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We present a quasi-common-path interferometer with a double field of view (FOV). The laser beam of an imaging system is separated into three parts using three mirrors; the first and second beams are used to image two different areas of a sample, while the third beam functions as a reference beam. The reference beam is prepared by making clear area in a sample and projecting it on an image sensor. A double FOV is obtained by Fourier domain multiplexing, whereby two interferometric images corresponding to two different areas of a sample are modulated with two different spatial carrier frequencies. The feasibility of this technique is experimentally demonstrated by imaging two different areas of a test target with a single image sensor.
© 2015 Optical Society of America
1. Introduction
Quantitative phase microscopy (QPM) is a non-contact, wide-field, high-resolution interferometric imaging technique that can measure the topographic or tomographic profiles of dynamic microscopic samples [1–11]. In order to obtain more information of highly dynamic objects, methods for simultaneous acquisition of multiple interference patterns are required. QPM employs Fourier domain multiplexing to record more than two holograms in a single measurement [12–14]. One application of multiplexing is removing index-thickness ambiguity in phase information. Tomographic deconvolution methods map refractive index of objects with variable illumination angles; however, in this technique, single-shot imaging is not possible [15–17]. Multiplexing methods have been investigated for simultaneously obtaining the thickness and the refractive index of a phase object by using multiple-wavelengths and the dispersion characteristics of a sample [18–22]. Another application of multiplexing is single-shot phase unwrapping to increase the measurable thickness or height of a sample. Many multiple-illumination [23–25] or multiple-wavelength [26–30] methods have been investigated to measure samples with large stepped height jumps by using Fourier domain multiplexing.
A field of view (FOV) that decreases with an increasing system magnification is another common problem in all optical microscopes. The FOV of a digital imaging system is mostly limited by the size of the detector array, which is much smaller than the observable FOV by an objective lens. The optical resolution R of a microscope is determined by the source wavelength λ and the numerical aperture (NA) of an objective lens. The Rayleigh criterion states that R = 0.61λ/NA [31]. Because the Nyquist sampling theorem requires at least two image pixels within R, the pixel pitch of a detector array must be less than m⋅R/2, where m is the magnification of the imaging system [32]. When the condition of the Nyquist sampling theorem is satisfied, the FOV or the distance on the object side of an imaging system covered by an image sensor becomes:
where N is the number of pixels of the image sensor along the horizontal or the vertical axis.As an example, consider a microscope working at a center wavelength of 550 nm with an objective lens of NA = 0.5 and an image sensor of 1280 × 720 pixels. For this microscope, the diffraction-limited resolution becomes R = 0.67 μm, and the horizontal and vertical FOVs are 429 μm and 242 μm, respectively. In this way, there exists a trade-off between the value of R and the FOV. In other words, Eq. (1) states that the resolution R of an imaging system is sacrificed in order to obtain a larger FOV for a given number of pixels N of the detector. A detector array with a larger number of pixels is therefore preferred in order to extend the FOV of an imaging system. There is the same trade-off relation between the resolution and the FOV in off-axis digital holographic microscopy.
Taking multiple images by mechanically translating the sample or image sensor can be used as a simple method to increase the FOV of an interferometric imaging system. However, a motorized stage may add extra vibrational noise to an interferometric imaging system. Furthermore, because there exists a time gap between images, this method is not appropriate for monitoring dynamic objects. An alternative method of increasing the FOV proceeds by taking many images simultaneously with a single detector array by using multiplexing algorithms. Recently, a novel multiplexing scheme with a double FOV has been proposed for QPI [33]. This technique is based on multiplexing two different fringe patterns on a single detector array in the Fourier domain by using two different carrier frequencies. Furthermore, this technique can be extended to obtain a triple FOV without losing its original resolution [34]. However, the complex and sensitive alignment of the pinhole used in this system (for producing the reference beam) can lead to potential problems in practical applications; moreover, this technique introduces the mechanical instability that is associated with many different beam paths. Common-path interferometer have improved temporal stability [35–41]; because, the reference beam and sample beam travel along the same path. In general, common-path interferometers are more immune to environmental vibrations than conventional separate-path interferometers.
