# Development of Simultaneous Dual-Resolution Digital Holography System
## 2 Principle and Method
Here, the blue, P-polarized object and reference beams interfere with one another, and the red, S-polarized object and reference beams interfere with one another. Through the design described above, the first two main goals of this research paper are been realized: (1) Two in-coherent holographic setups without cross-interference have been built in a multiplexed measurement setup, with each setup being independent and consisting of an object beam and reference beam which are split into two pairs of polarized beams; (2) the two optical channels have different fields of view, large and small. The red-S-polarized light goes through one more lens (L4) than the blue-P-polarized light, and the fields of view between the two channels can be adjusted by the focus length of L4. Next, we will discuss how to separate the measuring results from a single image recorded by one camera. The carrier frequency used to obtain the frequency shift on the Fourier domain can be introduced by adjusting the two reference beams to change the incident angle from the normal of the CCD plane [30]. The hologram of the large field of view is generated by interference between the blue polarized object beam and reference beam, which both represent P-polarized light. The hologram of the small field view is also obtained through interference between the red polarized object beam and the red reference beam, both S-polarized light. The holograms of the large and small fields of view are simultaneously recorded by the CCD camera in the form of an intensity difference. These two reference beams and two object beams' wave fronts can be represented by Equations (1)-(4):  
\[u_{r1} =|u_{r1}(x,y)|exp\Big{\{}i\Big{[}2\pi\Big{(}f_{1x}x+f_{1y}y\Big{)} \Big{]}\Big{\}} \tag{1}\] \[u_{r2} =|u_{r2}(x,y)|exp\Big{\{}i\Big{[}2\pi\Big{(}f_{2x}x+f_{2y}y\Big{)} \Big{]}\Big{\}}\] (2) \[u_{o1} =|u_{o1}(x,y)|exp[i\phi_{1}(x,y)]\] (3) \[u_{o2} =|u_{o2}(x,y)|exp[i\phi_{2}(x,y)] \tag{4}\]  
where \(u_{r1}\) is the wave front of the red polarized reference beam, \(u_{r2}\) is the wave front of the blue polarized reference beam, \(u_{o1}\) is the wave front of the red polarized object beam, and \(u_{o2}\) is the wave front of the blue polarized object beam. The values \(f_{1x}\), \(f_{1y}\), \(f_{2x}\), and \(f_{2y}\) represent the components of the spatial frequency. \(f_{1x}\) and \(f_{1y}\) are introduced by angle \(\alpha_{2}\)between the red polarized reference beam and the normal of the CCD plane, while \(f_{2_{x}}\) and \(f_{2_{y}}\) are introduced by angle \(\beta_{2}\) between the blue polarized reference beam and the normal of the CCD plane. These components can be expressed as Equations (5)-(8), where \(\lambda\) is the laser wavelength:  
\[f_{1_{x}}=\frac{sina_{2_{x}}}{\lambda} \tag{5}\]  
\[f_{1_{y}}=\frac{sina_{2_{y}}}{\lambda} \tag{6}\]  
\[f_{2_{x}}=\frac{sin\beta_{2_{x}}}{\lambda} \tag{7}\]  
\[f_{2_{y}}=\frac{sin\beta_{2_{y}}}{\lambda} \tag{8}\]  
The intensity image of both large and small fields of view can be recorded on the CCD camera simultaneously. \(u_{r1}\) and \(u_{r2}\), as well as \(u_{o1}\) and \(u_{o2}\), are non-coherent polarized lights. \(u_{r1}\) and \(u_{o1}\) are both S-polarized lights, represented by red lines, that interfere together to obtain the hologram for the small field of view, while \(u_{r2}\) and \(u_{o2}\) are P-polarized lights, represented by blue lines, that result in the hologram of the large field of view when they interfere with each other. The intensity recorded can be expressed as follows:  
Figure 2:0  
Figure 2:1  
Figure 2:2  
The \({}^{*}\) denotes the complex conjugate of \(u_{r1}\), \(u_{r2}\), \(u_{o1}\), and \(u_{o2}\). \(I_{S}\) and \(I_{L}\) are the intensity of the small field of view and the large field of view, respectively, while \(I\) is the total intensity recorded by the CCD camera. The intensity image is converted to the Fourier domain to obtain the phase information by applying a Fourier transform to \(I\).