## Abstract

Fresnel incoherent correlation holography (FINCH) is a self-interference incoherent digital holography technique. It possesses a higher lateral resolution than an equivalent incoherent imaging system. However, FINCH has lower axial resolution than the direct imaging systems with the same numerical aperture. A decade after the FINCH invention, a different incoherent holographic method named coded aperture correlation holography (COACH) was developed with improved axial resolution but with the same lateral resolution as direct imaging. In this study, we propose and demonstrate a variant of COACH called coded aperture with FINCH intensity responses (CAFIR) with an improved lateral resolution that is similar to the FINCH system while maintaining its high axial resolution similar to the direct imaging system. CAFIR is implemented with coded phase masks to generate an ensemble of quasi-randomly distributed FINCH-like responses. Point spread holograms and object holograms are recorded under identical conditions, and they are cross-correlated to obtain the image reconstruction. Imaging of a multiplane object is studied to compare the axial resolution of CAFIR with FINCH and direct imaging systems.

© 2021 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement

## 1. Introduction

Digital holography is an indirect imaging technique where holograms are first acquired using an image sensor, and then the image is reconstructed digitally by a computational algorithm [1–5]. Thus, digital holography is a two-step process that has some advantages over classical imaging. For example, a hologram can contain depth information of three-dimensional (3D) objects utilizing phase information encoded in the interference patterns between an object and the reference beams [2–6]. Digital holography, in general, is classified into coherent and incoherent digital holography (IDH) depending on the light source used for the object illumination [1–6]. Wave interference can be easily achieved with coherent light beams, but many imaging tasks are widely applicable only under incoherent illumination [3,4,6,7]. In general, imaging systems under incoherent illumination have a modulation transfer function (MTF) with a larger spatial bandwidth than coherent systems with the same aperture dimensions [8,9]. Hence, the incoherent image usually has a higher image resolution than the coherent image. IDH can be useful in several applications like biological imaging [2], fluorescence microscopy [10], aberration compensation [6], and synthetic aperture imaging [11–13]. IDH is realized either by optical scanning holography [14], or multiple view projection methods [15], or the self-interference techniques [4,16]. Fresnel incoherent correlation holography (FINCH) [10] and coded aperture correlation holography (COACH) [16,17] are two notable self-interference IDH techniques based on an on-axis system with a diffractive phase aperture. In general, FINCH has a superior lateral resolution and an inferior axial resolution compared to direct incoherent imaging and COACH systems [17,18]. Several IDH systems based on FINCH and COACH have been developed and studied in the past, which utilize their unique features [13,16–28]. In the present study, we propose a new imaging method that integrates advantages from both FINCH and COACH techniques such that this hybrid IDH system has the improved lateral resolution of FINCH with the same axial resolution of COACH.

To further understand the advancement proposed herein, we briefly summarize FINCH and COACH, while a more detailed analysis is introduced in the methodology section. FINCH has been developed as a single-channel self-reference IDH technique [10,27]. In FINCH, light emitted from an object point splits into two beams, each of which propagates through a different quadratic phase mask. The interference pattern between the two beams is recorded by a digital camera such that the sum of the entire interference patterns from all the object points is the obtained digital hologram. As a consequence of violating the Lagrange invariant [29–32], reconstructed images in FINCH can exhibit about 2 and 1.5 times better-resolving power than the equivalent coherent and incoherent imaging systems, respectively. On the contrary to the higher lateral resolution, FINCH exhibits lower axial resolution than direct imaging [17].

COACH was introduced nearly a decade later than FINCH in 2016 [17]. In the case of COACH, the emitted light from an object point splits into two beams, one of the beams is modulated by chaotic coded phase mask (CPM), and the other propagates without modulation. The two mutually coherent beams interfere on the camera plane, where the recorded hologram is obtained as the accumulated interference patterns contributed from the entire object points. The recorded holograms are digitally processed to reconstruct a 3D image of the 3D object. In comparison to FINCH, COACH has a higher axial resolution but a lower lateral resolution [17].

