THz spectroscopy experiments on nanostructures typically require the use of a substrate to support the sample being examined. It is therefore indispensable to precisely characterize the complex-valued and frequency-dependent refractive index of the substrate material. Its optical properties in the THz frequency range can be accessed unambiguously through THz time-domain spectroscopy (THz-TDS). In this study, we employ THz-TDS to probe the thermal effect on both the THz refractive index and optical thickness of common dielectric substrate materials Si, MgO, and Sapphire (Al2O3), broadly used in research and technology [1,2,3].

The phase refractive dynamics are examined via standard THz-TDS. Fig.1 (a)-(c) shows the frequency-dependent real part of the phase refractive index for the substrates in the temperature range of 10 to 300 K. Each substrate exhibits a noticeable reduction of the real part of the refractive index with decreasing temperature. Furthermore, group refractive index and optical thickness are analyzed with THz pulses in time-domain. The peak THz electric fields through dry air (reference) and a substrate (sample) define the times Δ𝑡₁₀ and Δ𝑡₂₁, as indicated in Fig.1 (d). The former corresponds to the time shift of THz fields between reference and sample, and the latter corresponds to the round-trip time of the THz pulse propagating in the sample. The group refractive index and the substrate thickness, shown in Fig.1 (e) and (f) respectively, can be determined independently from Δ𝑡₁₀ and Δ𝑡₂₁.