Abstract
The coherence holography offers an unconventional way to reconstruct the hologram where an incoherent light illumination is used for reconstruction purposes, and object encoded into the hologram is reconstructed as the distribution of the complex coherence function. Measurement of the coherence function usually requires an interferometric setup and array detectors. This paper presents an entirely new idea of reconstruction of the complex coherence function in the coherence holography without an interferometric setup. This is realized by structured pattern projections on the incoherent source structure and implementing measurement of the crosscovariance of the intensities by a singlepixel detector. This technique, named structured transmittance illumination coherence holography (STICH), helps to reconstruct the complex coherence from the intensity measurement in a singlepixel detector without an interferometric setup and also keeps advantages of the intensity correlations. A simple experimental setup is presented as a first step to realize the technique, and results based on the computer modeling of the experimental setup are presented to show validation of the idea.
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Introduction
Since its inception more than seventy years ago, holography offers a powerful tool for imaging and light synthesis^{1,2,3,4,5}. Availability of highquality array detectors and reconstruction algorithms has further revolutionized the holography, and optical reconstruction in the holography is replaced by the digital means, and technique is called digital holography (DH). The DH keeps inherent advantages of the holography concerning the complex amplitude distribution and additionally offers a simplified reconstruction of the hologram by a numerical means^{5}. Conventional DH records and reconstructs the complex wavefield by an optical field distribution itself. The phase information of the wavefield is an important physical parameter of the light. Among various methods to recover the phase information, the DH is a wellestablished technique for quantitative phase imaging (QPI). Various QPI techniques have been developed, and significant among them are inline, offaxis, and phaseshifting holography. In recent years, attempts have been made to improve the transverse spatial resolution in the DH by random^{6,7,8} and structured light illumination^{9,10,11,12}. Requirements to retrieve information from the selfluminous or incoherent object have also inspired new trends in the holography^{13,14,15,16,17,18,19}.
In a significant development, Takeda and coworkers have developed an unconventional holography called coherence holography (CH), where the information of the complex field is reconstructed as a distribution of the spatial coherence^{20}. Here, the hologram is reconstructed by an incoherent light illumination rather than by a coherent light. The CH has opened new research directions on recording and shaping the spatial coherence for applications such as spatial coherence tomography, profilometry, imaging, and coherence current^{21,22,23,24,25}. Principle of the CH is derived from the van CittertZernike theorem, which connects the incoherent source structure with a farfield spatial coherence of the light. The main task in the CH is to design an appropriate interferometer for the measurement of spatial coherence. These interferometers mainly employ secondorder correlation or fourthorder correlations measurement by the array detectors. In contrast to the interferometers based on the secondorder correlations, the fourthorder correlation, i.e., intensity interferometers are highly stable for the coherence measurement^{26}. Recently, Naik et al.^{27} made use of the fourthorder correlation in the hologram, and technique is called a photon correlation holography (PCH). Basic principle of the PCH is derived from a connection between the crosscovariance of the intensities with the modulus square of the Fourier spectrum of the incoherent structure. However, phase information of the spectrum is lost in the PCH in contrast to the CH and DH techniques. Recovery of complex field parameters in the CH and DH is possible by employing an interferometer setup which makes the system bulky and prone to external disturbances and instabilities^{2,22,24}. Recently, holographic methods based on interference of the coherence waves have been proposed to overcome the phase loss issue in the PCH experiments^{28,29,30,31,32,33}. However, the interferometric systems bring bulkiness in the experimental implementation and also require flexible control over the reference field to get the interference fringes in the crosscovariance function at the array detectors plane.
