## 2 Recording Method  
In this section, we describe our proposed shift-peristrophic multiplexing method. Figure 1 shows the conceptual illustration of shift multiplexing with spherical reference waves. In the figure, \(\mathbf{k_{s}}\), \(\mathbf{k_{r}}\), and \(\mathbf{k_{g}}\) are signal, reference, and grating vectors, respectively.  
Because spherical waves can be expressed as superimposed plane waves advancing in various directions, the shape of the grating varies by location. Therefore, multiplexing is enabled simply by displacing the medium. The required shift amount in the \(x\)-direction to suppress crosstalk \(\Delta x\) can be expressed as [21]:  
\[\Delta x=\frac{\lambda z_{0}}{L\theta_{s}} \tag{1}\]where \(z_{0}\) is the location of the reference wave source, \(L\) is the medium thickness, and \(\theta_{s}\) is the incident angle of the signal. In our previous studies [22, 23], shift selectivity in the \(y\)-direction was determined to be inferior compared with the \(x\)- and \(z\)-directions because the \(y\)-direction is parallel to the hologram grating.  
To improve the recording density by effective utilization of dynamic range, we combined shift multiplexing with spherical reference waves and peristrophic multiplexing. Figure 2 shows the principle of the peristrophic multiplexing. The relation between the wave vector of signal \(\mathbf{k_{s}}\), reference \(\mathbf{k_{r}}\), and the grating vector \(\mathbf{k_{g}}\) is also shown in the figure. If the medium is rotated by a minute angle, the Bragg condition is broken by the change in each wave vector, and the recorded hologram is not reproduced. In the event that a new hologram is recorded in the current state, each hologram is independently reproduced without crosstalk. The Bragg selectivity \(\Delta\theta\) of the peristrophic multiplexing with plane wave is expressed as [14]:  
\[\Delta\theta=\sqrt{\frac{2\lambda}{L}\cdot\frac{\cos\theta_{s}}{\sin\theta_{r} (\sin\theta_{s}+\sin\theta_{r})}} \tag{2}\]  
where \(\Delta\theta\) is the required rotation angle of the medium to suppress crosstalk, and \(\theta_{s}\) and \(\theta_{r}\) are the incident angles of the signal and reference, respectively.  
High recording density is expected using a recording method that combines such shift multiplexing with peristrophic multiplexing. The procedure of shift-peristrophic multiplexing is as follows. First,  
Figure 1: Principle of the peristrophic multiplexing.  
Figure 2: Principle of the peristrophic multiplexing.  
shift multiplexing is performed along the \(x\)-direction. The medium is then rotated by \(\Delta\theta\), and shift multiplexing is repeated: the procedure is shown in Figure 3. Figure 4 shows the recording method for the disk-shaped medium. An optical recording head moves in the direction of the shift multiplexing along the straight line shown in Figure 4. In addition, only the hologram diameter moves with the \(y\)-axis, and shift multiplexing is performed. The shift-multiplexed holograms are recorded at each \(y\)-axial track until the disk is full.  
These holograms occupy all of the disk space by rotating the medium by 10\({}^{\circ}\). This space is then overwritten with shift-multiplexed holograms. This procedure is executed for each angle, resulting in a series of composite holographic images, and can be fundamentally executed 18 times at intervals of 10\({}^{\circ}\). Figure 4 shows that a multiple shift also is performed along the \(x\)-axis after the medium is rotated by 10\({}^{\circ}\).