Edge optical scattering of two-dimensional materials

Rayleigh scattering has shown powerful abilities to study electron resonances of nanomaterials regardless of the specific shapes. In analogy to Rayleigh scattering, here we demonstrate that edge optical scattering from two-dimensional (2D) materials also has the similar advantage. Our result shows that, in visible spectral range, as long as the lateral size of a 2D sample is larger than 2 {\mu}m, its edge scattering is essentially a knife-edge diffraction, and the intensity distribution of the high-angle scattering in $\mathbf{k}$ space is nearly independent of the lateral size and the shape of the 2D samples. The high-angle edge scattering spectra are purely determined by the intrinsic dielectric properties of the 2D materials. As an example, we experimentally verify this feature in single-layer $MoS_{2}$, in which A and B excitons are clearly detected in the edge scattering spectra, and the scattering images in $\mathbf{k}$ space and real space are consistent with our theoretical model. This study shows that the edge scattering is a highly practical and efficient method for optical studies of various 2D materials as well as thin films with clear edges.

scattering signals, such as Raman scattering and photoluminescence, and possesses a much higher contrast than that of reflection and transmission images. Due to these unique advantages, Rayleigh scattering has become a powerful method for nanoparticle imaging [1] and disordered potential studies in two-dimensional (2D) electron gas [2,3].
Traditionally, it is often considered that Rayleigh scattering can only be applied to zerodimensional (0D) samples. But recently, it was successfully applied to one-dimensional (1D) materials in which the scattered light has the similar behaviors to 0D cases. One of the most successful applications is the optical studies of individual single-walled carbon nanotubes, such as chiral index identification [4][5][6], optical imaging [7], excitonic optical wires study [8], and interaction between nanotubes [9]. In principle, further generalizing the Rayleigh scattering method to 2D structures is infeasible, because 2D coherent dipoles excited by the incident light can only generate reflected and transmitted light. For this reason, to the best of our knowledge, the dielectric properties of 2D materials can only be measured by reflection or transmission spectra, such as ellipsometry [10][11][12][13] and differential reflection method [14]. Edges of 2D materials are considered to be another kind of structure other than 1D and 2D ones, which often possess significant and unique physical properties different from bulks [15][16][17][18][19]. Unfortunately, similar to 0D and 1D structures, the optical reflection and transmission measurements of the edges suffer from the low response compared with the background. However, the edge also has scattering signal because the broken translational symmetry provides effective momentum compensation necessary for the light scattering. In this paper, we show that the high-angle edge scattering signal is exclusively determined by the intrinsic dielectric properties of the sample's edge.
Furthermore, in visible spectral range, it is approximately independent of the lateral size and the shape of the 2D sample as long as its lateral size is larger than 2 μm (4, where  is the wavelength of the scattered light). This requirement makes edge scattering nicely suitable for almost all 2D samples that could be obtained in laboratories. We also demonstrate an experimental setup for high-angle edge optical scattering measurement in real space and k space. Under this setup, A and B excitons' resonance peaks are unambiguously resolved from the edge of monolayer MoS2 flakes.
Our results show that the edge scattering could be a powerful and convenient method for the optical studies of various 2D materials' edges as well as thin films with clear edges. It is also promising for a variety of characterizations in 2D systems such as grain/domain boundaries, phase separation/transition, interlayer coupling, topological edge states, heterojunctions, and so on, as long as the discontinuity of dielectric properties exists.

