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Abstract
Laser field-induced spatial self-phase modulation (SSPM) in WS2 and MoS2 atomic layer liquid suspensions displayed the diffraction profile of concentric rings at the far-field due to the coherent superposition of transverse wave vectors with characteristic spatial nonlinear phases. The evolution of the number of rings indicated the spatial alignment of anisotropic atomic layers in the liquid base solution. The intensity-dependent number of symmetric rings revealed the nonlinear refraction coefficients of MoS2 and WS2 atomic layers which were estimated to be ∼ −1.96 × 10−16 m2/W and ∼ −1.11 × 10−16 m2/W, respectively. The central interference profile and the diffraction pattern identified the negative polarity of nonlinear refraction. The vertically asymmetric diffraction ring indicates the phase distortion of optical field due to the heat convection.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
An optical field propagates through an optical material with its own characteristic spatial and temporal phase changes. The spatial phase of optical field includes linear and nonlinear phases. The linear phase of optical field in the medium changes with the linear refractive index, and the nonlinear phase of optical field is shifted by the characteristic nonlinear refractive index of optical material and the applied optical intensity. The nonlinear phase is mutated by the applied intensity, and the spatial nonlinear phase is modulated by the spatial distribution of intensity and the characteristic nonlinear refractive index of optical medium [1]. The coherent superposition of transverse wave vectors with characteristic spatial nonlinear phases display the diffraction profile of concentric rings at far-field [2]. The diffraction profile of spatial self-phase modulation (SSPM) reveals the nonlinear optical properties and the interaction characteristics of optical field and medium. The evolution of the number of rings implies the spatial alignment of anisotropic atomic layers in the liquid base solution. The intensity-dependent number of symmetric rings reveals the nonlinear refraction coefficients of atomic layers [3]. The central interference profile and the diffraction pattern identifies the polarity of nonlinear refraction. The asymmetric diffraction discloses the phase distortion of optical field due to the heat convection [4,5]. The evolution of self-phase modulation with the spatial alignment of anisotropic atomic layers and the evolution of vertical asymmetric diffraction due to thermal convection are characteristic nonlinear optical processes in atomic layer liquid suspension, which can’t be observed with a bare atomic-layer material.
The SSPM of carbon disulfide was observed by Callen et al in 1967 for the first time [6]. Recently, the nonlinear optical characterization of atomic layers with SSPM was reported [3,5,7] by multiple research groups. Wu and Zhao’s group [7] proposed a “wind chime” model to describe the emergence of electron coherence from the nonlocal domains of nanoflakes which were driven by the SSPM. They reported the bandgap-dependent nonlinearity of nanoflakes using the SSPM with 200-fs lasers and CW lasers at visible and near-IR spectra, and all-optical switching by the two-color phase modulation. G. Wang and J. Wang’s collaborators [3] characterized the nonlinear refractive index of atomic layers using the distortion of SSPM with a CW laser at 488 nm. Wu and Zhao’s group [8] articulated the third-order nonlinearity of MoSe2 flakes with SSPM and the threshold intensity as a function of photon energy, and the correlation between the third-order nonlinearity and the absorption of MoSe2 flakes. Jia and Xiang’s collaborators [5] reported the broadband optical nonlinearity at visible spectra, and all-optical switching two-color spatial cross-phase modulation (SXPM).
In this article, the spatial self-phase modulation in MoS2 and WS2 atomic layer liquid suspension was investigated by the SSPM using a pulsed laser which has a temporal pulse width of ∼1.5 ps and a repetition rate of ∼82 MHz in order to characterize the magnitude and the polarity of nonlinear refractions of atomic layers, the horizontal and vertical diffraction characteristics as a function of time duration of laser excitations, the phase modulation as function of input intensity, which provide the scientific and technological significances to pave the way for the all optical-switching based on the atomic layers.