This study presents an advanced version of COACH termed coded aperture with FINCH intensity responses (CAFIR) with a better lateral resolution than the regular COACH and better axial resolution than FINCH. CAFIR is an integration of FINCH and COACH in the sense that the point response of the CAFIR is a mix of the point responses of FINCH and COACH. On the one hand, the point response of one of the versions of COACH is an ensemble of randomly distributed dots [33,34], and on the other hand, the point response of FINCH is the interference between two different spherical waves [21]. Therefore, the present integration of FINCH and COACH means that the point response of CAFIR is an ensemble of randomly distributed patterns of interference between two different spherical waves. In the past, COACH and FINCH methods were already combined [35–37]. However, [35,36] do not guarantee that the lateral resolution of FINCH and axial resolution COACH can be achieved simultaneously. A brief explanation regarding this point is presented in the methodology section. In [37], authors suggest improvement of the lateral resolution in the proposed system, but they mentioned that further study is required regarding the axial resolution improvement yet to appear. In CAFIR, the FINCH-COACH integration always guarantees to violate the Lagrange invariant, and the randomness of the point response ensures the same depth of focus as of COACH. Thus, CAFIR preserves the lateral resolution of FINCH and the axial resolution of COACH. This study is presented in five sections. After the introduction, the next section presents the methodology of FINCH, COACH, and CAFIR to understand the theory of CAFIR that emerged from FINCH and COACH principles. Experiments and results are discussed in the third and fourth sections, respectively, followed by conclusions.

## 2. Methodology

#### 2.1 COACH as a generalization of FINCH

FINCH is a technique to capture two-dimensional (2D) digital holograms of incoherently illuminated 3D scenes based on the self-interference principle. Under incoherent illumination, two different spherical waves originated from the same point create an on-axis incoherent hologram. Since its invention, FINCH has undergone several improvements and optimizations, which led the technique to a point where it violates the Lagrange invariant property of imaging systems [29–32]. Thus, FINCH is inherently a super-resolution imaging system, compared with an equivalent standard system having a similar numerical aperture (NA). To further demonstrate this, let us consider the optimal scheme of FINCH [21] shown in Fig. 1(a), in which a wave from a point object is collimated by the lens *L _{0}* and modulated by two different diffractive lenses multiplexed on a spatial light modulator (SLM). Consequently, two spherical waves are generated such that one diverges from, and the other converges toward, two duplicated image points of the same object point. The two spherical waves interfere at the hologram plane, where we assume that they are as close as possible to a full overlap of their projected spots. This interference intensity is described as follows,

*r**=*(

*x,y*) is the vector of the hologram plane coordinates, and

*r*_{0}

*=*(

*x*

_{0}

*,y*

_{0}) is the vector of the off-axis displacement in the object plane.

*f*

_{0}is the focal length of the lens

*L*,

_{0}*λ*is the central illumination wavelength,

*z*is the gap between the SLM and the camera,

_{h}*I*

_{1},

*I*

_{2}are the intensities of the two duplicated image points, and

*z*

_{1},

*z*

_{2}are the distances from the hologram plane to the two duplicated image points.

*C.C.*stands for the Complex Conjugate term. Usually, the reconstruction process in FINCH is carried out by extracting one of the cross-terms (also known as an interference term) from the intensity of Eq. (1) using a phase-shift procedure, followed by the Fresnel backpropagation to the best in-focus plane. From the geometry of Fig. 1(a), it follows that under the condition of ${z_h} = {{2{z_1}{z_2}} / {({{z_2} - {z_1}} )}}$, there is maximal overlap between the two interfering beams [32]. Based on Eq. (1), the reconstructed off-axis point is positioned at a distance ${z_r} = {1 / D}{{ = {z_1}{z_2}} / {({{z_2} + {z_1}} )}}$ from the hologram at the height of ${{|{{{\boldsymbol r}_0}} |{z_h}} / {{f_0}}}.$ Therefore, the transverse magnification of the complete system is ${M_T} = {{{z_h}} / {{f_0}}}.$ On the other hand, the size of any reconstructed point is dependent on the NA and the magnification of the imaging system. To calculate the magnification of a single reconstructed on-axis point, we take the product of the magnification of the first imaging system, ${{({{z_h} - {z_1}} )} / {{f_0}}},$ and the magnification of the hologram reconstruction ${{{z_r}} / {{z_1}}}.$ Substituting the overlap condition into this product gives that the magnification of a single reconstructed on-axis point is ${M_o} = {{{z_h}} / {({2{f_0}} )}}$. Therefore, when two points are imaged by a direct imaging system and by FINCH with the same NA and magnification, the size of the points is the same in the two systems, but their separation is doubled in FINCH. The ratio ${{{M_T}} / {{M_o}}}$ is 1 in direct imaging systems and in all the systems that satisfy the Lagrange invariant, but ${{{M_T}} / {{M_o}}} = 2$ in FINCH that satisfies the overlap condition. In other words, the image resolution of FINCH is inherently better than any direct imaging system with the same NA. However, the inherent non-classical superior transverse resolution capability of FINCH comes at the price of low axial resolution [17].

