Reflexión electromagnética/Índice/Figura 3.6

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Figura 3.6 Se analiza el coeficiente de reflexión de Fresnel cuando se hace una polarización vertical usando dos materiales con diferente índice de permitividad. Magnitud de los coeficientes de reflección en función del ángulo de incidencia para ${ϵ}_r=4$, ${ϵ}_r=12$, utilizando ${\mathrm{\Gamma }}_{||}=\frac{E_r}{E_i}=\frac{{\eta }_{2}sin{\theta }_t-{\eta }_{1}sin{\theta }_i}{{\eta }_{2}sin{\theta }_t+{\eta }_{1}sin{\theta }_i}$ ${\mathrm{\Gamma }}_L=\frac{E_r}{E_i}=\frac{{\eta }_{{}_2}sin{\theta }_i-{\eta }_{1}sin{\theta }_t}{{\eta }_{2}sin\theta i+{\eta }_{1}sin{\theta }_t}$ la geometría en la figura 3.4. Polarización paralela (Campo Eléctrico en plano de incidencia) Polarizacion perpendicular (Campo Eléctrico no en plano de incidencia) Se analiza el coeficiente de Reflexión de Fresnel con una polarización horizontal se puede observar que el ángulo de incidencia para el cual la reflexión es del 100% es únicamente para 0° y luego disminuye progresivamente en ambos casos hasta que la transmisión sea muy alta con un ángulo de incidencia de 90°.

Figura 3.6 ${\mathrm{\Gamma }}_{||}=\frac{E_r}{E_i}=\frac{{\eta }_{2}sin{\theta }_t-{\eta }_{1}sin{\theta }_i}{{\eta }_{2}sin{\theta }_t+{\eta }_{1}sin{\theta }_i}$ ${\mathrm{\Gamma }}_L=\frac{E_r}{E_i}=\frac{{\eta }_{{}_2}sin{\theta }_i-{\eta }_{1}sin{\theta }_t}{{\eta }_{2}sin\theta i+{\eta }_{1}sin{\theta }_t}$ where is the intrinsic impedance of the i th medium (i = 1, 2), and is given by $\sqrt{\frac{{\mu }_1}{ϵ1}}$ , the ratio of electric to magnetic field for a uniform plane wave in the par-ticular medium. The velocity of an electromagnetic wave is given by $\frac{1}{\sqrt{\mu ϵ}}$, and the boundary conditions at the surface of incidence obey Snell’s Law which, referring to Figure 3.4, is given by $\sqrt{{\mu }_1{ϵ}_1}sin90-{\theta }_i=\sqrt{{\mu }_2{ϵ}_2}sin90-{\theta }_t$ The boundary conditions from Maxwell’s equations are used to derive equations (3.19) and (3.20) as well as equations (3.22), (3.23.a), and (3.23.b). ${\theta }_i={\theta }_r$ and, $E_r=\mathrm{\Gamma }*E_i$ $E_t=1+\mathrm{\Gamma }*E_i$ where $\mathrm{\Gamma }$ is either ${\mathrm{\Gamma }}_{||}$ or ${\mathrm{\Gamma }}_{LL}$, depending on polarization. For the case when the first medium is free space and ${\mu }_1$ $=$ ${\mu }_2$, the reflection coefficients for the two cases of vertical and horizontal polarization can be simplified to ${\mathrm{\Gamma }}_{||}=\frac{-{ϵ}_r\sin {\theta }_i+\sqrt{{ϵ}_r-{\cos }^2{\theta }_i}}{{ϵ}_r\sin {\theta }_i+\sqrt{{ϵ}_r-{\cos }^2{\theta }_i}}$ and ${\mathrm{\Gamma }}_1=\frac{\sin {\theta }_i-\sqrt{{ϵ}_r-{\cos }^2{\theta }_i}}{\sin {\theta }_i+\sqrt{{ϵ}_r-{\cos }^2{\theta }_i}}$ Figure 3.6 Parallel Polarization (Electric Field in the plane of incidence) Perpendicular Polarization (Electric Field not in plane of incidence)