1. Introduction

Phase-resolved imaging techniques have been extensively investigated, including methods for qualitative phase imaging (phase contrast microscopy and differential interference contrast microscopy) [1] and methods for quantitative phase measurement [2]. To study the dynamics of cells when they interact with one another and their environment, there is a need for quantitative phase measurement with high motion sensitivity and spatial resolution. However, high-resolution imaging with quantitative phase contrast remains challenging. In this study, we developed a complex optically computed phase microscopy (complex-OCPM) technology based on the unique combination of low coherence interferometry and optical computation, to perform high-resolution, quantitative phase measurement. With the capability to resolve sub-cellular structures in label-free cells, complex-OCPM is a promising technology for continuous observation of morphology and dynamics of bio-samples.

Complex-OCPM takes a unique approach for phase imaging. There are mainly two categories of techniques for quantitative phase imaging, depending on how the phase is extracted [2]. One category of technologies extracts the phase from spatial domain interferometric fringes, such as imaging systems based on off-axis interferometry [3]. The spatial resolution of these phase imaging systems is limited by the characteristics of imposed waveform (specifically the density of interferometric fringe). Another category of technologies extracts the phase from a time domain waveform, such as imaging systems based on phase-shifting interferometry [4]. Phase-shifting interferometry usually relies on mechanical scanning of the optical path length, and has a limited temporal resolution and phase stability. Optical coherence tomography (OCT) has also been used in quantitative phase measurement [5,6]. However, conventional OCT systems perform raster scanning prioritizing axial ($z$) dimension, leading to a low imaging speed in the transverse ($(x,y)$) plane. To address the trade-off between phase quantification and spatial localization, complex-OCPM directly measures the real and imaginary parts of the complex amplitude for the optical signal. The key innovations of complex-OCPM include direct measurement of complex amplitude ($a_{complex}=|a|e^{j\phi }$) of the optical field emerged from the sample ($E=a_{complex}e^{j(\omega t+kz)}=|a|e^{j\phi }e^{j(\omega t+kz)}$) and direct extraction of the phase ($\phi$) using the argument of the complex amplitude. In this study, we evaluated the performance of complex-OCPM imaging and demonstrated the capability of complex-OCPM to resolve sub-cellular structures in label-free cells.

2. Setup

The hardware implementation of complex-OCPM is similar to that of real-OCPM. The major difference between real-OCPM and complex-OCPM is the computation implemented optically and implemented in post-processing for phase extraction. Complex-OCPM linearly combines two real measurements obtained when the spatial light modulator (SLM) projects a sinusoidal waveform and its orthogonal waveform, assembles a complex signal, and extracts the phase that is the argument of the complex signal. In comparison, real-OCPM projects a sinusoidal waveform at the SLM to measure the real part of the complex reflectivity, and utilizes Hilbert transform to generate a complex signal from which the phase is extracted [7].

Details of the hardware setup of OCPM can be found in our previous publications [7,8]. To explain the principle of complex-OCPM, we show the system configuration in Figure 1. Briefly, the imaging platform consists of a low coherence interferometer and an optical computation module. The low coherent interferometer uses an LED (660nm central wavelength and 40nm bandwidth) as the broadband source. The system has a theoretical axial resolution of 7.2$\mu m$ (in water) and the experimentally characterized axial resolution is 8.4 $\mu m$. Several factors may contribute to the difference between theoretical and experimental axial resolution, including an effective bandwidth smaller than that of the light source and the nonlinearity of the SLM [7]. The reference arm and sample arm use a pair of water dipping lenses (40X Nikon CFI APO NIR Objective, NA=0.8) for sample illumination and signal/reference photon collection. Light exiting the interferometer is processed by the optical computation module which has a diffraction grating (1200 lines/mm), a lens, and a reflective SLM (Holoeye LC-R 720). The periodical groove structure of the grating is along x direction. Hence the grating does not alter light propagation along y direction. Signal photons from different y coordinates of the sample arrive at different rows of the SLM and eventually hit different rows of the camera’s sensing array. On the other hand, light originated from a specific x coordinate of the sample plane is dispersed by the the grating, and modulated by different pixels at the SLM. After SLM modulation, the light goes through the grating for the second time and is detected without spectral discrimination by the camera. The combination of SLM modulation and spectral integration optically computes the inner product between the interferometric spectrum and a Fourier basis function, which effectively selects the depth of imaging. The imaging platform establishes a one-to-one mapping between transverse coordinates in the sample plane and in the sensor plane, and utilizes optical computation to achieve depth resolved imaging.

