## Abstract

An important quest in optical imaging has been, and still is, extending the resolution of imaging systems beyond the diffraction limit. We propose a superresolution technique in which the image is first blurred by a scattering mask, and then recovered from the blurry data with improved resolution. We introduced a scattering mask into the space between the observed objects and the objective lens of a Fresnel incoherent correlation holography (FINCH) system to demonstrate the method. Optical waves, containing high spatial frequencies of the object, which are usually filtered out by the limited system aperture, were introduced into the system due to the scattering nature of the scattering mask. As a consequence, both the effective numerical aperture and the spatial bandwidth of the system were enlarged. The image resolution could therefore be improved far beyond the resolution limit dictated by the limited numerical aperture of the system. We demonstrated the technique using a modified FINCH system and the results were compared with other systems, all having the same aperture dimensions. We showed a resolution enhancement in comparison to conventional FINCH and regular imaging systems, with the same numerical apertures. The theoretical and experimental data presented here establishes the proposed method as an attractive platform for an advanced superresolution system that can resolve better than conventional imaging systems.

© 2017 Optical Society of America

## 1. INTRODUCTION

Imaging is the technique of recording a visual representation of an object [1–4]. One of the important goals in imaging is to be able to resolve as small as possible visual details of the object. However, in optical imaging, the barrier in achieving this goal is the diffraction limit i.e., the inability to collect all of the light diffracted from the object due to the limited aperture size of the imaging system [5]. Different techniques have been developed to surpass the diffraction limit [6–13]. These methods usually gain the resolving power at the expense of either the field of view (FOV) or the time resolution. In other words, either the resolving power is gained only in a relatively small area out of the entire object area [7], or relatively many images of the same object, under different conditions, are acquired over time [8–13].

Holography is an indirect method of imaging that provides the ability to record and reconstruct 3D information of an object [14]. An incoherent digital holography technique termed Fresnel incoherent correlation holography (FINCH) was invented in 2007 [15–17]. FINCH has been proved to resolve images better than a non-holographic lens-based imaging system with the same numerical aperture (NA) [17]. This peculiarity of FINCH arises due to its operation mode of recording holograms by a self-interference. Hence, FINCH breaks the classical limits governed by the Lagrange invariant condition [18] and enables, under specific conditions [17,18], resolution beyond the diffraction limit by theoretical factors of 2 and 1.5, in comparison with conventional coherent and incoherent systems with the same NA, respectively. Recently, an optical configuration of FINCH with structured illumination was demonstrated to resolve images beyond the resolution limit of FINCH dictated by the NA of the system [19].

In this study, we propose a new technique to gain the resolving power of the FINCH system beyond the diffraction resolution limit. In the proposed technique, a random-like coded phase mask (CPM) is introduced between the object and the FINCH system. The CPM, when displayed in front of the object, diffuses part of the incident light into the system, including the high spatial frequencies that are otherwise discarded by the system due to its limited NA. By capturing the otherwise-lost high spatial frequencies, the effective NA is increased beyond the physical NA of the system. Consequently, the effective spatial bandwidth of the system is enlarged, and smaller details of the object can be resolved. Since the proposed method is FINCH enhanced by the CPM, it is dubbed coded FINCH (C-FINCH).

Although resolution enhancement of FINCH is demonstrated herein, the current methods of hologram recording and reconstruction are similar to another technique of incoherent hologram recording, known by the term coded aperture correlation holography (COACH) [20]. Like FINCH, COACH can be classified as an incoherent self-reference holography system in which the beam radiated from the object is split into two beams. One of these beams passes through a CPM displayed on a spatial light modulator (SLM). From the SLM, the beam propagates to the sensor plane on which it interferes with another beam that comes from the same object but without being modulated by the CPM. The intensity distribution of the two-beam interference is stored in the computer as a digital hologram. At least two holograms must be recorded in this method. Before recording the hologram of an object, one should record at least one point spread hologram (PSH). The PSH is a hologram of a point source located in the center of the plane on which the object is displayed. Both the PSH and the object hologram are recorded under the same conditions and with the same CPM. The PSH is used in the digital reconstruction stage, such that the image is digitally reconstructed by correlating the object hologram with the PSH. These techniques of recording and reconstruction are also applied in the present C-FINCH method for superresolution.

There is also some similarity between C-FINCH and Kim’s work on imaging with an incoherent self-reference holography system through a scattering medium [21,22]. However, unlike Refs. [21,22] where the scattering medium is a given unchangeable disturbing element, in C-FINCH, the scattering medium is artificially synthesized and controllably changed, not only to image the object covered by this scattering medium but also to resolve the image with better image resolution than in the case without the scattering medium.

