The ability to control the light-matter interaction in the deep subwavelength regime has fascinating consequences for material sciences and a range of photonics technologies. The statistical properties of emerging fields strongly depend on both the source of radiation and specific characteristics of the material system. Here we consider the coherence properties in the proximity of surfaces illuminated by strongly randomized optical fields and demonstrate that the spatial extent of near-field correlations depends on the density of defects in two-dimensional crystalline lattices. We also show that this effect can be used as an efficient elastic scattering method to characterize the density of defects in two-dimensional crystalline materials. Systematic experiments demonstrate the relationship between the spatial coherence length of scattered light and a characteristic length associated with structural disorder in graphene. The fact that one single layer of atoms can modify properties of electromagnetic radiation could lead to new means of controlling light at subwavelength scales and has implications for the emerging field of two-dimensional materials.
© 2017 Optical Society of America
Conventional coherence theory describes universal properties of fields radiated by statistically homogeneous sources . In practice, however, sources have finite extents, their particular structure cannot always be neglected, and sometimes the field correlations are measured in the proximity of the source where the influence of evanescent components can be dominant. As a result, the coherence properties are non-universal [2,3]. Most interestingly, coherence properties can be related to the structural morphology of the medium emitting the radiation and, therefore, can act as distinctive characterization tools . For instance, in Refs. [3,5–7], it was shown that the extent of spatial coherence depends on the distance from the source of radiation and also on the field properties at its surface. Recently, a connection between the near-field speckle pattern and the internal structure of a randomly inhomogeneous medium has been revealed [8,9].
Here we explore a new paradigm: the effect of a monolayer of atoms on the coherence properties of an electromagnetic field that interacts with it. More specifically, we will demonstrate that the extent of spatial coherence is influenced by the presence of impurities or defects in two-dimensional crystalline lattices.
Defects in crystalline materials act as genuine Rayleigh scatterers. In two-dimensional structures, the inherently lower number of disturbance centers and their extremely weak scattering renders power measurements unrealistic. We will demonstrate that, even as we rely on elastic interactions, measurements of spatial coherence mitigate these limitations and permit characterization of the structural homogeneity of two-dimensional materials. The interaction with a partially coherent field provides an effective simultaneous excitation along an ensemble of -vectors, which creates ideal conditions for examining the coherent elastic scattering that originates within the interface.
We will exemplify this phenomenon on monolayer graphene. We chose this material for two reasons. First, most of its properties are rather well understood. Second, the role of defective sites in its crystalline structure has been extensively studied, and the defect characterization techniques are well-established from graphene synthesis to the assembling of graphene-based devices [10–14]. Techniques such as scanning  and transmission  electron microscopies, scanning probe microscopies such as atomic force microscopy  and scanning tunneling microscopy , have all been used for defect characterization.
The inelastic scattering of light by phonons (vibrational Raman scattering) is also extremely sensitive to the amount and type of defects. The sensitivity relates to the breaking down of momentum conservation in the scattering processes mediated by the broadband spectrum of spatial of frequencies (wave vectors, ) provided by the defects [19,20]. The relaxation of this momentum conservation-based selection rule gives rise to disorder-induced bands in the Raman signal, and the shape and relative intensity of these features provide both quantitative and qualitative information about the defects [19–24].
Although the inelastic scattering of light has been broadly employed to investigate defects in graphene [19,20], to the best of our knowledge, assessing the elastically (Rayleigh) scattered component has been, to date, out of reach. Since the defects are proven to act as important sources of extra momentum in the inelastic component of the scattered light, it is expected that momentum-related effects should be observed on the elastic part as well [19,20]. Although commercially available dispersive spectrometers are able to analyze broad ranges of temporal frequencies with high sensitivity, their resolution in the spatial-frequency domain permits only a limited -space analysis. In these instruments, the information is extracted solely from the quasi-monochromatic Rayleigh lines generated in laser-based spectroscopy. For these reasons, the Rayleigh signals are usually treated as unwanted features in light scattering spectroscopy.
