One of the principal challenges in biomedical research is imaging through turbid media [1]. Optical imaging methods tackling this problem strive to minimize the effect of the random multiply scattered light [2] or to control it optically or digitally [3]. The first group of methods increasingly utilize the coherence-gating effect [4]. This filtering phenomenon arises from a limited spatial and temporal coherence of illumination. Therefore, it is intrinsic to partially coherent optical systems. Imaging techniques that use the coherence gate for separating only the ballistic (scattered by the object only) light from the unwanted multiply scattered (non-ballistic) light include optical coherence tomography and microscopy [5], achromatic digital holography [6,7], Horn microscopy [8], or white-light diffraction tomography [9]. In comparison to the coherent one, partially coherent illumination provides numerous advantageous imaging properties such as higher spatial resolution, depth sectioning, and speckle noise reduction [7]. However, the use of coherent beams prevails owing to their convenient implementation and simple description allowing inverse modeling [10].

Partially coherent beams require the complete description of at least four-dimensional functions, making them less convenient for computational imaging [11]. The most common representations include cross-spectral density, mutual coherence function (MCF), and phase-space distributions [11]. Their experimentally challenging measurement is redeemed by being irreplaceable in illumination source characterization [12], crystal structure determination [13], three-dimensional object reconstruction [14], object tracking behind opaque structures [15], or quantification of light diffusion parameters [16]. Setting coherence gate to the position separating only the ballistic light makes imaging techniques [5–9] unable to measure complete MCF.

In this Letter, we demonstrate the principal importance of the complete MCF measurement for partially coherent interferometric imaging of objects with strongly scattering surroundings. We show that MCF describing the cross correlation between the object and reference beam comprises the image information formed by both the ballistic and non-ballistic light. In addition to the ballistic image, we extract from the complete MCF by coherence gate manipulation images formed by the non-ballistic light fractions. If the turbid medium is rather weakly scattering, the ballistic light contains almost unaltered specimen information. However, when the scattering is stronger, and the ballistic to multiply scattered light ratio is poor, the complementary non-ballistic light components can significantly improve insufficient ballistic imaging. Combining all of the scattered light components, we demonstrate high-quality quantitative phase imaging throughout the entire field of view (FOV), even in situations with a completely corrupted ballistic images.

To demonstrate the ideas mentioned above, we begin with a theoretical description of the MCF measurement in the optical setup used for the experimental part. We used a coherence-controlled holographic microscope (CCHM) [7], commercially available as Telight Q-Phase. The optical setup (see Fig. 1) consists of optically equivalent object and reference arms, with a scattering object present in the object arm. The diffraction grating (DG) is implemented according to principles proposed by Leith [6] in the reference arm to ensure a formation of an off-axis hologram in the interference plane (IP). The fundamental image properties depend on the parameters of condensers ($ C $) and objective lenses ($ O $) in couples with tube lenses (TL) characterized by the pupil functions (see Ref. [17]) ${P_{C}}$ and ${P_{O}}$, respectively. The axes $ x $, $ y $ of the object space Cartesian coordinate system lie in the object plane, and the axis $z$ points in the optical axis direction (see Fig. 1). The position of a point in the image plane is specified by the coordinates of the optically conjugated point $\textbf{q} = (x,y)$ in the object plane. A halogen lamp ${\cal S}$ filtered by an interference filter (IF, central wavelength 650 nm and 10 nm full width at half-maximum) provides thermal, spatially broad, and quasi-monochromatic illumination. It is imaged to the object focal planes of the $ C $ forming Köhler illumination. The transverse wave vector coordinates ${\textbf{K}_{\cal S}}$ of a plane wave behind $ C $ are proportional to the respective source point coordinates. For this reason, the source properties can be characterized by a function of ${\textbf{K}_{\cal S}}$. We use reduced wave vector notation $|\textbf{K}| = 1/\lambda$, where $\lambda$ is the wavelength of light, and $\textbf{K} = ({\textbf{K}_{t}},{K_z}) = ({K_x},{K_y},{K_z})$, where ${\textbf{K}_{t}}$ is the transverse wave vector, and ${K_z} = \sqrt {|\textbf{K}{|^2} - |{\textbf{K}_{t}}{|^2}}$.

