## Abstract

In this paper, we present a phase-shifting-free method to improve the resolution of digital holographic microscopy (DHM) under the structured illumination (SI). The SI used in the system is different from the traditional SI for it is free of the visible structure due to two illumination lights with orthogonal polarization states. To separate the recorded information and also retrieve the object phase, two reference beams with different carrier frequencies and orthogonal polarization states are adopted. The principle component analysis (PCA) algorithm is introduced in the reconstruction process. It is found that the modulated frequency of SI besides the quadratic phases of the imaging system can be easily removed with help of PCA. Therefore, phase-shifting is not required both in recording and reconstruction process. The simulation is performed to validate our method, while the proposed method is applied to the resolution enhancement for amplitude-contrast and phase-contrast objects imaging in experiments. The resolution is doubled in the simulation, and it shows 78% resolution improvement in the experiments.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

In recent years, DHM, as one of amplitude-contrast and phase-contrast imaging techniques, becomes more and more popular in the biomedical imaging [1–3]. Its spatial resolution is determined by the numerical aperture of imaging system and the illumination wavelength according to the Abbe’s theory [4]. From this theory, there are two ways to improve the resolution, which are using a short wavelength illumination and enlarging the numerical aperture. The deep ultraviolet illumination has been applied to DHM, obtaining the nanoscale imaging [5]. To enlarge the numerical aperture, the synthetic aperture digital holography [6–8], is an effective technique, but multiple holograms are required to reconstruct the superresolved image. Although the modified synthetic aperture digital holography can record one-shot hologram by using the wavelength [9], spatial [10], angular and polarization [11,12] multiplexing techniques, those setups are still complex to be built. Recently, a single-shot synthetic aperture DHM using angular-multiplexing and coherence gating technique is presented [13], however, its system has strict requirements on the optical path differences of any two beams. It is necessary to pursue a simpler method to DHM for resolution enhancement.

Structured illumination microscopy (SIM) [14–16], with the advantages of wide-field, long working distance, less photo-toxicity and flexibility, has been introduced to a quasi-common path DHM [17]. There is no doubt DHM under SI simplifies the synthetic aperture setup although half of object field has to be used as the reference wave. A Mirau microscope objective has been applied to SI-DHM with full object field [18]. The two proposed systems are simple in building, but they both require at least three holograms with phase-shifting of structured illumination in each orientation and phase-shift algorithms to reconstruct the resolution-enhanced image. To reduce the recording times, Ma et al. [19] have proposed a DHM system with the structured illumination generated by a grating, where two holograms per direction are required to obtain super-resolution image. Besides, a SI-DHM by using Fresnel’s biprism to control the frequency of structured illumination has been proposed [20], which also requires two holograms to improve the resolution in one orientation by an iterative algorithm. A free of phase-shifting SI-DHM scheme has been presented [21], but it needs prior knowledge of the distribution of SI and two holograms per direction to reconstruct the resolution-enhanced image. It is shown that the regular SI-DHM usually needs at least two holograms in each orientation, and a phase-shifting algorithm with the known phase-shift amounts and the accurate carrier frequency estimation of SI are required to respectively decompose and synthesize the spectrums. Phase-shifting limits the application in the field of real-time imaging.

To get the phase-contrast image of SI-DHM, the additional phase brought by SI and the quadratic phase aberration of microscopic imaging system have to be removed. Although the phase of the SI can be calculated before the experiment, it will increase the additional recording time. Compared with the phase of SI, the quadratic phase aberration can be compensated by physical [22,23] and numerical methods [24–26]. Zuo et al. [27,28] have presented a fast and automatic quadratic phase aberration compensation method in DHM based on the principle component analysis (PCA) algorithm. Due to its good performance, we think about utilizing the PCA algorithm to accurately shift the recorded information to the right position besides dealing with the quadratic phase.

