In the past, holography, and three-dimensional imaging (3D) techniques were mostly powered by coherent light sources since spatially incoherent illumination demanded cumbersome optical architectures such as rotational shearing interferometers, triangle interferometers and conoscopic holography¹. The need for such optical architectures is derived from the requirement that two-beam interference (TBI) was needed to compress 3D information in a two-dimensional (2D) hologram. The birth of active optical devices such as spatial light modulators and computer processing methods, eased the constraints of optical architectures of incoherent holography with TBI. One milestone in this research direction was the development of Fresnel incoherent correlation holography (FINCH), which is one of the widely used incoherent holography architectures today^2-4. In 1968, deconvolution based 2D coded aperture imaging (CAI) using randomly arranged pinholes was demonstrated by Dicke and Ables to overcome the low-intensity problem when using a single pinhole for X-ray and gamma-ray imaging^{5, 6}. However, the study was not explored beyond 2D imaging. In 2008, a 3D imaging technique based on CAI was demonstrated with wavelength as the third dimension⁷. Later, the CAI method was extended to 3D imaging and four-dimensional (4D) imaging with depth as the third dimension^8-10, and depth and wavelength as the third and fourth dimensions, respectively^{11, 12}. Therefore, CAI offers multidimensional imaging capabilities without TBI and with a minimum number of optical components, which makes them more attractive than FINCH-based techniques.

This research project began when we attempted the CAI method employing the benchtop Fourier transform infrared microspectroscopy (FTIRm) system at the Australian Synchrotron operating with a conventional Globar^TM IR source. This instrument, which comprises a Bruker Hyperion 3000 IR microscope and V70 FTIR spectrometer, is used in support of many applications at the Infrared Microspectroscopy (IRM) beamline, whereby the identification and analysis of functional groups within a sample enable a better understanding of diverse materials from diseased tissue to composite materials and foods to paintings^13-15. We applied the CAI method to the benchtop FTIR instrument at the IRM beamline by manufacturing scattering lenses on calcium fluoride (CaF₂) substrates. A matching 15× (NA = 0.4) Cassegrain objective lens (COL) and a 64×64 focal plane array (FPA) imaging detector (pixel size = 40 μm) were used. During this study, we noticed that COL exhibited interesting focal characteristics with sharp autocorrelation functions and low cross-correlation with respect to other planes (see Supplementary information Section 1, Section 2).

Let us ask this fundamental question – why is scattering based coded apertures preferred for 3D imaging? The reason is that the average speckle size formed by scattering is approximately the diffraction limited spot size¹⁶. Consequently, the autocorrelation function is sharp and about twice that of the size of the focal spot that can be obtained with a lens of the same NA¹. Secondly, the speckle patterns change with depth resulting in the capability to discriminate different planes along depth by cross-correlation. Therefore, for 3D imaging, a sharp autocorrelation and low cross-correlation along depth (SALCAD) is needed. The fields generated using scatterers can be classified as random SALCAD fields and the one from COL as a deterministic SALCAD field. Deterministic SALCAD fields require a lower photon budget than the random counterparts and the point spread functions (PSFs) can be calculated, unlike random SALCAD fields where PSFs are recorded⁹. So, this imaging concept can be considered as correlation holography when the SALCAD condition is satisfied and the object is illuminated by a spatially incoherent source. Recently, a modified approach was employed where sparse randomly arranged focal spots were generated instead of scattering to improve the signal to noise ratio¹⁷. However, this method is suitable only for imaging a single plane, and to image multiple planes, many such phase masks are needed to be spatially multiplexed. Fresnel zone aperture based 2D imaging systems have been reported in the past^18-20.

The FTIRm consists of collinear dual beams generated from two sources: (i) a standard white light source for bright-field visible light observation of the sample plane and (ii) a broadband Globar^TM IR source that allows the acquisition of mid-IR (MIR) spectral images in the range ~ 899 cm^–1–3845 cm^–1 as shown in Fig. 1(a). The FTIR single-beam spectrum obtained from the Globar^TM source is shown in Fig. 1(b) in which a total of 765 spectral channels equally separated by an average value of 3.85 cm^–1 were used. The MIR beam from the FTIR spectrometer is focused on the sample plane and the scattered signal is recorded by the FPA detector. With 15 × COL, the pixel pitch is ~2.7 μm in the sample plane and the field of view is around ~172 × 172 μm².