 

Fig. 1. The configuration of the complex-OCPM system. OBJ: objective; BS: beam splitter; PBS: polarized beam splitter; SLM: spatial light modulator.

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3. Theory

Consider a 3D spatial coordinate system ($x,y,z$) in which the sample is defined. The spatial resolution of real-OCPM is limited by the modulation waveform imposed by the SLM along $y$ dimension, while complex-OCPM has the capability to fully utilize the resolution offered by imaging optics. Unless specified, resolution referred subsequently indicates transverse resolution along $y$ dimension. Compared to a conventional optical microscope, the phase imaging capability of our imaging system does not alter light propagation along $x$ dimension and does not affect diffraction limited optical resolution along $x$ dimension.

Denote the optical field emerging from the sample as $E_s(x,y,k)=\sqrt {M_0(k)}\int a_{complex}(x,y,z)$ $e^{j2kz}\,dz$ ($k=2\pi /\lambda$) and denote the reference optical field as $E_r=\sqrt {M_0(k)}R_0$, assuming the reference is provided by a reflector located at $z=0$ with a constant reflectivity $R_s$. The SLM sees the superposition of reference light and sample light (intensity): $\beta |E_s+E_r|^2$, which is further written as Eq. (1) ($\beta$: system loss; $M_0(k)$: source spectrum; $a_{dc}(x,y)=|\int [a_{complex}(x,y,z)e^{jkz}]\,dz|^2+|R_0|^2$; $Re()$: to take the real part of a complex function).

In real-OCPM, the SLM modulates $i_{SLM}$ with a sinusoidal pattern $f_{cos}(y,k)$ [Eq. (2)], leading to $i_{cos,SLM}$ in Eq. (3). As illustrated in Fig. 1, $i_{cos,SLM}$ goes through the grating again and allows the camera to perform photon detection without spectral discrimination, as indicated by the integral with regard to $k$ in Eq. (4). To derive Eq. (4), we assume an impulse sample function located at depth $z_{cell}$: $a_{complex}(x,y,z)=\delta (z-z_{cell})a(x,y)$, $z_{SLM}=2z_{cell}$ and $a(x,y)=|a(x,y)|e^{j\phi (x,y)}$. Detailed derivations of Eq. (4) can be found in Supplementary Materials. In real-OCPM, the real measurement $i_{cos}(x,y)$ in Eq. (4) is further processed in a computer to extract the phase.

In complex-OCPM, an additional measurement $i_{sin}(x,y)$ is taken when the SLM projects the pattern $f_{sin}$ [Eq(5)]. Light modulated by the SLM is $i_{sin,SLM}$ [Eq. (6)] and the signal detected by the camera is $i_{sin}(x,y)$ [Eq. (7)]. DC component is taken when there is no interference phenomenon and is subtracted from $i_{cos}(x,y)$ and $i_{sin}(x,y)$ to assemble a complex measurement $i_{complex}(x,y)$ [Eq. (8)] for phase measurement.

