## Abstract

Unconventional systems that adopt the concept of ghost schemes have led to advancements in some imaging applications. However, their application in quantitative phase imaging remains a challenge. Here, we introduce a basis for quantitative phase imaging with ghost diffraction and demonstrate ghost diffraction holographic microscopy for complex-valued imaging. We achieve this by introducing an off-axis holography approach in the modified ghost diffraction system. We also realize a correlation hologram in the cross-correlation of intensities from two detectors in the modified ghost diffraction setup and digitally process the correlation hologram to image complex-valued objects. To generate experiment results, we use a modified interferometer setup and exploit the spatial statistics of the scattered field from a time-frozen pseudothermal light source. Finally, we evaluate the efficacy of the approach by simulation and follow that with experiments that demonstrate imaging of pure phase objects, planar transparencies, and resolution test targets, among others.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. INTRODUCTION

Diffraction imaging that exploits correlation features of the scattered optical field has been regarded as an area of interest with numerous unprecedented developments in fundamental and applied domains of atomic imaging [1,2], crystallography [3–5], holography [6,7], correlation imaging [8–10], imaging through turbid media [11–13], ghost diffraction (GD) and ghost imaging (GI) [14–19], etc. In these imaging and diagnosis approaches, correlation properties of the scattered field in terms of second-order, fourth-order, and higher-order correlation functions are utilized. Among these approaches, GI and GD perform coherent imaging with incoherent light by utilizing the cross-correlation of intensity fluctuations of the scattered field from the object and a field that has never interacted with the object. The conventional Hanbury Brown–Twiss (HBT) scheme [7,20] is capable of retrieving the diffraction pattern of an amplitude-only object illuminated by an incoherent source. On the other hand, the GD scheme provides flexibility in obtaining the diffraction pattern of both amplitude-only and complex-valued objects even though the source is incoherent [21–23]. The realization of spatially correlated beams to the classical regime from its quantum counterpart and then to computational approaches unfold the GI and GD approaches to more real-valued scenarios with enhanced imaging and resolution features [24–33]. Later, several innovative approaches employing advanced algorithms such as compressive sensing [34], deep learning [35], differential measurement [36,37], sequential deviation [38], etc., were applied to enhance image quality and reduce the complexity of GI approaches. Recently, intriguing aspects of GI have expanded to realms other than the usual optical domain with cutting-edge technological developments [39–43]. Despite all these achievements, substantial progress in the ghost framework is confined primarily to amplitude-only objects and amplitude-modulated parts of the complex-valued object.

The long standing problem of phase retrieval [44] in diffraction imaging has attracted growing interest in recent times with the development of advanced tools of optimization theory and information processing [45,46]. In view of this phase recovery challenge, a few ghost schemes have been developed with due focus on complex-valued imaging [23,47–51]. An early work towards this goal reported the retrieval of a diffraction pattern of a pure phase object using entangled-photon pairs [47], which was then demonstrated with classical incoherent light [48]. Algorithmic phase retrieval was employed in lens-less Fourier transform GI [49], and later a two-step phase retrieval method was proposed for the imaging of a complex-valued object [50]. In parallel with the intensity correlation approaches, a modified Young interferometer measuring the modulus and phase of the field correlation function in GD was demonstrated with a one-dimensional phase object [23]. Recently, microscopy with a ghost system was introduced for imaging of amplitude and phase objects in the presence of aberrations [51]. Additionally, some techniques employing interferometry in the object arm [52–54], Fourier-space filtering [55], and phase shifting digital holography [56], among others, have been demonstrated. The majority of studies on GI and GD focus on theories, require sophisticated measuring setups, and are restricted to either low spatially varying objects such as binary phase or specially designed phase objects. Moreover, quantitative phase imaging with ghost schemes demands special attention on measurements from time-frozen intensity pattern without resorting to time averaging. The requirement of time-frozen intensity recording is important to apply to GI schemes in the quantitative imaging of two- and three-dimensional objects in a real-time scenario.

