# Theoretical Zero-Thickness Broadband Holograms Based on Acoustic Sieve Metasurfaces
## 3 Results
### Hologram Realized by the Array of Identical Point Sources
\[\text{RMSE}=\sqrt{\sum_{i=1}^{l}\left[\sum_{j=1}^{l}\left[\frac{E\big{(}x_{i },y_{j},z_{0}\big{)}}{\text{MAX}(E)}-\frac{E_{\text{desired}}\big{(}x_{i},y_{ j},z_{0}\big{)}}{\text{MAX}(E_{\text{desired}})}\right]^{2}\right]/(l\cdot l-1)}\] where \(E\big{(}x_{i},y_{j},z_{0}\big{)}\propto p^{2}\big{(}x_{i},y_{j},z_{0}\big{)}\) and \(E_{\text{desired}}\big{(}x_{i},y_{j},z_{0}\big{)}\propto p^{2}_{\text{desired }}\big{(}x_{i},y_{j},z_{0}\big{)}\), respectively, represent the sound energy generated by the hologram and the sound energy of the desired image at \(\big{(}x_{i},y_{j},z_{0}\big{)}\) in the image plane; \(\text{MAX}(E)\) and \(\text{MAX}(E_{\text{desired}})\), respectively, represent the maximum values of \(E\) and \(E_{\text{desired}}\); \(I\) and \(J\), respectively, represent the numbers of samples along the \(x\)-direction and the \(y\)-direction. The RMSE is a sensitive method to show the profile differences between the sound energy generated by the hologram and the sound energy of the desired image and can be used to characterize the imaging quality in the image plane. The smaller the value of the RMSE, the closer the sound field of the image plane to the desired sound field. When the RMSE = \(0\), it means that the image generated by the hologram is the same as the desired image. 4. Perform a mutation operation (\(1\to 0\) or \(0\to 1\)) for a random gene \(A_{\text{a,b}}\) whose coordinate is \((x_{m,n},y_{m,n},0)\) in the parent chromosome \(A\) to generate the offspring chromosome \(A_{\text{n}}\). 5. Calculate the effect of this mutation for the sound field in the desired image plane using the Rayleigh-Sommerfeld theory, represented by \(p_{\text{mutated}}(x,y,z_{0})\). When the mutation operation is \(1\to 0\), according to Equation (1), \(p_{\text{mutated}}(x,y,z_{0})\) satisfies \[p_{\text{mutated}}(x,y,z_{0})=-\frac{\text{i}\rho_{0}c_{0}k_{0}Q_{0}}{2\pi} \frac{\text{e}^{-\text{i}k_{0}\sqrt{(x_{\text{a,b}}-x)^{2}+(y_{\text{a,b}}-y)^ {2}+z_{0}^{2}}}}{\sqrt{(x_{\text{a,b}}-x)^{2}+(y_{\text{a,b}}-y)^{2}+z_{0}^{2}}}\] When the mutation operation is \(0\to 1\), \(p_{\text{mutated}}(x,y,z_{0})\) satisfies \[p_{\text{mutated}}(x,y,z_{0})=\frac{\text{i}\rho_{0}c_{0}k_{0}Q_{0}}{2\pi} \frac{\text{e}^{-\text{i}k_{0}\sqrt{(x_{\text{a,b}}-x)^{2}+(y_{\text{a,b}}-y)^ {2}+z_{0}^{2}}}}{\sqrt{(x_{\text{a,b}}-x)^{2}+(y_{\text{a,b}}-y)^{2}+z_{0}^{ 2}}}\] (6) Superimpose this effect on the parent sound field \(p(x,y,z_{0})\) to get the offspring sound field \(p_{\text{n}}(x,y,z_{0})\) in the desired image plane, as follow \[p_{\text{n}}(x,y,z_{0})=p(x,y,z_{0})+p_{\text{mutated}}(x,y,z_{0})\]7. Calculate the value (RMSE) of the offspring fitness function \(F_{\text{n}}\) by \(p_{\text{n}}(x,y,z_{0})\) and \(p_{\text{desired}}(x,y,z_{0})\). 8. Compare the magnitudes of \(F_{\text{n}}\) and \(F\), and take the smaller one as the new parent fitness function. The chromosome with the smaller fitness function is taken as the new parent chromosome and the sound field with the smaller fitness function is taken as the new parent sound field. That is, if \(F_{\text{n}}<F\), then let \(F=F_{\text{n}}\), \(A=A_{\text{n}}\), \(p(x,y,z_{0})=p_{\text{n}}(x,y,z_{0})\); if \(F<F_{\text{n}}\), then let \(F=F\), \(A=A\), \(p(x,y,z_{0})=p(x,y,z_{0})\). If \(F\) is always less than \(F_{\text{n}}\) for a certain number (such as 100,000 times) of iterations, then record chromosome \(A\) and end the iteration. 9. If the iteration does not end, go back to step 4. Based on the desired image shown in Figure 2a, the overall map (as shown in Figure 2b) of the encoded PSs can be designed by the modified genetic algorithm at \(z_{0}=120\lambda_{0}\), where \(\lambda_{0}=c_{0}/f_{0}=10\) mm is the wavelength (the background medium is chosen as air with a sound velocity of \(c_{0}=340\) m/s and a density of \(\rho_{0}=1.2\) kg/m\({}^{3}\), and the radiation frequency is set at \(f_{0}=34\) kHz).