Brillouin spectroscopy based on the pump-probe technique can provide the intrinsic information of the medium, which has been widely used in sensing scenarios such as local temperature and strain monitoring [1,2], biomedical diagnostics [3,4], imaging [5], and material science [6]. The core of this technique is to determine the Brillouin spectrum with high efficiency, or rather the Stokes frequency shift of the medium, by acquiring the power gain/loss of the probe versus frequency scanning [7–9], as shown in Fig. 1(a). This pump-probe method is easier to operate in Raman spectroscopy, but it is still a challenge for present commercial photoelectric equipment for Brillouin spectroscopy [10]. More specifically, in contrast to the Raman spectrum, the frequency region of the Brillouin spectrum usually is several MHz to GHz, depending on the sample to be measured [11]. Consequently, it is difficult to separate the pump and probe using a frequency sorter (e.g., a dichroic mirror). Using a polarization sorter, such as a polarized beam splitter (PBS), is another common way to achieve a collinear pump-probe scheme, as shown in Fig. 1(b). Unfortunately, the pump stray light, comprising the elastic and inelastic scattering background, has almost the same polarization component as the output probe; thus, it can overwhelm the tiny variation of the probe power [12–14]. As a tradeoff, one has to use a weak pump power, which leads to a low signal-to-noise ratio (SNR) and long acquisition time. To overcome this issue, Remer et al. recently proposed a scheme that uses hot rubidium-85 vapor as a notch filter to highly absorb the background noise, and realized a high SNR and high-speed stimulated Brillouin scattering (SBS) spectroscopy measurements [15]. However, a pump wavelength first needs to match with an absorption line of the atom; then, the vapor can only remove the elastic scattering of the pump. The SBS noise, an amplified spontaneous Brillouin scattering of the pump, cannot be filtered out. The SBS noise arises from the interactions between the pump and incoherent phonons, and it will be the main background noise for the case of a strong pump and weak signal [16–18]. Therefore, to date, only a noncollinear pump-probe scheme can in principle realize a complete signal–noise separation because the signal and noise are naturally spatially separated [19–21]. In this scenario, however, one has to give up all the merits of a collinear and confocal system, such as high gain and simple operation.

Alternatively, in our previous work, we demonstrated a signal–noise spatial separation for collinear Brillouin pump-probe scheme [as shown in Fig. 1(c)], where the degree of freedom of the orbital angular momentum (OAM) was used to extract probe signals from the strong SBS noise [22]. Generally speaking, the incoherence SBS noise can be considered as the mixing of various spatial modes, including lower-order OAM modes, which can also be demultiplexed into Gaussian modes and mixed into Brillouin signals. Moreover, the demultiplexing process needs a far enough propagating evolution distance and a pinhole with an appropriate aperture such as a low-pass filter, which results in a large energy loss. To address these problems, in this work, we put forward what we believe, to the best of our knowledge, is a novel signal–noise spatial separation method for a collinear pump-probe scheme. By spiral phase filtering for the precoding signal (called spiral phase precoding), we achieve on-demand tailoring spatial distribution of Brillouin signals and hence, in principle, the signal spatial separation from the background noise without loss. In the following, we first present the theoretical approach and discuss the principle of spiral phase precoding involving higher-order vortex filtering; then we introduce this scheme into the pump-probe technique based on stimulated Brillouin amplification (SBA). Finally, the proof-of-principle work is verified by experimentation.

