## 2 Micrograting of a Micro Holographic Storage System  
### Vector Diffraction Theory for Micrograting  
Figure 1 shows a schematic diagram of the MHSS to demonstrate recording and reading principles. As shown in Figure 1a, two counter-propagating beams of circularly polarized light are focused onto a photopolymer using a quarter wave plate (QWP). The micrograting is created by interference between the signal and reference beams. In Figure 1b, the reading of the recorded micrograting is carried out by the reflection of reference beam. Using this micrograting configuration, we assume that the signal andreference beams have the same amplitude and waist, _i.e._, \(E_{sig}=E_{ref}\) and \(w_{sig}=w_{ref}\)[7]. To calculate the electric field in the photopolymer, we introduce vector diffraction theory which accounts for polarization illumination and the boundary conditions of multi-layer system such as lens-air-photopolymer configuration [12].Vector diffraction theory approaches the limits of the validity of scalar theory used to describe the illumination; additionally, this theory provides an exact solution when the NA exceeds 0.6 [12, 13, 14, 15]. For accurate treatment of a high NA MHSS, it is necessary to use a theoretical description based fully on vector diffraction theory.  
Figure 1: Recording and reading principles of micro holographic storage system (MHSS): (**a**) recording and (**b**) reading. The recording and reading beams propagate along the \(z\)-axis. The incident beam is circular polarized light.  
Figure 2: Notation scheme for the field vectors and coordinates of the optical system.  
The electric field structure of the optical system is given by Equation 1, the well-known diffraction integral. Note that the notations used for the field vectors and coordinates of the system are illustrated in Figure 2.  
\[\mathbf{E}(r)=\frac{i}{2\pi}\iint_{\Omega}\frac{\mathbf{a}(k_{x},k_{y})}{k_{z}}\,e^{ikr} dk_{x}k_{y} \tag{1}\]  
where the integration takes place over the exit pupil of the optical system, in terms of the spatial-frequency coordinates \(k_{x}\) and \(k_{y}\). The electric field at the exit pupil is effectively decomposed into plane waves having frequencies of \(k_{x}\), \(k_{y}\), and \({k_{z1}}^{2}={k_{1}}^{2}-{k_{x}}^{2}-{k_{y}}^{2}\). The relationship of the field distribution between the exit pupil and the entrance pupil is given by \(\mathbf{a}(k_{r},k_{\varphi})=f\sqrt{k_{z1}/k}\cdot\mathbf{M}\cdot E_{0}(k_{r},k_{ \varphi})\) with \(f\) corresponding to the focal length of the lens system, and the general lens matrix incorporating possible layer transitions, as derived and summarized in reference [12].  
The MHDS should be considered exact focal point shift due to dual wavelengths (green: recording and reading beams; red: focusing beam) and the refractive index of the photopolymer. Figure 3a,b demonstrate the focal shift associated with different wavelengths and differing refractive indices, respectively. Figure 3b shows the free space ray and refracted ray of the beam that is virtually focused at point \(O\). The exit pupil of the focusing system is situated in the first medium with a refractive index of \(n_{1}\). At \(z=z_{1}\), the beam is transmitted through an abrupt transition to a semi-infinite second medium with refractive index \(n_{2}\) that leads to a focal shift \(d_{f}\). To find the optimal focal position, we will use the [PERSON]'s theory for maximum diffraction efficiency and beam quality/minimum SA and micrograting size [16].