# Development of a Spatial Heterodyne Terahertz Raman Spectrometer with Echelle Gratings
## 2 Principles  
### Basic Principles  
A schematic diagram of the E-SHTRS is shown in Figure 1. When compared with the spatial heterodyne terahertz Raman spectrometer, the original 150 gr/mm gratings on the two interference arms were replaced with 36 gr/mm echelle gratings. The excitation light source used was a 532 nm laser, and the output beam was adjusted using a linear variable filter with a power range from 0 to 400 mW. Beam prism 1 divides the beam into two beams with a ratio of 9:1, where 10% of the beam is used for path adjustment of the instrument, and 90% of the laser light is purified by the bandpass filter (BPF) group. Next, beam collimation is carried out by the collimation lens group. The collimated beam is then reflected toward the objective lens by the Bragg notch filter (BNF) and reflector 3. The sample is placed at the focus of the objective lens. The beam that is incident on the sample will then excite Raman scattering light. The intensity of Raman scattering light can be expressed as:  
\[I_{R}=KP\sigma I_{L} \tag{1}\]  
where \(K\) is the proportional coefficient related to the temperature and other conditions; \(P\) is the number of scattering molecules; \(\sigma\) is the wavenumber of the incident light; and \(I_{L}\) is the intensity of the incident light. The Raman scattering light is reflected by reflector 3 and then passes through the BNF filter and the collimation lens group. During this process, a 550 nm short-pass filter is used to filter out light at wavelengths of less than 550 nm; the BNF filter is set to be tilted at a specific angle to the optical axis and is used to weaken the intensity of the Rayleigh scattering light, and the pinhole appendix is used to filter any superfluous stray light. After filtering and purification, the Raman scattering light is incident on beam prism 2, and is then divided into two light beams of equal intensity. These two light beams are incident on prism 1 and 2 on the two interference arms at a minimum deviation angle \(\theta_{P}\). \[\theta_{P}=arcsin[nsin(2/\alpha)] \tag{2}\]Here, \(\alpha\) is the angle of the prism and \(n\) is the refractive index of the prism. Subsequently, the two beams incident on two echelle gratings with a Littrow angle \(\theta_{L}\) to the optical axis. For the same incident wavefront, after being bundled through beam splitter 2, the angle between the two emitted wavefronts is \(2\gamma\); then, the grating equation for the two interference arms is:  
\[d(\sin\theta_{L}+\sin(\theta_{L}-\gamma))=m\lambda \tag{3}\]  
where \(m\) is the grating diffraction level, \(d\) is the grating constant, and \(\lambda\) is the incident wavelength. The resulting interference light is received by a charge-coupled device (CCD) detector after passing through the stop. When the incident light wavelength happens to be the Littrow wavelength, the diffracted light passing through the grating will return along its original path, the angle of the outgoing wavefronts of the two beams is zero, and the spatial frequency can be expressed as:  
\[f_{x}=2\sigma sin\gamma\approx 4(\sigma-\sigma_{L})tan\theta_{L} \tag{4}\]  
To suppress ghost lines, the grating on one interference arm can be rotated around the X-axis by \(\varepsilon/2\), and then, in the grating diffraction direction (i.e., the Z-axis direction), the intensity of the two-dimensional interference fringe can be expressed as:  
\[I(x)=\int_{0}^{\infty}B(\sigma)\{1+cos(8\pi(\sigma-\sigma_{L})xtan\theta_{L}+ \sigma y\varepsilon)\}d\sigma \tag{5}\]  
The optical path difference (OPD) between the two diffraction wavefronts is:  
\[OPD=4 xtan(\theta_{L}) \tag{6}\]  
\[OPD_{max}=4x_{max}tan\theta_{L}=2 Wsin\theta_{L} \tag{7}\]  
Here, \(W\) is the grating width. The spectral resolution can be expressed as:  
\[\delta\sigma=\frac{1}{2 OPD_{max}}=\frac{1}{4 Wsin\theta_{L}} \tag{8}\]  
Figure 1: Schematic diagram of the echelle grating spatial heterodyne terahertz Raman spectrometer.