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Abstract
We propose a compact and easy-to-align lateral shearing common-path digital holographic microscopy, which is based on a slightly trapezoid Sagnac interferometer to create two laterally sheared beams and form off-axis geometry. In this interferometer, the two beams pass through a set of identical optical elements in opposite directions and have nearly the same optical path difference. Without any vibration isolation, the temporal stability of the setup is found to be around 0.011 rad. Compared with highly simple lateral shearing interferometer, the off-axis angle of the setup can be easily adjusted and quantitatively controlled, meanwhile the image quality is not degraded. The experiments for measuring the static and dynamic specimens are performed to demonstrate the capability and applicability.
© 2017 Optical Society of America
1. Introduction
Digital holographic microscopy (DHM) is a powerful interference measurement technique that can simultaneously provide the quantitative amplitude and phase of a light wave carrying the information of a specimen [1,2]. Due to its non-invasiveness, label-free and wide field imaging capability, DHM has become an important tool in various research fields [3–6]. Most of the traditional DHM configurations are based on Mach–Zehnder or Michelson interferometers to form an off-axis geometry, which can retrieve the object information from a single-shot hologram [7,8]. In such off-axis interferometers, the object and reference beams travel along different geometries. Therefore, the retrieved quantitative phase information with high sensitivity is subject to the phase noise because of the environment effects such as the air fluctuations and mechanical vibrations. This leads to the lower temporal stability, which may impede the measurement of a sample with requirement for high stability [9]. Temporal stability will be improved if the object and reference beams propagate in a common-path geometry or through a set of identical optical elements [10]. Hence, common-path interferometers attract more and more attention due to their high temporal stability and compact features.
In recent years, common-path interferometers have been remarkably developed to improve the temporal stability of the systems [11–29]. As a whole, according to the origination of reference beam, common-path interferometers can be divided into two categories. One category is creating a separate uniform reference beam from one of the two identical object beams using a spatial filter [11–17]. For example, the diffraction phase microscopy uses a diffraction phase grating in conjunction with a 4f lens system and a spatial filter positioned in the Fourier plane to create a reference beam [11–13]. And the τ interferometer is constructed with a pinhole mirror, positioned in the Fourier plane of one of the two interference arms to create a reference beam [14,15]. The advantage of these common-path interferometers allows imaging the dense specimens. But, they require precise optical alignment due to spatial filtering. The other category is using the self-referencing techniques, in which a part of the object beam containing no sample information acts as the reference beam [18–29]. Although the second category of common-path interferometers may only image the sparse specimens, they require fewer optical elements and are easier to implement. For example, a single cube beam splitter is used in a nonconventional configuration to both split and combine a diverging spherical wave emerging from the microscope objective [18,19]. The Lloyd’s mirror configuration is used to fold a part of the object beam forming self-referencing [20]. In addition, several lateral shearing interferometers have been constructed by using the Modified Hartmann Mask, glass plate, Wollaston prism, etc [22–28]. However, these extremely simple and compact lateral shearing interferometers cannot adjust the separation angle of the two laterally sheared beams. In other words, the spatial frequencies of the interference fringes are not adjustable to optimally match the pixel size of a charge-coupled device (CCD). Recently, the flipping interferometry, a close-to-common-path interferometer, is proposed to fully control the off-axis angle and effectively avoid to suffer from ghost images [29]. It also needs half empty optical field of view, but it can image the relatively dense samples.
In this paper, we propose an alternative lateral shearing common-path DHM belonging to the second category of common-path interferometers, which is based on a slightly trapezoid Sagnac interferometer to create two laterally sheared beams and form off-axis geometry. In the trapezoid Sagnac interferometer, the two laterally sheared beams pass through a set of identical optical elements in opposite directions. The ring geometry of the trapezoid Sagnac interferometer shows high stability [30]. With the advantage of high stability, the trapezoid Sagnac interferometer has been used to generate arbitrary spatially variant polarization vector beams [31]. According to the laws of geometric optics, we quantitatively analyze the geometrical relationship of the two laterally sheared beams. Although it requires half empty optical field of view, the proposed setup, without spatial filtering in the Fourier plane, is easy-to-align, compact and highly stable. In addition, compared with highly simple lateral shearing common-path interferometers [22–28], the off-axis angle between the object and reference beams can be easily adjusted and quantitatively controlled to optimally match the pixel size of CCD in some degree. Meanwhile the image quality of the setup is not degraded.