In this paper, we demonstrate a novel approach for simultaneous quantitative phase and amplitude imaging, called GD holography (GDH), by utilizing the off-axis holography in GD. Applicability of the technique is also demonstrated in quantitative phase microscopy. To achieve this, a reference random field is mixed with the fields emanating from the GD system, and correlations of the intensities from two channels are measured. A holographic pattern appears in the correlation of intensities. The digital processing of this correlation hologram provides the complex-valued object in the desired plane. The approach utilizes a time-frozen field from a pseudothermal light source and considers the ergodicity in space (rather than in time). This permits to use space averages over the observation plane as a replacement of the ensemble averages in the evaluation of correlation structures [8]. Under such consideration, an average over numerous speckle spots within a single time-frozen speckle pattern can be used to replace the ensemble average over many different realizations of the speckles for a single point in the pattern.

## 2. METHODS

#### A. GDH Concept

A comparison of the HBT scheme, classical GD scheme, and the proposed technique is shown in Fig. 1. The HBT scheme [Fig. 1(a)] can retrieve the diffraction pattern of an amplitude-only object illuminated by an incoherent source utilizing the autocorrelation of the scattered field intensity at the detector plane. However, the classical GD scheme [Fig. 1(b)] has the potential to retrieve the diffraction pattern of a complex-valued object illuminated by an incoherent source. Let us consider a spatially incoherent source generated by impinging a coherent light on a rotating diffuser. The stochastic field emanating from the diffuser splits into two identical spatially correlated copies by a beam splitter. One copy of the scattered field interacts with a complex-valued object, which is considered the object field ${u_1}({\boldsymbol r})$, and the other non-interacted copy is considered the reference field, ${u_2}({\boldsymbol r})$. These separated and axially propagated stochastic fields reach the detector plane at a distance ‘$Z$’. The detector plane constitutes two independent detectors (D1 and D2) for the separate detection of object and reference field intensities. This conventional GD system is shown in Fig. 1(b), which utilizes the cross-correlation of the intensity fluctuations at these two detector planes for retrieval of the diffraction pattern of the object under consideration.

For a stochastic process that obeys Gaussian statistics, this cross-correlation of intensity fluctuations is expressed as [57] ${\boldsymbol G =}{| {{{\boldsymbol g}^{(G)}}({{\boldsymbol r}_1},{{\boldsymbol r}_2})} |^2}$, with ${{\boldsymbol g}^{(G)}}({{\boldsymbol r}_1},{{\boldsymbol r}_2})$ representing the field cross-correlation function; ${{\boldsymbol r}_1}$ and ${{\boldsymbol r}_2}$ denote the position vectors of spatial points at the detector plane. Thus, the conventional GD scheme will give only the modulus of the correlation function, i.e., the amplitude part of the ${{\boldsymbol g}^{(G)}}({{\boldsymbol r}_1},{{\boldsymbol r}_2})$. This creates the phase retrieval problem [44] and thus limits the execution of the conventional GD scheme in the case of a complex-valued object. To retrieve the complex correlation function and the complex-valued object, we utilize the holographic concept and introduce a reference random field, ${u_R}({\boldsymbol r})$, as shown in Fig. 1(c).

This independent reference random field superposes with the object and reference fields at the detector plane. The resultant optical fields reaching detectors D1 and D2 are, respectively, given by

