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Abstract
Perineuronal nets (PNNs) are important functional structures on the surface of nerve cells. Observation of PNNs usually requires dyeing or fluorescent labeling. As a network structure with a micron grid and sub-wavelength thickness but no special optical properties, quantitative phase imaging (QPI) is the only purely optical method for high-resolution imaging of PNNs. We proposed a Scattering Quantitative Interference Imaging (SQII) method which measures the geometric rather than transmission or reflection phase during the scattering process to visualize PNNs. Different from QIP methods, SQII method is sensitive to scattering and not affected by wavelength changes. Via geometric phase shifting method, we simplify the phase shift operation. The SQII method not only focuses on interference phase, but also on the interference contrast. The singularity points and phase lines of the scattering geometric phase depict the edges of the network structure and can be found at the valley area of the interference contrast parameter SINDR under different wavelengths. Our SQII method has its unique imaging properties, is very simple and easy to implement and has more worth for promotion.
© 2024 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
There are various network structures with specific functions in biological tissues. Observation of these structures is an important means to study biological processes and the mechanisms behind them. These network structures are usually transparent and exist at micron or even subwavelength scales, making them difficult to observe with conventional optical means. For example, perineuronal nets (PNNs) are network structures attached to the surface of nerve cells and are mainly composed of hyaluronic acid, chondroitin sulfate proteoglycans, etc. [1,2] PNNs have micron-scale grid dimensions and sub-wavelength thickness, and have no specific optical properties. PNNs is a type of specialized extracellular matrix in the central nervous system, acting as a protective barrier and regulates synaptic plasticity of neurons. In study of brain science, such as memory and learning mechanisms, as well as treatment of brain diseases and injuries, PNNs is an important research object. Observation of network structures such as PNNs usually is achieved using dyeing or fluorescent labeling methods. Purely optical methods that utilize the dispersion, birefringence and other optical properties of biological tissues are not suitable for PNNs, like polarized light microscopy (PLM) [3,4] and instant polarized light microscopy (IPOL) [5,6] method. Quantitative phase imaging (QPI) [7–10] is a universal choice for imaging transparent biological tissues. Interferometric QPI methods have a contradiction between the number of phase shifts and spatial resolution. According to the angle between the interference light and the reference light, the QPI method is divided into three types: off-axis interferometry [11–13], coaxial interferometry [14–17] and slightly off-axis interferometry [18–20]. Coaxial interference QPI methods usually have a higher spatial resolution but require longer modulation time and complex phase shifting components. Simplifying the implementation of phase shift is of great significance to promote the application of QPI method.
At the end of the last century, when the concept of geometric phase was widely discussed, the geometric phase shifter was proposed [21]. This method of phase shifting using polarization rotation simplifies the phase shifting work of the interferometry system. The characteristic of the geometric phase shifter is that the phase shift amount only depends on the rotation angle, not affected by wavelength [22,23]. It only needs to use an achromatic wave plate to work over a broad spectrum. Through reasonable arrangement of polarization channels, geometric phase shifters can be applied to interference imaging systems, such as Differential Interference Contrast (DIC) Microscopy [24] and Digital Holography Microscopy [25,26], making phase shifting more convenient and accurate. The principle of geometric phase not only provides a convenient phase shifting tool, but also gives a new understanding of light-matter interaction.
Light scattering is an inherent property of objects, and the distribution of the scattering field is often closely related to the structure of the scatterer. The observation method based on scattered field measurement is purely optical and can be applied to any microscopic network structure without markers or special optical properties. Under circularly polarized basis, light scattering produces circularly polarized components opposite to the incident polarization. Such components arise from the varying direction of light propagation, which changes the polarization vector projection in the observation plane. An additional GP is introduced to the opposite scattering components according to the changing trajectory of the polarization state on Poincaré sphere. This additional GP carries orbital angular momentum equal to the change in spin angular momentum to maintain the conservation of total angular momentum. This process is called spin-orbit Interaction (SOI) [27]. As an example, the scattering phase of nano-spheres [28] or nano-holes [29] is a 4$\pi $ phase vortex for electric field in XY directions and a 2$\pi $ phase vortex for electric field in Z direction. The phase vortex represents the rotational distribution of the scattering direction. In the scattering field of micro-nano structures, phase singularities appear at the center of orbital angular momentum and act as the structural skeleton of the scattering field. Phase singularity points and the phase-changing lines among them provide rich structural information of the scattering field and then describe the structural characteristics of the scatterers [30,31]. Since the GP has nothing to do with the optical path, the scattering phase is not wavelength-sensitive, which is different from the general QPI and PLM methods.
