## 2 Theory
### Generation of Nondiffracting Beams with CGH  
Although the ideal real axicon can effectively generate nondiffracting Bessel beams, the fabrication errors will influence the quality of nondiffracting beams. Therefore, an easy-to-process method is put forward. The method is loading the CGHs to the SLM to achieve the phase modulation of the incident beam. The hologram for simulating the positive axicon is present in Figure 2a, and Figure 2b is the central profile of the hologram. From Figure 2b, the profile can be looked at as an assembly of equally spaced grooves, and \(d\) and \(h\) are the period and the height of the grooves, respectively. Figure 2: (**a**) A hologram for realizing the positive axicon, (**b**) the central profile of the hologram. Phase delay is determined by the phase difference of the light passing through different gray level in the SLM. Therefore, the transmission function of a single groove is:  
\[t(r)=\exp[-ik(n_{1}-1)r\tan\theta_{\text{B}}]=\exp[-ik(n_{1}-1)r\frac{h}{d}] \tag{8}\]  
where \(r\) is the radial position, \(k=2\pi/\lambda\), \(\theta_{\text{B}}=\arctan\left(\text{h/d}\right)\) is the base angle, and \(n_{1}\) is the refractive index of the groove. In the single-groove condition, when the size of \(r\) is \(d\), the phase at this time is \(2\pi\), then:  
\[k(n_{1}-1)h=2\pi \tag{9}\]  
Substituting \(k=2\pi/\lambda\) into Equation (9), we can obtain the expression of groove height \(h\) that:  
\[h=\frac{\lambda}{n_{1}-1} \tag{10}\]  
Because the hologram has circular symmetry, we take half of it to analyze. The transmittance function of the hologram can be looked at as the convolution between \(t(r)\) and Dirac comb function \(comb(r)\), as shown in Figure 3[18]. \[R(r)=t(r)\otimes comb(r)=\sum_{m=0}^{N-1}\exp\left[-ik(r-md)(n_{1}-1)\frac{h} {d}\right] \tag{11}\]  
where \(\otimes\) is the convolution operator. If the SLM is illuminated with Gaussian beam, the electric field after \(N\) cycles behind SLM has the form:  
\[U(r)=A\exp\left(-\frac{r^{2}}{\omega^{2}}\right)\!\!R(r)=A\exp\left(-\frac{r^{ 2}}{\omega^{2}}\right)\!\!\sum_{m=0}^{N-1}\exp\left[-ik(r-md)(n_{1}-1)\frac{h} {d}\right] \tag{12}\]  
where \(A\) is the amplitude of the Gaussian beam, \(\omega\) is the incident beam waist, and \(N\) is equal to \(D/2d\). The hologram has circular symmetry, and, according to the theory of Fresnel diffraction, the diffraction field at distance \(z\) behind the SLM can be written as:  
\[ \begin{array}{ll}E_{1}(r_{1},z)&=\frac{A}{i\lambda z}\exp\left[ik\left(z+ \frac{r_{1}z^{2}}{2z}\right)\right]\\ &\times\!\!\int_{0}^{2\pi}\exp[-ik\frac{r_{1}}{z}\cos(\theta-\xi)\right]\!d\xi \\ &\times\!\!\int_{0}^{L/2}r\exp\left(ik\frac{k^{2}}{z}\right)\!\!\sum_{m=0}^{N-1 }\exp\left[-ik(r-md)(n_{1}-1)\frac{h}{d}\right]\exp\left(-\frac{r^{2}}{\omega^ {2}}\right)\!\!dr\end{array}  \tag{13}\]  
where \((r,\,\xi)\) are the polar coordinates in the plane \(z=0\), and \((r_{1},\,\theta)\) are the polar coordinates in the plane \(z\). Making use of:  
\[\int_{0}^{2\pi}\exp[ix(\varphi-\varphi_{0})]d\varphi=2\pi J_{0}(x) \tag{14}\]  
Figure 3: The principle of calculating \(R(r)\) with convolution. Equation (13) can be rewritten as:  
\[ \begin{array}{ll}E_{1}(r_{1},z)&=A\frac{2\pi}{H\Sigma}\exp\left[ik\left(z+\frac {r_{1}^{2}}{2z}\right)\right]\\ &\times\int_{0}^{L/2}rJ_{0}\left(k\frac{r_{1}r_{2}}{z}\right)\exp\left(-\frac{r _{2}^{2}}{\omega^{2}}\right)\sum\limits_{m=0}^{N-1}\exp\left[ik\left[\frac{r _{2}^{2}}{z}-(r-md)(n_{1}-1)\frac{h}{d}\right]\right]\!d\tau\end{array}  \tag{15}\]  
The stationary-phase method provides a good approximation to the value of the integral in Equation (15). A function for the phase part of the integral in Equation (15) is defined as:  
\[f(r)=\frac{r^{2}}{2z}-(r-md)(n_{1}-1)\frac{h}{d} \tag{16}\]  
\[f^{\prime}(r)=\frac{r}{z}-(n_{1}-1)\frac{h}{d}=0 \tag{17}\]  
According to Equation (17), the stationary point \(r_{\rm q}\) can be obtained as:  
\[r_{\rm q}=z(n_{1}-1)\frac{h}{d} \tag{18}\]  
Equation (15) can be expressed in the form:  
\[