# Research on Compressed Sensing Spectrometry Based on Electro-Optical Transmittance Coding
## 2 Principle  
Coding compression measurements are necessary processes in compressed sensing spectrometry. Compared with transmittance control devices, such as LCVR, FP or films, this study proposes the transverse electro-optic modulation of LiNbO\({}_{3}\) to realize transmittance control coding. The basic principle is shown in Figure 1. A X-cut LiNbO\({}_{3}\) electro-optic crystal is the key component of the coding modulator. Herein, incident light passes along the optical axis (z-axis), and the driving electric field is applied along the \(x\)-axis, which is perpendicular to the light direction. The electro-optic device is placed between two polarizers with orthogonal transmittance axes. The intensity of incident light is regulated, which presents transmittance regulation, and realizes intensity coding in the spectral dimension. In Figure 1, two transverse refractive index axes of the LiNbO\({}_{3}\) electro-optic crystal are formed at 45\({}^{\circ}\) around the X axis under the action of an electric field, and the value of the crystal's refractive index changes according to the applied electric intensity. When the  
Figure 1: Schematic of spectrometry using a LiNbO\({}_{3}\) electro-optical transmittance coding modulator. incident light passes through the electro-optical coding modulator between the polarizers, the optical transmittance can be described as [15]  
\[T_{(\lambda,V)}=I_{(\lambda,V)}/I_{0(\lambda)}\propto\sin^{2}(\delta_{(\lambda,V )}/2), \tag{1}\]  
where, \(\lambda\) is the wavelength of the incident light; \(V\) is the voltage of the external electric field applied to the electro-optical transmittance coding modulator; \(I_{0(\lambda)}\) and \(I_{(\lambda,V)}\) are the light intensity before and after passing through the modulator, respectively; \(\delta_{(\lambda,V)}\) is the phase retardation of two polarization components in the LiNbO\({}_{3}\) electro-optic crystal, and it changes with the change of the refractive index, which can be expressed as [16]  
\[\delta_{(\lambda,V)}=\frac{2\pi}{\lambda}(n_{x^{\prime}}-n_{y^{\prime}})L_{z}= \frac{2\pi}{\lambda}n_{0}^{3}\gamma_{22}\frac{L_{z}}{L_{x}}V. \tag{2}\]  
where, \(L_{x}\) and \(L_{z}\) are the length of the crystal in the \(x\)-axis and \(z\)-axis, respectively. The electric field applied in the x-direction is directly proportional to the voltage \(V\). According to Equations (1) and (2), when the size of the LiNbO\({}_{3}\) crystal is optimally designed, the spectral transmittance can be determined by the external electric field voltage \(V\) only. Therefore, the transmittance can be regulated by the modulator with different external electric field voltages. Multiple and different transmittance peaks can appear in the entire spectral range, and different external electric field voltages can realize the separation of transmittance peak wavelengths and the least overlap. This condition is conducive to the realization of compressed sensing spectrum measurements. As shown in Figure 1, the detector detected the light intensity each time, which is regulated by the electro-optic modulator. Each circumstance constitutes a special coding measurement in the process, and it can be described as  
\[Y_{j}=\int T_{(\lambda,V_{j})}X_{(\lambda)}d\lambda, \tag{3}\]  
\(X_{(\lambda)}\) is the spectral power distribution of the incident light. To perform compressed spectrum measurement analysis and inversion reconstruction, the light intensities detected by the detector are recorded in a series of discrete values. The measurement process is described in a matrix form as follows. \[y=\Phi x, \tag{4}\]  
The spectral signal is represented by \(x\), and the number of spectral channels is set to \(N\) such that \(x\) meets \(x\in R^{N\times 1}\). Multiple measured spectral signals are represented by \(y\), and the number of measurements is set to \(M\) such that \(y\) meets \(y\in R^{M\times 1}\). For compressed sensing measurements, \(M\) is much less than \(N\).