Another pioneer study that employs a random pattern to achieve superresolution is the turbid lens imaging (TLI) technique proposed by Choi *et al.* [23,24]. In TLI, the effective NA is increased, and thus the image resolution is improved due to scattering of waves with high spatial frequencies by the turbid media. The purpose of the CPM presented in our study is similar to the turbid media. However, TLI and C-FINCH are different in almost any other aspect. TLI operates only with coherent light, whereas C-FINCH works only with incoherent light. Additionally, the training process of the system and the reconstruction procedure are different, and apparently more efficient, in C-FINCH. Because of the use of incoherent light, the C-FINCH technique can be applied for fluorescence and other incoherent microscopy techniques. Moreover, unlike [23,24], in this study, we have investigated the behavior of C-FINCH under different degrees of scattering.

The advantages of the proposed method are: (1) A resolution enhancement of more than triple is achieved in the present experiment, but theoretically the system aperture can stop being a limiting factor of the image resolution. (2) Relatively short time of image acquisition is employed. (3) The imaging system is passive without emitting any signal or radiation toward the object, and it operates well with any kind of incoherent illumination (e.g., fluorescent illumination). (4) The process of capturing the images is completely motionless, performed from a single viewpoint and without any scanning. (5) The method can be applied to any imaging application that can be implemented by a FINCH system or by any other self-interference hologram recorder. (6) The resolution enhancement is controllable such that a user can adjust electronically between the parameters of resolving power versus the size of the FOV.

The paper consists of four sections. In the second section, the methodology of this study is presented. The experimental procedure is discussed in the third section. The comparison results of the proposed technique with regular imaging and FINCH are presented in the final section. In addition, detailed analysis of the geometry of C-FINCH is presented in Appendix A.

## 2. METHODOLOGY

The scheme illustrated in Fig. 1(a) describes the optical configuration of the C-FINCH. A pinhole is critically illuminated [25] by a spatially incoherent and quasi-monochromatic light source using the lens ${L}_{0}$. Thus, light diffracted from every point on the object plane can interfere only with itself, and it cannot interfere with light diffracted from a different point, enabling the implementation of the self-interference principle. The light emitted from an object point is incident on the CPM located at a distance of ${d}_{1}$ from the object plane. The CPM is computed by the Gerchberg–Saxton algorithm (GSA) [26], illustrated in Fig. 2. For every iteration, two constraints are defined in the spatial and frequency domains. In the CPM domain, the complex amplitude is constrained to be a pure phase because the CPM is a pure phase mask. In the spectrum domain, the constraint is such that the complex amplitude should have a uniform magnitude over the desired area of the spatial spectrum with the bandwidth of $2B=2{B}_{x}=2{B}_{y}$. This bandwidth controls the degree of scattering of the CPM. When $B$ is maximal, the CPM scatters the incident light along all the possible angles up to the angle $\pm \lambda /2\mathrm{\Delta}$, where $\lambda $ is the central wavelength of the illumination, and $\mathrm{\Delta}$ is the pixel size of the CPM. We define the ratio $\sigma =B/{B}_{\mathrm{max}}$ (where $2{B}_{\mathrm{max}}$ is the maximal spectral bandwidth) as the scattering degree, which is a value that can be varied between 0 and 1. It is shown later that the scattering degree $\sigma $ controls the trade-off between the resolution enhancement and the FOV size. The CPM on which a pure random phase (after $n$ iterations of the GSA) is produced with an effective diameter $D$ diffuses the incident light and acquires the high frequencies that are otherwise discarded by the limited NA of the system. The effective diameter is the diameter of the part of the CPM that participates in the imaging process, i.e., the light scattered from this part is introduced into the FINCH section of the system and recorded by the camera. A refractive lens ${L}_{1}$ is mounted at a distance of ${f}_{0}$ from the object plane to collect the light diffused by the CPM and projects the beams onto an SLM located right behind ${L}_{1}$. The polarization of the light diffracted by the pinhole is oriented at 45° with respect to the active axis of the SLM using a polarizer ${P}_{1}$. On the SLM, a diffractive lens with a focal length of ${f}_{d}$ is displayed. Since the SLM active axis is oriented at 45° with respect to the polarization of the incident light, two optical channels are created corresponding to the two orthogonal polarization directions. Hence, the SLM is employed as a beam splitter of two orthogonal polarization orientations. A refractive lens ${L}_{2}$ with a focal distance of ${f}_{o}$ is located behind the SLM, implementing the dual lens FINCH configuration [27].