2. Theory
The refractive index changes the characteristic nonlinear refraction coefficient and the applied optical intensity. The intensity-dependent refractive index n is [5],
where n0 is the linear refractive index, γ is the nonlinear refraction coefficient, $I({r,z} )$ is the spatially distributed input intensity, and Δn is the change in refractive index. The radial component of output electric field through the sample for the input electric field in the Gaussian profile is given by [9–14],
Fig. 1. Nonlinear phase (black curve, Eq. (3)) and transverse propagation wave vector (blue curve, Eq. (4)) as a function of radial position of r for (a) positive and (b) negative nonlinear refraction coefficients (γ).
The number of concentric rings depends on the maximum nonlinear phase shift $\Delta \varphi {}_{NL}(0 )= k\gamma I(0 )L = 2\pi N$. The number of diffraction rings as a function of applied intensity has a slope of $k\gamma L/2\pi$ [1,3],
Then, the nonlinear refraction coefficients of atomic layers can be estimated from the slope of the number of diffraction rings as a function of applied intensity.The far-field intensity with the Fraunhofer approximation of Fresnel–Kirchhoff diffraction is given by [9],
Fig. 2. Diffraction patterns at far-field due to the SSPM in optical medium with different nonlinear phase shifts $\Delta {\varphi _{NL}}(0 )$ of (a) 0π, (b) 1π, (c) 2π, (d) 5π, (e) 8π, and (f) 11π.
Fig. 3. Diffraction patterns at far-field due to the SSPM in optical medium with different nonlinear phase shifts $\Delta {\varphi _{NL}}(0 )$ of (a) 0π, (b) −1π, (c) −2π, (d) −5π, (e) −8π, and (f) −11π.
Figure 2 shows the diffraction patterns at the far-field due to SSPM in optical medium with different positive nonlinear phase shifts. The central peak of diffraction pattern is due to the positive nonlinearity or self-focusing property of an optical medium. The number of diffraction rings is increased as the maximum nonlinear phase shift is increased. The intervals between the outer rings are larger than those between the inner rings. No diffraction pattern for zero nonlinear phase shift is shown in Figs. 2(a) and 3(a). The diffraction pattern for $\Delta {\varphi _{NL}}(0 )= 2\pi$ exhibited the side wings. The diffraction pattern for $\Delta {\varphi _{NL}}(0 )\ge 2\pi$ displayed the number of interference patterns. The positive nonlinear phase with positive nonlinear refraction coefficient displayed a distinct central peak which was disappeared with the negative maximum nonlinear phase with negative nonlinear refraction coefficient. The width and strength of the outermost diffraction ring are larger due to the larger deformed wavefront for lesser nonlinear phase change at the margin of propagation wave [17]. The interference fringe is proportional to the derivative magnitude of the angular deformation of wavefront to the nonlinear phase change [17]. Figure 3 shows the diffraction pattern due to SSPM in optical medium with different negative nonlinear phase shifts. The suppression or disappearance of central peak indicates the negative nonlinearity or self-defocusing property of an optical medium.
In addition to the intensity-dependent nonlinear phase changes, the asymmetric diffraction rings due to the heat convection of optical medium characterizes the thermal-induced nonlinear refraction coefficient which is estimated by the ratio of distortion angle θD to the half-cone angle θH [5],
The Eq. (7) indicates that the larger thermal distortion of diffraction pattern has the larger nonlinear refraction change within the diffraction angle.
3. Materials and methods
The WS2 and MoS2 atomic layer liquid suspensions in 70% ethanol and 30% water were purchased from the Graphene Laboratory [18]. The concentration of WS2 atomic layer was ∼26 mg/L. The number of layers were ∼1-4 atomic layers with the lateral sizes of ∼50 - 150 nm. The concentration of MoS2 atomic layers was ∼18 mg/L. The number of layers were ∼1-8 atomic layers with the lateral sizes of ∼100 - 400 nm.