In this paper, we propose a single-shot recording system for SI-DHM, the extended high-frequency components can be recorded simultaneously without overlap by the polarization and angular multiplexing techniques. Subsequently, the PCA algorithm is applied to compensate the phase aberration as well as shift the spectrum to the right position. Thus, an accurate synthesized spectrum can be obtained without the known carrier frequency information of SI. That is to say, the phase-shifting is replaced with the polarization encoding and multiplexing techniques and spectrum-shifting is implemented by the PCA compared to the traditional SI-DHM. The principle of the proposed method is explained in Section 2. In Section 3 and Section 4, the simulation and the experiments are performed respectively to validate the proposed method. The paper is summarized in Section 5.

## 2. Principle

In general, the optical field distribution of the one dimensional (1D) two-beam structured illumination [29] can be written as

*O*is the complex amplitude of the sample. As a matter of convenience, the recording plane coincides with the image-plane of the imaging system. In the SI-DHM, the object beam will be usually a slight spherical wave even if there is a plane-wave design that uses the compensated lenses in setup without the precise calibration, which will introduce a quadratic phase factor [30]. The object wave-field in the recording plane is therefore given by

*x*and

*y*are the coordinates of the recording plane, and $M$ is the lateral magnification time of the imaging system. ${h}_{q}\left(x-{x}_{0},y-{y}_{0}\right)=h\left(x-{x}_{0},y-{y}_{0}\right){P}_{q}\left(x,y\right)$ is the point-spread function of the imaging system, where

*h*is the point-spread function without the quadratic phase factor of the imaging system, and ${P}_{q}\left(x,y\right)=\text{exp}\left[i\beta \left({x}^{2}+{y}^{2}\right)\right]$ is the quadratic phase factor in the image-plane, where the factor$\beta $actually describes the spherical phase curvature introduced by the imaging system. Thus, the Eq. (3) can be expressed as

First, the spectrum regions of the first two components of Eq. (9) are filtered out and relocated to the spectrum center based on the rough spectrum peaks, respectively. Both two spectrum regions are cropped for obtaining two cropped spectrums with the size of *a* × *b* (the whole spectrum with size of *A* × *B*). Second, the IFFTs of two cropped spectrums are performed to obtain the reconstructed object waves, which can be written as

_{${Q}_{1}\left(x\right)=\text{exp}\left(i{l}_{1}\cdot x\right){P}_{q}\left(x\right)$}and ${Q}_{2}\left(x\right)=\text{exp}\left(i{l}_{2}\cdot x\right){P}_{q}\left(x\right)$ are the phase aberrations with the cropped size, where ${l}_{1}$ and ${l}_{2}$ describe the residual tilt phase differences caused by the inaccurate spectrum centering. Third, the PCA algorithm is performed based on the singular value decomposition (SVD) [27]. The SVDs of the phases of ${O}_{R-}$ and ${O}_{R+}$ can be respectively expressed as where ${U}_{j}$ and ${V}_{j}$ (

*j*= 1, 2) are the orthonormal matrices with the sizes of

*a*×

*a*and

*b*×

*b*, and $\sum {}_{j}$ are the diagonal matrices consisting of the singular values, which can be written as

*min*is the minimum value of

*a*and

*b*. In general, the reconstructed phase of Eq. (10) or Eq. (11) without the phase aberration can be regarded as the small perturbation to the whole reconstructed phase distribution. So, the phase aberrations can be regarded as the dominant phase components of Eq. (10) or Eq. (11). The ${\sigma}_{j,1}$ is selected and the other singular values are set to zero, and the dominant components in ${Z}_{R-}$ and ${Z}_{R+}$ can be extracted, which are ${Q}_{1}$ and ${Q}_{2}$. Their conjugates, ${Q}_{1}^{*}$ and ${Q}_{2}^{*}$, are easily obtained. Thereby, two reconstructed object waves are compensated and added up, and the sum can be written as

To obtain an isotropic enlargement of the CTF, in general, the above same procedure need to be applied to three different orientations [32]. In fact, two orthogonal orientations are enough to show 2D resolution enhancement. In this paper, 2D structured illumination (along the ${x}_{0}$, ${y}_{0}$ direction) is applied to show the resolution improvement in 2D spatial domain.

