1. Introduction

Monolayer and multilayer two-dimensional graphene materials, which have only recently been implemented exhibits extraordinary electronic, mechanical and thermal properties, and hence have quickly become an extensive exploration field of world-wide research. The lateral dimensions of graphene materials can reach tens microns, but the thickness is however only in the order of single atomic scale. Owing to the optical diffraction occurred in conventional optical microscopes, their resolutions can hardly reach the level that meets the requirements of analyzing graphene’s dimension features and other physical characteristics such as edge shapes, wrinkles, defects, impurities and grain boundaries. Whilst non-optical techniques such as scanning tunneling microscope (STM) [1], high resolution transmission electron microscopy (HRTEM) [2], photon induced near-field electron microscopy (PINEM) [3] and atomic force microscopy (AFM) [4] are capable of characterizing graphene layer thickness as well as other structural characteristics, they all form image by point-scanning, and their visual field sizes are only from less than 1 nm to few microns, too small to meet highly needed measurement efficiency requirement.

To overcome the difficulties in optical resolution improvement, many efforts on optical microscopy have been made, including optimizing SiO2/Si substrates to improve contrast built on Fresnel reflection [5, 6], using contrast spectroscopy and improved refractivity to build empirical thickness-contrast relationship for variable thickness quantification [7–9]. Fluorescent quenching microscopy [4, 10], on the other hand, can provide much higher contrast and even possibly higher spatial and structural resolving power than normal microscopy. It however involves applying extra chemicals on thin graphene to produce fluorescence for imaging. As a consequence, samples under measurement (SUM) will be undesirably changed by, e.g., adding coating for optical fluorescent quenching, oxidizing underlying metals [11], modification liquids [12] to tune the refractivity to improve contrast and condensing micron droplets [13] around the graphene flakes to improve the visibility. These methods are therefore cumbersome, damage prone and the resolution improvement is still clearly limited by the optical diffraction effect. Although Raman spectroscopic imaging has the necessary optical contrast for graphene layer differentiation, it cannot avoid the above mentioned diffraction effect and has relatively lower lateral resolution [10]. Like those non-optical imaging techniques, all the optical methods mentioned above rely on point-scanning to form image and therefore the problem of low imaging efficiency remains unresolved.

Many resolution improvement researches have been done based on wide object field and far image field, one of the prominent advancements is to image the sub-nanometer sensitivity [14] and nanometer resolution [15] indirect vector parameters by modulating the variation of far field point spread function (PSF) with varying polarization status, such that the anisotropic spatial scattering and wavefront aberration can be quantified [16–18]. In these methods, the PSF is narrowed by modulation and filtering, so that higher image sensitivity and resolution are achieved [19–21]. We are here to report polarization (parameter) indirect microscopic imaging (PIMI), an indirect optical imaging technique for graphene inspection that is capable of reaching sub-100 nm super-resolution, breaking the diffraction limit, and has the potential to reach even higher resolving power.

2. Theoretical basis

Instead of utilizing the direct optical field intensity, in which the resolution is intrinsically limited by the optical diffraction, PIMI indirectly obtains the contrast built by the indirect optical wave parameters obtained from modulated optical field. In this paper these optical wave parameters are confined to polarization parameters that define polarization status, including polarization phase, orientation [22, 23], degree of polarizations, and depolarization field intensity [24]. In comparison with field intensity, polarization parameters are more sensitive to anisotropic structural variations such as the change of graphene layer overlapping, boundary defect characteristics, sheet wrinkles and structural difference between sheet and substrates. By establishing the relationships between the structural anisotropy and the far field intensity variation, we can indirectly obtain multiple images with contrasts built using the above polarization parameter varieties. All these images, though which can be used independently and have different spatial resolving power, are compensative to each other to reveal more detailed metrological and structural information about the SUM [22–24].