On the other hand, significant attempts have been made to develop computational imaging techniques such as singlepixel imaging with random and structured field illumination over the past few years^{34,35,36,37,38,39,40,41,42,43}. In contrast to using conventional cameras and twodimensional array detectors, singlepixel techniques make use of projection of light patterns onto a sample while a singlepixel detector measures the light intensity collected for each pattern. Therefore, stage of spatial sampling is moved from the camera to the programmable diffraction element where the structured patterns are loaded. Singlepixel imaging techniques have brought advantages such as use of a nonvisible wavelength or precise time resolution, which can be costly and practically challenging to realize as a pixilated imaging device. Recently, a combination of optical and computation channels has been developed for the reconstruction of the threedimensional (3D) amplitude object from a singlepixel detector, and technique is called hybrid correlation holography (HCH)^{41}. This technique makes use of crosscovariance of the intensity and is derived from the connection between complex coherence function and intensity correlation for Gaussian random field. A new scheme based on the recovery of the complexvalued object in a modified HCH scheme with an interferometric setup has been developed^{44}.
In this paper, we present a new technique for the reconstruction of the complex field within the framework of the PCH and present a new theoretical basis for the reconstruction in correlation holography. This approach equips correlation imaging with a complete wavefront reconstruction without an interferometric setup but keeping the advantage of the intensity correlation. For this purpose, a structured light illumination is projected on the incoherent structure, and a farfield spectrum is measured by a singlepixel detector. A complex Fourier spectrum from the intensities is successfully obtained from the fourstep phase shifting in the structured illumination. Although the Fourier spectrum measurement in the CH is based on the Hanbury BrownTwiss (HBT) approach with a singlepixel detector but active illumination strategy in the proposed technique helps to overcome the phase loss problem of the typical HBT approach. Applying twodimensional (2D) inverse Fourier transform (IFT) to the obtained spectrum yields the desired digital hologram. The phaseshifting illumination approach also brings the elimination of noise that is statistically the same. Moreover, such noninterferometric approach assisted with structured light illumination offers a new and stable method for imaging with incoherent light. A detailed theoretical foundation and implementation of the proposed technique in comparison to the CH and PCH are discussed below.
Basic principle
Basic principles of the CH and PCH have been discussed in detail in Ref.^{20,27}. However, for the sake of continuity and to connect with basic principle of the proposed technique, we briefly describe the CH and PCH. Figure 1a represents a coherent recording of the complex field of an object in the Fourier hologram H(r) . The hologram H(r) is a transparency that is read out with incoherent light, as shown in Fig. 1b for the CH. To describe the reconstruction process of the hologram in Fig. 1b, consider the complex field of light immediately behind the source as
where i denotes the imaginary unit and \(H(r)=H(r) \exp [i \delta (r)]\) with H(r) and \(\delta (r)\) being the amplitude transmittance of the hologram and deterministic phase of the readout light, respectively. The spatial vector at the source is \(r\equiv (x,y)\). The random phase inserted in the light path to destroy spatial coherence by the rotating ground glass (RGG) is represented by \(\phi (r)\) at a fixed time t. A lens in Fig. 1b with focal distance f is used to Fourier transform the randomly scatted light field from the source, and the complex field on the observation plane becomes
where \(k \equiv (k_{x},k_{y})\) is spatial frequency coordinate at the observation point. Twopoint correlation of the random field is characterized as
here \(\left\langle .\right\rangle\) represents the ensemble average which will be replaced by the temporal average in the experiment. The rotating ground glass is considered to produce an incoherent source, i.e. \(\langle \exp \left[ i\left( \phi \left( r_{2}\right) \phi \left( r_{1}\right) \right) \right] \rangle \equiv \delta \left( r_{2}r_{1}\right)\). Considering \(k_{2}=k\) and \(k_{1}=0\), Eq. (3) transforms into the van Cittert Zernike theorem as
where \(I(r)=H(r)^{2}\) is the source placed at the RGG plane and F(k) represents the Fourier spectrum of the incoherent source at the farfield. It is important to mention that Eq. (4) connects an incoherent source structure at the RGG plane with the farfield complex coherence function and the source structure I(r) can be a hologram or any real source structure^{30,45,46}. The complex spatial coherence function is used to record the incoherent object and develop a new kind of unconventional holography^{20,45}. The basic principle of the CH is described by Eq. (4) and therefore provides reconstruction of the object as the distribution of the complex coherence function. The random field intensity at the observation plane, at a fixed time t corresponding to one rotation state of the RGG, is represented as
The random intensity pattern I(k) is having no direct resemblance to reconstruction of the object. The crosscovariance of the intensities of the Gaussian random field is given as^{26}
