Interferenceless and motionless method for recording digital holograms of coherently illuminated 3D objects by coded aperture correlation holography system

Nathaniel Hai; Joseph Rosen

doi:10.1364/OE.27.024324

1. Introduction

Recording and reconstruction of realistic three-dimensional (3D) scenes using digital holography have many advantages compared to equivalent regular imaging techniques [1]. Compressing 3D data into 2D matrix by a single camera shot, while not compromising the spatial and time resolutions of the image acquisition, is an important benefit of holography [2–4]. Recently, an incoherent digital holography technique called interferenceless coded aperture correlation holography (I-COACH) has been developed. I-COACH possesses the same imaging qualities as an equivalent regular imaging system, in addition to the ability of 3D imaging by three [5], or two [6] camera shots, or even by a single [7,8] shot. In I-COACH, the light emitted from the object is modulated by a pseudorandom coded phase mask (CPM), and recorded as a hologram by a digital camera, without interfering with any other beam. This object hologram is then cross-correlated with a library of point spread holograms (PSHs), where each PSH is recorded for a point object positioned at a different z distance. The result of the cross-correlation is the 3D image of the original object. Due to its flexibility, I-COACH has motivated further improvements such as; extending the system’s field of view [9], imaging through partial aperture [10], imaging with resolution enhancement [11,12] and imaging through a scattering layer [13,14]. The lack of wave interference in the I-COACH system simplifies the optical configuration, and shows that the 3D information of the scene can be deduced from the magnitude of the CPM-modulated wave and not only from its phase. The interferenceless feature has encouraged us to introduce the I-COACH technique into the field of coherent imaging along with the accompanied above-mentioned benefits.

In this study, we present a new method dubbed interferenceless coherent coded aperture correlation holography (IC-COACH) for recording coherent holograms without wave interference. Following the proper calibration stage with a guidestar, a digital hologram of a 3D scene can be acquired from two camera shots of the laser-illuminated scene. The laser light emitted from the scene is modulated by CPM and recorded by the digital camera without interference with any reference wave. To the best of our knowledge, this work is the first demonstration of recording a digital hologram of coherently illuminated objects without two-beam interference.

Why is it hard to record coherent holograms without interference between object and reference waves, or in other words, what is the conceptual obstacle to copy the I-COACH method from incoherent to coherent illumination? To answer this, one should use the linear system theory of spatial optical systems. In general, incoherent systems are linear in the intensity distributions and hence the intensity response to any input object is a convolution between the object intensity function and the system point spread function (PSF). Basically, in I-COACH the reconstructed image is obtained by a process of decorrelation done on the intensity response of the object. On the other hand, in coherent illuminated systems the linearity exists in the complex amplitude functions and not in the intensities. That is to say that the intensity recorded by the digital camera is the magnitude square of the convolution between complex amplitude of the object and the system PSF. The recorded pattern is no longer a convolution between two real-valued functions like in the incoherent case. Because of the non-linearity of the magnitude square operation, a direct decorrelation cannot reconstruct the image. The method described in the next section provides a way to detour this conceptual obstacle, but of course not without penalties.

2. Methodology

Since the goal of the present study is to transfer the method of I-COACH to coherent light systems, let us first summarize the principles of I-COACH under incoherent illumination [5]. I-COACH is a technique of recording digital holograms without interference and reconstructing 3D images by a digital program. By the term “digital hologram” we mean that a digital camera records 2D matrix containing an image of a 3D scene, such that this image can be digitally reconstructed entirely from the 2D matrix. In a formal notation the 2D digital hologram ${H_{OBJ}}({\bar{r}} )$ of an object ${I_{OBJ}}({\bar{r},z} )$ is given by,

(1)$${H_{OBJ}}({\bar{r}} )= \int {{I_{OBJ}}({\bar{r},z} )} \ast t({\bar{r},z} )dz,$$

where $\ast $ is 2D convolution, $({\bar{r},z} )= ({x,y,z} )$ are the system coordinates and $t({\bar{r},z} )$ is the PSH of the recording system which can be bi-polar real [6,7] or general complex [5] function. The 3D PSHs are a-priori acquired in a guidestar calibration process in which each $t({\bar{r},{z_j}} )$ from the PSH library is obtained as a response to an object point at z_j. Each z_j plane of the 3D image ${I_{IMG}}({\bar{r},z} )$ is reconstructed by 2D correlation of the object hologram with a related-to-$t({\bar{r},z} )$ reconstructing function $R({\bar{r},z} )$ as follows,

(2)$${I_{IMG}}({\bar{r},{z_j}} )= {H_{OBJ}}({\bar{r}} )\otimes R({\bar{r},{z_j}} ),$$

where $\otimes$ is 2D correlation.

