Single-Shot Common-Path Off-Axis Dual-Wavelength Digital Holographic Microscopy Based on Two-Dimensional Grating Diffraction

We present a single-shot dual-wavelength common-path off-axis digital holographic microscopic (CO-DHM) imaging method based on two-dimensional grating diffraction. This method improves the utilization rate of the interference field under the limited photosensitive size of the camera, and further expands the original camera’s field of view (FOV). In addition, the mode of orthogonal carrier frequencies close to the diagonal direction can optimize the utilization of the camera’s spatial bandwidth. Compared with the traditional dual-wavelength CO-DHM using one-dimensional grating or prism beam splitting, this method effectively avoids the aliasing of high-frequency components of the +1-order spectrum of different wavelengths in the frequency domain. We provide quantitative phase imaging experiments for the full FOV of USAF resolution chart, onion epidermal cells and standard polystyrene beads. The results prove that the system can enlarge the interferometric FOV by nearly 74.0% without changing the imaging parameters, such as magnification and resolution, and can achieve high-precision quantitative phase imaging with only a single hologram.


INTRODUCTION
Digital holographic microscopy (DHM) combines traditional optical microscopy imaging technology and holographic imaging technology to quantitatively obtain the amplitude and phase distribution of the original object light in the field of microscopic imaging [1][2][3][4][5][6]. Digital holographic microscopy has the characteristics of no contact, no chemical damage, and three-dimensional dynamic imaging during the entire sample observation process. It has been widely used in life, medicine, environment, materials, manufacturing, microelectronics, and other fields [7][8][9][10][11][12]. Common-path off-axis digital holographic microscopy (CO-DHM) system based on grating diffraction, as a typical and simple DHM, can realize high-stability non-contact real-time dynamic monitoring of samples [13,14]. The system uses a classic optical microscope and is equipped with an additional diffraction module [15][16][17][18]. The object-reference light interference pattern is obtained on the camera plane by using grating beam splitting and the diffraction field spectrum filter technology, and then the complex amplitude distribution of the object light field is reconstructed by a computer [19]. In this way, three-dimensional quantitative information of the sample is obtained [20].
Recently, CO-DHM has been adapted to many types of light sources. Researchers have improved the resolution and spatial phase sensitivity of the CO-DHM system by using broadband light sources and structured light sources [21,22]. In 2014, Moham Reza Jafarfard et al proposed a dual-wavelength diffraction phase microscopy system to accurately measure the thickness and refractive index of samples [23]. In 2015, N.A. Talaikova et al. measured the refractive index of red blood cells using the dual-wavelength transmitted light and reflected light simultaneous irradiation mode [24]. In 2020, Manoj Kumar et al developed a dual-wavelength co-channel off-axis system based on prism splitting to measure the phase and refractive index of samples at one time [25]. However, the Fourier orders of different wavelengths in the spectrogram of the above schemes are in the same direction, and strict system parameter optimization is required to avoid aliasing of the high frequency components of the +1-order spectrum. If the selected area is too small during frequency domain filtering, High frequency components will be lost. To solve this problem, MIT Poorya Hosseini and others developed a dual-wavelength interference microscope in 2018, which provides single-lens off-axis interferometry by encoding two spectral images at orthogonal spatial frequencies, but the optical path uses two 4f systems [26]. The structure of the system is more complicated [27]. Behnam Tayebi et al developed a dualpinhole dual-wavelength CO-DHM in 2018, which maximized the optimization of two arbitrary wavelengths and crosstalk-free two-dimensional sampling schemes [28]. Compared with the previous CO-DHM, the field of view was expanded, but the system adopted two Gratings and two pinholes, it brings higher requirements to the stability of the system.
The size of the CO-DHM system's field of view (FOV) is limited by two key factors: the size of the object reference light interference spot size and the camera's photosensitive size. In recent years, several methods have been proposed in the field of digital holography to increase the recording FOV without changing the size of the camera sensor or loss of imaging resolution. One of them is the spatial phase scanning digital holographic imaging technology, which scans the sample on a reference light field, periodically obtains multiple frames of sample phase images, and finally uses algorithms to combine them into a large field of view holographic image. However, this type of method sacrifices the time resolution or the range of the dynamic area allowed in the recorded image [29]. The other is the off-axis interferometric field of view multiplexing technology, which optically multiplexes two off-axis interferometric fields of view collected from different areas on the sample to a single camera sensor, which can realize wide-area dynamic imaging without loss of spatial resolution. However, the currently reported digital holographic microscopy optical paths using this technology require complex device design and use more optical devices [30].
To solve the above problems, we propose a dual-wavelength CO-DHM based on a two-dimensional diffraction grating. This system can multiplex two FOVs into a single off-axis interferogram without using special optical elements. And it also can obtain wider FOV imaging without losing time resolution. The two FOVs share the dynamic measurement range of the camera and will not affect imaging parameters such as magnification and resolution. The interference area between the object light of two wavelengths and the reference beam is determined by the adjustable shearing distance that can be adjusted by controlling the axial position of the grating [31], and this distance determines the enlarged area of the field of view. Furthermore, the spatial bandwidth of the camera can be maximized by simply rotating the two-dimensional grating to form orthogonal carrier frequencies close to the diagonal direction. Quantitative phase imaging of samples can be achieved by placing the module as an additional unit on the exit port of the inverted microscope. To prove the high precision of the system, the large field of view quantitative phase imaging capability, the imaging results of the resolution plate, standard polystyrene beads and onion epidermal cell are provided.