In order to understand the effect of near field coupling on far field PSF, a model is developed to simulate the field distribution from an anisotropic point source illuminated with polarization-modulated light source. As is shown in Fig. 1, in this model we used gratings structure to fill in a single source point to generate strong structure anisotropy (Fig. 1 (a)), and through a standard microscopic light path reaching the image plane where ray tracing produces the Huygens PSF (one dimension section shown as any of the curves in Fig. 1 (b)). When the PSF sectioned along the illumination polarization direction, its field intensity-pixel position curve will vary its shape with both the near field anisotropy and the status of the illumination polarization. If the polarization status is modulated, one can sinusoidally formulate the intensity variation in each spatial point that the PSF covers. Reversely, one can use a known structure to check if the derived intensity variation follows the coupling principal. To make the right judgment, one can first fit the measured (or simulated) intensity variation to the formula (Fig. 1(c)) and use the fitting merit index, such as the adjusted-root-square (Adj-R-Square), to filter out those spatial points which have Adj-R-square smaller than the setting value, e.g. 0.95 (or 95%). Through a sum of various diffraction in the microscopic light path, there can be an abroad PSF that covers many (e.g., 60 in FWHM in Fig. 1) pixels of which only a few at the center (7 in Fig. 1) meet with the fitting criteria to the given formula, that are considered authentic points. The other points on the other parts of the PSF do not follow Eq. (1) and are considered as measurement noise that can be filtered out by the above fitting and filtering method. As a result, the PSF is narrowed and the diffraction limit is broken. In the approach to establish the correct formula for a right coupling effect between a known structure symmetry type (or some characteristics) and the selected polarization variation, one can use the Jones and Mueller formulas explained in the following paragraph and more details will be published in other papers by the authors soon. With the fact that materials structural symmetry type of a SUM, which is graphene in this occasion, is normally well defined and can be easily found by pre-measurement scanning (such as using conoscopic imaging), using the above diffraction limit broken method to improve indirect image resolution of graphene microscopy is well applicable. By accurately tracking and measuring the variation of modulated light field at the image plane, one can retrieve the polarization parameters at each graphene objective field point to form the indirect images reflecting the near-field morphology or structure characteristics.

 

Fig. 1 Illustration diagram of (a) an exaggerated round spot on an anisotropic SUM, (b) the variable Huygens PSFs from (a) under the illumination of various polarization, (c) the field-polarization data fitting to sinusoidal variation at pixels at the PSF shown in (b), and (d) the adjusted R-square value (for fitting merit indication) varying with pixel position, i.e., only a part of pixels (pixel47-52) qualify the best fit value (>0.95) and can be selected for being associated with the object point shown in (a), those pixels happened to be at the center of the unpolarized PSF (Unpol in b)

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Using the geometric optical coordination system, polarization orientations illustrated in Fig. 2 and the Jones model [14, 15, 22, 23], one can find that during polarization sweeping, the pixel intensity, I_i, depending on the vector sum of two wavefronts of different polarization states from the intrinsic structural anisotropy of the SUM, varies with input polarization angle α. This dependence can be described by Eq. (1) in a simplified form, where I_dp is the average intensities over all polarization status transmitted through or reflected from each field points of the SUM, so its value least affected by scatterings near field change of polarization status, a₀ = 1/2, a₁ = cos2φsinδ, a₂ = -sin2φsinδ and sinδ is the sine of phase difference between two orthogonal polarization states, which is usually small for single graphene perturbation and increases dramatically for increasing graphene layers, φ is the polarization angle of the slow axis.

 

Fig. 2 Diagram of the coordinate system, ray and polarization directions.

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The field intensities at individual pixel of the CCD matrix were recorded while sweeping the polarization which is achieved by rotating a linear polarization disc in a control module. The intensity variation with parameters, such as the polarization phase and angles were fit to Eq. (1) to derive the variation curve (Fig. 1(c) and Fig. 3(b)). From the derived curve, the values of the parameters such as I_dp, sinδ and φ can be extracted, and during this process, spatial filtering for resolution and sensitivity improvement illustrated in Fig. 1 is automatically carried. From the above polarization parameters, using the relationship between Jones’s and Mueller’s model, the Stokes parameters, S₀, S₁, S₂ and S₃ can be further calculated with Eq. (2) [25].

