## Abstract

Modern optical imaging possesses a huge information capacity whose corresponding space-bandwidth product (SBP) reaches tens of megapixels. However, despite the advances in optical and electronic devices, the SBP of an optical microscope is greatly limited, resulting in a reduced field of view or resolution of an image. In this paper, we exploit the Kramers–Kronig relations in digital holography to achieve high SBP imaging, demonstrating a complex amplitude image that can surpass the SBP of a bright-field image. The capability of the proposed method is demonstrated by imaging static samples and biological tissues. We successfully measure a 4.2-megapixel complex amplitude image whose bright-field counterpart exhibits 16.7 megapixels.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. INTRODUCTION

The space-bandwidth product (SBP) [1] of an imaging system describes the amount of information generated by the system. In microscopy, the SBP is defined by the product of the area of the field of view (FoV) and the area of the spatial frequency band of an image. Although the SBP of a modern microscope reaches up to tens of megapixels, the practical SBP is limited to only a few megapixels. This is because the number of pixels in a commercially available image sensor is much lower than the SBP the optical microscope can provide. Thus, the majority of optical information remains undetected, limiting the advance of microscopy for large-scale high-throughput imaging, which is essential for digital histology [2], drug discovery [3], and cell biology [4].

Recently, several methods have attempted to overcome the limitations of the SBP. Based on phase-retrieval algorithms, iterative methods such as Fourier ptychography [5,6], out-of-focus imaging [7], and pixel super-resolution [8] offer enhanced SBPs in imaging that exceed SBPs of detectors. However, there are certain prerequisites if these methods are to obtain correct images; Fourier ptychography requires images to have data redundancy in the frequency domain, out-of-focus imaging needs a weakly scattering sample and images at different axial positions, and the pixel super-resolution method requires precisely controlled subpixel shifts between images. Furthermore, since these methods enhance SBPs by combining information from multiple images, they require multiple measurements and have low SBPs per measurement.

An interesting approach related to this issue can be found in radio communication. Single-sideband modulation adopts a complex amplitude signal to reduce signal bandwidth [9], exploiting the fact that the bandwidth of a complex amplitude signal is smaller than that of a real-valued signal. Since the same principle holds in imaging [10], a similar strategy can be considered in optical microscopy; one can measure a complex amplitude image instead of an intensity image to obtain a higher SBP, because the bandwidth of a complex amplitude image is less than that of an intensity image at the same FoV.

Holographic imaging is an effective way to acquire a complex amplitude image with a microscope and to investigate transparent microscopic objects such as cells and tissues [11–13]. In particular, off-axis holography [14,15] enables single-shot measurement of a complex amplitude image. This is achieved with an off-axis interferogram that isolates the complex amplitude image from its twin image, and autocorrelation in the spatial frequency domain. However, as a result, an off-axis interferogram has a broad bandwidth, which is inefficient regarding the SBP of the complex amplitude image.

To improve the SBP of off-axis holographic imaging, various techniques have been suggested. For example, methods utilizing multiplexing [16–18] and aliasing [19] show efficient uses of the finite bandwidth of an image sensor. Alternatively, reductions in the interferogram bandwidth are achieved by subtracting the autocorrelation using multiple interferograms [20,21], or by suppressing it through numerical methods [22,23], speckle [24], a wavelet transform [25], Hilbert transform [26], or an optimization algorithm [27], at the cost of sample constraints. Nonlinear filtering methods based on the Fresnelet transform [28], the cepstrum [29], an iterative algorithm [30], and a local weighted least squares estimation [31] suppress the autocorrelation without imposing a condition on a sample. However, they require time-consuming iterative processes, *a priori* information about samples, or certain imaging conditions. The Fresnelet method requires a Fresnel hologram and different thresholding of Fresnelet coefficients depending on the sample. The cepstrum method and the iterative algorithm require a strong reference and a complex amplitude image to be confined within a quadrant of the spatial frequency domain. The least squares estimation needs modification in weighting windows and sampling conditions depending on local spatial frequency of an interferogram. Furthermore, none of the previous studies has demonstrated the capability of off-axis holographic imaging to surpass bright-field imaging regarding the SBP.