EXPERIMENTS AND METHODS
System and Principle Figure 1 shows the proposed experimental scheme of dualwavelength CO-DHM based on two-dimensional gratings. The experiment uses a single longitudinal mode He-Ne laser (λ R = 632.8 nm) and a semiconductor laser (λ G = 532 nm) as the illumination source. The two beams of light are combined by the beam splitting prism (BS) and irradiate the sample (S) at the same time. Among them, the spatial filter (SF) plays a role in filtering out the high frequency noise of the laser and collimating and expanding the beam. The two-dimensional grating (G) is placed behind the image plane (IP) to generate different diffraction orders that carry sample information. The light beam is collimated by the tube lens (TL) into parallel light and incident on the grating. On this basis, a 4f imaging module is added to image the sample image to the camera plane. Then, for each wavelength component in the Fourier plane of the 4f system, the 0th order and +1 order diffracted lights are extracted in two mutually orthogonal directions using filters. The design of the filter (after rotating 45°) is shown in Figure 1. The strongest 0-order diffracted light passes through the pinhole on the spatial filter and becomes an ideal uniform spherical wave as the reference beam in interference. The weaker +1-order diffracted light passes through a larger spatial filter hole separately. The outgoing beam is used as the object light. To ensure better fringe contrast, an attenuator with a transmittance of 50% is placed behind the 0-level filter hole on the spectrum plane. Finally, the +1-order object beam with the complete information of the sample will interfere with the plane reference light on the imaging surface of the CCD sensor to generate two holograms which are captured and recorded by the CCD sensor at one time.
Since the beams of two wavelengths have the same propagation mode in this module and are independent of each other, after the beam is filtered by the Fourier plane spectrum (as shown in Figure 2), only the 0-order and +1order light in a single direction can pass. For simplicity, we consider the light field distribution of a single wavelength in a one-dimensional plane. The transmittance function of a single Frontiers in Physics | www.frontiersin.org February 2022 | Volume 10 | Article 822714 wavelength through a two-dimensional diffraction grating can be simplified as Here, β 2π/ ∧; Λ is the grating period; c0, c1 are Fourier series. Figure 3 shows a schematic diagram of the 4f system module and coordinate system. The sample (S) is imaged on the image plane (IP) after being magnified by the microscope objective. In this module, (x O , y O ) are the coordinates of the IP plane; (x G , y G ) are the coordinates of the grating plane; (x L , y L ) are the coordinates of the first lens plane, (x F , y F ) are the Fourier of the 4f system Plane coordinates; (x I , y I ) are the coordinates of the camera plane. Here, we have derived Eqs 2-8 based on Figure 3 through the Fresnel diffraction integral formula. First, the complex amplitude of the object light field at the grating plane can be expressed as Among them, d is the distance between the grating and the image plane. A (x O ,y O ) is the complex amplitude of the sample image of the IP plane. After the diffraction grating, the complex amplitude of the light field in front of the first Fourier lens is expressed as  Substituting formula (1) and (2) into formula (3), it can be simplified to Taking into account the Fourier transform properties of the lens, after the light field passes through the first lens of the 4f system, the distribution of the complex light field in front of the Fourier