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Abstract
We present a novel single-shot four-wavelength quantitative phase microscopy (FW-QPM). Four lasers operating at different wavelengths are multiplexed with a pair of dichroic mirrors and a polarization beam splitter in a three-mirror quasi-common-path interferometer. After a single-shot interference pattern is obtained with a monochrome camera, four holograms of different wavelengths were demultiplexed from it in the frequency domain with polarization- and frequency-division multiplexing. Polarization-division demultiplexing scheme uses polarization dependent visibility changes in an interference pattern, and it plays a critical role in making only two interference patterns exist within a single quadrant in the frequency domain. We have used a single-mode optical fiber as a phase object sample and demonstrated that a measured single-shot interference pattern can be successfully demultiplexed into four different interferograms of different wavelengths with our proposed scheme.
© 2017 Optical Society of America
1. Introduction
Both amplitude and phase information of light waves can be obtained by using quantitative phase microscopy (QPM) [1–5] or digital holographic microscopy (DHM) [6, 7]. When these interferometric phase measurements are interpreted as optical path lengths of light, we may acquire three-dimensional (3D) or topological information of targets with nanometer-scale sensitivity. Because of this important property, QPM or DHM is considered as one of the most powerful tools for non-contact 3D optical imaging or microscopy [8–11]. The 3D structure of a sample cannot be properly retrieved from a single interferogram when there exist discreet path length jumps larger than the wavelength of the light source used for an interferogram [12,13]. Multiple interference patterns are normally taken using several lasers with different wavelengths to solve this problem. There is also a great need for taking multiple interferograms in full color holographic imaging for 3D displays, where at least three holograms with different wavelengths are sequentially taken [14]. In order to reduce the number of holograms to be recorded in these applications, single-shot multiple-wavelength holographic imaging techniques have been proposed [15–17]. Frequency-domain multiplexing (FDM) is a brilliant method that has recently been introduced for this purpose. In FDM information at different wavelengths is recorded in different areas and is filtered and separated in the frequency domain [18]. Sometimes an incorrectly estimated carrier frequency generates additional errors in a reconstructed 3D image measured by an FDM based system, and this problem can be resolved by calibrating a measurement system with an object of a known 3D structure [19]. M. Kim et al. have reported a dual-wavelength Mach–Zander interferometer-based technique that solves phase wrapping ambiguity in reconstructed phase images [20]. This method suffers from the complexity of its measurement setup and instability in measured phase data because of free space Mach–Zander interferometer-based arrangements in its experimental setup. Quantitative dispersion microscopy, proposed by D. Fu et al., has solved the instability problem by using a common-path interferometer with dual wavelengths [21]. However, this scheme also used a complex experimental setup based on a free space interferometer with multiple arms. Later, dual-wavelength diffraction phase microscopy with a monochrome camera was introduced [22]. This is a common-path technique that has high stability and simplicity in its structure. However, synchronous spatial filtering of two different wavelengths on the Fourier plane of its beam path is a difficult process, and the technique has not been further developed for triple-wavelength diffraction phase microscopy.
Triple-wavelength digital holography has recently been demonstrated based on various ingenious techniques, such as using the Doppler effect, color camera imaging, and spatial frequency multiplexing, among others [23–27]. T. Kiire et al. have reported a color digital holographic microscope by using a single monochrome image sensor [24]. This technique employed the Doppler effect to separate information at three different wavelengths. Single-shot imaging was not possible in this case. Several schemes for triple-wavelength digital holography have been proposed based on color cameras [26, 27]. In these approaches, the intrinsic Bayer filter of a camera sensor is used to record holographic information at different wavelengths. Since a repeated pattern of 2 × 2 pixels composed of two green, one red, and one blue pixels is used in the Bayer filter of a color image sensor, the field of view and the resolution of an interferogram with a color image sensor is lower than those with a monochrome sensor. Furthermore, the number of wavelengths that can be used in this scheme is restricted by the structure of a color camera and the Byer filter used in the camera. In order to measure large step heights, lasers with small wavelength differences are preferred to have a high dynamic-range in multiple-wavelength holographic interferometry [12, 13]. However, it is very difficult to take interferograms of two close wavelengths that are separated by less than a few tens of nanometers with this method. Mann et al. have reported a single-shot triple-wavelength digital holography technique that employs spatial frequency multiplexing [28]. The principle of this technique is the same as that of dual-wavelength holography, but its structure is much more complicated than that of dual-wavelength holography, because it is made of three independent Mach–Zander interferometers. N. Lue et al. have introduced another triple-wavelength holography setup composed of quasi-common-path interferometers for single-shot 3D imaging [29]. It uses a spatial frequency multiplexing scheme based on a double-mirror interferometer. This technique has various advantages, such as simplicity in operational arrangements, high vibrational stability, and an inexpensive setup with just one monochrome camera. However, the number of wavelengths that can be used in this method is restricted in practical experiments. This is because the crosstalk of information between neighboring wavelengths in the frequency domain is unavoidable as the number of different employed laser wavelengths increases.
