Nonlinear generation of hollow beams in tunable plasmonic nanosuspensions

Over the past few decades, colloidal suspensions and soft matter in general have attracted a great deal of interest due to their potential applications in a variety of fields ranging from life sciences to chemistry, physics, as well as photonics.^1–9 Of particular interest are synthetic and artificial materials that exhibit novel nonlinear optical properties such as dielectric nanoparticle suspensions.^10–15 Various synthetic materials with extraordinary nonlinear properties have been demonstrated for applications in photonic devices including optical modulators and optical switches.^16–21 Plasmonic nanosuspensions, for instance, have been proposed and tested as an ideal material candidate for applications in nonlinear optics.^22–30 Such colloidal systems are particularly interesting due to the fact that the optical response of suspended metallic nanoparticles is mediated by a localized surface plasmon resonance (SPR) and can be readily tuned through composition, size, and shape.^22,31–33 The tunable optical nonlinearity in plasmonic nanosuspensions exhibits superior performance, as compared to their dielectric counterparts, in supporting beam guidance and self-trapping.^{22,23,28,29,34}

In colloidal suspensions of metallic nanoparticles, robust soliton-like propagation has been observed due to an effective cubic–quintic-like plasmonic resonant nonlinearity.^26,34–37 The wavelength-dependent nonlinear response can be used to explore promising photonic applications based on controlling light by light. Interestingly, a plasmonic nanosuspension can be used as a tunable nonlinear medium in which the refractive index changes along the beam path, for instance, when a Gaussian-like beam of a particular wavelength is launched through it. As a result, a nonlinear phase shift occurs on another beam that is collinearly propagating. The process of phase shift in one beam modulated by intensity-dependent index changes in another beam is the well-known cross-phase modulation (XPM), which has been intensively investigated for controlling light-by-light technologies.^38–41 Yet, most of the previous endeavors have been focused mainly on the context of nonlinear optical fibers, liquid crystals, or conventional nonlinear materials.^42–47 One may wonder what advantages and new features could emerge when a plasmonic resonant nonlinearity comes to play a role.

On the other hand, “hollow” Gaussian beams have been extensively investigated due to their special intensity distribution and propagation characteristics^48,49 and have attracted much attention in the research fields of atom manipulation,^50–52 optical trapping,^53–55 and plasma physics.^56–59 The propagation and transformation dynamics of a hollow beam in nonlinear media have also been explored for potential applications in optical limiting⁶⁰ and high-energy laser propagation.⁶¹ Various methods for generating hollow beams have been proposed and demonstrated, such as employing spatial filtering, beam shaping, optical fibers, and the nonlinear interaction of photons with orbital angular momentum.^62–65 A natural question is if the nonlinear response of plasmonic suspensions to light excitations can bring about a new possibility for the generation of hollow beams. Specifically, can the XPM mentioned earlier be used to induce hollow beam formation in metallic nanosuspensions, taking advantage of their plasmonic resonant nonlinearity?

In this work, we propose and demonstrate nonlinear hollow beam generation in plasmonic nanosuspensions. We analyze and discuss wavelength-dependent nonlinear optical responses in two typical gold nanoparticle (nanosphere and nanorod) suspensions. Experimentally, we observe that a low-power probe beam propagating in colloidal gold nanosuspensions can be modulated and controlled by a pump beam at a different wavelength, resulting in an unexpected nonlinear reshaping of its initial intensity profile into a doughnut-like pattern. We find that the probe and pump wavelengths can be interchanged by using different gold nanosuspensions (for instance, we can have a green beam for the pump and an infrared beam for the probe for nanosphere suspension, but the same infrared beam for the pump and the green beam for the probe for nanorod suspension), yet the hollow pattern only appears in the probe beam , which does not exhibit a strong nonlinear response on its own. Furthermore, our experimental results are also corroborated by numerical simulations using coupled nonlinear Schrödinger equations (NLSEs) involving the XPM effect from the two copropagating optical beams.