In this study, we present a double-FOV, quasi-common-path phase microscope (DFQPM) that has high stability with respect to external vibrations. The proposed technique is based on the simultaneous recording of two FOVs on a single arrayed detector by using Fourier-domain multiplexing. We obtain highly stable fringe patterns by adapting a quasi-common-path interferometer design with a simple structure [39]. Interference patterns are obtained by placing three separately tilted mirrors on the beam path of an optical imaging system. Two mirrors are used to project two different areas of a transparent sample on a detector array, while the third is used to project a clear area of a sample on the detector array, thereby functioning as a reference beam. This common-path interferometer uses nearly identical paths for the reference and the sample beams, and we observe that it is quite robust to environmental vibrations, as compared with other interferometers. The beams belonging to different object areas are separated and modulated by tilting the mirrors. The two object beams and the reference beam are then combined to form interference patterns on the detector array, and the phase information of the different areas of the sample is subsequently demultiplexed or separated in the Fourier domain. The feasibility of the technique is demonstrated with a 1951 U.S. Air Force test target and polymer microspheres that measure 10 µm in diameter.
2. Double-field-of-view, quasi-common-path phase microscopy
The schematic diagram of our proposed DFQPM setup is shown in Fig. 1. A beam originates from a single-wavelength laser and passes through a sample, an objective lens, and a tube lens. Three mirrors are located after the tube lens in order to obtain three quasi-common path beams on an image sensor. The camera sensor is positioned at the focal plane of the tube lens. The mirrors need to be positions as close as possible to the tube lens of the imaging system. Since a part of an input beam through a sample is used as a reference beam, some part of a sample should be empty or uniform in structure. The size of the empty area imaged on the camera sensor should be large enough to cover the area of the image sensor [39, 40]. For low-concentration objects, one of the beams can be employed as a reference beam, while the rest function as two FOVs. The red arrow indicates the beam separation on the sample plane. This separation can be defined by the position of the mirrors after the tube lens. A camera sensor is placed at the focal plane of the tube lens in order to record the interference pattern of the beams. The size of the sensor is 4.7 mm by 3.5 mm with 640 by 480 pixels.
Fig. 1 Schematic diagram of (a) transmission double-field-of-view (FOV) phase microscopy and (b) reflective double-FOV phase microscopy. Red arrows indicate the position of the objects on the sample plane. R, M, BS, S, and CCD indicate the reference, mirror, beam splitter, sample and charge-coupled device camera, respectively.
Most of commercially available microscope objective lenses guarantee the flatness of an image on the image plane within 20 mm distance along the transverse direction, which is much larger than the physical sizes of most commercially available image sensors. Therefore, a practical scheme for increasing the FOV of a DHM is quite important. The modulation frequencies of the off-axis interferogram on the camera can be controlled by tilting the mirrors. It should be noted that different selections of the image area leads to different modulation frequencies. The schematic diagrams for two configurations of the FOV mirrors are shown in Figs. 2(a) and 2(b). If the left side of the second mirror lies in the plane of the first mirror, then the minimum spatial modulation frequency for two beams at the camera can be expressed as:
where q and λ are the modulation frequency and wavelength of the light source, respectively. θ is the angle between the norm of the second and first mirror, and can be expressed as:where t, l, and d are the distance between the two mirrors, the length of the CCD, and the distance between the first mirror and the CCD, respectively. The FOV observed by the objective lens and the highest spatial frequency of the camera limit the maximum distance t. Furthermore, when the two FOVs become closer, the spatial resolution at the edge decreases.The intensity of the interference pattern formed by the two sample beams and the reference beam at the sensor plane can be expressed as:
where ER, EFOV1, and EFOV2 are the electric field amplitudes of the reference, the first FOV, and the second FOV beams, respectively. qx and qy are the spatial frequencies of the off-axis interferometer. φsFOV1, φsFOV2, and φsFOV1-FOV2 are the phases of the first, second, and the phase of overlapping FOVs, respectively. φbFOV1, φbFOV2, and φbFOV1-FOV2 are their associated background phases, respectively. To extract the different phases of Eq. (4), the real image component of each phase is selected in the frequency domain of Eq. (4). The inverse Fourier transform of the filtered data provides complex interference data of each phase; for the first FOV, this can be written as [30]:Dividing this quantity by the complex interference data of the first FOV background, we can obtain:The same process for the second FOV and the overlapping component allows us to reconstruct their phases as: where isFOV1, isFOV2, ibFOV1, and ibFOV2 are complex interference data of the first and second FOVs.3. Experimental results
We performed experiments using a diode laser operating at wavelength of λ = 532 nm. The magnification of the objective lens was selected to be 40 × . Three mirrors coated with aluminum, located just after the tube lens, separated the beam into three beams, and subsequently reflected them to the camera.
