High-resolution reconstruction of spectrum-overlapped off-axis holography by deflecting reference beam of Gaussian symmetry

Digital holography (DH), as an interferometric imaging technique that uses the digital image sensor, such as a CCD to record and reconstruct the sample information numerically, has become a significant tool for quantitative phase imaging.^1,2 When an image sensor discretely samples the interferogram, its limited pixel size and number determine the available spatial bandwidth. According to the tilt angle between the object and reference beams, the DH systems are categorized into on-axis, off-axis, and slightly off-axis DHs. In the off-axis DH, when an appropriate tilt angle is selected, the +1-order, –1-order, and zero-order spectra are completely separated from each other in the spatial Fourier domain, enabling single-shot, real-time quantitative amplitude, and phase reconstruction.³ In this case, the bandwidth occupation of +1-order spectrum is limited below 1/4 spatial bandwidth of the CCD sensor, resulting in low bandwidth utilization and reconstruction resolution.⁴ On the contrary, when the selected tilt angle is inadequate, +1-order and −1-order spectra are completely separated but partially overlapped with the zero-order spectrum, resulting in image artifact and degradation of the reconstructed image quality. Furthermore, when the +1-order and −1-order spectra are just separated and fully overlapped with the zero-order spectrum, i.e., the slightly off-axis DH, the bandwidth occupation of +1-order spectrum can be maximized to 1/2, significantly improve the reconstruction resolution.^5,6 Therefore, in the case of a finite spatial bandwidth product (SBP) of CCD sensor, how to make full use of the SBP completely eliminate the overlapped zero-order spectrum and accurately extract the +1-order spectrum is a key to achieving artifacts-free, high-resolution image reconstruction in off-axis DH.

Many methods have been proposed to suppress or eliminate the zero-order spectrum of off-axis DH, mainly include (1) Iterative calculation of the object wave intensity method based on prerecorded reference wave intensity;^7–9 (2) Fourier space spectrum filtering or extracting methods;^10–12 and (3) multiple exposures subtraction methods.^6,13–18 Although the above-mentioned first and second kinds of method can eliminate most zero-order spectrum effectively in the off-axis DH, the reconstruction resolution is still low due to limited utilization of SBP. In the case of large or fully overlap between the ±1-order and zero-order spectra, these methods will fail. For the two-shot subtraction methods^6,13–16 based on the phase-shifting, the measurement accuracy is limited by the phase shift error of PZT and polarization elements or extracted phase shift error. Significantly, Kramers–Kronig relations are exploited to achieve high SBP imaging in slightly off-axis DH.¹⁹ This method requires a prerecorded image of the reference beam, and the intensity of the reference beam needs to be much stronger that of the object beam.

In this paper, we propose a high-resolution image reconstruction for the spectrum-overlapped off-axis digital holography. By utilizing the intensity symmetry of the Gaussian laser beam, two off-axis holograms are obtained by symmetrically deflecting the reference beam, and their frequency spectra are subtracted from each other to achieve zero-order spectrum elimination. Figure 1 shows the principle of eliminating the zero-order spectrum of the slightly off-axis DH system by symmetrically deflecting the Gaussian reference beam. The laser beam emitted by He–Ne laser is divided into an object beam and a reference beam by a beam splitter (BS1) after passing through a neutral density filter (NDF) and a 10× beam expander (BE1). Here, NDF is used to reduce the light intensity of the laser beam, so as to avoid the over-exposure of the CCD. Then, the object beam is reflected by a mirror M1 in turn to illuminate the specimen to be measured. The reference beam is expanded by another 5× beam expander (BE2) and interferes with the object beam on a CCD camera (1624 × 1228 pixels, pixel size of 4.4 μm) to generate a hologram. The deflection of the reference beam is controlled by a piezoelectric actuator (PA) loaded on M2. In order to ensure that the reference beam spot completely covers the entire photosensitive surface of CCD (being about 7.15 × 5.40 mm² in size) during the deflection process, the diameter of the reference beam is expanded to be about 50 mm through BE1 and BE2.

