Results and Discussion

Fig. 1 displays the experimental setup and typical SSPM patterns obtained from the flake domains of MoS₂ in a suspension (Methods). It can be seen from Fig. 1B that the sample is mainly composed of MoS₂ flakes with a thickness of 30 ∼ 55 layers. In Fig. 1 D–F we show the SSPM patterns obtained using 400-nm ultrafast laser pulses, 532-nm continuous wave (cw) laser beams, and 800-nm ultrafast laser pulses, respectively. The SSPM is a nonlinear optical process in that the phase (rather than the intensity) of the output beam depends on the input intensity (Methods).

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Fig. 1.

(A) Schematic lattice structure of MoS₂. (B) Scanning electron microscopy image of the MoS₂ flakes. (C) Schematic experimental setup. (D–F) The SSPM patterns generated using 400-nm, 532-nm, and 800-nm laser beams, respectively.

To unambiguously verify that we have observed SSPM, we performed intensity-dependence measurements. Fig. 2 shows that the number of rings N increases nearly linearly with the laser intensity, until the saturation is reached. By this linear dependence the optical nonlinear coefficient n₂, hence χ⁽³⁾, can be obtained directly (Fig. 2 and Methods). Because nearly all of the light is diffracted off the incidence direction and the Rayleigh scattering is low, ideally it is a purely coherent third-order nonlinear optical process. This pure coherence must originate from the coherence within the sample—the electron coherence, where the electronic wave function preserves a definite phase (see discussion below). In the case that photoexcited carriers become out of phase due to collisions, impurity scatterings, and boundary reflections, they reassume their coherence transiently (at a timescale of femtoseconds) under the external light pulse.

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Fig. 2.

Intensity dependence of ring numbers and bandgap dependence of χ⁽³⁾. The slopes directly illustrate the nonlinear coefficients χ⁽³⁾. The solid lines are guides to the eye. (Inset) The thresholds to observe the SSPM patterns at different photon energies.

Next we consider nonlocal electron coherence. In our sample, each flake is a domain. Charge carriers in any form (including photocarriers, excitons, free electron–hole pairs, and intrinsic electrons and holes) in different and the same domains are initially completely out of phase. Besides, each domain has an arbitrary orientation. Upon light interaction, photoinduced quasiparticles interact with the electric field (Fig. 3A) and assume its local phases.

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Fig. 3.

Wind-chime model and the emergence of electron coherence. (A, Left) Collective response of the MoS₂ flakes to the laser beam electric field, being aligned to form a “wind chime.” (A, Right) SSPM pattern formed. (B and 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).

We propose a wind-chime model to describe the emergence of electron coherence from the nonlocal domains driven by SSPM (Fig. 3A). Initially there exists an arbitrary orientation angle θ between a MoS₂ flake (thus the electrical polarization it contains) and the electric field. Due to energy relaxation, the electric field reorients the flakes so that each domain contains an axis parallel to the polarization of the external field, as if each domain is “hung” by a vertical “thread.” This image mimics that of a wind chime (Fig. 3A). Once the wind chime is formed, the electron coherence is completely set up within and among each of the different domains. This scenario depicts the emergence of nonlocal electron coherence as a complex collective behavior.

We first show that the model is reasonable. Assuming a circular disk shape for the MoS₂ flake domains, we derived (SI Text, section S1) that the rotation torque generated by the laser pulse M=∫|P×E|dV has a magnitude ofM=14sin 2θ(εr−1)εrε0E02πR2he−(2(ct−z)2/c2τ2),[1]where R is the disk radius, h is the disk thickness, and τ is the pulse width. Taking the Newton fluid with constant viscosity coefficient η and linear velocity v at the interface, the rotation torque V due to the viscous force isV=η∫0πdvdr(R⁡sin⁡φ)(ξ⋅2πR⁡sin⁡φ)(Rdφ)=πηΩξR3,[2]where Ω is the rotation velocity and ξ is the portion of the fluid globe that is rotating together with the disk. Hence the rotation angle accumulated due to each single light pulse is (SI Text, section S1)δθ=∫0TΩdt≈∫0∞Ω0e−(πηξR3/(JMoS2+JSolution))tdt=(εr−1)εr2TIh4ηξRcsin 2θ,[3]where T is the repetition period. The time needed for the pattern formation is (SI Text, section S1)T=εrπηξRc1.72(εr−1)Ih=0.3 s,[4]where we have used the values ε_r = 3.33 for MoS₂, η = 3.2 × 10⁻⁴ Pa∙s for acetone, R ∼ 1 μm, h ∼ 10 nm, I = 250 W/cm², and ξ ∼ 0.03 in our experiment. As a result, T does not depend on T. Thus, for both pulsed and cw excitations the results are identical, which has been verified in our experiment. To compare with the experiment, we recorded the formation processes of the ring patterns (see snapshots in Fig. 3 B and C). The ring number and diameter both increase monotonously with time until the maximum is reached after 0.20 s. This compares well with the predicted value by the wind-chime model. Thus, the observed time for the ring formation is a strong experimental support to the wind-chime model.