In this work, we developed a coaxial Scattering Quantitative Interference Imaging (SQII) method to observe the far-field scattering of biological network structures with micron-scale and subwavelength thickness. This method only requires simple modifications on commonly used transmission microscopes, using rotated linearly polarized illumination and performing circular polarization analysis on the transmitted light. The optical system and calculation principle of this method are very similar in form to some polarization microscopies [32], but the connotations are completely different. In past research, we found that some PLM systems have significant imaging results for samples without obvious polarization anisotropy. This phenomenon can be well explained by scattering geometric phase. We will show that our SQII can observe the morphology of PNNs or other similar non-anisotropic scattering structures.
2. Theory and method
Considering the transmission scattering process with linearly polarized illumination under a circular polarization basis, the transmission scattering process can be described by the following formula:
$S_R^ + $ and $S_R^ - $ are the RCP transmission scattering components under RCP and LCP illumination. If illumination is linearly polarized and RCP analysis is performed on the output field, the intensity of the measured image can be described by the following formula:
Figure 1(a) and 1(b) shows the graphical representations of Eqs. (1) and (2). $S_R^ - $ describes the SOI scattering process since its spin direction is inverted and $S_R^ + $ describes the non-SOI scattering process. 2α is an additional phase shift introduced by the rotating angle α of the linearly polarized illumination. Equation (2) describes the interference between these two circular polarized components. The rotation of the polarized illumination realizes the phase shift 2α, allowing us to calculate interference intensity $|{S_R^ + } |$, $|{S_R^ - } |$ and interference phase difference $\Delta {\varphi _R}$ through the principle of phase-shifting interferometry.
Fig. 1. (a) The transmission scattering model under circular polarization basis. (b) The coaxial scattering interference process with rotating linearly polarized illumination under RCP analysis.
The $S_R^ - $ component, with reversed spin angular momentum, usually arises from the polarization birefringent properties of the sample or changes in propagation direction during scattering. For the scattering situation, the changes in propagation direction will introduce a GP to $\varphi _R^ - $, in addition to the transmission phase (TP). $\Delta {\varphi _R}\textrm{} = \varphi _R^ + \textrm{} - \varphi _R^ - $ exactly represents the negative GP. The change of GP reflects the change in the scattering angle, which in turn reveals the gradient of the scattering structure. Meanwhile, the reversal of spin angular momentum leads to the transfer of angular momentum from spin to orbit, forming singularity points and phase vortices at the scattering vertices (for example: the top of nano-spheres) with uncertain scattering directions and 0 intensity. These points, along with the rapid phase change lines among them, constitute the skeleton of a scattered field and reflect the structural characteristics of the scatterer.
In SQII method, two interference components $S_R^ - $ and $S_R^ + $ is generated by the sample. The interference contrast is a sample-dependent parameter and is not artificially controlled. Both interference contrast and phase carry structural information of the sample. To calculating the interference contrast and phase through numerical Fourier analysis, consider ${E_0} = \; \sqrt 2 $, Eq. (2) can be rewritten as:
Where:The angle α of the polarizer is rotated through 180 degrees in N steps to fulfill the $2\pi $ phase shift. Compared with the Four-step phase shifting in most QPI methods, the phase shifting step N here needs to be increased to ensure accurate measurement under lower interference contrast (see Supplement 1). Parameters ${a_i}$ can be found from:
Interference contrast $SIN{D_R}$ and interference phase $\Delta {\varphi _R}$ can be found from:
Figure 2(a) shows the schematic diagram of our method. The simulation was performed under illumination at three wavelengths: 600nm (R), 524nm (G) and 455nm (B). These wavelengths are selected from the spectral peaks of our color SCMOS (acA4024-8gc, Basler). Simulation under circularly polarized illumination can directly obtain all the scattering components in the Eq. (1), and simulation under rotating linearly polarized illumination can simulate our SQII method and obtain measurement parameters $\Delta {\varphi _R}$ and $SIN{D_R}$. We choose the total-field scattered-field (TFSF) source to obtain the pure scattering field and add back the removed incident component manually. This helps reduce the influence of standing waves caused by simulation boundaries. The near-field simulation results are all transformed into far-field with a numerical aperture of 0.9, which simulated the imaging process of a 100X Olympus objective lens. To ensure that the far-field transformation can obtain sufficient angular spectrum information, the image size is 18 × 18${\mathrm{\mu} \mathrm{m}}$, slightly smaller than the simulation area.