where \(\Delta I(k)=I(k)\langle \Delta I(k)\rangle\) is the fluctuation of the intensities with respect to its average mean value. Eq. (6) highlights the basic principle of the PCH and is sketched in Fig. 1c, wherein the phase part of the coherence function is lost. To circumvent the issue of phase recovery with only measurement of the crosscovariance of the intensities, we here present a new technique called STICH. A basic principle of the STICH is represented in Fig. 1d and described as follows. A two dimensional (2D) structured illumination with its spatial frequency \((k_{x},k_{y})\) and initial phase \(\theta\) is projected on the RGG. This structured illumination is a sinusoidal pattern and is represented as
where a is an unmodulated term of the illumination pattern, and b represents the contrast. The light coming out of the structured transparency propagates through the RGG, which is used to mimic an incoherent light source. A hologram H is placed next to the RGG, as shown in Fig. 1d. Therefore, the instantaneous complex field immediately after the H is expressed as:
The instantaneous complex field at the singlepixel detector is represented as
where \(\Omega\) represents the illuminated area, w is a scale factor whose value depends on the size and the location of the detector, \(E_{n}\) represents the response of background illumination. The instantaneous random intensity at the singlepixel detector is given as
The random intensity variation from its mean intensity is calculated as
where the angular bracket \(\langle . \rangle\) denotes the ensemble average and \(\langle I_{\theta }\left( k_{x}, k_{y}\right) \rangle\) is mean intensity. The crosscovariance of the intensities is
The 4step phaseshifting approach allows extraction of the complex Fourier coefficient \(F(k_{x},k_{y})\) by combination of four crosscovariance fuctions of the intensities \(D_{0}\), \(D_{\pi /2}\), \(D_{\pi}\) , \(D_{3\pi /2}\) corresponding to the illumination patterns, and the fourier coefficient \(F(k_{x},k_{y})\) is represented as^{34}
The Fourier coefficient is expressed as
By computing Fourier coefficients F (i.e., the Fourier spectrum) using Eq. (14) for a complete set of \((k_{x},k_{y})\), the desired complex field distribution is reconstructed. A 4step phaseshifting sinusoid illumination plays an essential role in the proposed technique and helps to reconstruct the incoherently illuminated hologram in the CH. Eq. (13) can not only assemble the Fourier spectrum of the desired incoherent source structure but also eliminate undesired direct current (DC) terms.
Experimental design and algorithm
A possible experimental design for the proposed technique is shown in Fig. 2. A monochromatic collimated laser light is folded by a beam splitter (BS) and incident on a spatial light modulator (SLM). The SLM in Fig. 2 is considered to be a reflective type and loaded with the h number of sinusoidal patterns in a sequence.
The sinusoidal pattern displayed to the SLM is inserted into the incident beam, and subsequently, this structured light transmits through the BS and illuminates the RGG. The RGG introduces randomness in the incident structured light. As shown in Fig. 1a, a computergenerated hologram of an offaxis object is used as transparency and placed adjacent to the RGG. The structured pattern embedded in the stochastic field due to the RGG illuminates a hologram H(x, y). Scattering of the light through the RGG generates a stochastic field with the Gaussian statistics. The scattered light further propagates and is Fourier transformed by a lens L at the singlepixel detector plane D. Corresponding to the spatial frequency and initial phase of the loaded structured pattern, an instantaneous random field at the singlepixel detector is represented by \(E_{\theta }\left( k_{x}, k_{y}\right)\). The instantaneous signal at the detector is represented as \(E_{\theta }\left( k_{x}, k_{y}\right) ^{2}\). After the singlepixel measurement for a particular random phase mask, we stored the value in our personal computer (PC) for postprocessing. Due to the Gaussian statistics, the crosscovariance of the intensities is estimated at the singlepixel corresponding to different sets of random phase masks introduced by the RGG. The crosscovariance of the intensities at the singlepixel detector for a given frequency pair \((k_{x}, k_{y})\) and initial phase \(\theta\) is represented as \(D_{\theta }\left( k_{x}, k_{y}\right)\). For a complete set of Fourier coefficients, we illuminate the hologram H(x, y) by the structured patterns with full sets of spatial frequency \((k_{x}, k_{y})\). Each complex Fourier coefficient corresponding to that spatial frequency is extracted by using a 4step phaseshifting approach. The number of Fourier coefficients in the Fourier domain is the same as the number of pixels in the spatial domain.
The algorithm proposed in this paper is an iterative heuristic that aims to reconstruct complex object encoded into the hologram as the distribution of the complex coherence function. In contrast to the previously reported CH, here we report the use of the structured illumination at the RGG plane. This strategy makes reconstruction procedure completely different from previously developed reconstruction methods and also equips us to extract the complex field even from a singlepixel detector. This is a unique feature of our proposed technique that helps to recover the complex coherence without interferometry. The algorithm in our work is implemented using MATLAB and simulated on a personal computer. Figure 3 shows the steps of the algorithm, which are:

1.
A hologram of size \(N \times N\) pixels is taken as transparency.

2.
Construction of sinusoidal patterns and random phase masks :

(a)
Total M number of different random phase masks of same size of hologram are generated.

(b)
The 2D sinusoidal patterns of size \(N \times N\) pixels with initial phase \({\theta }=(0,\pi /2,\pi ,3\pi /2)\) are constructed by considering discretized spatial frequency space \(k<k_{x}, k_{y}<k\).

(a)

3.
Iterative steps:

(a)
First, a 2D sinusoidal pattern for that particular frequency pair \((k_{x}, k_{y})\) is taken. A singlepixel detector is used to sense the random light field ( matrix multiplication of hologram and sinusoidal pattern and random phase mask ), and corresponding random intensity is represented using Eq. (10).

(b)
In such a way, other intensity patterns at singlepixel detector corresponding to different sets of random phase masks introduced by the RGG are obtained. Random intensity variations from its mean intensity are calculated using Eq. (11). Crosscovariance of the intensity for the taken spatial frequency pair \((k_{x}, k_{y})\) is calculated using Eq. (12).

(c)
Calculation of each complex valued Fourier coefficient \(F\left( k_{x}, k_{y}\right)\) corresponding to the spatial frequency pair is simulated from four crosscovariance fuctions of the intensities \(D_{\theta }(\theta =0,\pi /2,\pi ,3\pi /2)\) ( see Eq. 14).
For the complete set of desired \(R\times R\) size Fourier coefficients, we need to iterate the above steps over the discretized spatial frequency space unless the last iteration is obtained. Total \(4R^{2}(=4 \times R \times R)\) number of sinusoidal patterns need to be projected, including the fourstep phaseshifting by an SLM shown in Fig. 2, on which the patterns are controlled by a personal computer (PC) directly. A Fourier coefficients \(F\left( k_{x}, k_{y}\right)\) is obtained with 4 measurements. So, basically, for fully sampling \(R \times R\) size Fourier coefficients consumes \(4 \cdot R^{2} \cdot M(=4 \times R \times R \times M)\) measurements of singlepixel detector to reconstruct the complex object.