2. Experimental setup and principle
Figure 1 shows the experimental setup of the lateral shearing common-path DHM based on a slightly trapezoid Sagnac interferometer. The light beam from a fiber coupled laser with wavelength of 532 nm is expanded and collimated by the beam expander BE and lens L1, respectively. The collimated beam passing through a polarizer P1 becomes linearly polarized at angle of 45°. The linearly polarized beam illuminates the specimen and subsequently magnified by a 5 × microscope objective MO. The magnified object beam is again collimated using the tube lens L2. Then, it goes through the trapezoid Sagnac interferometer module, which consists of a polarized beam splitter PBS and the mirrors M3, M4, and M5. In this module, the object beam is split into two counter-propagating orthogonal polarized (p- and s-polarized) beams. The two beams pass through a set of identical optical elements and finally are recombined by the PBS. To create the off-axis geometry, the angle of the recombined two beams can be quantitatively controlled by slightly rotating the PBS. Here, by properly adjusting the PBS, only the wavefront of the transmitted p-polarized beam will always keep parallel to the CCD target. As a result, a part of s-polarized beam containing no sample information acts as the reference beam and the p-polarized beam acts as the object beam so that the image quality is not degraded. Finally, the recombined two beams interfere with each other after passing through a polarizer P2, which is used to adjust the interference fringes visibility. The interference fringes are recorded by a CCD (The imaging Source DMK41BU02, 1280 × 960 pixels, and pixel size 4.65 μm × 4.65 μm).
Fig. 1 Experimental setup of the lateral shearing common-path DHM based on a slightly trapezoid Sagnac interferometer; BE: beam expander; L1, L2: lenses; P1, P2: polarizers; M1-M5: mirrors; MO: microscope objective; PBS: polarized beam splitter.
To more clearly understand the geometrical relationship of the two output light beams from the trapezoid Sagnac interferometer, we demonstrate the p- and s-polarized components, respectively. For simplicity, the multiple refraction of light beams at the air-PBS interfaces is not shown because it doesn’t ultimately affect the direction of the output light beams propagation from the PBS. Figure 2(a) shows the schematic of the p-polarized component, where θ1, θ2, and θ3 denote the angles between the p-polarized light beam and the mirrors M3, M4, and M5, respectively. θ is the angle of the output p-polarized light beam with the incident light beam and can be given by
Fig. 2 Schematic of light beams in the trapezoid Sagnac interferometer. (a) p-polarized component; (b) s-polarized component.
Figure 2(b) shows the schematic of the s-polarized component, where θ 0′ denotes the angle between the incident light beam and the dielectric coating of the PBS, θ 1′, θ 2′, and θ 3′ denote the angles between the s-polarized light beam and the mirrors M3, M4, and M5, respectively. θ ′ is the angle of the output s-polarized light beam with the incident light beam and can be given by
According to the laws of geometric optics [32], the angles between s-polarized light beam and three mirrors interfaces can be separately expressed as follows
Hence, the separation angle Δθ between the output p- and s-polarized light beams can be given by
Equation (6) indicates that only when (θ1 + θ2 + θ3-θ 0′) = π/2, the two output light beams will be parallel. As is known, when the PBS is slightly rotated, only the direction of s-polarized light beam propagation will change. As a result, the p-polarized light beam acts as the object wave so that the image quality is not degraded. Hence, the separation angle between the two output light beams from the trapezoid Sagnac interferometer can be adjustable and quantitatively controlled by slightly rotating the PBS.