The entire process of I-COACH described above is based on the linearity of incoherent systems with 2D intensity distributions. This linearity is expressed by the well-known equation ${I_{Out}}({\bar{r}} )= {I_{In}}({\bar{r}} )\ast {|{h({\bar{r}} )} |^2}$, where $h({\bar{r}} )$ is the coherent impulse response function [15]. ${I_{In}}({\bar{r}} )$ and ${I_{Out}}({\bar{r}} )$ are the system input and output intensity distributions, respectively. When one wants to transfer the I-COACH from incoherent to coherent regimes, he cannot directly use Eqs. (1) and (2), because in coherent systems the linearity is between 2D complex amplitudes and not between intensity distributions. The equation of the output intensity of a coherent system is ${I_{Out}}({\bar{r}} )= {|{{A_{In}}({\bar{r}} )\ast h({\bar{r}} )} |^2}$, where ${A_{In}}({\bar{r}} )$ is the input 2D complex amplitude satisfying the relation ${I_{In}}({\bar{r}} )= {|{{A_{In}}({\bar{r}} )} |^2}$ [15]. Therefore, the first goal of the proposed method is to satisfy the relation,

(3)$${|{{A_{In}}({\bar{r}} )\ast f({\bar{r}} )} |^2} \approx {|{{A_{In}}({\bar{r}} )} |^2} \ast q({\bar{r}} ),$$

at least for a broad set of input objects and for non-trivial functions $f({\bar{r}} )$ and $q({\bar{r}} )$. If Eq. (3) is valid for a broad family of input object, one can use Eqs. (1) and (2) with this family, and therefore in principle, one of the I-COACH techniques proposed in [5–8] can be applied.

To satisfy Eq. (3), each 2D object in the input scene is convolved with an array of isolated focal points spread chaotically over the entire camera plane. The distance between any two adjacent points should be larger than the image of the input object, such that between any two points the image can be displayed without any overlap between the image and any other point from the set. If this non-overlap condition is satisfied the following relation is valid,

(4)$$\begin{aligned}{\left|{{A_{In}}({\bar{r}} )\ast \sum\limits_k {{a_k}\delta ({\bar{r} - {{\bar{r}}_k}} )} } \right|^2} & = {\left|{\sum\limits_k {{a_k}{A_{In}}({\bar{r} - {{\bar{r}}_k}} )} } \right|^2}\\ &= {\sum\limits_k {a_k^2|{{A_{In}}({\bar{r} - {{\bar{r}}_k}} )} |} ^2} = {|{{A_{In}}({\bar{r}} )} |^2} \ast \sum\limits_k {a_k^2\delta ({\bar{r} - {{\bar{r}}_k}} )} , \end{aligned}$$

where δ(·) is the delta function of Kronecker and a_k’s are constants. Hence, Eq. (4) satisfies Eq. (3) for all the input objects that are small enough such that their images can be introduced between any two adjacent points without overlap. At this point, one realizes that the penalty for using IC-COACH is the limited field of view (FOV) of the system which is roughly smaller than the FOV of a regular imaging system with the same NA by a factor of N^0.5 where N is the number of the PSH dots. Another feature of IC-COACH deduced from Eq. (4) is that the system can image intensity only and not phase objects [16,17].

As mentioned above, the PSH is a group of dots, which means that each CPM replicates the object to an ensemble of identical images randomly spread over the camera plane. The CPMs are generated using a modified Gerchberg–Saxton algorithm (GSA) [18] by propagating an initial random phase from the CPM plane to its Fourier related plane and enforcing the desired intensity of dots randomly distributed over the camera plane, as is illustrated in Fig. 1. The intensity on the camera plane is constrained to be the chaotic distributed dots, whereas the phase is used as the rank of freedom. The constraint on the CPM plane is a phase-only matrix according to the character of the used spatial light modulator (SLM) on which the CPM is displayed. When the resulting CPM is displayed on the SLM, the square magnitude of the ensemble of dots becomes the system’s PSF such that an object in the input creates an ensemble of image replications over the camera.

Fig. 1. Flow chart of the modified GSA used for synthesizing the pseudorandom CPM that generates the ensemble of dot pattern on the camera plane.

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Since the image is reconstructed by a cross-correlation between the PSH and the system response to the tested object, the number of object replications on the camera plane has an influence on the image quality. Low density of replications in the ensemble will increase the signal to noise ratio (SNR) of the reconstructed image at the expense of high level of background noise on the reconstruction plane. Therefore, for given setup specifications the optimal number of replications in the ensemble should be adjusted experimentally.

