 3(c) and 3(d), the diagonal elements J11 and J22 have strong signals for both amplitude and phase, as expected. However, for the off-diagonal images J12 and J21, the amplitude images look very dim, and the phase images are very grainy; only the regions along the structural boundaries of the proximal tubules of the kidney show recognizable signals, particularly for the amplitude images for |J12| and |J21|. This implies that the birefringent materials are distributed with higher concentrations near the boundaries otherwise which cannot be visible in the normal phase images .
3.3 Matrix analysis for the polarization response of a tissue
Quantitative analysis for a Jones matrix of a tissue is a useful tool to study the molecular formation and physical properties in connective tissues, such as the concentration of fibrous collagen structures [26–28]. For more quantitative exploration for polarization response of a biological tissue, we apply the formalism of matrix analysis to the information acquired by our PSQPM.
First, we conducted imaging of a tendon tissue, which is known to have a relatively higher concentration of collagen fibers; collagen fibers have sufficient contrast for polarization measurements due to the intrinsic birefringence. We used a 200 μm-thick tissue slice extracted from a mouse-tail tendon as a sample, placing the tissue between a slide glass and a cover slip; then we took polarization-selective images of the tissue’s edge area. We then obtained a spatially resolved Jones matrix using Eq. (5). Figures 4(a) and (b) show the amplitude and phase images of the Jones matrix for the sample, respectively. The blank region without the sample has a uniform amplitude distribution in Fig. 4(a) and well-defined phase values in Fig. 4(b) in the diagonal but almost zero amplitude and associated fragmented phase distribution in the off-diagonal. This is because the free space has no polarization response. The region being occupied by the sample also shows amplitude and phase distributions that reflect the typical striped features of tendon tissues in the diagonal elements. Unlike the blank area, the sample region also has noticeable values that show similar characteristic structures in the off-diagonal elements in both amplitude and phase; this is because of the polarization interruption due to the anisotropy introduced by the collagen fibers in the sample. Particularly, for instance, |J21|2 refers to the fraction of light energy being transferred from the polarization state of (1 0)T at the input side to (0 1)T at the output side by the sample, and ∠J21 is the phase associated with that portion of light; otherwise, the photon migration between the orthogonal states is strictly forbidden. Thus, the Jones matrix presented in Figs. 4(a) and (b), especially the off-diagonal elements, shows that the tendon tissue has significant anisotropy that originated from the fibrous collagen structures.
Next, we performed the matrix diagonalization for the measured Jones matrix; for each point of the tendon image, we constructed a 2 × 2 complex matrix from the measured Jones matrix elements; then we obtained the eigenvectors and corresponding eigenvalues by the matrix diagonalization. In this analysis, the two directions of eigenvectors refer to the directions of the principal axes, through which no polarization interruption occurs as long as the input polarization is parallel to one of these principal axes. In addition, the phases of the eigenvalues represent the additional phases that the fields parallel to the corresponding principal axes obtain when they propagate through the sample. We calculated the phase retardation between the two principal axes from the phase difference of the two eigenvalues of the Jones matrix, and the result is depicted in Fig. 4(c). Unlike the empty area where the retardation is almost zero, the tissue region shows a certain amount of retardation, which verifies the existence of two refractive indices associated with the two principal axes, i.e., the birefringence.
Furthermore, we also investigated the directions of the two eigenvectors for the Jones matrix. Due to the complex nature of the measured Jones matrix, the eigenvectors are also represented with complex numbers for those components in general. This implies that the polarization states corresponding to these eigenstates are elliptical. We calculated the eccentricity of the elliptical polarization for one of the eigenvectors. In Fig. 4(d), the distribution is overlaid on top of the amplitude image of the tendon tissue; the eccentricity is 1 in the empty background region due to the linearity of the eigenvector in that free space. In contrast, the distribution in the tissue region has values slightly smaller than 1 for the weakly elliptical polarization. As seen in the eccentricity map for the eigenvector in Fig. 4(d), the effect of the birefringence in the tendon tissue is mild; nonetheless, PSQPM can detect it with sufficient sensitivity.
In summary, we demonstrated digital holographic microscopy for quantifying the information space containing the polarization responses of biological specimens. The information space retains a full Jones matrix as a sub-space at every point of the object image drawn in two dimensions. By employing the ISD and using the off-axis detection configuration, our method (PSQPM) is featured with the single-acquisition and multi-imaging capability, which enabled us to acquire the full polarization information with a single-shot measurement. With the nature of the complex field imaging of digital holographic microscopy, PSQPM can cover the full information space for constructing a polarization-sensitive map of an object.
With this method, we successfully obtained the complete polarization-sensitive maps of biological tissues that have low level of optical anisotropy with sufficient sensitivity. By applying the systematic analysis to the measured polarization response, the phase retardation between the two principal axes and the eccentricity of the elliptical polarization associated with the eigenstate were quantitatively explored. With the ability of single-shot acquisition and the verified measurement sensitivity, our method will facilitate the quantitative and precise measurements for the fast dynamics associated with small birefringence and optical anisotropy in biological specimens.
A grant of the Korea Health Technology R and D Project through the Korea Health Industry Development Institute (KHIDI), the Ministry of Health and Welfare, Republic of Korea (grant number: HI14C3477 and HI13C1501). The Original Technology Research Program for Brain Science through the National Research Foundation of Korea (NRF) funded by the Ministry of Science ICT and Future Planning (grant number: 2015M3C7A1029034).
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