## 3. Simulation

A numerical simulation of SI-DHM is performed to validate our proposed method, as shown in Fig. 1. A phase-modulated sample whose spatial period ranges uniformly from 2 μm to 14 μm is used as the simulated object. The quadratic phase function ${Q}_{\text{pha}}=\text{exp}\left[i\beta \left({x}^{2}+{y}^{2}\right)\right]$ is the quadratic phase aberration of DHM, where $\beta =1.2\times {10}^{\text{-}4}$*μm*^{−1}. The cut-off frequency of the DHM system is 12 *μm*^{−1}, which means its resolution is 12 *μm*. We first apply the horizontal (1D) SI to DHM for demonstrating the spatial resolution enhancement. To double the resolution of DHM, the carrier frequency of SI is set as same as the cut-off frequency of the system. The compound hologram is formed by interfering between the object wave and two reference waves with different carrier frequencies, as shown in Fig. 1(a). From the magnified hologram, two group patterns can be clearly seen. Its Fourier spectrum is shown in Fig. 1(b), from which we can see that two recorded object spectrums are separated completely. The distance between the zero-order and object spectrum is determined by the carrier frequency of the reference wave (${\alpha}_{-}$,${\alpha}_{+}$). The peak of the object spectrum is shifted from the aperture center by the carrier frequency ${f}_{0}$ of SI and we can see that the peaks of two object spectrums become not sharp. Even if the quadratic phase aberration remains unchanged in experiment and multiple phase-shift holograms are recorded to reconstruct phase image using phase-shift algorithm similar to regular SI-DHM [17], the accurate estimation of SI’s carrier frequency becomes difficult and the phase aberration will inevitably reside in the reconstruct phase image. Two object spectrums are filtered and synthesized based on the rough central frequencies, then the reconstructed phase image is obtained, as shown in Fig. 1(c). It can be seen from it that the object phase is overwhelmed by the quadratic phase aberration of imaging system and the residual tilt phase aberration. The PCA algorithm is adopted to eliminate these unwanted phase aberration and implement the spectrum-shifting based on the accurate central frequencies. We use a local high-resolution discrete Fourier transform [33] to calculate the central frequencies of two compensated spectrums, and the pixel difference between the integer pixel peak and the sub-pixel peak is zero when the up-sampling factor equals to sixteen. That is to say, the central frequency is precisely located in the integer pixel in Fourier domain, which makes spectrums synthesizing very easy and accurate. Thus, the horizontal synthesized spectrum can be obtained by simply adding two compensated spectrums, as shown in the inset of Fig. 1(d), from which we can see that the synthesized spectrum is enlarged along the horizontal orientation. The circle in the inset denotes the passband of the regular CTF. The reconstructed resolution-enhanced phase image without phase aberration by PCA algorithm is shown in Fig. 1(d), from which we can see that the vertical line pairs of the third group (with spatial period of 6 *μm*) can be resolved. For comparison, the reconstructed phase image by regular DHM is shown in Fig. 1(e), which shows that only the line pairs above the sixth group (with the spatial period of 12 *μm*) can be resolved. Comparing the Fig. 1(d) with Fig. 1(e), we can find that the horizontal resolution of SI-DHM using PCA algorithm is improved by a factor of two, but the vertical resolution of that deteriorates. This is because that 1D two-beam SI only extends the high-frequency information of object spectrum along the SI’ orientation while losing the high-frequency information perpendicular to the SI’ orientation, just as the synthesized spectrum in Fig. 1(d).

To obtain 2D resolution enhancement, the vertical SI is also applied to the DHM and another compound hologram is obtained, the above same procedure is used to obtain the vertical synthesized spectrum. Then the horizontal and vertical synthesized spectrums are added up to obtain the 2D synthesized spectrum, as shown in the inset of Fig. 1(f), which shows that the bandwidth of CTF is extended along two orthogonal orientations and the original object spectrum does not have the loss. The reconstructed 2D resolution-enhanced phase image is shown in Fig. 1(f). Comparing the Figs. 1(f) and 1(e), we can see that the 2D resolution enhancement is achieved. The phase distributions along the colorful lines of the third group line pairs in Figs. 1(d)-1(f) are plotted in their insets. As can be clearly seen from them, the resolution of SI-DHM using PCA algorithm is improved by a factor of two compared to the regular DHM.