The optical configuration of the IC-COACH system is the classical 4-f spatial filtering system shown in Fig. 2. A plane wave from a coherent source illuminates an object $O({\bar{r}_s})$ centered around the optical axis and the diffracted light is Fourier-transformed by lens L₁ onto the SLM plane. The spatial spectrum of the object is modulated by the pseudorandom CPM displayed on the SLM. The CPM is engineered by the GSA, such that it projects chaotic ensemble of pre-determined number of object replications onto the camera plane. As mentioned above, the constraint on the CPM is to avoid overlap between the replications on the camera plane, and thus in accordance with Eq. (4), the intensity over the camera is,

(5)$${I_k} = {|{O({{{{{\bar{r}}_R}} \mathord{\left/ {\vphantom {{{{\bar{r}}_R}} {{M_T}}}} \right.} {{M_T}}}} )} |^2} \ast \sum\limits_{i = 1}^N {a_i^2\delta ({{{\bar{r}}_R} - {{\bar{r}}_{k,i}}} )} ,$$

where ${\bar{r}_R}$ stands for the transverse coordinates on the camera plane, M_T is the system lateral magnification which is equal to f₂/f₁, N is the number of image replications in the ensemble and k is the exposure number (k = 1,2). Bi-polar PSH and object holograms are generated in order to reduce the background noise usually obtained by cross-correlation between two positive functions [6,7]. Two camera exposures, corresponding to two independent CPMs are recorded by the camera and subtracted, one from the other, to yield bi-polar PSH and object holograms. The image reconstruction is obtained by cross-correlation between the object hologram, H_OBJ and the phase-only filtered version of the PSH, H’_PSH, as the following [6]

(6)$$\begin{aligned} {I_{IMG}} & = {H_{OBJ}} \otimes H{^{\prime}_{PSH}}\\ & = {|{O({{{{\bar{r}}_R}} \mathord{\left/ {\vphantom {{{{\bar{r}}_R}} {{M_T}}}} \right.} {{M_T}}}) \ast {H_{PSF}}} |^2} \otimes H{^{\prime}_{PSH}}\\ & = {\left|{O({{{{\bar{r}}_R}} \mathord{\left/ {\vphantom {{{{\bar{r}}_R}} {{M_T}}}} \right.} {{M_T}}}) \ast \left[ {\sum\limits_{i = 1}^N {{a_i}\delta ({{\bar{r}}_R} - {{\bar{r}}_{1,i}})} - \sum\limits_{i = 1}^N {{b_i}\delta ({{\bar{r}}_R} - {{\bar{r}}_{2,i}})} } \right]} \right|^2} \otimes H{^{\prime}_{PSH}}\\ & = {|{O({{{{\bar{r}}_R}} \mathord{\left/ {\vphantom {{{{\bar{r}}_R}} {{M_T}}}} \right.} {{M_T}}})} |^2} \ast \left[ {\sum\limits_{i = 1}^N {a_i^2\delta ({{\bar{r}}_R} - {{\bar{r}}_{1,i}})} - \sum\limits_{i = 1}^N {b_i^2\delta ({{\bar{r}}_R} - {{\bar{r}}_{2,i}})} } \right] \otimes H{^{\prime}_{PSH}}\\ & = {{\mathfrak{F}}^{ - 1}}\left\{ {{\mathfrak{F}}[{{{|{O({{{{\bar{r}}_R}} \mathord{\left/ {\vphantom {{{{\bar{r}}_R}} {{M_T}}}} \right.} {{M_T}}})} |}^2}} ]{\mathfrak{F}}\left[ {\sum\limits_{i = 1}^N {a_i^2\delta ({{\bar{r}}_R} - {{\bar{r}}_{1,i}})} - \sum\limits_{i = 1}^N {b_i^2\delta ({{\bar{r}}_R} - {{\bar{r}}_{2,i}})} } \right]\exp ( - j\varphi )} \right\}\\ & = {{\mathfrak{F}}^{ - 1}}\{{{\mathfrak{F}}[{{{|{O({{{{\bar{r}}_R}} \mathord{\left/ {\vphantom {{{{\bar{r}}_R}} {{M_T}}}} \right.} {{M_T}}})} |}^2}} ]|h |\exp (j\varphi )\exp ( - j\varphi )} \}\\ & = {|{O({{{{\bar{r}}_R}} \mathord{\left/ {\vphantom {{{{\bar{r}}_R}} {{M_T}}}} \right.} {{M_T}}})} |^2} \ast {{\mathfrak{F}}^{ - 1}}\{{|h |} \}\approx {|{O({{{{\bar{r}}_R}} \mathord{\left/ {\vphantom {{{{\bar{r}}_R}} {{M_T}}}} \right.} {{M_T}}})} |^2}, \end{aligned}$$

where ${\mathfrak{F}}\{ \}$ stands for 2D Fourier transform and the Fourier transform of the PSH is denoted as,