 

Fig. 3 Illustration diagram of (a) modulated polarization parameter imaging system, (b) measured points and fitted curve of single pixel intensity variation.

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Where E_x and E_y are the X and Y optical field components. Stokes parameters, S₀, S₁, S₂, S₃ along with the depolarization intensity, I_dp, phase difference, sinδ, and polarization angle, φ can be retained to each pixel coordinates in the sensor space to form individual parameter maps, or indirect images as we call them, with the contrasts not built by the optical intensity but the spatial polarization parameters related to near field graphene structural characteristics.

3. Experimental details

Based on the above theory, we design an indirect polarization microscope with modules including the conventional microscopic light path, polarized illumination, polarization control of the incident light, optical image field detection matrix (CCD) and the control and analysis interface.

As is illustrated in Fig. 3(a), in PIMI measurement, under the narrow-line width (< 20 nm) 532 nm illumination, the polarization sweeping [14, 15, 22–25] and the CCD sensor matrix imaging were synchronized by automatic positioning and moving control. To reduce the systematic error, the system was carefully calibrated.

In the actual experiments, PIMI system illustrated in Fig. 3 was built by using an Olympus reflection microscopic system BX51M as the basic optical microscopic path, and inserting into it with polarization-modulation mechanics with the angle precision of 0.05 degree. The optical intensity data collection monochrome CCD in the system was manufactured by Basler (piA2400-17gm) with pixel resolution of 3.45 micron and output dynamic range of 12 bits.

The measurements were performed for a variety of graphene samples including chemically prepared and CVD made graphene on different substrates including glass slides, oxidized silicon and silicon wafers. The measurements were carried out with both bright and dark illumination formats. The polarization status of the microscopic illumination was modulated by a mechanically controlled linear polarizer, and therefore could be precisely tuned over the whole 2π, where half circle was to cover all possible polarization spatial angle range, full circle for system error elimination.

For a comparison between polarization parameter indirect imaging and conventional direct microscopic imaging, we show a set of resultant images of chemically prepared graphene flakes on glass slide in Fig. 4. In the case of bright field reflection microscopy, graphene produced stronger reflection than glass substrate. The more layers a graphene sheet had, the stronger the reflection was, and the brighter the features in the direct images were. However, from the experiments we observed that, (1) contrasts between different layers of thickness in direct (I₀) image (Fig. 4(a)) was still too weak to visualize the layer by layer difference; (2) the polarization field average (I_dp) image (Fig. 4(b)) could provide the visibility to differentiate even a single layer; (3) sensitivity to variation of graphene features in direct image was also much weaker than that in phase difference (sinδ) image (Fig. 4(c)), only the latter led to many thinner flakes, residual particles, edges, wrinkles etc. visible; and (4) compensatively, with the help of the angle of slow axis polarization (ϕ) image (Fig. 4(d)), we could further determine the orientation of flakes with different average atomic arrangements through those multilayer mesh-structured graphene sheets.

 

Fig. 4 Bright (a to d) and dark (e to f) field images taken by the modulated polarization parameter imaging system using 100x objective, direct grey scale image of a chemically prepared graphene overlapped to 1 to 6 layers on a glass slide (a, e), its indirect polarization parameter images of polarization averaged intensity I_dp (b, f), polarization phase difference (c, g) and polarization angle of slow axis ϕ to horizontal axis (d, h)

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As also shown in Fig. 4 on the same graphene sample, in the case of dark field microscopy, optical field collected by microscopic objective were mostly from flake edges, wrinkles and impurity particles of graphite residuals. Direct dark imaging did not collect enough scattering light for building up necessary contrast (Fig. 4(e)) so that feature miscount had been inevitable. On the contrary, the indirect polarization field average image (Fig. 4(f)) picked up much more scattering features from multilayer graphene, also, phase difference contrast image (Fig. 4(g)) resulted from collecting phase changes during polarization scan contained stronger image contrast. Since most of the flake edges, particle residuals and wrinkles were structurally anisotropic, scatterings were directional and polarization status were spatially abruptly variable. The sensitive phase difference image, with resolution being much higher than the direct gray scale image, made more particles visible, and contained not only the scattered features but also their scattering format which was considered as useful information for retrieving the shape and molecular arrangement characteristics in even higher resolution [19–21]. With even higher sensitivity than phase difference to anisotropic field variation, polarization angle contrast imaging picked up more near field scattering spectra to form the image (Fig. 4(h)) filled with characteristic scattering noise, which was however highly desirable for scattering modeling and edge characterization.