In a collinear Brillouin pump-probe scheme, to realize the signal–noise separation in the transverse plane at the output port, the key is reshaping the intensity distribution of the signal and noise, and the ideal situation is to use a lossless operation. The spiral phase precoding is inspired from the spiral phase contrast (SPC) imaging, which has been demonstrated as a powerful tool for edge detection in image processing by a vortex structure filter [23]. In this technique, exerting a different OAM to signal photons in the Fourier space can change the path of photon propagation, and then change the spatial position of photons, or rather the spatial structure of the image. Figure 2(a) shows the generic setup of SPC imaging with a typical Fourier spatial filter (i.e., spiral phase plate, SPP). The output of the object function $E_{\mathrm{in}}(r,\phi )$ after the spiral phase filtering system can be written as $E_{\mathrm{out}}^{\ell }(r,\phi )={\mathcal{F}}^{-1}\{\mathcal{F}\{E_{\mathrm{in}}(r,\phi )\}\exp (i\ell \phi )\}$, where $\ell$ is the order (i.e., topological charge) of SPP filter and the azimuth angle $\phi$ corresponds to phase values in a range from 0 to $2\pi$. This equation can also be described as the convolution (represented by the symbol $\otimes$) of the object function $E_{\mathrm{in}}(r,\phi )$ with the kernel $K_{\ell }(r,\phi )=i\exp (i\ell \phi )/(2\pi r^2)$ [24] [that is, $E_{\mathrm{out}}^{\ell }(r,\phi )=E_{\mathrm{in}}(r,\phi )\otimes K_{\ell }(r,\phi )$]. This indicates that the spatial structure of the output image can be tailored by changing the input image or filter. It has been demonstrated that inserting an SPP with $\ell =1$ or $\ell =2$ in the Fourier space can generate the spatial distribution of the brightness enhancement of edges or areas with curved edges [25–27]. Compared to other edge enhancement methods, such as numerical spatial differentiation and dark-field imaging, this technique has the advantage of being near lossless [28,29]. To further discuss the influence of the higher-order filtering on the spatial structure of the imaging, we present a method of the separation of variables and numerical iterative. Thus, the output imaging via spiral phase filtering with $\ell =2$ can be specified as $$\begin{gathered} E_{\mathrm{out}}^2(r,\phi )={\mathcal{F}}^{-1}\{\mathcal{F}\{E_{\mathrm{in}}(r,\phi )\}\exp (i2\phi )\} \\ ={\mathcal{F}}^{-1}\{\mathcal{F}\{E_{\mathrm{in}}(r,\phi )\}\exp (i\phi )\}\otimes {\mathcal{F}}^{-1}\{\exp (i\phi )\} \\ =E_{\mathrm{out}}^1(r,\phi )\otimes K_1(r,\phi ). \end{gathered}$$(1)

On this basis, we can obtain the output after filtering with any higher order ($\ell \ge 2$), which then can be described in the form $$E_{\mathrm{out}}^{\ell }(r,\phi )=E_{\mathrm{out}}^{\ell -1}(r,\phi )\otimes K_1(r,\phi ).$$(2)Equation (2) shows that the output light field after spiral phase filtering with $\ell$ order can be obtained by the convolution of the output via $\ell -1$ order filtering and the first-order kernel function. More specifically, the calculation process of the higher-order SPP Fourier filtering is equivalent to performing multiple spiral phase filtering of $\ell =1$ on the initial injected image light field step by step.

Here, we choose a phase pattern that consists of a pentagram, square, and circle, as shown in Fig. 3(b0), to demonstrate selective highlighting of higher-order vortex filtering and spatial reshaping of Brillouin signals. The phase pattern contains rich information such as curves, straight lines, and corners so that we can clearly observe the highlighting and tailoring of the spatial structure of the light field. More generally, another phase pattern can be used according to different scenarios. The phase jump on the edge of each pattern is set to $\pi$. Figure 3(a0) shows the initial Gaussian profile. Figures 3(a1) and 3(a2) depict the simulation of the standard isotropic edge enhancement with $\ell =1$, and the brightness of the contours of curve and angle with $\ell =2$, respectively. Figures 3(a3)–3(a6) show the simulation results of higher-order reshaping on the spatial structure of the input Gaussian beam, with $\ell =3$, 4, 5, 6. Figures 3(b1)–3(b6) show the simulation results of the output phase profiles after spiral phase filtering with $\ell =1$, 2, 3, 4, 5, 6. One can see a solid spot and corresponding flat phase appearing in the center of the beam and phase profiles for $\ell =5$, respectively, as shown in Figs. 3(a5) and 3(b5). This is mainly because the spectrum of the original phase pattern has an OAM component of $\ell =-5$, which can be specifically studied by OAM spectrum analysis [30]. Figures 3(c0)–3(c4) show the experimentally observed transverse profiles for $\ell =0$, 1, 2, 4, 6, with a continuous light source. Both the simulation and experimental results confirm our prediction.