Digital holography based on the multiwavelength spatial-bandwidth-extended capturing-technique using a reference arm (Multi-SPECTRA) was lately proposed [30], in which intentional aliasing is introduced during the data acquisition by an imaging sensor to obtain an extended use of space in the frequency domain. In Multi-SPECTRA, the angle between the reference and object waves for one of the two lasers is carefully arranged such that either the horizontal or vertical carrier frequency becomes larger than the Nyquist frequency of an imaging sensor. This causes one of the carrier frequencies of the two lasers to be located in different quadrants of the frequency domain, owing to aliasing in its digitizing process. As a result, data occupies all the four quadrants in the frequency domain, and the number of wavelengths can be increased in an interferogram without crosstalk. Although this great idea has been experimentally demonstrated with dual-wavelength holography, there is no report about extending the concept toward the usage of three or four wavelengths. Furthermore, it requires high modulation frequencies and it cannot be employed for measuring two holograms with small difference in their wavelengths.
In this paper, we have presented a single-shot four-wavelength quantitative phase microscopy (FW-QPM) technique with four different lasers. The method can use more spatial frequencies in the 2D frequency domain. Since the space in the frequency domain is effectively used, the number of possible lasers or wavelengths in a single-shot interferogram could be increased from two to four. The experimental setup for this single-shot quadruple laser holography is based on the triple-mirror interferometer that we proposed previously [31, 32]. In this case, the beam from a bright field microscope is split into three parts: one object beam containing the sample information, and two plane-wave reference beams that have different propagation directions with each other. We used a polarization-mode demultiplexing scheme combined with a frequency-domain demultiplexing method to separate interference patterns at four different wavelengths. Since four interferograms at different wavelengths are recorded simultaneously by a single monochrome camera, the data acquisition speed is determined by the measurement speed of the imaging sensor. The feasibility of this technique is demonstrated by measuring the refractive index profile of a fiber with four lasers of different wavelengths with a single exposure. The spatial resolution of an image is determined by its spectral contents in the frequency domain. Since each wavelength component is demultiplexed in the frequency domain with a proper frequency-filtering window, the spatial resolution of each reconstructed image is reduced as we add more wavelengths to a single-shot QPM system.
2. Polarization-multiplexed four-wavelength quantitative phase microscopy
Figure 1 illustrates the principle of our proposed single-shot four-wavelength quantitative phase microscopy method. Figure 1(a) shows a typical distribution of data in the frequency domain when an interferogram is measured by a double-mirror interferometer with three lasers [29]. The red, purple, and yellow circles represent frequency components of the three different lasers. As the angles between an object and reference beam are all the same for the three wavelengths in this interferometer, all six circles in the first and third quadrants are on one line and are located close to each other in the frequency domain. This makes it difficult to separate the information of one wavelength from that of the other two. Because the spacing between any two circles depends on the wavelength difference between two lasers, the wavelengths of the lasers need to be changed to adjust the spacing between the circles. Figure 1(b) shows a distribution of frequency-domain data when four lasers are used instead of three in a double-mirror interferometer. The circles are located even closer to each other, such that overlapping of information between neighboring wavelengths begins to occur. There are several possible solutions to solve this problem. The first is to increase the spatial bandwidth of a measurement system by decreasing the pitch size of its camera.
