Coded Aperture Correlation Holography (COACH) - A Research Journey from 3D Incoherent Optical Imaging to Quantitative Phase Imaging

Abstract

Coded aperture correlation holography (COACH) combines incoherent digital holography with coded aperture imaging. COACH is also a method to record incoherent digital holograms of three-dimensional object scenes. Still, COACH can be used for several other incoherent and coherent optical applications. In this chapter, we survey the prime landmarks on the topic of COACH from two major perspectives: architectures and applications of the various systems. We explore the main configurations of hologram recorders in the COACH systems. For each design, we describe some of the recent implementations of these recorders in optical imaging. We conclude the chapter with general ideas on this technology.

1. Introduction

Imaging by optical waves has been known in the technology world for centuries [1]. For most of this time, imaging has been direct in the sense that images recorded on the eye retina, photographic film, or electronic sensor have been replicas of the observed scenes. However, the computing revolution of the second half of the twentieth century has opened many possibilities for indirect rather than direct imaging. In indirect imaging, a modified version of the observed scene is transferred from the image sensor to the computer to process and reconstruct the image of the original scene. One of the indirect imaging methods is coded aperture imaging, proposed in the sixties for X-ray imaging [2, 3, 4] and later adapted to the visible light using coded phase-masks [5] instead of an array of randomly distributed pinholes used in X-ray imaging [4].

Digital holography [6, 7, 8] can also be classified as indirect imaging, although it is special in the sense that the pattern recorded by the image sensor is an interference pattern between two light beams. At least one of the beams originates from the object. However, in the case of incoherent digital holography by self-interference, both interfering beams originate from the object [7]. In 2016, the two different concepts of coded phase-aperture imaging and incoherent digital holography were combined into a new indirect imaging method, dubbed coded aperture correlation holography (COACH) [9]. COACH merges the merits of these two different imaging modalities and enables three-dimensional (3D) imaging with interesting and unexpected features. More specifically, COACH is an electro-optical technique to record digital holograms of two- and three-dimensional scenes, where at least part of the light from the object passes through a coded phase-mask. COACH was initially proposed as an additional method to record incoherent digital holograms without scanning and evolved in several different directions. The COACH concept was inspired by several previous methods and systems [2, 3, 4, 5, 10, 11, 12] and has already stimulated several studies since then [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33]; some of them are mentioned in the following. This chapter provides an overview of research activities in the technology of COACH done by several researchers in the field.

About a year after the invention of COACH, a simpler version of it was proposed. This version operates without two-wave interference and can demonstrate some applications, such as 3D imaging. The modified version was called interferenceless COACH (I-COACH) [15]. Usually, the I-COACH is preferred whenever an application can be performed by both COACH and I-COACH with the same quality. This rule of thumb is reasonable because the calibration of a single wave system, such as I-COACH, is simpler, and its noise immunity is higher than that of an incoherent interferometer, such as COACH. However, not all the applications successfully implemented by COACH can be executed by I-COACH, and examples are given in the following.

COACH and I-COACH were initially invented for 3D imaging of incoherently illuminated scenes. Recently, the concept of coherent COACH with and without two-wave interference has been examined [34, 35, 36]. A central application of coherent digital holography is quantitative phase imaging (QPI) [37], and hence in the following, we review different ways to implement QPI using COACH [35, 36].

This review consists of six main sections. The development of incoherent COACH and I-COACH architectures, with different modalities and characteristics, are reviewed in the following two sections. We describe the coherently illuminated I-COACH and COACH techniques in the fourth and fifth sections, respectively. The concluding section summarizes the review.

2. Incoherent COACH

Incoherent COACH belongs to the family of self-interference digital holography systems [38]. The general optical configuration of these systems is shown in Figure 1. The flow of information starts from the light emitted from each object point in the upper part of Figure 1. The light propagates toward a beam splitting unit and is split into two waves. Each wave is modulated differently by a modulation component. The two waves originate from the same object point and hence are mutually coherent, although the light emitted, or reflected, from the object is spatially incoherent. Therefore because of the mutual coherence, the two waves with different wavefronts interfere at the sensor plane. The image sensor accumulates the entire interference patterns of all the input points to an incoherent hologram. A single hologram, or several acquired holograms, are introduced into a digital computer, where the various operations of the digital processor are schematically shown in the lower part of Figure 1. In the case of several holograms, they are superposed into a single digital hologram. Finally, the image of the object is reconstructed from the processed hologram by an appropriate numerical algorithm.

