2. Equation of small-scale self-focusing of spatially partially coherent beams and the complex screen method to synthesize spatially partially coherent beams

The analyzed spatially PCBs were quasi-monochromatic with a temporal coherence length that was much larger than the spatial coherence length. Here we focus on laser-induced breakdown with nanosecond pulse duration in the high peak power Nd:glass laser system. The response time of the Nd:glass was significantly shorter than the pulse duration and the coherence time, allowing the self-focusing to be steady-state self-focusing.

The nonlinear wave equation in paraxial approximation^[ Reference Bespalov and Talanov^8] has the following form:

(1) $$\begin{align}{\nabla}_{\perp}^2E+2 jk\frac{\partial E}{\partial z}=-{k}^2\left(\frac{n_2{\left|E\right|}^2}{n_0}\right)E,\end{align}$$

where $\left|\frac{\partial^2E}{\partial {z}^2}\right|<<k\left|\frac{\partial E}{\partial z}\right|$ ; ${\nabla}_{\perp}^2$ is the transverse Laplacian; ${n}_0$ denotes the linear refractive index; ${n}_2$ is the nonlinear refractive index; $k$ is the wave vector in the medium. Equation (1) includes the diffraction term ${\nabla}_{\perp}^2E$ and the nonlinear term $-{k}^2\left(\frac{n_2{\left|E\right|}^2}{n_0}\right)E$ . This equation serves as a foundation for investigating the SSSF of spatially PCBs.

The modulated optical field $E$ can be described by linear superposition of a strong background field $T$ (infinite plane waves with arbitrary spatial coherence) with finite small-scale perturbation fields^[ Reference Brown^2]:

(2) $$\begin{align}E=T\left(x,y,z=0\right)\left(1+\sum \limits_i{u}_i(z){e}_i\left(x,y\right)\right),\end{align}$$

where ${u}_i(z){e}_i\left(x,y\right)<<1$ , ${u}_i(z)=a(z)+ ib(z)$ , and both $T$ and $E$ satisfy Equation (1)^[2]. Substituting Equation (2) into Equation (1), we obtain Equation (3):

(3) $$\begin{align}&{\nabla}_{\perp}^2e\left(x,y\right)\times Tu(z)+2 jkTe\left(x,y\right)u{(z)}_z+2{k}^2\left(\frac{n_2{\left|T\right|}^2}{n_0}\right)\nonumber\\ &\times T\times \mathit{\operatorname{Re}}\left(u(z)e\left(x,y\right)\right)=-2u(z)\left({T}_xe{\left(x,y\right)}_x+{T}_ye{\left(x,y\right)}_y\right),\end{align}$$

where $u{(z)}_z=\frac{\partial u(z)}{\partial z}$ , ${T}_x=\frac{\partial T}{\partial x}$ and ${T}_y=\frac{\partial T}{\partial y}$ . The left-hand side of Equation (3) is consistent with the SSSF equation of coherent theory. The right-hand side of Equation (3) is the underlying physics that produces the difference in SSSF between spatially PCBs and coherent beams, where $e{\left(x,y\right)}_x,\ e{\left(x,y\right)}_y$ and $u(z)$ are all associated with the modulation. Here, ${T}_x$ and ${T}_y$ are proportional to $\sqrt{G\left({v}_x,{v}_y\right)}$ , and $G\left({v}_x,{v}_y\right)$ represents the power spectrum. When $G\left({v}_x,{v}_y\right)\to \delta \left({v}_x,{v}_y\right)$ , Equation (3) is reduced to the equation for coherent beams. Thus, the SSSF is related to the light intensity and to the power spectrum (which indicates the spatial coherence of the beams) of the optical field for spatially PCBs.

We then introduced the CS method to synthesize spatially PCBs with different spatial coherence. The methodology is presented as follows.

Figure 1. The correlation functions ${\mu}_\mathrm{Gauss}$ and ${\mu}_\mathrm{Bessel}$ are shown in (a) with different $\sigma$ . The corresponding power spectra ${G}_\mathrm{gauss}$ and ${G}_\mathrm{circle}$ are shown in (b). Here, $\sigma ={\sigma}_\mathrm{Gauss}$ are 0.5 and 0.227 mm, respectively.

