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Structured transmittance illumination coherence

holography

Aditya Chandra Mandal1,2, Tushar Sarkar1, Zeev Zalevsky3, and Rakesh Kumar Singh1,*

1

Laboratory of Information Photonics and Optical Metrology, Department of Physics, Indian Institute of Technology

(Banaras Hindu University), Varanasi, 221005, Uttar Pradesh, India

2Department of Mining Engineering, Indian Institute of Technology (Banaras Hindu University), Varanasi, 221005,

Uttar Pradesh, India

3Bar-Ilan University, Faculty of Engineering and Nano Technology Center, Ramat-Gan, Israel

*krakeshsingh.phy@iitbhu.ac.in

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 cross-covariance of the intensities by a single-pixel detector. This technique, named

structured transmittance illumination coherence holography (STICH), helps to reconstruct the complex coherence from the

intensity measurement in a single-pixel 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.

Introduction

Since its inception more than seventy years ago, holography offers a powerful tool for imaging and light synthesis

1–4

. The

availability of high-quality 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

4

. 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 well-established technique for quantitative phase imaging (QPI).

Various QPI techniques have been developed, and signiﬁcant among them are in-line, off-axis, and phase-shifting holography.

In recent years, attempts have been made to improve the transverse spatial resolution in the DH by random

5–7

and structured

light illumination

8–11

. Requirements to retrieve information from the self-luminous or incoherent object have also inspired new

trends in the holography12–18.

In a signiﬁcant development, Takeda and co-workers 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

19

. 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

20–24

. Principle of the CH is derived from the van Cittert-Zernike theorem, which

connects the incoherent source structure with a far-ﬁeld 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 second-order

correlation or fourth-order correlations measurement by the array detectors. In contrast to the interferometers based on

the second-order correlations, the fourth-order correlation, i.e., intensity interferometers are highly stable for the coherence

measurement

25

. Recently, Naik et al.

26

made use of the fourth-order 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 cross-covariance 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

arXiv:2109.10326v1 [physics.optics] 21 Sep 2021

and DH is possible by employing an interferometer setup which makes the system bulky and prone to external disturbances

and instabilities

2,21,23

. Recently, holographic methods based on interference of the coherence waves have been proposed

to overcome the phase loss issue in the PCH experiments

27–32

. 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

cross-covariance function at the array detectors plane.

On the other hand, signiﬁcant attempts have been made to develop computational imaging techniques such as single-pixel

imaging with random and structured ﬁeld illumination over the past few years

33–41

. In contrast to using conventional cameras

and two-dimensional array detectors, single-pixel techniques make use of projection of light patterns onto a sample while a

single-pixel 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. Single-pixel imaging

techniques have brought advantages such as use of a non-visible 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 three-dimensional (3D) amplitude object from a single-pixel detector, and

technique is called hybrid correlation holography (HCH)

40

. This technique makes use of cross-covariance 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 complex-valued object in a modiﬁed HCH scheme with an interferometric setup has been

developed42.

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 far-ﬁeld spectrum is measured by a

single-pixel detector. A complex Fourier spectrum from the intensities is successfully obtained from the four-step phase shifting

in the structured illumination. Although the Fourier spectrum measurement in the CH is based on the Hanbury Brown-Twiss

(HBT) approach with a single-pixel detector but active illumination strategy in the proposed technique helps to overcome the

phase loss problem of the typical HBT approach. Applying two-dimensional (2D) Fourier transform (IFT) to the obtained

spectrum yields the desired DH. The phase-shifting illumination approach also brings the elimination of noise that is statistically

the same. 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.

