Speckle-Correlation Holographic Imaging: Advances, Techniques, and Current Challenges

Abstract

The imaging modalities of correlation-assisted techniques utilize the inherent information present in the spatial correlation of random intensity patterns for the successful reconstruction of object information. However, most correlation approaches focus only on the reconstruction of amplitude information, as it is a direct byproduct of the correlation, disregarding the phase information. Complex-field reconstruction requires additional experimental or computational schemes, alongside conventional correlation geometry. The resurgence of holography in recent times, with advanced digital techniques and the adoption of the full-field imaging potential of holography in correlation with imaging techniques, has paved the way for the development of various state-of-the-art approaches to correlation optics. This review article provides an in-depth discussion of the recent developments in speckle-correlation-assisted techniques by focusing on various quantitative imaging scenarios. Furthermore, the recent progress and application of correlation-assisted holographic imaging techniques are reviewed, along with its potential challenges.

1. Introduction

In recent years, there has been significant progress in correlation-assisted imaging modalities across various fields, such as astronomy, industrial metrology, communication, medical imaging, holography, and turbid media imaging [1,2,3]. Unlike conventional optical imaging techniques, correlation imaging modalities use the correlation of random intensity patterns to reconstruct or characterize object information. Many correlation-assisted approaches in coherent or partially coherent imaging or characterization schemes make use of the information-carrying properties of speckle patterns for the development of advanced optical devices [4,5]. Since the invention of the laser and subsequent advancements in its applications, the generation of granular patterns known as speckles has garnered significant interest due to the challenges and possibilities these random intensity patterns present. The presence of speckle patterns complicates many coherent imaging techniques, such as synthetic aperture radar (SAR) imaging [6], ultrasound imaging [7], optical coherence tomography (OCT) [8,9], and digital holography (DH) [10,11], among others. Consequently, the scientific community has developed various methods to reduce speckle to enhance imaging quality [12,13,14]. Additionally, several investigations into the fundamental properties of speckle patterns have been conducted, with diverse applied interests, paving the way for exploiting the randomness of speckle patterns as a versatile tool in various science and engineering applications. Reflecting wide research efforts, numerous techniques have been demonstrated by utilizing the characteristic features of speckle patterns, including imaging through random media [15], speckle photography [16,17,18,19], speckle interferometry [20,21], speckle holography [22,23], speckle metrology [23,24,25,26,27,28], speckle contrast imaging [29,30,31], speckle optical tweezers [32], speckle correlation techniques [33,34,35,36,37], correlation holography [38,39,40], and ghost imaging [41,42,43,44], among others.

Among the diverse utilizations of speckle pattern characteristics, correlation-assisted imaging and characterization techniques have attracted considerable interest in recent years due to their dominant imaging features, including higher resolution, robustness in turbid environments, and flexible integration with advanced computational techniques. The extensive utilization of correlation measurements began six decades ago with the demonstration of the Hanbury Brown–Twiss (HBT) effect [45,46], which employs intensity fluctuations at two different detectors to measure the angular size of stars. Subsequently, correlation techniques have extended beyond photons to include electrons, subatomic particles, and matter waves, resulting in several significant advancements. The randomness of speckle has been explored through theoretical investigations that emphasize the statistical nature of emergent light from disordered media under coherent light illumination, and later with the introduction and validation of the memory effect [47,48,49]. The combination of the information-conserving nature of speckles and the mathematical tool of correlation has advanced numerous imaging techniques, such as imaging through scattering medium, ghost imaging, intensity correlation imaging, and correlation holography. Several techniques for non-invasive imaging through scattering media have been developed in the past decades by exploiting the autocorrelation of the intensity distribution of the speckle pattern and using inverse reconstruction techniques with iterative computational approaches [15,50,51,52,53]. Additionally, ghost imaging and ghost diffraction modalities utilize object illumination with a random beam, and the respective correlation of intensity fluctuations detected by a bucket or single pixel detector with a spatially varying detector for various high-resolution and remote-access imaging applications [54,55,56].

While most correlation techniques rely on temporal statistical optics [3], the last decade has witnessed the theoretical introduction and experimental advancements in the emerging technology of spatial correlation holography [38,39,57,58,59,60]. This approach leverages the randomness and spatial statistical features of the associated speckle field. Unconventional holography techniques, such as coherence holography [38] and photon correlation holography [40], have been successfully demonstrated by reconstructing three-dimensional information from spatial field correlation and spatial intensity correlation, respectively. Subsequently, spatial correlation holography techniques have been extended to the vectorial regime through the development of vectorial coherence holography [61] and Stokes holography [62]. Furthermore, the potential of correlation holography methods has been applied in various imaging and characterization domains in recent years. These applications include complex coherence function retrieval [63,64], imaging through scattering media [65,66,67,68,69], ghost diffraction imaging [70,71], phase-shifting holography [72,73,74,75], synthesis and control of coherence-polarization elements [76,77], measurement of generalized Stokes parameters [78,79], polarization recovery through scattering media [80], synthesis of orbital angular momentum modes [81,82], and Stokes correlation [83,84], among others. Notably, earlier imaging applications related to speckle correlation techniques focused on reconstructing the amplitude information of an object from the spatial intensity correlation function, often excluding the inherent phase information. In contrast, recent advancements in speckle correlation techniques have incorporated the complex-field reconstruction capability of holography in combination with spatial correlation to retrieve the complex-field information of the object.