The cross-spectral density (CSD) function can be expressed as follows:

(4) $$\begin{align}W\left({\boldsymbol{r}}_1,{\boldsymbol{r}}_2,z\right)=<T\left({\boldsymbol{r}}_1,z\right){T}^{\ast}\left({\boldsymbol{r}}_2,z\right)>.\end{align}$$

The brackets represent the time average over the response time of the medium, $\boldsymbol{r}=\widehat{\boldsymbol{x}}x+\widehat{\boldsymbol{y}}y$ . If the statistical properties of spatially PCBs are of the Schell-model type, we obtain the following:

(5) $$\begin{align}W\left({\boldsymbol{r}}_1,{\boldsymbol{r}}_2,z=0\right)={E}_{\mathrm{c}}\left({\boldsymbol{r}}_1\right){E_{\mathrm{c}}}^{\ast}\left({\boldsymbol{r}}_2\right)\mu \left({\boldsymbol{r}}_1-{\boldsymbol{r}}_2\right),\end{align}$$

where ${E}_{\mathrm{c}}\left(\boldsymbol{r}\right)$ denotes the coherent part of the beam. According to the condition proposed by Gori et al. ^[ Reference Gori and Santarsiero^43, Reference Gori, Ramírez-Sánchez, Santarsiero and Shirai^44] for devising a genuine correlation function of PCBs, $\mu$ can be expressed as follows:

(6) $$\begin{align}\mu \left({\boldsymbol{r}}_1-{\boldsymbol{r}}_2\right)=\iint \sqrt{G\left({\boldsymbol{v}}_1\right)}\sqrt{G\left({\boldsymbol{v}}_2\right)}\delta \left({\boldsymbol{v}}_1-{\boldsymbol{v}}_2\right)\times {e}^{-j{\boldsymbol{r}}_1{\boldsymbol{v}}_1}{e}^{-j{\boldsymbol{r}}_2{\boldsymbol{v}}_2}\mathrm{d}{\boldsymbol{v}}_1\mathrm{d}{\boldsymbol{v}}_2,\end{align}$$

where $\boldsymbol{v}=\widehat{\boldsymbol{x}}{v}_x+\widehat{\boldsymbol{y}}{v}_y$ , and $\delta \left({\boldsymbol{v}}_1-{\boldsymbol{v}}_2\right)$ is the Dirac function, which can be expressed as follows:

(7) $$\begin{align}\delta \left({\boldsymbol{v}}_1-{\boldsymbol{v}}_2\right)=<R\left({\boldsymbol{v}}_1\right)R{\left({\boldsymbol{v}}_2\right)}^{\ast }>\hspace{-1pt},\end{align}$$

where $R\left(\boldsymbol{v}\right)$ is a random complex function whose real and imaginary parts are independent, with unit variances and standard normal distributions. When $R\left(\boldsymbol{v}\right)$ is refreshed, a new random optical field $T$ is generated. Substituting Equations (6) and (7) into Equation (5), the CSD function can be rearranged as follows:

(8) $$\begin{align}W\left({\boldsymbol{r}}_1,{\boldsymbol{r}}_2,z=0\right)\approx \frac{1}{N}\sum \limits_{n=1}^N{T}_n\left({\boldsymbol{r}}_1\right){T}_n^{\ast}\left({\boldsymbol{r}}_2\right),\end{align}$$

with the following:

(9) $$\begin{align}T\left(\boldsymbol{r}\right)={E}_{\mathrm{c}}\left(\boldsymbol{r}\right)\times \varphi \left(\boldsymbol{r}\right),\end{align}$$

(10) $$\begin{align}\varphi \left(\boldsymbol{r}\right)=\iint \sqrt{G\left(\boldsymbol{v}\right)}R\left(\boldsymbol{v}\right){e}^{-i2\pi \boldsymbol{rv}}\mathrm{d}\boldsymbol{v}.\end{align}$$

The CSD function can be described by Equation (8) to obtain sufficient optical fields. Different $T\left(x,y,z\right)$ can be obtained by changing the spatial coherence lengths and line shapes of the correlation function $\mu$ . Using the Fourier transform, we can compute the corresponding power spectrum $G\left({v}_x,{v}_y\right)$ based on the spatially partially coherent optical fields $T\left(x,y,z\right)$ .