Figure 1.

Recording and reconstruction of holograms in a general self-interference digital holography system.

COACH was proposed as a generalized case of Fresnel incoherent correlation holography (FINCH) [10, 11, 12], a well-known technique of recording holograms, which also belongs to the self-interference systems. In FINCH, a quadratic phase-mask modulates at least one of the two waves. In COACH, on the other hand, the quadratic phase-mask of FINCH is substituted by a diffractive chaotic phase-aperture. The initial goal of COACH was like FINCH, that is, to acquire a hologram of the 3D observed scene illuminated by quasi-monochromatic spatially incoherent light. COACH’s optical scheme is depicted in Figure 2. The light from an object is split into two beams, and only one of the object beams is modulated by the chaotic mask termed coded phase-mask (CPM). The modulated beam is coherently interfered with the unmodulated object beam due to their common origin. Because COACH is on-axis system, it needs a phase-shifting procedure and complex hologram synthesis [39]. That means that three holograms are recorded, each of which with the CPM multiplied by a different phase-constant. The three holograms are superposed digitally in the computer such that the result is a complex-valued hologram. This digital hologram is reconstructed into a single image without the twin image and the bias term.

Figure 2.

Schematic diagram of coded aperture correlation holography (COACH). SLM - spatial light modulator.

Unlike other well-known incoherent hologram recorders, such as FINCH [10, 11, 12, 40] and Michelson-interferometer-based incoherent holographic systems [41, 42, 43, 44], COACH does not have a defined image plane where the wavefront can numerically propagate from the hologram to the reconstruction plane. Hence, COACH has different recording and reconstruction procedures. In other words, COACH consists of a two-step recording procedure: a one-time calibration and then imaging. In the initial stage of the calibration, one illuminates a moving pinhole along the optical axis, and the image sensor records a point spread hologram (PSH) for every axial location of the pinhole. The set of PSHs is accumulated in a library for later use in the imaging stage. Following the calibration process, an object hologram is recorded under the same restrictions and with the coded apertures as the PSH acquisition. The 3D image of the observed scene is reconstructed by a two-dimensional (2D) cross-correlation between the object hologram and the corresponding elements of the PSH library.

Although FINCH influenced the COACH structure, COACH has different features than FINCH. The image reconstruction has been modified to 2D cross-correlations with guidestar responses instead of the Fresnel back-propagation of FINCH [10, 11, 12, 40]. Compared to FINCH, COACH has better axial resolution but worse lateral resolution [9, 45]. However, the main difference is that COACH can do the same holographic 3D imaging without two-wave interference [15]. Nevertheless, several applications can only be performed by a version of the original COACH with two-wave interference. One of such applications is a one-channel-at-a-time incoherent synthetic aperture imager [46], summarized next.

2.1 One-channel-at-time incoherent synthetic aperture

An interesting application for COACH is incoherent imaging with synthetic aperture (SA). SA is a familiar super-resolution method and a conventional technique in astronomy to accomplish image resolution beyond the diffraction limit dictated by the physical aperture [47] of the telescope. Since its invention a century ago [48], incoherent SA imaging was usually realized by at least two optical channels operating simultaneously. The wave interference between two incoming light beams, both originated from the same object, was recorded over time from several viewpoints within the SA region. Then, the interference intensity patterns were processed to produce an image of the object with a resolution equivalent to complete SA [22, 48]. A single-channel SA is possible for cases of imaging systems with coherent light [49], but astronomical imaging is usually done with incoherent light sources. A solution to this double-channel problem of SA incoherent imaging is the lately proposed incoherent single-channel SA technique termed one-channel-at-time incoherent synthetic aperture imager (OCTISAI) [46].

