2.1.

Optical Configuration

Figure 1 illustrates the optical configuration of the proposed common-path DHM system using a biprism. The sample is mounted onto a translation stage for focusing and is illuminated by a low-power collimated laser diode module (CPS532, Thorlabs) of wavelength 532 nm and spectral bandwidth $<1\mathrm{nm}$. The light scattered by the sample is imaged by a system composed of an infinity-corrected $40\times /0.65\mathrm{NA}$ microscope objective (MO) and an achromatic tube lens (TL). The lateral magnification of the system is $M=-44.44\times$ since the focal lengths of TL and MO are 200 and 4.5 mm, respectively. It is important to mention that the proposed system operates in the telecentric regime since the aperture stop of the MO is located at the front focal plane of the TL (e.g., object focal plane). A Basler $\mathrm{acA}5472\text{-}17\text{-}\mu \mathrm{m}$ CMOS sensor ($5472\times 3648\mathrm{pixels}$, ${2.4\text{-}\mu \mathrm{m}}^2$ pixel size) is placed at the image plane of the imaging system (e.g., back focal plane of the TL) allowing the acquisition of in-focus images. Therefore, for reconstructing phase images of thin samples, such as the one shown in this work, there is no need to apply refocusing algorithms in the reconstruction process.

Fig. 1.

Proposed common-path digital holographic microscopy based on an FB. (a) Schematic of the system, (b) experimental interference FOV, and (c) optimization of the space bandwidth of the camera by rotating the biprism.

The key element of the proposed technique is the insertion of an FB³⁵ between the TL and the sensor plane. Thus, at any observation plane behind the FB, there is a coherent superposition between the two split waves generated by the biprism. Remember that sinusoidal fringes are generated in the region of superposition, the rhombus-shape region with a maximum FOV, located at a distance $s_{\max }=(L/4)\tan \delta$ from its vertex, equal to half of the biprism’s lateral extension, ${\mathrm{FOV}}_{\max }=L/2$. Importantly, in this common-path DHM system, one only reconstructs the phase information that it is encoded within the fringes’ FOV (e.g., rhombus-shaped region). The fringes’ FOV at the sensor plane depends on the position of the FB ($s$), its refringence angle ($\delta$), and its lateral extension ($L$). The closer the FB is to the sensor, the narrower the fringes’ FOV is. Figure 1(b) plots the experimental fringes’ FOV for an FB with $\delta =5\mathrm{deg}$, $L=20\mathrm{mm}$, and refractive index $n=1.5197$ at a wavelength of 532 nm. As expected, the experimental data follows the trend of the theoretical prediction: the fringes’ FOV varies linearly with the FB’s position. The maximum fringes’ FOV equal 9.684 mm (error $<4\%$ compared with the theoretical prediction) is found when the relative FB position is $\mathrm{\Delta }{s}_{\text{max}}=77.63\mathrm{mm}$ (e.g., the position between the TL and the FB is $s_{\text{max}}=s_0+\mathrm{\Delta }{s}_{\text{max}}$). At that position ($s_{\text{max}}$), the fringes’ FOV occupies 74% of the whole sensor area, overcoming the main disadvantage of the FB-based DHM reported in Ref. 36.

It is important to highlight that the biprism generates two mutually coherent point sources that superimpose in amplitude to produce a high-contrast sinusoidal interference pattern beyond the FB. The fringes’ contrast in the sinusoidal interference pattern is directly related to the degree of temporal coherence of the real source. In other words, the use of a source with a broad spectral bandwidth produces a dramatic decay in the visibility of the fringes, resulting in a reduced fringes’ FOV. Therefore, in our FB-based DHM system, the illuminating source should have a spectral bandwidth ($\mathrm{\Delta }\lambda$) no greater than 10 nm.³⁷

Since our DHM system operates in the telecentric regime, the FB is illuminated by a plane wave, thus the fringes’ frequency in the FB-based DHM shown in Fig. 1(a) is independent of the biprism’s position, $p^{\prime }=1/2u_0$, where $u_0=(n-1)\tan \delta /\lambda$ $(n,\delta )$ are the biprism’s refracting index and refringence angle, respectively. The theoretical and experimental value of the fringes’ periods are equal to 6.08 and $7\mu \mathrm{m}$, respectively. Clearly, the experimental value is close to the theoretical value. Note that for this case, the off-axis condition is satisfied because the fringes’ frequency ($u_m=1/p\mathrm{n}=0.1429\mu {\mathrm{m}}^{-1}$) is higher than $3u_c/2=0.0412\mu {\mathrm{m}}^{-1}$ where $u_c=\mathrm{NA}/(\lambda M)=0.0275\mu {\mathrm{m}}^{-1}$ the cut-off frequency of the imaging system (e.g., maximum spatial frequency resolvable by the system) expressed as a function of $M$ and the NA of the system. As shown in Fig. 1(c), there is no overlay between the diffraction orders in the Fourier domain, consequently, our proposed system operates in a single-shot off-axis regime. In off-axis DHM, a better optimization of the finite space bandwidth of the sensor is achieved when the $\pm 1$ orders are placed along the diagonal of the sensor’s space bandwidth, allowing their best allocation with no overlapping. In the proposed FB-based common-path DHM system, one can ensure the best optimization of the sensor bandwidth by rotating the FB by 45 deg [Fig. 1(c)].