One way to address the poor axial resolution of FINCH is to integrate the chaotic CPM of COACH such that it still guarantees the violation of Lagrange invariant in a way that improves the lateral resolution. The COACH configuration, in which at least one of the two apertures of the system is chaotic, is shown in Fig. 1(b). In this COACH system, the intensity distribution at the camera plane in response to a point in the origin of the input plane takes the form of interference between two types of beams; One, which is modulated by the CPM displayed on the SLM, and another, which is not affected by the SLM and remains collimated. Hence the intensity on the camera plane is,

*A*is a constant representing the collimated wave, and

*H*(

**) is the wave arriving at the camera modulated by the CPM. In order to extract the interference term $A^\ast H(\mathbf{r})$ from the pattern of Eq. (2), the modulated signal is further multiplied by a phase constant three times (**

*r**θ*

_{k}_{ }= [0, 2π/3, 4π/3]) in a standard on-axis phase-shifting procedure [17,18]. Since

*H*(

**) contains information on the lateral and axial distribution of the point response, it is termed point spread hologram (PSH). The intensity of an incoherently illuminated 2D object can be regarded as a collection of laterally shifted points as follows,**

*r**O*(

**)=Σ**

*r**(*

_{i}a_{i}δ

*r-r*_{0,i}), with different intensities

*a*. Since the light source is spatially incoherent, interference occurs only between waves emitted from the same object point. Based on the shift-invariance property of the system, the cross-term of the interference after the phase-shift procedure, in the case of the input object

_{i}*O*(

**), is $\sum\limits_i {{A_i}^\ast H({\boldsymbol r} - {M_t}{{\boldsymbol r}_{0,i}}})$.**

*r**A*is a constant of the

_{i}*i*

^{th}point response, and

*M*

_{t}=z_{h}/f_{0}is the lateral magnification. This sum contains the information of the 3D scene and therefore is termed object hologram. In order to reconstruct the image of the object, the object hologram should be cross-correlated with the PSH, as follows,

*δ*-like function, approximately is equal to 1 around the origin and small negligible values elsewhere. Since the beam modulated by the CPM is diffracted to a chaotic pattern, the cross-correlation of Eq. (3) is sensitive to axial translations of the object point. Therefore, COACH has inherent 3D holographic imaging capability. This capability is accessible by a one-time guide-star calibration process, in which different PSHs of the system are recorded for a point object placed at different axial planes along the range of interest. All the PSHs are cross-correlated with the recorded object hologram, and the axial value of the PSH that reconstructs the best in-focus image indicates the axial location of the object. Importantly, the lateral and axial resolutions of a regular COACH system [17] are determined by the corresponding correlation distances. Since these distances are governed by the system aperture size, the resolving capability of COACH is equivalent to a direct imaging system with the same NA. Consequently, the COACH framework is expected to have a better axial and worse lateral resolution than FINCH. In the next subsections, a holographic system integrating the superior resolving capability of the two methods, i.e., the lateral resolution of FINCH and axial resolution of COACH, is proposed.