#### B. Simulation Results

To illustrate the proof of the principle, we simulate the GD system with pure phase and real-valued objects. The simulation is implemented by considering a light source of wavelength ‘$\lambda = 632.8\; {\rm nm} $’ with a beam size of 4.5 mm illuminating the diffuser, 18 mm distance between the diffuser and the object, 420 mm distance between the diffuser and the detector, and with a pixel size of 5.5 µm in the detector plane. The reference random field is created by illuminating an independent diffuser with an off-axis source of size 0.8 mm. The simulation is performed for a pure phase sample of a helical wave front $T({{{\hat{\boldsymbol r}}_1}}) = \exp({\rm i}{m}\phi)$ with ‘$ m $’ representing the topological charge and ‘$\phi $’ the azimuthal angle. The phase profile of ‘$m = 1$’ with a size of 4.5 mm diameter is shown in Fig. 2(a). The correlation of intensity fluctuations from the two detectors results in a ghost correlation hologram as expressed in Eq. (5), which is shown in Fig. 2(b). Formation of a fork pattern in the correlation hologram clearly indicates the encoding of the phase object in the correlation hologram. By applying the Fourier transform operation to the retrieved correlation hologram in a way similar to conventional off-axis holography, the complex field correlation function is retrieved at the detector plane. The retrieved amplitude and phase of the complex correlation function corresponding to the helical mode with ‘$m = 1$’ are demonstrated in Figs. 2(c) and 2(d), respectively. The amplitude structure is constructed with a dark core surrounded by an intensity value, and the phase profile shows a phase variation of zero to 2π.

Likewise, to demonstrate the ability of the approach in a real-valued case, we used a triangular aperture (0.5 mm aperture size and 2 mm separation) as another sample. The random field illuminating the aperture generates off-axis propagating waves with linear phase variations. Superposition of these off-axis propagating waves creates a network of vortices in the complex coherence function [61]. The triangular aperture used for simulation and the retrieved ghost correlation hologram in the GDH system are represented in Figs. 2(e) and 2(f), respectively. Results of the ghost correlation hologram show an array of the fork fringe. The complex correlation function is retrieved from this hologram, and is represented in Figs. 2(g) and 2(h). This retrieved complex correlation function shows an array of dark core and the helical phase structure corresponding to the array of null points in the distribution.

#### C. Experimental Design

A schematic sketch of the GDH experimental setup is shown in Fig. 3(a). The system consists of three modules: a conventional GD part (part I), an outer reference arm that provides a random field (part II), and the recording and digital processing module (part III). A He–Ne laser source (25-LHP-928-230, Melles Griot) of wavelength 632.8 nm is used as the coherent source beam for the GDH system. A non-polarizing beam splitter (BS1) splits the beam into two parts, which act as the source beams for parts I and II of the system. Part I of the system constitutes a conventional GD system, where the coherent beam impinges on a rotating diffuser (GG1) and creates a pseudothermal source for the sample illumination. A polarization beam splitter (PBS1) splits the stochastic field emanating from GG1 into orthogonal polarized components, where the horizontal component illuminates an object (S) positioned 18 mm from GG1 as the closest separation in the experimental setup due to the existing optical elements, and the vertical component acts as the reference beam for the conventional GD system. These beams are combined by PBS2 and allow to move in a common path. Polarized components are utilized in the imaging system for the development of a common path simultaneous recording of the ghost object and reference fields. The reflected beam from BS1 acts as the source beam for part II, which generates a reference random field by allowing the beam to pass through an off-axis microscope objective (MO:10X/0.25NA, Thorlabs Inc.) and a diffuser GG2. This reference random field from part II combines with the orthogonally polarized scattered fields from part I of the system using a beam splitter BS2. These fields were recorded in the transmitted and reflected arms of BS2 by CCD1 and CCD2, respectively. The polarizers (P1 and P2) are used for the projection and superposition of corresponding polarization components from different arms with maximum visibility. Two CCD cameras are synchronized and connected with a computer system for the simultaneous recording of the scattered fields. These CCDs and the interconnected computer system play the role of a digital processing module (part III). Intensity images are recorded using two identical CCD cameras (Prosilica GT1910 from Allied Vision having total pixels of 1920 × 1080 with a pixel pitch of 5.5 µm). The ground glass diffusers (DG10-220-MD and DG10-600-MD), BS (BS013), PBS (PBS251), and polarizer (LPVISC100) from Thorlabs Inc. are the optics used for the implementation of the experimental system.

