J. Cent. South Univ. (2019) 26: 2295-2306

DOI: https://doi.org/10.1007/s11771-019-4174-8

A review on spatial self-phase modulation of two-dimensional materials

ZHANG Xue-jun(张学军), YUAN Zhen-hua(袁振华), YANG Rui-xin(杨瑞欣),

HE Yi-lin(何奕林), QIN Ying-lin(秦应霖), XIAO Si(肖思), HE Jun(何军)

Hunan Key Laboratory of Super Microstructure and Ultrafast Process, School of Physics and Electronics, Central South University, Changsha 410083, China

Central South University Press and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Abstract: Self-diffraction appears when the strong laser goes through two-dimensional material suspension, and this spatial self-phase modulation (SPPM) phenomenon can be used to measure nonlinear optical parameters and achieve optical switch. At present, the mechanism of SPPM is still ambiguous. The debate mainly focuses on whether the phenomenon is caused by the nonlinear refractive index of the two-dimensional material or the thermal effect of the laser. The lack of theory limits the dimension of the phase modulation to the radius of the diffraction ring and the vertical imbalance. Therefore, it is urgent to establish a unified and universal SSPM theoretical system of two-dimensional material.

Key words: nonlinear optics; spatial self-phase modulation; two-dimensional materials

Cite this article as: ZHANG Xue-jun, YUAN Zhen-hua, YANG Rui-xin, HE Yi-lin, QIN Ying-lin, XIAO Si, HE Jun. A review on spatial self-phase modulation of two-dimensional materials [J]. Journal of Central South University, 2019, 26(9): 2295-2306. DOI: https://doi.org/10.1007/s11771-019-4174-8.

1 Introduction

Since the American physicist Theodore H. Maiman developed the world's first ruby laser in 1960, the generation of lasers has led to the rapid development of nonlinear optics [1, 2]. From the development track of nonlinear optics, the development law can be summarized as follows: the light source is turned from CW laser to ultra-fast pulse light; the time scale is from static to transient [3-6]. There is a special branch of nonlinear optics, spatial self-phase modulation (SSPM), when a high- intensity laser is incident into a homogeneous medium, a diffracted pattern of beam divergence is observed at a distance, and its phase is modulated by its own light intensity. SSPM is very mature in the field of liquid crystal [7-9]. Researchers use light field or electromagnetism field to control the orientation of liquid crystal molecules, regulate their self-diffraction properties, and achieve spatial phase modulation effects [10-15].

Since NOVOSELOV et al [16] first obtained the two-dimensional (2D) graphene after mechanical stripping, the outstanding photoelectric and mechanical properties of 2D materials have attracted wide interest of researchers [17-21]. Recent studies have shown that 2D materials have excellent nonlinear optical properties and ultra-fast dynamics [22-27]. SSPM is mostly used to study the nonlinear refractive index of 2D materials [28-30]. However, the SSPM mechanism of two-dimensional material is still unclear. The debate mainly focuses on whether it is dominated by the nonlinear refractive index of 2D materials or by the thermal effects of lasers. Many different interpretations have been present. This paper collates research related to SSPM of 2D materials that have been published, and divides existing self-phase modulation mechanisms of 2D materials into three types: 1) the Gaussian distribution of laser light intensity directly; 2) the thermal effect of laser high light intensity or the microbubbles; 3) the orientation of the 2D material regulated by the light filed. The lack of mechanism makes the SSPM of the 2D material limit to two dimensions of the modulation diffraction ring radius (or ring number) and the vertical imbalance. Therefore, there is an urgent need to establish a unified and universal theoretical system of 2D material’s SSPM.

2 SSPM effect dominated by nonlinear refractive index of two-dimensional material

In 2011, WU et al [31] reported SSPM of graphene discrete solutions. Researchers believe that this phenomenon results from the third-order optical nonlinear characteristics caused by high light intensity. Graphene adjusts nonlinear refractive index by energy band structure changing light intensity. Equations describing the light intensity and the nonlinear refractive index are given as follows:

                               (1)

               (2)

where n, n₀ and n₂ are respectively the index, linear refractive index and nonlinear refractive index; I is the light intensity; △ψ is the phase shift of the laser beam; r is the radial coordinate; λ is the laser wavelength in vacuum; L is the total propagation length contributing to the SSPM; I(r, z) is the intensity distribution; N is the total number of rings.