2.2.

Performance Characterization

To evaluate the performance of the proposed FB-based common path DHM system, the FB was placed to provide the highest fringes’ FOV, thereby ensuring that the maximum sample’s FOV was reconstructed. We first used calibrated phase objects, including the USAF target and the star pattern Quality Project Management (QPM) target (Benchmark Technologies). For both targets, we arbitrarily chose the ones whose nominal heights were 150 nm. Figure 2 shows the reconstructed phase images provided by our system. Note that the measured optical thickness ($t$) in Fig. 2 is related to the measured phase ($\varphi$) from the reconstruction method via $\varphi =2\pi n_{g}t/\lambda$, where $n_g=1.52$ is the refractive index of the glass as provided by the manufacturer. Figure 2(c) plots the optical thickness of the USAF target (top panel) and the star target (bottom panel). The USAF profile is computed along the vertical direction [marked by the red arrow in Fig. 2(a)] through the horizontal lines of group 9. For the star target, the profile was computed through the dashed pink line shown in Fig. 2(b). The measured thickness ($\mathrm{mean}\pm \mathrm{standard}\mathrm{deviation}$) for USAF and star targets, shown in Fig. 2(c), are $(144\pm 12)\mathrm{nm}$ and $(131\pm 18)\mathrm{nm}$, respectively. Within the experimental errors, the agreement between the experimental measurement and the manufacturer’s specification [e.g., 150 nm, light gray rectangle in Fig. 2(c)] is significantly high, verifying the accuracy of the proposed FB-based common-path DHM system. We have also used the USAF image to evaluate the spatial lateral resolution limit of the proposed DHM. From Fig. 2(a), we estimate the shortest resolvable distance to be equal to 775 nm [9-3 element, marked in Fig. 2(c)], which agrees with the theoretical resolution limit for coherent imaging systems,³⁸ $\lambda /\mathrm{NA}=818\mathrm{nm}$. Thus, our DHM system operates at the diffraction limit.

Fig. 2.

Evaluation of the FB-based common path DHM system using calibrated phase targets: (a) USAF and (b) star. 3-D reconstructed image of the targets in terms of the optical thickness. (c) Optical thickness profile of the USAF and the star target. For the USAF, the profile is taken through the horizontal lines of group 9 [direction marked by the red arrow in (a)]. For the star, the profile is taken along the radial coordinate shown in (b) (marked by the dashed pink line). The scale bar in panels (a) and (b) is $20\mu \mathrm{m}$.

The main advantage of common-path DHM systems over two-path systems is their high temporal stability. To test the temporal stability of our system, we recorded 633 holograms during 2.5 min at 4.22 fps. It is known that the stability measurement is sensitive to the experimental implementation of the system. The proposed FB-based common-path DHM system is mounted upon a floating optical table using no optical cage components. From each recorded hologram, we estimated the corresponding phase map and measured the mean value of the phase over an area of $13.96\times 13.96\mu {\mathrm{m}}^2$ known-to-be-flat region. Figure 3(a) shows the phase difference map (${\varphi }_{t+\mathrm{\Delta }t}-{\varphi }_t$) between two frames taking at times ($\mathrm{\Delta }t$) 1.25 and 2.5 min apart. For this FOV, the obtained phase value during the 2.5 min was $2.2765\pm 0.0003\mathrm{rad}$ ($\mathrm{mean}\pm \text{standard deviation}$). From the small value of the standard deviation, we conclude that our DHM system is relatively isolated from ambient fluctuations. Alternatively, other quantitative assessments of the fluctuation can be performed by evaluating the standard deviation of the phase at each pixel, $\epsilon (\varphi )$. Since the phase and thickness values are linearly related, one can estimate the thickness fluctuation with $\epsilon (t)$ being the standard deviation of the thickness from the standard deviation of the phase $\epsilon (\varphi )$ via $\epsilon (t)=[\epsilon (\varphi )\lambda ]/(2\pi n_g)$. Figure 3(b) shows the histogram of the thickness fluctuation for the two time periods. The mean value of the graph represents the thickness stability of our system, which is equal to 1.31 nm for 1 min and 2.03 nm for 2.5 min. These values confirm that the stability of the proposed common-path DHM system is within the range of a few nanometers, which enables high-resolution investigations of live specimens. Although our current implementation does not present subnanometer stability as other implementations,^14,15 the stability can be improved using a more stable laser source and more robust implementation with cage-system assemblies. Future work will improve the system’s stability to achieve subnanometer stability.

