# Dispersion of the Retardation of a Photoelastic Modulator
## 2 Principle Analysis of the Dispersion of the PEM  
The photoelastic modulation effect is a type of artificial birefringence phenomenon, the discovery of which has led to the development of a series of optical modulation devices. The most commonly used PEMs are the two-dimensional symmetric shape PEMs with large aperture and low static residual birefringence, using a piezoelectric actuator mounted on a photoelastic crystal. Typically, a \(-18.5^{\circ}\) X-cut quartz crystal plate is chosen as the piezoelectric actuator, and there are several types of isotropic materials, such as fused silica, CaF\({}_{2}\), ZnSe, and monocrystalline silicon, which can be used as the photoelastic crystal, enabling the extension of the spectral range from UV to terahertz. The structures and vibration forms of the PEMs are shown in Figure 1.  
The frequency of the length stretching vibration of the piezoelectric actuator matches with the natural frequency of the vibration of the photoelastic crystal. Based on the inverse piezoelectric effect, the mechanical vibration in the photoelastic crystal is excited and maintained by the piezoelectric actuator, which typically forms a stress standing wave and results in a periodic variation of the  
Figure 1: Schematic diagram of two-dimensional symmetric photoelastic modulators (PEMs) with stretching and contraction vibrations. birefringence, while the phase and polarization states of the incident light are modulated by the variable birefringence. When a sinusoidal voltage (\(V=V_{0}\sin 2\pi ft\), where \(V_{0}\) is the voltage amplitude and \(f\) is the resonant frequency) is applied to the piezoelectric actuator, the stress standing wave in the photoelastic crystal can be expressed as [15]  
\[T_{(x,y,z,t)}=T_{0}\cos\frac{\pi x}{L}\cos\frac{\pi y}{L}\sin(2\pi ft-\varphi), \tag{1}\]  
where \(L\) is the length of the photoelastic crystal, and spatial variables x and y extend from \(-L/2\) to \(L/2\). The amplitude of the stress, \(T_{0}\), is proportional to the \(V_{0}\) and can simply be defined as \(T_{0}=kV_{0}\), \(k\) is a coefficient related to the inverse piezoelectric effect of the actuator and the mount between the actuator and the photoelastic crystal, and \(\varphi\) is the phase delay between the vibration in the photoelastic crystal and the driving voltage. Further, the stress induced by the birefringence in the photoelastic crystal can be expressed as  
\[ \begin{array}{c}{n_{x}}^{\prime}={n_{0}}-\frac{1}{2}{n_{0}}^{3}(-P_{11}T_{( x)}+P_{12}T_{(y)})\\ {n_{y}}^{\prime}={n_{z}}^{\prime}={n_{0}}-\frac{1}{2}{n_{0}}^{3}(-P_{12}T_{(x) }+P_{11}T_{(y)})\end{array} \Bigg{\}}, \tag{2}\]  
where \(n_{0}\) is the refractive index of the isotropic material, while \(P_{11}\) and \(P_{12}\) are the photoelastic constants. When the light beam propagates in the PEM along with the z-axis, the phase and polarization states of the incident light are periodically modulated along the x and y-axes by the variable birefringence. The modulation retardation of the PEM can be described as  
\[\delta=\frac{2\pi}{\lambda}({n_{y}}^{\prime}-{n_{x}}^{\prime})d=\frac{\pi}{ \lambda}{n_{0}}^{3}(P_{12}-P_{11})(T_{(x)}+T_{(y)})d, \tag{3}\]  
where \(\lambda\) is the wavelength of the incident light, and \(d\) is the thickness of the photoelastic crystal. According to the Equations (1) and (2), Equation (3) can be rewritten as  
\[\delta=\frac{\pi}{\lambda}{n_{0}}^{3}d{V_{0}}(P_{12}-P_{11})(\cos\frac{\pi x} {L}+\cos\frac{\pi y}{L})\sin(2\pi ft-\varphi). \tag{4}\]  
Thus, we can define the retardation of the PEM as \(\delta=\delta_{0}\sin(2\pi ft-\varphi)\), and the amplitude of the PEM retardation \(\delta_{0}\) can be further written as \(\delta_{0}=N_{(\lambda)}T_{(V_{0})}\) using the method of the separation of variables, where \(N_{(\lambda)}\) is defined as \(N_{(\lambda)}=\frac{n_{0}^{3}}{\lambda}\), which can be attributed to the dispersion of the PEM retardation.