# Structured Light Patterns Work Like a Hologram
## 4 The Synthesis of Holograms and Structured Fringe Patterns with the Same Period  
To simplify calculation procedures, the consideration will be limited to the case of the holograms of focused images, when the plane (\(x_{0},y_{0}\)) with the object of holography (Figure 1) is transferred to the plane (\(x_{1},y_{1}\)), in which it would be necessary to perform Fresnel transformation, not at the stage of hologram synthesis but only later, when obtaining the image restored by the holograms. It is difficult to create material holograms in this way since the material object overlaps the reference beam with its shadow, and computer holograms are quite possible. A hologram with a period of 3 pixels per fringe, synthesized using the above-described method, is shown in Figure 3.  
Similarly, by transforming the depth of the 3D object (Figure 2d) to the size of \(\lambda/2\) and increasing the fringe frequency (Figure 2c) by their median multiplexing to 3 pixels per one fringe, we made the pattern of structured fringes with a period equal to the period of the carrier frequency at the hologram (Figure 4).  
To compare the results of diffraction from structures in Figures 3 and 4, Fourier transforms describing Fraunhofer diffraction were calculated, along with Fresnel transformations from each structure. The results for both the Fresnel and Fourier transforms are described below. The results of our further investigation, with the help of both transforms, turned out to be the same. For this reason, we show only the images obtained using the Fresnel transform below because they provide a more obvious illustration of image restoration at the final distance from the hologram and the same distance from the structured light fringe pattern. Both cases, namely the Fourier and Fresnel diffraction, correspond to image restoration according to the scheme shown in Figure 1 into the zero and plus-minus first diffraction orders for the corresponding hologram types.  
Figure 4: The patterns of structured fringes over the depth map from Figure 2d with a period of 3 pixels and the depth reduced to \(\lambda/2\): (**a**) vertical structured fringes are projected onto the mask, (**b**) the fringes superposed on the mask are stitched on the texture.  
Figure 3: Hologram of focused images: (**a**) from the object—depth map; (**b**) from the full 3D object—depth map with superposed texture.  
The results obtained in the calculation of the Fresnel transform from the hologram (Figure 3) are shown in Figure 5a,b. Here, Figure 5a corresponds to the amplitude component of the Fresnel transform, while Figure 5b corresponds to the phase component. The observation distance of Fresnel diffraction was chosen to provide reliable spatial separation between all three diffraction orders as shown in Figure 1.  
The results of similar calculations from the structured light fringe pattern (Figure 4) are presented in Figure 6a,b where Figure 6a corresponds to the amplitude component of the Fresnel transform, while Figure 6b corresponds to the phase component.  
One can see that the diffraction patterns at the array of structured fringes (Figure 6) and the hologram (Figure 5) are very similar to each other. A more detailed investiga  
Figure 5: Fresnel spectrum from the hologram shown in Figure 3: (**a**) the amplitude component of the spectrum with a colorbar from 0 to 100%; (**b**) the phase component of the spectrum with a colorbar from \(-\pi\) to \(+\pi\).  
Figure 6: Fresnel spectrum from the structured fringe pattern shown in Figure 4: (**a**) the amplitude component of the spectrum with a colorbar from 0 to 100%; (**b**) the phase component of the spectrum with a colorbar from \(-\pi\) to \(+\pi\).  
tion of the structure of the obtained holograms under substantial magnification confirms this similarity.