For a Gaussian beam, the intensity I(0, z) at the center is twice the average intensity I measured in the experiment.where a is the 1/e² beam radius. The total number of rings N is determined by for a given laser intensity.

                            (3)

In 2015, SHI et al [32] reported ultra-fast broadband SSPM of topological insulator Bi₂Te₃ discrete solution, and they calculated the corresponding nonlinear refractive index by using the mechanism of nonlinear refractive index change caused by Gaussian light intensity. Figure 1 shows SSPM patterns observed at different wavelengths. Figure 2 shows intensity distribution of experimental results in Figure 1(c) and the corresponding theoretical simulation results. From Figure 2, the experimental results fit theoretical results well.

In 2015, XIAO et al [33] reported dynamic self-diffraction phenomenon of femtosecond pulsed laser in MoS₂ dispersions and data analysis and simulation. The formation speed of self-diffraction phenomenon relates to the laser intensity, solution viscosity and the concentration of MoS₂ nanometer fragments, which can be proved via the completion time and final observation and analysis of the three stages of self-diffraction (see Figure 3). What’s more, the simulation method based on Gerchberg- Saxton algorithm was original.

In 2016, ZHANG et al [34] first reported the SSPM phenomenon of black phosphorus nanosheets, finding that black phosphorus has the strongest photosensitivity in the 2D materials that have been reported up to now. In June of the same year, LI et al [35] reported the SSPM phenomenon of MoS₂ with regular orientation of MoS₂ with regular orientation. It is believed that the orientation alignment caused by non-local electron coherence may enhance the nonlinear optical properties of 2D materials. In 2017, LI et al [36] reported to realize the three-phase all-optical switch using the SSPM effect of Bi₂Se₃ nanosheets. In 2018, JIANG et al [37] reported that the coupling TiO₂ was used to wavelength-dependently modulate the SSPM phenomenon of graphene oxide and realized that long-wave SSPM was stronger.

3 SSPM effect dominated by thermal effect of laser light intensity

In 2018, WANG et al [38] reported that they observed fast distorted self-diffraction rings in the vertical direction by horizontally incident graphene discrete solution with a focused continuous (CW) laser (see Figure 4). They believed that the thermal effect of laser light intensity leads to the thermal- like lens effect in the discrete solution of 2D materials, which leads to SSPM. Therefore, the distortion of self-diffraction is mainly caused by the non-axisymmetric thermal convection resulting from the strong laser thermal effect, and the relative change of nonlinear refractive index can be determined by the ratio of the distortion angle and the half-cone angle. It can be obtained from the following formulas that the nonlinear refractive index is caused by heat convection induced by laser:

                              (4)

                        (5)

For Gaussian beams θ_H it can be expressed as:

                               (6)

n_2e before and after the change can be expressed as:

                         (7)

where n_e, n_0e and n_2e are respectively refractive index, linear refractive index and nonlinear refractive index; I is the incident laser intensity. is the corresponding phase shift of the laser beam after passing through the graphene dispersion, and its effective optical path is L; C is a constant θ_H is distortion angle; θ_D is the half-cone angle.

Figure 1 SSPM patterns observed at different wavelengths:(The collapse time is less than 1 s. Reprinted with permission from Ref. [32]. Copyright (2015) American Institute of Physics)

Figure 2 Axial intensity distribution (a) and corresponding simulated intensity distribution (b) in Figure 1(c) (Reprinted with permission from Ref. [32]. Copyright (2015) American Institute of Physics)

Figure 3 Three-stage (expanding, collapsing and unbalancing) SSPM dynamics (a) and phase distribution restored using GS algorithm (b) (Reprinted with permission from Ref. [33]. Copyright (2015) The Optical Society)