Fig. 3.

Stability results for our FB-based DHM system. (a) Phase difference map (${\varphi }_{t+\mathrm{\Delta }t}-{\varphi }_t$) between two frames in a time period of 1.25 and 2.5 min apart and (b) histogram of the thickness for these time periods.

The advantages of combining high speed, resolution, accuracy, and stability of the FB-based DHM system are demonstrated by imaging a monolayer of human U87 glioblastoma cells, which represent the human primary glioblastoma cell line that is commonly used in brain cancer research. The commercially available U87 glioblastoma cells (American Type Culture Collection, Rockville, Maryland, USA) are used as cancerous samples. Cells are plated on glass coverslips and inserted in a tissue culture incubator for 24 h. Then cells on covered glasses are rinsed for 5 min with phosphate buffer saline and fixed in 10% buffered formalin for 18 h at 4°C. Finally, cells are mounted on glass slides using ProLong Gold as a mounting medium. Figure 4 shows the three-dimensional (3-D) pseudocolor phase images. As a convention, we have assumed that the phase value of the glass should be equal to zero. Thereby, we have estimated the average phase value over areas free of cells (e.g., glass) and subtracted it from the total reconstructed phase value. This procedure has been applied to the unwrapped phase map using the algorithm described in Ref. 39. Clearly, Fig. 4 shows a high-contrast phase image that renders structural details in the cells distinguishing some organelles. Although this is a proof-of-concept study, this result may open the door for extending our QPI system to any biology essay.

Fig. 4.

3-D pseudocolor quantitative phase images of human U87 glioblastoma cells using the proposed system. The image area is $97.21\times 140.41\mu {\mathrm{m}}^2$.

2.3.

Simplified Polarization-Sensitive FB-based DHM System

As previously mentioned, some biological organelle and extracellular matrix components are birefringent; that is, the brightness of their image varies with the polarization state of the illuminating beam of light. To provide PS measurements (e.g., a PS-DHM system), the sample imaged by our FB-based common-path DHM system is now illuminated with linear polarized light whose plane-of-vibration is varied in a controlled way. The polarization state of the illuminating beam changes by rotating a linear polarizer inserted before the sample holder [see Fig. 5(a)]. Note that one can measure the birefringence retardance by illuminating the sample with linearly polarized light and recording the transmitted wavefront without any further analyzer polarizer.²³ Again, we image U87 glioblastoma cells since it has been demonstrated that the glioblastoma cells present higher anisotropy than normal cells,⁴⁰ validating the proper use of these cells for PS imaging. Figure 5(b) shows the two-dimensional (2-D) phase information from U87 glioblastoma cells illuminated with two different polarization angles, 0 deg and 130 deg. In these maps, the areas that are enclosed by the dashed lines reveal details quite differently. Some features clear for the 130-deg phase map are hardly seen for the 0-deg phase map, which confirms the PS behavior of the glioblastoma cells. To obtain the retardance map²⁴ [e.g., $\mathrm{\Delta }\varphi ={\varphi }_e-{\varphi }_o=2\pi (n_e-n_o)d/\lambda$ where $n_e$ and $n_o$ are the refractive indexes of the extraordinary and ordinary waves, and $d$ is the thickness of the cells], we obtain the maximum and minimum values of the whole set reconstructed phase image at each pixel. Example phase maps for the extraordinary (e.g., maximum) and the ordinary (e.g., minimum) behavior are displayed in Fig 5(c). The subtraction of these unwrapped maps provides the retardance image, also displayed in Fig. 5(c). Note that on the retardance image, the contrast created is specific to the PS behavior of the sample since the cells and parts without any anisotropy are no longer visible on this image. The color scale bar in the retardance image corresponds to retardance values between 0.4 and 0.9 rad. Note that our phase sensitivity is three orders of magnitude smaller (0.0003 rad), guaranteeing that any divergence on the retardance image is due to differences of the anisotropy of the samples. Although a more rigorous research study would require the analysis of many more glioblastoma cells, our results should be regarded as a proof-of-concept study of the PS capability achieved by our system. Future work will involve the evaluation of the birefringence sensing using a benchmark birefringence target with known thickness $d$ and birefringence or a spatial light modulator to mimic one as in Ref. 22. After this validation, further studies using several biological specimens will also be performed to evaluate the potential of this system for biological imaging.

Fig. 5.

Quantitative phase images of human U87 glioblastoma cells using the proposed PS DHM system: (a) optical configuration of the simplified PS FB-based DHM system, (b) 2-D pseudocolor normalized phase maps at different polarization states (0 deg and 130 deg), and (c) ordinary and extraordinary phase maps and the retardance map.