Ⅱ Theoretical result
The theoretical model considers a ribbon sample with width of 2a ( Fig. 1(a)) and infinite length, which is placed on a flat substrate. The thickness of the ribbon is less than onetenth of the incident light wavelength and thus has no effect to the scattering in this model. A parallel beam of light with normal incidence to the substrate covers the sample entirely. The difference between the dielectric constants of the sample and the substrate leads to the corresponding discontinuity of the dipole field. We consider the contribution of the dipole field of the sample to the outgoing light in two parts. One part of the contribution is equal to the effect of the substrate, which includes reflected and transmitted light. The rest part of the contribution is the scattered light from the sample, which is related to the difference between the dipole field of the sample and the substrate, as indicated in Fig. 1(a). Far-field radiation generated by this contribution of the dipole field of the sample can be analogized to single-slit diffraction with slit width of 2a. When the width of the ribbon sample is infinite, the edge scattering observed at an edge of the sample is equivalent to the knife-edge diffraction. In order to simplify the calculation, we first consider the dipole moments of the sample and the substrate as a scalar. The distribution of far-field radiation in k space can be derived from the Fourier transform of the dipole field. The strength distribution of the difference between the dipole field of the sample and the substrate can be written as a scalar field function D(x), where A is the complex amplitude of the field. The dipole field is translational invariant in y direction, thus is y independent. If the dielectric constant of the sample is much larger than that of the substrate, = ( − ) ≈ , where and are the dielectric constants of the sample and the substrate respectively, E is the complex amplitude of the incident light field, and is the electric susceptibility of the sample. When the light is normal incident to the sample, the momentum of the dipole field in x-y plane is zero and A is independent of the spatial coordinates. Fourier transform of D(x) is which is plotted in Fig. 1(b). Since the far-field radiation intensity is proportional to the square of the dipole oscillation frequency and its intensity, the distribution of the electric field strength in k space, E(k), is proportional to F(k) where is the angular frequency of the electric field, is a constant related to dipole radiation. It can be considered as a sine oscillation function with 1/k amplitude modulation. Sin(ka)/k function is plotted in Fig.1(b) while a=5 and 10μm. When k is equal to 0, there is a main peak with the amplitude of Aa/π, which contributes to back and forward scattering. The other peaks are slightly affected by the sample width parameter a, including the part of our attention that generates high-angle scattering.
Intuitively, high-angle scattering intensity does not depend on the lateral size.
Especially when a is large enough, F(k) becomes a high-frequency oscillation in k space and therefore can be replaced by a mean value. The intensity of high-angle scattering is given by the integration of the range (k1, k2) where k1, k2>0, where sin 2 (ka) is approximately 1/2. It is clearly that ( 1 , 2 ) after approximation is independent of the lateral size 2a. Taking the integration interval as (10μm -1 , 20μm -1 ), we compare the numerical integrated result without approximation with the integration after approximation in Fig. 1(c). As a increases, the intensity without approximation gradually converges to the intensity after approximation. When a>1μm, the difference between two results is less than 5%. This means that the intensity of high-angle scattering gradually become independent of the lateral size 2 a as a increases.
Next we demonstrate that the far-field high-angle scattering is mainly induced by the edge. It can be obtained by its high-angle real space image, that is the inverse Fourier For the finite integration interval (k1, k2), there is no analytical solution to Eq. (5).
Integrating over the range (10μm -1 , 20μm -1 ), we give the numerical integration of  Fig. 1(e) that the phase distribution of ( 1 , 2 ) ( − ) around the edge is independent of the lateral size 2a. Therefore we can conclude that the dipole field in real space ( 1 , 2 ) ( ) around the straight edge is weakly affected by the lateral size and shape of the sample. This means that in practical applications, similar to Rayleigh scattering method, it is generally not necessary to consider shape and size factors when extracting the dielectric information from the high-angle scattering spectra. In the theoretical model, we considered the scenario of high-angle scattering under normal incidence. However, for convenience, we use grazing incident beam in experiment so that the collection of the high-angle scattering can be almost normal, as illustrated in Fig. 2(a). The incident beam is in the x-z plane, with incidence angle θ After a variable substitution = + 0 , the Fourier transform of ( ) can be the same as Eq. (2). Since the collection range of objective lens is (− 0 . . , 0 . . ), the integrating interval of the high-angle scattering intensity as Eq. (4) becomes ( 0 − 0 . . , 0 + 0 . . ) after the variable substitution. In practice the incident plane is fixed in the x-z plane, however, the flakes are randomly oriented on the substrate so there is always a non-zero angle between each edge of those flakes and the y axis as shown in Fig. 2(d). Therefore, the scattering obtained above should be further generalized. Considering an edge with an infinite length which has a non-zero angle α to the y axis, the edge scattering in k space must be perpendicular to the edge in real space, which is similar to one-dimensional Rayleigh scattering in Ref. [8]. The wave vector of the reflected light in the ( , ) plane is ( 0 , 0), regardless of the orientation of the edge. Fig. 2(e) shows the illustration of the scattering distribution in k space. Because the collection area of the objective lens is the area of a circle with center on the origin and radius 0 . ., the integrating interval of the high-angle scattering intensity is in this configuration. And the distance between the origin and scattering line is = 0 . Fourier transformation analysis of the 2D dipole field, can also give this result, which is further derived in Supplemental Material [20]. The integrating interval of the high-angle scattering intensity in Eq. (7) also indicates that the angle |α| between the edge and the y axis must be less than αc=arcsin(N.A./sinθ), otherwise the edge scattering will be out of the objective lens collecting range, as shown in Fig. 2(e). Then, If the electric susceptibility of the sample is anisotropic, is a complex tensor = ( ).
Here we discuss the scenario where the material has a uniaxial crystal structure, which is the case for a wide range of 2D materials are uniaxial crystals, such as MoS2. Their electric susceptibility can be written as the complex tensor below, where the coordinate system is defined as in Fig. 2 where ′ ( , ) = ( , )/| |. It's easy to obtain the spectra of | | 2 from the edge scattering excited by s wave. Deriving out is more complicated, because the edge scattering excited by p wave is determined by two complex numbers and out . Therefore, in principle and can be derived from the experiment data with the scattering distribution in k space described in Eq. (19).