The IR light produced by the Globar^TM source is spatially incoherent and so the light emitted from one object point does not interfere with the light emitted from another point, instead their intensities add up. As a result, this FTIRm system is a shift-invariant imager, linear in intensity, i.e., PSF and the intensity distribution obtained for an object $O$ can be expressed as $OH=O\otimes PSF+\sigma$, where $\sigma$ is the noise and ‘ $\otimes$’ is a 2D convolutional operator (see Supplementary information Section 1, Section 2). Therefore, if the $PSF$ is known for all axial shifts Δz, then the 3D image of $O$ can be reconstructed by solving the above equation. For instance, object $O=O\left(z_1\mathrm{,}{\lambda }_{\text{a}}\right)+O\left(z_2\mathrm{,}{\lambda }_{\text{a}}\right)+O_{\text{T}}$, which absorbs at ${\lambda }_{\text{a}}$ at z₁ and z₂ while $O_T$ refers to non-absorbing layers. The reconstruction of the chemical image of $O$ at plane $z_1$ is given as $O_R\left({\lambda }_{\text{a}}\right)=OH*PSF\left(z_1\right)=O\left(z_1\mathrm{,}{\lambda }_{\text{a}}\right)\otimes PSF\left(z_1\right)*$$PSF\left(z_1\right)+O\left(z_2\mathrm{,}{\lambda }_{\text{a}}\right)\otimes PSF\left(z_2\right)*PSF\left(z_1\right)+N$, where ‘ $*$’ is a 2D correlational operator and N is the reconstruction noise. The reconstructed image is $O\left(z_1\mathrm{,}{\lambda }_{\text{a}}\right)$ sampled by the autocorrelation function $PSF\left(z_1\right)*PSF\left(z_1\right)$, which is a Delta-like function with a minimum width ~1.22λ/NA added to $O\left(z_2\mathrm{,}{\lambda }_{\text{a}}\right)$ sampled by $PSF\left(z_1\right)*$$PSF\left(z_2\right)$, which is a blurred version of $O\left(z_2\mathrm{,}{\lambda }_{\text{a}}\right)$.

The $PSF$s were recorded using a 15 μm pinhole for Δz = 0 to 250 μm in steps of 25 μm. The lateral resolution of the system is therefore ~ 15 μm. The spectral data cubes $PSF(x\mathrm{,}y\mathrm{,}\nu )$ for Δz = 0, 125 μm and 250 μm along 14 averaged (~100 channels) spectral channels are shown in Fig. 1(c–e), respectively, whose non-changing images along ν indicate the suppression of spectral aberrations. The data structure of the output generated by the OPUS 8.0 software was transformed into cube data for processing in MATLAB. The COL generates sharp annular intensity distributions with four peaks, which are responsible for the sharp autocorrelation functions as shown in Fig. 1(f–h). Only a slight increase in the width of the autocorrelation functions was noticed for larger values of Δz. The depth cube data $PSF(x\mathrm{,}y\mathrm{,}\Delta z)$ was generated from recorded PSFs for Δz = 0 to 250 μm and averaged over all the 765 spectral channels as shown in Fig. 1(i). The spatial and spectral cubes were combined, cross-correlated with the central spatio-spectral point (2307 cm^–1, 125 μm) as shown in Fig. 1(j). It is seen that the FWHM along the depth axis is sharp at ~50 μm with mild fluctuations along the spectral axis. The above spatio-spectral characteristics of the FTIRm system are highly desirable for 3D chemical imaging.

The next important aspect of imaging is the reconstruction technique. It is well-established from previous studies that the PSF is not the optimal reconstruction function by cross-correlation²¹, and a non-linear reconstruction (NLR) method was introduced by Rosen in which the magnitudes of the spatial frequency spectrum of the $PSF$ and $OH$ are tuned in powers of α and β respectively in $R=|{\mathcal{F}}^{-1}$$\text{exp}\left[-\text{iarg}\left(\widetilde{OH}\right)\right]\left\}\right|$ until minimum entropy was obtained (see Supplementary information Section 3)²². Another approach is the Lucy-Richardson algorithm (LRA), which consists of a forward convolution between the initial guess solution and the PSF, which is compared with the recorded OH and the ratio is backward cross-correlated with the PSF and the result is multiplied to the initial guess. This process is iterated until the maximum likelihood object function is reconstructed^{23, 24}. The (n+1)^th reconstructed image is given as $R^{n+1}=R^n\left\{\frac{OH}{R^n\otimes PSF}PSF\text{'}\right\}$, where $PSF\text{'}$ is the 180 degrees rotated version and the complex conjugate of $PSF$. In this study, it was noticed that neither NLR nor LRA were optimal; however, when the linearity was broken of the backward cross-correlation of LRA by the NLR approach, a rapid convergence and significantly better estimation was obtained. The schematic of the Lucy-Richardson-Rosen algorithm (LRRA) is shown in Fig. 2(a) (see Supplementary information Section 3).