3. Experiment Results
We first use the traditional positive resolution target to demonstrate the working principle of the proposed setup. In the setup, a part of s-polarized beam containing no sample information acts as the reference beam and the p-polarized beam acts as the object beam. The digital holograms are recorded in the imaging plane using the CCD. Figs. 3(a1)-3(a3) show the digital holograms recoded at different off-axis angles by slightly rotating the PBS and the recorded holograms size is 1280 × 960. The corresponding spatial spectra are shown in Figs. 3(b1)-3(b3), respectively. As can be seen in Figs. 3(b1)-3(b3), the off-axis angles between theobject and reference beams are gradually increasing. Therefore, the off-axis angle of the setup can be adjustable to optimally match the pixel size of CCD in some degree. By means of angular spectrum method [33], the corresponding reconstructed intensity images are shown in Figs. 3(c1)-3(c3). Although the proposed setup needs half empty optical field of view and may only image sparse samples, as shown in Figs. 3(c1)-3(c3), the overlap between the duplicated images may be avoided by increasing the off-axis angle within a certain range [24]. In other words, the proposed setup can change the lateral shearing distance by adjusting the off-axis angle. In addition, the p-polarized light beam acts as the object beam so that by slightly rotating the PBS, the reconstructed intensity image remains unmoved and the image quality is not degraded.
Fig. 3 Experiment results for demonstration that the off-axis angle of the setup is adjustable by slightly rotating the PBS. (a1)-(a3) Digital holograms recoded at different off-axis angles; (b1)-(b3) Spatial spectra corresponding in 3(a1)-3(a3); (c1)-(c3) Reconstructed Intensity images corresponding in 3(a1)-3(a3).
In order to test the performance of the setup, we measure the pits ablated by high power laser irradiation on the silica glass surface. First, we record a background hologram used to eliminate the background phase noise and the tilt phase associated with the off-axis geometry. Then, the second hologram is recorded for the glass surface with a particular pit in the field of view. The complex amplitudes are numerically reconstructed by means of angular spectrum method [33]. By comprehensive use of double-exposure method and phase unwrapping algorithm, the phase difference Δφ between the two reconstructed object waves can be obtained, as shown in Fig. 4(a). Then, the pit depth can be calculated as h = Δφλ/(2πΔn), where Δn denotes the refractive index difference between the silica glass and air. Considering that the magnitude of the refractive index changes induced by the laser irradiation is about the order of 10−3, here Δn is 0.46 [34,35]. Figure 4(b) shows the corresponding 3D topography of the pit. Figure 4(c) shows the 1D depth profile along the horizontal line crossing the center of the pit. For the sake of comparison, the experiment results of the second pit are shown in Figs. 4(d)-4(f). Compared with the demonstrated experiment results of the two pits, the shape and depth of the two pits are obviously different. Hence, the proposed common-path DHM can provide an effective method to quantitatively visualize the effects of the laser irradiation.
Fig. 4 Measurement results of the two pits ablated by high power laser irradiation on the silica glass surface. (a), (d) Phase images of the first and second pits; (b), (e) 3D topographies of the first and second pits; (c), (f) 1D depth profiles along the horizontal line crossing the center of the first and second pits.
Next, we demonstrate the feasibility of the setup for dynamic measurement. The dynamics of water evaporation is meaningful and has wide applications, such as microfluidics, self-cleaning surfaces, spray cooling, and so on [36]. So, we use a tiny deionized water droplet as the measured specimen. The experiment is implemented at the temperature of 22°C. A background hologram is first recorded at a portion of the glass substrate containing no droplets to eliminate the background noise carried by all the subsequent recorded holograms. The subsequent holograms are recorded at a rate of 15 frames/s and a total of 90 holograms are obtained during the water evaporation process. Figure 5 shows the experiment results, in which Figs. 5(a)-5(d) give the reconstructed phase images of the water droplet during the evaporation process at t = 0 s, 2 s, 4 s, 6 s, respectively, and Fig. 5(e) displays the 1D thickness profiles along the horizontal line crossing the center of the water droplet corresponding in Figs. 5(a)-5(d). To more directly demonstrate the dynamics of water evaporation, Visualization 1 gives the 3D topography variation of the water droplet during the whole evaporation process.
