Optics of thin films in the soft x-ray and vacuum UV regions

Conclusions

The final results of this paper are expressed by equations (4.21)–(4.27). We can see that the formulae for the complex reflectivities r_R, r_L and transmittivity t of a stack are formally the same as those deduced in [12] for the visible light region with the following differences:

• i) The formulae of this paper were deduced for S-polarization only,

• ii) The coefficients r_j, t_j are given not by the classical Fresnel formulae but by eqs. (4.18) and (4.19.

• iii) The Snell law is replaced by the dispersion equation (for the j-th film) (2.11b), (4.2).

In [4] a more precise dispersion equation has been deduced

$$\cos \psi - cos\theta _{00} = - 2\pi (ak)^2 \frac{{\sin \theta _{00} }}{{\theta _{00} }}\chi _s - \pi (ak)^4 S_{2\chi _s^2 } \left( {2\pi \chi _s + \frac{{\cos ^2 \gamma _0 }}{2}} \right).$$

((5.1))

Since in the soft X-ray and vacuum UV regions ¦ε-l¦ < 0.1 holds, the difference between (2.11b) and (5.1) is negligible. For the same reason we can put in a good approximation χ_s = χ = (ε - l)/4π.

Our final formulae are based on the one-ray approximation (2.7) (4.1ab). In conformity with our paper [1] we conclude that formulae (4.21)–(4.27) are valid for λ ≧ 5a, whereas their application for shorter wavelengths claims some deeper analysis.