Fig. 1 2D frequency domain of: (a) three-wavelength double-mirror interferometer, (b) four-wavelength double-mirror interferometer, (c) four-wavelength double-reference triple-mirror interferometer, (d) four-wavelength quantitative phase microscope using a color-polarized double-reference triple interferometer.
The second solution is to increase the optical magnification to reduce the object bandwidth in the frequency domain. This decreases the field of view of an imaging system. Because these two solutions require changing either the physical dimensions of an imaging array or the wavelengths of lasers used for the measurements, these are not practical in most cases. The data shown in Figs. 1(a) and 1(b) exist only in the first and the third quadrants of the two-dimensional (2D) frequency-domain space, and overlapping of information between neighboring wavelengths can be reduced if all four quadrants in the frequency domain can be utilized. The third solution to reduce the crowdedness of information in Fig. 1(b) is to relocate two out of four circles from the first and third quadrants to the second and forth quadrants. Tahara et al. introduced a brilliant method that can move one of the carrier frequencies of two lasers to different quadrants in the frequency domain by using aliasing in a digitizing process [30].
However, the physical dimensions of an imaging array and the wavelengths of the lasers need to be carefully matched in this scheme for practical applications. Previously, we demonstrated multiple-mirror interferometry, which can effectively use all four quadrants in the 2D frequency domain [31, 32]. If we use a similar triple-mirror interferometer with one object beam and two reference beams, we can cause interferogram data to occupy all four quadrants of the frequency domain. Figure 1(c) shows typical frequency-domain data of four lasers measured by a triple-mirror quasi-common-path interferometer with two reference beams. The four circles in the first quadrant represent four spatial carrier frequencies determined by the angle between the sample beam and one of the two reference beams, while the positions of the other four circles in the second quadrant are determined by the angle between the sample beam and the other reference beam. As both reference lights contain all four color lasers, crosstalk between neighboring wavelengths is still present.
If a color filter that passes only two laser lights (green and violet circles) is put on the beam path of the first reference beam, while another color filter that passes other two laser lights (red and blue circles) is added on the other beam path, we obtain a less crowded distribution, as shown in Fig. 1(d). However, in practice, it is not easy to find correct color filters that satisfy our requirements, and we used two orthogonally aligned polarizers to remove unwanted carrier frequency components in the frequency domain. The experimental setup of our proposed scheme is illustrated in Fig. 2. It is based on the multiple-mirror quasi-common-path interferometer we introduced previously [31, 32]. The red and blue lasers are put together to form a single beam by a dichroic mirror, and another dichroic mirror is used to add the green and violet lasers together. The two combined beams are multiplexed by a polarizing beam splitter (PBS).
Fig. 2 Four-wavelength quantitative phase microscope made with four lasers. LR, LB, LG, and LV represent lasers with red, blue, green, and violet colors, respectively. MO and Mr are mirrors that reflect the object and reference beams, respectively. DM: dichroic mirror, PBS: polarizing beam splitter, M: mirror, S: sample, OL: objective lens, TL: Tube lens, PF: polarization filter, CCD: charge-coupled device.
Figure 3 shows the beam shapes and polarization configurations in the experimental setup. Figure 3(a) indicates that the polarization directions of the red and blue lasers are horizontal, while the polarization directions of the green and the violet lasers are vertical. The combined beam has both horizontal and vertical polarization states and passes through the sample, objective, and tube lenses. Figure 3(b) shows a polarization filter located just after the tube lens. In the polarization filter, two perpendicularly aligned polarizers are positioned for the two reference beams. These two polarizers work as color filters. There is no polarization filter for the object beam. Figure 3(c) shows three mirrors positioned just after the polarization filter. Each mirror is attached to a kinematic mount and is adjusted by controlling the position of modulation frequencies in the frequency domain manually.
Fig. 3 (a) Polarization directions of the four lasers used in our setup. The red and blue lasers are horizontally polarized, while the green and violet lasers are vertically polarized. (b) The configuration of a polarization filter for one object beam and two reference beams. The color of the lasers and the polarization directions of each part of the filter are indicated. (c) The position of three mirrors just after the polarization filter; two mirrors for the two reference beams reflect two clear parts of the sample (without an object), and one mirror for the object beam reflects phase-modulated light by an object. R, G, B, and V indicate red, green, blue, and violet lasers, respectively. The gray arrows indicate the polarization state. MO and Mr are mirrors for the object and reference beams.