As in many other COACH systems, the CPM of OCTISAI is synthesized using a modified version of the Gerchberg-Saxton algorithm (GSA) [50]. Then, the CPM is divided into N (in the following example N = 64) equal parts for the SA implementation. The optical setup of the OCTISAI experiment is shown in Figure 3 and described next. The system is first calibrated by collimating the light diffracted from a pinhole, where the collimating lens L₁ mimics the far-field imaging condition. A polarizer P₁ polarizes the collimated light to be oriented at 45⁰ regarding the active orientation of a spatial light modulator (SLM). The SLM is used as the display on which the CPMs of OCTISAI and all other systems in this chapter are displayed. Only a partial area of the SLM is used at a time, and all other parts are activated in a raster scan mode. Because of the polarization angle, the light is split into two orthogonal linear polarizations beyond the SLM. The CPM modulates one polarized wave, and the other wave passes the SLM without any change. Beyond the polarizer P₂, also oriented at 45⁰ to the SLM’s active axis, both beams have the same orientation enabling to record a pattern of interference between the two beams. The interference pattern between the modulated and unmodulated beams is captured by the image sensor. Three phase-shifted PSHs for the input point object (pinhole) are recorded for every partial aperture at each position in the SA region. Then, three phase-shifted object holograms are captured for the input object with the same phase-apertures as before in the calibration. Next, using the digital computation capabilities, the entire PSH parts are stitched together into one synthetic PSH. The parts of the object hologram are also processed into one synthetic object hologram by a similar procedure. The final image with the enhanced resolution is obtained by a 2D cross-correlation between the two synthetic holograms.

Figure 3.

The tabletop experimental setup for one-channel-at-time incoherent synthetic aperture imager (OCTISAI) inside the blue rectangle, BS₁ and BS₂ – beamsplitters, CMOS camera - Complementary metal-oxide-semiconductor camera, L₀₁, L₀₂, and L₁ - refractive lenses, LED₁ and LED₂ - identical light-emitting diodes, P₁ and P₂ – polarizers, SLM - spatial light modulator, and USAF - United States Air Force resolution target. Adapted from [46].

The complete experiment of OCTISAI is extensively described in [46], and here we briefly describe only the main results. In the experiment, a collection of PSHs was produced using three CPMs, each having a phase-shift exp(iθ_j), where θ_1,2,3 = 0^o, 120^o, and 240^o. A pinhole of 25 μm diameter was positioned in the input. After the PSH creation, group 3, element 1 of the United States Air Force (USAF) negative resolution chart, replaced the pinhole. We recorded the three object holograms with the same three CPMs used for the PSHs. The synthetic object holograms and PSHs were produced by stitching respective partial holograms and superimposing corresponding synthetic three-intensity responses. Finally, the object hologram was cross-correlated with the phase-only filtered version of the synthetic PSH. The outcome of this cross-correlation is the final reconstructed image. The COACH images related to the partial and complete apertures are shown in Figures 4(a1) and (a2), respectively. For comparison, Figures 4(a3) and (a4) show the corresponding images of direct imaging with a setup of a single lens and similar numerical apertures. The stitched holograms after the superposition are shown in Figure 4(b). Figure 4(c) presents the reconstructed images for OCTISAI with various area sizes of the SA holograms. Figures 4(c1) and (c2) are produced using the central eight, horizontally [4(c1)] and vertically, [4(c2)] stitched partial holograms, respectively. Figures 4(c3)–(c6) show the reconstruction results with 2 × 2, 4 × 4, and 6 × 6 central sub-holograms, and the entire 64 sub-holograms. The resolution enhancement by raising the number of stitched partial holograms is demonstrated. Comparing Figure 4(c6) with Figures 4(a1) and (a3), one can conclude that OCTISAI’s images have higher resolution than the images taken with a limited aperture in both techniques of COACH and direct imaging.

Figure 4.

(a₁, a₂) COACH reconstructed images and (a₃, a₄) direct images of limited aperture (a₁, a₃) and full aperture (a₂, a₄), magnitude and phase of (b₁, b₂) PSH and (b₃, b₄) object holograms of the complete SA, (c₁-c₆) reconstructed images after stitching of (c₁) 8 central horizontal sub-holograms, (c₂) eight central vertical sub-holograms, (c₃) 2 × 2, (c₄) 4 × 4, (c₅) 6 × 6 central sub-holograms, and (c₆) full 64 (8 × 8) sub-holograms. Adapted from [46].

