For over 70 years ago, holography has played an important role in non-destructive testing, quantitative imaging[
Laser & Optoelectronics Progress, Volume. 58, Issue 10, 1011011(2021)
Correlation Holography with A Single-Pixel Detector: A Review
Correlation holography uses incoherent light to reconstruct holograms. This technique reconstructs objects as distributions of two-point coherence function rather than using optical fields, as in conventional holography. The basic principle of correlation holography is derived from the van Cittert--Zernike theorem and relies on the similarity between the optical field and the coherence functions. Experimental implementation of the correlation holography techniques requires a field or intensity interferometer, and fringe analysis and cross-covariance measurement in these interferometers require a conventional camera with array detectors. With the availability of digitally controlled diffractive elements, it is possible to replace the incoherent light source, such as a rotating ground glass, with a digital source loaded with the random patterns in sequence. Such strategies ease the burden on the detector and allow for correlation holography with a single-pixel detector (SPD) to be used. This review paper discusses a close connection between digital holography and correlation holography. The principles of correlation holography with the SPD are reviewed in detail, and the advantages of using digital sources to mimic incoherent illumination in the correlation holography are examined in the context of three-dimensional and complex field imaging.
1 Introduction
For over 70 years ago, holography has played an important role in non-destructive testing, quantitative imaging[
Lohman proposed using holography for polarized light by extracting the vector nature of the light’s wavefront[
The availability of array detectors and reconstruction algorithms has further revolutionized holography. Optical reconstruction of holograms has become supplanted by DH. This technique preserves unique features of a hologram, such as the complex amplitude distribution, and provides a simple reconstruction method[
A conventional detector with a large number of pixels candigitally capture an image of fine interference fringes, and a computer with high computational efficiency can numerically reconstruct a hologram from the captured image[
Single-pixel imaging, presented by Duarte et al. [
Several correlation techniques have been proposed for imaging[
Since these proposals, attempts have been made to develop various types of correlation holography for light structuring and imaging[
Recently, we developed hybrid correlation holography (HCH) using an SPD to reconstruct 3D and complex-valued objects. Here, the hybrid provides a combination of optical and computational channels to image objects using an SPD[
This paper first introduces the concept of generalized holography and holography using an SPD. A comparison of DH and correlation holography is then made. The different architectures of correlation holography, such as CH, VCH, and PCH, are briefly discussed help understand HCH. After a detailed discussion about correlation holography, the possibility of replacing thermal light sources with a digital source is discussed in the context of the HCH using an SPD for 3D and complex field imaging.
2 Digital holography
DH is based on interferometry and can simultaneously provide amplitude and phase distributions of a target. In 1967, Goodman and Lawerence[
To explain the basic principle of the DH, we present a generalized framework with polarized light. A coherent optical field emerging from the object and reference is explained as
where Tm(r1) represents object transmittance function for the orthogonal polarization components of the light and m=x,y stands for the orthogonal polarization components. Subscript O and R indicate object and reference fields, respectively. The spatial coordinates at the source and the recording (hologram) plane are represented by r1 and r2, respectively. A complex field distribution at the recording plane is indicated as
Polarization fringesalong the recording plane are derived using Eq. (3) and are given as
where sn(r2), n=0--3 are the Stokes parameters of the recording plane and represent polarization[
Consider a situation where the polarization is either uniform or ignored. This situation corresponds to a well-established DH, where a recording of only intensity modulation, i.e., I(r2)=
where the intensity of the object and reference fields are given by terms IO and IR. The complex field of the object is represented by
Figure 1.A typical experimental geometry based on the Mach-Zehnder geometry for an off-axis holography system
In some interferometric systems,the object and reference beams propagate the same path, which is known as common-path interferometry. Such experimental geometry is widely investigated to overcome the issue of stability of off-axis holography while maintaining its advantages[
Figure 2.(a) Sketch of a cyclic lateral shearing interferometer, the arrows denote the central ray path; (b) sketch of the experimental arrangement for quantitative phase cyclic interferometer. The object is placed at one-half of the optical field of view (FOV). The interferometer section makes two laterally shifted and collinearly propagating non-collimating wavefronts for interference at the CCD plane. CCD: charged coupled device[86]
