OCT uses interferometric detection to provide high-resolution volumetric imaging of biological samples in vivo [1]. Since its original invention in time-domain [2], researchers improved the OCT speed performance by employing the Fourier-domain (FD) [3] and tunable lasers [4] to reach video-rate acquisition [5]. Though OCT technology exhibited rapid development in recent years, there are still crucial issues. For instance, most scanning OCT systems feature low NA objectives to increase the axial imaging range by extending the depth of field. As a result, the lateral resolution becomes poor compared to axial resolution, making the in vivo cellular-level imaging challenging.

An improved transverse resolution without compromising the depth of field is provided by the full-field (FF)-OCT. FF-OCT uses wide-field illumination and parallel interferometric detection [6]. This approach was originally developed in time-domain with spatially incoherent light sources (LED or thermal sources). Such FF-OCT was shown to provide nearly isotropic resolution below 1 µm [7]. However, an attempt to further boost the FF-OCT imaging speed with FD detection that uses tunable lasers resulted in a cross talk problem. The spatial coherence of the laser produces coherent artifacts—the so-called cross-talk-generated noise, which induces the speckle pattern and hampers the image resolution [8]. Though FD-FF-OCT, supported by numerical phase correction, reveals cellular features of the retina, cross talk prevents visualization of the deeper retinal layers like choroid  [9].

Recently, we developed the novel approach to suppress the coherent cross talk in FD-FF-OCT, which we call spatiotemporal optical coherence (STOC) manipulation [10,11]. In STOC, we modulate the phase of light incident on the sample in time with a set of phase patterns generated by the spatial phase modulator (SPM). Then, the phase-modulated three-dimensional (3D) reconstructions are averaged incoherently to achieve the cross-talk-free volumes. Lately, we used a liquid crystal spatial light modulator (LC SLM) as the SPM to image the 1951 USAF resolution test chart covered by biological tissue ex vivo [12]. In parallel, we developed a cross-talk-free FD-FF-OCT system that employed a high-speed deformable membrane as an SPM [13]. We already applied it to image human skin [13] and retina [14] in vivo. In this Letter, we show that cross-talk-free FD-FF-OCT can be described with the same formalism that we previously used for STOC with the only difference being that we use the coherent instead of incoherent averaging [10]. Moreover, we show that such an approach makes it possible to explain the behavior of a stable phase within imaging depth, which in turn allows using the algorithms for digital correction of optical aberrations [9].

Figures 1(a) and 1(b) sketch the two optical setups for STOC imaging [12,13]. The setups are based on the FD-FF-OCT configuration that uses SPM for phase modulation. The SPM can be placed in either the sample arm [Fig. 1(a)] or before the interferometer [Fig. 1(b)]. We call the former sample arm-only modulation (SAM), and the latter both arm modulation (BAM). In both cases, the SPM sequentially displays uncorrelated phase patterns, each of which is active during the whole laser sweep [Fig. 1(c)]. We record such modulated interferometric images with a two-dimensional (2D) camera and apply standard OCT data processing.

 

Fig. 1. Two implementations of the STOC imaging in the form of FD-FF-OCT that utilizes SPM. (a) Mach–Zehnder interferometer (MZI) with the SPM placed in the sample arm. (b) Linnik interferometer (LI) with the SPM located before interferometer. (c) In STOC, we modulate the FD-FF-OCT signal in time with a set of $M$ (largely) uncorrelated phase patterns. (d) We can average the resulting signal (${U_1},{U_2}, \ldots {U_M}$) incoherently or coherently. However, only in the LI arrangement, the phase relation between the two arms is preserved, and we can benefit from the coherent averaging to improve the contrast, ${{C}_M}$ of the final image (e). Abbreviations: SPM, spatial phase modulator; BS, beam splitter; ${{C}_M}$, Michelson’s contrast.

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Specifically, we average the resulting 3D reconstructions on either the magnitude- (incoherent averaging) or amplitude-basis (coherent averaging) [Fig. 1(d)]. However, when we use SAM [Fig. 1(a)], we distort the phase relation between the reference and sample fields. Thus, in this mode, we can only use incoherent averaging. In the second case, when we use BAM modulation, the phase relation between the two arms is preserved since the signal in both arms is altered by the same phase patterns. These phase patterns create speckle fields, which only interfere when the path difference in both arms is matched to within the speckle correlation length [15]. Hence, the coherent averaging is possible and, as explained below, we achieve improved image contrast, compared to the incoherent averaging.

 

Fig. 2. Two STOC-manipulated phasors with relative phase shifts of (a) ${\Delta }\varphi = 0$ and (b) ${\Delta }\varphi = \pi $.