Fig. 2. (a) Schematic diagram of our SQII system. (b) 2-D and 3-D PNNs-like network structure model. The maximum thickness of 3D model corresponding to maximum brightness in 2D maps is 200nm. The size of the structure and simulation area is 20 × 20${\mathrm{\mu} \mathrm{m}}$ (c) Longitudinal section of the FDTD simulation model based on the nerve cell surface structure. The illumination is from bottom to top.
Theoretically, the SQII method can work in both transmission and reflection modes. However, for most scattering materials and structures, the intensity of backscattering is much lower than that of forward scattering, and cannot obtain meaningful measurement results. Therefore, we only discuss the transmission SQII method.
We used the Hessian-based Frangi Vesselness filter [33] to obtain net structures with true randomness from stained cerebral cortex nerve cell section samples, as shown in Fig. 2(b). The scale of this structure is similar with that of PNNs. We use this kind of structure in the FDTD simulation to study the scattering characteristic of PNNs. Figure 2(c) shows the refractive index structure of the model without dispersion, which was set according to the surface material structure of nerve cell samples.
3. FDTD simulation
Figure 3(a) shows the phase $\varphi _R^ - $ of the SOI scattering component $S_R^ - $ at three illumination wavelengths. $\varphi _R^ - $ was extracted directly from the field results under LCP TFSF illumination. The removal of the incident plane wave, without changes in the spin angular momentum, obviously will not affect the SOI scattering component $S_R^ - $. So that the phase $\varphi _R^ - $ and intensity $|{S_R^ - } |$ of the scattered field are also the phase and intensity of the total field. The non-SOI scattering component $S_R^ + $ is disturbed by the standing wave resulted from the simulation boundaries, and cannot obtained directly from the simulation. As shown in Fig. 3(b), we added back the incident plane wave and calculated the $\Delta {\varphi _R}$(GP) and $SIN{D_R}$ parameters from the interference intensity maps according to equation (3–8). Phase shifting step N is 10, and the superposition of interferograms reduces the impact of standing wave patterns. The transmission phase of such tiny network structures is very small and flat, so we have $\Delta {\varphi _R}\textrm{} = \varphi _R^ + \textrm{} - \varphi _R^ - \; \approx C - \varphi _R^ - $. The singularity points and rapid phase change lines are almost the same in $\Delta {\varphi _R}$ and $\varphi _R^ - $, with opposite phase gradient, as shown in Fig. 4(b). Dark stripes in $|{S_R^ - } |$, corresponding to low scattering intensity, should represent the phase changing lines and singularity points in GP. While $|{S_R^ + } |$ is relatively large and nearly constant, the low value of $|{S_R^ - } |$ will lead to the low value of $SIN{D_R}$, as shown in Fig. 3(b) (see Supplement 1). Figure 3(c) shows the enlarged parameter images of the red dotted box areas in 3b, as well as the original network structure image at the corresponding position. Phase changing lines and singularity points in GP, as well as dark stripes in $SIN{D_R}$, correspond to the edge contours of the network structure. Singularity points are located at the intersection points of the network structure, as shown in Fig. 3(c). Simulation results indicate that the skeleton structure of the scattering field is related to the edge characteristics of the network structure, which in turn reflects the structure and morphology of the network structure. This makes our SQII method different from common QPI methods that image sample thickness or height.
Fig. 3. (a) $\varphi _R^ - $ and $|{S_R^ - } |$ parameters and the of scattered field obtained by the simulated field data. The images are 18 × 18${\mathrm{\mu} \mathrm{m}}$. (b) $\Delta {\varphi _R}$ and $SIN{D_R}$ parameters obtained through a simulated scattering interference process. The raw value range of $SIN{D_R}$ is 0 to 1, corresponding to the grayscale range of 0 to 255 for 8-bit images. (c) Enlarged images of areas in b outlined by red dotted line. The area size is 4.8 × 3.6${\mathrm{\mu} \mathrm{m}}$. The same area in the 20 × 20${\mathrm{\mu} \mathrm{m}}$ 2D network structure is also placed for comparison. The red dot lines mark the dark lines in $SIN{D_R}$, the white dot lines mark the phase changing lines and singularity points in $\Delta {\varphi _R}$. These figures are corresponding to the structure edges of the network, as shown by the yellow dot lines in the 2D network structure.