19,26

. However, for the sake of continuity and to

connect with basic principle of the proposed technique, we brieﬂy describe the CH and PCH. Fig. 1(a) represents a coherent

recording of the complex ﬁeld of an object in the Fourier hologram

H(r)

. The hologram

H(r)

is read out with the incoherent

light, as shown in Fig. 1(b). To describe the reconstruction process of the hologram in Fig. 1(b), consider the complex ﬁeld of

light immediately behind the hologram as

E(r) = H(r)exp[iφ(r)] (1)

where

i

denotes the imaginary unit and

H(r) = |H(r)|exp[iδ(r)]

with

|H(r)|

and

δ(r)

being the amplitude transmittance of the

hologram and deterministic phase of the readout light, respectively. The spatial vector at the source is

r≡(x,y)

. The random

phase inserted in the light path to destroy spatial coherence by the rotating ground glass (RGG) is represented by

φ(r)

at a ﬁxed

time

t

. A lens in Fig. 1(b) 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

E(k) = ZE(r)exp[−i k ·r]dr =ZH(r)exp[iφ(r)]exp[−i k ·r]dr (2)

where

k≡(kx,ky)

is spatial frequency coordinate at the observation point. Two-point correlation of the random ﬁeld is

characterized as

W(k1,k2) = hE∗(k1)E(k2)i

=ZZ H∗(k1)H(k2)hexp[i(φ(k2)−φ(k1))]iexp[−i(k2·r2−k1·r1)]dr2d r1

(3)

here

h.i

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.

hexp[i(φ(r2)−φ(r1))]i ≡ δ(r2−r1)

, where

k2=k

and

k1=0

.

2/10

(a) (b)

(c) (d)

Figure 1. (a) Represents coherent recording of complex ﬁeld of an object located at Z=0 , Lis a lens with focal distance f

(b) Represents the reconstruction process of the hologram with incoherent light, where

L

is a lens with focal distance

f

, RGG is

rotating ground glass,

H

is hologram ; (c) Represents the PCH setup: intensity correlation holography scheme; (d) Represents

the STICH setup, where SL is structured light pattern, His hologram, Dis single-pixel detector

Therefore, Eq. 3transforms into the van Cittert Zernike theorem as

F(k) = ZI(r)exp[−i k ·r]dr (4)

where

I(r) = H∗(r)H(r)

is the source hologram placed at the RGG plane and

F(k)

represents the Fourier spectrum of the

incoherent source at the far-ﬁeld. The basic principle of the CH is described by Eq. 4and 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 tcorresponding to one rotation state of the RGG, is represented as

I(k) = |E(k)|2(5)

The random intensity pattern

I(k)

is having no direct resemblance to reconstruction of the object. The cross-covariance of

the intensities of the Gaussian random ﬁeld is given as

h∆I(k)∆I(0)i=|F(k)|2(6)

where

∆I(k) = I(k)− h∆I(k)i

is the ﬂuctuation of the intensities with respect to its average mean value. Eq. 6highlights the

basic principle of the PCH and is sketched in Fig. 1(c), wherein the phase part of the coherence function is lost. To circumvent

the above-mentioned issue on recovery of complex phase with the only measurement of the cross-covariance of the intensities,

we present a new technique called STICH. Basic principle of the STICH is represented in Fig. 1(d) and described as follows. A

two dimensional (2-D) structured illumination with its spatial frequency

(kx,ky)

and initial phase

θ

is projected on the RGG.

This structured illumination is a sinusoidal pattern and is represented as

P

θ(x,y;kx,ky) = a+b·cos(kxx+kyy+θ)(7)

where

a

is an un-modulated 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. 1(d). Therefore, the instantaneous complex ﬁeld immediately after the His expressed as:

Eθ(x,y;kx,ky) = H(x,y)exp(iφ(x,y))(a+b·cos(kxx+kyy+θ)) (8)

The instantaneous complex ﬁeld at the single-pixel detector is represented as

Eθ(kx,ky) = En+wZZΩ

Eθ(x,y;kx,ky)dxdy (9)

3/10

where

Ω

represents the illuminated area,

w

is a scale factor whose value depends on the size and the location of the detector,

En

represents the response of background illumination. The instantaneous random intensity at the single-pixel detector is given as

Iθ(kx,ky) = |Eθ(kx,ky)|2(10)

The random intensity variation from its mean intensity is calculated as

∆Iθ(kx,ky) = Iθ(kx,ky)− hIθ(kx,ky)i(11)

where the angular bracket

h.i

denotes the ensemble average and

hIθ(kx,ky)i

is mean intensity. The cross-covariance of the

intensities is

Dθ(kx,ky) = h∆Iθ(kx,ky)∆Iθ(kx,ky)i(12)

The 4-step phase-shifting approach allows each complex Fourier coefﬁcient

F(kx,ky)

to be obtained by every four responses

corresponding to the illumination patterns, i.e.