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We validated the above approach experimentally, as shown in Fig. 1(e). To this end, we used LC SLM as SPM and imaged the 1951 USAF resolution test chart covered with a tailored microdiffuser (with 1° angle). When there is no phase modulation—STOC is disabled (STOC OFF column)—we get a distorted sample image due to the photon paths deflected by the diffuser. Then, we enabled STOC manipulation to modulate phase with $M\; = \;128$ patterns derived from the Hadamard matrix. For both hardware configurations, shown in Figs. 1(a) and 1(b), we suppress image distortions by STOC manipulation with the incoherent averaging [second column in Fig. 1(e)]. However, corresponding images contain an additive offset from an incomplete rejection of the cross talk noise. To overcome this issue, we use BAM and coherent averaging [third column in Fig. 1(e)]. On the other hand, for coherent averaging in SAM, we do not see the sample image but only residuals and a regular grid from the Hadamard phase patterns.

To explain the difference between the incoherent and coherent averaging, we use phasor analysis. Figure 2 depicts two phasors ${U_1},{U_2}$ of the STOC-manipulated signal with relative phase shifts of $\Delta \varphi = 0$ and $\Delta \varphi = \pi $, respectively,

We can now see that in both cases, the cross-terms (containing a product $U_s^{*}{U_n}$) cancel out. Hence, we suppress the coherent image distortions induced by the diffuser. However, the additive noise, ${I_n}$, is not removed in case of incoherent averaging. As shown previously in Fig. 1(e), this offset ${(I_n)}$ reduces the image contrast. For that reason, the resolution chart features appear gray instead of black. Coherent averaging (${I_{\rm coh}}$) overcomes the unwanted offset. Namely, we enhance the signal by a factor of 2, and ${I_n}$ is not present. As a result, the USAF features appear black as expected [bottom right cell in Fig. 1(e)]. Such a reduction of the noise floor is consistent with the complex averaging in scanning OCT systems [16].

To quantify image contrast improvement, we apply the Michelson’s formula, ${C_M} = ( {{I_{\max}} - {I_{\min}}} )/( {{I_{\max}} + {I_{\min}}} )$ to pixels at the edge of the black square, located between the eighth and ninth USAF groups. The contrast values, depicted in Fig. 1(e) show that coherent averaging as implemented in BAM enhances the contrast nearly 2.3 times with respect to the unmodulated case. On the contrary, the incoherent averaging provides only 1.26 times better contrast for both SAM and BAM.

In practice, we use $M$ 2D phase modulation patterns, ${{\boldsymbol \varphi }_m} = {\varphi _m}( {x,y} )$, and images are detected by a 2D array of pixels [${{\boldsymbol U}_{s,n}} = {U_{s,n}}( {x,y} )]$. Assuming the one-to-one correspondence between SPM and detector pixels, we extend the above derivations to matrix averaging. We denote matrices with bold symbols, and $x,y$ are discrete indices running from 1 to $N$, where $N$ is the detector side length (we assume square detector). After averaging $M$ phasors ${{\boldsymbol U}_m} = {{\boldsymbol U}_s} + {{\boldsymbol U}_n}{e^{i{{\boldsymbol \varphi }_m}}}$. with an additive noise term, we obtain

Coherent averaging thus enhances the signal term by a factor of $M$ (the number of phase patterns), which agrees with previous studies [17]. In contrast, the incoherent averaging does not enhance the signal and contains an additive offset (${{\boldsymbol I}_n}$). However, in both cases, the cross talk noise is suppressed. Most importantly, the coherent averaging does not alter the phase of the signal component (${U_s}$), if the lateral stability is preserved during the measurement.

We can utilize that to correct for geometrical aberrations in the postprocessing [9]. We show this here on images acquired with the previously described system [14] that employed a rapid deformable membrane as the SPM to generate pseudo-random phase patterns and had a lens inserted in the sample arm to form a Linnik interferometer. The camera implicitly performed coherent averaging since the membrane displayed many uncorrelated interference patterns during an acquisition time (15 µs) of a single frame. To induce geometrical aberrations, the objective lens was shifted from the optimal focus position by 150 µm producing a distorted OCT image, shown in the left column of  Fig. 3.

 

Fig. 3. Numerical phase correction compensates for the defocus aberration in STOC imaging to achieve nearly the same resolution as in-focus. All images were acquired with FD-FF-OCT and STOC manipulation.