3. Interferenceless incoherent COACH

As mentioned above, interferenceless coded aperture correlation holography (I-COACH) was published in 2017 [15] as a simpler configuration of the earlier proposed COACH [9]. Both systems spatially modify incoherent light by chaotic phase-masks. However, unlike COACH, I-COACH records holograms without two-beam interference. I-COACH is an incoherent 3D imaging method in which the image is digitally obtained by numerical 2D cross-correlation between the hologram of the object and the library of PSHs. The PSHs are recorded once in the calibration mode of the system, before the imaging stage, as shown in Figure 5. The same chaotic CPMs modulate the light waves in both the calibration and imaging stages. The modulated light is recorded by a digital camera after propagating in the free space. I-COACH system without two-beam interference can produce similar results as COACH because the intensity point-response of I-COACH on the sensor plane is highly sensitive to the axial location of the input point. Mathematically, the high sensitivity means that the cross-correlation between two intensity responses for two points located at two different axial positions is much smaller than the autocorrelation of each response [15]. Thus, the entire object points can be reconstructed in the 3D image space using 2D cross-correlations between a multi-point object and the library PSHs. The early configuration of I-COACH [15] has been developed into different systems with various architectures and with a variety of algorithms [16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33], each with strengths and weaknesses. The typical design of I-COACH is shown in Figure 5, where the same physical setup is schematically depicted in two modes of operation. The upper scheme shows the calibration process, in which the system collects a library of PSHs acquired for an object point positioned at different axial locations. When the library is completed, the same setup works in the imaging mode shown in the lower part of Figure 5. An incoherently illuminated 3D object replaces the single point in the system’s input. The object intensity response recorded by the sensor is 2D cross-correlated with each PSH of the library. The assembly of cross-correlation results is the desired reconstructed 3D image. This general scheme describes most I-COACH types and has been the basis for developments that have evolved since 2017 [16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31]; one example of an I-COACH application is depth-of-field engineering, briefly described next.

Figure 5.

Schematic diagram of interferenceless coded aperture correlation holography (I-COACH). The upper scheme refers to the calibration mode, whereas the lower describes the imaging mode.

3.1 Depth-of-field engineering

Long depth-of-field (DOF) in imaging systems has been important for many applications [51]. Generally, the DOF is dictated by the numerical aperture of the optical system. Reducing the numerical aperture extends the DOF, but it also unfavorably decreases the lateral image resolution of the system. Several methods have been advanced to extend the DOF of the optical system [51, 52, 53, 54, 55, 56, 57, 58, 59] with a minimal resolution decrease. Still, the complicated experimental and computational requirements have stimulated a search for simpler methods. This subsection reviews a new technique proposed first in [60] to engineer the DOF of imaging systems. DOF engineering is done by integrating radial quartic phase-functions (RQPFs) [61, 62] into the incoherent I-COACH shown schematically in Figure 6. The phase-mask displayed on the SLM of Figure 6 is a fusion of three separate phase-masks. The first is the chaotic CPM generated by GSA with the constraints of sparse dots on the camera plane [63] and a constant magnitude on the SLM plane. The second element is a positive diffractive lens used to fulfill the 2D Fourier relations of the GSA between the planes of the SLM and camera. The focal length of the diffractive lens f is determined such that each object is imaged on the camera. In other words, for the distance object-SLM (d_OS) and SLM-camera (d_SC), the three lengths satisfy the imaging equation $1/f=\left(1,/,d_{OS}\right)+\left(1,/,d_{SC}\right)$. The distance object-SLM is chosen as the distance from the center of the object space to the SLM. The third mask is the above-mentioned RQPF implemented to extend the DOF as desired. The RQPF with the phase-function $\text{exp}\left[i,2,\pi ,{\left(r,/,p\right)}^4\right]$ stretches the DOF of the sparse dots created by the CPM, where p is the modulation parameter controlling the length of the DOF, and r is the radial coordinate on the SLM plane. Near the back focal point of the diffractive lens on the camera plane, the RQPF generates sword beams with an almost constant intensity along a controlled propagation distance and a relatively narrow beam-like shape in any transverse plane [61, 62]. The 3D location and the length of the DOF can be determined by changing the parameters of the RQPF and the focal length of the lens. Multiplexing various threesomes of phase-masks (diffractive lens, CPM, and RQPF) with different modulation parameters can create various focusing curves. For instance, an imaging system that can image objects in two non-connected sub-volumes in the object space. In this example, the entire objects inside these sub-volumes remain in focus, while the images outside these sub-volumes are blurred and seen out-of-focus. The unusual DOF enables to image targets in specific sub-volumes simultaneously (or successively), whereas objects in other sub-volumes are blurred. Moreover, the engineered DOF allows to transversely shift an image from one volume relative to another image from another volume. Mutual transverse shifts of sub-volumes can avoid overlap between images when one object is behind or in front of another object.