In this review, we first provide a theoretical introduction to laser speckles and discuss the significance of speckle correlation, highlighting its implementation in various applied imaging scenarios. Next, we present an in-depth analysis of ensemble averaging through spatial averaging and examine the advantages of integrating digital holography with speckle correlation techniques. Furthermore, we review recent advancements in speckle correlation holography (SCH), including novel designs, emerging applications, and key challenges, with a focus on quantitative complex amplitude imaging. While the diverse applications of speckle correlation approaches extend beyond the scope of this review, we concentrate on the latest progress in speckle correlation-assisted holographic systems aimed at overcoming the limitations of quantitative complex amplitude imaging.

2. Laser Speckles

2.1. Origin of Speckles

The construction and investigation of continuous wave laser sources by Maimann in 1960 revealed the diverse applications of imaging and characterization techniques utilizing the coherent nature of the light source [85,86]. Besides the unique features of a coherent source, a chaotic granular pattern is observed when objects are viewed under highly coherent light. In subsequent years, this granular appearance was identified and termed ‘speckle,’ which originates when coherent light illuminates a rough surface with dimensions comparable to the optical wavelength of the light source [87,88]. Thus, speckles are interference patterns seen at the observation plane, resulting from the superposition of dephased wavelets arising from different scattering spots on the rough scattering surface [5]. Figure 1a and Figure 1b respectively illustrate the generation of speckle patterns corresponding to scattering on rough reflective and transmissive surfaces under coherent light source illumination. The speckle pattern is composed of numerous bright and dark spots that exhibit behavior akin to a random walk, resulting from the constructive and destructive interference of wavelets arriving at the observation plane. Speckle patterns in both reflective and transmissive rough surfaces arise from random interference effects—governed by height-induced phase shifts in reflection and refractive phase distortions in transmission. The selection between reflective and transmissive speckle techniques involves careful consideration of angular sensitivity, stability, and polarization characteristics under coherent illumination. Reflective scatterers produce speckle patterns with high angular sensitivity and partial polarization retention, making them ideal for precision applications like metrology, speckle interferometry, and non-destructive testing [4,28]. In contrast, transmissive scatterers generate speckle patterns with reduced angular sensitivity due to volumetric scattering and strong depolarization in thick media arising from multiple scattering. These properties make transmissive systems better suited for laser projection and biomedical imaging applications [89,90]. Initially, in applications involving coherent light, speckle patterns were considered as random phenomena that adversely affected the quality and resolution of imaging techniques such as SAR imaging, ultrasound imaging, and optical coherence tomography [8,91,92,93]. However, subsequent research on memory effects [49,94,95,96,97] and the statistical properties of speckle patterns [87,88,98] has revealed a broad spectrum of applications for speckles.

2.2. Speckle Correlation

The investigations on the statistical properties of the speckle pattern by Goldfischer [52], Goodman [87,99], and Dainty [47,88], etc., unveil the possibilities of utilizing the randomness of the speckle pattern for various applications. Later years witnessed several investigations on coherent wave propagation through rough scattering layers and explorations of the characteristics of speckle patterns [88,100]. Further, the late 1980s saw the discovery and establishment of the speckle memory effect [94,101], which celebrates the predictable behaviors or the information-preserving nature of the random speckle pattern. The retention of spatial information on the amplitude and phase of the incident wave on reflection or transmission through rough scattering layers provides the flexibility needed to comprehend the scattering medium. The random speckle pattern at an observation plane in the positive Z-direction is a coherent contribution of wavelet superpositions arising from an ensemble of scattering spots in the rough scattering layer. Therefore, the field distribution at the observation plane could be interpreted as the random walk phenomenon, similarly to the well-known classical problem of the random walk [5], which is expressed as:

$$E\left(\mathit{r}\right)={\displaystyle \sum _{k=1}^N\left|E_k\left(\widehat{\mathit{r}}\right)\right|}\exp \left(i{\varphi }_k\right)$$

(1)

where $\left|E_k\left(\widehat{\mathit{r}}\right)\right|$ and ${\varphi }_k$ are the amplitude and phase distribution, corresponding to the kth scattering point in the scattering medium, $N$ is the total number of scattering points in the scattering layer, and $\mathit{r}$ and $\widehat{\mathit{r}}$ are the position coordinates at the observation plane and scattering medium, respectively.