Unlike regular dual lens FINCH, the two spherical waves propagating beyond ${L}_{2}$ in C-FINCH are distorted due to the scattering nature of the CPM. The two scattered waves propagate to distances of ${f}_{0}$ and ${f}_{1}$ from the lens ${L}_{2}$ and create two scattered images. A second polarizer ${P}_{2}$ oriented at an angle of 45°, with respect to the active axis of the SLM, enables interference between the two scattered waves. An image sensor located at a distance of ${z}_{h}$ from ${L}_{2}$ is used to record the hologram. The bias terms and the twin image are canceled from the hologram using the phase shift method, similar to [15]. This phase-shift process involves displaying three diffractive lenses, each with a focal length ${f}_{d}$ but with three phase constants of ${\theta}_{1,2,3}=0,2\pi /3$, and $4\pi /3$. The three recorded raw holograms are stored in the computer and are superposed according to Eq. (2). Consequently, a complex hologram of the point object, denoted as ${H}_{\mathrm{PSH}}$, is obtained.

The hologram acquisition is repeated for the object, with the same system arrangement and the same CPM as in the case of ${H}_{\mathrm{PSH}}$, by replacing the pinhole by the object exactly at the same axial location. Three holograms are recorded again corresponding to the three phase masks displayed on the SLM, and the complex hologram resulting from their superposition is denoted ${H}_{\mathrm{OBJ}}$. The image of the object is reconstructed by cross-correlating ${H}_{\mathrm{OBJ}}$ with a phase-only filtered version of ${H}_{\mathrm{PSH}}$ in order to improve the signal-to-noise ratio (SNR) [28]. Note that it is sufficient to record the ${H}_{\mathrm{PSH}}$ only once as part of the training of the system, and the same ${H}_{\mathrm{PSH}}$ can be used to reconstruct an unlimited number of object holograms (located at the same axial distance).

Next, we describe the process of the hologram acquisition and reconstruction more rigorously in order to evaluate the various parameters of the imaging. As mentioned, there are two optical elements displayed on the SLM corresponding to the two different linear polarizations: a diffractive lens with a focal length ${f}_{d}$, and a constant phase. Therefore, in the absence of the CPM [see Fig. 1(b)], there are two imaging systems in the same optical channel corresponding to the two orthogonal polarizations. The images of the object are obtained at two different planes, ${S}_{1}$ and ${S}_{2}$, at the respective distances of ${f}_{0}$ and ${f}_{1}=1/[1/{f}_{d}+1/{f}_{0}]$ from the lens ${L}_{2}$. The camera plane is located between ${S}_{1}$ and ${S}_{2}$, such that there is a perfect overlap between the interfering beams on the camera plane. As explained in Ref. [27], this condition of beam overlap guarantees maximum resolving power in the case of classical FINCH. Although this condition is not valid for C-FINCH, we fulfill this condition to compare the imaging results of C-FINCH with that of optimal FINCH. The overlap condition imposes that the distance between the lens ${L}_{2}$ and the camera is, ${z}_{h}=2{f}_{o}{f}_{1}/({f}_{o}+{f}_{1})$.

When the CPM is introduced into the system [Fig. 1(a)] at a distance of ${d}_{1}$ from the object plane, and the system is illuminated from the object plane by a single point located at $({x}_{s},{y}_{s})$ with the amplitude of $\sqrt{{I}_{j}}$, the two image planes of the point object (without the CPM) become the Fourier planes of the CPM [14]. Moreover, the centers of these Fourier distributions are located at the images of the point object. In other words, on plane ${S}_{1}$, around the point $(-{x}_{s},-{y}_{s}){f}_{1}/{f}_{o}$, there is a Fourier transform of the CPM (multiplied by a quadratic phase function) with a diameter of about $\lambda {d}_{1}\sigma {f}_{1}/({f}_{o}\mathrm{\Delta})$. On plane ${S}_{2}$, around the point $(-{x}_{s},-{y}_{s})$, there is another Fourier transform of the CPM (multiplied by a different quadratic phase function) but with a diameter of about $\lambda {d}_{1}\sigma /\mathrm{\Delta}$. The camera is placed between the planes ${S}_{1}$ and ${S}_{2}$, a relatively short distance from both of them. Therefore, based on geometrical considerations, we can assume that on the camera plane, beyond the polarizer ${P}_{2}$, there is a sum of two distributions; both are centered on the point $(U,V)=(-{x}_{s},-{y}_{s}){z}_{h}/{f}_{o}$, and both have the same diameter of about ${D}_{I}=\lambda {d}_{1}\sigma {z}_{h}/({f}_{o}\mathrm{\Delta})$. Thus, the intensity distribution recoded by the camera is

