In the reflection mode measurement, the sample was placed in front of a reference mirror. It was assumed that all media are uniform, isotropic and non-magnetic, therefore, the internal dispersion can be ignored, as shown in Fig.1. The thickness of terahertz wave is d and the refractive index is ${{\tilde n} _1}$ in this Figure. The total reflected signal is the sum of the electric field intensity reflected by the 1st, 2nd, 3rd reflections and so on. According to Fresnel’s equation, the reflection and transmission coefficients can be correlated with the refractive indices of i and j media as follows[7]:
The first reflected terahertz wave is generated by the interface between media 0 and 1, which can be expressed as:
The terahertz wave reflected in the second time would be obtained by the lower surface of medium 1 and the upper surface interface of the metal mirror, which can be expressed as:
The terahertz wave reflected in the third time is also generated by the signal making two roundtrips within medium 1, which can be expressed as:
The terahertz wave reflected at the Nth time can be expressed as:
Among them, ${{p}}({{f}},l)$ represents the phase-shift of terahertz waves through a medium with thickness of d:
The final reflected terahertz wave can be expressed as:
The terahertz wave was completely reflected when the surface of the sample was smooth. While, the terahertz wave was reflected in all directions in the case of a rough surface. Hence, the established reflection model shown in Eq.(8) cannot be applied to study the reflection from a rough surface.
When establishing a model, the influence of interface roughness should be considered. Since the reflection spectrum can be affected by the reflected signals from rough and smooth surfaces, the influence of roughness parameters has to be considered to achieve the reflection spectrum more in line with the actual situation of the sample.
It is assumed that the reflection between the sample and metal mirror is a total reflection, hence, the reflection coefficient ${r_{12}} = {\rm{ - }}1$ (Eq.8) can be expressed:
From Fresnel relation, Eq.(9) can be changed into a function of refractive index, Although the Fresnel formula is the premise of a smooth sample-surface, usually the existence of certain surface roughness may cause the THz wave incident to the crude produced in diffuse reflective surface. Thus, the surface roughness of the scattering effect depends on the relatively long wavelengths and height of the surface. Therefore, as the wavelength decreases (the frequency increases), the surface becomes rougher to the incident wave, resulting in more extensive scattering.
The relationship between the reflected signals on the rough and smooth surfaces can be described by the Kirchhoff approximation:
where ${R_{\rm{rough}}}$ is the reflection of the rough plane in a specular direction, ${R_{\rm smooth}}$ is the mirror reflection of a smooth plane, σ is the root mean square roughness of a rough surface, ν is the velocity of terahertz wave in the sample, and c is the speed of light.
Equation(10) is used to modify the reflection signal of a rough surface to include the influence of roughness parameters, which can compensate the signal attenuation and the scattering generated by the rough surface in the subsequent parameter extraction and data processing.