Additionally, the introduction and advancements of correlation techniques [37,65,102] in the intensity distributions of speckle patterns make it possible to determine how well information is preserved as wavelets propagate to the observation plane. This is achieved by second- or higher-order correlations of the speckle field and intensity at two points in space or time. The two-point intensity correlation of the speckle patterns at the observation plane provides additional information on intensity or field correlation functions, which is expressed as:

$$C_I\left({\mathit{r}}_1,{\mathit{r}}_2\right)={〈\Delta I\left({\mathit{r}}_1\right)\Delta I\left({\mathit{r}}_2\right)〉}_E$$

(2)

where ${〈\dots 〉}_E$ represents the ensemble average, $I\left(\mathit{r}\right)={\left|E\left(\mathit{r}\right)\right|}^2$ is the instantaneous intensity distribution of speckle patterns, and $\Delta I\left(\mathit{r}\right)=I\left(\mathit{r}\right)-〈I\left(\mathit{r}\right)〉$ represents the fluctuations in the intensity values at specific points with respect to the average value. The conceptual quantity of the ensemble average is physically implemented in application scenarios by making use of the temporal averaging by assuming that the random field is stationary and ergodic in time or by considering the spatial averaging by assuming that the random field is stationary and ergodic in space [40,58]. Figure 2 demonstrates the conceptual schematic of the generation of the speckle pattern from a rotating diffuser, and the physical realization of the spatial intensity correlation with the concepts of temporal averaging and spatial averaging.

2.3. Snapshot Speckle Correlation

In coherence theory and statistical optics, statistical investigations of stochastic optical fields that fluctuate in time or space are associated with the concept of the ensemble average. In most statistical scenarios, the realization of the conceptual quantity of the ensemble average is physically implemented by utilizing the time average with the assumption that the stochastic optical field is stationary and ergodic in time [3]. However, the last decade witnessed the emergence of spatial statistical optics as a counter concept of temporal statistical optics, where the ensemble average is replaced with the spatial average, based on the assumption of spatial stationarity and spatial ergodicity for the stochastic optical fields [57,58,60]. Several futuristic theoretical and experimental advancements have emerged in recent years by taking advantage of the spatial averaging concept as an effective replacement of ensemble averaging in practical imaging and correlation synthesis scenarios. In addition, the utilization of spatial averaging in speckle-assisted applications presents the advantage of the development of snapshot detection techniques in combination with digital speckle correlation to retrieve and use the intensity and field correlation functions. Moreover, the inclusion of spatial averaging-assisted speckle correlation in holography, quantitative phase imaging, imaging through scattering medium, ghost imaging, etc., offer the possibility of real-time implementation of imaging systems with robust phase stability in any sort of environment.

The illumination of a coherent light source on a ground glass diffuser generates random speckle patterns, as shown in Figure 2. The speckle field distribution at a far field region is expressed as:

$$E\left(\mathit{r}\right)={\displaystyle \frac{\mathit{exp}\left(ikz\right)}{i\lambda z}}{\displaystyle \int E\left(\widehat{\mathit{r}}\right)\exp \left(i{\phi }_g\left(\widehat{\mathit{r}}\right)\right)}\exp \left[{\displaystyle \frac{ik}{2z}}\left({\left|\mathit{r}\right|}^2-2\mathit{r}.\widehat{\mathit{r}}+{\left|\widehat{\mathit{r}}\right|}^2\right)\right]d\widehat{\mathit{r}}$$

(3)

where $\lambda$ is the wavelength of light source; $k={\displaystyle \raisebox{1ex}{$2\pi $}\!\left/ \!\raisebox{-1ex}{$\lambda $}\right.}$ is the wave number; ${\phi }_g(\widehat{\mathit{r}})$ is the random phase introduced by the ground glass diffuser; $z$ is the propagation distance from the diffuser to the detector plane; and $\widehat{\mathit{r}}$ & $\mathit{r}$ are the spatial coordinates at the diffuser plane and observation plane, respectively. The spatial field correlation of the speckle field at the observation plane is given by:

$$\begin{array}{cc}\hfill W\left({\mathit{r}}_1,{\mathit{r}}_2\right)& ={〈E^{*}\left({\mathit{r}}_1\right)E\left({\mathit{r}}_2\right)〉}_S\hfill \\ & ={\displaystyle \int E^{*}\left(\widehat{\mathit{r}}\right)E\left(\widehat{\mathit{r}}\right)}exp\left[-i{\displaystyle \frac{2\pi }{\lambda z}}\left({\mathit{r}}_1-{\mathit{r}}_2\right)\widehat{\mathit{r}}\right]d\widehat{\mathit{r}}\hfill \end{array}$$

(4)

where ${〈\dots 〉}_S$ represents the spatial average, and $E^{*}\left(\widehat{\mathit{r}}\right)E\left(\widehat{\mathit{r}}\right)=I\left(\widehat{\mathit{r}}\right)$ is the intensity distribution of the source structure at the diffuser plane. Furthermore, the two-point intensity correlation or the fourth-order correlation of the speckle patterns detected at the observation plane is expressed in terms of the field or second-order correlation [32], as follows:

$$C_I\left({\mathit{r}}_1,{\mathit{r}}_2\right)={〈\Delta I\left({\mathit{r}}_1\right)\Delta I\left({\mathit{r}}_2\right)〉}_S={\left|W\left({\mathit{r}}_1,{\mathit{r}}_2\right)\right|}^2$$

(5)