To explore the optical properties of plasmonic nanosuspensions, it is crucial to have a good understanding of the response of individual nanoparticles to an optical field. SPR dominates the optical response of metallic nanoparticles and depends critically on their size, shape, and composition, as well as the optical properties of the irradiating beam laser, including the wavelength, polarization, and intensity.^22,31,66 In previous studies, the SPR of independent metal nanoparticles induced by an external electromagnetic field has been characterized by polarization, absorption, and scattering cross sections, whose spectral evolutions associated with these physical parameters reach their maximum value when the field wavelength approaches the SPR frequency. In addition, it is possible to achieve polarization contrast in specified spectral regions by selecting the proper polarization-dependent composite film comprised of metal nanoparticles.^67–69 Furthermore, plasmonic nanosuspensions illuminated by a light with a frequency near the SPR give rise to changes in the refractive index occurring in the proximity of the beam path. By appropriately engineering the main components of plasmonic colloids, we can straightforwardly tune the associated wavelength-dependent nonlinear optical responses, thus controlling the flow of light in soft-matter systems. In what follows, we shall investigate the nonlinear response in colloidal suspensions of two exemplary plasmonic gold nanoparticles (see Fig. 1) and demonstrate experimentally their suitability as a nonlinear medium to control light by light.

Besides, we also obtained experimentally the extinction spectra of two nanosuspension samples under examination. The combined absorption-extinction measurements are often used to characterize the optical properties of plasmonic nanosuspensions, where the extinction spectrum is the sum of the absorption and scattering of light by the metal nanoparticles.^22,76 The instrument used for measurement is a broad-spectrum spectrophotometer (Hitachi U-4100), and the characterization is performed by illuminating the two samples with incoherent white light. Measurements are presented in Figs. 1(a4) and 1(b4), and they refer to normalized results for the case of a colloidal suspension of nanospheres and nanorods, respectively. According to these measurements, we found that the gold nanosphere in suspension manifests an intense extinction effect near the wavelength of 532 nm. This result is consistent with the value previously obtained in simulations, and it is also indicative of the fact that a 532 nm pump leads to a significant nonlinear response for this soft matter. Meanwhile, the pairs of peaks appearing in the extinction spectrum of the gold nanorod suspension can be attributed to the transverse and longitudinal SPR modes, highlighting that the overall wavelength-dependent extinction effect of the plasmonic nanosuspension is the collective manifestation of an individual nanorod. One can see clearly that the peak value of the extinction cross section related to the longitudinal mode is much larger than that of the transverse mode, and as a consequence, the gold nanorod exhibits a strong nonlinear response near the wavelength of 1064 nm. Therefore, for the sample filled with gold nanorods, we use the laser wavelength of 1064 nm as the pump beam for our experimental characterization.

A schematic representation of the experimental setup used for observing the hollow-beam formation due to nonlinear beam coupling in plasmonic suspensions is shown in Fig. 2. A 30 mm-long quartz cuvette filled with a 5 ml mixture of the aqueous gold nanosuspensions (XFNANO Producer: No. XFJ6 7440-57-5 for the nanosphere suspension and No. XFJ61 7440-57-5 for the nanorod suspension; whose uniform concentration of these samples is about 0.05 mg/ml) is inserted in the optical path. When the sample containing gold nanosphere suspensions is examined, the continuous-wave (CW) laser operating at λ = 532 nm (green) acts as the pump beam, while the other CW laser source at the NIR wavelength (λ = 1064 nm) serves as the probe. On the contrary, the pump and probe beams are switched when the sample with gold nanorod suspensions is tested. In order to induce a plasmonic resonant optical nonlinearity, the pump beam is focused inside the cuvette directly in the plasmonic solution, and its focal point position is about 10 mm from the front surface of the cuvette. The pump power can be varied from 0 to 300 mW. Meanwhile, the probe beam is injected collinearly with the pump, and its input power is fixed at 5 mW for all sets of measurements. Before being focused on the sample, the collinear pump and probe beams are combined by a dichroic mirror. Due to the very low power, the probe beam does not experience nonlinear self-action itself in the plasmonic suspension. The probe intensity pattern at the output face of the sample is then measured by means of a notch filter in order to remove the high-intensity pump light. A CCD camera is employed to record the output intensity profiles of the probe beam for various input pump powers. In addition, as the notch filter obstructing the pump beam is replaced, the output intensity pattern of the pump beam when the hollow beam is generated can also be measured. Both nonlinearly-induced hollow beams and probe-beam guidance are verified by simply recording their corresponding output pattern images.