The SPC imaging has been explored in second-order nonlinear optical processes (e.g., second harmonic generation) to achieve a visible edge enhancement with invisible illumination [31–33]. Different from previous studies, we show, for the first time, to the best of our knowledge, a new mechanism of the SPC technique in an SBA-based pump-probe scheme. The nonlinear interaction occurs between the pump and precoding probe signal. This probe beam refers to the angular spectrum of Brillouin signal loading a specific image, with synchronous processing of spiral phase filtering in a Fourier space, as shown in Fig. 2(b). Behind the SPP a further set of two relay lenses is arranged such that the signal angular spectrum is imaged in the Brillouin amplifier cell (BA cell). Hence, the process of nonlinear energy transfer for weak signals can be simplified with the dot product operation as $$\mathcal{F}\{E_{\mathrm{in}}(r,\phi )\}·e^{i\ell \phi }·\exp [g_0\frac{P_P}{A}(L-z)],$$(3)where $P_P$ is the pump power, $g_0$ is the nonlinear gain coefficient, and $A$ is the area of the beams. It can be seen that the amplified signal angular spectrum grows exponentially with the distance, and $L$ is the total interaction length. Additionally, note that the procedure of spiral phase filtering and nonlinear amplification is a parallel processing in the Fourier space. After lens ${\mathrm{L}}_2$ “undoes” the Fourier transform, as shown in Fig. 2(b), we obtain the filtered and amplified output signal $E_{\mathrm{out}}^{\mathrm{SBA}}(r,\phi )$ in the image plane, which can be given by $$E_{\mathrm{out}}^{\mathrm{SBA}}(r,\phi )=E_{\mathrm{in}}(r,\phi )\otimes K_{\ell }(r,\phi )·\exp \left(g_0\frac{P_P}{A}L\right).$$(4)We can see from Eq. (4) that the Brillouin signal via spiral phase precoding has been amplified, accompanying the formation of the filtered image with a tailorable spatial structure. Meanwhile, the background noise can also be focused into smaller speckle spots near the optical axis by the lens ${\mathrm{L}}_2$. Therefore, the signal can be spatially separated from the noise by choosing a suitable topological charge and image structure.

Figure 4 shows the schematic illustration of this proof-of-principle configuration. The pump beam is produced by a pulsed Nd:YAG laser, which has an output at 532 nm after frequency doubling, a pulse width of 7.8 ns, a repetition rate of 1 Hz, and a linear polarization Gaussian profile. A 6.9 ns Gaussian-pulse beam with a Stokes frequency shift is used as the signal beam, which is generated by the SBS in the same nonlinear medium as the BA cell (the generation setup of the Stokes signal is omitted in Fig. 4). Thus, the phase-matching condition can be satisfied in the SBA process to ensure high gain. We choose ${\mathrm{CS}}_2$ as the nonlinear medium; its Brillouin frequency shift is about 7.7 GHz at room temperature and wavelength of 532 nm [34]. The combination of a half-wave plate (${\mathrm{HWP}}_1$ or ${\mathrm{HWP}}_2$) and polarized beam splitter (${\mathrm{PBS}}_1$ or ${\mathrm{PBS}}_2$) is used to control the polarization and intensity of the two beams, respectively. A desired phase jump image is loaded to the signal via a spatial light modulator (SLM) (PLUTO-2-NIR-011, HOLOEYE Photonics AG, Berlin, Germany). Then, the first lens ${\mathrm{L}}_1$ carries out the Fourier transform so that we obtain the angular spectrum of the precoding signal in its back-focal plane. The SPP [VPP-1c, RPC Photonics (now part of Viavi Solutions), Henrietta, NY, USA] with various orders in different zones is located in Fourier plane, which acts as the vortex filters. The filtered angular spectrum of the signal is imaged into the BA cell by a group of relay lenses consisting of two lenses ${\mathrm{L}}_2$ and ${\mathrm{L}}_3$. The diameter of the 150 mm long BA cell is 25.4 mm. The pump and angular spectrum of the signal interact in the center of the BA cell. The amplified angular spectrum can be reflected from the ${\mathrm{PBS}}_3$, and it “undoes” Fourier transform by the fourth lens ${\mathrm{L}}_4$. Besides, background noises such as the SBS noise are focused into smaller speckle spots through the lens ${\mathrm{L}}_4$. We can filter out the noise by a customized high-pass filter, which is generic quartz plate with a block in the center and the antireflection coating on both sides. All lenses have a focal length of 200 mm. We employ a CCD camera (BC106N-VIS, Thorlabs, Newton, NJ, USA) to record the output images, and an energy meter (919E-0.1-12-250, Newport Corp., Irvine, CA, USA) to measure the output beam energy.

