# Non-Destructive Porosity Measurements of 3D Printed Polymer by Terahertz Time-Domain Spectroscopy
## 3 Results  
### Comparison between Solid and 3D Printed Materials  
Figure 3 shows the effective refractive index and loss coefficient of two of the 3D printed samples, together with that of the solid PETG material. Note that the refractive index of solid PETG at THz frequencies is higher than in the visible (1.57 at 589.3 nm), as is the case in many materials, due to the contribution of ionic polarizability. Note also that solid PETG has moderately strong THz absorption that rises with frequency following a power law (Figure 3b inset). Strong absorption is caused by the presence of polar components in the PETG molecule, especially the OH and CO groups. The power law dependence of absorption arises from the glassy nature of PETG [49]. There are two significant differences between the solid material and 3D printed samples: in the solid, the refractive index is significantly higher, and the loss coefficient is slightly lower.  
The effective refractive index of 3D printed samples is lower than that of solid material due to the porosity present in the printed samples. Porosity, defined as the volume fraction of air in the material (\(p\)), causes a decrease in the effective refractive index \(n_{\mathit{eff}}\), as compared with that of solid \(n_{\mathit{solid}}\), due to the presence of air voids, with \(n_{\mathit{air}}\cong 1\). Using a linear approximation, the effective refractive index can be obtained as [50]:  
\[n_{\mathit{eff}}=p+(1-p)\;n_{\mathit{solid}} \tag{3}\]  
The porosities of all 3D printed samples were calculated and are analysed below in relation to the printer settings.  
The loss coefficient of 3D printed samples is higher than that of solid material due to the contributions of scattering and diffraction. On the one hand, porosity gives rise to scattering because the air voids act as scattering centers. Scattering loss manifests as a featureless loss edge that increases with frequency, typically according to a power law. On the other hand, porosity also reduces absorption loss, as less material is present in the beam path. The two effects counteract each other, such that the total is determined by the porosity, the material absorption, and the size of the voids. In the case of the PETG samples studied here, the effects closely counter-balance, so that the excess loss is small in all cases. This is because PETG is a polar polymer with moderately strong THz absorption. The excess loss \(\alpha_{exc}\) can be calculated by comparing the absorption in solid material \(\alpha_{\mathit{solid}}\) with effective loss in the sample \(\alpha_{\mathit{eff}}\):  
\[\alpha_{exc}=\alpha_{\mathit{eff}}-(1-p)\;\alpha_{\mathit{solid}} \tag{4}\]  
In addition to scattering, all 3D printed samples show a diffraction loss feature, seen as a weak broad peak. This can be explained as arising from the regular variation of the refractive index in the printed structure, which forms a Bragg grating [51] with a period \(H\) equal to the height of the printing layer. The Bragg frequency is then given by \(\
u_{0}=c/(2n_{\mathit{solid}}H)\). The Bragg feature occurs at 0.9 THz for layers of 100 \(\upmu\)m, at 0.6 THz for layers of 150 \(\upmu\)m, and at 0.3 THz for layers of 300 \(\upmu\)m. Since Bragg reflection increases with refractive index variation, the intensity of the diffraction feature reflects the degree of inhomogeneity in the sample. The intensity of the Bragg feature can be taken as excess loss at the local maximum \(\alpha_{exc,\mathit{peak}}\). Using Equation (4), the excess peak diffraction loss values of all 3D printed samples were calculated and are discussed below in relation to the printer settings.