(a)
Results
Here, we present computational results processed using MATLAB for the validation of our proposed experimental design. The phase information of the wavefield is a critical parameter to examine a complex field of an object. Figure 4a–d show both amplitude and phase distributions of objects, letter “P” and number “3” respectively, which are directly reconstructed from DH. Applying twodimensional Fourier transform on the DH brings out three spectra: a nonmodulating central DC term, the desired offaxis spectrum, and its conjugate. Location of an offaxis spectrum is governed by carrier frequency as shown in recording of DH in Fig. 1a. The unwanted DC term containing highfrequency content is digitally suppressed in Fig. 4 to highlight the objects located in an offaxis position. Figures 5 and 6 represent the reconstructed complex fields from holograms using the STICH technique at different numbers of random phase masks (M) and the quality of reconstruction depends on the value of M. In order to examine the effect of M on reconstruction quality of STICH, we evaluate visibility (\(\nu\)) and reconstruction efficiency (\(\eta\))^{47} for three different numbers of \(M = (200, 500, 1000)\) and results are given in Table 1. Amplitude and phase distributions of number “3” are shown in Fig. 5a–f for \(M = (200, 500, 1000)\). Similarly for letter “P”, Fig. 6a–f show the amplitude and phase distributions for \(M = (200, 500, 1000)\). In both object’s reconstruction central DC terms are digitally suppressed as shown in Figs. 5 and 6. The central DC in the reconstruction appears because of use of an offaxis hologram as the transparency. The visibility of a target reconstruction is defined as the degree to which it can be distinguished from background noise. It is calculated as the ratio of the average image intensity level in the signal region to the average background intensity level. Here Otsu’s method^{48} is used as a global threshold to identify the signal region.
In order to reconstruct complex fields of size \(100\times 100\) from hologram of size \(200\times 200\) using STICH, structured illumination patterns are generated according to Eq. (7), where \(a=0.5\), \(b=0.5\), spatial frequencies range is \(2.5\le (k_{x},k_{y})\le 2.5\) at steps of 0.0505.
The calculated visibility value for Fig. 4a,c are 127.6 and 64.7. The calculated visibility and reconstruction efficiency value for Fig. 5a–c and for Fig. 6a–c are given in Table 1. (I) and (II) respectively. The upper part of conjugate phase distributions in Figs. 5 and 6 are highlighted with white color ring. From Table 1 and Figs. 5 and 6, it can be seen that reconstruction quality improves with increase of value of M. Number of sinusoidal patterns required to be projected for the reconstruction of complex coherence function can be reduced using the compressed sensing approach^{49}. Moreover, automated coupling of the SLM with the singlepixel detector is expected to simplify the experimental implementation of the proposed work.
Conclusion
In conclusion, a new technique entitled STICH is presented to reconstruct the complex coherence from the intensity measurement with a singlepixel detector and without an interferometric setup. This brings the advantages of compatibility in the reconstruction of complex fields in the correlationbased imaging system. An experimental configuration and computational model of it are described to validate our idea. We have demonstrated the reconstructions of complex fields of objects at different random phase masks and the quality of reconstruction depends on the value of number of random phase masks used to realize the thermal light source. Furthermore, the proposed technique is capable to image the incoherent source at the scattering plane from the reconstructed complex coherence function. This incoherent source may appear as a hologram or nonhologram prior to the random scattering. Therefore, basic principle of structured light illumination for the coherence holography is expected to provide new direction on the holography with incoherent light and imaging through scattering medium.
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Acknowledgements
The work is supported by the Science and Engineering Research Board (SERB) India CORE/2019/000026 and Department of Biotechnology (DBT) BT/PR35557/MED/32/707/2019. T.S acknowledges support from University Grant Commission (UGC), India for his scholarship.
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A.C.M. conceived of idea and build the theoretical basis, experimental design, and completed simulation and preparation of manuscript. T.S. involved in experimental design, simulation, preparation of manuscript. Z.Z. provided advice and assistance, reviewing and editing the work. R.K.S. was involved in supervision, formulation of research goals and aims, funding acquisition, reviewing, and editing.
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Mandal, A.C., Sarkar, T., Zalevsky, Z. et al. Structured transmittance illumination coherence holography. Sci Rep 12, 4564 (2022). https://doi.org/10.1038/s41598022086034
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DOI: https://doi.org/10.1038/s41598022086034
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