# Spectral Shaping of a Superluminescent Diode for Terahertz Cross-Correlation Spectroscopy
## 3 Results and Discussion
### Simulation Results  
To support the interpretation of the experimental results from the previous section, the system has been simulated in MATLAB with a simple numerical model. The electric field \(E_{SLD}\) emitted by the SLD is modeled as band-limited white noise. The structuring of the optical spectrum is considered by numerically filtering the signal with a periodic ideal bandpass filter whose passband-width and period are matched to the experimentally implemented shapes. It was shown by [PERSON] et al. that the detected complex terahertz spectrum \(S_{THz}(\omega)\) in a MLLD-driven THz-CCS system is proportional to the magnitude \(\mid\)\(S_{II}(\omega)\)\(\mid\) of the Fourier transform of the intensity autocorrelation function \(\mathrm{R_{II}}(\tau)\) of the optical signal multiplied with the system's complex transfer function \(H_{Sys}(\omega)\)[15]. It is reasonable to assume that this model is also valid in a SLD-driven THz-CCS-system. The intensity autocorrelation function \(\mathrm{R_{II}}(\tau)\) is numerically determined from the optical intensity \(I_{SLD}(t)=\left|E_{SLD}(t)\right|^{2}\). The resulting terahertz spectrum can then be derived from:  
\[S_{THz}(\omega)=H_{Sys}(\omega)\cdot\left|S_{II}(\omega)\right| \tag{2}\]  
\(S_{THz}(\omega)\) is the complex terahertz spectrum. \(H_{Sys}(\omega)\) is the complex transfer function (system dependent). \(\mid\)\(S_{II}(\omega)\)\(\mid\) stands for the magnitude of the Fourier transform of the intensity autocorrelation function \(\mathrm{R_{II}}(\tau)\). The transfer function was measured with a commercially available continuous-wave terahertz frequency-domain spectroscopy system (Toptica TeraScan 1550). The simulated photocurrent in the time-domain is calculated by inverse Fourier transform of \(S_{THz}(\omega)\). To compare the measurement with the simulation, the same time range relative to the maximum of the simulated photocurrent as in the measurement is considered. Moreover, the same signal processing and analysis steps are applied to the simulated data as to the measurement data. The comparison of the measurement and simulation can be seen in Figure 8 for the parameters \(P_{THz}\), App, and DR, respectively. To effectively compare the effects of the structuring of the optical spectrum, the data for \(P_{THz}\), App, and DR are normalized to the measurements for the 2 THz reference bandpass structure. The solid black line in Figure 8 represents the value for the 2 THz reference measurements. The simulation shows a good agreement with the measurement results. For the feature periods in the range from 0.03-0.9 THz, the simulation results match the experiment nearly perfectly for \(P_{THz}\) with a feature width of 20 GHz, while it shows a \(\sim\)33% difference in the 10 GHz feature width case. The graph for \(\Delta\)pp shows again a strong deviation for filter periods bigger than 0.8 THz. However, the rest of the data match the measurement results quite well. In Figure 8c, the DR is plotted and the deviation between measurement and simulation is similar for both feature widths and lies around 5% lower than the measurement data. These results correspond to the fact that there are effects in a THz-CCS system which are not included in our model. However, we can predict the DR of a THz-CCS system with this numerical approach with a 5% accuracy.