Few imaging properties can be deduced from this analysis. Equation (7) indicates that the lateral magnification of C-FINCH is ${M}_{T}={z}_{h}/{f}_{o}$. The minimum resolved size on the image plane is the correlation length of the cross-correlation of Eq. (7), which is given by the smallest element that can be created on the camera plane. This smallest feature size cannot be deduced directly from Eq. (7) but is estimated by the following reasoning. Since planes ${S}_{1}$ and ${S}_{2}$ are in Fourier relations with the CPM, with the scaling factors of $\lambda {d}_{1}$ and $\lambda {d}_{1}{f}_{1}/{f}_{o}$, respectively, the relation between the camera plane and the CPM can be approximated to a Fourier relation with the scaling factor of $\lambda {d}_{1}{z}_{h}/{f}_{o}$. For an effective diameter of the CPM of $D$, the smallest feature size that can be measured on the camera plane has the size of $\lambda {d}_{1}{z}_{h}/(D{f}_{o})={M}_{T}\lambda /(2\text{\hspace{0.17em}}\mathrm{tan}\text{\hspace{0.17em}}{\theta}_{m})$, where ${\theta}_{m}$ is the maximal diffraction angle of the central point object recorded by the camera (see Appendix A). Therefore, the minimum resolvable size on the object plane is $\lambda /(2\text{\hspace{0.17em}}\mathrm{tan}\text{\hspace{0.17em}}{\theta}_{m})$. This result confirms that the CPM extends the effective NA of the C-FINCH by scattering light into the system, which is otherwise discarded by the limited aperture size of the FINCH system. The maximal diffraction angle ${\theta}_{m}$ of the central source point recorded by the camera is given in Appendix A by Eq. (A1) as follows,

Another important feature of imaging is the FOV. Assuming the FOV in the object domain without the presence of the CPM has dimensions of ${V}_{0}\times {V}_{0}$, then on the camera, taking into consideration the magnification factor, the FOV has dimensions of ${M}_{T}({V}_{0}\times {V}_{0})$. As the intensity distribution of any point source increases at the camera plane with the increase of the degree of scattering, the area of the hologram becomes larger than that of the regular FINCH. The increment in the area of the hologram is additive since, according to Eq. (5), the C-FINCH hologram is a convolution of ${H}_{\mathrm{PSH}}$ with the image, without the CPM, at the camera plane. Therefore, the size of the C-FINCH hologram is the sum of the FINCH hologram with the size of ${H}_{\mathrm{PSH}}$. Consequently, from the FOV of FINCH, one should subtract the size of ${H}_{\mathrm{PSH}}$ projected to the object plane. As mentioned, when the CPM is illuminated by a point source from the object plane, the dimensions of the intensity distribution on the camera are about ${D}_{I}\times {D}_{I}$, where ${D}_{I}=\lambda {d}_{1}\sigma {z}_{h}/({f}_{o}\mathrm{\Delta})$. Therefore, the FOV size of C-FINCH at the object plane is ${V}_{c}\times {V}_{c}$, where