A numerical algorithm based on Fourier fringe analysis is implemented to reconstruct the object information from the recorded hologram. The first two terms in Eq. (5) represent the DC component due to the irradiance of the individual object and reference field. The last two terms in Eq. (5) are the off-axis and spectrum terms in the frequency space, and these two terms are important to recover the complex field of the object. A digital 2D Fourier transform operation implemented on the recorded hologram separates various desired terms in the Fourier space. The numerical steps to process the hologram under the Fourier fringe analysis method are shown in
Figure 3.Steps highlighting Fourier fringe analysis of the recorder hologram
3 Polarization digital holography
Polarization, together with the amplitude and phase, describes the complete optical field of the target object. Colomb et al.[
where O=
Figure 4.An experimental setup to record the polarization hologram. BS is a beam splitter, HWP is a half-wave plate, M is mirror, P is polarizer, QWP is a quarter-wave plate, CCD is charged coupled device [14]
4 Coherence holography: scalar to vectorial domains
Conventional holography records and reconstructsa 3D image as an optical field distribution. As mentioned, Takeda et al. developed CH[
The recording process in coherence and CH is the same, but reconstruction is implemented using incoherent light illumination (rather than coherent). In CH (without polarization), a coherently recorded hologram is illuminated with incoherent light, and the object information is reconstructed as the distribution of the coherence function. Therefore, CH provides a 3D image as a spatial distribution of spatial correlation between a pair of points. An interferometer scheme is required to experimentally measure the spatial distribution of the complex coherence function. In the principle of CH, moving ground glass creates stationary quasi-ergodic time fluctuations, which grant one to supplant the ensemble average with time average, or integration time over the detector response time.
To provide a generalized principle of the correlation holography, we start our discussion with polarized light and consider CH as a special case where polarization is ignored. VCH is based on holograms for two orthogonal polarization complements, e.g., x and y.
To introduce the reconstruction of these holograms by an incoherent light source, a schematic representation of the reconstruction is shown in
where Hm(r1) is a fixed transmittance of the hologram and ϕm(r1,t) denotes a random phase introduced by the diffuser at a time t. The RGG in
Figure 5.Recording process in the vectorial coherence holography. (a) Hologram of Ex component of the light coming from the object; (b) hologram of Ey component of the light emanating from the object. Two holograms are used to reconstruct desired images in the coherence-polarization matrix of a light field [68]
Figure 6.Formation of orthogonally polarized holograms at the diffuser plane and reconstruction of these holograms by an incoherent light source. Instantaneous random fields at the observation plane, for two orthogonal polarization components, are represented by Ex(r2) and Ey(r2), and the random field is characterized by the coherence-polarization matrix W(r2,r2+Δr)
Characterization of the random light field is possible using a 2×2 coherence--polarization matrix. Under consideration of incoherent illumination, elements of the coherence--polarization matrix are represented as
Eq. (9) represents the connection between the incoherent source at the diffuser plane and the coherence function at the observation plane using a Fourier transform relation. The recorded object information in
5 Intensity correlation holography
Significant progress and developments in correlation optics led to the emergence of numerous fundamental and practical applications. Important results in the coherence optics are the van Cittert--Zernike theorem and the Hanbury Brown and Twiss (HBT) approach. A correlation function of the fourth order, with respect to complex amplitude, is used in the HBT approach, and this approach became a preferred method to analyze random fields. The HBT approach makes use of the relationship between second-order and fourth-order Gaussian random fields. This provides a simple and stable experimental method to characterize the correlation parameters. The approach offered a novel insight into statistical optics and in the development of highly stable and unconventional imaging systems, such as astronomical imaging, PCH, and ghost diffraction. In early astronomical imaging, second-order correlation was widely applied for imaging astronomical objects. The HBT approach, based on intensity correlation, introduced a new direction in astronomical and unconventional imaging. Since the correlation is examined electronically after measurement of instantaneous intensities, the intensity correlations are rather simple and free from instability due to external disturbances.