In the following, we will demonstrate that -space spectroscopy can be developed where the Rayleigh component of scattered light is used directly to investigate the influence of defects on two-dimensional lattices, such as graphene. The technique relies on illuminating the monolayer from underneath with a quasi-thermal monochromatic field (uniform -vectors distribution) and recoding the intensity of the transmitted radiation with a scanning near-field optical microscope (SNOM) using an aperture probe. Based on measurements of spatial coherence, this scheme practically enables for a -space analysis without the need of angle-dependent measurements. The approach considers the statistical properties of radiation in the near-field regime, where the defects provide a broadband source for extra momentum, which influences the -vectors distribution in the scattered field and, thus, modifies the spatial coherence properties of the emerging field. The information about the defect density is extracted from the spatial extent of the intensity-intensity correlations measured right above the graphene surface.
2. Experimental Procedure
An optical image (micrograph) of a graphene flake sitting on a glass cover slip is shown in Fig. 1(a). The graphene was produced following the mechanical exfoliation method reported in Ref. . Raman spectroscopy was used to monitor the quality of the sample, as detailed in Fig. 1(b). The Raman spectrum of the freshly produced graphene [bottom spectrum in Fig. 1(b)] reveals that the flake is a pristine monolayer without detectable defects.
To probe the elastic light scattering in the proximity of the graphene surface, the monolayer was illuminated through the supporting cover glass by a pseudo-thermal light source created at the output port of an integrating sphere (Thorlabs–IS200-4–Ø2”, 12.5 mm port diameter), which effectively scrambled the radiation from a 532 nm continuous wave laser radiation (100 mW power and spectral linewidth of ). The sample was positioned atop the exit port of the integrating sphere, and the near-field measurements of the resulting speckle patterns were performed using a Nanonics MultiView 4000 NSOM system, working in standard collection mode, as schematically depicted in Fig. 2(a). Near-field intensity distributions were collected across regions using a Cr-Au coated pulled silica fiber probe with a 150-nm aperture diameter. The probe was attached to a quartz tuning fork and operated in intermittent contact while the optical signal was detected using a photon multiplying tube (PMT; Hamamatsu, H7421).
3. Coherence Properties of Optical Near Fields
A typical speckle image (high-resolution intensity map) recorded over a area of pristine graphene is shown in Fig. 2(b). Similar measurements were performed over more than twenty different regions, which practically covers the whole sample area, and to ensure appropriate ensemble average. Subsequently, the second-order intensity-intensity correlation was evaluated from the recorded intensity distributions. Using standard properties of Gaussian random variables, the field correlation function can be expressed in terms of the intensity correlation function. Following this procedure, the average size () of the speckle distribution was extracted from the radially averaged full width at half-maximum (FWHM) of the autocorrelation peak. For the pristine graphene, we found . The same procedure applied to the field measured on the surface of bare glass results in an average speckle size .
To explore the effects of defective structures, we generated point defects in the same graphene piece through oxygen-plasma etching . For that, the samples were exposed to oxygen plasma in a reactive ion etching (RIE) system, model Plasma-Therm 790 SERIES. An oxygen flow rate of 5 sccm diluted by 50 sccm of helium flow was used. Gradual levels of defect densities were obtained by increasing the etching time, and the amount of defects were monitored by Raman spectroscopy. The Raman spectra of the sample obtained after distinct steps of oxygen-plasma etching are shown in Fig. 1(b). The intensities were normalized by the amplitude of the first order bond-stretching G band (). The disorder-induced D and D’ bands (at and , respectively) are absent (undetectable) in the Raman spectrum taken before the oxygen-plasma etching starts (confirming that the starting sample was pristine), but they become more and more intense as defects are introduced by the oxygen-plasma etching process. Following previously established protocols [21,22], the average distance between point defects () was extracted from the ratio ( stands for intensity, and the subscripts D and G label the respective bands). The experimentally obtained values of are plotted as a function of in the inset of Fig. 1(b). The values of were obtained from the solid line, which is a previously established model that links to [21,22]. It should be noticed that, although the first three steps of plasma etching generated defects in the sample, the graphene was not amorphous yet. This conclusion is based on the fact that the respective data points in the versus plot fall on the right side of the theoretical peak curve indicating that, although the sample has structural defects, most of the net crystalline structure is still preserved. Alternatively, the Raman spectrum obtained after the fourth step of oxygen-plasma etching (20 s) clearly shows that the ratio recoiled [see top spectrum in Fig. 1(b)], indicating that the sample was in an advanced degree of amorphization.