(7)$${\mathfrak{F}}\left\{ {\sum\limits_{i = 1}^N {a_i^2\delta ({{\bar{r}}_R} - {{\bar{r}}_{1,i}})} - \sum\limits_{i = 1}^N {b_i^2\delta ({{\bar{r}}_R} - {{\bar{r}}_{2,i}})} } \right\} = |h |\exp (j\varphi ).$$

The last approximation of Eq. (6) is valid under the assumption that the reconstructing function ${{\mathfrak{F}}^{ - 1}}\{{|h |} \}$ is a sharply peaked function, which is correct as long as the distribution of the PSH’s dots is random. The sharpness of the reconstructing function depends upon the number and the density of the PSH’s dots. Therefore, the parameters of the PSH must be optimized experimentally against the reconstruction SNR as demonstrated in the next section. The sharpness of the reconstructing function is further increased due to the bi-polarity of the PSH which appears also in a random order. Note that the fourth equality of Eq. (6) is due to the sparsity of the PSH dots across the camera and due to the non-overlap condition which enables to consider this coherently illuminated system as linear with intensity functions. An illustration of the entire process of IC-COACH including the three main stages of calibration, object recording and image reconstruction is given in Fig. 3.

Fig. 2. Optical configuration of IC-COACH.

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Fig. 3. Flow chart illustrating the suggested IC-COACH method.

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So far, the analysis has been applied for 2D imaging of the transverse plane at the front focal plane of L₁. For analyzing the imaging of a multi-plane 3D object, the 3D image of an object point is modeled as a Gaussian beam. If the object point is located at the front focal plane of L₁, the image on the camera plane is a cross-section with the Gaussian waist with the minimal size. In other words, the spots with the diameter of the Gaussian waist are the in-focus points approximated to the ensemble of delta functions. Object points outside the front focal plane are replicated as ensemble of Gaussian spots each of which is wider than the width of the Gaussian waist. Cross-correlation of these spots with the phase-only filtered version of the Gaussian spots, always yields relatively a wide image spot instead of the delta-like image obtained for the in-focus object points. In order to image other axial planes, one must create an ensemble of dots for each imaged axial plane. This multiple-plane PSH can be created by multiplexing several independent CPMs, each of which should yield in-focus ensemble of dots at different axial plane. The multiplexing can be either temporal or spatial multiplexing, whereas the former increases the acquisition time and the latter increases the reconstruction noise. To minimize the acquisition time, we demonstrate in this study spatial multiplexing of CPMs for imaging two different axial planes. Spatial multiplexing of more axial planes and time multiplexing will probably be investigated in the future. Although reconstructing a 3D scene with more than two axial planes is not demonstrated experimentally in the current work, it should be noted that the number of the reconstructed planes in a 3D scene using spatial multiplexing is limited by the object size, the number of PSH dots and the active area of the sensor. The main advantage of the spatial multiplexing is in its speed. Acquisition of the PSH is done offline in a calibration process before the object recording, and therefore should not be considered in the duration of the imaging process. To create the bi-polar object hologram two exposures are needed, and this reduces the system frame rate to half of the SLM refresh frequency, which for standard SLMs is 60 Hz. Acquisition of 30 frames per second can be regarded as a real-time operation and this frame rate is the same regardless of the amount of multiplexed CPMs.

For imaging of P axial planes, the bi-polar object hologram is given by,

(8)$${H_{OBJ}} = \sum\limits_{q = 1}^P {\left\{ {{{|{{O_q}({{{{\bar{r}}_R}} \mathord{\left/ {\vphantom {{{{\bar{r}}_R}} {{M_T}}}} \right.} {{M_T}}})} |}^2} \ast \sum\limits_{p = 1}^P {\left[ {\sum\limits_{i = 1}^N {{a_{p,i}}{g_{q,p}}({{\bar{r}}_R} - {{\bar{r}}_{1,p,i}})} - \sum\limits_{i = 1}^N {{b_{p,i}}{g_{q,p}}({{\bar{r}}_R} - {{\bar{r}}_{2,p,i}})} } \right]} } \right\}} ,$$

where q and p are the indexes of the axial planes, g_q,p’s are Gaussian functions with various diameters according to distances between the indexes q and p, and for the axial plane q = p g_q,p≈δ. Reconstructing the axial plane l is performed by correlating the bi-polar object hologram with the phase-only filtered version of the ensemble of dots corresponding to l axial plane, as follows,