P

0

,

P

π/2

,

P

π/2

,

P

3π/2

. The response

Dθ(θ=0,π/2,π,3π/2)

are used to obtain

the Fourier spectrum of H, i.e. F(kx,ky)as

[D0(kx,ky)−Dπ(kx,ky)] + i·Dπ/2(kx,ky)−D3π/2(kx,ky)

=2bwRRΩH(x,y)·exp[−i(kxx+kyy)] d xdy (13)

The Fourier coefﬁcient is expressed as

F(kx,ky) = RRΩH(x,y)·exp[−i(kxx+kyy)] dxdy

=1

2bw [D0(kx,ky)−Dπ(kx,ky)] + i·Dπ/2(kx,ky)−D3π/2(kx,ky)(14)

By computing Fourier coefﬁcients

F

(i.e., the Fourier spectrum) using Eq. 14 for a complete set of

(kx,ky)

, the desired complex

ﬁeld distribution is reconstructed. A 4-step phase-shifting sinusoid illumination plays an essential role in the proposed technique.

Eq. 13 can not only assemble the Fourier spectrum of the desired hologram image 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 gratings 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. 1(a), a computer-generated hologram of an off-axis

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 single-pixel detector plane

D

.

Corresponding to the spatial frequency and initial phase of the loaded structured pattern, an instantaneous random ﬁeld at

the single-pixel detector is represented by

Eθ(kx,ky)

. The instantaneous signal at the detector is represented as

|Eθ(kx,ky)|2

.

After the single-pixel measurement for a particular random phase mask, we stored the value in our personal computer (PC)

for post-processing. Due to the Gaussian statistics, the cross-covariance of the intensities is estimated at the single-pixel

corresponding to different sets of random phase masks introduced by the RGG. The cross-covariance of the intensities at the

single-pixel detector for a given frequency pair

(kx,ky)

and initial phase

θ

is represented as

Dθ(kx,ky)

. For a complete set of

Fourier coefﬁcients, we illuminate the hologram

H(x,y)

by the structured patterns with full sets of spatial frequency

(kx,ky)

.

Each complex Fourier coefﬁcient corresponding to that spatial frequency is extracted by using a 4-step phase-shifting 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 approach and also equips us to extract the complex ﬁeld even from a single-pixel 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. Fig. 3 shows the steps of the

algorithm, which are:

1. A hologram of size N×Npixels is taken as transparency.

4/10

Figure 2. Experimental conﬁguration: BS- beam splitter, SLM- spatial light modulator, RGG- rotating ground glass, T-

transparency, L- lens with focal distance f, D- single-pixel detector, PC- personal computer.

2. Construction of sinusoidal patterns and random phase masks :

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

(b)

The 2-D sinusoidal patterns of size

N×N

pixels with initial phase

θ= (0,π/2,π,3π/2)

are constructed by

considering discretized spatial frequency space −k<kx,ky<k.

3. Iterative steps:

(a)

First, a 2-D sinusoidal pattern for that particular frequency pair

(kx,ky)

is taken. A single-pixel 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 single-pixel 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. Cross-covariance of the intensity for the taken spatial frequency pair

(kx,ky)

is calculated using Eq.

12.

(c)

Calculation of each complex valued Fourier coefﬁcient

F(kx,ky)

corresponding to the spatial frequency pair is

simulated from four responses Dθ(θ=0,π/2,π,3π/2)( see Eq. 14).