The excitation source for the SSPM of atomic layers was a pulsed laser at ∼800 nm with a temporal pulse width of ∼1.5 ps and a repetition rate of ∼82 MHz. The thickness of optical medium (L) and the effective length were 10 mm and 7.7 mm, respectively. The beam waist at the focal point was ∼40 µm and the Rayleigh length was ∼9.6 mm. The diffraction profile of self-phase modulation was displayed on the white paper screen at the far-field, and the image on the screen was captured by a CCD USB 2.0 image camera (DCU223C, Thorlabs, Inc.).
The nonlinear refraction coefficients of MoS2 and WS2 atomic layers were characterized by the number of diffraction rings as a function of applied intensity at the fixed time duration of excitation. The polarity of nonlinear refraction was identified by the central interference and diffraction profiles at the far-field. The evolution of number of rings revealed the spatial alignment of anisotropic atomic layers in the liquid base solution. The asymmetric diffraction analyzed the thermal phase distortion of optical field.
4. Experiment results and analysis
Figure 4 shows the optical absorption spectra of WS2 and MoS2 atomic layer liquid suspension in 70% ethanol and 30% water. The weak characteristic absorption peaks of WS2 and MoS2 atomic layers at the visible region are assigned to the distinct exciton transitions of A and B exciton peaks which are related to the spin-orbital coupling at the K/K’ position in momentum space in the first Brillouin zone. The broad bands of absorption at the higher energies, which are called as C and D exciton peaks, are most likely linked to the band nesting near the Γ-Λ and Γ-M positions [19,20].
Fig. 4. Absorption spectra of MoS2 (blue) and WS2 (black) atomic layers, and optical spectrum of excitation laser (red).
Figures 5(a) and (b) display the laser-induced diffraction rings of MoS2 and WS2 atomic layer liquid suspension for the different time durations of laser excitations which has a temporal pulse width of ∼1.5 ps and repetition rate of ∼82 MHz. The applied input intensity was ∼0.7 GW/cm2. Figures 5(c) and (d) show the horizontal radius and upper vertical radius of diffraction rings as a function of time duration of laser excitations at the applied intensity of ∼0.7 GW/cm2, respectively. The change of diffraction ring size within ∼0.48-second time duration of laser excitation is attributed to the spatial alignment of anisotropic atomic layers responding to the excitation field [1,3,21–23]. After ∼0.48-second time duration of laser excitation, the Fig. 5 displays two different characteristics with the horizontal and the upper vertical diffractions as a function of time duration of laser excitations. The stable horizontal diffraction profile is due to the coherent superposition of transverse wave vectors with characteristic spatial nonlinear phases, and the reduction of upper vertical diffraction profile is due to the heat convection of atomic layer liquid suspension. Figure 6 displays the diffraction patterns at far-field due to the SSPM for the base solution (70% ethanol and 30% water), the MoS2 atomic layer liquid suspension, and the WS2 atomic layer liquid suspension at the applied intensity of ∼0.7 GW/cm2. Figure 6(a) does not show the distinct diffraction pattern of self-phase modulation for the solvent at the highest input intensity of experiment condition and the different time durations of excitation, while Fig. 6(b) and (c) display the distortions of upper vertical diffractions for the MoS2 atomic layer and the WS2 atomic layer liquid suspensions. It indicates that the stable horizontal diffraction profile of SSPM is originated from the coherent superposition of transverse wave vectors with characteristic spatial nonlinear phases of atomic layers, and the thermal distortion of upper vertical diffraction is due to the heat convection of atomic layers in the liquid suspension.
Fig. 5. Diffraction patterns at far-field due to the SSPM in (a) MoS2 atomic layer liquid suspension and (b) WS2 atomic layer liquid suspension for different time durations of laser excitation at the applied intensity of ∼0.7 GW/cm2. (c) Horizontal radius and (d) upper vertical radius of diffraction ring as a function of time duration of laser excitations at the applied intensity of ∼0.7 GW/cm2.