A diagram for intensity correlation holography is shown in
Figure 7.Experimental scheme for intensity correlation holography[79]
In
Instantaneous intensity pattern is given as
and the cross-covariant of the intensity for the Gaussian random field is given as
where <ΔI(r2)>=I(r2)-<I(r2)> represents the fluctuation of intensities over its mean value.
Using relation of the complex coherence established in the previous section, the cross-covariance function becomes
Amplitude distributions of the object encoded into the hologram are reconstructed as a distribution of the cross-covariance function, and the phase information is lost. This differentiates PCH from CH and DH, which provides a complex coherence function or complex optical field distribution. Naik et al. [
6 Correlation holography with a single-pixel detector: HBT approach
Correlation imaging methods, such as ghost imaging, diffraction imaging, and intensity correlation holography, are based on the intensity correlation. The evaluation of photon coincidence or intensity correlation helps recover the object information in ghost imaging. Two photodetectors are used to realize correlation measurement: one detects the non-interacted light field of the object to be reconstructed with high spatial resolution, and the other, known as a bucket detector, collects the interacted light field of the object.
Computational ghost imaging with a bucket detector has also been proposed in recent years. In this method, randomness is inserted into a light beam in a controlled manner using a digital device and supplanted a conventional thermal source with the SLM assisted source. As a separate methodology, intensity correlation-based holography reconstructs the 3D structure of the amplitude object from the intensity correlations. This has been implemented using a connection between the cross-covariance and the second-order correlation of the Gaussian random fields, as shown in Eq. (13).
In a separate development, we developed an HCH technique using an SPD.This method can numerically reconstruct 3D objects by estimating the intensity correlations of an SPD and digitally propagated intensity patterns at the far-field planes. The SPD records the incoming light field after interaction with the object, and the light fields in the digital channel do not pass through the object. Here, hybrid reveals the combination of optical and computational channels. A comparison of intensity correlation holography and HCH is shown in
Figure 8.(a) Denotes intensity correlation holography scheme, S is a spatial filter, L is a lens, RG is rotating ground glass, T is transparency; (b) HCH with SPD setup, BS is a beam splitter, SLM is a spatial light modulator, P is a polarizer, c is a correlator, D is a single-pixel detector [79]
The recorded intensity pattern at the camera plane is given as
where F indicates a 2D Fourier transform, T(r1) represents transmittance (or reflectance) of the object at the RGG plane, and ϕ(r1) is the phase inserted by the diffuser. Note that the intensity at different z planes around the detector position can be observed using the paraxial propagation. The cross-covariance of the intensity can be expressed as
where parenthesis <·> denotes the ensemble average and the intensity fluctuation is ΔI(r2)=I(r2)-<I(r2)>. This equation states that transparency encodes the information and determines the intensity co-variance.
In HCH, a sequence of random phasesis introduced in the laser light to mimic an incoherent source using a computer-controlled spatial light modulator. An SPD is introduced in the optical channel. The digitally stored random phase is numerically propagated in the digital channel at z. Finally, the cross-covariance of the intensities from the two channels is determined. An experimental scheme for the HCH is represented in
A spatially filtered collimated coherent beam enters the BS. The beam reflected from BS illuminates the reflective type SLM, which loads the random phases into the laser beam. The light from the SLM reflects back and propagates through the BS and illuminates a transparency Tm(r1) for m=x,y polarization components. To generalize the principle for the vectorial case, we establish the theory for coherent polarized objects wherein consideration of two orthogonal polarization components is enough. The orthogonal components are obtained by incorporating the Jones matrix P(θ) of a polarizer as
where P(θ)=
The intensity in the digital channel can be similarly represented. Therefore, the intensity correlation between the optical and digital channel, at orientation θ=0, is given as
where <·> indicates ensemble average. The intensity correlation for orientation of polarizer θ=π/2 is given as
Using the angular spectrum method for propagation from the source to the arbitrary observation plane in the digital channel, the second-order correlation from two channels is given as
Term Emo(0) represents the complex field at the SPD at the location (0,0). For the incoherent source, i.e.,