Ⅲ Experimental result
In in the scattering images in most of our cases. Fig. 3(a) shows the typical optical microscope image of the samples. The corresponding edge scattering image is shown in Fig. 3(b), besides the greatly enhanced contrast, one can also find that no more than one edge in each flake is "bright". To further study the scattering distribution in k space, an edge of a specific flake is studied by the Fourier transform system with the help of an aperture in real space. The optical path diagram is detailed in Supplemental Material [20]. Fig. 3(c) shows the intensity distribution image in real space of the edge scattering from a single flake after the aperture, and Fig. 3(d) is the corresponding intensity distribution image in ̃ space, where ̃= ⃗⃗ / 0 . It is clear that the scattering lines in  The edge scattering spectra can be directly collected by a spectrometer. An example spectrum is shown in Fig. 4(a), with the optical path factors, such as the incident light spectra and the optical loss spectra of the optical elements been corrected. It is clear that A and B exciton peaks are unambiguously resolved in both s wave and p wave scattering spectra. The | ( )| 2 is shown in Fig. 4(b), which is derived from the s wave scattering spectrum in Fig. 4(a) with the scattering function in Eq. (19). between two edges of a sample is less than the optical diffraction limit, those two edges can still be clearly resolved with the larger kx, such as in evanescent wave excitation [21]. Therefore in this case, the edge scattering could also have a super resolution beyond the optical diffraction limit, which could be implemented to resolve ultranarrow gaps of 2D materials.
In conclusion, we theoretically demonstrate that the high-angle edge optical scattering of 2D materials is independent of its lateral size and shape, and hence the dielectric

Section B: Fourier transform of two-dimensional residual dipole field
The dipole field of the ribbon, Fig S2(a), is written as is the same with that in main text, is incident angle of oblique incident angle of exciting laser. If we use new coordinate ( , )  Section E: Corrected spectra of scattered light with exciting laser spectrum, instrument optical loss, object lens collecting range and dipole polarization.
The spectra, detected by CCD2 should be corrected by the laser beam spectrum and optical loss spectra of optical element in optical path that are shown in Fig. S5 (a) and (b) respectively. To extract | ( )| 2 spectrum, angle distribution correction factor should be calculated first. S wave scattering intensity collected by the object lens is