Fig. 6. Diffraction patterns at far-field due to the SSPM for (a) the base solution (70% ethanol and 30% water), (b) the MoS2 atomic layer liquid suspension, and (c) the WS2 atomic layer liquid suspension at the applied intensity of ∼0.7 GW/cm2.
The number of diffraction rings as a function of applied peak intensity for (a) MoS2 and (b) WS2 atomic layer liquid suspensions at the time duration of 0.48 seconds are shown in Fig. 7. The number of rings were linearly increased as the input intensity was increased. The nonlinear refraction coefficients of MoS2 and WS2 atomic layer liquid suspensions were characterized using the slope $({k\gamma {L_{eff}}/2\pi } )$ for the number of diffraction rings as a function of applied peak intensity. The nonlinear refraction coefficients of MoS2 and WS2 atomic layer liquid suspensions were estimated to be ∼-1.96×10−16 m2/W and ∼-1.11×10−16 m2/W, respectively.
Fig. 7. Number of diffraction rings at the far-field due to the SSPM in atomic layers as a function of applied peak intensity for (a) MoS2 and (b) WS2 atomic layer liquid suspension at the time duration of ∼0.48 seconds. Diffraction rings at the far-field due to the SSPM in (c) MoS2 and (d) WS2 atomic layer liquid suspensions for different applied intensities.
The two-dimensional diffraction profiles of WS2 and MoS2 atomic layer liquid suspension has no distinct central peak of interference as shown in Fig. 8 which indicates the negative nonlinear refraction of atomic layers. Figures 8(a) and (b) show the best fittings of Eq. (6) to the diffraction patterns from the MoS2 and WS2 atomic layer liquid suspensions which revealed the maximum nonlinear phase shifts of −16π and −10π, respectively. All other parameters are the experiment conditions which include Io∼0.7 GW/cm2 for the applied peak intensity, D∼2.3 m for the distance from the sample position to the screen, λ∼800 nm for the laser excitation wavelength, zo∼9.6 mm for the Rayleigh length, and wo∼40 µm for the beam waist at focal point. The negative nonlinear phase shift $\Delta \varphi {}_{NL}(0 )= k\gamma I(0 ){L_{eff}}$ indicates the negative nonlinear refraction coefficients of MoS2 and WS2 atomic layers [19]. The characteristic diffraction patterns are shown in the Figs. 9(a) and (b) for the positive and negative maximum nonlinear phase shifts with the same magnitudes of 16π and 10π for the Figs. 8(a) and (b), respectively, which confirms the negative polarity of nonlinear refractions for the MoS2 and WS2 atomic layer liquid suspensions.
Fig. 8. Two-dimensional diffraction pattern (blue color) at the far-field due to the SSPM in (a) MoS2 and (b) WS2 atomic layer liquid suspensions. Theoretical fittings (red color) of (a) maximum nonlinear phase shift $\Delta {\varphi _{NL}}(0 )$=−16π for MoS2 atomic layers and (b) maximum nonlinear phase shift $\Delta {\varphi _{NL}}(0 )$=−10π for WS2 atomic layers. The fitting parameters include ko = 7.85×106 m−1, w(z) = 4.01×10−4 m, Io = 0.70 GW/cm2, and D = 2.3 m.
Fig. 9. Diffraction simulations at the far-field due to the SSPM in (a) MoS2 atomic layers with the maximum nonlinear phase shifts $\Delta {\varphi _{NL}}(0 )$ of −16π and 16π; and WS2 atomic layers with maximum nonlinear phase shifts −10π and 10π. The fitting parameters include ko = 7.85×106 m−1, w(z) = 4.01×10−4 m, Io = 0.70 GW/cm2, and D = 2.3 m.