#### 2.2 Sparse COACH

An important advantage of COACH systems is the capability to engineer the CPM according to the desired applications. This flexibility has been used to implement multiple types of COACH-based systems suitable for various imaging tasks. Improved spatial [34] and temporal [38] resolutions, extended field-of-view [39], partial aperture imaging with improved characteristics [40], and noise suppression [33] are few examples of CPM engineering to enhance specific imaging qualities. The common concept of all these systems is the use of sparsely and randomly distributed focal points over the hologram plane as the response to a point in the system input. Synthesizing the CPM is carried out by a modified Gerchberg and Saxton algorithm (GSA) [41], in which the intensity distribution on the camera is forced to some desired pattern, and the phase distribution is used as the degree of freedom. On the side of the CPM, the constraint is dictated by the nature of the SLM on which the CPM is displayed. The iterative transfer back and forth from the CPM to the camera planes is usually done by Fourier and inverse Fourier transforms. It is important to note that due to the diluted total intensity of the constraint matrix that contains, in some cases, a few light dots, there might be a lack of enough degrees of freedom. Thus, the algorithm convergence to a unit-amplitude CPM, dictated by a phase SLM used in COACH setups, is not always guaranteed. To overcome this obstacle, we increase the degrees of freedom in the constraint matrix by padding the central window of interest with a negligible constant intensity frame. The constraint matrix is therefore given by,

*ρ**is the size of the window at the hologram plane,*

_{0}

*r**is the location of the*

_{i}*i*-th focal point,

*a*and

*ε*are constants satisfying

*a>>ε*. Upon defining the constraint matrix according to Eq. (4), the Parseval [9] theorem is now satisfied, and the algorithm convergence is more likely achieved. The main benefit of sparse COACH is the ability to control the signal-to-noise ratio (SNR) and the visibility of the reconstructed image through sparsity and complexity of the PSHs [33,42]. However, in the current study, we employ the sparse response of COACH in order to merge the imaging merits of FINCH and COACH into a single holographic system. As already mentioned in the introduction, the COACH-FINCH hybrid system was already demonstrated in the past [35]. Nonetheless, in [35], the combination of the two methods was achieved by a weighted sum of the two characteristic masks, diffractive lens for FINCH and CPM for COACH, into a single phase-only mask so that the resolution values of the hybrid system have been averaged. In [36], similar to [35], a randomly multiplexed bifocal diffractive lens (RMBDL) was designed. But in [36], unlike most self-interference techniques instead of SLM, a fabricated RMBDL using electron beam lithography was used to study a hybrid FINCH-COACH system to reduce the cost, size, and weight of the imaging system. Unlike [35], the method was a single shot. As shown next, in the proposed CAFIR apparatus, the combination is done by granting the sparse COACH a response of a FINCH-type self-interference mechanism so that neither resolution type is compromised.

#### 2.3 CAFIR: COACH with the lateral resolving power of FINCH

Since the PSH of sparse COACH is composed of randomly distributed, diffraction-limited focal points, one can think of the CPM as a random ensemble of spatially translated FINCH-type diffractive lenses having focal length *z _{h}*-

*z*

_{1}. Next, we update the constraint matrix to contain the same group of randomly distributed points, all are focused on a plane at a distance

*z*+

_{h}*z*

_{2}from the SLM. Effectively another ensemble of translated FINCH-type diffractive lenses with different focal length is added, which is coaxial with the previous ensemble. To achieve the FINCH-type interference, the image sensor should be placed at a distance of

*z*from the SLM such that a full overlap between the multiple spots of the two series of interfering spherical waves occurs. This description is shown schematically in Fig. 1(c), and the intensity response to a point source located at the front focal plane of

_{h}*L*out of the optical axis at

_{0}**, can be described as,**

*r*_{0}

*r**(*

_{j}=*x*) is the translation of the

_{j},y_{j}*j*-th two focal points, one at

*z*

_{1}from and in front of the camera plane and the other at

*z*

_{2}from and beyond the camera plane.

*N*is the number of randomly distributed focal points at each focal plane. In Eq. (5), we assume that only the pairs of dots from coaxial diffractive lenses create interference patterns, and the interferences between non-coaxial waves are negligible, an assumption that can be guaranteed by increasing the space between neighbor dots. By capturing three intensity patterns with different phase shifts between the two sets of dots, one can extract the cross-terms of Eq. (5) given by,

*D*and

*B*are the same constants as are given below Eq. (1). Note that Eq. (6) highlights that the inherent superior lateral resolution of FINCH is preserved in CAFIR as long as the full overlap condition between the two interfering spherical beams is satisfied for all the

*N*pairs at the image sensor plane.