plane is The transmittance function of lens L 1 is expressed by Eq. 6. A filter is placed on the Fourier plane of the lens L 1 to allow the +1-order light to pass through and further lowfrequency filtering of the 0-order light, while other levels are blocked. Therefore, on the image plane, the +1-order object light that contains all the information of the original image and the 0-order reference light that only carries the DC content of the original image can be obtained [31]. Omitting the constant exponential coefficient, the complex amplitude distribution of the object light field on the Fourier plane is Where F represents the Fourier transform of the function.
where M 4f −f 2 /f 1 , which defines the magnification of the 4f system. According to Eq. 10, for a certain system, the interference angle α is a constant and has nothing to do with the position of the grating G, but is only controlled by the grating period and the focal lengths of the lenses L 1 and L 2 . From the geometric relationship, the shearing distance between the reference beam and the object beam on the camera plane is l αd As shown in Figure 4, the red and green circular areas respectively represent the light intensity distribution of the object light carrying all the sample information at the corresponding frequency of the camera plane, and the area inside the yellow circle represents the object reference light interference area. The shearing distance l is the distance between the center of the object light beam and the center of the reference light, and this distance determines the position of the interference area between the object light and the reference light in the image plane. It can be seen from Eq. 11 that the shearing distance can be adjusted by controlling the axial position of the grating. When the two-dimensional grating is located in the x O y O plane (d = 0), the clipping distance l is 0. And the red and green object images are overlapped on the camera plane in Figure 4, and the camera captures The field of view corresponding to the two wavelength beams is the same. When the distance d increases, one of the two fields of view produces a displacement of l 1 in the x direction, and the other produces a displacement of l 2 in the y direction. This configuration produces a multiplex of two linear, orthogonal, off-axis interference holograms, corresponding to two sample fields of view. Therefore, two object light fields can be digitally extracted from a single multi-channel hologram. The beams of each wavelength of the dual-wavelength CO-DHM are split by a two-dimensional grating to generate corresponding shearing fields in mutually orthogonal directions. The expansion rate of the field of view compared to the original camera field of view area can be expressed as Where l 1 and l 2 are the shearing distances corresponding to the light fields of different frequencies. w is the lateral width of the photosensitive surface of the camera.
Filter Structure Design To design the filter diaphragm reasonably, the following three key factors must be considered: One is the distance between the pinhole and the center of the rectangular diaphragm, Through grating diffraction calculation, we can get Δy Where Δx and Δy are the physical distances between the 0-order and the +1-order spectrum of green and red light on the Fourier plane, and λ G and λ R are the wavelengths of green and red light. Second, the size of the rectangular aperture should be The design is large enough to ensure that the weaker +1-order diffracted beams pass through in the orthogonal direction individually and completely. The optimal side length dimension (L) of the rectangular diaphragm design can be expressed as:

Phase Reconstruction Method
In the numerical reconstruction, the Fourier transform is performed on the image digital hologram recorded by the CCD to obtain the separable +1-order spectrum [32]. The original light field distribution can be obtained by performing frequency domain filtering on the +1-order spectra at different positions of the spectrogram, and then undergoing inverse Fourier transform. And calculate the arctangent of the complex amplitude of the object light wave to obtain the wrapped phase [33]. Here, the filtering method used in this article is the classical spectral filtering method. And the least squares unwrapping algorithm based on transverse shearing (LS-LS) is used to demodulate the wrapping phase. Among them, the phase compensation method in which the background optical complex amplitude is divided by the original light field is used to eliminate the additional phase factor and other background noise of the reference light wavefront. The distortion compensation method is detailed in Ref. [34]. It is worth pointing out that the camera only needs to collect a hologram to obtain background information of two wavelengths.

EXPERIMENTAL RESULTS
In order to verify the large field of view quantitative imaging capability of the system, a dual-wavelength CO-DHM optical path based on a two-dimensional diffraction grating as shown in Figure 1 was constructed. The focal lengths of the lenses L 1 and L 2 used in the experiment are 6 cm, and all lenses are achromatic to minimize chromatic dispersion [35]. The actual horizontal magnification of the 4f module is M 4f = 1. The object beam is collimated by the tube lens into parallel light and incident on the two-dimensional Ronchi grating (Λ = 8.6 µm). The two-dimensional grating is located behind the front focal plane of the first lens (d = 3 cm) to generate diagonally orthogonal carrier frequencies. The CMOS (Basler acA 2500-14 µm, sensor for-mat: 2,592 × 1944 pixels, pixel size of 2.2 µm) is located at the back focal plane of the 4f system to record the interference hologram. The camera's dynamic range (typical value) is 60.6 dB. In our experiment, the microscope is equipped with a 40×/0.60NA objective lens. For the design of the filter, the diameter of the 0-order filter hole is 50 µm, and the two 1order rectangular holes should be large enough to ensure that the +1order light of this wavelength passes completely, and the +1-order light of another wavelength is filtered out.

1951-USAF Resolution Chart Imaging
Through the quantitative phase imaging of the 1951-USAF resolution chart (positive chart), the system's ability to obtain two holograms at one time and expand the field of view is verified. Figure 5A shows two resolution plate holograms with different fields of view recorded by a CCD camera at the same time, and Figure 5B shows an enlarged view of the fringes orthogonal to each other by magnifying the part of the hologram in the red rectangular frame. Figure 5C shows the spatial Fourier transform image of the hologram. The ±1 order spectrum of two wavelengths (λ 1 = 632.8 nm and λ 2 = 532 nm) are separated and marked with corresponding color circles. And the wrapped phase is obtained by performing frequency domain filtering on the +1 spectrum at different positions of the spectrogram. Figure 5D show the reconstructed intensity map after red light frequency domain filtering. Figure 5E show the reconstructed intensity map of the green light. Finally, we get the final field of view size by stitching the pixels of the intensity map of two different resolution versions of the field of view, as shown in Figure 5F. Here, we selected the 1944 × 2,592 (camera full field of view) picture of the reconstructed resolution board illuminated by the green light source, and the (1,438 × 2,592) image size of the resolution board illuminated by the red light source. The image area (506 × 2,596) is the overlapping part of the two fields of view. From the geometric relationship, it can be calculated that the final field of view is 74.0% larger than that of the original camera.

Onion Epidermal Cell Imaging
In addition, we verified the ability of this technology to quantitatively image large scale biological cells by imaging the epidermal cells of plant onion. Figure 6A shows the multiplexed off-axis interference hologram recorded during the exposure of a single CCD camera; Figure 6B shows the spatial Fourier transform of the hologram; Figures 6D,E show the reconstructed intensity map, phase map and height map of λ 1 = 532 nm; Figures 6F,G show the reconstructed phase map and height map of λ 2 = 632.8 nm; Here, the image stitching method based on SIFT feature detection is used to reconstruct the image [36]. Figures 6C,H6 are the intensity map and height map obtained after the two fields of view are stitched together. It is worth noting that it is unreasonable if the spliced picture is directly regarded as a complete single reconstructed image. This is manifested in two aspects: one is for the restored intensity image, because the light intensities of the two wavelengths are different. And the camera's sensitivity to different monochromatic light results in the difference in the natural intensity of the samples recorded by the CCD in the two fields of view. The solution adopted in this work is: adjust the attenuator in the optical path to approximate the same light intensity and numerical adjustment of intensity map brightness; the second is shown on the height map, this method is effective when there is no sample at the stitching boundary. But for large-size biological samples, even if the inherent height of the sample is the same (the height calculation formula can refer to Ref. [37]), it still remains at the stitched boundary with a slight difference. Because the sample has a different dispersion, refractive index and absorption for a specific wavelength of laser light. Here is only to show that the mentioned method is effective for the final expansion of the imaging field of view.