Figure 6.

Optical scheme of the depth-of-field engineering system. DL - diffractive lens, CPM - coded phase-mask, RQPF - radial quartic phase-function. Adapted from [60].

Back to Figure 6, the incoherent light source critically illuminates the observed 3D scene using a lens L₀. In this scheme, an object volume is defined as the volume along z for which the DOF is extended, such that each object inside the volume produces an in-focus image in the output. Off-axis sub-volumes in Figure 6 indicate images of on-axis objects that are reconstructed out of the z-axis in the output due to an additional linear phase-mask attached to the other three-phase-masks (diffractive lens, CPM, and RQPF). The light emitted from the object scene is modulated by the combination of the phase-masks displayed on the SLM. For any point inside the object volume, the intensity recorded by the camera is distributed in the form of chaotically sparse dots. Like other I-COACH schemes, this dot pattern is used as the PSH, which reconstructs the image of any object by cross-correlation with the object hologram. In previous demonstrations of I-COACH [15, 16], the PSH has been recorded experimentally by illuminating a pinhole positioned at the system input. However, in [60], the PSH is digitally computed based on the known experimental parameters. The object reconstruction is done by a nonlinear cross-correlation [21] between the computed PSH and the object hologram.

Next, we show results of only a single object volume. In other words, the following I-COACH has extended DOF compared to direct imaging with the same numerical aperture. More complicated examples of DOF engineering can be found in Ref. [60]. The proposed technique is verified by an experimental setup like the scheme in Figure 6. Unlike typical I-COACH systems, two targets are positioned in two separate channels of the experimental setup such that they are located at each end of the object volume [60]. Two LEDs with refractive lenses separately illuminated the object in each channel. Both objects are from the USAF transmission resolution chart. In one of the channels, the object is element 6 of Group 2, and in the other channel, element 1 of Group 3 is used as the object. The targets are located at distances of 24 and 26 cm from the SLM, respectively. Light from the two targets was combined by a beamsplitter and projected on the SLM. The gap between the SLM and the camera was 22 cm. Direct images of the objects [shown in Figures 7(a) and (b)] on the camera plane were achieved by displaying only a single diffractive lens on the SLM with the focal length that satisfies the imaging equation, each for a different object in its own depth. For the I-COACH system, the PSH was computed using the optimal CPM that yielded ten randomly distributed dots on the camera plane. The reconstructed images of I-COACH are shown in Figure 7(c). In the case of direct imaging, it is clear from Figures 7(a) and (b) that the axial gap between the targets was too large to focus both targets at the same time. On the other hand, in the case of the I-COACH with the engineered DOF of 3 cm, the reconstructed images of Figure 7(c) show that both targets are in focus without any resolution decrease.

Figure 7.

(a, b) Direct images of the objects with two different lenses and (c) Reconstructed images from a single hologram using depth-of-field engineering. Adapted from [60].

4. Interferenceless coherent COACH

Optical recording of digital holograms with coherent light traditionally involves interference between object and reference waves, complicating the image acquisition [39]. With the coherent I-COACH, the concept of the coded aperture is adapted from the area of incoherent holography to record digital holograms of three-dimensional coherently illuminated scenes without two-wave interference or any kind of scanning. In addition to the obvious advantages of combining interferenceless holographic systems with coherent light, the proposed method enables relatively rapid image acquisition made possible by its inherent high signal-to-noise ratio (SNR). In [34], the I-COACH method was implemented for generating coherent holograms without interference between reference and object waves. The technique, called interferenceless coherent coded aperture correlation holography (IC-COACH), creates a bi-polar digital hologram of a 3D scene from two camera shots where the scene is illuminated by coherent laser light. The 3D image of the observed scene is reconstructed from the hologram by a deconvolution-like process.