Here, we propose and experimentally demonstrate a general method for high SBP holographic imaging exploiting the Kramers–Kronig (KK) relations. To retrieve a complex amplitude image, we introduce a complex analytic function whose real part is described by an off-axis interferogram. Then the KK relations recover the complex analytic function, from which the complex amplitude image is retrieved. The proposed method is guaranteed by the analyticity of the complex function that does not impose a constraint on a sample. Importantly, the analyticity holds even when the complex amplitude image is completely overlapped with its autocorrelation in the frequency space of the interferogram. Thus, the proposed method requires a greatly reduced interferogram bandwidth, achieving high SBP imaging beyond what has been demonstrated in conventional off-axis holographic imaging. We demonstrate the capability of the proposed method by imaging various samples, including a resolution target, polystyrene beads, human breast tissue, and mouse brain tissue. Furthermore, by maximizing the capability of the method with anamorphic imaging, we successfully measured a 4.2-megapixel complex amplitude image whose bright-field counterpart exhibits an SBP (16.7 megapixels) greater than the SBP of the detector (12 megapixels).

## 2. PRINCIPLE

#### A. Conventional Off-Axis Holographic Imaging

In conventional off-axis holographic imaging, a reference beam $R(\mathit{r})$ interferes with a sample beam $S(\mathit{r})$ with a slight tilt angle $\theta $. The reference beam is a quasi-plane wave with a wavelength of $\lambda $, which is expressed as $R(\mathit{r})={R}_{0}{e}^{i2\pi {\nu}_{R}\xb7\mathit{r}}$, where ${\nu}_{R}$ is the spatial frequency vector of $R(\mathit{r})$ with $|{\nu}_{R}|=\mathrm{sin}\text{\hspace{0.17em}}\theta /\lambda $. Then the two beams form an interferogram, $I(\mathit{r})={|S+R|}^{2}$, whose Fourier transformation can be written as

where $\tilde{A}(\nu )=\mathcal{F}[{|S|}^{2}+{|R|}^{2}]$, $\tilde{B}(\nu )=\mathcal{F}[{R}_{0}S]$, and $\mathcal{F}$ indicates Fourier transform. In Eq. (1), $\tilde{B}(\nu -{\nu}_{R})$ is centered at ${\nu}_{R}$ due to the spatial modulation of $R(\mathit{r})$; conventional off-axis interferometry utilizes this translation to isolate $\tilde{B}(\nu -{\nu}_{R})$ from $\tilde{A}(\nu )$.The signal isolation is achieved by choosing a large ${\nu}_{R}$ to avoid overlaps among the three terms in Eq. (1). The exact condition for ${\nu}_{R}$ can be determined by considering the pupil function of an imaging system. For an ideal imaging system having a numerical aperture of an objective lens of ${\mathrm{NA}}_{\mathrm{obj}}$ and a transverse magnification of $M$, the pupil function $\tilde{P}(\nu )$ can be expressed as

#### B. Kramers–Kronig Holographic Imaging

The KK relations connect the imaginary and real parts of a complex function that is analytic in the upper half-plane (UHP) [33,34]. For example, a function $f$($\omega $) that is analytic in the UHP of $\omega $ satisfies

The analyticity condition for $\chi (\mathit{r})$ can be found by rewriting it using a Taylor expansion,

where $|\beta (\mathit{r})|<1$ is assumed. Since $\chi (\mathit{r})$ is a power series of $\beta (\mathit{r})$, the analyticity of $\chi (\mathit{r})$ is guaranteed by the analyticity of $\beta (\mathit{r})$ [35,36]. If we assume an off-axis configuration where $R(\mathit{r})={R}_{0}{e}^{-i2\pi {\nu}_{R}\xb7\mathit{r}}$, $\beta (\mathit{r})$ is expressed as## 3. EXPERIMENTAL RESULTS