(9)$$\begin{aligned} {I_{IMG}} & = {H_{OBJ}} \otimes H^{\prime}_{PSH,l}\\ & = \sum\limits_{q = 1}^P {\left\{ {{{|{{O_q}({{{{\bar{r}}_R}} \mathord{\left/ {\vphantom {{{{\bar{r}}_R}} {{M_T}}}} \right.} {{M_T}}})} |}^2} \ast \sum\limits_{p = 1}^P {\left[ {\sum\limits_{i = 1}^N {{a_{p,i}}{g_{q,p}}({{\bar{r}}_R} - {{\bar{r}}_{1,p,i}})} - \sum\limits_{i = 1}^N {{b_{p,i}}{g_{q,p}}({{\bar{r}}_R} - {{\bar{r}}_{2,p,i}})} } \right]} } \right\}} \otimes H^{\prime}_{PSH,l}\\ & = \sum\limits_{q = 1}^P {{{|{{O_q}({{{{\bar{r}}_R}} \mathord{\left/ {\vphantom {{{{\bar{r}}_R}} {{M_T}}}} \right.} {{M_T}}})} |}^2} \ast \left[ {\sum\limits_{i = 1}^N {a_{q,i}^{}\delta ({{\bar{r}}_R} - {{\bar{r}}_{1,q,i}})} - \sum\limits_{i = 1}^N {b_{q,i}^{}\delta ({{\bar{r}}_R} - {{\bar{r}}_{2,q,i}})} } \right]} \otimes H^{\prime}_{PSH,l}\\ & + \sum\limits_{q = 1}^P {\sum\limits_{p \ne q}^P {{{|{{O_q}({{{{\bar{r}}_R}} \mathord{\left/ {\vphantom {{{{\bar{r}}_R}} {{M_T}}}} \right.} {{M_T}}})} |}^2} \ast \left[ {\sum\limits_{i = 1}^N {{a_{p,i}}{g_{q,p}}({{\bar{r}}_R} - {{\bar{r}}_{1,p,i}})} - \sum\limits_{i = 1}^N {{b_{p,i}}{g_{q,p}}({{\bar{r}}_R} - {{\bar{r}}_{2,p,i}})} } \right]} } \otimes H^{\prime}_{PSH,l}\\ & \approx {|{{O_l}({{{{\bar{r}}_R}} \mathord{\left/ {\vphantom {{{{\bar{r}}_R}} {{M_T}}}} \right.} {{M_T}}})} |^2} + \sum\limits_{q \ne l}^P {{{|{{O_q}({{{{\bar{r}}_R}} \mathord{\left/ {\vphantom {{{{\bar{r}}_R}} {{M_T}}}} \right.} {{M_T}}})} |}^2} \ast {g_{q,l}}({{\bar{r}}_R})} \,. \end{aligned}$$

In the last approximation of Eq. (9), it is assumed that $H^{\prime}_{PSH,l}$ is dominant enough to select only a single in-focus image from one plane (plane l) and P-1 out-of-focus images from the other not-l planes. Therefore, the result is a reconstruction of the object in the l axial plane with out-of-focus images from the other axial planes.

As mentioned above, in the present study the multi-plane CPMs are spatially multiplexed on a single SLM, although there are still two camera shots with two different SLM masks to create the bi-polar holograms. There are various ways to spatially multiplex the CPMs, some of them probably will be compared in the future. Herein, we choose a simple method of allocating every successive n×n SLM pixels to a CPM of the different axial plane in a chessboard-like pattern. Each CPM is created by an independent run of the GSA. Since the camera plane is the Fourier plane of the SLM, the size of each square on the SLM assigned to a CPM should be chosen between two limitations. The upper limit is given by the spatial bandwidth of the CPM, i.e. d<λf₂/A, where A is the PSH size (the size of the area containing the entire dots) and d is the chessboard pitch. The lower limit is governed by the crosstalk between adjacent squares on the SLM that can create an unwanted diffraction pattern on the camera plane. In this study n = 5 satisfies these limits. Recall that the GSA is iterated between the SLM plane and its Fourier plane, the Fourier relation does not longer exist for objects positioned out of the front focal plane of L₁, because these out-of-plane points illuminate the SLM with a spherical rather than a plane wave. This problem is easily solved by adding the phase of an appropriate diffractive lens to the corresponding CPM in order to compensate for the illuminating spherical wave. Thus, after adding a different compensating diffractive lens to each pair of CPMs of the corresponding axial plane, the Fourier relation between SLM and camera planes is satisfied for the entire CPMs. In the next section we describe a few experiments with the new IC-COACH method.