From the direct examination of the output intensity profiles of the probe beam for every experimental setting, the formation of hollow patterns in gold nanosuspension is observed to take place over a wide range of pump powers, from 30 to 250 mW. As seen in Figs. 3 and 4, our experimental results conducted in an aqueous suspension containing either gold nanospheres with a 40 nm average diameter or gold nanorods with a 6.5 aspect ratio demonstrate the generation of hollow beams under different wavelengths and power conditions.

In gold nanosphere suspensions, the pump beam is focused on the plasmonic sample at different input powers and experiences plasmonic resonant optical nonlinearity, while the collinear NIR probe beam is fixed at a low power of 5 mW to prevent nonlinear self-action. Experimental results in Figs. 3(b1)−3(b6) show clearly that the initial Gaussian-like intensity pattern of the NIR probe beam converts into a hollow-shaped beam after passing through the gold nanosphere suspension, especially under intense pump irradiations. In the linear regime of the pump beam, the low-power probe undergoes normal diffraction and does not exhibit an hollow pattern, as shown in Fig. 3(b1). However, as the pump power increases above a certain threshold value, nonlinear coupling takes place in the suspensions, and the probe beam propagating collinearly with the pump is guided and starts to deform and differ from the fundamental Gaussian pattern at the output, as shown in Fig. 3(b2). For higher pump powers, the probe-beam intensity progressively evolves into a hollow pattern. In particular, the hollow pattern appears for values higher than 30 mW, with both the ring radius and the hollow size varying with the pump power. Meanwhile, the output intensity patterns of the pump beam are also recorded by means of a notch filter [see Figs. 3(a1) and 3(a2)]. Only two output intensity patterns measured at two specific powers are presented for convenience in the illustration. For a relatively high power, one can see that the increasing tendency of the output pump size indicates a negative or self-defocusing response, although no hollow pattern is observed for the pump itself; this is a nonlinear effect that is mainly caused by a plasmon resonance. The self-defocusing nonlinearity of nanoparticle suspension under continuous light illumination has been verified by Z-scan experiments.^70–72 

Subsequently, the formation of nonlinearly-induced hollow beams is also observed in nanorod suspensions (see Fig. 4). In this case, the power threshold at which the hollow shaped pattern appears increases with respect to that in the nanospheres suspension. Comparing Figs. 3(b2) and 4(b3), the necessary pump power to create an evident hollow-beam pattern is significantly different for nanosphere and nanorod suspensions. This difference is mainly attributed to the nonlinear optical properties of the samples as well as a slight difference in the waist sizes of the two probe beams. However, when only the NIR pump beam is present, the output intensity patterns are consistent with previously obtained experimental results for gold nanosphere suspension. The dark-region size of the induced hollow beam is power dependent in the plasmon nanosuspension. In addition, we have also carried out additional experiments by using a gold nanorod suspension with a plasmon resonance at 810 nm (off from both pump and probe wavelengths) for a direct comparison, as shown in Fig. S3 in the supplementary material, thus highlighting the need for a plasmonic enhancement to initiate the nonlinear effect. At increasing pump powers, the dark region of the hollow beam becomes larger, indicating that gold nanosuspensions have a tunable nonlinear response within a certain range. We note that tunable hollow beams have demonstrated superior stability and efficiency in particle trapping and manipulation applications.^55,77 Here, the dimensions of the hollow beam can be modulated according to the pump power as well as the size of the particle, thus offering great potential for particle manipulation and non-invasive diagnosis.^53,78

