# Static terahertz time-domain spectroscopy

Terahertz time-domain spectroscopy, or **THz-TDS** in short, is an experimental method for the measurement of the optical properties of a sample in the terahertz frequency range.

This text will give a detailed graphical example of the steps involved in a typical analysis of raw data from a THz-TDS experiment. THz-TDS can be carried out in many configurations, including transmission, reflection, attenuated total reflection. Up to a certain point in the analysis the procedure is very similar for all configurations, namely the recording of a reference signal and a sample signal. The final step of the analysis then depends on the configuration.

The optical properties of a material is most often represented by the loss (absorption) and index of refraction. In order to measure both these quantities, we need a method which detects both the **amplitude** and **phase** of the light which has interacted with the sample. This is exactly what THz-TDS does. In this sense THz-TDS is a direct competitor to the technique known as dispersive Fourier Transform Infrared spectroscopy.

## Time-domain data

The raw data in a THz-TDS measurement consists of a time trace of the electric field of the THz signal. We record a trace without sample (the reference pulse) and then with the sample in the beam path (the sample pulse). An example of such a pair of pulses is shown in Figure 1 below.

The pulses are both of very short duration – less than a picosecond. You will notice that the sample signal has a lower amplitude, arrives at the detector at a later time, and has additional wiggles following the main part of the pulse. All these modifications to the pulse shape are related to the optical properties of the sample – in this case a thin disk of testosterone powder.

## From time to frequency

The next step in the THz-TDS procedure is to perform an analysis of the frequency content of the two signals. This is done by a Fourier transformation of each of the signals. The amplitude and phase of the reference and sample signals. The result is shown in Figure 2.

## Complex transmission function

Now the frequency-resolved transmission and amplitude of the electric field is calculated by taking the ratio of the sample to reference spectrum and the phase difference between the sample and reference signals, with a result shown in Figure 3. The experiment shown here is a transmission experiment; if the measurement had been carried out in reflection, the data in Figure 3 would represent the field reflection amplitude and phase, respectively.

## Calculation of the optical constants of the sample

Based on the complex transmission function in Figure 3, the dielectric properties of the sample can be calculated. This step requires knowledge of the sample; its thickness, is it placed between windows or sitting in free space. With this knowledge the complex transmission function can be related to for instance the absorption coefficient and the index of refraction of the sample. Figure 4 shows the extracted absorption coefficient, in absolute units of 1/cm, as well as the index of refraction. The phase difference in Figure 3 gives the index of refraction of the material, and with this knowledge, the Fresnel reflection losses at the interfaces can be taken into account, and the true absorption coefficient can be calculated.

Notice the gray curve in the left panel of Figure 4. This curve indicates the largest absorption coefficient that can be measured at a given frequency on this particular sample. The largest measurable absorption is determined by the frequency-dependent dynamic range of the reference signal. If the signal gets attenuated to a level smaller than the noise floor of the experiment, the apparent absorption will saturate at the gray curve in Figure 4.

The absorption coefficient and the index of refraction is one representation of the optical properties of the sample. Other standard representations are the dielectric function (or permittivity) and the conductivity of the sample. It is relatively easy to shift between the different representations of the dielectric properties, and the properties of the sample typically determines which representation is chosen. The absorption coefficient and index of refraction are typically used for the description of propagation of electromagnetic fields in matter. The permittivity is typically used to represent the dielectric properties of liquids and glasses, and the conductivity is typically used to describe samples with free charge carriers.

In Figure 5 we show the same data as in Figure 4, but this time represented as the permittivity of the sample. The complex permittivity is calculated as the square of the complex index of refraction.

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