## 3. EXPERIMENTAL PROCEDURE

To demonstrate the resolution capabilities of C-FINCH and to compare its performance with that of FINCH and regular imaging, the experimental setup shown in Fig. 3 was constructed with the ability to switch electronically between the various imaging systems without altering the setup and without any mechanical movements. Two identical lasers (Helium–Neon lasers $\lambda =632.8\text{\hspace{0.17em}}\mathrm{nm}$), denoted as ${\mathrm{LS}}_{1}$ and ${\mathrm{LS}}_{2}$, were spatially filtered by two spatial filter modules, ${\mathrm{SF}}_{1}$ and ${\mathrm{SF}}_{2}$, respectively. The back focal plane of the lens ${L}_{0-1}$ is located on the object plane (OP), so that the pinhole of ${\mathrm{SF}}_{1}$ is imaged on the OP to ensure the same axial distance for both the object and the pinhole. By this way, the first channel illuminated by ${\mathrm{LS}}_{1}$ is employed as the PSH channel, and the second channel illuminated by ${\mathrm{LS}}_{2}$ is employed as the object channel. The light emitted by ${\mathrm{LS}}_{2}$ is converted to a spatially incoherent source by transmitting the light through a rotating diffuser denoted as RD. The lens ${L}_{0-2}$ is used to project the source on the object in a mode of critical illumination. The Iris located in front of ${\mathrm{SF}}_{2}$ controls the size of the illuminated area on the OP, according to the required FOV. The beam splitter ${\mathrm{BS}}_{1}$ is used to combine the light from the two illumination channels. A negative resolution chart (1951 USAF resolution test chart) is mounted at the OP, and the pinhole (5 μm in diameter) is imaged on it from channel 1. The pinhole is imaged on the center of the observed region on the object. The light from the pinhole is transferred into the system through a transparent region of the chart. Beyond the OP the light is polarized by a polarizer ${P}_{1}$ oriented at 45° with respect to the active axis of ${\mathrm{SLM}}_{2}$. From ${P}_{1}$, the light is reflected by a beam splitter ${\mathrm{BS}}_{2}$ on a phase only ${\mathrm{SLM}}_{1}$ (Holoeye PLUTO, $1920\times 1080\text{\hspace{0.17em}}\mathrm{pixels}$, 8 μm pixel pitch, phase-only modulation) employed as the CPM, which is located at an axial distance of ${d}_{1}=125\text{\hspace{0.17em}}\mathrm{mm}$ from the OP. ${\mathrm{SLM}}_{1}$ is also rotated by 45° with respect to ${\mathrm{SLM}}_{2}$ so that its active axis is collinear with the incident polarized light. By doing so, maximum modulation of the incident light is guaranteed. Using ${\mathrm{SLM}}_{1}$ for displaying CPM allows electronic control of the CPM produced by the mentioned GSA. The light modulated by the CPM is incident, through the lens ${L}_{1}$ on ${\mathrm{SLM}}_{2}$ (same as ${\mathrm{SLM}}_{1}$), on which the three diffractive lenses (${f}_{d}=520\text{\hspace{0.17em}}\mathrm{mm}$) with three different phase shift values, $\theta =0,2\pi /3$, and $4\pi /3$, are displayed. The refractive lens ${L}_{1}$ (${f}_{0}=250\text{\hspace{0.17em}}\mathrm{mm}$) implements both lenses ${L}_{1}$ and ${L}_{2}$ of Fig. 1. As a result, the light scattered by ${\mathrm{SLM}}_{1}$ is collected by ${L}_{1}$ mounted close to ${\mathrm{SLM}}_{2}$ in order to transfer maximum scattered light to ${\mathrm{SLM}}_{2}$. As a result, the modulated beam converges to the distance ${f}_{1}=168.8\text{\hspace{0.17em}}\mathrm{mm}$ from the lens ${L}_{1}$. The system aperture is determined by the diffractive lens displayed on ${\mathrm{SLM}}_{2}$, which is limited in the present experiment by a rectangular mask of $280\times 280\text{\hspace{0.17em}}\mathrm{pixels}$ ($2.24\times 2.24\text{\hspace{0.17em}}\mathrm{mm}$). From ${\mathrm{SLM}}_{2}$ the light is reflected by ${\mathrm{BS}}_{2}$ and meets a second polarizer ${P}_{2}$, oriented at 45° with respect to the active axis of ${\mathrm{SLM}}_{2}$. Because of the orientation of ${P}_{2}$, the modulated and unmodulated beams reflected by ${\mathrm{SLM}}_{2}$ can interfere with each other. The image sensor (GigE vision GT Prosilica, $2750\times 2200\text{\hspace{0.17em}}\mathrm{pixels}$, 4.54 μm pixel pitch) located at the hologram plane at a distance of ${z}_{h}=201.6\text{\hspace{0.17em}}\mathrm{mm}$ from ${\mathrm{SLM}}_{2}$ (determined according to the mentioned overlap condition) records the holograms. The polarization direction of ${P}_{2}$ is perpendicular to the orientation of ${P}_{1}$, and therefore the incident light arriving directly through ${\mathrm{BS}}_{2}$ from the OP is blocked. The holograms of the pinhole and the object were recorded for the three phase shift values and superposed according to Ref. [15] to create complex holograms for the object and the pinhole.

The space between adjacent pixels and the fill factor of ${\mathrm{SLM}}_{1}$ yields a relatively intense unscattered beam, reducing the effect of the scattered light. To cancel the effect of the unscattered beam, the same two CPMs were displayed on ${\mathrm{SLM}}_{1}$ with phase shifts of $\varphi =0,\pi $, and the resulting holograms are superposed. As a result, six exposures in total are needed for recording each of the object and the pinhole complex holograms. To improve the SNR of the reconstructed image and to reduce the effects of other noise sources (e.g., stray light, CCD noise), five sets of holograms were recorded for both the pinhole and the object with five different independent CPMs. The resulting five complex reconstructed images from the correlation of the corresponding ${H}_{\mathrm{PSH}}s$ and ${H}_{\mathrm{OBJ}}s$ were averaged to improve the SNR [28].