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To correct for the defocus, we use digital aberration correction (DAC), which proceeds as follows. The complex data of each sample layer, ${U_l}( {x,y} )$, is 2D-Fourier transformed to obtain the spatial spectrum $ {\tilde U_l}( {{k_x},{k_y}} ) $. Then, ${\tilde U_l}( {{k_x},{k_y}} )$ is multiplied by $\exp [ {i\alpha Z_2^0( {{k_x},{k_y}} )} ]$, where $\alpha $ is an adjustable parameter, and $Z_2^0$ denotes the Zernike polynomial corresponding to defocus (with OSA/ANSI index of 4). The resulting product ${\tilde U_l}( {{k_x},{k_y}} )\exp [ {i\alpha Z_2^0( {{k_x},{k_y}} )} ]$ is 2D-inverse-Fourier transformed to obtain phase-corrected data ${U_{l,{\rm corr}}}( {x,y} )$. We then calculate the image sharpness metric (kurtosis) $\xi ( \alpha )$ on ${| {{U_{l,{\rm corr}}}( {x,y} )} |^2}$. This process is continued for various $\alpha $ until we optimize $\xi ( \alpha )$. The resulting, optimized USAF image appears in the second column of Fig. 3. Digital correction improves spatial resolution. The corrected image has almost the same resolution as the in-focus, reference image, depicted in the last column of Fig. 3.

When imaging simple objects like the USAF resolution test chart, the noise offset can be subtracted, and the image contrast can be stretched. However, this becomes an issue for complex objects, especially when imaging through the scattering layer. To show this, we imaged lens tissue with and without STOC manipulation, shown in Fig. 4. Again, the images were acquired with induced defocus. When STOC is disabled, the OCT image is significantly corrupted by the cross talk noise. The features of the sample like fibers are barely seen because of the strong influence from noise term ${I_n}$. However, we can significantly improve OCT image contrast by enabling STOC, which in practice is carried out by activating the deformable membrane or any other fast SPM in the FD-FF-OCT system. Nevertheless, fibers are still blurred due to the defocus aberration.

 

Fig. 4. Removal of cross talk noise and optical aberrations in images of lens tissue. Cross talk is removed by STOC (left column), whereas defocus aberrations are removed computationally (right column).

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Fig. 5. STOC imaging of the photoreceptor layer of the human retina in vivo (Visualization 1, Visualization 2, Visualization 3, and Visualization 4). (a) The cross-section (B-scan) with a green dashed rectangle indicating the IS/OS layer, which was corrected computationally to reveal photoreceptor cones (b) that were otherwise invisible (c). Red dashed curves divide the regions, where STOC was ON and OFF.

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Notably, we can correct for that using our DAC algorithm. As shown in the second column of Fig. 4, this algorithm works well on the noisy data (STOC OFF row). The blurred fibers become sharp, and we can see previously invisible features. Most importantly, the same approach can be applied to a STOC manipulated signal to improve not only the image contrast but also the sharpness of the sample features. This proves that STOC manipulation with coherent averaging implemented by using a fast SPM before the interferometer preserves the signal phase.

Finally, we imaged retina in vivo of a healthy 44-year-old volunteer with the FD-FF-OCT system [14]. To simultaneously record data with and without STOC manipulation, the SPM was shifted laterally so that only a part of the image was phase-modulated and the remaining part was unaffected because the membrane there was acting as a flat mirror. In the modulated region, SPM displayed many pseudo-random phase patterns within the integration time of a single camera frame (15 µs). Thus, the camera implicitly performed coherent averaging. However, since the phase evolved continuously from one pattern to another, it was difficult to estimate the exact number of uncorrelated patterns in the way done for LC SLM above. To further improve the signal-to-noise ratio, we recorded 10 volumes, each of which was captured within $\sim{9}\;{\rm ms}$. As shown in Fig. 5(a) we can image deeper into the retina when STOC is ON due to the cross talk suppression. Then, in postprocessing, we applied the DAC algorithm, extended to higher-order Zernike polynomials, and integrated intensity used as the sharpness metric. We estimated post-DAC resolution to $\sim 4.7\;\unicode{x00B5} {\rm m}$. Finally, we registered (spatially aligned) and incoherently averaged phase-corrected volumes to reveal photoreceptor mosaic [Fig. 5(b)], which is not visible without DAC [Fig. 5(c)]. Importantly, when STOC is ON, photoreceptors are better visible.

In summary, by extending FD-FF-OCT with a spatial phase modulator (SPM), we could average the resulting data incoherently or coherently. The SPM can be placed either in the sample arm or before the interferometer. We demonstrated that coherent averaging is suitable in the latter configuration because the phase relation between the two interferometer arms is preserved. Consequently, the phase of the useful signal is maintained after the modulation and the coherent averaging allowing correction of phase errors in postprocessing. We employed this to depict the IS/OS layer of the human retina in vivo, revealing the photoreceptor mosaic, the primary sensing element of the human visual system. Our results can thus pave the way for FD-FF-OCT to in vivo cellular-level noninvasive volumetric imaging.