To understand the evolution from incoherent to coherent I-COACH, we briefly summarize the principles of incoherent I-COACH first. Generally, an incoherent I-COACH hologram denotes a 2D function containing an image of a 3D scene, such that the image can be digitally reconstructed from the 2D function. Mathematically, the 2D digital hologram of a 3D object is given by,

$$H_{\mathit{OBJ}}\left(\overline{r}\right)=\int I_{\mathit{OBJ}}\left(\overline{r};z\right)\ast p\left(\overline{r};z\right)\mathit{dz},$$E1

where $\ast$ is 2D convolution at each z plane, $\overline{r}=\left(x,y\right)$ are the transverse coordinates, and $p\left(\overline{r};z\right)$ is the PSH of the recording system, which can be a general complex [15] or bi-polar real [16] function. The library of PSHs is a priori acquired in a calibration process with a guidestar, in which each $p\left(\overline{r};z_j\right)$ from the PSH library is computed as a response to an object point at z_j. Once the library is ready, and an object hologram is recorded, 2D cross-correlations between the object hologram and each PSH from the library reconstruct each z_j plane of the 3D image. This computation process is based on the linearity of incoherent optical systems with 2D intensity signals expressed by the following familiar convolution,

$$I_{\mathit{\text{Out}}}\left(\overline{r}\right)=I_{\mathit{\text{In}}}\left(\overline{r}\right)\ast {\left|h\left(\overline{r}\right)\right|}^2,$$E2

where $h\left(\overline{r}\right)$ is the coherent point spread function of the optical system. $I_{\mathit{\text{In}}}\left(\overline{r}\right)$ and $I_{\mathit{\text{Out}}}\left(\overline{r}\right)$ are the system input and output intensities, respectively. In contrast to incoherent, coherent optical systems are linear in corresponding to 2D complex amplitudes, and they obey the relation,

$$I_{\mathit{\text{Out}}}\left(\overline{r}\right)={\left|A_{\mathit{\text{In}}}\left(\overline{r}\right)\ast h\left(\overline{r}\right)\right|}^2,$$E3

where $A_{\mathit{\text{In}}}\left(\overline{r}\right)$ is the input 2D complex amplitude fulfilling the equation $I_{\mathit{\text{In}}}\left(\overline{r}\right)={\left|A_{\mathit{\text{In}}}\left(\overline{r}\right)\right|}^2.$ Because of the nonlinearity of Eq. (3), the implementation of the I-COACH concept in the coherent system is possible only for special cases. Hence, the coherent processor should be adapted in such a way that can satisfy the relation,

$${\left|A_{\mathit{\text{In}}}\left(\overline{r}\right)\ast h\left(\overline{r}\right)\right|}^2\approx {\left|A_{\mathit{\text{In}}}\left(\overline{r}\right)\right|}^2\ast q\left(\overline{r}\right),$$E4

where $A_{\mathit{\text{In}}}\left(\overline{r}\right)$ represents a broad set of input objects, and we assume that $h\left(\overline{r}\right)$ and $q\left(\overline{r}\right)$ are nontrivial functions. Eq. (4) is satisfied if $h\left(\overline{r}\right)$ is a set of points distributed over the camera plane such that the gap between any two points is wider than the size of $A_{\mathit{\text{In}}}\left(\overline{r}\right)$ [34]. The pattern of the random points on the camera plane is considered the system’s PSH. Hence, the CPM replicates the object to a set of the same images chaotically distributed over part of the camera plane. Such CPMs are created by a modified version of GSA [50], in which iterative transformations between the CPM’s plane and the spectral plane are done with suitable constraints at each plane. The constraint at the CPM’s plane is a constant magnitude distribution because the CPM is displayed on a phase-only SLM. In the spectral plane, which is also the camera plane, the intensity is constrained to be in a shape of randomly distributed dots over all or part of the plane.