The same setup in Fig. 3 is used to implement the conventional FINCH and the regular imaging systems for the purpose of comparison with C-FINCH. Switching between the various systems is done electronically by displaying various phase masks on the SLMs as the following: in the case of the conventional FINCH, ${\mathrm{SLM}}_{1}$ is inactivated and hence is considered as a constant phase plate. To implement a regular imaging system, ${\mathrm{SLM}}_{1}$ is operated as a planar mirror, and a diffractive lens satisfying the imaging condition from the object to the CCD planes is displayed on ${\mathrm{SLM}}_{2}$.

## 4. RESULTS AND DISCUSSION

The comparison results of C-FINCH for different values of $\sigma $, regular FINCH, and regular imaging are shown in Fig. 4. To demonstrate the resolution enhancement, two different experiments were conducted, each with a different purpose. In the first experiment, the resolution enhancement in C-FINCH was studied by analyzing the improvement in the visibility of the reconstructed images for various scattering degrees, as shown in Figs. 4(a1)–4(a9) and 4(b1)–4(b9). In the second experiment, we looked for the smallest lines on the chart that can be still resolved for each value of the scattering degree $\sigma $. It can be noted that in the case where $\sigma =0$, the imaging is actually done by a regular FINCH. Figures 4(a1)–4(a9) demonstrate the increase of visibility due to the improvement in the resolution. The graphs presented in Figs. 4(b1)–4(b9) show the intensity profile along the dashed red lines of Figs. 4(a1)–4(a9). Figures 4(c2)–4(c9) and 4(d2)–4(d9) show the smallest lines that can be resolved and the magnitude of the complex PSH, for various scattering degrees, respectively.

Visibility is used as a measure of resolution according to the Rayleigh resolution criterion in which the capability to resolve two points is dependent on the visibility of the two-point image, i.e., the ratio of the difference between the peak and the valley (between the points of the image) and the sum of the peak and the valley. It is evident from the visibility plots that the resolution of the C-FINCH is better than FINCH and regular imaging. It can also be noted that with the increase in the scattering degree, the resolution is improved. However, as expected, with a gradual increase in the resolution, there is a gradual loss of FOV. Thus, C-FINCH is a superresolution method that enhances the resolution more with the increase of the scattering degree of the CPM, at the expense of the FOV size reduction. Figure 5(a) presents the plot of the visibility values measured from Figs. 4(b2)–4(b9) versus the eight different values of the scattering degree.

In the second experiment, we investigate the smallest resolvable element in the case of the eight scattering degrees. Thus, the spatial bandwidth increase can be estimated. Figure 5(b) presents a comparative graph of the cutoff frequency ${\nu}_{c}$ calculated theoretically by $\mathrm{sin}{\theta}_{m}/\lambda $, where $\mathrm{sin}{\theta}_{m}$ is given by Eq. (A1) (presented as the dash linear graph), where all the distances in Eq. (A1) have been corrected for the path through the ${\mathrm{BS}}_{2}$. The experimental measurements from Figs. 4(c2)–4(c9) are presented in Fig. 5(b) by the blue rings.

As mentioned, to improve the SNR of the final images presented in Fig. 4, five complex reconstructed images corresponding to five different uncorrelated CPMs were averaged. To estimate the SNR improvement, an object in the form of a square is selected. FINCH and C-FINCH (with scattering degree of 0.44) were implemented, and the SNR of a single reconstruction and average of five complex reconstructions were calculated as shown in Fig. 6. This averaging technique is only a tool to enhance the SNR and is not a strict requirement of the C-FINCH technique.

C-FINCH is different from other diffraction-limited superresolution methods [8–13] in the sense that to improve the resolution, it does not acquire more visual information of the same scene. The spatial bandwidth of the image is enlarged, but a single acquired hologram contains the entire enlarged spatial spectrum of the object. Therefore, the overall information amount of C-FINCH is identical to that of FINCH with the same aperture size. The space-bandwidth product (SBP) is known as a measure of information quantity. Because C-FINCH acquires a single complex hologram with the same FINCH setup, the SBP of both systems is identical. Since it is evident theoretically and experimentally that the system bandwidth of C-FINCH is larger than of FINCH, the FOV size (“S” in the SBP) of C-FINCH must be smaller, as has indeed been demonstrated both theoretically and experimentally herein.