#### A. Holographic Imaging With and Without a Spectral Overlap

To demonstrate the capability of the proposed method, various samples are imaged, under the condition where $\tilde{A}(\nu )$ and $\tilde{B}(\nu -{\nu}_{R})$ severely overlap in the frequency space; $|{\nu}_{R}|={\mathrm{NA}}_{\mathrm{obj}}/\lambda M+\u03f5$ with $\u03f5>0$. Interferograms of a 1951 United States Air force (USAF) resolution target (#59-153, Edmund Optics, Barrington, New Jersey, USA) and a 10-μm-diameter polystyrene bead (72986, Sigma-Aldrich, Missouri, USA) are imaged. The experimental setup is shown in Fig. S2. Polystyrene beads are immersed in an index-matching oil with a refractive index of 1.56. Next, complex amplitudes are reconstructed by the proposed method and by the conventional off-axis method, respectively. For comparison, an identical interferogram per sample is used for the two methods. To remove the complex amplitude of illumination, each complex amplitude image of a sample is divided with that measured in the absence of a sample throughout the experiment.

The experimental results are shown in Fig. 1. In the proposed method, the USAF resolution target shows uniform amplitudes and phases at the position of patterns [Figs. 1(a) and 1(b)]. The result of the polystyrene bead shows uniform amplitude and a spherical phase distribution that are consistent with the known shape of the sample [Figs. 1(c) and 1(d)]. The polystyrene bead has a refractive index of 1.575 (at $\lambda =632.8\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{nm}$), resulting in a phase delay of 2.73 rad, which is consistent with the result [see inset in Fig. 1(d)].

However, in the conventional method, the amplitude and phase of both samples are severely distorted [Figs. 1(e)–1(h)]. The USAF resolution target exhibits stronger noise than for the case of the polystyrene bead, because of the broad spectrum of $\tilde{A}(\nu )$. This is expected from an off-axis interferogram whose fringe pattern has low spatial frequency [Fig. 1(i)]. Due to weak spatial modulation, $\tilde{A}(\nu )$ and $\tilde{B}(\nu -{\nu}_{R})$ are severely overlapped [Fig. 1(j)]. In such a condition, only the analyticity in Eq. (8) is satisfied, not the separation condition in Eq. (3). Thus, the proposed method can reconstruct correct complex amplitude images, while the conventional off-axis method cannot.

To investigate the validity of the proposed method in a conventional condition, the two methods are compared when $\tilde{A}(\nu )$ and $\tilde{B}(\nu -{\nu}_{R})$ are separated in the frequency space; $|{\nu}_{R}|>3N{A}_{\mathrm{obj}}/\lambda M$. Again, the USAF resolution target and a 10-μm-diameter polystyrene bead are used for imaging. The experimental results of the two different methods are shown in Fig. 2. The upper row corresponds to the result of the proposed method [Figs. 2(a)–2(d)], and the bottom row corresponds to the result of the conventional off-axis method [Figs. 2(e)–2(h)]. Both methods show clear holographic images of the USAF resolution target and polystyrene bead. Again, the measured phase value of the polystyrene bead agrees with the expected value of 2.73 rad [see insets in Figs. 2(d) and 2(h)]. More importantly, the results of the two different methods agree with each other. It confirms that the proposed method reconstructs identical information with the conventional off-axis field retrieval method. In this case, the off-axis interferogram has a fringe pattern with high spatial frequency [Fig. 2(i)]. The strong spatial modulation of the reference beam separates $\tilde{B}(\nu -{\nu}_{R})$ from $\tilde{A}(\nu )$ [Fig. 2(j)]. Consequently both the separation condition in Eq. (3) and the analyticity condition in Eq. (8) are satisfied.