From Eq. (2), the multiplexed spectrum image F₃ contains four spectral components. If the four spectral components are separated from each other, any +1-order spectrum can be extracted from F₃ accurately. However, to ensure that the four spectral components do not overlap with each other, the following two conditions should be met: the distance between the centers of two +1-order spectra should be larger than the spatial bandwidth ω₊₁ of a single +1-order spectrum, and ω₊₁ cannot exceed half of the spatial bandwidth of the CCD sensor. In the off-axis and the slightly off-axis DH systems, the center offset and the spatial bandwidth ω₊₁ of the +1-order spectrum can be, respectively, changed by adjusting the angle between the object beam and reference beam and the recording distance. Here, separation of the four spectral components can be achieved by controlling the deflection angle of the reference beam and the distance between the specimen and CCD. Therefore, the proposed method is applicable to all the off-axis and the slightly off-axis DH systems satisfying the above-mentioned conditions.

Figures 1(a)–1(c) show the incident angles between two beams and corresponding spectrum distributions of the slightly off-axis holograms with ω₊₁ equal to half of CCD bandwidth. Here, the black rectangle represents the photosensitive surface of CCD, and the pink circle represents the reference beam spot. Initially, the object beam O and the reference beam R₀ are set to be obliquely and vertically incident on the CCD surface, respectively, and the recording distance is adjusted to enable ω₊₁ to be half of the CCD bandwidth. The ±1-order spectra are distributed along 135° direction in the spatial Fourier domain, as shown in Fig. 1(a). Then, by deflecting the reference beam to left and right with an angle of θ and −θ sequentially with respect to the x axis, the ±1-order spectra will move to the vertical center and left/right sides (under-sampling), respectively, as shown in Figs. 1(b) and 1(c). As shown in Fig. 1(d), by subtracting the hologram spectra F₂ from F₁, the zero-order spectrum can be eliminated, and the four ±1-order spectra without overlapping can be obtained, each occupying 1/2 of the CCD bandwidth.

Obviously, this maximum deflection angle θ vary with the size of CCD surface. The larger the size of CCD is, the smaller angle θ will be. In the setup shown in Fig. 1, the theoretical θ is about 6.93°. However, in the experiment, when the spectrum bandwidth of ω₊₁ is adjusted to be half of the CCD bandwidth, the initial angle between the object and reference beam is limited to 2.06°. Therefore, the actual deflection angle is only 2.06°, which can be easily realized by a piezoelectric actuator.

The framework of the proposed method is shown in Fig. 2. Here, due to the influences of environmental disturbances, laser power fluctuations, and deflecting angle errors of the reference beam in real application, the intensities of the zero-order terms of two holograms are slightly different,²⁰ resulting in a small amount of residual zero-order spectrum. To solve this problem, we adopt region recognition to further eliminate the residual zero-order spectrum. As shown in Fig. 2, the multiplexed spectrum F₃ is binarized to generate a mask B₁; the foreground region closest to the spectrum center in B₁ is identified as the residual zero-order spectral region B₂; the spectrum F₄ is obtained by using B₂ to remove the residual zero-order spectrum in F₃, and then the +1-order spectral component OR^* in F₄ is extracted; finally, OR^* is numerically reconstructed by angular spectrum algorithm to obtain the specimen amplitude and phase maps.