In our experiment, we first investigated the intensity distribution of the amplified output and compared it with the corresponding input precoding signal profile. The average energies of the pump and input signal are set to 0.5 mJ and 10 nJ, with the same diameter of 1.5 mm, respectively. The CCD is set to a fixed exposure. Figures 5(a1)–5(a3) show the intensity profiles of the signal without vortex filtering [Fig. 5(a1)] and with vortex filtering of $\ell =1$, 2 [Figs. 5(a2) and 5(a3)], respectively. We can see that the input signal with the original Gaussian profile is reshaped after the SPP is inserted; in other words, the tailorable spatial structure of input signal via spatial phase precoding is clearly done. At the same time, nonlinear energy transmission occurs between the pump and signal angular spectrum. Figure 5(b1) illustrates the amplified output signal profile without spiral phase filtering. Figures 5(b2) and 5(b3) show the highlighting tailored spatial structure of the amplified signal with spiral phase filtering of $\ell =1$, 2, respectively. Particularly noteworthy is that the observed output profile shows the desired structure and retains all the characteristics and key information of the original phase image. Here, the background noise is much weaker than the amplified signal due to the weak pump, so it is difficult to observe in the output image. These results indicate that the nonlinear interaction and spiral phase precoding are processed in parallel and, more importantly, the signal angular spectrum can remain high fidelity during nonlinear amplification in the Fourier space. Because the phase matching is realized in the nonlinear interaction, the frequency shift of the output signal after spiral phase precoding remains the same as the Brillouin shift of the medium.

Then, to demonstrate the signal–noise separation method, we need to increase the pump energy to generate a more noticeable noise. Figure 6 shows the output mixed beam of the amplified signal and noise when the pump and input signal are set to 5 mJ and 1 μJ, respectively. In this case, the SBS noise is the main component of the background noise. We can see that the noise can be focused to a smaller speckle spot after passing through lens ${\mathrm{L}}_4$, and completely separated from the signal profile, as shown in the dotted red circles of Figs. 6(a) and 6(b), respectively. It is easy to filter out the noise with a customized high-pass filter to improve the SNR of the Brillouin signal. The SNR can be determined by the energy ratio of the amplified signal to the background noise detected by the energy meter [17,35]. The SNRs without and with spiral phase precoding of $\ell =1$ are $\sim 4.7\mathrm{dB}$ and $\sim 43\mathrm{dB}$, respectively, under the condition of the pump of 5 mJ and the input signal of $5\times {10}^{-12}\mathrm{J}$. These results confirm that the signal–noise separation is indeed verified in our optical system, indicating that the scheme can spatially separate the signal from the background noise, and have a high SNR. Furthermore, the phase pattern loaded on the signal can be flexibly adjusted to meet different application requirements.

Finally, we experimentally study the energy transfer between the pump and signal angular spectrum during nonlinear amplification. Figure 7(a) shows the dependence of the energy conversion efficiency [$\eta =(E_{\mathrm{Sout}}-E_{\mathrm{Sin}})/E_P$] on the pump energy ($E_P$) for the input signal energy ($E_{\mathrm{Sin}}$) of 0.1 mJ, where the $E_{\mathrm{Sout}}$ represents output signal energy. It can be seen that the $\eta$ will exceed 40% for $E_P>6\mathrm{mJ}$. The $\eta$ tends to be saturated as a result of the stronger competition from the pump self-SBS, and then can be improved by introducing a double-stage or laser beam combination amplification stage or employing special nonlinear liquids [20,35,36]. Moreover, we also achieve a high-gain amplification for a weak Brillouin spectrum signal, as seen in Fig. 7(b). A signal amplification factor [SAF, $(E_{\mathrm{Sout}}-E_{\mathrm{Sin}})/E_{\mathrm{Sin}}$] of $\sim 1.6\times {10}^7$ is obtained when the signal is reduced by about $5\times {10}^{-12}\mathrm{J}$. The pump energy $E_P$ is chosen as 5 mJ.

We can also see from Fig. 7 that $\eta$ and SAF remain slightly smaller when a higher-order vortex filter is used at the same conditions. This phenomenon ascribes to a poor overlap of the Gaussian profile pump and the zero-order Fourier component of the input signal image, which typically contains a major amount of the total signal intensity in the Brillouin amplification. The zero-order Fourier component of the image field attaches a helical phase through the SPP in the Fourier space, thus forming a donut profile. The beam with the higher-order helical phase has stronger diffraction effects, so the beam sizes change relatively rapidly as the propagation distance increases, resulting in a lower $\eta$ and SAF in a nonlinear interaction [37,38].

In summary, we have demonstrated on-demand tailoring spatial distribution of Brillouin spectrum signals by introducing the spiral phase precoding into the pump-probe scheme. We take a complex phase pattern as an example to study the selective highlighting via higher-vortex spiral phase filtering and spatial reshaping of Brillouin signals. The proof-of-principle experiments show that the intensity distribution of the Brillouin signal can be tailored and separated from the background noise by the ingenious Fourier transform. Furthermore, this method offers the advantages of high efficiency and high gain in Brillouin amplification. We believe our work provides a practical way toward quasi-noise-free nonlinear interactions involving pump-probe scanning detection.