Incoherent imaging systems are linear with respect to the wave’s intensity. Therefore, upon placing an object at the front focal plane of *L _{0}*, the pattern captured by the image sensor is the 2D convolution of the object intensity function

*O*(

**) with the system’s point spread function of Eq. (5). Applying the phase-shift procedure on three recorded patterns yields the interference term given by the convolution**

*r**H*(

**)=**

*r**O*(

*r*/*M*)*

_{t}*h*(

**), where the sign ‘*’ stands for 2D convolution.**

*r**H*(

**) is regarded as the complex-valued object hologram. Reconstruction of the object’s image is done by a cross-correlation between**

*r**H*(

**) and**

*r**h*(

**) as follows,**

*r**α*is a real number between -1 and 1, which is chosen by the desire to suppress the noise on the reconstructed image. The approximation of the last line of Eq. (7) is valid more for

*h*(

**) having almost constant Fourier magnitude, which can be approximated more easily if the focal points of the PSH are randomly distributed. Similar to conventional COACH systems, the capability of the 3D reconstruction is achieved by cross-correlating the object hologram with the reconstructing function**

*r**h*(

**) corresponding to the desired transverse plane. By using Eq. (7) for this task, our proposed CAFIR system is expected to have an axial resolution that is similar to conventional direct imaging system together with the superior lateral resolution of FINCH.**

*r*## 3. Experiments

The experimental setup for the CAFIR is presented in Fig. 2. It is a two-channel digital holographic optical setup. Both illumination channels are mounted with light-emitting diodes (LEDs) (Thorlabs LED635L, 170 *mW*, λ = 635 *nm*, Δλ = 15 *nm*) as incoherent light sources. A 25 µ*m* pinhole was used as a point object. 23 lines per *mm* (*lp/mm*) and 20 *lp/mm* negative National Bureau of Standards (NBS 1963A Thorlabs) resolution charts were used as the objects. The objects were mounted in separate channels and were illuminated by LEDs in critical illumination conditions through the lenses *L _{01}* and

*L*in the respective channels. For a single plane experiment, only one channel was illuminated at a time. The light diffracted from the object was incident on the collimating lens

_{02}*L*of the focal length

_{0}*f*

_{0}= 20

*cm*. Prior to all the experiments with the objects, a library of point response holograms was recorded for different axial locations by placing the pinhole at different distances from the lens

*L*, with reference to the focal point of

_{0}*L*. The light collimated by the lens

_{0}*L*was polarized by the polarizer

_{0}*P*oriented along the active axis of the SLM (Holoeye PLUTO, 1920 × 1080 pixels, 8 µm pixel pitch, phase-only modulation). Two quadratic phase functions with different focal length

*f*

_{1}=

*z*

_{h}-z_{1}

*= 32*

_{ }*cm*and

*f*

_{2}=

*z*

_{h}+z_{2 }

*=*49.27

*cm*were displayed on the SLM covering 1080×1080 pixels. Each quadratic phase function is displayed on a chessboard grid, one is on the pixels that the sum of their indices is odd and the other is on the pixels that the sum of their indices is even. To the two quadratic phase functions, we added the phase of CPM computed by the GSA. The light modulated by the two quadratic functions and the CPM was focused to the ensemble of dots at two different axial locations but with full overlap at the image sensor (Hamamatsu ORCA-Flash4.0 V2 Digital CMOS, 2048 × 2048 pixels, 6.5 μ

*m*pixel pitch, monochrome) placed at a distance

*z*= 38.8

_{h}*cm*from the SLM. For the multiplane experiment, two channels were simultaneously illuminated. The two different resolution charts were placed at two different distances from the lens

*L*and object holograms were recorded by a computer.

_{0}## 4. Results

In the first experiment, we tested the best optimal value of *f*_{1} and *f*_{2} for superior image quality under observed reconstructed images. The study was conducted by varying Δ*f=*(*f*_{1}-*f*_{2}) and keeping *z _{h}*=38.8

*cm*as a fixed parameter in the entire experiment.

