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.
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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 , fluorescence microscopy , aberration compensation , and synthetic aperture imaging [11–13]. IDH is realized either by optical scanning holography , or multiple view projection methods , or the self-interference techniques [4,16]. Fresnel incoherent correlation holography (FINCH)  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 .
COACH was introduced nearly a decade later than FINCH in 2016 . 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 .
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 . 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 , 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.
2.1 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  shown in Fig. 1(a), in which a wave from a point object is collimated by the lens L0 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,
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,
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  and temporal  resolutions, extended field-of-view , partial aperture imaging with improved characteristics , and noise suppression  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) , 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,
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 zh-z1. Next, we update the constraint matrix to contain the same group of randomly distributed points, all are focused on a plane at a distance zh+z2 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 zh 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 L0 out of the optical axis at r0, can be described as,
Incoherent imaging systems are linear with respect to the wave’s intensity. Therefore, upon placing an object at the front focal plane of L0, the pattern captured by the image sensor is the 2D convolution of the object intensity function O(r) 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 H(r)=O(r/Mt)*h(r), where the sign ‘*’ stands for 2D convolution. H(r) is regarded as the complex-valued object hologram. Reconstruction of the object’s image is done by a cross-correlation between H(r) and h(r) as follows,
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 µm 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 L01 and L02 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 L0 of the focal length f0 = 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 L0, with reference to the focal point of L0. The light collimated by the lens L0 was polarized by the polarizer 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 f1 = zh-z1 = 32 cm and f2 = zh+z2 = 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 μm pixel pitch, monochrome) placed at a distance zh = 38.8 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 L0 and object holograms were recorded by a computer.
In the first experiment, we tested the best optimal value of f1 and f2 for superior image quality under observed reconstructed images. The study was conducted by varying Δf=(f1-f2) and keeping zh=38.8 cm as a fixed parameter in the entire experiment. f1 was changed from 30 to 35 cm, and f2 was changed accordingly by the formula f2=zhf1/(2f1-zh) 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 . 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.
In the calibration stage, PSHs were captured by placing a pinhole at the focal length of L0 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 f1 of the quadratic phase function from 30 to 35 cm, and for zh = 38.8 cm, are shown in Figs. 4(a1)–4(a6) for FINCH, and 4(b1)-4(b6) for CAFIR. The interval of f1 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 f1 is required to be found. The value f1=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 f1=32 cm. The MSE was calculated with the direct image as the reference image, as shown in image 4(c) . 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 Ф1,2,3=0, 2π/3, and 4π/3 for CAFIR and FINCH at f1=32 cm are shown in Figs. 5(a1)–5(a3), 5(b1)–5(b3) and 5(c1)–5(c3), 5(d1)–5(d3), 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(a4), 5(a5), 5(b4), 5(b5) and 5(c4), 5(c5), 5(d4), 5(d5) 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 (Imax - Imin)/(Imax + Imin) for three bright and dark middle grating lines, where Imax is maximum, and Imin 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.
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(a1)–8(a5) present direct imaging results. In Figs. 8(a1), 8(a2), 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(a4), 8(a5), 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(a3), 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(b1)–8(b5) and 8(c1)–8(c5), respectively. Comparing Fig. 8(b1) with Figs. 8(a1), 8(c1), 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.
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.
The authors declare no conflicts of interest.
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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