We then discuss the emergence of the ac electron coherence. Writing the electronic wave function at r_A as ψA(rA)=ρA(rA)eiϕ(rA), with ψA∗(rA)⋅ψA(rA)=ρA(rA) being the local electron density, the phase ϕ(rA)=k⋅rA−ω t+ϕ0(rA) of the electronic wave function is completely determined by the external light field (Fig. 3A). The wave function oscillates at a frequency of ∼10¹⁴ Hz (i.e., optical frequency). Classically, this is a forced oscillation, where electrons have to assume the enforced local phase by the external field, even surviving scatterings and collisions. Here ϕ​0(rA) represents the phase lag in response to the external field. It is not the initial random phase, which is wiped off by dissipation. The same applies to the electronic wave functions at r_B in a different (or the same) flake. Because the two wave functions are both coherent with the light wave, they are coherent to each other too. This is a dynamic or ac electron coherence, which is different from the commonly known steady-state or dc electron coherence in transport measurements. The electrons do not move far away from their equilibrium positions. After the wind-chime formation the electron coherence becomes stronger because more electrons get correlated and fewer collisions get involved, owing to the aligned geometry. At the far field, the intensity is I(rC)=E∗(rC)⋅E(rC), where E(r_C) is the electric field of the light wave at r_C. It becomes E(rC)=ς(χ(3))Ψ(rC), with Ψ(rC)=ei(k⋅rC−ωt+ϕ 0)[ρA(rA)e−i k⊥⋅rA+ρB(rB)e−i k⊥⋅rB] being the optical phase except for a normalization factor. Here k⊥=dΔϕ/dr is the wavevector generated due to the nonlinear response term of n₂ (Methods). The exact coefficient ς(χ(3)) is determined by the optical nonlinearity χ⁽³⁾, which is comprehensively determined by the carrier properties in the material.

We found that SSPM can be used to measure easily the χonelayer(3) value for many of the layered quantum materials (we have also observed SSPM in dispersion of MoSe₂ flakes and obtained the χ⁽³⁾ value). In other coherent methods (e.g., THG or FWM), the signal is much weaker than the input beam (25). Instead, in SSPM the diffracted beam is strong, owing to the intrinsic electron coherence. Significantly, SSPM can be used to measure χ⁽³⁾ from near IR to UV, compared with THG working only for the region where the 3 ω signal can be easily detected. This is essential for materials with a relatively large gap. In this sense we have found a powerful and ubiquitous (in some cases unique) way to measure χ⁽³⁾ for many of the 2D gapped quantum materials.

To quantitatively investigate the gap-dependent property, we obtained χ⁽³⁾ values for excitations at multiple wavelengths (Fig. 4A). For example χonelayer(3) at 532 nm is 1.6 × 10⁻⁹ electrostatic units (e.s.u.) (i.e., 2.23 × 10⁻¹⁷ m²/V²; Methods), which is orders of magnitude larger than that of conventional semiconductors, owing to the high carrier density and mobility in 2D layered materials. In parallel we calculated the absorption as a function of wavelength for few-layer MoS₂, where the dipole transition is proportional to the occupation probabilities of valence and conduction bands. As shown in Fig. 4A, the experimental results and theoretical calculations compare well. A sharp change in χ⁽³⁾ and threshold values can be seen at 1.72 eV (722 nm), which is exactly the gap value of MoS₂. This gap-dependent property is a manifestation of the electronic band structure of MoS₂. Thus, SSPM can be used to investigate band structures of layered quantum materials and can be conveniently compared with other experimental methods such as absorption and photoluminescence. The SSPM method applies especially to those materials where large single crystals are not available and absorption is hard to measure.

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Fig. 4.

Bandgap-dependent χ⁽³⁾ and I_th and the SSPM mechanisms. (A) Bandgap dependence of χ⁽³⁾ (solid spheres) and I_th (open circles) for ultrafast (blue) and cw (red) excitations, respectively. The calculated photoabsorption spectrum of MoS₂ is also shown (black and red curves) for comparison. The below-gap (purple) and above-gap (orange) regions are marked. (B) Scenarios of the SSPM formation mechanisms: Ⓐ Single-photon absorption-induced SSPM; Ⓑ two-photon absorption-induced SSPM; Ⓒ indirect-bandgap phonon-assisted SSPM. The bandgap structure is adopted from ref. 2. (C) Two-photon absorption leads to SSPM of the fundamental beam.