3. Simulations

As Equation (3) is unsolvable analytically, we used the split-step Fourier method with the following parameters: ${n}_0=1.5$ , ${n}_2=1.5\times {10}^{-13}\left(\mathrm{esu}\right)$ , $\lambda =1.053\;\unicode{x3bc} \mathrm{m}$ , ${I}_0=10\;\mathrm{GW}/{\mathrm{cm}}^2$ , propagation distance $L=2\;\mathrm{cm}$ . The computational grid of $512\times 512$ points corresponds to a physical size of $2\;\mathrm{cm}\times 2\;\mathrm{cm}$ . The number of CSs was set as $1.1\times {10}^5$ and the spatial frequency $f$ was accompanied by a small-scale modulation at $z=0\;\mathrm{cm}$ , where ${a}_0=0.01$ , ${b}_0=0$ and $e\left(x,y\right)=\cos \left(2\pi \cdot f\cdot x\right)$ . The MI gain coefficient of the SSSF was obtained by calculating the degree of modulation of the intensity distribution through the transmission. The amplitudes are satisfied as follows:

(11) $$\begin{align}\frac{u(z)}{u(0)}=\frac{{e}^{gL}+{e}^{- gL}}{2}.\end{align}$$

We simulated the impact on the MI gain coefficient $g$ by varying the spatial coherence lengths and line shapes of the correlation function $\mu$ . Figure 1(a) demonstrates the 1D distributions of Gaussian correlation functions ${\mu}_\mathrm{Gauss}={e}^{-\left(\frac{x^2+{y}^2}{2{\sigma}^2}\right)}$ , where $\sigma$ is the spatial coherence length and the Bessel correlation function ${\mu}_\mathrm{Bessel}={J}_1\left(2\pi \boldsymbol{vr}\right)$ . We defined the spatial coherence length ${\sigma}_\mathrm{Bessel}$ as the value of the first zero of the Bessel function. Figure 1(b) demonstrates the corresponding power spectra, ${G}_\mathrm{gauss}$ and ${G}_\mathrm{circle}$ , which satisfy the following:

(12) $$\begin{align}{G}_\mathrm{gauss}\left(0,0\right)={G}_\mathrm{circle}\left(0,0\right),\end{align}$$

(13) $$\begin{align}\iint {G}_\mathrm{gauss}\left({v}_x,{v}_y\right)\mathrm{d}{v}_x\mathrm{d}{v}_y=\iint {G}_\mathrm{circle}\left({v}_x,{v}_y\right)\mathrm{d}{v}_x\mathrm{d}{v}_y.\end{align}$$

Equations (12) and (13) indicate that both power spectra exhibit the same maximum values and equal energy. Thus, we use ${\sigma}_\mathrm{Gauss}$ uniformly to refer to $\sigma$ in the following.

We set $f=10$ , 15, 20, 25, 30, 35, 40, and $42\;{\mathrm{cm}}^{-1}$ . In Figure 2(a), $\mu ={\mu}_\mathrm{Gauss}$ ; in Figure 2(b), $\mu ={\mu}_\mathrm{Bessel}$ , and the simulations are displayed as data points, each point representing a result of the MI gain coefficient $g$ at the modulation frequency. The simulation results for the MI gain coefficient $g$ of the coherent beams fit well with the curves established using the analytical formula (solid line), indicating the correctness of the simulation. For the PCBs, the MI gain coefficients $g$ are consistently lower than those of the coherent beams, yet they exhibit a similar trend. Thus, we postulate that this is comparable to a reduction in the input intensity when injecting coherent beams. The remaining lines in Figure 2 represent the analytical gain curves of the MI gain coefficient $g$ of coherent beams with different injection intensities. As expected, the curves match the data points perfectly at $10\;\mathrm{GW}/{\mathrm{cm}}^2$ and other injection intensities, verifying the accuracy of the prediction. Thus, the formula for $g$ in coherent theory is extended to the following^[ Reference Boyd, Lukishova and Shen^45]:

(14) $$\begin{align}g=\frac{\left|K\right|}{2k}\sqrt{\frac{2\pi {I}_0}{P_\mathrm{cr}}\alpha -{\left|K\right|}^2},\end{align}$$

Figure 2. Analytical gain curves (lines) corresponding to different input intensities: (a) simulation results of the MI gain coefficient $g$ at different spatial coherence lengths when $\mu ={\mu}_\mathrm{Gauss}$ , ${I}_1=9.51\;\mathrm{GW}/{\mathrm{cm}}^2$ , ${I}_2=9.12\;\mathrm{GW}/{\mathrm{cm}}^2$ , ${I}_3=8.64\;\mathrm{GW}/{\mathrm{cm}}^2$ , ${I}_4=8.17\;\mathrm{GW}/{\mathrm{cm}}^2$ ; (b) simulation results of the MI gain coefficient $g$ at different spatial coherence lengths when $\mu ={\mu}_\mathrm{Bessel}$ , $I'_1=9.67\;\mathrm{GW}/{\mathrm{cm}}^2$ , $ I_2'=9.41\;\mathrm{GW}/{\mathrm{cm}}^2$ , $ I_3'=9.12\;\mathrm{GW}/{\mathrm{cm}}^2$ , $ I_4'=8.93\;\mathrm{GW}/{\mathrm{cm}}^2$ .

Figure 3. Variations in $\alpha$ and B-integral with respect to different light densities and correlation functions.

where $K=2\pi \cdot f$ ; ${P}_\mathrm{cr}=\frac{\lambda^2c}{32{\pi}^2{n}_2}$ is the critical power; $\alpha$ represents the decrease factor, $\alpha =1$ for coherent beams and $\alpha <1$ for spatially PCBs. Referring to Equation (14), we deduce the fastest growing frequency ${K}_\mathrm{max}=\sqrt{\frac{\pi {I}_0}{P_\mathrm{cr}}\alpha }$ and the maximum gain coefficient ${g}_\mathrm{max}=\frac{\pi {I}_0}{2{kP}_\mathrm{cr}}\alpha$ for PCBs. Equation (14) illustrates that the suppression effect of spatially PCBs for SSSF can be equated to a reduction in the injected light intensity or an increase in the critical power of coherent beams. Figure 3 compares the effect of the line shape and the spatial coherence length of the correlation function on the decrease factor $\alpha$ and the B-integral for the same incident light intensity; it also compares the effect of the incident light intensity on the decrease factor $\alpha$ and the B-integral with the same spatial coherence. Experience has shown that the B-integral must be less than approximately 2 to avoid unacceptable small-scale modulation growth^[ Reference Nakano, Miyanaga, Yagi, Tsubakimoto, Kanabe, Nakatsuka and Nakai^46]. Thus, we set the B-integral of the coherent beams to 2 and $B={g}_\mathrm{max}L$ without considering the gain of the medium. It is obvious from Figure 3 that the $\alpha$ corresponding to the Gaussian correlation function is smaller than that corresponding to the Bessel correlation function. This is because the spatially PCBs of the Gaussian correlation function have a wider power spectrum range when energy is conserved, implying that the larger the spatial divergence angle of the beams, the more rapidly the B-integral decreases and the better the suppression of SSSF. For the same spatial coherence length, the B-integral decreases more when the incident light intensity is low, which means that to achieve the same SSSF suppression effect, the spatial coherence of the beam is much higher at a low light intensity than at a high light intensity. For the same spatial coherence, $\alpha$ decreases nonlinearly with incident intensities, which means the coupling between the spatial coherence and the incident light intensity of spatially PCBs. This conclusion is consistent with the physical nature revealed by Equation (3). To characterize the reference value of the spatial coherence of the beam, we used the ratio $\beta$ of the period of the fastest growing modulation to the spatial coherence length. For example, with an incident light intensity of $10\;\mathrm{GW}/{\mathrm{cm}}^2$ , the B-integral decreasing rate began to accelerate significantly when $\beta$ was greater than 1.13, corresponding to a spatial coherence length of 0.26 mm. This result aligns consistently with diffraction by apertures illuminated with spatially PCBs^[ Reference Shore, Thompson and Whitney^47].