The spatio-spectral aberrations of the system have been studied²⁵. A cross shaped object (150 μm × 150 μm) and four random pinholes (50 μm in diameter) ablated on chromium-gold layer coated calcium fluoride substrate were used as test objects. The visible light image and IR image when Δz = 0 are shown in Fig. 2(b) and 2(c) respectively. The IR image when Δz = 100 μm is shown in Fig. 2(d). The PSF recorded for Δz = 100 μm is shown in Fig. 2(e). The reconstruction results with LRA (150 iterations) (Supplementary Video 1), NLR (α = 0, β = 0.6) and LRRA (20 iterations, α = 0.2, β = 1) (Supplementary Video 2) are shown in Fig. 2(f–h), respectively. The recorded IR images for Δz = 0, 150 μm and 200 μm are shown in Fig. 2(i), 2(j)and 2(l), respectively, and the reconstructions using LRRA for Δz = 150 μm and 200 μm are shown in Fig. 2(k) and 2(m), respectively. The RMSE and entropy for LRA, NLR and LRRA are compared for the cross object and LRRA was found to perform significantly better than both LRA and NLR as shown in Fig. 2(n).

The experiment was then repeated on a bundle of silk fibers. The absorption spectrum of silk is shown in Fig. 3(a) and the chemical image (Δz = 0) is shown in Fig. 3(b). The reconstruction results of Fig. 3(b) using LRRA with PSFs recorded at Δz = 25 μm, 50 μm and 75 μm are shown in Fig. 3(c–e), respectively. The image directly recorded at Δz = 50 μm is shown in Fig. 3(f). Comparing Fig. 3(d) and 3(f), a good overlap can be observed. A single strand of silk fiber was imaged as shown in Fig. 3(g) whose varying blur indicates varying depth. It was reconstructed using a PSF recorded at Δz = 50 μm as shown in Fig. 3(h) and the corresponding direct image is shown in Fig. 3(i). A polymer test pattern, prepared on a CaF₂ substrate, using a chrome-on-glass United States Air Force (USAF) 1954 target as a lithography mask was examined next, whose absorption spectrum is shown in Fig. 3(j). The direct imaging and reconstruction of object ‘3’ from the USAF target using LRRA are shown in Fig. 3(k) and 3(l), respectively. In some cases, it was necessary to adjust the magnification of the PSFs before reconstruction to achieve optimal reconstruction and for some cases, synthetic PSFs generated from recorded PSFs were used for reconstruction (see supplementary information Section 4).

In conclusion, we demonstrated 3D imaging without TBI using deterministic fields. Unlike scattering based SALCAD fields, where the PSFs have to be recorded at all axial planes, with deterministic SALCAD fields it is possible to calculate the PSFs from the complex amplitude of the optical modulator. Consequently, the reconstruction in this case mimics that of conventional incoherent holography. Furthermore, we have invented a new reconstruction method by introducing non-linearity into the LRA using the NLR approach, which performs significantly better than both LRA and NLR. The method and reconstruction have been demonstrated in the FTIRm chemical imaging system by introducing necessary data structure conversions (see Supplementary information Section 5). We believe that the approach can be easily adapted to power sensitive areas such as astronomical, fluorescence and biomedical imaging. Above all, we also believe that this research work will benefit the users of synchrotron based FTIRm technique to be able to investigate multiple samples during the limited beamtime in the future which was the motivation that led to this research work. The proposed method opens a new direction where deterministic fields and structured light can be engineered for rapid 3D imaging with a low photon budget, high lateral, and axial resolutions.

The evolution of incoherent holography over the years from the complicated architectures such as rotational shearing interferometer²⁶, conoscopic holography²⁷, and FINCH^{2, 3}, to this version of holography with deterministic SALCAD fields using Lucy-Richardson-Rosen algorithm is interesting. We believe that the proposed method will compete with the existing coherent holography methods such as MIR digital holography and holographic interferometry using quantum cascade laser²⁸, single shot Raman holography²⁹, and label-free second harmonic phase imagers³⁰.

This research was undertaken on the IRM beamline at the Australian Synchrotron (Victoria, Australia), part of ANSTO (Proposal ID. 15775, Reference No. AS1/IRM/15775 and Proposal ID. M17333, Reference No. AS2/IRM/17333). This work was performed in part at the Swinburne's Nanofabrication Facility (Nanolab). Funded by European Union’s Horizon 2020 research and innovation programme under grant agreement No. 857627 (CIPHR).

The authors declare no competing financial interests.

Supplementary information for this paper is available at https://doi.org/10.29026/oes.2022.210006