Figure 10(a) shows the schematic sketch of the half-cone and distortion angles of diffraction profiles at the far-field where the half-cone angle was measured at the excitation time duration at 0.48 seconds and the distortion angle of diffraction profile was measured at the excitation time duration at 2.0 seconds. Figures 10(b) and (c) include the distortion and half-cone angle ratio of ${\theta _D}/{\theta _H}$ as a function of applied peak intensity for (b) MoS2 and (c) WS2 atomic layers, respectively. Figures 10(d-k) and (l-m) show the images of diffraction profiles for different applied peak intensities for MoS2 and WS2 atomic layers, respectively. The distortion radius for the upper half of diffraction profile is due to the laser-induced heat convection atomic layers in liquid base solution [3,4,24,25]. The laser-induced temperature fields was proven by the simulation [26], and the laser-induced heat convection of atomic layers in base solution has an analogy with the heat convection due to a heating wire in liquid [27]. The change in nonlinear refraction coefficient due to thermal effect was estimated by Eq. (7) as shown in Figs. 10(b) and (c) [1,3–5]. The changes in nonlinear refraction coefficients for MoS2 and WS2 atomic layers were ∼0.7×10−16 m2/W and ∼0.4×10−16 m2/W at the applied intensity of ∼0.7 GW/cm2, respectively, using the Eq. (7), while the nonlinear refraction coefficients of MoS2 and WS2 atomic layers were estimated to be ∼-1.96×10−16 m2/W and ∼-1.11×10−16 m2/W, respectively, using the Eq. (5).
Fig. 10. (a) Schematic sketch of the half-cone and distortion angles of SSPM, (b) half-cone and distortion angles as a function of applied peak intensity; the ratio of ${\theta _D}/{\theta _H}$ (left y-axis) and change in nonlinear refraction coefficient (right y-axis) as a function of applied peak intensity for (b) MoS2 and (c) WS2 atomic layer liquid suspension, respectively; (d-k) and (l-m) represent the half-cone and distortion radii of MoS2 and WS2 atomic layer liquid suspension for different applied peak intensities, respectively.
5. Conclusion
The laser intensity-induced SSPM in MoS2 and WS2 atomic layer liquid suspension displayed the concentric diffraction rings at far-field. The formation of concentric diffraction rings is due to the coherent superposition of transverse wave vectors. The liquid suspension of atomic layers provided a unique material system to characterize the cubic nonlinearity with the SSPM in atomic layers for photonic applications which can’t be achieved with bare atomic layers. The temporal evolution of diffraction ring morphology indicated the spatial alignment of atomic layers, maximum diffraction rings at the intermediate time, and thermal distortion of the upper vertical ring of SSPM at the longer time duration of laser excitation. The vertically asymmetric diffraction ring indicates the phase distortion of the optical field due to heat convection. The number of rings as a function of applied peak intensity revealed the magnitude of nonlinear refraction coefficients. The slope of the number of rings as a function of applied peak intensity revealed the nonlinear refraction coefficients of ∼-1.96×10−16 m2/W and ∼-1.11×10−16 m2/W for MoS2 and WS2 atomic layers, respectively. The central interference profile and the diffraction pattern identified the polarity of nonlinear refraction. The evolution of self-phase modulation with the spatial alignment of anisotropic atomic layers and the evolution of vertical asymmetric diffraction due to the thermal convection are characteristic nonlinear optical processes in atomic layer liquid suspension, which can’t be observed with a bare atomic-layer material. Therefore, the nonlinear optical characteristics of atomic layer liquid suspension offer the distinct scientific and technological significances to pave the way for the all optical-switching based on the atomic layers.
Funding
Army Research Office (W911NF-15-1-0535); National Aeronautics and Space Administration (NNX15AQ03A).
