# Stimulated Thermal Scattering in Two-Photon Absorbing Nanocolloids under Laser Radiation of Nanosecond-to-Picosecond Pulse Widths

^{1}

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## Abstract

**:**

^{−1}. We develop the first order-in-perturbation model of the four-wave mixing two-photon absorption-assisted SRMS process which shows that at nanosecond pulses, amplification is predominantly due to the thermal-induced coherent oscillations of polarization while the slow temperature wave acts also as a dynamic spatial grating which provides a self-induced optical cavity inside the interaction region. At a picosecond pulse width, according to our model, the spectral overlap between pump and signal pulses results in formation of only the dynamic spatial temperature grating, and we succeeded at recovering the linear growth of the spectral shift with the pump power near the threshold.

## 1. Introduction

^{−1}which are close to the SBS spectral shifts. We provide a comprehensive theoretical analysis of 2PA SRMS in both the nanosecond and picosecond pump pulse width regimes. We treat it as a nonlinear four-wave mixing process. At nanosecond pulse width, the pump and the backward and forward signal waves are spectrally well-separated, and the two-photon absorption process results in two kinds of temperature perturbations (and coupled to them density perturbations), one of which is the spatially homogeneous coherent oscillation at the beat frequency, and the second is the slow temperature wave propagating at a velocity close to the speed of sound. We show that the amplification effect in the nanosecond interaction regime is provided predominantly by coherent thermal oscillations, while the slow temperature wave acts also as a dynamic spatial grating which provides a self-induced optical cavity inside the interaction region. At picosecond pulse width, spectra of the pump and backward signal pulses overlap, coherent polarization oscillation disappears, and only the dynamic temperature grating provides the scattering and amplification effects.

## 2. Materials and Methods

#### 2.1. Preparation of Stable Ag and ZnO Nanoparticle Suspensions

#### 2.2. Measurements of the Two-Photon Absorption Coefficient of Laser Radiation in Nanoparticle Colloids

^{−4}cm/GW and $7.2\times {10}^{-2}\mathrm{cm}/\mathrm{GW}$, respectively [5]. This discrepancy prompted us to clarify this value and measure nonlinear 2PA coefficient of Ag suspension at the 100 ps pulse duration. To do so, we performed the measurements of nonlinear transmission of 100 ps laser pulses through the cell with 0.005 M Ag nanoparticle colloid, i.e., the dependence of transmission through the cell energy of the laser pulse on its energy at the front edge of the cell.

^{−1}we fit the measured transmissions of the 100 ps laser pulse with $\beta =6.2\times {10}^{-2}\mathrm{cm}/\mathrm{GW}$ which is close to the value measured at the 10 ns laser pulse width in [5].

#### 2.3. Nanosecond 2PA SRMS Experimental Scheme

^{4+}crystal and operating at TEM

_{00q}mode. After amplification and frequency doubling, it provides us with nearly Fourier-transform-limited laser radiation at the wavelength of 0.527 μm, the pulse energy up to 20 mJ, and the 12 ns FWHM pulse width corresponding to 40 MHz spectral width.

_{1}providing the caustic length of ~1 cm in air. The 2PA SRMS emerges through four-wave mixing between two (forward and backward) pump waves and two (forward and backward) coupled signal waves.

#### 2.4. Picosecond 2PA SRMS Experimental Scheme

## 3. Results

#### 3.1. Experimental Results of 2PA SRMS in the Nanosecond Time Domain

#### 3.2. Experimental Results of 2PA SRMS with 100 ps Laser Pulse Width

^{−1}which is almost as high as for the SBS frequency shift but has the opposite sign (in Ag nanoparticle colloid in toluene, SBS reveals as the Stokes shifted signal while 2PA SRMS reveals as the anti-Stokes signal). For the points at sufficiently high pump pulse energies in the right-hand side of Figure 11, the signal and the pump spectra almost overlap with each other but can be rather well separated on interferogram due to sufficiently high power of the backscattered signal.