Using the same indirect super resolution optical microscopic imaging system shown in Fig. 3, graphene flakes on different substrates, from glass slides to oxidized silicon substrates, overlapping from 1 to 3 layers, had also been imaged and analyzed. Figure 5 showed comparative results between direct and indirect imaging of graphene flakes prepared by micromechanical cleavage and transferred to a Si wafer with a SiO₂ cap on top. It was clearly evident (Fig. 5 (a)) that, at the same noise level, both sensitivity and contrast of polarization parameters such as I_dp were a few to more than 10 times higher than that of field intensity. As a result, layer numbers, which are difficult to differentiate in normal direct optical microscopy (the I₀₀ curve in Fig. 5 (a)), can be easily counted for in the contrast variation curve of the polarization parameter alone line pixel positions in an indirect super resolution image (the I_dp curve in Fig. 5 (a)). The validation of the layer counting by the indirect polarization imaging had been confirmed by high contrast [7, 8] Raman spectroscopic analysis taken on the same sample. As shown in Fig. 5 (b), from monolayer, through bilayer to trilayer, the measurement made by indirect super resolution microscopy was exactly identical to what were measured by the contrast variation built by spectroscopic intensity [7].

 

Fig. 5 Imaging analysis result of chemically prepared 1-3 layers graphene flakes on silicon substrate,(a) polarization parameter I_dp and direct image intensity I₀₀ pixel variation curves through the line intersecting graphene flakes of different overlapping thickness; (b) measured Raman spectra of the flakes on the same sample, its peak intensity variation confirms number of overlapping layers.

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4. Discussion

As described in previous section, PIMI obtains optical images indirectly by analyzing optical intensity dependence on polarization parameters, and then extracts sample’s structural or compositional information. During measurements, the incident light with spectrum line width (<20 nm) is modulated by a polarizer with a precision of 0.05 degree, and as such the errors due to the system arrangement are greatly suppressed. As we have shown, PIMI enables one to obtain optical images with much higher resolution than that obtained by conventional optical microscopes. In a conventional optical microscope, its resolution is ultimately limited by the system’s diffraction. PIMI however adopts modulated scanning on one or more polarization parameters, such as the angle of polarization perpendicular to the optical axis, while continually measuring the optical field strength variation at all image point represented by pixels in the CCD matrix. The field variation is then recorded and fitted with the impact of near field polarization status at the corresponding sample points with anisotropic structures. It has been shown [16–21, 25, 26] that this local polarization status, at the near field right after the graphene SUM, will vary with the incident polarization and determine the shape of the PSF (Fig. 1 (b)). As can be seen in the above image retrieval procedure through fitting (Eqs. (1) and (2), and Fig. 1(c)) and filtering (setting fit merit level in Fig. 3(d)), those otherwise too broadly spread spatial light [21] in the PSF has been made unrelated to the near field source point. As a result, the PSF effectively shrinks [18–20], or in the other word, diffraction limit is broken and super resolution is achieved. Consequently, the measurement sensitivity and resolving power to the local structural change increases, e.g., it can to its best take only about one to two CCD pixels, i.e., about 30-70 nm in image space to sense the edge of a monolayer graphene (I_dp curves in Fig. 5(a) and Fig. 6(b), and S₀ curve in Fig. 6(c)). Otherwise, it took rather more than 30 pixels, i.e., about 900-1500 nm in direct imaging (Fig. 5(a) I₀₀ curve and Fig. 6(a)) to effectively sense the edge.