In experimental implementation and digital processing, spatial averaging is achieved by maintaining a fixed distance ${\mathit{r}}_2-{\mathit{r}}_1$, which is considered as the detector pixel pitch while systematically translating a defined window $I^k\left(x,y\right)$ across the entire recorded intensity distribution. The two-point intensity correlation $C_I\left({\mathit{r}}_1,{\mathit{r}}_2\right)$ is obtained by correlating $\Delta I^k\left(x,y\right)\Delta I^k\left(0,0\right)$ for different realizations of the speckle patterns, which is expressed as:

$$C_I\left({\mathit{r}}_1,{\mathit{r}}_2\right)={\displaystyle \frac{{\displaystyle \sum _{k=1}^M\left[\Delta I^k\left(x,y\right)\Delta I^k\left(0,0\right)\right]}}{M}}$$

(6)

where $M$ is the number of different realizations resulting from the pixel-by-pixel movement of the defined matrix $I^k\left(x,y\right)$. Furthermore, it is important to note that the two-point intensity correlation described in Equation (5) represents the modulus square of the field correlation function. As a result, the realization of intensity correlation measurements only provides amplitude information of $W\left({\mathit{r}}_1,{\mathit{r}}_2\right)$, while phase information remains inaccessible. The retrieval of quantitative phase information, particularly when combined with amplitude data, remains a fundamental challenge in most diffraction-based imaging systems. However, recent advances in computational methods and the resurgence of holographic techniques have opened new avenues for addressing this quantitative imaging problem.

The integration of correlation techniques with laser speckle patterns and the adoption of holographic schemes unlocks unique opportunities by leveraging the speckle patterns’ inherent information-carrying properties across various applications. Over the past two decades, significant advancements have been made in speckle correlation methods, evolving from the foundational HBT intensity interferometer to diverse fields such as holography, microscopy, and polarimetry. Before delving into the development of SCH techniques, Figure 3 presents a timeline of key milestones in correlation holographic approaches utilizing the randomness of the speckle pattern. This overview highlights major research trends and summarizes critical progress in speckle correlation-assisted holography.

3. Correlation Holographic Imaging

The field of statistical optics has experienced remarkable progress since the realization of the HBT effect in astronomy, where interference is observed through the correlation of intensity fluctuations over time [45,46]. Over the years, much of the research in statistical optics has centered on correlation phenomena involving stochastic optical fields that vary randomly in time. However, the advent of coherence holography [38] and photon correlation holography [40], which integrate holography with statistical optics by emphasizing stochastic optical fields that vary randomly in space, has introduced the concept of spatial statistical optics [39,57,58,59,60]. In recent years, several groundbreaking demonstrations in statistical optics have emerged, particularly in imaging and synthesis, leveraging second-order or fourth-order correlation functions of random fields. Early approaches in this domain utilized intensity correlation to reconstruct object information from the spatial covariance function, but these methods were limited to recovering only the amplitude, leaving the phase information unresolved. In contrast, the past decade has seen the development of innovative schemes capable of simultaneously recovering both amplitude and phase information [63,65,70]. These advancements have been achieved by combining off-axis speckle-assisted holography with intensity correlation techniques, marking a significant leap forward in the field.

Figure 4a represents the conceptual schematic of the speckle field-assisted correlation holographic imaging system. The approach utilizes a reference independent random speckle field to implement the speckle holography system by superposing the field with the speckle field generated by the object located in the diffuser arm. In the off-axis speckle holography scheme, the reference random field is generated by off-axis point source illumination on an independent reference diffuser. The correlation function associated with this off-axis reference random field is given by,

$$W^R\left({\mathit{r}}_1,{\mathit{r}}_2\right)={\displaystyle \int circ\left(\raisebox{1ex}{$\widehat{\mathit{r}}-{\mathit{r}}_s$}\!\left/ \!\raisebox{-1ex}{$a$}\right.\right)}\exp \left[-i{\displaystyle \frac{2\pi }{\lambda z}}\left({\mathit{r}}_1-{\mathit{r}}_2\right)\widehat{\mathit{r}}\right]d\widehat{\mathit{r}}$$

(7)

where ‘$a$’ is the radius of the off-axis point source, and ‘${\mathit{r}}_s$’ is the shift from the center for the generation of off-axis point, $\lambda$ is the wavelength of light source, and $z$ is the propagation distance from the reference scattering medium to the detector plane. The intensity distributions of the random speckle patterns generated from the object diffuser and the reference diffuser are shown in Figure 4b and Figure 4c, respectively. The intensity distribution of the superposed field is represented in Figure 4d. Subsequently, a digital speckle correlation assisted with the concept of spatial averaging delivers the correlation hologram shown in Figure 4e. Unlike the conventional hologram [107,108,109], where the superposition occurs between the reference and object beam, in SCH the hologram formation could be considered as the superposition of the correlation functions corresponding to the object and refence random fields [63]. Therefore, the correlation hologram is expressed as:

$$C_H\left({\mathit{r}}_1,{\mathit{r}}_2\right)={\left|W^O\left({\mathit{r}}_1,{\mathit{r}}_2\right)+W^R\left({\mathit{r}}_1,{\mathit{r}}_2\right)\right|}^2$$

(8)

Here $W^O\left({\mathit{r}}_1,{\mathit{r}}_2\right)$ is the object information-encoded field correlation function, and $W^R\left({\mathit{r}}_1,{\mathit{r}}_2\right)$ is the reference field correlation function. The reference speckle field generated from the off-axis point source illumination on the reference diffuser creates the reference correlation function $W^R\left({\mathit{r}}_1,{\mathit{r}}_2\right)$, which acts as a covering support to $W^O\left({\mathit{r}}_1,{\mathit{r}}_2\right)$ for creating a correlation hologram when ‘$a$’ is very small, i.e., a point source.