Figure 3 shows reconstructed phases of a reflective 1951 U.S. Air Force test target that were obtained by the configuration shown in Fig. 1(b). Figures 3(a)–3(c) show 2D images that were generated from the phase of the overlapped, first, and second FOV. Figure 2(d) shows the 3D image with a double FOV.
Fig. 3 2D reconstructed phase of (a) overlapping field of view (FOV), (b) the first FOV, and (c) the second FOV. (d) 3D reconstructed FOV. The pseudo color scale bar shows the phase in radians, whereas the white scale bars indicate a length of 10 µm.
Figure 4 shows the experimental results for the 10 µm polymer microspheres inside a medium with a refractive index of 1.55 using a transparent DFQPM (Fig. 1(a)). Figures 4(a) and 4(b) show interference patterns of the object and a logarithmic scale of the 2D intensity pattern of its Fourier transform, respectively. The real image components of the three states are separated by employing a filter for each phase. Figure 4(c) shows the 2D reconstructed image using the phase of the overlapping FOVs. Figure 4(d) shows the 2D position of the two FOVs; the background image is obtained by scanning the surrounding areas.
Fig. 4 (a) An experimentally obtained 2D interference pattern of the 10 µm polymer microspheres belonging to different imaging areas. (b) 2D intensity pattern of the Fourier transform of (a) with a logarithmic scale. (c) Reconstructed 2D pseudo color phase of the two overlapping fields of view (FOVs). (d) The 2D positions of the two FOVs on the sample plane.
Figure 5(a) shows the 3D image of the beads that was reconstructed by two FOVs component. Although Fig. 4(a) shows the overlapping objects, the reconstructed image in Figs. 5(b) and 5(c) show that two FOVs are reconstructed without overlap.
Fig. 5 Reconstructed 3D image of the (a) overlapping fields of view (FOVs), (b) the first FOV, and (c) the second FOV. The pseudo color scale bar shows the phase in radians, whereas the white scale bars indicate a length of 10 µm.
To obtain the spatial and temporal stabilities of our proposed DFQPM we repeatedly measured 250 phase images without a sample. Measured phases in each double FOV image with are supposed to be 0 radian in this case. There are 614,400 (2 X 640 X 480) pixels in each measured DFQPM image. We have calculated standard deviation of 614,400 pixels for each image. Figure 6(a) shows the histogram of these standard deviations for 250 measured images. It shows that the most of the measured images have standard deviation of 2.3 milliradian, which can be considered as the spatial stability of our DFQPM system. We have also calculated the standard deviation of 250 repeated measurements for each pixel. Figure 6(b) shows the distribution of these standard deviations for 614,400 pixels. It shows that the most of pixels have a standard deviation of 4.7 milliradian, and we can take this as the temporal stability of our DFQPM.
Fig. 6 (a) Phase standard deviation distribution for each of the 250 images, (b) phase standard deviation distribution of 250 images for each pixel.
4. Conclusion
We have presented DFQPM as a quasi-common path, single-shot, simple technique for increasing the FOV in off-axis interferometric methods. Three different parts of the object beam are separated into three beams by tilting three mirrors. Two beams that include different FOVs of the object provide two sample beams; meanwhile, the remaining beam, which possesses no object information, operates as a reference beam. The phase information of the different FOVs is obtained by separating the phase of the overlapping FOVs in the frequency domain. The experimental results demonstrate the feasibility of our proposed technique. The maximum FOV of a conventional camera sensor is doubled by using DFQPM for a 1951 U.S. Air Force reflective test target, along with transparent polymer microspheres in a medium. We expect that this technique can be used in concert with other interferometric techniques.
Acknowledgments
This work was financially supported by the MEST through the National Research Foundation of Korea (Grant No. 2012R1A4A1029061) and by the Ministry of Education, Science, and Technology of Korea through the BK21 program’s financial support of the Institute of Physics and Applied Physics at Yonsei University.
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