Before validate the performance of the proposed method, the elimination effects of the zero-order spectrum were evaluated by using two holograms of a USAF 1951 resolution test target recorded by in a DH setup constructed according to Fig. 1. As shown in Figs. 3(a)–3(c), the power weights (sum of squares of intensity) of 160 × 50 pixels areas centered on the zero-order spectra of images of F₁, F₃, and F₄ were calculated, which are 2.2921 × 10¹⁵, 8.3023 × 10¹¹, and 1.1380 × 10¹¹, respectively. It can be seen that through direct subtraction of two spectra, most zero-order spectrum is eliminated and the suppression ratio of power weights achieves 3.62 × 10⁻⁴. After performing the region recognition, the suppression ratio further improves to 4.96 × 10⁻⁵. This indicates that the proposed method can suppress the zero-order spectrum very effectively, which is beneficial for the subsequent accurate extraction of the desired +1-order spectral component OR^*.

Then, three different samples were measured to verify the effectiveness of the proposed symmetrical deflection of the reference beam (SDRB) method. First, in the DH setup, the reconstruction results of the USAF 1951 resolution test target, respectively, obtained by the SDRB method, the traditional off-axis holography method (OAH), and the subtraction method based on reference wave polarization adjustment (RWPA)¹⁸ were compared. For the OAH experiment, the recording distance was set as 188.5 mm to avoid overlapping of the ±1-order spectra with the zero-order spectrum. For slightly off-axis DH experiments of RWPA and SDRB, as the ±1-order spectra can partially or completely overlap with the zero-order spectrum, the recording distance was reduced to 55.7 mm to make the spatial bandwidths of the ±1-order spectra as large as possible. After extracting the +1-order spectra with three method, same angular spectral diffraction, phase unwrapping algorithm,²¹ and phase aberration compensation²² were performed to reconstruct the sample.

Figures 4(a1)–4(a3) show the hologram spectra obtained by OAH, RWPA, and SDRB, respectively, where the white rectangles represent the spatial filtering windows for extracting the +1-order spectra. From Fig. 4(a1), the spatial bandwidth of the +1-order spectrum obtained by OAH is 1/4 of the CCD bandwidth, and the spatial bandwidth utilization (SBU) is 9.82%.²³ In Figs. 4(a2) and 4(a3), the spatial bandwidths of the +1-order spectra obtained by RWPA and SDRB can be close to 1/2 of the CCD bandwidth, and their SBU can reach 39.27%.

To qualitatively and quantitatively compare the reconstructed results obtained by our method and other methods, the profile intensity curves of the group 4 element 4 and group 5 element 6 are added to the intensity maps shown in Figs. 4(c1)–4(c3). The visibility values ²⁴ and the lateral resolutions of the group 4 element 4 region (35 × 80 pixels) and the group 5 element 6 region (11 × 27 pixels) are measured and summarized in Table I. This indicates that RWPA and SDRB can identify the group 5 element 6, corresponding to the actual resolution of 8.77 μm, while OAH cannot achieve the resolution of 8.77 μm due to poorer visibility.

As shown in Fig. 4(c2), some local fringe noises can be observed in the reconstructed intensity map of RWPA, which is caused by the residual zero-order spectral component during the polarization adjustment. Compared with OAH, the SBU and the actual resolution of SDRB have been improved by 4 and 2.5 times (22.10 μm/8.77 μm), respectively. Figures 4(d1)–4(d3) are the phase maps reconstructed from Figs. 4(a1)–4(a3), where the phase standard deviations of the white rectangle background areas are 0.2093, 0.1897, and 0.1168 rad, respectively. It demonstrates that the proposed SDRB method has strong noise suppression ability.

Second, a digital holographic microscopy (DHM) setup was constructed by adding a 4× microscope objective (MO) with 0.10 NA between the specimen and BS2, and an ovarian slice was measured to further evaluate the performance of the proposed method. Figure 5(a1) shows the 2D structure of the ovarian cells observed by an optical microscope (VHX-6000; 100×). Figure 5(a2) is the initial hologram spectrum of the ovarian cells acquired in the slightly off-axis DHM experiment. In the hologram spectra shown in Figs. 5(b1) and 5(b2), the white rectangles represent the spatial filtering windows for extracting the +1-order spectra.