References
1. S. D. Durbin, S. M. Arakelian, and Y. R. Shen, “Laser-induced diffraction rings from a nematic- liquid-crystal film,” Opt. Lett. 6(9), 411–413 (1981). [CrossRef]
2. L. Cheng, Z. Zhang, L. Zhang, D. Ma, G. Yang, T. Dong, and Y. Zhang, “Manipulation of a ring-shaped beam via spatial self- and cross-phase modulation at lower intensity,” Phys. Chem. Chem. Phys. 21(14), 7618–7622 (2019). [CrossRef]
3. G. Wang, S. Zhang, X. Zhang, L. Zhang, Y. Cheng, D. Fox, H. Zhang, J. N. Coleman, W. J. Blau, and J. Wang, “Tunable nonlinear refractive index of two-dimensional MoS2, WS2, and MoSe2 nanosheet dispersions,” Photonics Res. 3(2), A51–A55 (2015). [CrossRef]
4. G. Wang, S. Zhang, F.A. Umran, X. Cheng, N. Dong, D. Coghlan, Y. Cheng, L. Zhang, W.J. Blau, and J. Wang, “Tunable effective nonlinear refractive index of graphene dispersions during the distortion of spatial self-phase modulation,” Appl. Phys. Lett. 104(14), 141909 (2014). [CrossRef]
5. Y. Jia, Y. Shan, L. We, S. Dai, D. Fan, and Y. Xiang, “Broadband nonlinear optical resonance and all-optical switching of liquid phase exfoliated tungsten diselenide,” Photonics Res. 6(11), 1040–1047 (2018). [CrossRef]
6. W. R. Callen, B.G. Huth, and R. H. Pantell, “Optical Patterns of Thermally Self-Defocused Light,” Appl. Phys. Lett. 11(3), 103–105 (1967). [CrossRef]
7. Y. Wu, Q. Wu, F. Sun, C. Cheng, S. Meng, and J. Zhao, “Emergence of electron coherence and two-color all-optical switching in MoS2 based on spatial self-phase modulation,” Proc. Natl. Acad. Sci. 112(38), 11800–11805 (2015). [CrossRef]
8. W. Wang, Y. Wu, Q. Wu, J. Hua, and J. Zhao, “Coherent Nonlinear Optical Response Spatial Self-Phase Modulation in MoSe2 Nano-Sheets,” Sci. Rep. 6(1), 22072 (2016). [CrossRef]
9. L. Deng, K. He, T. Zhou, and C. Li, “Formation and evolution of far-field diffraction patterns of divergent and through self-focusing and self-defocusing media,” J. Opt. A: Pure Appl. Opt. 7(8), 409–415 (2005). [CrossRef]
10. A. G. Nibu, S. S. Silpa, S. N. Ardra, M. M. Anisha, and D. Ambika, “Effect of viscosity on the far-field diffraction pattern of spatial SPM,” Opt. Appl. 47, 591–599 (2017).
11. R Karimzadeh, “Spatial SPM of a laser beam propagating through liquids with self-induced natural convection flow,” J. Opt. 14(9), 095701 (2012). [CrossRef]
12. R. Zamiri, R. Parvizi, A. Zakaria, A. R. Sadrolhosseini, G. Zamiri, M. Darroudi, and M. S. Husin, “Investigation on nonlinear-optical properties of palm oil and silver nanoparticles,” J. European Optical Society Rap. Pubs. 7, 12020 (2012). [CrossRef]
13. E. V. G. Ramirez, M. L. A. Carrasco, M. M. M. Otero, S. C. Cerda, and M. D. I. Castillo, “Far field intensity distributions due to spatial SPM of a Gaussian beam by a thin nonlocal nonlinear media,” Opt. Express 18(21), 22067 (2010). [CrossRef]
14. Q. Zhang, X. Cheng, Y. Zhang, X. Yin, M. Jiang, H. Chen, and J. Bai, “Optical limiting using spatial self-phase modulation in hot atomic sample,” Opt. Laser Technol. 88, 54–60 (2017). [CrossRef]