Figure 4 Schematic of half-cone angle and distortion angle (a), initial perfect circular at 0.35 s (b), distorted SSPM diffraction ring pattern until its stabilization (c) (Reprinted with permission from Ref. [38]. Copyright (2018) American Institute of Physics)

In 2017, SHAN et al [39] used CW laser to study the SSPM phenomenon of graphene, and believed that the direct cause was thermal lens effect caused by laser, which had little relationship with the type of 2D materials. The SSPM effect caused by laser thermal effect was studied in materials such as MoS₂, graphene oxide and Bi₂Si₃. SHAN et al [40] focused on measuring various new 2D materials TaSe₂ using CW lasers. SSPM effect is due to thermal effects of discrete bodies, such as NbSe₂ and graphene oxide. Other two-dimensional materials of graphene in 2D materials, such as MoS₂ [41], graphene oxide [40, 42], Bi₂Si₃ [43], TaSe₂ [44], 2D tellurim [45] and graphdiyne [46]. Materials can also produce SPPM effect caused by laser thermal effects. JIA et al focused on the use of CW lasers to measure various new 2D materials GeSe [47], NbSe₂ [48], MXene [49] and other composite materials [50] through theoretical analysis on 2D materials that the generation of the SSPM effect is due to thermal effects.

In 2018, SUN et al [51] believed that pulsed laser heating produced bubbles of similar size in water without 2D materials, and each bubble became a source of scattering. The high intensity laser field places the bubbles in the proper position so that they can interfere constructively. This interpretation has something to do with the early interpretation of the Z-scan results for 2D material solutions (the optical limiting of early 2D material suspensions is considered to be nonlinear scattering caused by microbubbles). SUN et al [52, 53] also studied the SSPM effect caused by thermal effects in pure solvents and solids. ABEYWICKREMA  et al [54] also believed that the generation of thermal bubbles leads to the emergence of the SSPM phenomenon.

4 Nonlocal electron coherence of laser beam intensity

In 2015, WU et al [55] reported the broad- band SSPM phenomenon in MoS₂ dispersions, and first proposed a wind-chime model to account for the emergence of nonlocal electron coherence caused by laser beam intensity in 2D material sheets. Due to laser-induced electron coherence, these flakes are rotated and aligned to be along the light polarization direction. Figure 5 shows the mimetic diagram of a wind chime. The light wave is coherent with the wave functions A and B, forming a coherent state of the wave function C in the far field.

In 2016, WANG et al [56] reported the SSPM phenomenon of MoSe₂ suspension solution, compared the SSPM phenomenon of graphene, and continued to study the non-local electronic coherence and wind bell-like arrangement mechanism.

5 Modulation and application of SSPM

For lack of theory system, the modulation only focuses on two dimensions of diffraction ring number and vertical imbalance degree, while further study is needed. The current theoretical research is mostly limited to the study of system parameters, such as imaging distance of intrinsic physical mechanism [57, 58].

Specifically, for the modulation of the diffraction ring number, the number of self- diffraction rings increases accordingly as the illumination intensity increases. The nonlinear refractive index and the third-order polarizability of the corresponding 2D material can be obtained via the relationship between the intensity of the illumination and the number of self-diffracting rings. This method to measure the third-order polarizability has been widely used (see Figure 6). However, in the case where the mechanism of SSPM is not cleared, the corresponding nonlinear refractive index results from high laser intensity directly or the accumulation of the laser thermal effect, or it has nothing to do with the material at all, there is not a conclusive answer.

Figure 5 Wind-chime model and emergence of electron coherence (left), collective response of MoS₂ flakes to laser beam electric field, being aligned to form a “wind chime” (right (a)), SSPM pattern formed (b, c) (Snapshots of the pattern formation at 473-nm and 532-nm laser beam excitations, respectively. The whole process takes~0.20 s, which is consistent with the wind-chime model prediction (main text). Reprinted with permission from Ref. [55]. Copyright (2015) PNAS)

For a vertical imbalance degree, it is generally believed that thermal effect caused by the accumulation of laser radiation leads to thermal convection, which makes the self-diffraction rings deform up and down, as shown in Figure 7.