The optical configuration of the IC-COACH system of [34] shown in Figure 8 is based on the classical 4-f spatial filtering system, with the SLM positioned at the Fourier domain and the camera at the image plane. In this setup, the spatial spectrum of the object is modulated by the CPM displayed on the SLM. The CPM was produced by the GSA to duplicate the input object over an ensemble of points randomly distributed at the camera plane. Two different chaotic CPMs are sequentially displayed on the SLM to create two different random sets of replications of the object. These two sets are subtracted from each other to produce a bi-polar object hologram. The ability of IC-COACH to image multi-plane objects is accomplished by multiplexing on the SLM, a few independent CPMs, each of which yields an in-focus different set of dots on a different transverse plane. Each ensemble of out-of-focus dots becomes focused on the camera plane for a point object positioned at the corresponding transverse plane. Lastly, the desired transverse image of the observed 3D scene is reconstructed by cross-correlation between the object hologram and the corresponding PSH. Figure 9 shows the reconstructed images for two different planes and two different gaps between the object planes, forming two different multi-plane scenes. Experimental demonstration for imaging diffusely reflective objects also appears in [34], making the IC-COACH system suitable for processing speckle images obtained by coherent illumination.

Figure 8.

Experimental setup of interferenceless coherent coded aperture correlation holography (IC-COACH) with two independent illumination channels. BS_1,2,3: Beamsplitters, M_1,2: mirrors, and SLM: spatial light modulator. Adapted from [34].

Figure 9.

Image reconstructions of different object planes obtained by a correlation with the corresponding PSH in IC-COACH. Adapted from [34].

5. Coherent COACH with two-wave interference

IC-COACH described in the previous section is an adaptation of the incoherent I-COACH to the case of coherent illumination. This system is capable of imaging 3D scenes holographically, but it cannot do phase imaging of any kind. To enable QPI of transparent objects, COACH has been integrated with a Mach-Zehnder interferometer [35, 36]. QPI, in general, is done by capturing the wavefront passing through thin transparent objects and converting it to an optical thickness map of the examined objects. This method is useful for many applications, including label-free biological cell imaging [64, 65] and nondestructive quality tests [66, 67].

Like the previous demonstrations of I-COACH [34, 63], the image of the observed object is projected to randomly and sparsely distributed replications over the camera plane. As before, the replications are obtained by a pseudorandom CPM synthesized by modified GSA [50]. The CPM is displayed on a phase SLM in the configuration of the coherent sparse COACH (CS-COACH) shown in Figure 10. The image sensor records the interference pattern between the waves of the image replications and of a reference tilted plane wave as follows:

Figure 10.

Optical configuration of coherent sparse COACH. MO: Microscope objective, BS_1,2,3: Beamsplitters, M₁: Mirror, and SLM: Spatial light modulator. Adapted from [36].

$$I\left(x,y\right)=|O\left(x,y\right)exp\left[\mathit{j\varphi }\left(x,y\right)\right]\ast \sum _{i=1}^N\delta \left(x-x_i,y-y_i\right){+R\cdot exp\left[j\frac{2\pi }{\lambda }\left(xsin{\theta }_x+ysin{\theta }_y\right)\right]|}^2,$$E5

where O(x, y) is the object amplitude, and ϕ(x, y) is its phase, R is the reference wave amplitude, λ is the illumination wavelength, N is the number of image replications, (x_i, y_i) are the displacement values of the i-th replica from the camera origin, and (θ_x, θ_y) are the angles between the object and reference waves in the x-z and y-z planes, respectively. It should be noted that off-axis holography is used to acquire holograms by a single camera shot. A digital filtering process in the spatial frequency domain eliminates the bias term and the twin image from the recorded intensity pattern. The processed object hologram is:

$$H_{\mathit{OBJ}}\left(x,y\right)=R^{\ast }\cdot O\left(x,y\right)exp\left[\mathit{j\varphi }\left(x,y\right)\right]\ast \sum _{i=1}^N\delta \left(x-x_i,y-y_i\right).$$E6