#### B. Wide Field of View Imaging

The biggest advantage of the proposed method is the enhancement in the SBP. Due to the reduced bandwidth of an interferogram, a large amount of information can be measured with a detector having a finite bandwidth and a fixed number of pixels. As an experimental demonstration, wide FoV imaging is conducted with human breast tissue. The tissue is prepared with the standard procedure of formalin fixation and paraffin embedding. For imaging, an objective lens with an NA of 0.4 is adopted (UPLSAPO $10\times 2$, Olympus, Tokyo, Japan). The total magnification of the setup is kept at 10, identical to the magnification of the objective lens.

The measured quantitative phase image is shown in Fig. 3(a). The interferogram is imaged when $\tilde{A}(\nu )$ and $\tilde{B}(\nu -{\nu}_{R})$ are completely overlapped [Fig. 3(b)]. The FoV of the image is $1.32\times 0.88\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{mm}$ with a diffraction-limited size of 0.96 μm. Insets (i, ii) clearly show fine structures of the tissue. The SBP of the complex amplitude image is 1.46 megapixels [the area of the FoV, $1.16\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\mathrm{mm}}^{2}$, multiplied by the area of spatial frequency band, $\pi {(\mathrm{NA}/\lambda )}^{2}$]. Figure 3(b) shows the Fourier spectrum of the measured off-axis interferogram. The conventional off-axis method cannot be adopted in this condition, because the diameter of $\tilde{A}(\nu )$ ($0.25\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\mathrm{\mu m}}^{-1}$) takes 78% of the lateral bandwidth of the detector ($0.32\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\mathrm{\mu m}}^{-1}$), making it impossible to avoid the spectral overlap.

The SBP of the proposed method can be further improved by introducing anamorphic imaging. The bandwidth required for the proposed method in the direction perpendicular to the modulation is half the bandwidth required in the modulation direction (see Supplement 1 for details). By magnifying the image differently in $x$ and $y$ directions, the SBP can be further increased. As a proof of concept, wide FoV imaging in the anamorphic condition is demonstrated. First, the experimental setup is modified as in Fig. S2(b). The anamorphic imaging is achieved with two sets of $4\text{-}f$ telescopic imaging arrays using cylindrical lenses. The same objective lens as in the case of human breast tissue is adopted. The total magnification of the setup is 8.333 and 4.166 in $x$ and $y$ directions, respectively. Mouse brain tissue, prepared according to the protocol in [38], is imaged. The aberration induced by the cylindrical lenses has been numerically compensated (see Supplement 1).

The measured quantitative phase image is shown in Fig. 4. We found a diagonal fringe in the phase originating from the surface reflection between two coverslips, due to variation in optical thickness across the wide FoV [Fig. 4(a)]. This artifact is suppressed with the assumption that the sample does not exhibit a structure larger than 100 μm (details in Supplement 1).

The corrected image is shown in Fig. 4(b). The FoV of the images is $1.58\times 2.11\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{mm}$ with a diffraction-limited size of 0.96 μm. The SBP of the measured complex amplitude image is 4.2 megapixels. Figure 4(c) shows the Fourier spectrum of the off-axis interferogram. The term $\tilde{B}(\nu -{\nu}_{R})$ completely occupies the vertical bandwidth of the detector, and $\tilde{A}(\nu )$ exceeds the bandwidth of the detector. Consequently, the conventional off-axis method suffers from spectral overlap (see Supplement 1). In addition, bright-field microscopy will not be free from aliasing at this imaging condition. In Fig. 4, however, bright-field microscopy is unsuitable because the sample is a 3-μm-thick unstained brain tissue slice. Thus, the sample is highly transparent, which can be visualized with quantitative phase images, not with intensity images. On the contrary, the proposed method enables correct retrieval of a complex amplitude image. Furthermore, the bright-field counterpart of the measured complex amplitude image has an SBP of 16.7 megapixels. Considering that the SBP of the detector is 12 megapixels, the amount of information measured in the proposed method exceeds not only that of the conventional off-axis method, but also that of bright-field microscopy. In Fig. 5, the SBPs per measurement of various methods are shown as a function of the pixel count of a detector. It is assumed that the pixel count of a detector becomes a bottleneck to SBPs. Each line indicates the maximum available SBP per measurement for a given detector pixel count. Thus experimental values lie below their corresponding lines. Iterative high-SBP techniques [5] exhibit relatively low SBP per measurement because of data redundancy and multiple measurements. Instead, they take advantage of simple setups without the need of interferometry. Data points of references are calculated using the following values. For out-of-focusing imaging pixel super-resolution (OFI-PSR) [7], $4.6\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\mathrm{mm}}^{2}$ FoV, 0.3 NA at 532 nm, five images, and the pixel count of 1.45 million are used. For Fourier ptychography [5], $120\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\mathrm{mm}}^{2}$ FoV, 0.5 NA at 632 nm, 137 images, and the pixel count of 23 million are used. For pixel super-resolution, $24\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\mathrm{mm}}^{2}$ FoV, 0.5 NA at 500 nm, 36 images, and the pixel count of five million.