Working with a programmable SLM with a variable degree of scattering enables one to make the trade-off between the image resolution and the FOV. However, the current technology of SLMs with 8 μm pixel size (and a relatively limited effective area of $8.19\times 15.4\text{\hspace{0.17em}}\mathrm{mm}$) does not enable implementation of C-FINCH in microscopes due to the relatively low scattering angle. Nevertheless, the experiments presented in Refs. [23,24] prove that microscopic scattering layers between the specimen and the microscope objective can improve the image resolution of the microscope.

Like COACH [20,28], the present C-FINCH can be extended to 3D imaging by creating a library of PSHs, i.e., a set of digital holograms of a single point located on the optical axis at different Z distances from the imaging system. An object hologram of a multi-plane object can be reconstructed plane by plane using different pre-recorded PSHs from the library. Figure 7 presents an example of 3D imaging implemented using C-FINCH. Four reconstructed images were recorded by the C-FINCH with a scattering degree of 0.22, whereas the mentioned resolution chart was located in two different planes. Plane 1 and Plane 2 are located at 250 and 245 mm from the lens ${L}_{1}$, respectively. By using the setup presented in Fig. 3, the object and the pinhole are shifted along the axial direction by 5 mm. Two holograms and two PSHs were recorded in the two mentioned axial locations. Only when the object holograms are correlated with the appropriate PSH recorded at the same axial distance is a relatively high quality reconstructed image obtained. In case the object hologram is correlated with a different PSH recorded at a different axial distance, the reconstruction is blurred due to the relatively short axial correlation length. Hence, according to this result, we can conclude that C-FINCH has the capability of 3D imaging. For reconstructing images of 3D objects at different axial planes, a library of pre-recorded PSHs should be prepared *a priori*. However, this PSH library needs to be created only once and can be used any number of times to reconstruct the images of any objects. Other feature of C-FINCH that has not been dealt with in the present study but that is expected to provide an additional benefit is the improved axial resolution. This benefit is expected based on our studies of COACH [20,28] and will be investigated in the future.

## APPENDIX A

In this appendix, marginal ray tracing analysis is discussed to understand the performance of C-FINCH with respect to regular FINCH and regular imaging. Figure 8 shows the marginal ray tracing in the presence (solid lines) and absence (dashed lines) of CPM. It seems that in the presence of CPM, light rays (solid lines) emerging with larger angles of diffraction are collected by the system aperture in comparison with the case without the CPM. All the following analysis is done under the assumption of small angles.

The angle ${\theta}_{\sigma}$ represents the angle between the marginal ray that meets the CPM and the diffracted ray that enters into the system aperture due to scattering and is dictated by the spectral bandwidth $2B$ that is chosen in the GSA process as follows: ${\theta}_{\sigma}=\lambda \xb7B$. From the geometry of the setup, the maximal angle diffracted from the object and still acquired by the system (denoted as ${\theta}_{m}$ in Fig. 8) is given as: ${\theta}_{m}={\mathrm{tan}}^{-1}[(2({f}_{0}-{d}_{1})\mathrm{tan}({\theta}_{\sigma}-{\theta}_{m})+w)/2{d}_{1}]$.

Taking into consideration that $w\ll {f}_{0}$, the angle of the marginal ray can be simplified as

## Funding

Israel Science Foundation (ISF) (1669/16, 439/12).

## Acknowledgment

We thank Dr. Roy Kelner for the useful discussions.

## REFERENCES

**1. **M. Gu, ed., *Advanced Optical Imaging Theory* (Springer-Verlag, 1999), Vol. 75.

**2. **P. Nisenson and C. Papaliolios, “Detection of Earth-like planets using apodized telescopes,” Astrophys. J. **548**, L201–L205 (2001). [CrossRef]

**3. **R. Carriles, D. N. Schafer, K. E. Sheetz, J. J. Field, R. Cisek, V. Barzda, A. W. Sylvester, and J. A. Squier, “Invited review article: imaging techniques for harmonic and multiphoton absorption fluorescence microscopy,” Rev. Sci. Instrum. **80**, 081101 (2009). [CrossRef]

**4. **A. R. Kherlopian, T. Song, Q. Duan, M. A. Neimark, M. J. Po, J. K. Gohagan, and A. F. Laine, “A review of imaging techniques for systems biology,” BMC Syst. Biol. **2**, 74 (2008). [CrossRef]

**5. **E. Hecht, *Optics* (Pearson Education, 2002).

**6. **B. O. Leung and K. C. Chou, “Review of super-resolution fluorescence microscopy for biology,” Appl. Spectrosc. **65**, 967–980 (2011). [CrossRef]