However, thermal convection does make the thermal lens deform in 2D dispersions and then cause vertical imbalance, or thermal convection does cause the uneven concentration of 2D materials, which indirectly causes the uneven nonlinear refractive index, thereby imbalance? If the latter, the deformation of self-diffraction can be used to study the subtle concentration changes, which is worthy a further study.

Figure 6 SSPM of graphene flakes to use relationship between number of SSPM rings and incident laser intensity to calculate third-order polarizability (CW laser was used in this test). (Reprinted with permission from Ref. [58]. Copyright (2016) American Institute of Physics)

Figure 7 Fast distorted self-diffraction rings in vertical direction by horizontally incident graphene discrete solution by laser. Researcher believed vertical imbalance mainly results from non-axisymmetric thermal convection (Reprinted with permission from Ref. [59]. Copyright (2014) American Institute of Physics)

For the application of SSPM, owing to uncleared mechanism, it is used as a black box, still in its infancy [60-66]. For examples, all-optical switching and wavelength switch were achieved via SSPM of few layers SnS₂ [67]. Using non-local electronic coherence theory, in 2015, WU et al [55] proposed the application of two-color all-optical switch.

Researchers used CW laser of 473 nm and 532 nm as the control light and controlled light, respectively, successfully achieved that weak light was used to modulate strong light as shown in Figure 8, where the ratio of weak light to strong light can reach 1:60, using SSPM of MoS₂ as the core mechanism. This study can be further explored, because some important parameters of optical switch, such as response time, response wavelength, have not been given or optimized in detail or simulated [67-70]. In addition, the use of vector light field for SSPM is also available [71-74].

6 Conclusions

From the above reports, the SSPM study on 2D materials has entered a critical stage. If the SSPM is dominated by the thermal effects of strong laser, the study should be biased collective effect with a response time of the second level; if the SSPM is dominated by the nonlinear optical characteristics of the 2D material itself, then this effect can be extended to ultrafast dynamics and nonlinear optics of 2D materials. Thus, it is urgent to establish a unified and universal theoretical system of SSPM of 2D material to confirm the self-diffraction mechanism of the collective or monomer in 2D materials dispersion solution to achieve full modulation and give a unique self-diffraction application. In general, the SSPM is dominated by the thermal effects, if CW laser is used or the optical absorption is huge in resonance absorption wavelength; the SSPM is dominated by the nonlinear optical characteristics of the 2D material itself, if the femtosecond pulsed laser is used as a light source with high transient intensity but low heat effect. This review will benefit to fully reveal the self-diffraction dynamics mechanism of 2D materials, expand the application range of 2D materials in the field of SSPM, and provide basic theory and technology for finally realizing the application of 2D materials in SSPM.

Figure 8 CW laser of 473 nm and 532 nm as control light and controlled light, respectively, successfully achieved that weak light was used to modulate strong light, using SSPM of MoS₂ as the core mechanism:(Reprinted with permission from Ref. [55]. Copyright (2015) PNAS)

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(Edited by YANG Hua)

中文导读

二维材料的空间自相位调制研究进展

摘要：当强激光入射二维材料离散体时，透过的激光会出现自衍射环，这是一种空间自相位调制现象，可以用于测试非线性光学参数和实现光开关。现在，关于空间自相位调制现象的机理仍有争论，主要聚焦在是二维材料的非线性折射率主导还是激光的热效应主导。理论的缺失限制了相位调制的维度，只能调控衍射环的半径和竖直方向的失衡。因此，有必要建立一个普适的空间自相位调制理论体系。

关键词：非线性光学；空间自相位调制；二维材料

Foundation item: Project(6187031976) supported by the National Natural Science Foundation of China

Received date: 2019-08-27; Accepted date: 2019-09-20

Corresponding author: XIAO Si, PhD, Associate Professor; Tel: +86-18673123070; E-mail: sixiao@csu.edu.cn; ORCID: 0000-0002- 7172-3607