In Fig. 4A the experimental results compare well with the calculated absorption along x and y, but not that along the z direction (SI Text, section S2). This is strong evidence that, in our experiment, light propagation is perpendicular to the x or y but not the z axis, which confirms the validity of the wind-chime model. Note that this is an even stronger restriction than the wind-chime model. Considering the plasmonic interaction due to free carriers (26), it is indeed possible that the collective behavior leads to a “frozen” wind chime, where all domains become parallel to each other and perpendicular to the direction of laser propagation. The gap-dependent nature displayed in Fig. 4 shows again that SSPM is mainly due to the various charge carriers.

The below-gap SSPM has never been reported before. Three mechanisms might be responsible for it (Fig. 4 B and C): Ⓐ single-photon SSPM with Fermi distribution smearing, Ⓑ two-photon SSPM, and Ⓒ phonon-assisted indirect-gap SSPM. For Ⓐ, it can be seen that Fermi smearing relevant for significant photoabsorption occurs only at zones near the H and K points in momentum space. For Ⓑ, the carriers generated through two-photon absorption build up an electrical polarization, which diffracts not only those photons above the gap, but also those below the gap (Fig. 4B). As such, even though it is a two-photon nonlinear process, one expects a linear power dependence relation because the change in refractive index is proportional to the light intensity I. Heuristically, the excited carriers diffract both excitation photons (i.e., two output photons). In comparison there is only one output photon in second harmonic generation and two-photon photoluminescence. Two-photon SSPM is the most likely mechanism responsible for below-gap SSPM. For Ⓒ, the indirect gap photon excitation requires assistant phonon injection (Fig. 4C), which usually has a very small probability. In-gap defect states are relatively sparse (resonant in-gap states are more sparse) and localized (whereby charge carriers cannot move freely and scattering occurs frequently, which ruins the acquired uniform phase), so they are unlikely a major reason for SSPM.

We further demonstrate that, in forming SSPM, free carriers generated by photons at one wavelength can diffract photons at another wavelength, as if they were generated by the latter photons. The experimental setup is illustrated in Fig. 5C, which demonstrates a two-color (473 nm and 532 nm) SSPM. When we fix the power of one beam (even below threshold) and increase that of the other, the ring number and diameter for each beam both increase simultaneously. The brightness decreases, with the total intensity remaining unchanged, and energy conservation is preserved (Fig. 5D). In Fig. 5B the ring number of the 473-nm pattern exhibits linear dependence on the sum of the two beam powers, although the intensity of the 473-nm beam is fixed. The same is true for the 532-nm ring (SI Text, section S3). Once the coherence is correlated, each beam experiences the SSPM—as if the free carriers are created by the beam itself. Fig. 5A is an overall illustration of this property. This indistinguishability helps explain why in two-photon SSPM one can still have linear dependence and still obtain a relatively large χ⁽³⁾.

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Fig. 5.

All-optical switching based on SSPM. (A and B) Dependence of the 473-nm beam ring number on the sum intensity. (C) Schematic of the switch. (D) Generating the 473-nm pattern by increasing the 532-nm beam intensity. Both ring numbers (and diameters) increase simultaneously, with fixed 473-nm intensity below threshold.

Significantly, our experiment is to our knowledge the first demonstration of all-optical switching based on SSPM. Because the switching beam is controlling the phase of the other one, a small change in intensity will result in the shift of the strong signal pattern in real space, where a detector is placed. The detector can be made to detect the whole rings. Thus, a weaker beam can switch on and off a stronger beam (27, 28). Quantitatively taking the intensity of the signal beam (strong beam) at 473 nm to be I_strong = 120 W/cm², the intensity of the control beam (weak beam) at 532 nm needs only to be I_weak = 2.0 W/cm² to change the phase of the strong beam by π (i.e., ΔN = 0.5, Fig. 2). Hence to make an all-optical switch with full contrast ratio, I_weak:I_strong = 1:60. This is characteristic of weak-control–strong all-optical switching, which is apparently cascade possible (27, 28).

It is interesting that in the experiment intraband electron coherence has been correlated between photocarriers of 473-nm and 532-nm excitations. We note that similar coherence is also induced among the holes, the excitons, and the free electron–hole pairs. The two-color switching can further enable high-contrast-ratio operations, because in two-color operation optical filters can be used to block the signal beam. Two-color switching also makes it possible for multichannel mixed switching and thus enables very broad-band optoelectronic devices. We leave additional details of the SSPM-based all-optical switching to future investigations. In summary, our experiment demonstrates an all-optical switching based on SSPM, with the advantages including a weak-control–strong, cascade-possible, high-contrast-ratio, room-temperature device; broadband functioning; condensed state; and particularly broad-band-accessible quantum materials.