Effective Medium Theory for Calculating Reflectance from Metal--Dielectric Multilayered Structure
An effective medium theory (EMT) for calculating optical reflectance from a surface of metal-and-dielectric multilayered structures (MDMS) has been described. MDMS is a strongly-anisotropic optical medium of which the dispersion surface is cylindrical for transverse-magnetic (TM) polarized light. A...
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| Published in: | Japanese Journal of Applied Physics Vol. 51; no. 4; pp. 042202 - 042202-9 |
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| Main Authors: | , , |
| Format: | Journal Article |
| Language: | English |
| Published: |
The Japan Society of Applied Physics
01-04-2012
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| Online Access: | Get full text |
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| Summary: | An effective medium theory (EMT) for calculating optical reflectance from a surface of metal-and-dielectric multilayered structures (MDMS) has been described. MDMS is a strongly-anisotropic optical medium of which the dispersion surface is cylindrical for transverse-magnetic (TM) polarized light. A coefficient of reflection has been derived by applying the dispersion surface of MDMS to the phase-matching condition and the boundary conditions at the interface between an isotropic medium and the MDMS. The reflectance calculated by this anisotropic effective medium theory has agreed with the results by the finite-difference time-domain (FDTD) method, even for oblique incidence. Moreover, transmittance and reflectance from a finite thick MDMS layer are also derived by using the effective medium theory. |
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| Bibliography: | Reflection and refraction of light at the surface of an MDMS. MDMS is a semi-infinite body with an interface on the $x$--$y$ plane. The plane of incidence is parallel to the $x$--$z$ plane. The metal and dielectric layers of MDMS are parallel to the $y$ axis. An MDMS and its dispersion surface. (a) A schematic of MDMS with period $\Lambda$. Thicknesses of metal layers are $a$. (b) A cylindrical dispersion surface of MDMS for TM-polarized wave. Cross sections of dispersion surfaces normalized by $\omega/c$ for TE and TM waves. (a) Real parts of the normalized propagation vectors, and (b) imaginary parts of them. The structure of MDMS is illustrated in (a). The copper layers are 0.09 μm thick and the dielectric layers are 0.21 μm thick. The wavelength of light is 1.5 μm in a vacuum. Phase matching condition of the reflection and refraction. (a) A schematic of reflection and refraction of a light wave, and (b) dispersion surfaces of Region 1 (Isotropic) and Region 2 (MDMS). Calculated reflectance from a surface of MDMS with respect to the angle of incidence $\theta_{\text{in}}$. (a) Reflectance for the period of $\lambda/20$. The solid curve is reflectance calculated with the effective medium theory. Solid circles are results by the FDTD method. The dotted curve is the reflectance for an isotropic medium with refractive index of $n_{\scale70%\tbox{eff}}$ ($=1.82$). The dashed curve is the reflectance calculated by applying the effective refractive index $(n_{\xi}{}^{2}+n_{\scale70%\tbox{eff}}{}^{2})^{1/2}$ to Fresnel's equation of reflection. (b) Reflectance calculated by EMT and FDTD method for different periods of the structure ($\Lambda=0.075$, 0.15, and 0.30 μm). Calculated reflectance from a surface of MDMS with a tilt angle of 30°. Solid curve is reflectance calculated with the effective medium theory. Solid circles are results by the FDTD method. The dotted curve is the reflectance for an isotropic medium with the refractive index of $n_{\scale70%\tbox{eff}}$ ($=1.82$). The dashed curve is the reflectance calculated by applying the effective refractive index $(n_{\xi}{}^{2}+n_{\scale70%\tbox{eff}}{}^{2})^{1/2}$ to Fresnel's equation of reflection. A schematic of a finite-thick MDMS. Region 1 and Region 3 are isotropic media with refractive indices of $n_{1}$ and $n_{3}$, respectively. Region 2 is MDMS with the thickness of $d$. The light was is incident from Region 1. Phase-matching condition at the lower interface. (a) A schematic of reflection and refraction of the light wave, and (b) dispersion surfaces of Region 2 (MDMS) and Region 3 (isotropic). Calculated transmittance and reflectance for a finite-thick MDMS as a function of the angle of incidence. The MDMS is a 0.2-μm-thick copper grating with a period of 0.075 μm. $T$ is transmittance calculated by the effective medium theory (EMT) and the rigorous coupled wave analysis (RCWA). $R$ is reflectance. Calculated transmittance and reflectance for the finite-thick MDMS as a function of the thickness $d$. (a) Transmittance $T$ calculated by the effective medium theory (EMT) and the rigorous coupled wave analysis (RCWA). (b) Reflectance $R$ by (EMT) and (RCWA). Calculated transmittance and reflectance for the 0.2-μm-thick MDMS as a function of the angle of incidence. (a) The structure period is 0.15 μm, and (b) the structure is 0.30 μm. $T$ and $R$ are transmittance and reflectance, respectively. (EMT) is the results of the effective medium theory, and (RCWA) is the results of the rigorous coupled wave analysis. |
| ISSN: | 0021-4922 1347-4065 |
| DOI: | 10.1143/JJAP.51.042202 |