4. Implementation

The discovery and advancements in the HBT effect [45,46,110,111,112], along with the van Cittert–Zernike theorem [113,114,115,116,117], paved the way for the emergence of several futuristic and unconventional correlation and coherence synthesis techniques. These techniques leverage second-order and fourth-order correlation functions to achieve novel imaging and measurement capabilities. Subsequently, the integration of holography into speckle correlation techniques provided a breakthrough in addressing the challenges of complex amplitude reconstruction in intensity correlation-based HBT systems within the spatial domain [40,63,64]. In recent years, by harnessing the information-carrying properties of speckle patterns and the complex-field reconstruction capabilities of digital holography, a range of innovative approaches have been developed. These include correlation holographic HBT systems [63,72,75,103], complex-field imaging through scattering media [67,68,73,74], ghost diffraction imaging [56,70,71,82], and other cutting-edge methodologies that push the boundaries of optical imaging and coherence analysis [77,78,81,83,118].

4.1. Correlation Holographic HBT Scheme

The progress in HBT intensity interferometers, which utilize the correlation of speckle patterns, has opened new avenues for applying correlation functions in various control and synthesis applications within statistical optics. The characteristic properties of speckle patterns are measured by analyzing the intensity correlation of the intensity fluctuations observed at the far field. According to the van Cittert–Zernike theorem, the two-point intensity correlation at the far field is linked to the intensity distribution of the incoherent source through a Fourier transform relationship [113,115,117]. Additionally, the moment theorem establishes that the intensity correlation corresponds to the modulus square of the field correlation function [40,119]. As a result, the measurement of intensity correlation effectively yields a field correlation function, capturing only the amplitude information of the complex correlation function while leaving the phase information unresolved. The loss of phase information in the complex-field correlation function restricts the applicability of HBT-based systems for full-phase space imaging [40], limiting their use in three-dimensional image reconstruction, biomedical microscopy, phase microscopy, and other related applications. Consequently, HBT imaging schemes often require supplementary computational or experimental methods to address phase retrieval challenges. Recent advances in optimization theory, digital signal processing, and machine learning have opened new avenues for overcoming these limitations, enabling more robust quantitative phase recovery. Over the past decade, various techniques, such as higher-order correlation, iterative phase retrieval algorithms, and deep learning architecture, have been developed to reconstruct the complex correlation function [64,105,120,121]. Meanwhile, the inherent ability of holography to capture both amplitude and phase has been successfully leveraged to advance correlation-based imaging techniques, further expanding the applicability of correlation approaches to quantitative phase recovery scenarios. By combining holography with HBT intensity interferometers, these developments have significantly enhanced the ability to extract comprehensive coherence information, paving the way for more sophisticated applications in optical imaging and analysis. Figure 5 represents the comparison of a conventional spatial correlation-assisted HBT interferometer with techniques utilizing the combination of intensity interferometer and holography scheme.

In the context of retrieving the complex correlation function from intensity correlation, Singh et al. introduced a holographic scheme integrated with an HBT-based intensity interferometer [63]. In this setup, an independent reference random field is superimposed with the field from a conventional intensity interferometer, as illustrated in Figure 5b. The experimental implementation employs a Mach–Zehnder interferometer and a digital correlation strategy that utilizes spatial averaging to analyze speckle correlations from snapshot-recorded speckle patterns. Central to this approach is the retrieval of a correlation hologram, as explained in Equation (8), followed by the application of a Fourier transform-based fringe analysis procedure to recover the complex correlation function. Figure 6a showcases experimental results corresponding to a fork hologram with a topological charge of three at the scattering layer. This technique enables the simultaneous and well-resolved recovery of both amplitude and phase distributions of the complex correlation function at the detector plane. The successful retrieval of the complex correlation function opens up significant opportunities for applying this method in various imaging and characterization scenarios that rely on correlation functions. Furthermore, the versatility of this approach is demonstrated through its application in quantitative phase imaging of diverse phase structures through scattering layers [66], highlighting its potential for advanced optical analysis and imaging in complex media.

