Reflectance and Transmittance of a Layered Structure for a Normally Incident Electromagnetic Wave
DOI:
https://doi.org/10.15407/ujpe70.12.844Keywords:
reflectance, transmittance, optics of layered structures, Fresnel equationsAbstract
Simple analytic expressions have been obtained for the reflection and transmission coefficients in the case of a linearly polarized monochromatic planar electromagnetic wave normally incident on a layered structure consisting of planar dielectric layers. The refractive indices of the layers may be complex-valued. In contrast to the standard transfer-matrix method, in which implicit expressions for the reflectance and transmittance are obtained, the proposed one provides explicit expressions for those parameters in terms of the layers’ thicknesses and the coefficients in the Fresnel formulas for the layer interfaces. As an example, the reflection coefficient is determined for the normal incidence of an electromagnetic wave on a semi-infinite periodic structure formed by two continuously alternating dielectric layers.
References
1. Hung-Shan Chen, Yu-Jen Wang, Chia-Ming Chang, YiHsin Lin. A polarizer-free liquid crystal lens exploiting an embedded-multilayered structure. IEEE Photon. Technol. Lett. 27, 899 (2015).
https://doi.org/10.1109/LPT.2015.2399932
2. Anyong Qing, Yizhe Zhao, Zhiyong Zhang. Microwaves, Millimeter Wave and Terahertz Liquid Crystals (Springer, 2024).
3. J.M. Vigoureux. Polynomial formulation of reflection and transmission by stratified planar structures. J. Opt. Soc. Am. A 8, 1697 (1991).
https://doi.org/10.1364/JOSAA.8.001697
4. M-S. Tomaˇs. Recursion relations for generalized Fresnel coefficients: Casimir force in a planar cavity. Phys. Rev. A 81, 044104 (2010).
https://doi.org/10.1103/PhysRevA.81.044104
5. A. Pitelet, E. Mallet, E. Centeno, A. Moreau. Fresnel coefficients and Fabry-Perot formula for spatially dispersive metallic layers. Phys. Rev. B 96, 041406(R) (2017).
https://doi.org/10.1103/PhysRevB.96.041406
6. B. Busson. Sum-frequency generation at interfaces: A Fresnel story II. Analytical ex-pressions for multilayer systems. J. Chem. Phys. 159, 034706 (2023).
https://doi.org/10.1063/5.0151003
7. J.C. Torres-Guzm'an, A. D'ıaz-de-Anda, J. Arriaga. Exact closed forms for the transmittance of electromagnetic waves in one-dimensional anisotropic periodic media. J. Phys. A: Math. Theor. 57, 205201 (2024).
https://doi.org/10.1088/1751-8121/ad4077
8. S. Luhn, M. Hentschel. Analytical Fresnel laws for curved dielectric interfaces. J. Opt. 22, 015605 (2020).
https://doi.org/10.1088/2040-8986/ab5c42
9. A. Yariv, P. Yeh. Optical Waves in Crystals. Propagation and Control of Laser Radiation (John Wiley and Sons, 1984) [ISBN: 0-471-09142-1].
10. J.J. Morales, M.J. Nuevo. A general method for calculating reflection and transmission coefficients in multiple slabs. Am. J. Phys. 59, 1140 (1991).
https://doi.org/10.1119/1.16625
11. W.B. Jones, W.J. Thron. Continued Fractions. Analytic Theory and Applications (Addison-Wesley Publishing Company, 1980) [ISBN: 978-0-521-30231-9].
12. M. Zahn. Electromagnetic Field Theory (John Wiley and Sons, 1979) [ISBN: 978-0-471-02198-8].
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