These mirrors form a quasi-common-path three-beam interferometer. Two mirrors at the bottom work as two reference beams reflecting clear areas of a sample (without an object). The mirror at the top makes an object beam that has phase-modulated information from an object. A CCD sensor with 480 × 640 pixels is placed at the focusing plane of the tube lens. The three reflected beams form an interference pattern on the image plane. Consequently, the Fourier transform of an obtained interferogram from our setup shown in Fig. 2 looks like that illustrated in Fig. 1(d). In order to have proper image of an object, the object mirror needs to be larger than the camera sensor, and the beam from the central part of the object mirror should be aligned to the center of the CCD. The objective lens and the other lens in the setup must be achromatic, to reduce the chromatic aberration caused by different laser wavelengths. Considering that two beams with orthogonal polarization states cannot make an interference pattern, the optical intensity pattern of four lasers on the camera plane for our proposed technique can be expressed as
where r and O stand for the reference and object waves, respectively. R, G, B, and V represent the red, green, blue, and violet wavelength lasers. Here, we considered a one-dimensional interferogram along the x-axis for simplicity. P and S indicate two perpendicular polarization states: P and S polarization states. q represents spatial modulation or carrier frequency and φ is the phase of an object to be measured. For example, is the intensity of the P-polarized violet laser in the reference beam, and is the intensity of the S-polarized blue laser in the object beam. We ignored any background phases associated with the field curvatures of the object and two reference beams. Since a spatial carrier frequency is determined by the incident angle between an object and reference beam on the camera plane, it can be expressed as:where λ is the source wavelength and θλ is the angle between the two beams. Therefore, the modulation or the carrier frequency qλ, which is the center position of each circle in the Fourier-transformed interferogram data illustrated in Fig. 1, is controlled by tilting the reference mirror θλ. Here, the tilted angles for the red and blue lasers are same, and the angles for the green and violet lasers are identical: θR = θB, θG = θV. When we take the 2D Fourier transform of Eq. (1), the object information measured by each laser is localized in the frequency domain near the corresponding modulation frequency of each wavelength described in Eq. (2). The interferogram of each laser can be retrieved by applying an appropriate filter in the frequency domain on one of the two sidebands of each modulation frequency. The radius or size of a filter is decided by considering the numerical aperture of the objective lens, the magnification of the imaging system, and the wavelength of a modulation frequency. The inverse Fourier transform of filtered frequency-domain data is a complex function, which can be written as:where n represents the polarization state (P or S) of a laser. Dividing this equation by the modulation function of and taking its argument, we can obtain the phase information of an object for a given wavelength λ.3. Results and discussions
Four different lasers operating at 406, 488, 532, and 632 nm wavelengths were used in our experiments. In order to demonstrate the feasibility of our proposed FW-QPM, we have measured the thickness profile of a single-mode optical fiber with an 8-µm core and 125-µm cladding. It also has a depressed cladding of diameter 20 µm. An objective lens with a numerical aperture of 0.3 and a magnification of 10X was used for imaging. Figure 4(a) shows raw data of a recorded interferogram for the fiber sample. A magnified view of the interferogram clearly shows some of the four spatial carrier frequencies. The Fourier transform of the interferogram is shown in Fig. 4(b). The two bright spots in the first quadrant are for the green and violet lasers of wavelength 532 and 406 nm, respectively. According to Eq. (2), the information for the laser with a longer wavelength is closer to the origin. The other two bright spots in the second quadrant are for the blue and red lasers of wavelength 488 and 632 nm, respectively. The information of the red laser is closer to the origin in this case. This image is similar to the schematic shown in Fig. 1(d). Even though the wavelength difference between the green and blue lasers is only 44 nm, it is clear that they can be well separated in the frequency domain in our method. The interferogram of each wavelength is separately filtered from the Fourier-transformed data shown in Fig. 4(b).
Fig. 4 (a) Raw interferogram data for an optical fiber with 20 µm core size and 125 µm cladding diameter. (b) 2D Fourier-transformed data of the raw interferogram.