A typical intensity map corresponding to a defective graphene layer is shown in Fig. 2(c) and, as can be seen, the average speckle size appears to be smaller than in the case of pristine graphene. The average speckle size was evaluated for all samples, and the results are summarized in Fig. 3 as a function of the inverse of . We emphasize that the data were obtained from the same sample (albeit treated with different exposure times of oxygen-plasma etching) to effectively cancel any possible influence from additional structural variations such as wrinkles or roughness. Moreover, all intensity maps were recorded using the same near-field probe. The data at correspond to the starting sample of pristine graphene. It can be readily observed that, initially, the speckle size decreases when the density of defects rises. Note that increasing the defect density above a certain limit should eventually saturate this reduction of measured coherence length (speckle size). This can be explained by the finite dimension of each defective site, which can be attributed to the coherence length photo-excited electrons in graphene [27,28]. For sparse defect densities, each defective site acts independently to diminish the measured size of the speckle. However, when the effective defect sizes start to overlap at high densities, the sample becomes homogenously amorphous and the speckle size increases back to the values corresponding to optically homogenous media close to that of pristine graphene. Therefore, it is clear that the scattering at the defective sites in graphene modifies rather significantly the extent of near-field spatial coherence.
The results summarized in Fig. 3 can be understood as a modification of the angular spectrum of transmitted radiation. The strong scattering inside the integrating sphere effectively randomizes the electromagnetic field and creates at the exit port an emerging field with uniform angular spectrum typical to a Lambertian source. After passing through a dielectric interface, this uniform -space distribution is modified due to the angular dependence of the corresponding Fresnel transmission coefficients. In general, the effect of a dielectric slab will be to suppress the contributions of -vectors at small angles from the interface and, therefore, to increase the transversal correlation length. When additional isotropic scatterers are distributed across the surface, the -vector distribution tends to recover its initially isotropic shape because of the angularly uniform contributions from all the Rayleigh scattering centers. At very high defect densities, the monolayer acts again as a homogeneous structure with an effective dielectric constant as clearly seen in Fig. 3.
Nevertheless, the measurements are performed near the interface, at distances , where the field distribution is more complex. Here, the non-universal properties of spatial coherence above an interface are expected to reveal the characteristic length scales of the morphology of a medium . An effective description considers the interface as a homogenous, planar, statistically stationary source of radiation that is characterized by a field-field correlation function , which extends over a characteristic distance . Above the interface, the homogeneous and inhomogenous components of the field-field correlation function, , both depend on but evolve differently with distance . This is illustrated in Fig. 4, where the width of the normalized correlation function, evaluated at , is plotted as a function of the inverse of the extent of field–field correlations in the plane of the source. Following the model outlined in Ref. , we consider a Gaussian source with its spectral degree of coherence of the form . The sharp decay of this function suggests that measurements of spatial coherence should be quite sensitive to variations in the field distribution across the interface. This is the reason why, in our experiments, the measured speckle size in Fig. 3 depends significantly on the average distance between point defects , the length scale that determines the span of field-field correlations in the plane of the sample. Although we cannot provide an exact quantitative relationship between a measured average distance between two point defects and a generic model describing a statistically stationary source of radiation, the decays of the spatial coherence lengths are evidently similar. This suggests that the model captures the main features of the experimental behavior and, therefore, one can safely conclude that the two quantities and are essentially the same.
We note that, in fact, we do not directly measure . However, the strong field randomization produced by the integrating sphere effectively generates a statistically stationary field obeying circular Gaussian statistics. In this case, high-order correlation functions factorize into products of second-order correlations and, therefore, determining the spatial extent of the degree of spatial coherence amounts practically to a measurement of the intensity-intensity spatial correlation function . This is exactly the correlation that can be inferred from intensity maps, such as the ones illustrated in Fig. 2. Of course, a direct comparison between the measured speckle size and the estimations based on the coherence model outlined here should also include corrections due to the point spread function (PSF) of the NSOM probe. This is illustrated in Fig. 3, where the predictions of this model corrected for the experimental PSF (dotted line) make a good description of the dependence between the measured speckle size and the average distance light between point defects in the graphene monolayer.