Following optical computation, real-OCPM and complex OCPM detect $i_{cos}(x,y)$ and $i_{complex}(x,y)$ respectively, and utilize different post processing methods to extract the phase. For real-OCPM, 2D Fourier transform ($\mathfrak {F}$) is applied to $i_{cos}(x,y)$ (DC subtracted), leading to $I_{cos}(f_x,f_y)$ [Eq. (9)] that is the linear combination of $A(f_x,f_y+\alpha _y)$ and $A^*(-f_x,-f_y+\alpha _y)$ [Fig. 2(a)] where $A(f_x,f_y)$ represents the Fourier transform of $a(x,y)$. Band-pass filtering is applied to $I_{cos}(f_x,f_y)$ to obtain $A(f_x,f_y+\alpha _y)$. Afterwards, we shift $A(f_x,f_y+\alpha _y)$ in spatial frequency domain to obtain $A(f_x,f_y)$ and apply inverse Fourier transform ($\mathfrak {F}^{-1}[A(f_x,f_y)]$) to obtain $a(x,y)$ ($a(x,y)=|a(x,y)|e^{j\phi (x,y)}$) and its phase $\phi (x,y)$ [7]. For complex-OCPM, 2D Fourier transform ($\mathfrak {F}$) is applied to $i_{complex}(x,y)$, leading to $I_{complex}(f_x,f_y)$ that is expressed as Eq. (10). Notably, $I_{complex}(f_x,f_y)$ is a linear combination of $\mathfrak {F}(i_{cos}(x,y))$ [Fig. 2(b)] and $\mathfrak {F}(i_{sin}(x,y))$ [Fig. 2(c)]. Such a linear combination eliminates the term $A^*(-f_x,-f_y+\alpha _y)$ in $I_{complex}(f_x,f_y)$ [Eq. (10) and Fig. 2(d)]. As a result, $I_{complex}(f_x,f_y)$ does not require spatial frequency domain filtering and $A(f_x,f_y)$ can be directly obtained by spectrally shifting $I_{complex}(f_x,f_y)$.

 

Fig. 2. (a) $I_{cos}(f_x,f_y)$ where $A(f_x,f_y+\alpha _y)$ and $A^*(-f_x,-f_y+\alpha _y)$ do not overlap; (b) $I_{cos}(f_x,f_y)$ where $A(f_x,f_y+\alpha _y)$ and $A^*(-f_x,-f_y+\alpha _y)$ overlap; (c) $I_{sin}(f_x,f_y)$ where $A(f_x,f_y+\alpha _y)$ and $A^*(-f_x,-f_y+\alpha _y)$ overlap; (d) $I_{complex}(f_x,f_y)$.

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Figure 2 illustrates how complex-OCPM achieves a higher spatial resolution compared to real-OCPM. The principle of complex-OCPM is also demonstrated using experimental data (Supplementary Materials Figure S2). For real-OCPM, the resolvable bandwidth of the sample (the half bandwidth of the signal $\Omega _s$, as indicated by the radius of the circle) directly determines the spatial resolution and $\Omega _s$ has to be smaller than $\alpha _y$. In other words, the spatial resolution of real-OCPM is limited by the sinusoidal modulation along y direction (specifically the value of $\alpha _y$). As shown in Fig. 2(a), when $\Omega _s<\alpha _y$, $A(f_x,f_y+\alpha _y)$ and $A^*(-f_x,-f_y+\alpha _y)$ do not overlap, and $A(f_x,f_y)$ can be accurately recovered by filtering $I_{cos}(f_x,f_y)$. In comparison, when $\Omega _s>\alpha _y$, $A(f_x,f_y+\alpha _y)$ and $A^*(-f_x,-f_y+\alpha _y)$ overlap [Fig. 2(b)] and $A(f_x,f_y)$ cannot be obtained by filtering $I_{cos}(f_x,f_y)$. On the other hand, $\alpha _y$ is determined by the need to sample the sinusoidal modulation [Eq. (3)] above Nyquist rate using SLM pixels: $\alpha _y<\pi M/(L_{SLM})$ ($L_{SLM}$: the dimension of the SLM pixel; $M$: the magnification from the sample plane to the SLM plane). Assume $A(f_x,f_y)$ has a Gaussian envelop (in $f_y$ dimension), let $2\Omega _s$ be the spectral range beyond which $A(f_x,f_y)$ is significantly less (<5%) than its peak value, and define the transverse resolution of real-OCPM $\delta r_{real}$ as the full width half maximum (FWHM) of the system’s point spread function (in y dimension). We have $\delta r_{real}=\frac {9.6\sqrt {2ln2}}{\Omega _s}$. Given $L_{SLM}=20\mu m$, $M=80$ and the fact that $\Omega _s<\alpha _y$, $\delta r_{real}>\frac {9.6\sqrt {2ln2}}{\alpha _y}>\frac {11.3}{\pi M/(L_{SLM})}\approx 0.90\mu m$, which is consistent with experimental result of 0.98$\mu m$ ([7]). This value is inferior to diffraction limit: $\delta r_{diff}=0.61\lambda _0/NA=0.56\mu m$. Unlike real-OCPM, the spatial resolution of complex-OCPM is not limited by the modulation pattern (the value of $\alpha _y$ in $f_{cos}(y,k)$ and $f_{sin}(y,k)$), because it assembles a complex measurement [Eq. (8)] and eliminates spatial frequency domain complex conjugate symmetry [Fig. 2(d)] [9]. Hence the spatial resolution of complex-OCPM is only limited by the imaging optics.