## 3. EXPERIMENTAL RESULTS

#### A. GDH Results

Experimental results corresponding to the simulation shown in Fig. 2 are demonstrated in Fig. 4. The pure phase of helical mode with ‘$m = 1$’ is introduced in the experiment by illuminating a vortex phase plate (VPP) with a random field. The first row in Fig. 4 corresponds to the phase sample with ‘$m = 1$’, and the second row corresponds to the triangular aperture. Figures 4(a) and 4(b) are recorded raw intensities of object and reference fields at the CCDs. Similarly, the recorded intensities for triangular aperture are shown in Figs. 4(f) and 4(g). The ghost correlation holograms for the respective cases [Figs. 4(c) and 4(h)] are successfully retrieved from the cross-correlation of intensity fluctuations. The amplitude and phase results of the complex correlation function from the retrieved ghost correlation hologram on digital analysis [59] (see Supplement 1) are shown in Figs. 4(d), 4(e), 4(i), and 4(j) for the respective cases. A good quantitative match between simulation and experimental results confirms the validity of our approach in retrieval of the complex field correlation function, and thereby the realization in quantitative imaging of a complex-valued object.

To assess the performance of our system in complex-field imaging, we validate our approach with a pure phase object, planar transparency, etc. The complex field correlation functions are retrieved from the respective ghost correlation holograms by utilizing the Fourier transform method [59]. By employing a digital beam propagation approach based on the angular spectrum method, the complex-valued object at the specific plane is recovered [58]. The corresponding complex-valued information of different objects are demonstrated in Fig. 5. Figures 5(a) and 5(e) represent the amplitude and phase distributions for the phase sample with ‘$m = 1$’. To demonstrate more imaging scenarios, we used two Chinese characters “HUA” (size ${4.4}\;{\rm mm} \times {4.4}\;{\rm mm}$) and “QIAO” (size ${3.6}\;{\rm mm} \times {3.6}\;{\rm mm}$) printed on a transparency sheet with bright letters and dark background (see Supplement 1). Recovered amplitude distributions are demonstrated in Figs. 5(b) and 5(c), and the corresponding phase distributions are shown in Figs. 5(f) and 5(g).

To check the applicability in more imaging scenarios and to comment on the quantitative imaging potential of the proposed imaging system, we used a negative 1951 USAF resolution test target (R3L3S1N, Thorlabs) as our sample. A quantitative analysis on the resolution of the GDH system is demonstrated by the random field illumination in different groups and elements of the test target. The experimental results for group 1 element 5 in the target are shown in Figs. 5(d) and 5(h) as amplitude and phase distribution, respectively. A quantitative analysis is performed to obtain the minimum resolvable features with the GDH system, and the reconstructed experimental results are demonstrated in the first row of Fig. 6. The system has a good resolving ability up to group 0 element 4 of the test target, as demonstrated in Fig. 6(d), which corresponds to 1.41-line pairs/mm with a line width of 355 µm. Features smaller than this size start to deteriorate with the current demonstration system, as represented in Figs. 6(e) and 6(f), which correspond to group 0 element 6 and group 1 element 2.

#### B. GDHM Results

Furthermore, on considering the significance of the approach with micrometer-sized phase sensitive samples, the GDH system is converted into GD holographic microscopy. Inclusion of an identical microscopy configuration in the ghost object and reference field [as shown in Fig. 3(b)] in part I of the experimental configuration in Fig. 3(a) (see Supplement 1) converts the GDH to a GDHM system. A varying configuration with a different microscope objective and tube lens combination is implemented according to the size of the sample area under consideration. In experimental demonstrations, we used microscope objectives with ${40\times\!/0.65\; {\rm NA}}$, ${60\times \!/0.85\;{\rm NA}}$, and ${100\times \!/1.25\;{\rm NA}}$ from Thorlabs Inc. and tube lenses of focal lengths 100 mm and 200 mm. The second row in Fig. 6 demonstrates the GDHM system image reconstruction for various groups in the USAF resolution test target. A detailed comparison of the GDH and GDHM systems with different combinations of microscopy configurations is described in Table 1. Figures 6(g)–6(l) represent image reconstruction corresponding to groups 4–7 of the USAF test target by using different microscopy configurations. The GDHM system with the demonstrated configuration can resolve group 7 element 1 of the USAF test target with good resolution, which corresponds to 128-line pairs/mm with a line width of 3.9 µm.