 

Fig. 1. Optical setup of the coherence-controlled holographic microscope: ${\cal S}$, light source; A, aperture; IF, interference filter; L, relay lens; BS, beam splitters; M, mirrors; $ M_m $, movable mirrors; C, condensers; O, objective lenses; TL, tube lenses; DG, diffraction grating; OL, output lenses; IP, interference plane.

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From the time-averaged intensity measurements in IP, we digitally retrieve the cross-correlation information between object-scattered and reference light by a carrier removal in the Fourier plane [7]. The cross-correlation term dependent on the transversal displacement $\Delta \textbf{q} = (\Delta x,\Delta y)$ and relative time-delay $\tau$ of the object-scattered and reference field can be understood as their MCF $\Gamma (\textbf{q},\textbf{q} - \Delta \textbf{q},\tau)$. Therefore, interferometric imaging for a given time-delay $\tau$ and transverse displacement $\Delta \textbf{q}$ is a partial MCF measurement. Modulus and phase image for particular $\Delta \textbf{q}$ and $\tau$ is obtained as modulus and argument of $\Gamma$, respectively. The acquisition of MCF (complex-valued images) for all accessible $\Delta \textbf{q}$ and $\tau$ makes the measurement complete. The complete MCF is available to every partially coherent interferometric system, where the reference-field position and optical path difference can be manipulated. In our system, it is controlled by the reference arm elongation and objective shift (see Fig. 1). We further show that the most common arrangement of imaging setups fixed at $\tau = 0$ and $\Delta \textbf{q} = (0,0)$ correspond to ballistic light imaging, while the MCF at $\Delta \textbf{q} \ne (0,0)$ carries significant amounts of information about the non-ballistic imaging processes. As the coherence-gating effect is mainly related to the low spatial coherence in transmitted-light systems, while temporal coherence has the influence of the second order [18], we discuss only manipulation with $\Delta \textbf{q}$ in this Letter. Given a complete spatial source incoherence, the expression for the measured MCF for $\tau = 0$ has, according to Eq. (4.4-31) in Ref. [19], the form

Light propagation within the system is derived similarly to Ref. [17] using the multi-slice model of a volumetric specimen [20]. We assume no backscattering and only paraxial propagation. The object is formed by a set of layers with spatial frequency spectra ${T_n}({\textbf{Q}_n})$ in axial positions ${z_n}$ of the object space, where $n \in N = \{A \ldots - 2, - 1,0,1,2 \ldots B\}$. Negative $n$ refers to ${z_n} \lt 0$, positive $n$ refers to ${z_n} \gt 0$, and $n = 0$ denotes the in-focus layer. The propagation in object path is described by

Equation (3) also clarifies the CGF setting for the ballistic and non-ballistic imaging mode. Measuring MCF for CGF position $\Delta \textbf{q} = (0,0)$ corresponds to $|{\textbf{Q}_n}| = 0$, $\forall n \ne 0$, i.e., to the ballistic light imaging. It is evident that non-ballistic combinations of ${\textbf{Q}_n}$ that satisfy $|\sum {z_n}{\textbf{Q}_n}/K| \approx 0$ also affect the ballistic image. This effect is negligible only for weakly scattering surroundings. Otherwise, the image will not be a perfect representation of the object in focus. The same also applies for MCF measurement for $\Delta \textbf{q} \ne (0,0)$, the non-ballistic light imaging. Both ballistic and non-ballistic imaging obey the same equation, Eq. (3), and differ only by the parameter $\Delta \textbf{q}$, so we treat them equally.