Fig. 4. (a) RGB averaged $SIN{D_R}$ maps obtained by the simulated scattering interferometry experiment. The image size is 18 × 18${\mathrm{\mu} \mathrm{m}}$. (b) $\varphi _R^ - $, $SIN{D_R}$ and $\Delta {\varphi _R}$ curves plotted along the same location marked by the orange arrow in a. The plot length is 4.8${\mathrm{\mu} \mathrm{m}}$. There is an absolute phase difference between the curves introduced by calculation and measurement. The general trend of phase changes, or the gradient distribution of phases at different wavelengths, is consistent.
The scattering GP, which outlines the sample structures, is not sensitive to wavelength difference. The pattern skeleton of $\Delta {\varphi _R}$(GP) and $SIN{D_R}$ maps keep their shape and position when the illumination wavelength changes. This provides a way to reduce image noise and improve image quality through experiments under multi-wavelength. We averaged the $SIN{D_R}$ images at the three FDTD simulation wavelengths. Figure 4(a) shows the RGB averaged $SIN{D_R}$ images with higher contrast and stability. Figure 4(b) shows the $\varphi _R^ - $, $SIN{D_R}$ and $\Delta {\varphi _R}$ curves distributed along the 4.8${\mathrm{\mu} \mathrm{m}}$ orange arrow in Fig. 4(a). The intensity curve of the averaged $SIN{D_R}$ is clearer and more stable, and the intensity valley points (marked by vertical dotted lines in three curves) in $SIN{D_R}$ exactly mark the position of the maximum scattering phase change rate. Through FDTD simulation, we demonstrate our SQII method and prove its theoretical feasibility for observing PNNs-like network structures. The wavelength-insensitive property of the scattering geometric phase means that the SQII method does not require strictly monochromatic illumination, which avoids the loss of illumination light and reduce the imaging noise. It also allows us to combine measurements at multiple wavelengths to further improve the imaging quality.
4. Experiment results and discussion
We built up a SQII system according to the schematic diagram in Fig. 1(a) based on an Olympus BX51 microscope. The phase shift step N was 10 in the experiment. A slice sample of cerebral cortex cells was measured through a 100x Olympus achromatic plan objective with 0.9NA. Figure 5(a) shows a 55.5 × 37${\mathrm{\mu} \mathrm{m}}$ raw RGB transmission image of the sample, taken by the color SCMOS acA4024-8gc. An interferogram of the upper surface of the sample was obtained with reflection DIC method for reference. The sliced cells were covered with other tissue structures and solid solutes, making it difficult for common QPI methods to work. The SQII method based on scattering process is less affected by this. The measurement error mainly comes from the slight difference in half mirror’s reflectivity for o-light and e-light, and angle deviation of the quarter-waveplate (see Supplement 3). The experimental system uses a mechanical modulator, and needs 5-10 seconds in total for the 10 step phase shifting and imaging. The original interferograms obtained in the experiment can be found in Supplement 2 and in Visualization 1.
Fig. 5. (a)Raw transmission image of a Cerebral cortex slice sample and a reflective DIC image of the sample upper surface. The image area is 55.5 × 37${\mathrm{\mu} \mathrm{m}}$ and the enlarged area is 18.5*18.5${\mathrm{\mu} \mathrm{m}}$. (b) RGB channel $SIN{D_R}$ and $\Delta {\varphi _R}$ maps corresponding to the enlarged area in a. The dynamic range of the R and G channel $SIN{D_R}$ images have been adjusted for better display. (c) Enlarged $SIN{D_R}$, $\Delta {\varphi _R}$ and DIC images corresponding to the area with a red dotted border in b. Note that the red and blue pseudo-boundaries outlined by the white dashed line represents a smooth slow change in phase from 2$\pi $ to 0, not fast phase change.
Figure 5(b) shows the enlarged $SIN{D_R}$ and $\Delta {\varphi _R}$ maps of RGB channels in the 18.5 × 18.5 $\mu m$ area with a red dotted border in Fig. 5(a). The halogen lamp of our system has very little blue spectral component. The signal of the B channel is weak and the signal-to-noise ratio is low. As a result, the variation of intensity with phase shift is no longer sinusoidal, the measured $\Delta {\varphi _R}$ (GP) is blurry, and the $SIN{D_R}\; $ loses its connotation with scattering contrast (see Supplement 2). Compare Fig. 5(b) and Fig. 5(a), patterns in $SIN{D_R}$ of B channel are mainly related to the overall thickness and surface morphology of the sample.