**7. **E. Betzig and J. K. Trautman, “Near-field optics: microscopy, spectroscopy, and surface modification beyond the diffraction limit,” Science **257**, 189–195 (1992). [CrossRef]

**8. **M. J. Rust, M. Bates, and X. Zhuang, “Sub-diffraction-limit imaging by stochastic optical reconstruction microscopy (STORM),” Nat. Methods **3**, 793–796 (2006). [CrossRef]

**9. **M. G. L. Gustafsson, “Surpassing the lateral resolution limit by a factor of two using structured illumination microscopy,” J. Microsc. **198**, 82–87 (2000). [CrossRef]

**10. **J. R. Hassig, “Digital off-axis holography with synthetic aperture,” Opt. Lett. **27**, 2179–2181 (2002). [CrossRef]

**11. **V. Mico, Z. Zalevsky, P. Garcia-Martinez, and J. Garcia, “Single step superresolution by interferometric imaging,” Opt. Express **12**, 2589–2596 (2004). [CrossRef]

**12. **G. Indebetouw, Y. Tada, J. Rosen, and G. Brooker, “Scanning holographic microscopy with resolution exceeding the Rayleigh limit of the objective by superposition of off-axis holograms,” Appl. Opt. **46**, 993–1000 (2007). [CrossRef]

**13. **Y. Kashter and J. Rosen, “Enhanced-resolution using modified configuration of Fresnel incoherent holographic recorder with synthetic aperture,” Opt. Express **22**, 20551–20565 (2014). [CrossRef]

**14. **J. W. Goodman, *Introduction to Fourier Optics* (McGraw-Hill, 1968).

**15. **J. Rosen and G. Brooker, “Digital spatially incoherent Fresnel holography,” Opt. Lett. **32**, 912–914 (2007). [CrossRef]

**16. **J. Rosen and G. Brooker, “Non-scanning motionless fluorescence three-dimensional holographic microscopy,” Nat. Photonics **2**, 190–195 (2008). [CrossRef]

**17. **J. Rosen, N. Siegel, and G. Brooker, “Theoretical and experimental demonstration of resolution beyond the Rayleigh limit by FINCH fluorescence microscopic imaging,” Opt. Express **19**, 26249–26268 (2011). [CrossRef]

**18. **J. Rosen and R. Kelner, “Modified Lagrange invariants and their role in determining transverse and axial imaging resolutions of self-interference incoherent holographic systems,” Opt. Express **22**, 29048–29066 (2014). [CrossRef]

**19. **Y. Kashter, A. Vijayakumar, Y. Miyamoto, and J. Rosen, “Enhanced super resolution using Fresnel incoherent correlation holography with structured illumination,” Opt. Lett. **41**, 1558–1561 (2016). [CrossRef]

**20. **A. Vijayakumar, Y. Kashter, R. Kelner, and J. Rosen, “Coded aperture correlation holography–a new type of incoherent digital holograms,” Opt. Express **24**, 12430–12441 (2016). [CrossRef]

**21. **M. K. Kim, “Adaptive optics by incoherent digital holography,” Opt. Lett. **37**, 2694–2696 (2012). [CrossRef]

**22. **M. K. Kim, “Incoherent digital holographic adaptive optics,” Appl. Opt. **52**, A117–A130 (2013). [CrossRef]

**23. **Y. Choi, T. D. Yang, C. Fang-Yen, P. Kang, K. J. Lee, R. R. Dasari, M. S. Feld, and W. Choi, “Overcoming the diffraction limit using multiple light scattering in a highly disordered medium,” Phys. Rev. Lett. **107**, 023902 (2011). [CrossRef]

**24. **Y. Choi, C. Yoon, M. Kim, W. Choi, and W. Choi, “Optical imaging with the use of a scattering lens,” IEEE J. Sel. Top. Quantum Electron. **20**, 61–73 (2014). [CrossRef]

**25. **D. J. Goldstein, *Understanding the Light Microscope: A Computer Aided Introduction* (Academic, 1999) Chap. 1.

**26. **R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik **35**, 227–246 (1972).

**27. **B. Katz, J. Rosen, R. Kelner, and G. Brooker, “Enhanced resolution and throughput of Fresnel incoherent correlation holography (FINCH) using dual diffractive lenses on a spatial light modulator (SLM),” Opt. Express **20**, 9109–9121 (2012). [CrossRef]

**28. **A. Vijayakumar, Y. Kashter, R. Kelner, and J. Rosen, “Coded aperture correlation holography (COACH) system with improved performance,” Appl. Opt. **56**, F67–F77 (2017). [CrossRef]