## 4. DISCUSSION AND CONCLUSION

The experimental design is capable of simultaneous detection of the scattered fields at the two detector planes by synchronizing the detectors and recording the intensities at an instant of time. In the proposed scheme, a wide-sense spatial stationarity is considered at the detector plane, and thereby the ensemble average is replaced by the spatial average [62]. The advantages of spatial averaging have been exploited recently in several correlation imaging techniques with due importance in spatial statistical optics [7,8,62–64]. Despite the substantial significance of the spatial average in efficiency over the time average, the approach has only very few executions in GI. In our experimental implementation and digital processing, spatial averaging is performed by fixing the distance $\Delta {\boldsymbol r =}{{\boldsymbol r}_1} - {{\boldsymbol r}_2}$ and by varying the pixel positions ${{\boldsymbol r}_1}$ and ${{\boldsymbol r}_2}$ in both detectors. The spatial averaging is implemented by moving a $200 \times 200$ window from the sample arm (i.e., intensity ${I_1}({{\boldsymbol r}_1})$ at CCD1) over the entire intensity image ${I_2}({{\boldsymbol r}_2})$ in CCD2. Moving the sample arm window provides the flexibility of covering the spatial Fourier spectrum like the detection at various detector positions and consequently improves the effective bandwidth. A single snapshot recording at two detectors is enough to retrieve the ghost correlation hologram, as many spatial points are involved in the digital process of averaging. This provides an additional advantage of a high-speed configuration and possible implementation to more beneficial applications.

The imaging quality of the GDHM is determined by a set of factors—first, the validity of the assumptions of Gaussian statistics and realization of the spatial stationarity of the stochastic field at the observation plane. The successful image recovery in the proposed approach indicates that the assumptions were valid in the proposed imaging system. Second, the spatial resolution of the imaging system is not only governed by the numerical aperture of the microscopy configuration but also depends on the size of the speckle grains illuminating the object. For faithful image recovery, the condition of the speckle-grain size being less than the object spatial structure is required. In addition, other factors influencing imaging quality include the speckle grain size at the CCD plane and the extent of spatial stationarity to replace the ensemble averaging by space averaging. The extent of spatial stationarity over the observation plane affects the imaging field of view [8]. In our imaging system, an aperture of specific dimensions is placed at the rotating diffuser plane to control the beam size and thereby ensure the sampling criterion of speckle grain size greater than two pixels of the CCD. Finally, our approach is also alienated from any iterations and convergence issues.

In conclusion, we have developed a completely new approach on GD and implemented it into the domain of microscopy for imaging of spatially varying complex-valued objects. The imaging potential of the technique is established both theoretically and experimentally by imaging various macroscopic and microscopic samples. Execution of the spatial averaging to retrieve the ghost correlation hologram provides an additional advantage of imaging dynamic samples from a time-frozen intensity pattern. Finally, the provision of a new extension to the GD by holography is expected to open new dimensions in holography, microscopy, and tomography, and in harnessing the correlation function for specific applications.

## Funding

National Natural Science Foundation of China (11674111); Distinguished Young Scientific Research Talents Plan in Universities of Fujian Province (2018J06017); Fundamental Research Funds for the Central Universities (ZQN-PY209); Science and Engineering Research Board (EMR/2015/001613, SB/OS/PDF-265/2016-17).

## Disclosures

The authors declare no conflicts of interest.

See Supplement 1 for supporting content.

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