15. I. C. Khoo, J. Y. Hou, T. H. Liu, P. Y. Yan, R. R. Michael, and G. M. Finn, “Transverse self-phase modulation and bistability in the transmission of a laser beam through a nonlinear thin film,” J. Opt. Soc. Am. B 4(6), 886–891 (1987). [CrossRef]
16. Y. Shan, J. Tang, L. Wu, S. Lu, X. Dai, and Y. Xiang, “Spatial self-phase modulation and all-optical switching of graphene oxide dispersions,” J. Alloys Compd. 771, 900–904 (2019). [CrossRef]
17. J. J. Wu, S. H. Chen, J. Y. Fan, and G. S. Ong, “Propagation of a Gaussian profile laser beam in nematic-liquid crystals and the structure of its nonlinear diffraction rings,” J. Opt. Soc. Am. B 7(6), 1147–1157 (1990). [CrossRef]
18. J. N. Coleman, U. Khan, K. Young, A. Gaucher, S. De, R.J. Smith, I. V Shvets, S. K. Arora, G. Stanton, H. Kim, K. Lee, G.T. Kim, G. S. Duesberg, T. Hallam, J. J. Boland, J. J. Wang, J. F. Donegan, J. C. Grunlan, G. Moriarty, A. Shmeliov, R. J. Nicholls, J. M. Perkins, E. M. Grieveson, K. Theuwissen, D. W. Mccomb, P. D. Nellist, and V. Nicolosi, “Two-Dimensional Nanosheets Produced by Liquid Exfoliation of Layered Materials,” Science 331(6017), 568–571 (2011). [CrossRef]
19. T. Neupane, S. Yu, Q. Rice, B. Tabibi, and F. J. Seo, “Third-order optical nonlinearity of tungsten disulfide atomic layer with resonant excitation,” Opt. Mater. 96, 109271 (2019). [CrossRef]
20. D. Kozawa, R. Kumar, A. Carvalho, K. Kumar Amara, W. Zhao, S. Wang, M. Toh, R.M. Ribeiro, A.H. Castro Neto, K. Matsuda, and G. Eda, “Photocarrier relaxation pathway in two-dimensional semiconducting transition metal dichalcogenides,” Nat. Commun. 5(1), 4543 (2014). [CrossRef]
21. R. Wu, Y. Zhang, S. Yan, F. Bian, W. Wang, X. Bai, X. Lu, J. Zhao, and E. Wang, “Purely coherent nonlinear optical response in solution dispersions of graphene sheets,” Nano Lett. 11(12), 5159–5164 (2011). [CrossRef]
22. Z. Wang, “Alignment of graphene nanoribbons by an electric field,” Carbon 47(13), 3050–3053 (2009). [CrossRef]
23. G. Wang, S. Higgins, K. Wang, D. Bennett, N. Milosavljevic, J. J. Magan, S. Zhang, X. Zhang, J. Wang, and W. J. Blau, “Intensity-dependent nonlinear refraction of antimonene dispersions in the visible and near-infrared region,” Appl. Opt. 57(22), E147–E153 (2018). [CrossRef]
24. T. Myint and R.R. Alfano, “Spatial phase modulation from permanent memory in doped glass,” Opt. Lett. 35(8), 1275–1277 (2010). [CrossRef]
25. R. G. Harrison, L. Dambly, D. Yu, and W. Lu, “A new self-diffraction pattern formation in defocusing liquid media,” Opt. Commun. 139(1-3), 69–72 (1997). [CrossRef]
26. Y. Wang, Y. Tang, P. Cheng, X. Zhou, Z. Zhu, Z. Liu, D. Liu, Z. Wang, and J. Bao, “Distinguishing thermal lens effect from electronic third-order nonlinear self-phase modulation in liquid suspensions of 2D nanomaterials,” Nanoscale 9(10), 3547–3554 (2017). [CrossRef]
27. C. M. Vest, M. L. Lawson, and A. Arbor, “Onset of Convection near a Suddenly Heated Horizontal Wire,” Int. J. Heat Mass Transfer 15(6), 1281–1283 (1972). [CrossRef]