The phase profile of the sample for each wavelength is obtained from Eq. (3). Figures 5(a)-5(d) show 2D pseudocolor phase profiles of the sample optical fiber obtained by the four lasers. We have used the unwrapping algorithm described in [33], Antonopoulos et al., to obtain each phase profile. We have used the unwrapping algorithm described in [33], Antonopoulos et al., to obtain each phase profile. Figure 5(e) shows overlapped phase profiles for all four wavelengths. If we know the refractive index difference between the sample and the surrounding medium at a given wavelength, we can obtain the thickness profile of a phase object with:
Fig. 5 (a) 2D pseudocolor phase images of the fiber sample for (a) violet, (b) blue, (c) green, and (d) red lasers. The color bar shows the phase in radians, and the white scale bar represents 50 µm. (e) Overlapped cross-sectional phase profiles of the fiber for four different wavelengths.
where t(x) is the thickness profile, is the phase profile, and Δnλ(x) is the refractive index difference profile between the object and the medium at a given wavelength λ. For a single-mode fiber, the refractive index difference Δnλ is constant within the core, the cladding, and the depressed cladding areas. When we know the structure of the sample fiber or the thickness of each material in the fiber, we can calculate the refractive index differences between each material, which are important parameters in determining the characteristics of an optical fiber. Since we obtain the phase profiles at four different wavelengths, the dispersion characteristics of a sample can be obtained as well. The data acquisition speed of our proposed single-shot multiple wavelength interferometry is 110 frames per second, which is the frame rate of the image sensor used in our experiment.
Apart from the many advantages of our proposed FW-QPM, there are some limitations. First of all, it is practically very difficult to increase the number of lasers beyond four, because of crowdedness or overlapping of data in the frequency domain. Second, our method cannot be used for a birefringent object, where the polarization state of light is not conserved. Another limitation of FW-QPM is that the field of view of our imaging system defined by the objective lens used in the system must be larger than three times the size of the camera sensor. This is because of the three mirrors used our setup after the objective lens, shown in Fig. 2 and 3.
A densely distributed sample without a clear part cannot be properly measured with FW-QPM. This is because the reference beams in our proposed FW-QPM are made from the clear part of a sample, which is without any structure. Finally, a low-coherence light source with a coherence length of less than a few millimeters may not be suitable for this method. Since the two reference beams are tilted at different angles, the path length difference between an object beam and one of the two reference beams can be larger than the coherence length of the light source. In this case, the interference pattern can be present only within a small area of the image sensor.
4. Conclusions
We have demonstrated a novel single-shot four-wavelength quantitative phase microscopy technique that can retrieve four holograms of different wavelengths from a single monochrome interference image. This was possible by making the data to be sparsely distributed in all four of the quadrants in the frequency domain with a combination of polarization and frequency-division multiplexing techniques. In order to show the feasibility of this scheme, a single-shot interference pattern was measured with a three-mirror quasi-common-path interferometer for a depressed-core single-mode optical fiber sample. Fine tuning of the incident angles of two reference beams toward an arrayed image sensor in the three-mirror quasi-common-path interferometer was carried out to place the center positions of the four interferograms as far as possible from each other in the frequency domain. Instead of using bandpass filters, we used polarization-division multiplexing by using two orthogonally aligned polarizers in our FW-QPM, which causes only two interferograms of different wavelengths to exist in one quadrant in the frequency domain. Since there is no need for a bandpass color filter or a Bayer filter in front of an image sensor, higher-resolution images can be obtained with monochrome cameras with more pixels. Another advantage of this polarization-division multiplexing scheme is that there is no restriction in selecting the wavelength of a laser associated with the availability of a certain color filter. The experimental results demonstrate the feasibility of our proposed FW-QPM. We believe that our novel demonstration of compact single-shot four-wavelength interferometry with the usage of polarization- and frequency-division multiplexing can be adapted to many other imaging applications, such as confocal microscopy, structured imaging, and 3D holographic imaging.
Funding
National Research Foundation of Korea (2012R1A4A1029061, 2013R1A1A2062448, 2017R1A2B2003808, 2017R1A2B4003950); Ministry of Science, ICT and Future Planning of Korea (CAMM-2014M3A6B3063712, IITP-2017-2016-0-00464); Ministry of Trade industry & Energy of Korea (10062417); Ministry of Education Science and Technology of Korea (BK21 program).
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