## 4. Experiments and results

The proposed SI-DHM recording system is a modified Mach-Zehnder interferometer, as shown in Fig. 2, where He-Ne laser (with the wavelength λ = 632.8 nm) is used as the light source. After being expanded and collimated (BEC), the light is divided into two beams by a beam splitter (BS1). One beam served as the object beam is reflected by a mirror (M2) and illuminates the phase grating (G, with the spatial frequency of 300 Lp/mm). A filter (F) is inserted at the Fourier plane of a 4f system consisting of lens L1 and lens L2, which blocks the 0th order diffraction light from the grating and makes the polarization states of ± 1st orders diffraction lights mutually orthogonal. The structure of the filter is shown in the inset of Fig. 2, where we can see that rotating the polarizer films denoted by the blue disks can make the polarization states of ± 1st orders light mutually orthogonal. The double-headed arrow denotes the orientation of polarization states. The focal lengths of L1 and L2 are both 100 mm. The carrier frequency of SI is determined by the grating frequency and the magnification time of the 4f system. The object (O) then is magnified by the imaging system composed of a microscope objective (MO) with NA = 0.25 and the tube lens (TL) with focal length 200 mm. The other beam going through BS1 is further divided into two reference beams by the polarized beam splitter (PBS). The relative carrier frequency of two reference beams in spatial-frequency domain is decided by the adjustment on the horizontal and vertical angles of PBS and M3; the relative carrier frequency between two reference beams and the object beam is decided by the adjustment on the horizontal and vertical angles of BS2. In this way, the carrier frequencies (${\alpha}_{-}$,${\alpha}_{+}$) of two reference beams can be adjusted freely. The interference pattern between the magnified object beam and two reference beams is recorded by the CCD detector with 1280 × 960 pixels (pixel size: 3.75 × 3.75*μm*). Three samples are applied to perform the experiments for demonstrating the proposed SI-DHM system and the reconstruction algorithm. The theoretical resolution limit is given by 0.77*λ*/NA [34] in regular DHM system, which is 513 Lp/mm. To show how much the resolution improve utilizing the USAF 1951 resolution test target, we add a aperture in front of the tube lens to restrict the resolution of regular DHM.

We use the USAF 1951 resolution test target, as the amplitude-contrast sample, for showing how much the spatial resolution of the proposed system improves. Similar to the simulation, only the horizontal (1D) SI is applied to illuminate the sample. The intensity distribution and Fourier spectrum of the recorded compound hologram are shown in Figs. 3(a) and 3(b). From the red zoomed square area in Fig. 3(a) or the spectrum components in Fig. 3(b), we can see that there are two group interference patterns in the compound hologram. The amplitude images reconstructed by the SI-DHM using the direct spectrum synthesizing, the regular DHM and the SI-DHM using the PCA algorithm are shown in Figs. 3(c)-3(e) respectively, where the sections included in the red squares are zoomed to show the details. From the Fig. 3(d), we can see that the Element 4 Group 8 (362 Lp/mm) above the green line can be completely resolved. That is to say, the actual resolution of the regular DHM is limited to 362 Lp/mm by the aperture. It is shown that there is the severe distortion in Fig. 3(c) even if the spatial resolution is improved to some degree. However, the Fig. 3(e) shows a resolution-improved amplitude image without distortion compared to the Fig. 3(c). Besides, from the Fig. 3(e), we can find that the resolution only is improved along the SI’s orientation. In order to better show the 2D resolution enhancement of the proposed method, two shots is needed and the 2D resolution-enhanced amplitude image is obtained by the proposed method, as shown in Fig. 3(g). From this figure, we can see that the unresolved Element 3 Group 9 (645 Lp/mm) in regular DHM can be completely resolved by SI-DHM utilizing the PCA algorithm. It shows that the enhanced resolution limit (645 Lp/mm) basically reaches at the expected resolution limit, that is, 662 Lp/mm. The horizontal plots along the blue line in Fig. 3(d) and the red line in Fig. 3(e) are shown in Fig. 3(f). The horizontal and vertical plots along the blue lines in Fig. 3(d) and the red lines in Fig. 3(g) are shown in Figs. 3(h) and 3(i) respectively. From them, it is clear to see that the spatial resolution of regular DHM obtains 78% resolution enhancement by the SI-DHM based on the PCA algorithm.