This hologram includes several randomly distributed replications of the object over the image plane. Like the procedure explained in [34, 63], the reconstruction of the object’s complex amplitude is performed by 2D cross-correlation between the object hologram H_OBJ and the PSHs. Figures 11(a) and (b) show the phase-image of polystyrene microspheres (FocalCheck, 6 μm diameter) with the proposed CS-COACH method. For comparison purposes, the phase-images extracted from a regular Mach-Zehnder interferometer using conventional off-axis holography are shown in Figures 11(c) and (d). It is apparent that the image of CS-COACH has higher SNR than the conventional technique. This advantage is attributed to the averaging procedure over several replications accompanied by the reconstruction using cross-correlation. Noise reduction is one of the several advantages of CS-COACH in comparison to open-aperture equivalent systems. Another advantage presented in Ref. [36] is extending the field-of-view (FOV) of the imaging system. Extended FOV realized with the same focal length of the microscope objective and without sacrificing the image resolution is an important advantage in microscopy.

Figure 11.

Reconstructed phase images of polystyrene microspheres and the phase cross-sections using (a)-(b) CS-COACH and (c)-(d) conventional off-axis holography. Units of the left panel color bars are radian. Adapted from [36].

6. Discussion and summary

For all its forms, COACH is a rapidly evolving technology because of the desire to enhance the resulting images and due to the new applications supported by the method. Any technology of imaging is expected to be as quickly as possible with the least camera shots. While the early version of I-COACH [15] operated with three camera shots taken under three independent CPMs, the number of shots and CPMs was decreased to two in [16]. By multiplexing two CPMs in space instead of time as before [15, 16], a single-camera shot was applied in Ref. [18]. I-COACH [19] and CS-COACH [36] with extended FOV were demonstrated by calibrating the systems with extended PSHs beyond the conventional FOV. The numerical reconstruction procedure was changed in [21] by substituting the ordinary linear cross-correlation with new nonlinear cross-correlation optimized to yield a correlation distribution with the lowest entropy. A different nonlinear cross-correlation with other cost-function in the optimization process was employed in [26, 30]. Some of the noise on the resulting images in the early versions [15, 16] appeared because of the low-intensity level per pixel of the PSH on the sensor plane. This difficulty was treated in [63] by imposing a PSH with the structure of sparse dots of light intensity distributed chaotically inside a limited region. The same problem was differently solved in [30] with PSHs of a ring shape. The electro-optical calibration in the upper part of Figure 5 was changed by a pure digital technique of synthesizing the library of PSHs in the computer [68]. Lateral resolution can be considered one of the holy grails of optical imaging. Improving the lateral resolution by I-COACH has been treated in [23, 45, 69, 70] by different approaches. Usually, I-COACH’s lateral and axial resolutions are the same as those of lens-based imaging systems with the same numerical aperture. The methods of [23, 45, 69, 70] improve the lateral resolution beyond the diffraction limit enforced by the finite numerical aperture of optical systems. In [23], resolution-enhanced images of the observed objects are reconstructed by a nonlinear cross-correlation between object holograms and PSHs. In [69, 70], a CPM displayed on the SLM was introduced between the object and the input aperture of a regular lens-based imager. Thus, the effective numerical aperture was increased beyond the characteristic numerical aperture of the imaging system. The effective numerical aperture and the improved resolution limits can be tuned by altering the scattering degree of CPMs [69, 70]. Other applications of COACH and I-COACH and their context in a frame of systems with dynamic diffractive phase-apertures are reviewed in [17, 71, 72].

To conclude this review, we note that COACH for all its modes is based on the extension of the resources available for imaging in a few ways. First, the real-valued aperture function of ordinary direct imaging is replaced with the complex-valued aperture function of COACH. Second, the COACH aperture is modified over time in the multiple-shot versions. Finally, an additional stage of digital processing is integrated with the optical system. These additional resources add to the COACH system new capabilities and unique features. Even though I-COACH is a simpler form of COACH and thus is preferred for many 3D imaging projects, there are some unusual applications in which COACH with two-beam interference is required. Incoherent synthetic aperture imagers [20, 22], the hybrid FINCH-COACH system [45], and quantitative phase-imagers [35, 36] are characteristic examples of systems that two-beam interference is necessary for their operations. However, other applications can be implemented successfully on I-COACH; some are presented herein others might be proposed in the future.