where z is the propagation distance and kz=
Therefore, the 3D object is reconstructed as a distribution of the cross-covariance. As an example, consider a case of a scalar object, i.e., Tx(r1)=Ty(r1)=T(r1). Using propagation kernel and coherent beam propagation, 3D transparency placed at different distances can be considered, as shown in
and is shown in
Figure 9.A transparency, shown on the right-hand side, with 3D information of objects located at three different z planes [79]
Figure 10.Reconstructed objects at three different longitudinal distances (z) from the focal planes [79]
The production of HCH is described as follows. The SLM inserts a set of random phase structures ϕnl(M) into the laser on the nl pixel, where M stands for a total number of the random phases inserted by the SLM. These random phases appear to follow
The size of each random pattern displayed on the SLM is 400×400. The cross-covariance is estimated by correlating the intensities from two channels as explained earlier. A result of the cross-covariance for transparency is given in
Figure 11.Change in reconstruction quality with an increasing number of random phases in the HCH
7 Correlation holography with a single-pixel detector for complex field
Recently, we have developed a new technique to reconstruct the complex field within the HCH framework[
Figure 12.(a) Coherence waves interference setup with the correlation of the intensity detected by a CCD, L is the lens, D is a pinhole, BS are beam splitter, M are mirrors, T is transparency, MO is microscope objective, RGG is rotating ground glass; (b) optical channel in the HCH with a single-pixel detector, BD is beam displacer, PBS is a polarization beam splitter, SLM is the spatial light modulator, CA is a circular aperture, c is correlator, D is single-pixel detector; (c) architecture for digital propagation of the random fields and correlation of single point intensity with two-dimensional propagated intensity[80]
The random intensity pattern at a fixed time t in the optical channel is represented as
where
Considering independent random light fields in two arms of the MZI, the coherence function is
where <
where ΔIc(Δr) and ΔIo(Δr) represent the intensity fluctuations in the digital and the optical channels, respectively.
Consider the complex light field at the transparency as
where Tn(r1) is the transmittance function, which may be real or complex value. Tn(r1) represents an incoherent hologram source structure at the diffuser plane. The coherence function at the observation plane is the Fourier transform of the incoherent source at the diffuser plane through the van Citttert--Zernike theorem. To apply the holography in Eq. (28), we use a reference coherence,
A coherent beam loaded with exp
In the developed method, we obtain the cross-covariance coming from optical and digital channels.
The cross-covariance and reconstructed objects are shown in
Figure 13.Recovery of the complex field in a single-pixel modified HCH approach for two different transparencies. Results for an off-axis hologram of the object “1” is used as a transparency, (a) cross-covariance of the intensities, (b) amplitude of the object, and (c) a phase of the object; a spiral phase plate is used complex transparency and results are (d) cross-covariance of the intensity, (e) amplitude of the vortex field, and (f) a phase of the vortex field. The interpolated portion of these results are shown in corner of each figure [80]
In the second case, a spiral plate is inserted into the setup. The light emanating from the spiral phase plate is a complex field andis represented as rlexp(ilϕ), where r and ϕ are spatial and azimuthal coordinate and l is the azimuthal mode. For the unit azimuthal index, the cross-covariance and reconstructed complex-valued object are shown in Figs. 13 (d)--(f). A piled-up phase variation of 2π around the point of singularity in the vortex reveals the topological charge in
8 Conclusion
We have discussed different types of correlation holography and reviewed the principle of correlation holography with an SPD. As highlighted in the beginning, we first introduced the idea of the correlation holography with a generalized theoretical base covering scalar and vectorial aspects of the light. Subsequently, these theoretical bases and discussions on the correlation holography were used to review the development of the correlation holography with the SPD. In the context of available resources on digital holography, this review concentrates mainly only on the correlation holography with an SPD for 3D and complex field imaging.
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Sarkar Tushar, Chandra Mandal Aditya, Ziyang Chen, Jixiong Pu, Kumar Singh Rakesh. Correlation Holography with A Single-Pixel Detector: A Review[J]. Laser & Optoelectronics Progress, 2021, 58(10): 1011011
Category: Imaging Systems
Received: Mar. 2, 2021
Accepted: Apr. 23, 2021
Published Online: May. 28, 2021
The Author Email: Rakesh Kumar Singh (krakeshsingh.phy@iitbhu.ac.in)