## Abstract

Laser field-induced spatial self-phase modulation (SSPM) in WS_{2} and MoS_{2} 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 MoS_{2} and WS_{2} atomic layers which were estimated to be ∼ −1.96 × 10^{−16} m^{2}/W and ∼ −1.11 × 10^{−16} m^{2}/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 MoSe_{2} 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 MoSe_{2} 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 MoS_{2} and WS_{2} 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],

*n*is the linear refractive index,

_{0}*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],

*z*position [13],

*z*

_{o}is the Rayleigh length,

*α*is the absorption coefficient,

*L*is the length of optical medium,

*k*is the wavenumber vector. The output field of Eq. (2) includes the amplitude terms and the exponent phase of $\varphi ({z,r} )= {\varphi _L}({z,r} )+ {\varphi _{NL}}({z,r} )$. The intensity-dependent nonlinear phase ${\varphi _{NL}}({z,r} )$ is [15],

*L*[3,5,16], where

*L*

_{1}is the distance from the exit surface of sample to the focal point,

*L*

_{2}is the distance from the input surface of sample to the focal point,

*L*

_{2}-

*L*

_{1}=

*L*is the thickness of optical medium,

*k*is the wavenumber vector of base solution, and $\Delta \varphi {}_{NL}(0 )$ is the maximum nonlinear phase shift. If the maximum nonlinear phase shift $\Delta \varphi {}_{NL}(0 )= k\gamma I(0 ){L_{eff}} \ge 2\pi$, the diffraction pattern at far-field will be displayed. The spatial change of nonlinear phase is the transverse propagation wave vector which is given by [2],

_{s}*r*and

_{1}*r*, the interference between the different phases of an optical wave occurs. If the nonlinear spatial phase shift $\Delta {\varphi _{NL}} = {\varphi _{NL}}({{r_1}} )- {\varphi _{NL}}({{r_2}} )$ of an optical wave has an even or odd integer number of

_{2}*mπ*, the local phases of an optical wave will have the coherent superposition of in the form of constructive or destructive interference that provides the concentric diffraction rings at far-field.

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],

*R > 0*) at the far-field and the different positive and negative maximum nonlinear phase shifts. The radius of curvature is given by $R(z )= z({1 + {{({{{{z_o}} \mathord{\left/ {\vphantom {{{z_o}} z}} \right.} z}} )}^2}} )$ for the beam waist $w(z )= {w_o}\sqrt {({1 + {{({{z \mathord{\left/ {\vphantom {z {{z_o}}}} \right.} {{z_o}}}} )}^2}} )} = {w_o}\sqrt {R(z)/({R(z) - z} )}$ where

*w*is the beam waist at the focal point, and

_{o}*z*

_{o}is the Rayleigh length [13]. The material-independent simulation parameters for the far-field intensity of Eq. (6) are I

_{o}∼0.7 GW/cm

^{2}for the applied peak intensity, D ∼2.3 m for the distance from the sample position the screen, λ ∼800 nm for the wavelength of excitation source, z

_{o}∼9.6 mm for the Rayleigh length, w

_{o}∼40 µm for the beam waist at the focal point, and $\Delta {\varphi _{NL}}(0 )$ for the different maximum nonlinear phase shift.

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 *θ _{D}* to the half-cone angle

*θ*[5],

_{H}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 WS_{2} and MoS_{2} atomic layer liquid suspensions in 70% ethanol and 30% water were purchased from the Graphene Laboratory [18]. The concentration of WS_{2} 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 MoS_{2} 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 MoS_{2} and WS_{2} 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 WS_{2} and MoS_{2} atomic layer liquid suspension in 70% ethanol and 30% water. The weak characteristic absorption peaks of WS_{2} and MoS_{2} 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].

Figures 5(a) and (b) display the laser-induced diffraction rings of MoS_{2} and WS_{2} 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/cm^{2}. 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/cm^{2}, 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 MoS_{2} atomic layer liquid suspension, and the WS_{2} atomic layer liquid suspension at the applied intensity of ∼0.7 GW/cm^{2}. 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 MoS_{2} atomic layer and the WS_{2} 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.

The number of diffraction rings as a function of applied peak intensity for (a) MoS_{2} and (b) WS_{2} 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 MoS_{2} and WS_{2} 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 MoS_{2} and WS_{2} atomic layer liquid suspensions were estimated to be ∼-1.96×10^{−16} m^{2}/W and ∼-1.11×10^{−16} m^{2}/W, respectively.

The two-dimensional diffraction profiles of WS_{2} and MoS_{2} 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 MoS_{2} and WS_{2} 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 I_{o}∼0.7 GW/cm^{2} 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, z_{o}∼9.6 mm for the Rayleigh length, and w_{o}∼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 MoS_{2} and WS_{2} 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 MoS_{2} and WS_{2} atomic layer liquid suspensions.

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) MoS_{2} and (c) WS_{2} atomic layers, respectively. Figures 10(d-k) and (l-m) show the images of diffraction profiles for different applied peak intensities for MoS_{2} and WS_{2} 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 MoS_{2} and WS_{2} atomic layers were ∼0.7×10^{−16} m^{2}/W and ∼0.4×10^{−16} m^{2}/W at the applied intensity of ∼0.7 GW/cm^{2}, respectively, using the Eq. (7), while the nonlinear refraction coefficients of MoS_{2} and WS_{2} atomic layers were estimated to be ∼-1.96×10^{−16} m^{2}/W and ∼-1.11×10^{−16} m^{2}/W, respectively, using the Eq. (5).

## 5. Conclusion

The laser intensity-induced SSPM in MoS_{2} and WS_{2} 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} m^{2}/W and ∼-1.11×10^{−16} m^{2}/W for MoS_{2} and WS_{2} 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).

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