## 4. DISCUSSION AND CONCLUSION

We have demonstrated a method for high-SBP off-axis holographic imaging. Exploiting the KK relations, the proposed method retrieves an exact complex amplitude without imposing any constraint on a sample. It efficiently utilizes the interferogram bandwidth to provide a 3.34- to 4-fold increase in the SBP compared to the conventional method. Furthermore, with anamorphic imaging, the bright-field counterpart of the measured complex amplitude exhibits an SBP greater than the SBP of the detector.

The proposed method is fundamentally different from previous methods for zero-order suppression under the spectral overlap in off-axis geometry. The proposed method requires a single interferogram. This is in contrast to the previous method, where multiple interferograms are required [20,21,24,39,40]. In addition, the proposed method provides an analytic expression for the complex amplitude without imposing a constraint on a sample. Thus, the retrieval process is simple, and the measured complex amplitude is exact. Existing methods require assumptions for a sample, such as uniform intensity [22,26], weak phase [23,41], weak phase gradient [25], or slowly varying complex amplitude [27,42], to guarantee correct measurement of complex amplitudes.

Despite similarities with the nonlinear filtering method [29], the proposed method does not require a complex amplitude image to be confined within a quadrant of the frequency space. On the other hand, the proposed method requires the analyticity condition of the KK relations, which allows a reduced interferogram bandwidth that is less than that demonstrated in the previous study. Importantly, the proposed method does not require a bandwidth that is more than the size of a pupil function in the direction perpendicular to modulation. This feature enables a further enhanced SBP by adopting anamorphic imaging. Also, by generalizing the principle to three dimensions, e.g., using $\chi (x,y,{\nu}_{z})$ and its analyticity in UHP of ${\nu}_{z}$, the proposed method can provide a framework that can embrace the different principles for off-axis [29] and on-axis [43] configurations. The proposed method requires an image of the reference beam in addition to an off-axis interferogram. Nevertheless, the image of the reference beam needs to be measured once. Thus, the proposed method effectively becomes a single-shot method with a stable setup. This becomes particularly useful for large FoV image stitching or time-lapse imaging, where a large number of images are used. It should also be noted that the proposed method is available with spatially coherent light.

The proposed method can be combined with other techniques, such as synthetic aperture microscopy [44], optical diffraction tomography [45–47], and common-path off-axis interferometers [48–50]. We envision that the proposed method will benefit holographic imaging with an enhanced SBP and become an effective tool for 3D imaging, high-throughput histopathology, and large-scale studies of cells and tissue.

## Funding

National Research Foundation of Korea (NRF) (2015R1A3A2066550, 2017M3C1A3013923, 2018K000396, 2018R1A6A3A01011043); KAIST; Tomocube; BK21+ program.

## Acknowledgment

We thank Seonggeun Kim for the helpful discussion. We also thank Lunit Inc. for providing the breast tissue slide.

See Supplement 1 for supporting content.

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