3. Experiments and results

IC-COACH system was experimentally tested by the optical setup shown in Fig. 4. Beam from HeNe laser (AEROTECH λ = 632.8 nm, Max. output power 25 mW) was split by a beam-splitter (BS₁) into two paths for illuminating two different objects. Note that there is no wave interference between the channels and the two-channel arrangement exists mainly to separately record the PSH and object hologram as fast as possible with minimum changes in the setup. Another use of the two channels is in the experiment of 3D multi-plane object, where the two planar objects are displayed each in a different channel, only to keep the transverse gap between the objects as small as possible. Construction of 3D scene using two transmissive planar objects separated by some axial gap has enabled us to flexibly examine the three-dimensional capabilities of the IC-COACH system. Two identical lenses L₁ and L₂ (f₁= 400 mm, D = 50.8 mm) were used in the 4-f system, where a reflective phase-only SLM (Holoeye PLUTO, 1920 × 1080 pixels, 8 µm pixel pitch) was positioned at their common focal plane. The CPMs designed by the GSA were displayed on the SLM and a digital camera (Thorlabs 8051-M-USB, 3296 × 2472 pixels, 5.5µm pixel pitch, monochrome) located at the back focal plane of L₂ recorded the incident light intensity.

Fig. 4. Experimental setup of IC-COACH with two independent illumination channels.

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The first stage of the experiment is to determine the optimal number of dots needed to image a given object. In general, too many dots reduce the intensity of each dot which might reduce the SNR of the recoded holograms. On the other hand, too few dots simplify the PSH and hence the background noise and the intensity of the out-of-focus images might be increased. Several CPMs yielding various number of dots were tested with 30 µm diameter pinhole in one channel to record first the PSH. Later, with the National Bureau of Standards chart (NBS 1963A Thorlabs) in the other channel, the object hologram was recorded. Two intensity patterns, corresponding to two independent CPMs displayed on the SLM (both projecting equal number of dots), were recorded for the pinhole and for the 18 lp/mm element of the NBS chart, separately by blocking each time the light from the other channel. This procedure was repeated for different number of sparsely distributed dots on the camera plane by changing the CPM pair on the SLM, where the non-overlap condition was always met. The dots placement is done by creating a virtual circle of non-dot zone around each already-placed dot. The radius of the non-dot zone equals to the diameter of the smallest circle that contains the object to be imaged. This way, the position of the next dot to be placed is randomly selected from all the sensor area excluding the non-dot zone of already-placed dots, and still the algorithm ensures that there is no overlap between two adjacent object replications.

Subtraction of the two exposures (for specific number of dots) of each channel generated two bi-polar holograms H_PSH for the pinhole and H_OBJ for the NBS object. The image of the object was digitally reconstructed by cross-correlation of the object hologram H_OBJ with a phase-only filtered version of the pinhole hologram H’_PSH. The tested object was chosen to be as small as possible in order to avoid overlap between the replications even for high number of dots. On the other hand, the same object was yet resolved by the system, which its minimal resolved size was ∼36 µm (0.61λ/NA), and the system’s NA was ∼0.011. Figure 5 depicts the reconstructed images for different number of dots on the camera plane, from 10 to 90 dots. In order to conduct a reliable comparison between the different reconstruction images we quantify the quality of each image by: (1) its SNR and (2) the average visibility of the gratings along the horizontal and vertical red dashed lines in Fig. 5(a). The result below each reconstructed image in Fig. 5 stands for the product of these criteria which determines the optimal reconstructed image. Comparison between the different reconstructions indicates that for the current IC-COACH system and for the given object, the best number of dots to be projected on the camera from the CPM is between 30 to 50 dots [Figs. 4(c)–4(e)]. For these CPMs, the product of the reconstructed image SNR (left number) with the average horizontal and vertical visibility (right number) yields higher score than others. Moreover, this quantitative comparison between the reconstruction images demonstrates the inherent trade-off between projecting high or low numbers of dots on the camera plane. Too many replications of the input signal reduce the SNR level in the reconstruction images as shown in Figs. 5(f)–5(i), while PSH generating an ensemble of too few dots reduces the PSH complexity and blurs the image reconstruction as demonstrated in the visibility values [Figs. 5(a) and 5(b))]. It should be also noted that decreasing the number of dots with compromising the SNR and visibility extend the FOV. This experiment is a manifestation of the method’s robustness in terms of the CPM engineering, which is not subjected to any constrains other than the non-overlap between image replications condition. Due to the random location of each image replication, of course within the limit of the non-overlap condition, once the number of optimal dots on the camera was determined there is no need to further optimize the CPM and system is ready for use.

Fig. 5. (a)-(i) Image reconstructions of IC-COACH for various number of dots from 10 to 90 in interval of 10 dots. The product of SNR multiplied by the average of the horizontal and vertical visibilities for each reconstructed image is indicated by red below the images. Dashed red lines in Fig. 5(a) indicate the paths of calculating the visibilities for all the images of Figs. 5(a)–5(i).