## 4. Discussion

#### 4.1. Model of the Nonlinear Medium Response under 2PA SRMS

_{j}and α

_{j}are the number density and the polarizability of molecules and nanoparticles of all types j contained in a colloidal solution.

#### 4.2. Theory of 2PA SRMS through Four-Wave Mixing in Nanosecond Regime

_{NL}. These terms represent two types of resonance temperature fluctuations: the first kind (the first and the second terms in (15)) corresponds to the slow temperature wave $~\mathrm{exp}[\pm i(\mathsf{\Omega}t+qz)]$ with the beat frequency $\mathsf{\Omega}={\omega}_{+}-{\omega}_{0}={\omega}_{0}-{\omega}_{-}$ and the wavenumber $q=2{k}_{0}\approx {k}_{+}+{k}_{0}\approx {k}_{0}+{k}_{-}$ propagating in the negative direction of the axis z along with the anti-Stokes signal. This is a dynamical thermal space grid emerging as a result of the interference between the counter-propagating pump and signal waves. It provides efficient self-consistent distributed feedback through the Bragg reflection of amplified waves [26]. The second kind of temperature perturbation (the third and the fourth terms in Equation (15)) $~\mathrm{exp}[\pm i(\mathsf{\Omega}t+\mathsf{\Delta}kz)]$ oscillates also at the beat frequency Ω but the wavenumber is $\mathsf{\Delta}k=|{k}_{\pm}-{k}_{0}|$. Note that at our experimental conditions with the measured frequency shifts of ~100 MHz, one can estimate $\mathsf{\Delta}kL<<1$. In fact, these terms describe homogeneous coherent temperature oscillations which provide coherent amplification, i.e., coherent energy transfer, from pump to signal waves in the four-wave mixing process.

#### 4.3. Theory of Picosecond 2PA SRMS Backward Signal Amplification

_{0}(x) is the Bessel function of zero order. Finally, calculating the nonlinear signal amplitude, we find

## 5. Conclusions

^{−1}which are close to the SBS spectral shifts. The scattering efficiency (signal-to-pump ratio of pulse energies) is measured to be very high reaching ~50% in Ag and even above in ZnO nanocolloids, which make the 2PA SRMS very prospective for phase conjugation in the picosecond domain. We extended our four-wave mixing model of 2PA SRMS into the picosecond time domain. In contrast with the nanosecond domain, where the pump and the forward and backward scattered signals are well separated in frequencies, the spectra of pump and signal waves significantly overlap at picosecond 2PA SRMS. As a result, only the dynamic temperature grating provides the scattering and amplification effects making the picosecond 2PA SRMS more akin the parametric amplification process.

## Author Contributions

## Funding

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**The results of Ag (

**a**,

**b**) and ZnO (

**c**,

**d**) nanoparticles characterization: (

**a**,

**c**) characteristic size measurements using the correlation spectroscopy method processed with the use of DynaLS software; (

**b**,

**d**) scanning transmission electron microscope patterns along with the diameter values extracted for several nanoparticles.

**Figure 2.**Nonlinear transmission of the 100 ps 0.532 μm laser radiation in the 5 cm-thick layer of Ag nanoparticle suspension measured in two domains of pulse energies, (

**a**) below 1 mJ, (

**b**) $1\xf74.5$ mJ. Experimental points are given by rectangles. Normalized output radial intensity profiles are shown below the plots: (

**c**) low-distorted near-Gaussian profile at low pulse energies and (

**d**) heavily distorted profile at $1\xf74.5$ mJ pulse energies. Green lines in (

**a**,

**b**) show the fitting according to Equation (2).

**Figure 3.**Experimental setup for nanosecond four-wave mixing SRMS generation. It contains: 5 cm interaction cell (C), lenses with the focal lengths 7.5, 5.5, and −15 cm (F

_{1−3}, respectively), total reflection mirrors (M), the glass wedge (W), the 4 mm aperture (D), the neutral density filter (NF), and two photodiodes PD of ~1 ns resolution. Typical oscilloscope traces of the pump and Anti-Stokes signal pulses are shown at correspondent PD in the right upper corner. The direction map of the pump pulses and scattering signals is shown below the cell C.