4. Results

We processed real and complex OCPM data using Matlab 2022a in a Mac computer (Macbook Pro, with Apple M1 CPU and 8GB RAM). The processing time for complex-OCPM (1024 by 1024) was tested to be 0.157s, and the processing time for real-OCPM (1024 by 1024) was tested to be 0.163s. Signal processing time of complex-OCPM is slightly shorter because real-OCPM requires spatial domain filtering. The image acquisition time of complex-OCPM is twice as long as that of the real-OCPM, because the camera has to take to two measurements to assemble a complex measurement. Currently, the camera is operated at a frame rate of 8Hz and the frame rate for complex-OCPM imaging is 4Hz. The frame rate is limited by the exposure time, as a sufficiently large exposure time is needed to collect enough signal photons.

To demonstrate the advantage of complex-OCPM in spatial resolution and validate its quantitative phase imaging capability, we projected sinusoidal patterns to the SLM (Eq. (2) and Eq. (5) with $\alpha _y \approx 0.82\mu m^{-1}$) and imaged a resolution target with reflective bars (Reflective Variable Line Grating Target, Thorlabs) that was not a phase object. We intentionally chose to project sinusoidal patterns with low frequency along $y$ dimension (small $\alpha _y$) to highlight the resolution advantage of complex-OCPM. Figure 3(a) shows $i_{cos}(x,y)$ (black), $i_{sin}(x,y)$ (green), $|A(x,y)|$ obtained from real-OCPM, and $|A(x,y)|$ obtained from complex-OCPM ($x=0$ and DC was subtracted from $i_{cos}(x,y)$ and $i_{sin}(x,y)$). The period of the modulation waveform seen from the green curve or the black curve is approximately $8\mu m$ which is consistent with the value of $\alpha _y$ ($2\pi /\alpha _y\approx 8\mu m$). The real-OCPM signal (the blue curve) fails to show the sharp transition from high reflectivity region to low reflectivity region (and vice versa). This is because a small $\alpha _y$ leads to a low spatial resolution. In comparison, the complex-OCPM signal (the red curve) clearly shows a sharper edge between regions with different reflectivities, as the spatial resolution of complex-OCPM is independent of $\alpha _y$. The phase images ($\phi (x,y)$) are shown in Fig. 3(b) (real-OCPM) and 3(c) (complex-OCPM). Similar to amplitude signal, the phase image obtained from complex-OCPM [Fig. 3(c)] has higher spatial resolution compared to real-OCPM phase image [Fig. 3(b)]. To validate the capability of complex-OCPM in phase quantification, we averaged the phase values within regions of interests (ROIs) indicated by the boxes in Fig. 3(c). ROI1 in the black box corresponded to an elevated area with chrome deposition and ROI2 in the red box corresponded to the glass substrate. We converted the phase to optical path length ($d=\lambda _0\phi /(4\pi )$), and estimated the height of the chrome deposition to be $d_{chrome}=d_{ROI1}-d_{ROI2}=122nm$ which is consistent with the value provided by the manufacturer (120nm). For real-OCPM, the same value was obtained.

 

Fig. 3. (a) $i_{cos}$ (black), $i_{sin}$ (green), $|A|$ obtained from real-OCPM (blue), and $|A|$ obtained from complex-OCPM (red); (b) real-OCPM (phase); (c) fcomplex-OCPM (phase).