In conclusion, we have demonstrated a novel phenomenon: an essentially transparent atomic monolayer is capable of modifying the coherence properties of radiation transmitted through it. We have demonstrated a direct relationship between the statistical properties of the optical near fields and the structural properties of two-dimensional materials. This demonstration can lead to new means for controlling the complex properties of optical radiation at subwavelength scales.
We have shown that spatial coherence measurements in the proximity of two-dimensional interfaces constitute a rather general, elastic scattering approach to establish the presence and assess the density of defects in two-dimensional crystalline materials. This is notoriously difficult for transparent materials where the amount of scattered power is minimal and the so-called Rayleigh scattering is most of the time impractical. We have shown that a relative assessment of the spatial coherence of the incident and the scattered fields is sufficient to determine the density of defects, without resorting to elaborate inelastic measurements.
Finally, the use of the coherence degree of freedom in characterization protocols is largely unexplored. Our findings provide a simple and robust way to access structural properties that are traditionally probed only by inelastic scattering of light.
Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) (PDE201525/2015-1); Cátedra CEPSA.
The authors thank Dr. Sergey Sukhov for valuable discussions. L. G. C. acknowledges financial support from CNPq. F. S. B. acknowledges financial support from Cátedra CEPSA.
1. W. H. Carter and E. Wolf, “Coherence properties of Lambertian and non-Lambertian sources,” J. Opt. Soc. Am. 65, 1067–1071 (1975). [CrossRef]
2. A. Dogariu and R. Carminati, “Electromagnetic field correlations in three-dimensional speckles,” Phys. Rep. 559, 1–29 (2015). [CrossRef]
3. A. Apostol and A. Dogariu, “Spatial correlations in the near field of random media,” Phys. Rev. Lett. 91, 093901 (2003). [CrossRef]
4. C. Henkel, K. Joulain, R. Carminati, and J.-J. Greffet, “Spatial coherence of thermal near fields,” Opt. Commun. 186, 57–67 (2000). [CrossRef]
5. H. Roychowdhury and E. Wolf, “Effects of spatial coherence on near-field spectra,” Opt. Lett. 28, 170–172 (2003). [CrossRef]
6. R. Carminati, “Subwavelength spatial correlations in near-field speckle patterns,” Phys. Rev. A 81, 053804 (2010). [CrossRef]
7. A. Apostol and A. Dogariu, “Coherence properties near interfaces of random media,” Phys. Rev. E 67, 055601 (2003). [CrossRef]
8. V. Parigi, E. Perros, G. Binard, C. Bourdillon, A. Maître, R. Carminati, V. Krachmalnicoff, and Y. D. Wilde, “Near-field to far-field characterization of speckle patterns generated by disordered nanomaterials,” Opt. Express 24, 7019–7027 (2016). [CrossRef]
9. R. R. Naraghi, S. Sukhov, and A. Dogariu, “Disorder fingerprint: intensity distributions in the near field of random media,” Phys. Rev. B 94, 174205 (2016). [CrossRef]
10. F. Banhart, J. Kotakoski, and A. V. Krasheninnikov, “Structural defects in graphene,” ACS Nano 5, 26–41 (2011). [CrossRef]
11. O. V. Yazyev and Y. P. Chen, “Polycrystalline graphene and other two-dimensional materials,” Nat. Nanotechnol. 9, 755–767 (2014). [CrossRef]
12. L. Vicarelli, S. J. Heerema, C. Dekker, and H. W. Zandbergen, “Controlling defects in graphene for optimizing the electrical properties of graphene nanodevices,” ACS Nano 9, 3428–3435 (2015). [CrossRef]
