## 2 Virtual OAP Mirror Based on LCoS-SLM
### Modeling and Simulation  
The LCoS-SLM is an ideal phase-only modulation component. It can only change the phase of input light without changing its amplitude. With this important feature, LCoS-SLM can be used as an optical compensator, and it could achieve equivalent optical surface function with a specially designed phase loaded.  
The optical design of the light path needs to convert the parallel input beam into a focused beam with the help of LCoS-SLM. This function is the same as an off-axis parabolic mirror (Figure 3). In this model, the LCoS-SLM worked as a virtual off-axis parabolic surface, and its role is to compensate for the optical path difference (OPD) between the input plane wavefront and output spherical wavefront. Therefore, the problem is to calculate the compensation phase.  
Assuming that the LCoS-SLM is placed directly facing the input light, according to the off-axis angle and the focal length of the OAP model, the calculation of the compensation phase is done by ray tracing. However, the calculated frequency would exceed the limit of the LCoS-SLM spatial frequency and lead to poor optical performance. According to the Nyquist-Shannon sampling theorem, the phase sampling interval should be less than the reciprocal of the largest space-frequency sampling function of LCoS-SLM. The pixel size of LCoS-SLM determines the maximum of its spatial frequency of the phase  
Figure 3: Off-axis parabolic mirror (OAP) model as an off-axis parabolic mirror.  
Figure 2: The scheme to apply the light spots array generated by LCoS-SLM in an off-axis optical alignment.  
map, and it affects the system's ability to project the focus light spot, but its size could not be shrunk. If the calculated phase-frequency overcomes the LCoS-SLM, the focus spot could not be generated properly. Therefore, the LCoS-SLM needs to be rotated by a precision rotation stage in advance to the design off-axis angle and recalculate the OPD and phase.  
The entire process is shown in Figure 4. According to the OAP model in Figure 3, the reflected focal length (RFL) of the OAP is defined as \(L\), the off-axis angle as \(\theta\), the parent focal length as F and the off-axis magnitude as \(D\). The value of parameters \(L\) and \(\theta\) can be assigned from the wanted focus spot position and the rotation angle of LCoS-SLM. The off-axis magnitude can be calculated as:  
\[D=L\sin\theta \tag{1}\]  
With these parameters, the parent focal length of the OAP could be calculated by:  
\[F=0.5(1+1/\cos\theta)D \tag{2}\]  
and the mathematical expression of the OAP model is:  
\[Z_{\text{OAP}}=C\Big{[}(X+D)^{2}+Y^{2}\Big{]} \tag{3}\]  
where \(C=\frac{1}{4}\)\(F\). According to Equation (3), the phase of virtual OAP1 could be got. The calculated phase map has over dense phase fringe on the left part. Obviously, the virtual OAP1 model is not suitable for optical experiments.  
The coordinates of the virtual OAP2 is obtained from the virtual OAP1 model with the following RT transformation [20]:  
\[(x_{2},y_{2},z_{2})^{T}=R(x_{1},y_{1},z_{1})^{T}+T \tag{4}\]  
where \(R\) and \(T\) mean the rotational matrix and translation matrix, respectively, with rotational and translation calculations of the coordinate system of the virtual parabolic surface, the phase map of LCoS-SLM can find a suitable solution within the spatial frequency limit.  
Moreover, the phase map could be calculated, as well. The density of the phase fringe has been greatly reduced, which is be beneficial to the better performance of the LCoS-SLM. According to  
Figure 4: OAP phase recalculation model generated by coordinate transformation.  
the above model, the LCoS-SLM can work as a virtual OAP, and its reflective focus length can be redesigned easily and quickly by programming.