Fig. 5 Measurement results of a tiny deionized water droplet during evaporation process. (a)-(d) Reconstructed phase images at t = 0 s, t = 2 s, t = 4 s, t = 6 s, respectively; (e) 1D thickness profiles along the horizontal line crossing the center of the water droplet corresponding in Figs. 5(a)-5(d). The 3D topography variation during the whole evaporation process can be seen in Visualization 1.
Then, to demonstrate the capability of the proposed setup on biomedical applications, we image living HeLa cells, one of the human cervical cancer cell lines. The HeLa cells submerged in phosphate buffered saline solution are attached to the culture plate before measurement. First, we replace the 5 × microscope objective of the setup with a 50 × microscope objective for imaging HeLa cells. A background hologram is recorded without HeLa cells in the field of view. Then, the hologram with a HeLa cell in the field of view is recorded. By comprehensive use of double-exposure method and phase unwrapping algorithm, the quantitative phase image of the HeLa cell without labeling is shown in Fig. 6(a) and the corresponding 3D phase image is shown in Fig. 6(b).
Finally, in order to character the high temporal stability of the proposed common-path DHM, the proposed setup and a DHM based on a traditional Mach–Zehnder interferometer are separately performed without the samples in the field of view and under the same conditions. A series of holograms are recorded at the rate of 15 frames/s for 30 seconds without any vibration isolation. The phase distributions are reconstructed by means of angular spectrum method. Then the phase difference distributions are calculated by comparing the reconstructed phase distributions to that of the first recorded hologram. 10000 random pixel points in the same area of every phase difference distributions are selected to calculate the standard deviation. Figure 7(a) shows the histogram of these standard deviation of the proposed setup indicating that a mean fluctuation is 0.011 rad. The histogram of these standard deviation of the DHM based on a traditional Mach–Zehnder interferometer, with a mean fluctuation 0.106 rad, is shown in Fig. 7(b). In other words, the proposed setup shows about one-order of magnitude improvement in temporal stability over the traditional Mach–Zehnder interferometer configuration.
Fig. 7 Temporal stability of the proposed setup and DHM based on a traditional Mach–Zehnder interferometer. (a) Histogram of standard deviation of the proposed setup; (b) Histogram of standard deviation of the DHM based on a traditional Mach–Zehnder interferometer.
4. Conclusions
We have demonstrated a compact and easy to implement lateral shearing common-path DHM based on a slightly trapezoid Sagnac interferometer. In this interferometer, the two laterally beams pass through a set of identical optical elements in opposite directions and thus have nearly the same optical path difference. Compared with a DHM based on a traditional Mach–Zehnder interferometer, the proposed setup shows about one-order of magnitude improvement in temporal stability, with a mean fluctuation 0.011 rad. Based on theory and experiment analysis, the off-axis angle between the object and reference beams can be easily adjusted and quantitatively controlled to optimally match the pixel size of CCD in some degree, meanwhile the image quality is not degraded. Although the proposed setup may only image sparse samples, the overlap between the duplicated images may be avoided by increasing the off-axis angle of the setup within a certain range. We perform the experiments of measuring the pits ablated by laser irradiation, water evaporation process and HeLa cells to demonstrate the capability and applicability of the setup. It has significant application potential for dynamic process measurement, especially for monitoring live biological cells.
Funding
National Natural Science Foundation of China (NSFC) (11634010, 61405164, 61127011, 11404262).
Acknowledgments
We thank Sheng Liu for his helpful discussions on the laws of geometric optics. We also thank Yamin Wang from the National Engineering Research Center for Miniaturized Detection Systems, College of Life Science, Northwest University, for providing HeLa cells.
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