13. W. Zhao, Y. Wang, Z. Wu, W. Wang, K. Bi, Z. Liang, J. Yang, Y. Chen, Z. Xu, and Z. Ni, “Defect-engineered heat transport in graphene: a route to high efficient thermal rectification,” Sci. Rep. 5, 11962 (2015). [CrossRef]
14. T. V. Alencar, M. G. Silva, L. M. Malard, and A. M. de Paula, “Defect-induced supercollision cooling of photoexcited carriers in graphene,” Nano Lett. 14, 5621–5624 (2014). [CrossRef]
15. Z.-J. Wang, G. Weinberg, Q. Zhang, T. Lunkenbein, A. Klein-Hoffmann, M. Kurnatowska, M. Plodinec, Q. Li, L. Chi, R. Schloegl, and M.-G. Willinger, “Direct observation of graphene growth and associated copper substrate dynamics by in situ scanning electron microscopy,” ACS Nano 9, 1506–1519 (2015). [CrossRef]
16. A. W. Robertson and J. H. Warner, “Atomic resolution imaging of graphene by transmission electron microscopy,” Nanoscale 5, 4079–4093 (2013). [CrossRef]
17. G. López-Polín, C. Gómez-Navarro, V. Parente, F. Guinea, M. I. Katsnelson, F. Pérez-Murano, and J. Gómez-Herrero, “Increasing the elastic modulus of graphene by controlled defect creation,” Nat. Phys. 11, 26–31 (2015). [CrossRef]
18. J. C. Koepke, J. D. Wood, D. Estrada, Z.-Y. Ong, K. T. He, E. Pop, and J. W. Lyding, “Atomic-scale evidence for potential barriers and strong carrier scattering at graphene grain boundaries: a scanning tunneling microscopy study,” ACS Nano 7, 75–86 (2013). [CrossRef]
19. R. Beams, L. G. Cançado, and L. Novotny, “Raman characterization of defects and dopants in graphene,” J. Phys. Condens. Matter 27, 83002 (2015). [CrossRef]
20. A. C. Ferrari and D. M. Basko, “Raman spectroscopy as a versatile tool for studying the properties of graphene,” Nat. Nanotechnol. 8, 235–246 (2013). [CrossRef]
21. L. G. Cançado, A. Jorio, E. H. M. Ferreira, F. Stavale, C. A. Achete, R. B. Capaz, M. V. O. Moutinho, A. Lombardo, T. S. Kulmala, and A. C. Ferrari, “Quantifying defects in graphene via Raman spectroscopy at different excitation energies,” Nano Lett. 11, 3190–3196 (2011). [CrossRef]
22. M. M. Lucchese, F. Stavale, E. H. M. Ferreira, C. Vilani, M. V. O. Moutinho, R. B. Capaz, C. A. Achete, and A. Jorio, “Quantifying ion-induced defects and Raman relaxation length in graphene,” Carbon 48, 1592–1597 (2010). [CrossRef]
23. J. Ribeiro-Soares, M. E. Oliveros, C. Garin, M. V. David, L. G. P. Martins, C. A. Almeida, E. H. Martins-Ferreira, K. Takai, T. Enoki, R. Magalhães-Paniago, A. Malachias, A. Jorio, B. S. Archanjo, C. A. Achete, and L. G. Cançado, “Structural analysis of polycrystalline graphene systems by Raman spectroscopy,” Carbon 95, 646–652 (2015). [CrossRef]
24. A. Eckmann, A. Felten, A. Mishchenko, L. Britnell, R. Krupke, K. S. Novoselov, and C. Casiraghi, “Probing the nature of defects in graphene by Raman spectroscopy,” Nano Lett. 12, 3925–3930 (2012). [CrossRef]
25. Y. Huang, E. Sutter, N. N. Shi, J. Zheng, T. Yang, D. Englund, H.-J. Gao, and P. Sutter, “Reliable exfoliation of large-area high-quality flakes of graphene and other two-dimensional materials,” ACS Nano 9, 10612–10620 (2015). [CrossRef]
26. M. C. Prado, D. Jariwala, T. J. Marks, and M. C. Hersam, “Optimization of graphene dry etching conditions via combined microscopic and spectroscopic analysis,” Appl. Phys. Lett. 102, 193111 (2013). [CrossRef]
27. C. Casiraghi, A. Hartschuh, H. Qian, S. Piscanec, C. Georgi, A. Fasoli, K. S. Novoselov, D. M. Basko, and A. C. Ferrari, “Raman spectroscopy of graphene edges,” Nano Lett. 9, 1433–1441 (2009). [CrossRef]
28. R. Beams, L. G. Cançado, and L. Novotny, “Low temperature Raman study of the electron coherence length near graphene edges,” Nano Lett. 11, 1177–1181 (2011). [CrossRef]