Standard Polystyrene Beads Imaging
Finally, we imaged standard polystyrene beads to verify the imaging accuracy of our system. The standard polystyrene microspheres have a diameter of (9.8 ± 0.2)μm and a refractive index of 1.59. Use a dropper to take a small number of microspheres and drop them on the glass slide. After the original ethanol solution has evaporated, use the refractive index matching solution to immerse the microspheres. Finally, put the standard sample into the stage and adjust the focus to make it image on the plane of the CCD camera. Figures 7A-D show the phase images of the microspheres with λ 1 = 632.8 nm and λ 2 = 532 nm, respectively. The red rectangular frame identifies the area as the overlapping area of different fields of view. Figures 7E,F respectively show the height images of the microspheres with λ 1 = 632.8 nm and λ 2 = 532 nm. The background noise of phase map and height map is caused by impurities in the medium, laser speckle and coherent parasitic fringe noise. Figure 8 shows the one-dimensional phase and height reproduction of the same bead corresponding to the   Figure 8B, the longitudinal diameter of the microspheres illuminated by the red light source is 9.85 µm; the longitudinal diameter of the microspheres illuminated by the green light source is 10.00 µm. The main source of the errors in the measurement of the two wavelengths is due to the slight difference in refractive index of the test sample at different frequencies [23]. Here, we assume that the refractive index of the sample at different frequencies is the same; the second is the error caused by manual selection of the center position of the microbead.

CONCLUSION
In summary, we propose a two-wavelength CO-DHM system based on two-dimensional grating diffraction to improve the utilization of the interference field under the limited photosensitive size of the camera. This scheme can simultaneously perform two-wavelength and large field of view quantitative phase imaging. And compared with one-dimensional gratings and prisms as light-splitting elements, the use of two-dimensional gratings to split light to rotate 45°to form orthogonal carrier frequencies in diagonal directions can optimize the use of the spatial bandwidth product of the camera. Experimental results show that this technology can enlarge the common-path off-axis interference FOV by nearly 74.0% without changing the imaging parameters, such as magnification and resolution. It is worth pointing out that by optimizing the grating and 4f system lens parameters to the maximum, there is still room for improvement in this value to a certain extent.
At the same time, because the camera FOV and the acquisition frame rate are interchangeable, this technology can increase the camera's ability to acquire the frame rate by using fewer pixels without reducing the original reconstructed FOV [38]. By providing this module with conventional inverted microscopy, a broader FOV or faster collection rate can be provided for a given camera, thereby achieving greater and fastmoving sample imaging.
Of course, this solution also has all the advantages of singlelens multi-wavelength CO-DHM. It brings higher stability to the system by adopting the common-channel structure and multi-wavelength multiplexing method. By selecting the overlap area of the FOV of different wavelengths, the thickness, refractive index and dispersion of the sample can be measured at the same time (as mentioned in Refs. [4,23]). This technology opens up a new way for the integration of versatility, high stability, wide field of view three-dimensional biological quantitative imaging technology.

DATA AVAILABILITY STATEMENT
The raw data supporting the conclusion of this article will be made available by the authors, without undue reservation.

AUTHOR CONTRIBUTIONS
Conceptualization, HW, and JW; methodology, GM, QZ, ZD, and JW; software, QZ, and XW; validation, XW, YG; formal analysis, JW, and HW; investigation, JW, XW; resources, HW, and WW; data curation, HW; writing-original draft preparation, JW; writing-review and editing, XW, JW, and HW; visualization, JW, and WW; supervision, HW; project administration, HW; funding acquisition, HW. All authors have read and agreed to the published version of the manuscript.