It is instructive to demonstrate the principles first on a simplified case of imaging through a single scattering layer. We show that when scattering by the surrounding medium is strong, non-ballistic light carries image information of the same quality as the ballistic light and that the images formed are complementary. We demonstrate this claim on the case of quantitative phase imaging of a phase resolution target through an opaque etched glass plate separated by a coverslip, which defines the ${z_1}$ distance. For this experiment, we used 20x objectives with ${\rm NA} = 0.5$, and the illumination aperture was ${{\rm NA}_{C}} = 0.5$. The layer of interest was placed in the focus of the objective and imaged through the strongly scattering layer. We chose the scattering strength of the etched layer such that a bright-field image with partially coherent illumination, see Fig. 2(a), does not show any part of the resolution target. More importantly, not even coherence gated ballistic light provides high-enough quality throughout the whole FOV, see ballistic modulus and phase images in Figs. 2(c) and 2(d). We have measured the complete MCF with [purple curve in Fig. 2(b)] and without (orange curve) the scattering layer, and we display the normalized modulus $|{\gamma _{R}}| = |{\Gamma _{R}}|/|{\Gamma _{\max }}|$ for a point $ R $ [see Fig. 2(c)] against $\Delta x$ in Fig. 2(b). Here, $|{\Gamma _{\max }}|$ stands for the highest possible value of $|\Gamma |$ in CCHM. In general, as the modulus drops, the quality of the image derived at that point of the MCF degrades. There are two reasons, the MCF modulus is in relationship to the root mean square error of phase measurement through inverse proportion (see Chap. 9.4.4 in Ref. [21]), and phase singularities occupy near-zero modulus areas [22]. Therefore, the MCF modulus serves as an indicator of phase quality. Regarding the curves in Fig. 2(b), one can expect to acquire some information only in $\Delta x$ points with a high modulus. It is evident that when imaging only a single planar object the complete MCF measurement (orange curve) does not add any information to the ballistic mode. However, when we image through at least one additional strongly scattering layer, non-ballistic parts of the MCF contain valuable information. We adapt Eq. (3) for this case to facilitate further discussion:

 

Fig. 2. Results for imaging of the phase resolution target through an etched glass plate. (a) Bright-field ballistic imaging mode. (b) Normalized modulus of measured MCF against $\Delta x$ at point $ R $. (c), (e) Modulus and (d), (f) phase images derived from MCF for $\Delta \textbf{q} = (0,0)\;\unicode{x00B5}{\rm m}$ (ballistic light image) and $\Delta \textbf{q} = (- 2,0)\;\unicode{x00B5}{\rm m}$ for imaging through the etched glass plate. (g) Synthetic quantitative phase image created by a combination of non-ballistic images. Objectives used $20{\rm x}/{\rm NA} = 0.5$, ${{\rm NA}_{C}} = 0.5$.

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This equation suggests that each part of the MCF acquired for different $\Delta \textbf{q}$ can be interpreted as an image ${t_0}$ of the resolution target shifted by $\Delta \textbf{q}$. The scattering on the etched layer manifests itself as modulation by the inverse Fourier transform of the spatial frequency spectrum ${T^\prime _1}({\textbf{Q}_1})$. CGF limits this spectrum only to scattering vectors satisfying relation ${\textbf{Q}_1}{z_1}/K \approx - \Delta \textbf{q}$. The produced modulation appears to be random with high and low modulus areas. The stronger coherence-gating effect reduces more the range of contributing ${T^\prime _1}({\textbf{Q}_1})$ spatial frequencies, and, therefore, the size of the high-modulus areas increases. This reduces the number of phase singularities and makes the additive phase from out-of-focus scattering more uniform. As the CGF allows a finite range of scattering layer components to contribute, the image is blurred. The blur is limited by CGF Airy disk radius, which defines the transversal spatial resolution. The CGF width (0.8 µm) can be measured when no object is present; see peak width in Fig. 2(b). We can conclude that the change of the imaging mode to non-ballistic light generally does not affect the size of high-quality phase areas or the resolution. Therefore, neither of the imaging modes is inferior.