For the complete set of desired

R×R

size Fourier coefﬁcients, we need to iterate the above steps over the discretized

spatial frequency space unless the last iteration is obtained. Total

4R2(= 4×R×R)

number of sinusoidal patterns need

to be projected, including the four-step phase-shifting by an SLM shown in Fig. 2, on which the patterns are controlled

by a personal computer (PC) directly. A Fourier coefﬁcients

F(kx,ky)

is obtained with 4 measurements. So, basically,

for fully sampling

R×R

size Fourier coefﬁcients consumes

4·R2·M(= 4×R×R×M)

measurements of single-pixel

detector to reconstruct the complex object.

5/10

Figure 3. Proposed STICH algorithm steps.

Results

Here, we present computational results processed using MATLAB for the validation of our proposed experimental setup.

The phase information of the waveﬁeld is a critical parameter to examine a complex ﬁeld of an object. Fig 4.(a),(c) and

(b),(d) show both amplitude and phase distributions of objects, letter “P” and number “3” respectively, which are directly

reconstructed from DH. Applying two-dimensional Fourier transform on the DH brings out three spectra: a non-modulating

central DC term, the desired spectrum, and its off-axis local and unit conjugate. Location of off-axis spectrum is governed by

carrier frequency as shown in recording of DH in Fig. 1(a). The unwanted DC terms containing high-frequency content are

digitally suppressed in Fig 4 to highlight the objects located in off-axis position. Fig 5 and Fig 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 it. In order to examine the effect of

M

on reconstruction quality of STICH, we evaluate visibility

(

ν

) and reconstruction efﬁciency (

η

)

43

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 5. (a)-(c) and (d)-(f) for

M= (200,500,1000)

. Similarly

for letter “P”, Fig 6. (a)-(c) and (d)-(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 Fig 5 and 6. The central DC in the reconstruction appears

because use of off-axis 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 method44 is used as a global threshold to identify the signal region.

In order to reconstruct complex ﬁelds of size

100 ×100

from hologram of size

200 ×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≥(kx,ky)≤2.5

at steps of 0.0505.

6/10

(a) (b)

(c) (d)

Figure 4. Recovery of complex ﬁelds without RGG from holograms of objects, letter “P” and number “3” (a) Amplitude

distribution of letter “P” (b) Phase distribution of letter “P” (c) Amplitude distribution of number “3” (d) Phase distribution of

number “3”.

The calculated visibility value for ﬁg. 4.(a) and (c) are 127.6 and 64.7. The calculated visibility and reconstruction efﬁciency

value for ﬁg. 5. (a)-(c) and for ﬁg. 6.(a)-(c) are given in Table 1. (I) and (II) respectively. The upper part of conjugate phase

distributions in Fig. 5 and Fig. 6 are highlighted with white color annular rings. From Table 1 and Fig 5 and Fig 6, it can be

seen that reconstruction quality improves with increase of value of M.

7/10

(a) (b) (c)

(d) (e) (f)

Figure 5. Recovery of the complex ﬁelds using STICH for three different Mvalues: an off-axis hologram of number “3” is

used as transparency (a), (d) Amplitude and phase distribution of object for

M=200

; (b), (e) Amplitude and phase distribution

of object for M=500; (c), (f) Amplitude and phase distribution of object for M=1000.

(a) (b) (c)

(d) (e) (f)

Figure 6.

Recovery of the complex ﬁelds using STICH for three different

M

values: an off-axis hologram of letter “P” is used

as transparency (a), (d) Amplitude and phase distribution of object for M=200; (b), (e) Amplitude and phase distribution of

object for M=500; (c), (f) Amplitude and phase distribution of object for M=1000.

8/10

Conclusion

In conclusion, a new technique entitled STICH is presented to reconstruct the complex coherence from the intensity measure-

ment with a single-pixel detector and without an interferometric setup. This brings the advantages of compatibility in the

reconstruction of complex ﬁelds in the correlation-based imaging system. A experimental conﬁguration and computational

model of it is 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. The proposed technique is expected to provide new direction on the coherence holography and imaging

through scattering medium.