To confirm that the observed nonlinear response is the result of a surface plasmon resonance, we did a series of comparison experiments. First, the same experiment is carried out in a pure aqueous solution without immersing any gold nanoparticles, and in this case, no appreciable nonlinear action on either the probe or the pump beam is observed, thus excluding the influence of the solution itself. Second, we carried out the same experiment by using a gold nanorod suspension with a plasmon resonance off both the pump and the probe wavelengths. In this latter case, the output intensities of the probe beam (again with fixed low power at 5 mW) do not exhibit a pronounced hollow pattern generation as the NIR pump power increases, indicating that the nonlinear response is mediated by the plasmonic resonant absorption.

In summary, we have demonstrated tunable hollow beam generation in nonlinear plasmonic nanosuspensions, illustrating an example that plasmonic gold nanosuspensions represent a versatile platform for controlling the flow of light in soft-matter systems. We have shown the wavelength-dependent nonlinear optical response of gold nanosphere and nanorod suspensions, and then we observed that a low-power probe beam can be reshaped and controlled by a high-power pump beam over 30 mm-long propagation through the plasmonic nanosuspensions. Furthermore, we have illustrated the cross-phase modulation effect of two beams in the nonlinear medium, which indicates that plasmonic nanosuspensions have great potential for all-optical limiting and switching. We envision that more exotic structured beams can also be designed and created in a variety of optical systems with plasmonic resonant interactions. These results will not only add insight into nonlinear optics with plasmonic nanosuspensions but may also offer an effective plasmonic platform for all-optical signal-processing technologies and devices.

The supplementary material includes additional experimental results in pure aqueous solution without immersing any Au nanoparticle, other conditions for nonlinear beam propagation in different plasmonic nanosuspensions for direct comparisons, as well as data from calculations for a single gold nanorod particle.

We acknowledge Trevor Kelly and Huizhong Xu for insightful discussion.

This research was funded by the National Key R&D Program of China (Grant No. 2022YFA1404800), the National Natural Science Foundation of China (Grant Nos. 12134006 and 11922408), and the Natural Science Foundation of Tianjin (Grant No. 21JCJQJC00050). D.B. acknowledges support from the 66 Postdoctoral Science Grant of China and the National Natural Science Foundation (Grant No. 12250410236). R.M. acknowledges support from NSERC and the CRC research program. J.Y. acknowledges support from NSF grant (DMS-1910282).

The authors have no conflicts to disclose.

J.Z., D.L., and D.B. contributed equally to this work.

Jingyan Zhan: Conceptualization (equal); Data curation (equal); Methodology (equal); Software (equal); Validation (equal); Visualization (equal); Writing – original draft (equal). Denghui Li: Conceptualization (equal); Investigation (lead); Software (equal); Supervision (lead); Validation (equal). Domenico Bongiovanni: Formal analysis (equal); Writing – original draft (equal). Yinxiao Xiang: Validation (equal). Shengyao Chen: Formal analysis (equal). Yujie Zhang: Formal analysis (equal). Liqin Tang: Writing – review & editing (equal). Daohong Song: Writing – review & editing (equal). Jianke Yang: Writing – review & editing (equal). Roberto Morandotti: Supervision (supporting); Writing – review & editing (supporting). Zhigang Chen: Funding acquisition (equal); Project administration (equal); Resources (equal); Supervision (equal); Writing – review & editing (equal). All authors contributed to the editing of the manuscript.

All source data that support the plots within this paper and other findings of this study are available from the corresponding authors upon reasonable request.