The R and G channel images have stronger signal and are less affected by noise. Compared with B channel image, R and G channel images contain detailed features of scattered fields related to the sample structure. It is hard to identify the singularity points and phase changing lines directly in GP images. The phase distribution measured experimentally, distracted by the complex background of actual samples, is not as regular as the simulation. But the dark stripes in the $SIN{D_R}$ parameters can still be clearly identified. The complex distribution of the geometric phase increases the difficulty of phase interpretation in actual experiments, so the interference intensity is of great significance for scattering quantitative interferometry.
In the DIC image, the surface structure distribution is relatively uniform. However, in the upper right corner of the images Fig. 5(b), the higher $SIN{D_R}$ values indicate strong scattering in this area, which was not reflected in the DIC image. Several $SIN{D_R}$ dark stripes, GP change lines and singular points was found, as shown in Fig. 5(c), which do not belong to the surface structure of the sample, compared with reflective DIC results. This situation shows that there are strips or mesh like structures with strong scattering properties inside the sample, such as PNNs.
In Fig. 5(b) and 5(c), by averaging the R and G channel $SIN{D_R}$ parameters, a more stable and clear network structure distribution is obtained. Due to poor signal quality, the B channel data cannot reflect the scattering characteristics and therefore does not participate in the averaging. Table 1 reflects the information entropy of the $SIN{D_R}$ image of each RGB channel in Fig. 5(b). Information entropy is a parameter that evaluates the richness of image information, including image noise [34]. Therefore, higher information entropy usually means greater image noise, such as the B channel $SIN{D_R}$. The R and G channels have lower entropy values and noise. The information entropy of averaged R and G channel $SIN{D_R}$ decreased, which shows that the scattering field features in these two channels $SIN{D_R}$ images are consistent. The superposition process highlights the features and reduces the noise. The average information entropy of the RGB averaged $SIN{D_R}$ does not further decreased. This shows that B channel $SIN{D_R}$ has high noise level but lack scattering feature, and should be excluded.
Table 1. Information entropy of ${\boldsymbol SIN}{{\boldsymbol D}_{\boldsymbol R}}$ imagesa
The consistency of the scattering features in the R and G channel images proves that the experimental results are indeed caused by scattering phase instead of propagation phase or the polarization phase. Theoretically, treating the sample as a material with polarization birefringence properties can also explain the experimental phenomena under monochromatic conditions. However, polarization birefringence properties are sensitive to wavelength changes and cannot explain the consistency of phase distribution under different colors. Our SQII method provides an effective means for observing tiny biological scattering structures from the principle of scattering geometric phase.
5. Conclusion
The phase singularity points, where the phase gradient is infinite, act as the skeleton of microstructures’ scattered fields. Measuring the singularity points and the gradient large value lines of the scattering geometric phase can help the observation of many kinds of microstructures. Through simulation, we proved that the singular points and gradient large value lines of the scattered GP $\Delta {\varphi _R}$ outline the edge of the PNNs-like network structure, although there might be a drift in the position. The position of these points and lines can also be located by the valley area of interference contrast $SIN{D_R}$. We proposed a SQII method based on geometric phase shifter to measure the interference geometric phase $\Delta {\varphi _R}$ and interference contrast $SIN{D_R}$. The SQII method uses the geometric phase shifter to simplify the phase shifting operation. SQII method is only sensitive to scattering features and can be used on sliced samples with structural disturbances, which the QPI method is not applicable to. The image quality and stability of $SIN{D_R}$ parameters are further improved by averaging the $SIN{D_R}$ in RG channels since the singularity of the scattered field caused by the structure is not sensitive to wavelength. The experimental results of the brain cortical cell section sample confirm the existence of scattering features in $SIN{D_R}$ and $\Delta {\varphi _R}$ images, and preliminarily prove the practicality of our method for the observation of micro-network structures like PNNs. The stability of results under different color channels indicated that the experiment cannot be explained by polarization birefringence of the sample, but explained by the scattering interference process and the geometric phase. This is where the theoretical value of our SQII method lies.
Our SQII method is easier to deploy, reduces the cost of the microscopy imaging system, and can contribute to the development of histology and brain science. However, current work is no thorough enough. Detailed performance of our method requires further calibration and research. The impact of non-ideal lighting conditions and unexpected scattering from complex structures within the sample on the measurement results also shall be explored in the future.
Funding
National Key Scientific Instrument and Equipment Development Projects of China (No.61827814); National Natural Science Foundation of China (No. 62105155); Ministry of Science and Technology collaborative project (B17023); Natural Science Foundation of Jiangsu Province (BK20210326).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
Supplemental document
See Supplement 1 for supporting content.
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