References

1. Gabor, D. A new microscopic principle. Nature 161, 777–778 (1948).

2. Leith, E. N. & Upatnieks, J. Reconstructed wavefronts and communication theory. JOSA 52, 1123–1130 (1962).

3.

Yamaguchi, I. Phase-shifting digital holography. In Digital Holography and Three-Dimensional Display, 145–171

(Springer, 2006).

4.

Hendry, D. Digital holography: Digital hologram recording, numerical reconstruction and related techniques, u. schnars, w.

jueptner (eds.), springer, berlin (2005),(164 pp.+ ix.,$ 69.95), isbn: 354021934-x (2006).

5. Park, Y. et al. Speckle-ﬁeld digital holographic microscopy. Opt. express 17, 12285–12292 (2009).

6.

Vinu, R., Chen, Z., Pu, J., Otani, Y. & Singh, R. K. Speckle-ﬁeld digital polarization holographic microscopy. Opt. letters

44, 5711–5714 (2019).

7. Choi, Y. et al. Dynamic speckle illumination wide-ﬁeld reﬂection phase microscopy. Opt. letters 39, 6062–6065 (2014).

8.

Gao, P., Pedrini, G. & Osten, W. Structured illumination for resolution enhancement and autofocusing in digital holographic

microscopy. Opt. letters 38, 1328–1330 (2013).

9.

Ma, J., Yuan, C., Situ, G., Pedrini, G. & Osten, W. Resolution enhancement in digital holographic microscopy with

structured illumination. Chin. Opt. Lett. 11, 090901 (2013).

10.

Micó, V., Zheng, J., Garcia, J., Zalevsky, Z. & Gao, P. Resolution enhancement in quantitative phase microscopy. Adv. Opt.

Photonics 11, 135–214 (2019).

11. Gao, P. et al. Recent advances in structured illumination microscopy. J. Physics: Photonics (2021).

12. Rosen, J. & Brooker, G. Digital spatially incoherent fresnel holography. Opt. letters 32, 912–914 (2007).

13.

Tahara, T., Kanno, T., Arai, Y. & Ozawa, T. Single-shot phase-shifting incoherent digital holography. J. Opt.

19

, 065705

(2017).

14.

Bouchal, P., Kapitán, J., Chmelík, R. & Bouchal, Z. Point spread function and two-point resolution in fresnel incoherent

correlation holography. Opt. express 19, 15603–15620 (2011).

15. Rosen, J. & Brooker, G. Fluorescence incoherent color holography. Opt. Express 15, 2244–2250 (2007).

16.

Quan, X., Matoba, O. & Awatsuji, Y. Single-shot incoherent digital holography using a dual-focusing lens with diffraction

gratings. Opt. letters 42, 383–386 (2017).

17.

Watanabe, K. & Nomura, T. Recording spatially incoherent fourier hologram using dual channel rotational shearing

interferometer. Appl. optics 54, A18–A22 (2015).

18.

Rosen, J., Brooker, G., Indebetouw, G. & Shaked, N. T. A review of incoherent digital fresnel holography. J. Hologr.

speckle 5, 124–140 (2009).

19. Takeda, M., Wang, W., Duan, Z. & Miyamoto, Y. Coherence holography. Opt. express 13, 9629–9635 (2005).

20.

Duan, Z., Miyamoto, Y. & Takeda, M. Dispersion-free optical coherence depth sensing with a spatial frequency comb

generated by an angular spectrum modulator. Opt. express 14, 12109–12121 (2006).

21.

Takeda, M., Wang, W., Naik, D. N. & Singh, R. K. Spatial statistical optics and spatial correlation holography: a review.

Opt. Rev. 21, 849–861 (2014).

22.

Naik, D. N., Ezawa, T., Singh, R. K., Miyamoto, Y. & Takeda, M. Coherence holography by achromatic 3-d ﬁeld correlation

of generic thermal light with an imaging sagnac shearing interferometer. Opt. express 20, 19658–19669 (2012).

23.

Soni, N. K., Vinu, R. & Singh, R. K. Polarization modulation for imaging behind the scattering medium. Opt. letters

41

,

906–909 (2016).