The polystyrene particle-cluster (with particle diameter of 1.97 μm) also has been used to validate the 2D resolution enhancement of the proposed method. In order to obtain the 2D resolution enhancement, 2D structured illumination is applied to the regular DHM (with the resolution power of 362 Lp/mm). Thus, two compound holograms are recorded to reconstruct the resolution-enhanced images. The reconstructed amplitude images by the regular DHM, the SI-DHM using direct spectrum synthesizing and the SI-DHM using the PCA are shown in Figs. 4(a), 4(c) and 4(e), respectively, and the corresponding reconstructed phase images are shown in Figs. 4(b), 4(d) and 4(f), respectively. We can see that, from Figs. 4(c) and 4(d), although the resolution improvement can be seen, there are the severe phase aberrations caused by the quadratic phase aberration and inaccurate spectrum synthesizing in the reconstructed complex amplitude. In contrast, the Figs. 4(e) and 4(f) show the undistorted resolution-enhanced complex amplitude. Furthermore, it is shown that the polystyrene particle-cluster in Figs. 4(a) and 4(b) turn into the distinguishable polystyrene particles. The amplitude distributions along the red lines in Figs. 4(a) and 4(e) are plotted in Fig. 4(g), and the phase distributions along the red lines in Figs. 4(b) and 4(f) are plotted in Fig. 4(h). It is clear that the unresolved amplitude and phase of polystyrene particle can be completely resolved by the SI-DHM based on the PCA algorithm compared to the regular DHM.

The label-free cat neurons sample has been applied to validate the proposed method in the field of biological sample. Under the second experiment conditions, the cat neurons are imaged by the proposed SI-DHM system. The reconstructed phase images using the regular DHM and the SI-DHM by the PCA are shown in Figs. 5(a) and 5(b). From the red zoomed square area, we can see that the ambiguous dendrite and the unresolved neuron cells in Fig. 5(a) become distinguishable in Fig. 5(b). The plots along the red lines in the red squares of Figs. 5(a) and 5(b) are shown in Fig. 5(c), which shows that two adjacent neuron cells unresolved in the regular DHM can be well resolved in the SI-DHM by the PCA.

From the above experiment results, we can find that there are some artefacts in the reconstructed images obtained by the proposed method. In our opinion, they are caused by the background noises, which comes from the non-uniform illumination light, the flaws of optical element and the stains of samples, and so on. From the Eq. (6) and Eqs. (16)-(17), we can see that the synthesized 1D CTF is composed of two displaced regular CTFs. That is to say, the synthesized 2D CTF is the superposition of four regular CTFs. So, the background noises of the reconstructed images by the proposed method are the superposition of multiple background noises within the regular CTFs, which makes the reconstructed image by the proposed method have more background noises than the regular DHM image. In order to reduce the background noise from the source, we can make the illumination more uniform and keep the optical elements clean, and etc. Furthermore, we can subtract the reconstructed images from the background image, which is obtained by reconstructing the hologram that does not contain the sample information using the proposed method. By this way, the artefacts of reconstructed image can be avoided.

## 5. Conclusion

In conclusion, a resolution enhancement method with SI-DHM by the polarization encoding and angular multiplexing techniques based on the PCA algorithm in single-shot per direction has been presented. The simulation experiment has been performed to initially validate the proposed method, and then three experiments using the different samples have been performed to validate the application in the amplitude- and phase- contrast sample imaging and the biological sample imaging. The two dimensional amplitude and phase resolution enhancement has been obtained by recording only two holograms without phase-shift algorithm. Additionally, the fast undistorted high resolution amplitude- and phase- contrast imaging has been achieved by applying the PCA algorithm in the reconstruction process, which shows the potential applications in the high resolution real-time imaging.

## Funding

National Natural Science Foundation of China (NSFC) (61775097, 61605080, 61465005); National Key Research and Development Program (2017YFB0503505); Open Foundation of Key Lab of Virtual Geographic Environment (Nanjing Normal University), Ministry of Education (2017VGE02).

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