*f*

_{1}was changed from 30 to 35

*cm*, and

*f*

_{2}was changed accordingly by the formula

*f*

_{2}

*=z*

_{h}f_{1}/(2

*f*

_{1}-

*z*) in order to satisfy the overlap condition. On the SLM, two quadratic functions were multiplexed by a chessboard method and multiplied by the chaotic CPM designed by GSA shown in Fig. 3 [41]. CPM was synthesized with the constraint of five randomly distributed Fresnel zone patterns on the camera plane. Thus, five single-pixel dots were randomly placed inside the constraint area of 100 × 100 pixels at the Fourier plane of the GSA. After fifty iterations, the generated phase pattern was used as the CPM. The chessboard was spread over 1080 × 1080 pixels of the SLM, and each chessboard cell occupies a single alternate SLM pixel consisting of one of the two quadratic phase functions with two different focal lengths. Note that instead of a chessboard pattern, the quadratic functions can be distributed randomly [4,10] at the cost of more background noise over the reconstruction plane. It is important to note that generating the random dots pattern based on multiple Fresnel lenses instead of the CPM generated by the GSA is not recommended for two main reasons. First, due to technical and optical power considerations, the generated mask should modulate only the phase. Second, the CPM maintains the random aperture feature of COACH, which benefits the axial resolution of CAFIR.

_{h}In the calibration stage, PSHs were captured by placing a pinhole at the focal length of *L _{0}* in channel-1. Then in the imaging stage, object holograms were recorded, putting the NBS resolution chart in the other channel. In this experiment, the object plane coincides with the axial location of the pinhole. The object hologram was cross-correlated with PSH to reconstruct the image. The reconstruction results with phase-only filtering (POF,

*α*= 0) for varying focal length

*f*

_{1}of the quadratic phase function from 30 to 35

*cm*, and for

*z*= 38.8

_{h}*cm*, are shown in Figs. 4(a

_{1})–4(a

_{6}) for FINCH, and 4(b

_{1})-4(b

_{6}) for CAFIR. The interval of

*f*

_{1}was equal to 1

*cm*for both FINCH and CAFIR. Comparing the systems, the reconstructions with CAFIR are more stable than with FINCH. For FINCH experiments optimal value of

*f*

_{1}is required to be found. The value

*f*

_{1}=32

*cm*gives better reconstruction results for FINCH and CAFIR as concluded from minimum mean square error (MSE) shown in Fig. 4(c). Hence, all the further experiments are conducted with

*f*

_{1}=32

*cm*. The MSE was calculated with the direct image as the reference image, as shown in image 4(c) [11]. The direct image was captured by placing a single lens of focal length equivalent to 38.8

*cm*, as is shown in Fig. 4(d). A detailed illustration of all the holograms and reconstruction results, with the optimal focal length, are presented in the next paragraph.

The intensity patterns of point-object and object are captured with phases equal to *Ф*_{1,2,3}*=*0, 2π/3, and 4π/3 for CAFIR and FINCH at *f*_{1}=32 *cm* are shown in Figs. 5(a_{1})–5(a_{3}), 5(b_{1})–5(b_{3}) and 5(c_{1})–5(c_{3}), 5(d_{1})–5(d_{3}), respectively. The phase-shifting technique was used to eliminate the twin image and the zeroth-order term. Phase and magnitude of superimposed hologram are shown in Figs. 5(a_{4}), 5(a_{5}), 5(b_{4}), 5(b_{5}) and 5(c_{4}), 5(c_{5}), 5(d_{4}), 5(d_{5}) for PSHs and object holograms and for CAFIR and FINCH, respectively. The direct image of the equivalent system is shown in Fig. 5(e) for comparison. The image of CAFIR was reconstructed by cross-correlating superposed object hologram and phase-only filtered version of PSH. The FINCH images were obtained by back-propagating to the in-focus plane. The CAFIR reconstruction can be further improved with a linear correlation between PSH and object holograms as shown in Fig. 5(h) with different values of the filter *α* in the range [−1,1] and steps of 0.2. From the results, we can conclude that CAFIR with linear filtering of *α*= −0.2 gives a better reconstruction result than POF.