9/10

24.

Wang, W. & Takeda, M. Coherence current, coherence vortex, and the conservation law of coherence. Phys. review letters

96, 223904 (2006).

25. Goodman, J. W. Statistical optics (John Wiley & Sons, 2015).

26.

Naik, D. N., Singh, R. K., Ezawa, T., Miyamoto, Y. & Takeda, M. Photon correlation holography. Opt. express

19

,

1408–1421 (2011).

27.

Singh, R. K. & Sharma M, A. Recovery of complex valued objects from two-point intensity correlation measurement.

Appl. Phys. Lett. 104, 111108 (2014).

28.

Singh, R. K., Vyas, S. & Miyamoto, Y. Lensless fourier transform holography for coherence waves. J. Opt.

19

, 115705

(2017).

29.

Kim, K., Somkuwar, A. S., Park, Y., Singh, R. K. et al. Imaging through scattering media using digital holography. Opt.

Commun. 439, 218–223 (2019).

30.

Somkuwar, A. S., Das, B., Vinu, R., Park, Y. & Singh, R. K. Holographic imaging through a scattering layer using speckle

interferometry. JOSA A 34, 1392–1399 (2017).

31.

Chen, L., Singh, R. K., Chen, Z. & Pu, J. Phase shifting digital holography with the hanbury brown–twiss approach. Opt.

Lett. 45, 212–215 (2020).

32.

Chen, L., Chen, Z., Singh, R. K., Vinu, R. & Pu, J. Increasing ﬁeld of view and signal to noise ratio in the quantitative

phase imaging with phase shifting holography based on the hanbury brown-twiss approach. Opt. Lasers Eng.

148

, 106771

(2022).

33.

Zhang, Z., Ma, X. & Zhong, J. Single-pixel imaging by means of fourier spectrum acquisition. Nat. communications

6

,

1–6 (2015).

34.

Martínez-León, L. et al. Single-pixel digital holography with phase-encoded illumination. Opt. express

25

, 4975–4984

(2017).

35.

Horisaki, R., Matsui, H., Egami, R. & Tanida, J. Single-pixel compressive diffractive imaging. Appl. Opt.

56

, 1353–1357

(2017).

36.

Edgar, M. P., Gibson, G. M. & Padgett, M. J. Principles and prospects for single-pixel imaging. Nat. photonics

13

, 13–20

(2019).

37.

Shin, S., Lee, K., Baek, Y. & Park, Y. Reference-free single-point holographic imaging and realization of an optical

bidirectional transducer. Phys. Rev. Appl. 9, 044042 (2018).

38. Hu, X. et al. Single-pixel phase imaging by fourier spectrum sampling. Appl. Phys. Lett. 114, 051102 (2019).

39. Ota, K. & Hayasaki, Y. Complex-amplitude single-pixel imaging. Opt. letters 43, 3682–3685 (2018).

40. Singh, R. K. Hybrid correlation holography with a single pixel detector. Opt. letters 42, 2515–2518 (2017).

41.

Gibson, G. M., Johnson, S. D. & Padgett, M. J. Single-pixel imaging 12 years on: a review. Opt. Express

28

, 28190–28208

(2020).

42.

Chen, Z., Singh, D., Singh, R. K. & Pu, J. Complex ﬁeld measurement in a single pixel hybrid correlation holography. J.

Phys. Commun. 4, 045009 (2020).

43.

Hillman, T. R. et al. Digital optical phase conjugation for delivering two-dimensional images through turbid media. Sci.

reports 3, 1–5 (2013).

44.

Otsu, N. A threshold selection method from gray-level histograms. IEEE transactions on systems, man, cybernetics

9

,

62–66 (1979).

Acknowledgements

The work is supported by the Science and Engineering Research Board (SERB) India- CORE/2019/000026. T.S acknowledges

support from University Grant Commission (UGC), India for his scholarship.

Author contributions statement

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 involved in supervision, formulation of research goals and aims, funding acquisition,

reviewing, and editing.

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