## 2 Theory  
Figure 1 shows the theoretical model used in this study. A monopole primary source is located at \(\mathbf{r}_{\mathrm{p}}=(x_{\mathrm{p}},y_{\mathrm{p}},z_{\mathrm{p}})\) in the free field with a strength of \(q_{\mathrm{p}}\) and a secondary source is located at \(\mathbf{r}_{\mathrm{s}}=(x_{\mathrm{s}},y_{\mathrm{s}},z_{\mathrm{s}})\). The error plane with a dimension of \(L\times L\) is placed at \(z=0\), and both the primary and secondary sources are at the same side of it. The target direction of noise reduction is defined by the wavevector \(\mathbf{k}=(k_{x},k_{y},k_{z})=(k\ \mathrm{sin}\ \theta\ \mathrm{cos}\ \varphi,k\ \mathrm{sin}\ \theta\ \mathrm{sin}\ \varphi,\)\(k\ \mathrm{cos}\ \theta)\) with \(0\leq\theta<\pi/2\), \(0\leq\varphi<2\pi\), where \(k\) is the wavenumber; \(k_{x}\), \(k_{y}\), and \(k_{z}\) are the components of the wavevector in the \(x\), \(y\), and \(z\) directions, respectively. A far-field point in the target direction, \(\mathbf{r}_{0}=(x_{0},y_{0},z_{0})\), is used to evaluate the directivity control performance. To reduce the noise in the target direction, the sound pressure at the far-field point \(\mathbf{r}_{0}\) can be directly adopted as the cost function [20]  
\[J_{1}\ =\ \left|\frac{\mathrm{j}\omega\rho_{0}q_{\mathrm{p}}\mathrm{e}^{- \mathrm{j}\left|\mathbf{r}_{0}-\mathbf{r}_{\mathrm{p}}\right|}}{4\pi\left| \mathbf{r}_{0}-\mathbf{r}_{\mathrm{p}}\right|}+\frac{\mathrm{j}\omega\rho_{0} q_{\mathrm{s}}\mathrm{e}^{-\mathrm{j}\left|\mathbf{r}_{0}-\mathbf{r}_{\mathrm{s}} \right|}}{4\pi\left|\mathbf{r}_{0}-\mathbf{r}_{\mathrm{s}}\right|}\right|^{2} \tag{1}\]  
Figure 1: Schematic diagram of the theoretical model. where j, \(\omega\) and \(\rho_{0}\) are the unit imaginary number, angular frequency and air density, respectively. The optimized strength of the secondary source can be obtained by minimizing Equation (1), which is  
\[q_{\rm s,1}\ =\ -\frac{|{\bf r}_{0}-{\bf r}_{\rm s}|}{|{\bf r}_{0}-{\bf r}_{ \rm p}|}q_{\rm p}{\rm e}^{-{\rm j}{\rm k}(|{\bf r}_{0}-{\bf r}_{\rm p}|-|{\bf r}_ {0}-{\bf r}_{\rm s}|)} \tag{2}\]  
In the study of Fourier acoustics, the angular spectrum method is employed to decompose a complex sound field into a superposition of plane waves from different directions; therefore, the angular spectrum in target direction can also be adopted as the cost function in active directivity noise control. The angular spectrum through the \(z=0\) plane along the direction of \({\bf k}\) is [13]  
\[ \begin{array}{rl}P_{\rm p}({\bf k},0)&=\iint\limits_{S}p(x,y,0){\rm e}^{{\rm j }(k_{x}x+k_{y}y)}{\rm d}S\\ &=\frac{|{\rm i}\omega p_{0}q_{\rm p}}{4\pi}{\rm e}^{{\rm j}(k_{x}x_{\rm p}+k_ {y}y_{\rm p})}\iint\limits_{S}\frac{{\rm e}^{-{\rm j}\sqrt{x^{2}+y^{2}+z_{\rm p }^{2}}}}{\sqrt{x^{2}+y^{2}+z_{\rm p}^{2}}}{\rm e}^{{\rm j}(k_{x}x+k_{y}y)}{ \rm d}S\end{array}  \tag{3}\]  
where \(S\) and \(p(x,y,0)\) represent the infinite integration plane at \(z=0\) and sound pressure at a point \((x,y,0)\) in the plane. According to Ref. [21], Equation (3) is consistent with the formula for calculating the sound pressure at \((x,y,z_{\rm p})\) generated by an infinite plate in the \(z=0\) plane. By extending the relevant conclusion in Ref. [13] to the complex domain, Equation (3) can be written as  
\[P_{\rm p}({\bf k},0)\ =\ \frac{\rho_{0}c_{0}q_{\rm p}}{2\cos\theta}{\rm e}^{{ \rm j}{\bf k}{\bf r}_{\rm p}}. \tag{4}\]  
The angular spectrum of the secondary source in the direction of \({\bf k}\), i.e., \(P_{\rm s}({\bf k},0)\), can also be calculated referring to Equation (4).