Comparing Figs. 5(a2), 5(b1), and 5(b2), the zero-order spectral components decrease significantly in sequence. Figures 5(c1) and 5(c2) show the intensity maps of the ovarian cells reconstructed by RWPA and SDRB, respectively. The fringe-shaped imaging artifacts caused by the overlapping of the residual zero-order spectrum with the +1-order spectrum can be clearly observed in the enlarged red rectangle inset of Fig. 5(c1) but are invisible in Fig. 5(c2). Figures 5(d1) and 5(d2) show the phase maps of the ovarian cells reconstructed by RWPA and SDRB, respectively. Similarly, the fringe-shaped phase artifacts covering the ovarian cells are evident in the enlarged red rectangle inset in Fig. 5(d1) but are almost undetectable in Fig. 5(d2). The phase standard deviations of the background regions marked by the white rectangles are 0.7830 and 0.4157 rad, respectively. These results indicate that the phase map reconstructed by SDRB has a lower noise level and a flatter background region, which is attributed to SDRB's better ability to eliminate the zero-order spectrum.

At last, a microlens array (LBTEK, MLAS10-F05-P150-AB) was measured using the slightly off-axis DHM setup with a 10× MO. Figures 6(a1) and 6(a2) are the hologram spectra obtained by RWPA and SDRB, respectively. Obviously, RWPA still retains some zero-order spectral components, while SDRB eliminates almost all the zero-order spectral components, as shown by the enlarged red rectangle insets in Figs. 6(a1) and 6(a2). Consequently, massive fringe-shaped imaging artifacts appear in the RWPA-reconstructed intensity map [Fig. 6(b1)] but are not apparent in Fig. 6(b2). Figure 6(c1) is the 3D profile of the microlens array reconstructed by RWPA with localized phase artifacts and deformations. Figure 6(c2) is the 3D profile reconstructed by SDRB with smooth and artifact-free microlens surfaces. In addition, the profile curves obtained by RWPA and SDRB are extracted along the black lines in Figs. 6(c1) and 6(d1) and compared with the measurement result of a white light interferometer (WLI) (Zygo, NewView 9000), as shown in Figs. 6(d1) and 6(d2). It can be seen that the profile curve obtained by SDRB matches better with that obtained by WLI.

In summary, we have proposed a spectrum-overlapped off-axis holography reconstruction method. Using the intensity symmetry of the Gaussian laser beam, the two holograms recorded by symmetrically deflecting the reference beam have the same intensity. By the subtraction of the two hologram spectra, most of the zero-order spectral components can be eliminated. Combining region recognition to further automatically locating and removing the residual zero-order spectral components, the +1-order spectrum can be extracted accurately. Experimental results show that SDRB can not only suppress the imaging artifacts caused by the zero-order spectrum overlapping but also achieve high spatial bandwidth utilization of DH and DHM systems. Being achievable the maximum utilization of half-space bandwidth of the CCD sensor, the proposed method can realize high-resolution phase quantitative measurement, which will have important applications in off-axis/slightly off-axis DH/DHM systems.

Compared with the typical two-shot subtraction phase-shifting methods, such as pi-phase and random phase, which need to accurately control or calculation of the phase-shifting amount and record two holograms for each measurement, the proposed method has significant advantages: (1) it does not need to precisely control the deflection angle of reference beam. This is because the deflecting angle error only makes the intensities of the zero-order terms of two holograms slightly different, which can be easily solved by later region recognition; (2) the reference hologram (H₁) is only required initially. During the measurement, only one measurement hologram (H₂) is needed, and the +1-order spectrum of the measurement hologram can be extracted in real time, achieving single-shot measurement. Therefore, the proposed method is applicable to the dynamic or real-time measurements.

We wish to acknowledge the National Natural Science Foundation of China (NSFC) (No. 52035015), the Zhejiang Science and Technology innovation leading talent project (No. 2022R52052), and the Natural Science Foundation of Zhejiang Province (No. LQ23E050020).