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Next, the axial resolution of IC-COACH was investigated and compared to an equivalent system of regular lens-based imaging. In this experiment the pinhole played the double role as the tested input object as well as the PSH generator. Therefore, the 30 µm diameter pinhole was placed at the axial plane of object 2 while the object 1 channel was blocked, and a bi-polar PSH of the system was recorded, followed by recording an object hologram of the same pinhole at different axial planes in a range of 50 mm (25 mm backward and forward from the front focal plane of L₁). The PSH was cross-correlated with the point object hologram from the different axial planes. Figures 6(a)–6(c) illustrates the acquired bi-polar holograms for three different planes (without translation, and with translation of 5 mm and 10 mm, respectively) and the respective reconstructions of the point object are illustrated in Figs. 6(d)–6(f). The projected number of dots was chosen to be 75 in order to demonstrate the expansion of the Gaussian beam waist as the pinhole is translated to different planes other than L₁ front focal plane. As expected, the quality of the reconstructed image is gradually impaired as the object distance from the front focal plane is increased. An equivalent experiment was carried out for regular imaging system by displaying a constant phase mask on the SLM and capturing the system response for the input point object at the various axial planes. The average intensities of the 9 central pixels (3×3 pixels square obtained from the Airy disc size on the camera plane) at each axial distance were measured in order to quantify the axial resolution of the IC-COACH in comparison to the regular imaging system. Figure 6(g) depicts the normalized average intensity of the reconstructed and imaged peaks along the different axial planes. The plots confirm that IC-COACH axial resolution is as good as the axial resolution of the equivalent regular imaging system. In addition, based on scalar diffraction theory for the current system specifications, the calculated axial resolution is about 10.5 mm (2λ/NA²). This figure is close to the experimental result of about 9 mm axial resolution implied from the FWHM of the plot in Fig. 6(g).

Fig. 6. (a)-(c) Bi-polar object holograms (not all the dots are shown because of sampling limitations) and (d)-(f) the respective object reconstructions for point object located at different axial planes at (d) 0, (e) 5 and (f) 10 mm relative to L₁ front focal plane. (g) Normalized intensity of the central peak of the point image as a function of the point object translation from the front focal plane of L₁. The red line indicates the reconstructed image by IC-COACH and the black line is for the regular lens-based imaging system. Negative translation values correspond to the axial location greater than the focal distance of L₁.

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Demonstration of the 3D imaging capabilities of the IC-COACH system was initiated by a calibration process in which the system response to the 30 µm diameter pinhole was recorded at two different axial planes. A bi-polar PSH for each axial plane was generated from two independent CPMs by subtraction of one recorded intensity from the other. Overall there are 4 independent CPMs in this process, each of the two pairs for a different axial location composes the bi-polar PSH by one CPM of the positive dots and the other for the negative dots. The two CPM pairs for the two axial planes are spatially multiplexed to a single mask on the SLM by allocating each successive 5×5 SLM pixels square to the different pair. Once the calibration process is over and the PSHs for the axial planes are recorded, the system is ready for 3D object imaging. The 16 lp/mm element of the NBS chart was placed at plane P₁, whereas the 18 lp/mm element from another NBS chart was placed at plane P₂ and was translated to different axial locations. A bi-polar 3D object hologram was acquired by subtracting two images from the same two multiplexed CPMs of the calibration process. Note that regardless the number of observed axial planes, the digital camera is always exposed only twice for the positive and the negative parts of the object hologram. Reconstruction of the desired plane from the dual-plane scene is done by cross-correlating the object hologram with a phase-only filtered version of the PSH corresponding to the desired reconstructed plane. Figure 7 illustrates the various stages of the image reconstruction process of the scene using IC-COACH. The two upper rows are the camera acquisition of the positive and negative parts of the object hologram. The third row is the bi-polar object hologram and the two lower rows of Fig. 7 are the image reconstruction at the two different axial planes. The three columns of Fig. 7 indicate on three different gaps between the dual planes of Δz = 10, 15 and 20 mm. The importance of the bi-polar object holograms is emphasized in the two lower rows and the three upper rows of Fig. 7. First, a cross-correlation between bi-polar functions reduces the background noise typically obtained from a cross-correlation between positive functions. Second, the subtraction between two intensities cancels the undesired diffraction orders from the SLM shown clearly in the first two rows but almost disappear in the third row. The last two rows of Fig. 7 show that IC-COACH is able to successfully reconstruct the 3D object and to differentiate between the two axial planes by using the PSH of the corresponding plane. Moreover, it can be noted that as the separation distance between the planes, Δz, is increased the image reconstruction of the out-of-focus object is more blurred.

Fig. 7. Image reconstruction process of IC-COACH for different 3D object realizations indicated by the separation between two planes Δz. Upper two rows are the intensity on the image sensor for positive and negative CPMs. The third row is the bi-polar object hologram obtained by subtracting the second row from the first row. The lower two rows show the 3D image reconstruction at different planes obtained by a correlation with the corresponding PSH.