The visibility and the image resolution can be evaluated by looking at a cross-section of the images along with one of the gratings in the figures. Figure 6 shows plots of such cross-sections for the direct image, FINCH, and CAFIR presented in Figs. 5(e), 5(f) and 5(g). The average visibilities of FINCH, CAFIR, and DI were found at 0.87, 0.77, and 0.49, respectively. Average visibility is calculated by averaging (*I _{max}* -

*I*)/(

_{min}*I*+

_{max}*I*) for three bright and dark middle grating lines, where

_{min}*I*is maximum, and

_{max}*I*is minimum intensity of the consecutive bright and dark grating lines. From Fig. 6, FINCH has better visibility than CAFIR and DI. As CAFIR is reconstructed from a randomly distributed pattern, the overall signal compared to the background is lower than the FINCH, which affects the visibility of the CAFIR reconstructed image. We can observe that FINCH and CAFIR have better visibility than direct imaging. However, the visibility of FINCH is slightly better than CAFIR.

_{min}In the next experiment, the axial resolution of CAFIR in comparison to FINCH and direct imaging was studied. The graphs in Fig. 7 for a reconstructed point object are plotted for CAFIR and FINCH along the *z*-axis with 1 *mm* step and up to 51 *mm*. The PSHs were recorded by placing a 25-micron pinhole at different axial distances and then cross-correlated them with a PSH at the central position. The plot of direct imaging was obtained by moving the camera along the *z*-axis and measuring the direct image for each *z* location. Based on Fig. 7, one can conclude that the depth of field of the CAFIR is similar to that of direct imaging, whereas the axial resolution of FINCH is poorer than the other two imaging methods.

The last set of experiments consists of two-plane direct and holographic imaging experiments. Where the two individual NBS charts are placed in two separate channels, one in channel-1 and the other in channel-2, with axial separation varying from -10 to 10 *mm* in intervals of 5 *mm.* The central axial position in the range of [-10, 10] *mm* is a reference position where the pinhole is placed, and PSHs were recorded. The object hologram is obtained by moving two objects to an axial different location and reconstruction is obtained by cross-correlating the object hologram with the measured PSH. Figures 8(a_{1})–8(a_{5}) present direct imaging results. In Figs. 8(a_{1}), 8(a_{2}), the camera is focused on the stable object 20 *lp/mm* of NBS inverted resolution chart while object 23 *lp/mm* moves by 5 *mm* step in a forward direction. In the Figs. 8(a_{4}), 8(a_{5}), the camera is focused on the stable object 23 while object 20 moves backward away from the system by 5 *mm* step. In Fig. 8(a_{3}), both objects are located at the same axial distances. FINCH images were obtained by the backpropagation technique, whereas CAFIR holograms were reconstructed using the cross-correlation technique. The reconstruction images for FINCH and CAFIR are shown in Figs. 8(b_{1})–8(b_{5}) and 8(c_{1})–8(c_{5}), respectively. Comparing Fig. 8(b_{1}) with Figs. 8(a_{1}), 8(c_{1}), we can see that the out-of-focus object 23 *lp*/*mm* is more prominent for the case of FINCH than direct image and CAFIR. The same conclusions can be drawn from the other axial positions of the images as well.

## 5. Conclusion

In this study, we propose a novel IDH system titled CAFIR that integrates two systems, COACH and FINCH, such that their advantages are maximized. CAFIR incorporates the superior axial resolution of COACH with the superior lateral resolution of FINCH. CAFIR can violate the Lagrange invariant like FINCH in a way that the transverse magnification of the gap between every two points is up to twice larger than the magnification of each point. This kind of violation is the reason for the enhanced lateral resolution of FINCH and CAFIR over regular COACH and direct imaging. In the aspect of the axial resolution, the experimental investigation indicates that CAFIR preserves the axial resolution of direct imaging, which is higher than of FINCH. The CAFIR system has additional advantages along with the great advantage of super lateral resolution with optimal axial resolution. The cross-correlation with the optimal power value of α helps to improve the reconstructed image significantly. This study shows that there are cases in technology where combining two systems can yield a system with the best features of the parent systems instead of their average features.

## Disclosures

The authors declare no conflicts of interest.

## Data availability

Data underlying the results presented in this study are not publicly available at this time but may be obtained from the authors upon a reasonable request.

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