 

Fig. 6 Image quality and resolution analysis result of direct (a), and PIMI I_dp (b) and S₀ (c) images from a monolayers graphene in the sample.

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To make the image visibility comparable, this paper applies orientation certainty level (OCL) and mean square error (MSE), root means square error (RMSE) calculation [27, 28] to the direct and indirect images. We have found that even if the monolayer graphene, with the thickness at 0.34 nm, is too thin to accumulate enough near field structural perturbation to the polarization status, and indeed the image contrasts of some of the polarization parameter such as phase difference (sinδ) and Stokes 1 to 3 (S₁ to S₃) are roughly the same as the direct imaging image (I₀₀), however, under the same noise level, the image contrast and the orientation certainty level of one or two outstanding parameters such as the polarization averaged intensity (I_dp) and total degree of polarization (S₀) have about 3 times more contrast than direct imaging image. According to Eq. (1) and (6), I_dp and S₀ may combine the sensitivity of sinδ, φ and all Stokes parameters, and null off their data noise during curve fitting, so that at the critical image position such as the edge of a monolayer graphene, the mean square noise is about one order smaller. Low noise PIMI data also helps to achieve the above mentioned higher edge resolution (Fig. 6). The high sensitivity and resolution in PIMI also increase the easiness of imaging, which explains why we can image the details of thin graphene sheets on both glasses and silicon substrates without the need of special treatment or care on them (on glass in Fig. 4 and on silicon in Fig. 5). Further image resolution analysis should consider building up an image data platform so that image quality index such as sensitivity, spatial resolution, composition and structural characteristic resolution can be compared more authentically.

Because of the improvement in resolution and sensitivity, PIMI images, in replacement of the lower resolving power direct gray scale image (Fig. 7(a)), can compensatively provide grain boundary analysis on graphene flakes made by CVD deposition. On the upper part of Fig. 7, the images of a graphene on a glass substrate are shown, where CVD grains with proximately hexagonal shape adjoin together to form a relatively larger flake, and many of the flakes adjoin to form the large sheet. With all 4 images, we can compensatively found the edges of the flake, which contain grains of different layers, particles of broken graphene or chemical residuals. More importantly, with PIMI, some of the boundaries between grains (about 2 μm) can now be clearly seen. On the bottom of each image, the line variation plot between corresponding parameter and pixel position, confirms that the gray scale intensity image, I₀₀ (Fig. 7(a)), fail to reveal any difference between grains, but I_dp and S₀ curves (Fig. 7(b) and 7(c)) show evidently the location of the grain boundaries and also a continual variation of the thickness from the center to the boundary of each grain. It is also worthwhile to pointing out that polarization angle image (Fig. 7(d)), which is considered sensitive to atomic arrangement change, picks up all structural or orientation differences for particles and flakes but not the grain boundary, which might just indicate that the structure between grains remained unchanged. Further investigation on the grain’s inner structure and structure changes at the boundary need to be conducted with improved optics, including with reduced chromatic aberration (narrowing bandwidth (currently 20 nm)) and less environmental vibration (currently not isolated) and more parallel experiment of other high resolution image techniques.

 

Fig. 7 Line pixel variations of direct gray scale image I₀₀ (a), PIMI images of polarization average intensity I_dp (b), total degree of polarization S₀ (c), angle of polarization ϕ (d), and the I_dp at a grain boundary (e) along the intersection lines across CVD made graphene grains in corresponding images (above the curves).

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5. Conclusion

In conclusion, we have developed a new method that utilizes polarization parameters in a conventional wide field reflection microscope with additional polarization modulation control and imaging data analysis, to indirectly image and analyze graphene flakes. The PIMI with the resolution improvement beyond the diffraction limit can conveniently provide up to 10 times more resolving power on graphene flakes than that of a conventional microscope. This method also has much higher sensitivity to graphene overlapping layers, edges, impurity particles, wrinkles and also grain boundaries. By using the PIMI technique, we were able to visualize graphene and make further thickness and structural analysis regardless of substrate types.