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We quantitatively evaluated the transverse resolution of complex-OCPM by imaging a 1951 USAF resolution target. We projected sinusoidal patterns [Eq. (2) and Eq. (5)] with $\alpha _y=7.2\mu m^{-1}$ to the SLM. Determined by the value of $\alpha _y$, the spatial resolution of real-OCPM was approximately $1.6\mu m$ ($\delta r_{real}>\frac {9.6\sqrt {2ln2}}{\alpha _y}$), while complex-OCPM image could achieve a higher spatial resolution. This is illustrated in Fig. 4. Figure 4 (a) and 4(b) show amplitude image and phase image of complex-OCPM. Figure 4(c) is the ROI (phase image) showing the $9^{th}$ group elements of the resolution target. Individual bars of the $3^{rd}$ element in the $9^{th}$ group are clearly discernible, suggesting a resolution higher than 0.78$\mu m$ and higher than the value limited by $\alpha _y$ ($1.6\mu m$). The smallest structure of the resolution target has a dimension of 0.78$\mu m$, which limits our capability to directly assess the transverse resolution of complex-OCPM. Nevertheless, we were able to manually measure the rising and falling edge of the signal corresponding to the square in Fig. 4(c) and the resolution along $x$ and $y$ dimensions were estimated to be 0.55 $\mu m$, consistent with diffraction limit ($0.61\lambda /NA \approx 0.5\mu m$). Figure 4(d) and 4(e) show amplitude image and phase image of real-OCPM. The bars of the $3^{rd}$ element in the $8^{th}$ group are discernible, while the bars of the $4^{rd}$ element in the $8^{th}$ group are not discernible, suggesting a spatial resolution around $1.55\mu m$, consistent with the value determined by $\alpha _y$. Figure 4(f) is the ROI (phase image) showing the $9^{th}$ group elements of the resolution target where all the horizontal bars cannot be observed because of insufficient spatial resolving power.

 

Fig. 4. 1951 resolution target, complex-OCPM image: (a) amplitude; (b) phase; (c) phase ROI; real-OCPM image: (d) amplitude; (e) phase; (f) phase ROI.

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We demonstrated complex-OCPM imaging on cultured C2C12 cells. The C2C12 cell line is a subclone of myoblasts and features a radial branching morphology consisting of long fibers. The cells were cultured in Dulbecco’s Modified Eagle Medium (DMEM) supplemented with 10% fetal bovine serum, 1% penicillin / streptomycin at 37$^{\circ }$C in a humidified 5% $CO_2$ incubator. Figure 5(a) shows the real-OCPM image of a C2C12 cell, while Fig. 5(b) shows the complex-OCPM image of the same cell . In Fig. 5(a) and 5(b), regions with a significantly larger phase retardation correspond to cell nucleus that has a higher refractive index. Complex-OCPM reveals the fibrous morphology surrounding the cell nucleus with finer details compared to real-OCPM. Figure 5(c) and 5(d) compare the ROI in the vicinity of the red arrows in Figure 5 (a) and (b). The fibers appear thinner in complex-OCPM because of its higher spatial resolution. Figure 5(e) and 5(f) compare the ROI in the vicinity the black arrows in Fig. 5(a) and 5(b). Two distinct fibers are visible in complex-OCPM image [Fig. 5(f)], as indicated by the arrows with solid and dashed lines, while these two fibers are not discernible in real-OCPM image [Fig. 5(e)]. The morphology of C2C12 cells are validated in comparison with confocal microscopy in Supplementary Materials Figure S2.

 

Fig. 5. C2C12 cells: (a) real-OCPM image; (b) complex-OCPM image; ROI near the red arrow: (c) real-OCPM and (d) complex-OCPM; ROI near the black arrow: (e) real-OCPM and (f) complex-OCPM.

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5. Conclusion and discussion

In conclusion, we developed and validated a complex-OCPM imaging technology that directly measures the complex field reflectivity of the sample. Instead of extracting the phase from a local (spatially or temporally) waveform, complex-OCPM extracts the phase as the argument of a complex number for each pixel of the image and allows phase resolved measurement to fully utilize the spatial resolution of the imaging optics.

The high spatial resolution of complex-OCPM may result in a compromised phase sensitivity. The sensitivity of phase measurement is fundamentally limited by the signal to noise ratio (SNR) of the intensity measurement and is determined by the number of signal photons under shot noise limit [10]. The higher spatial resolution of complex-OCPM implies fewer photons contributing to individual resolution cells. This can be seen from our experimental results where complex-OCPM images have a lower apparent SNR compared to real-OCPM images. To improve phase sensitivity in our future work, we can synthesize the phase image by taking more measurements with different SLM patterns and utilizing more photons to render the signal at individual pixels. Alternatively, we can impose SLM patterns that are adaptive to the characteristics of the sample and reconstruct the image using appropriate prior knowledge to improve phase measurement sensitivity.