**Figure 4.**Experimental setup for 2PA SRMS of 100 ps pulses. There are two interaction geometry options, (

**a**) sharp focusing of pump radiation into the interaction cell of 3 cm length with the lens of 5 cm focal length, and (

**b**) mild focusing into short 1 cm interaction cell with the lens of 21.5 cm focal length, providing the four-wave mixing process due to the backward reflection of the pump pulse by the rear side of the cell. The upper part in (

**b**) shows the scheme of Fabry–Perot spectral measurements: Pr—waveguide prism; F

_{4-5}—objectives transcribing the edge of the prism on the CCD matrix plane.

**Figure 5.**Typical Fabry–Perot spectra in the case of nanosecond pulses: pump (

**lower part**), backscattered Brillouin and Rayleigh signals (

**upper part**).

**Figure 6.**Spectral shifts of the anti-Stokes backward SRMS signal vs the pump pulse energy in the nanosecond pulse interaction regime, experimental points are given by triangles.

**Figure 7.**Efficiency of backward SRMS reflection vs. the pump pulse energy in the nanosecond interaction regime, which is measured as the peak-to-peak ratio of PD signals. Experimental points are shown by rectangles. Threshold pump pulse energy is ~0.3 mJ, the slope is ~0.4.

**Figure 8.**SRMS backscattering light spectra initiated by 100 picoseconds pulses in Ag nanoparticle colloids in toluene (

**a**) and ZnO nanoparticle colloids in water (

**b**,

**c**). The results are given for the following experimental schemes: (

**a**) four-wave mixing scheme at the mild focusing case (Figure 4b with F

_{2}= 21.5 cm): (

**b**) sharp focusing scheme (Figure 4a with F

_{0}= 5 cm); (

**c**) scheme of Figure 4a with the replaced lens of F

_{0}= 15 cm focal length.

**Figure 9.**Backscattering efficiency for 0.003 M Ag nanoparticle solution in toluene for the 100 ps pump laser pulse, the threshold is ~0.1 mJ. Experimental points are shown by rectangles.

**Figure 10.**Reflectivity (signal-to-pump ratio of pulse energies) of ZnO nanoparticle solution in water vs. pump pulse energy measured for the Stokes shifted (red circles) and anti-Stokes shifted backscattered radiation (black squares) from 100 ps pump laser pulses.

**Figure 11.**Anti-Stokes 2PA SRMS spectral shifts vs. the pump pulse energy measured in Ag nanoparticle colloid solution (0.003 M) under 100 ps 0.532 μm laser radiation.

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**MDPI and ACS Style**

Erokhin, A.I.; Bulychev, N.A.; Parkevich, E.V.; Medvedev, M.A.; Smetanin, I.V. Stimulated Thermal Scattering in Two-Photon Absorbing Nanocolloids under Laser Radiation of Nanosecond-to-Picosecond Pulse Widths. *Nanomaterials* **2022**, *12*, 2567.
https://doi.org/10.3390/nano12152567

**AMA Style**

Erokhin AI, Bulychev NA, Parkevich EV, Medvedev MA, Smetanin IV. Stimulated Thermal Scattering in Two-Photon Absorbing Nanocolloids under Laser Radiation of Nanosecond-to-Picosecond Pulse Widths. *Nanomaterials*. 2022; 12(15):2567.
https://doi.org/10.3390/nano12152567

**Chicago/Turabian Style**

Erokhin, Alexander I., Nikolay A. Bulychev, Egor V. Parkevich, Mikhail A. Medvedev, and Igor V. Smetanin. 2022. "Stimulated Thermal Scattering in Two-Photon Absorbing Nanocolloids under Laser Radiation of Nanosecond-to-Picosecond Pulse Widths" *Nanomaterials* 12, no. 15: 2567.
https://doi.org/10.3390/nano12152567