In another development, a phase-shifting technique is incorporated along with the correlation-assisted speckle holography system to circumvent the challenges associated with the implementation of an off-axis holography scheme [72]. The utilization of off-axis geometry adds space-bandwidth limitations and restricts the field of view due to the presence of a direct current (dc) and conjugate spectrum in the spatial frequency spectrum. A conceptual representation of the phase-shifting-assisted HBT-based holography scheme capable of phase reconstruction is shown in Figure 5c. The technique is demonstrated with the phase-shifting procedure by utilizing a spatial light modulator (SLM) in combination with a digital speckle correlation procedure to retrieve the respective correlation holograms. The phase difference of the sample and the carrier is expressed as:

$$\Delta \phi =ta{n}^{-1}\left[{\displaystyle \frac{{C^{\prime }}_{H2}\left({\mathit{r}}_1,{\mathit{r}}_2\right)-{C^{\prime }}_{H4}\left({\mathit{r}}_1,{\mathit{r}}_2\right)}{{C^{\prime }}_{H3}\left({\mathit{r}}_1,{\mathit{r}}_2\right)-{C^{\prime }}_{H1}\left({\mathit{r}}_1,{\mathit{r}}_2\right)}}\right]$$

(9)

where $\Delta \phi ={\phi }_s-{\phi }_c$ with ${\phi }_s$ and ${\phi }_c$ represents the phase information corresponding to the sample and the carrier, ${C^{\prime }}_{Hn}\left({\mathit{r}}_1,{\mathit{r}}_2\right)$ the Fourier transforms correlation holograms with $n=1\mathrm{to}4$, and $ta{n}^{-1}$ is the inverse of the tangent function.

The experimentally demonstrated phase reconstruction results corresponding to the phase-shifting-assisted HBT-based holography system are shown in Figure 6b. Later, a polarization phase-shifting scheme is integrated with the HBT-based holography scheme, as shown in Figure 5d, to control the overlap issues of dc and cross terms in the spatial frequency spectrum, thereby increasing the field of view and signal-to-noise ratio of the speckle holography system [73,75]. The approach exploits the polarization-dependent spatial modulation and the characteristic features of SLM to achieve phase-shift modulation in only one of the orthogonal polarization components. The inclusion of the polarization-assisted phase-shifting scheme in HBT-based off-axis speckle holography system enhances the reconstruction quality and field of view in comparison to other approaches.

4.2. Imaging Through Scattering Media

The light transport through scattering media and the respective imaging of objects obscured in the scattering media always pose several technical challenges [15,122]. A variety of cutting-edge approaches have been demonstrated in the last few decades to efficiently recover the object information behind the scattering layer, including the optical conjugation method [123,124,125], using ballistic light [126,127], wavefront shaping techniques [122,128], deconvolution methods [129], the time-gated method [130], auxillary reference object scheme [131], speckle correlation method [50,53,132], bispectrum analysis [133], and shower curtain method [134], etc. Among the various demonstrated approaches, the potential of speckle correlations is successfully exploited in the trail-blazing innovations by Bertolotti et al. [50,53] and Katz et al. [132]. These techniques utilize the autocorrelation of the speckle field and advanced phase retrieval techniques for the recovery of the object. Furthermore, non-invasive imaging with a single-shot lensless detection scheme without the requirement of wavefront shaping or time-gating is demonstrated for imaging through scattering media and around the corners [65,135]. Even though most of the demonstrated techniques deliver outstanding reconstruction abilities, the approaches are limited by the inefficacy of full-phase space imaging due to the loss of phase information. To overcome the compex amplitude recovery challenges through scattering media, some futuristic techniques have been demonstrated, exploiting the concept of SCH by investigating light transport and imaging through scattering media [67,68,73,74]. A conceptual diagram of the imaging technique utilizing the SCH technique is shown in Figure 7a. The technique utilizes a dual holographic scheme, where the object hologram is formed behind the scattering layer and a correlation hologram is generated at the detector plane. Conceptual representations of the formations of an off-axis hologram and an inline hologram behind the scattering layer are shown in Figure 7b and Figure 7c, respectively. The presence of a scattering layer in the path of the hologram-encoded beam scrambles the hologram information into random speckle patterns, where the hologram information is not directly recoverable. Therefore, the technique utilizes the SCH scheme by creating an off-axis reference random field and superimposing it with the speckle field from the object arm, which results in the generation of a correlation hologram, as discussed in Section 3.

The experimental scheme developed by Somkuwar et al. [68] demonstrated the capability of the technique by considering the formation and recovery of a Fourier hologram and an off-axis hologram behind the scattering layer. The developed imaging system utilizes the speckle holography system and digital speckle correlation using spatial averaging to recover the complex correlation function, as described in Equation (8). The hologram information generated at the scattering layer is recovered by using the retrieved complex correlation function at the camera plane. The recovery of the hologram at the scattering layer provides the advantage of reconstructing the complex amplitude information corresponding to the object and, consequently, the three-dimensional (3D) information. By utilizing the digital procedure involving the Fourier transform method of fringe analysis [136], the complex amplitude information corresponding to the object is reconstructed at the respective planes. Figure 8 represents the experimental results of the imaging system, where a custom-designed 3D object is used to demonstrate and validate the technique. Figure 8a represents the 3D object used for the experimental realization of the imaging system. The correlation hologram retrieved by digital speckle correlation and the recovered Fourier hologram at the scattering medium are shown in Figure 8b and Figure 8c, respectively. The reconstructed 3D amplitude and phase information corresponding to the object are represented in Figure 8d and Figure 8e, respectively. Despite the technique delivering a robust solution to 3D complex amplitude imaging through a scattering medium, the need for an off-axis hologram formation behind the scattering layer confines the execution of the approach to several imaging scenarios.