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Finally, as illustrated in Fig. 8, the experimental setup of Fig. 4 was slightly modified in order to image a 2D reflective object of white ink printed on a black inkjet photo paper using the suggested IC-COACH system. Since the dimensions of the reflective object are different than that of the previously examined NBS charts (approximately twice bigger), the number of PSH dots should be optimized for the current object dimensions. To this end, several CPM pairs differ by the number of projected PSH dots were examined by recording a PSH followed by recording an object hologram. Image reconstructions, accomplished by a cross-correlation of the object hologram with the phase-only filtered version of the PSH, of the letter ‘B’ for various number of dots are shown in Figs. 9(a)–9(j). The optimal reconstructed image in terms of SNR (noted for each reconstructed image below the object) is given by Fig. 9(b), where the number of dots used is 8. Figures 9(k)–9(m) illustrates the image reconstructions using IC-COACH of the letters ‘B’, ‘G’ and ‘U’ for the optimal number of dots (N = 8), whereas regular imaging shots under incoherent white illumination of the same objects are shown in Figs. 9(n)-(p). Regular imaging shots were captured using a commercial camera of Samsung Galaxy S6 Edge smartphone (with f/1.9, f = 4.3 mm lens and 5312 × 2988 pixels, 1.12 µm pixel pitch sensor). One can notice the coherent nature of the IC-COACH method implied by the speckle pattern presented on the images of IC-COACH in comparison to the more uniform images from the regular imaging. Speckle noise in digital-holography-based coherent imaging systems is a well-studied research area [19] and there are many works aiming at de-noising this phenomenon using variety of approaches [20]. On the other hand, there are many examples of using speckles as a useful tool in variety of applications [21–23]. We believe that IC-COACH can provide a reliable, fast and easy to assemble optical system for further study the phenomena of speckle noise in coherent imaging using digital holography.

Fig. 8. Experimental setup of IC-COACH for imaging reflective objects.

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Fig. 9. (a)-(j) Image reconstructions of reflective objects using IC-COACH for varying number of replications projected on the camera from 4 to 40, with interval of 4 replications. (k)-(m) Image reconstruction of different reflective objects using the CPM of optimal number of replications (N = 8). (n)-(p) Lens-based images under incoherent white illumination of the three different reflective objects.

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4. Summary and conclusions

We have proposed and experimentally tested a novel coherent holographic imaging system using an on-axis interferenceless digital holography technique. The proposed IC-COACH system has the capability to record a digital hologram of a 3D scene without wave interference and by only two camera shots. In principle, a single camera shot might be enough by use of non-linear reconstruction [8], a technique that probably will be tested on IC-COACH in the future. A successful image reconstruction by cross-correlation between the object hologram and a phase-only filtered version of the PSH is achieved by creating ensemble of randomly and sparsely distributed replications of the input object on the camera. The intensity pattern on the camera plane is obtained by modulating the input wave with a pseudorandom CPM designed by the GSA. Similar to its incoherent counterparts, 3D imaging with IC-COACH also involves an offline calibration process with a guidestar, in which the system responses to a point object located on the optical axis are recorded along the axial extent of the imaged 3D scene. In addition to the explicit advantages of recording coherent holograms in an on-axis configuration, without the requirement of any interferometer, an important benefit of the proposed method is the relatively high SNR along the process. The source light is not split to object and reference beams but entirely illuminates the object. Then, the object light is split to optimal number of beams such that every image replica is much higher than the noise level of the camera on one hand, and on the other hand there are enough replicas to guarantee a relatively low reconstruction noise. Some of the other advantages of I-COACH listed in [8–12] might also be implementable in IC-COACH. These features place IC-COACH as a promising method for fast coherent 3D imaging tasks that can capture dynamic scenes.

The proposed new type of holograms releases users from the need to interfere waves, simplifies the optical setups for recording holograms, make them more robust and make their calibration easier. For applications of optical imaging, the observed 3D scene can be compressed to 2D digital hologram by only two camera intensity shots. In the computer the 2D hologram can be reconstructed to a 3D image of the original observed scene. The high acquisition rate and the simplicity of the technique make it attractive for further research and development.

Funding

Israel Science Foundation (ISF) (1669/16); Ministry of Science, Technology and Space.

Acknowledgment

This study was done during a research stay of JR at the Alfried Krupp Wissenschaftskolleg Greifswald.

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Interferenceless and motionless method for recording digital holograms of coherently illuminated 3D objects by coded aperture correlation holography system

Abstract

1. Introduction

2. Methodology

3. Experiments and results

4. Summary and conclusions

Funding

Acknowledgment

References

Cited By

Figures (9)

Equations (9)

Optics Express