10.6: Example- Copper

The real and imaginary parts \(\left(\mathrm{n}_{\theta}+i \kappa_{\theta}\right)=\sqrt{\epsilon_{\mathrm{r}}-\sin ^{2} \theta}\) have been plotted in Figure (10.6.8) as a function of the angle of incidence, θ, for room temperature copper and for a wavelength of λ= 0.5145 microns (see Table(10.1)). As can be seen from the figure, the angular dependence of the indices n_θ, \(\kappa\)_θ is not very pronounced. For a lossy material such as copper that has a complex dielectric constant the reflectivity, E_R/E₀, is complex; that is, the phase shift between the incident wave and reflected wave electric vectors is neither 0^◦ (in phase) nor 180^◦ (out of phase). The real and imaginary parts of the reflectivity have been plotted in Figure (10.6.9) as a function of the angle of incidence for S-polarized 0.5145 micron light incident on room temperature copper; the absolute value of the reflectivity has been plotted in Figure (10.6.10).

Figure \(\PageIndex{8}\): The dependence of the complex index of refraction, n_θ + i\(\kappa\)_θ, upon angle of incidence of the incident wave calculated for copper at room temperature and for an incident wavelength of λ = 0.5145 µm. The normal component of the wave-vector in copper is given by \(k_{z}=\left(\frac{\omega}{c}\right)\left(n_{\theta}+i \kappa_{\theta}\right)\). At this wavelength the relative dielectric constant for copper is \(\epsilon_{r}\)=(-5.34+i6.19), n=1.19, and \(\kappa\)=2.60.

Figure \(\PageIndex{9}\): The real and imaginary part of the reflectivity of copper, E_R/E₀, as a function of angle of incidence for a wavelength λ= 0.5145 µm and S-polarized radiation. Copper at room temperature; \(\epsilon_{r}\)= (-5.34 + i6.19), n=1.19, and \(\kappa\)=2.60.

Similarly, the real and the imaginary parts of the ratio H_R/H₀ have been plotted in Figure (10.6.11) as a function of the angle of incidence for P-polarized 0.5145 micron light incident on copper; the absolute value of this ratio is shown in Figure (10.6.12). The reflection coefficient for P-polarized radiation is given by R_P = E_R/E₀ but this is very closely related to the ratio H_R/H₀ because E₀ = Z₀H₀ and E_R = −Z₀H_R, where Z₀= 377 Ohms, the impedance of free space. Notice that the real part of the reflectivity for P-polarized light vanishes at an angle of incidence of approximately 69^◦ ; the phase of the reflected light at that angle is shifted by 90^◦ relative to the incident light. The phase shift between reflected and incident light is much less pronounced for S-polarized light; approximately 15^◦ for an angle of incidence of 69^◦ .

Figure \(\PageIndex{10}\): The absolute value of the reflectivity of copper, | E_R/E₀ |, as a function of angle of incidence for a wavelength λ= 0.5145 µm and S-polarized radiation. Copper at room temperature; \(\epsilon_{r}\)=(-5.34 + i6.19), n=1.19, and \(\kappa\)=2.60.

Figure \(\PageIndex{11}\): The real and imaginary parts of the complex ratio HR/H0 for copper as a function of angle of incidence for a wavelength λ= 0.5145 µm and for P-polarized radiation. Copper at room temperature; \(\epsilon_{r}\)=(-5.34 + i6.19), n=1.19, and \(\kappa\)=2.60.