Subsequently, a more efficient and applicable method utilizing an in-line holographic scheme was introduced by Vinu et al. [67], where the hologram formation behind the scattering layer occurs due to the holographic diffraction of the object upon illumination with a coherent light source. The method employs a single-shot recording scheme and digital speckle correlation procedure to retrieve the complex correlation function, and, consequently, the in-line hologram recovery behind the scattering medium. In addition, the technique makes use of the computational phase retrieval method for the simultaneous reconstruction of amplitude and phase information from the recovered in-line hologram by removing the twin-image artifacts. Figure 9 represents the experimental results corresponding to the imaging through the scattering medium employing a digital in-line holographic scheme. The retrieved correlation hologram by digital speckle correlation and the recovered in-line hologram behind the scattering layer are shown in Figure 9b and Figure 9c, respectively. The simultaneously reconstructed amplitude and phase distribution of the object at various planes behind the scattreing medium are shown in Figure 9d and Figure 9e, respectively. In addition, the capability of the reference speckle field-assisted speckle correlation method in quantitative phase imaging through scattering medium is demonstrated by Singh et al. [66]. An extension to these approaches, by integrating the phase-shifting and polarization phase-shifting, was manifested by Li et al. [73,74]. The adoption of phase-shifting and polarization phase-shifting approaches presents the advantage of increased field of view, robust phase recovery, a high signal-to-noise ratio, better stability, etc., in quantitative phase imaging (QPI), by suppressing the reduntant terms in the spatial frequency spectrum in the digital reconstruction process.

4.3. Holographic Ghost Diffraction Imaging

The advancements in ghost imaging (GI) and ghost diffraction (GD) techniques have facilitated the development of cutting-edge imaging and characterization methods, including ghost spectroscopy, ghost tomography, GI with atoms, electron GI, terahertz GI, X-ray GI, temporal GI, ghost holography, ghost nanoscopy, and more [42,43,44,54,137,138]. Since GI relies on the correlation of intensity fluctuations, many state-of-the-art approaches face challenges in phase recovery, which is a byproduct of intensity correlation techniques. Over the past few decades, several innovative methods have been demonstrated the ability to recover complex amplitude information, building on principles such as GD schemes, interferometry, phase-shifting holography, advanced phase retrieval algorithms, and Fourier-space filtering, among others [139,140,141,142,143].

Inspired by the simultaneous complex amplitude reconstruction capability of SCH techniques, Vinu et al. introduced a holographic GD scheme capable of single-shot quantitative complex amplitude imaging [56,70]. A conceptual representation of this technique is illustrated in Figure 10, where a conventional GD scheme is integrated with a speckle field-assisted holography system. This approach superimposes an off-axis random reference field from an independent diffuser onto the random fields generated by the conventional GD scheme. Additionally, the technique leverages the spatial stationarity and ergodicity of the random fields at the detector plane to enable time-frozen recording of the diffraction intensity distribution, resulting in a snapshot holographic GD system. Compared to conventional GD schemes, the holographic GD (HGD) scheme employs two spatially resolved detectors and utilizes spatial averaging-assisted correlation of intensity fluctuations. The ghost correlation of intensity fluctuations and the retrieval of ghost correlation hologram are expressed as:

$$\mathit{G}\left({\mathit{r}}_1,{\mathit{r}}_2\right)=〈\Delta I_1({\mathit{r}}_1)\Delta I_2({\mathit{r}}_2)〉={\left|{\mathit{g}}^{(\mathrm{G})}({\mathit{r}}_1,{\mathit{r}}_2)+{\mathit{g}}^{(\mathrm{R})}({\mathit{r}}_1,{\mathit{r}}_2)\right|}^2$$

(10)

where $I_1(\mathit{r})$ and $I_2(\mathit{r})$ are the intensity distributions at the two spatially resolved detectors with $\Delta I(\mathit{r})=I(\mathit{r})-〈I(\mathit{r})〉$, ${\mathit{g}}^{(\mathrm{G})}({\mathit{r}}_1,{\mathit{r}}_2)$ and ${\mathit{g}}^{(\mathrm{R})}({\mathit{r}}_1,{\mathit{r}}_2)$ are the field correlation functions corresponding to the ghost arm and the off-axis reference arm, respectively. This design enhances the system’s efficiency, making it particularly suitable for real-time imaging applications. The complex amplitude reconstruction results, corresponding to various objects like pure-phase objects, transmission-type objects, and standard USAF 1951 resolution test targets, etc. are shown in Figure 11a–h, respectively.

Moreover, to enhance the applicability of the approach across diverse imaging scenarios, the HGD system has been extended to the microscopy domain by integrating appropriate microscopy configurations into the ghost diffraction arms. Experimental results obtained using the microscopy-assisted HGD system, applied to a standard USAF 1951 resolution test target with different group elements, are presented in Figure 11i–l. These results, along with the corresponding quantitative analysis, demonstrate the system’s capability for complex-valued object imaging and highlight its potential for real-time imaging applications in various practical scenarios.

