# Terahertz Raman Measurements Using a Spatial Heterodyne Raman Spectrometer
## 2 Theory  
### Basic Theory  
The spatial heterodyne THz Raman spectrometer is depicted in Figure 1. A 532 nm laser beam passes through lens 1 and is linearly varied from 0 to 140 mW using a linear variable filter, and is then incident into beam splitter 1. It is divided into parts with a ratio of 1:9. One part (10%) is used for the dimming path. The other part (90%) is incident to the bandpass filters (BPFs) for laser purification; after the noise is removed, this part is reflected to the collimating lens through mirror 2 and then reflected by mirror 3 and incident to the BNF. It is then reflected from the filter to mirror 4, and then reflected to the objective lens and focused to the sample through the objective lens. The Raman signal generated by the laser beam incident on the sample is reflected to the BNF by mirror 4, and then incident to the BNFs through the collimating lens and the filter. A pinhole aperture is inserted into the collimating lens to remove stray light. To achieve maximum inhibition of Rayleigh light, the BNFs are inclined at an angle to the optical axis and face opposite to each other to avoid interference between them. After selective filtering, the signal is incident into beam splitter 2 and is divided into two equal beams. The two beams are incident into prism 1 (2) on their interference arms at the minimum deviation angle \(\theta_{P}\) and then incident into grating 1 (2). \[\theta_{P}=\arcsin\Bigl{(}n\cdot\sin\Bigl{(}\frac{\alpha}{2}\Bigr{)}\Bigr{)} \tag{1}\]The refractive index of the prism is \(n\), and \(\alpha\) is the apex angle of the prism. The gratings are tilted at a fixed angle (the Littrow angle) to the optical axis so that light at the Littrow wavelength exits the interferometer parallel to the optical axis. The interference between the two beams results in wavenumber-dependent shearing. Light from the beam splitter passes through the aperture and is incident to the imaging optics attached to the detector, resulting in Fizeau-type interference fringes that are received by the detector. The grating equations of the two interference arms are:  
\[\sigma(\sin(\theta_{L}\pm\gamma))=mG_{1,2} \tag{2}\]  
where \(\sigma\) is the wavenumber of the incident light, \(\theta_{L}\) is the Littrow angle of the grating, \(m\) is the diffraction order, \(G_{1,2}\) are the groove densities of the gratings, and \(2\gamma\) is the angle between the two output wavelengths (\(2\gamma=0\) for the Littrow wavelength). The Littrow wavenumber is \(\sigma_{0}\). In the first-order approximation, the angle between two output wavefronts of any wavelength is derived as:  
\[2\gamma=4\text{tan}\theta_{L}.\frac{(\sigma-\sigma_{0})}{\sigma} \tag{3}\]  
The spatial frequency of the interference is:  
\[f_{x}=2\sigma sin\gamma\approx 4(\sigma-\sigma_{0})\text{tan}\theta_{L} \tag{4}\]  
The Fourier relation between the interferogram and the incident spectrum is:  
\[I(x)=\int_{0}^{\infty}B(\sigma)[1+\cos(2\pi x\cdot 4\text{tan}\theta_{L}\cdot( \sigma-\sigma_{0}))]d\sigma \tag{5}\]  
The spectrum of the reconstructed image can be obtained through an inverse Fourier transform. The spectral resolution of the system depends on the parameter [19] of the grating on the interference arm:  
\[\delta\sigma=\frac{1}{4W\text{sin}\theta_{L}} \tag{6}\]  
where \(W\) is the grating width. According to the sampling theorem, the spectral range of the system depends on the number of pixels of the detector:  
\[\Delta\sigma=\frac{N}{2}\cdot\delta\sigma \tag{7}\]  
Figure 1: Optical structure of the spatial heterodyne THz Raman spectrometer. The relationship between spectral resolving power \(R\) and maximum field of view \(\Omega_{\text{max}}\) is:  
\[R=\frac{2\pi}{\Omega_{\text{max}}} \tag{8}\]  
Therefore, to achieve high resolution and high throughput at the same time, we place a prism with a certain apex angle on the interference arm.