We also use this experiment to demonstrate the complementary nature of non-ballistic images. Consider, for example, a small region of the FOV marked by the yellow dashed ring in Figs. 2(c)–2(f). One can see that information about the object is missing in the ballistic light [see the zero modulus in Fig. 2(c) and the random phase in Fig. 2(d)]. The shape of the complete MCF in Fig. 2(b) indicates that the information might be present for particular $\Delta \textbf{q} \ne (0,0)$ with a higher modulus. We show the image acquired for $\Delta \textbf{q} = (- 2,0)\;\unicode{x00B5}{\rm m}$, see Figs. 2(e) and 2(f), to support this statement. The modulus is higher in the circled area, and the phase information is present. These findings suggest that a combination of images acquired for different $\Delta \textbf{q}$ can provide information about the whole FOV.

We designed a method to reconstruct a high-quality synthetic image throughout the whole FOV from the measured MCF. We acquired images for an $11 \times 11$ grid of $\Delta \textbf{q}$ centered at the ballistic position with the sampling of 1 µm, based on the Airy disk radius of CGF to avoid overlap. Adapting our preliminary work [23], we start by shifting images to their correct positions based on the CGF position $\Delta \textbf{q}$. Then, corresponding small regions of all 121 images are phase-corrected for linear phase distortions with the objective to maximize the modulus of their coherent superposition. Then, the phase quality is after superposition significantly improved. A full FOV image is stitched together from smaller overlapping areas producing a high-quality quantitative phase image, and the result is displayed in Fig. 2(g). The phase quality is consistent throughout the FOV and significantly improved in comparison to the ballistic light phase image in Fig. 2(d). The spatial resolution is also in agreement with the prediction. The smallest resolvable elements of the resolution target in Fig. 2(g) are 0.81 µm wide, which corresponds to the CGF width. It is important to note that coherent averaging of 121 ballistic images in this static case reduces low-contrast hologram noise but does not deal with the phase jumps and singularities, which are the main cause of image degradation. When the scattering medium is highly dynamic, averaging a high ballistic image number can provide similar image improvement as in Fig. 2(g) [24].

To verify the feasibility of our approach for imaging through turbid biological media, we carried out an experiment with a slab of chicken breast instead of the etched glass layer. The chicken breast was cut into a slab of thickness from 1 to 2 mm and squeezed between two coverslips separated by 0.8 mm thick ring. The chicken muscle tissue has a transport mean free path of 1.25 mm, mean free path of 43.7 µm, and an anisotropy factor of 0.965 [25]. One can see that neither ballistic [Figs. 3(a) and 3(b)] nor non-ballistic [Figs. 3(c) and 3(d)] images sufficiently reproduce the phase resolution target. We employed the same procedure as for the imaging through the etched glass plate. Even in this case it was possible to reconstruct a full FOV synthetic image [Fig. 3(e)] with high phase quality and spatial resolution.

 

Fig. 3. Results for imaging through a slab of chicken breast. (a), (c) Modulus and (b), (d) phase images derived from the MCF for $\Delta \textbf{q} = (0,0)\;\unicode{x00B5}{\rm m}$ and $\Delta \textbf{q} = (3,3)\;\unicode{x00B5}{\rm m}$. (e) Synthetic quantitative phase image created by a combination of non-ballistic images. Objectives used $20{\rm x}/{\rm NA} = 0.5$, ${{\rm NA}_{C}} = 0.3$.

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In this Letter, we first demonstrated the equivalence between the complete MCF measurement and coherence-gated imaging for all accessible coherence gate positions in partially coherent interferometric systems. In the case of imaging through strongly scattering media, we showed that the coherence gate manipulation enables imaging by fractions of non-ballistic (multiply scattered) light and that it carries complementary information to the ballistic image. Finally, we proved the immense value of using coherence-gated non-ballistic light by proposing a method for significant improvement of quantitative phase imaging quality through strongly scattering media. Our outcomes benefit not only CCHM but a wide range of partially coherent interferometric systems. A minor alteration of imaging systems [6–10] would allow them to measure the complete MCF and extend their imaging capabilities in biomedical research.