Very recently, Vinu et al. demonstrated an extension to the HGD system by integrating the phase-shifting method [71], which solves some phase imaging challenges in the ghost holography system like space-bandwidth limitation, limited field of view, complex imaging design, etc. The technique utilizes an SLM-assisted fast-switching temporal phase-shifting procedure in the HGD system, which delivers the potential for direct QPI and possible utilization of the technique in slow dynamic measurements. The phase difference between the object and the carrier in the phase-shifting-assisted HGD system is expressed as:

$$\Delta \phi =ta{n}^{-1}\left[{\displaystyle \frac{G_2^T({\mathit{r}}_1,{\mathit{r}}_2)-G_4^T({\mathit{r}}_1,{\mathit{r}}_2)}{G_3^T({\mathit{r}}_1,{\mathit{r}}_2)-G_1^T({\mathit{r}}_1,{\mathit{r}}_2)}}\right]$$

(11)

where $\Delta \phi ={\phi }_o-{\phi }_c$, with ${\phi }_o$ and ${\phi }_c$ representing the phase map of the object and the carrier, respectively; $ta{n}^{-1}$ is the inverse of tangent function; and $G_n^T({\mathit{r}}_1,{\mathit{r}}_2)$ is the Fourier transform of the ghost correlation holograms with $n=1\mathrm{to}4$. Figure 12 represents the experimental results of a phase-shifting-assisted HGD system, while Figure 12a–d represents the retrieved correlation holograms corresponding to the four phase-shifting conditions. Figure 12e represents the phase object (vortex phase pattern of topological charge 3) used for experimental demonstration purposes. The reconstructed phase information utilizing the phase-shifting procedure and the corresponding filtered phase information corresponding to a vortex with a topological charge of 3 are shown in Figure 12f and Figure 12e, respectively.

5. Conclusions and Outlook

Advances in theoretical and experimental breakthroughs in SCH introduce a novel holographic technique that integrates speckle correlation and digital speckle-assisted holography. This method enables quantitative complex amplitude reconstruction across diverse imaging scenarios, independent of environmental conditions. By generating correlation holograms, SCH retrieves the complex correlation function through analysis of speckle intensity distributions from superimposed holographic arms. This allows for simultaneous amplitude and phase recovery, offering the flexibility needed to apply advanced computational techniques—such as digital propagation algorithms—for reconstructing complex object information under varying imaging conditions. A key innovation of SCH is its use of spatial averaging as an efficient alternative to ensemble averaging in digital speckle correlation. This breakthrough facilitates single-shot imaging, making SCH highly suitable for real-time and in-vivo applications. Recent advancements in SCH have opened exciting possibilities for imaging and characterization through the controlled synthesis and utilization of spatial field and intensity correlation functions.

Table 1. Representative speckle correlation methods and their applications.

Basic Principle	Method	Applications
Spatial field correlation (second-order correlation)	Coherence holography [38]	3D 1 object reconstruction [144], imaging through diffuser [145]
Spatial intensity correlation (fourth-order correlation)	Photon correlation holography [40]	3D 1 object reconstruction using intensity correlation [135], holographic correloscopy for imaging through scattering medium [65]
	Holographic spatial HBT system [63]	Complex-valued object imaging through scattering layer [67,68,74], QPI 1 through scattering medium [66], helicity estimation of coherent vortex beam [81]
	Ghost diffraction holography [70]	Quantitative ghost microscopy [70], phase-shifting-holographic ghost diffraction imaging [71]
Polarized vectorial field correlation (CP-matrix elements, GSPs)	Vectorial coherence holography [61] Stokes holography [62] Higher-order Stokes correlation [104]	Polarization speckles [79], determination of CP 1 matrix elements and GSPs 1 [78,118], Stokes holography-HBT scheme [103], OAM 1 spectrum estimation [146,147]

¹ 3D: three-dimensional, CP: coherence-polarization, GSPs: generalized Stoke parameters, QPI: quantitative phase imaging, OAM: orbital angular momentum.

summarizes key technological developments in SCH, along with their specific applications.

While recent advances have demonstrated the versatility of SCH in conventional imaging, scattering media imaging, and ghost imaging, etc., by maintaining robust phase stability, several critical challenges remain. Although SCH-based ghost diffraction imaging enables the development of a ghost holographic microscopy system with real-time quantitative complex amplitude reconstruction capability, its application to biomedical imaging remains limited by resolution constraints. Breaking the diffraction limit for super-resolution imaging will be crucial for translating this technology into biomedical laboratories. While SCH has proven effective for scalar complex amplitude imaging, its potential in the vectorial regime remains largely unexplored, with only preliminary theoretical and experimental studies conducted to date [78,84,111,148,149]. The key challenge is achieving image reconstruction through thick scattering media, as current methods primarily address thin scattering samples. Future developments must either mitigate multiple scattering effects or exploit them through higher-order correlations in polarized vectorial fields. Furthermore, Stokes correlation and investigations on generalized two-point polarization-based correlation parameters are expected to offer new insights for polarimetry and the design of advanced polarization microscopy systems. Substantial opportunities emerge from integrating advanced computational techniques such as optimized phase retrieval algorithms, compressive sensing, and deep learning architectures, etc., to enhance the efficiency and compactness of the SCH system [105,150]. Early progress in compressive correlation holography [106] and virtual reference light-assisted imaging through scattering media [121] suggests promising avenues for future innovation.