# Dragon Notes

UNDER CONSTRUCTION
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## Oblique Incidence [Diel]

Perpendicular Polarization

When obliquely incident on a boundary separating two dielectric media, a perpendicularly polarized EM-wave described in terms of its $$E$$- and $$H$$-field components ${\overrightarrow{\boldsymbol{E}}}_i\left(x,z\right)=\widehat{\boldsymbol{y}}E_{i0}e^{-j{\beta }_1\left(x{\mathrm{sin} \left({\theta }_i\right)}+z{\mathrm{cos} \left({\theta }_i\right)}\right)},$ ${\overrightarrow{\boldsymbol{H}}}_i\left(x,z\right)=\left[\boldsymbol{-}\widehat{\boldsymbol{x}}{\mathrm{cos} \left({\theta }_i\right)}+\widehat{\boldsymbol{z}}{\mathrm{sin} \left({\theta }_i\right)}\right]\frac{E_{i0}}{{\eta }_1}e^{-j{\beta }_1\left(x{\mathrm{sin} \left({\theta }_i\right)}+z{\mathrm{cos} \left({\theta }_i\right)}\right)}.$ will have reflected field components described as ${\overrightarrow{\boldsymbol{E}}}_r\left(x,z\right)=\widehat{\boldsymbol{y}}E_{r0}e^{-j{\beta }_1\left(x{\mathrm{sin} \left({\theta }_r\right)}-z{\mathrm{cos} \left({\theta }_r\right)}\right)},$ ${\overrightarrow{\boldsymbol{H}}}_r\left(x,z\right)=\left[\widehat{\boldsymbol{x}}{\mathrm{cos} \left({\theta }_r\right)}+\widehat{\boldsymbol{z}}{\mathrm{sin} \left({\theta }_r\right)}\right]\frac{E_{r0}}{{\eta }_1}e^{-j{\beta }_1\left(x{\mathrm{sin} \left({\theta }_r\right)}+z{\mathrm{cos} \left({\theta }_r\right)}\right)},$ and the transmitted field components described by ${\overrightarrow{\boldsymbol{E}}}_t\left(x,z\right)=\widehat{\boldsymbol{y}}E_{t0}e^{-j{\beta }_2\left(x{\mathrm{sin} \left({\theta }_t\right)}-z{\mathrm{cos} \left({\theta }_t\right)}\right)},$ ${\overrightarrow{\boldsymbol{H}}}_t\left(x,z\right)=\left[-\widehat{\boldsymbol{x}}{\mathrm{cos} \left({\theta }_t\right)}+\widehat{\boldsymbol{z}}{\mathrm{sin} \left({\theta }_t\right)}\right]\frac{E_{t0}}{{\eta }_1}e^{-j{\beta }_2\left(x{\mathrm{sin} \left({\theta }_t\right)}+z{\mathrm{cos} \left({\theta }_t\right)}\right)}.$ Reflected and transmitted wave amplitudes are then related to the incident wave’s via $\boxed{{\mathrm{\Gamma }}_{\bot }=\frac{E_{r0}}{E_{i0}}=\frac{{\eta }_2{\mathrm{cos} \left({\theta }_i\right)}-{\eta }_1{\mathrm{cos} \left({\theta }_t\right)}}{{\eta }_2{\mathrm{cos} \left({\theta }_i\right)}+{\eta }_1{\mathrm{cos} \left({\theta }_t\right)}}}\ \boxed{{\tau }_{\bot }=\frac{E_{t0}}{E_{i0}}=\frac{2{\eta }_2{\mathrm{cos} \left({\theta }_i\right)}}{{\eta }_2{\mathrm{cos} \left({\theta }_i\right)}+{\eta }_1{\mathrm{cos} \left({\theta }_t\right)}}}$
Parallel Polarization

For the case of parallel polarization, the corresponding wave field descriptions are ${\overrightarrow{\boldsymbol{E}}}_i\left(x,z\right)=\left[\widehat{\boldsymbol{x}}{\mathrm{cos} \left({\theta }_i\right)}-\widehat{\boldsymbol{z}}{\mathrm{sin} \left({\theta }_i\right)}\right]E_{i0}e^{-j{\beta }_1\left(x{\mathrm{sin} \left({\theta }_i\right)}+z{\mathrm{cos} \left({\theta }_i\right)}\right)},$ $\ {\overrightarrow{\boldsymbol{H}}}_i\left(x,z\right)=\widehat{\boldsymbol{y}}\frac{E_{i0}}{{\eta }_1}e^{-j{\beta }_1\left(x{\mathrm{sin} \left({\theta }_i\right)}+z{\mathrm{cos} \left({\theta }_i\right)}\right)};$ ${\overrightarrow{\boldsymbol{E}}}_r\left(x,z\right)=\left[\widehat{\boldsymbol{x}}{\mathrm{cos} \left({\theta }_i\right)}+\widehat{\boldsymbol{z}}{\mathrm{sin} \left({\theta }_i\right)}\right]E_{r0}e^{-j{\beta }_1\left(x{\mathrm{sin} \left({\theta }_r\right)}-z{\mathrm{cos} \left({\theta }_r\right)}\right)},$ $\ {\overrightarrow{\boldsymbol{H}}}_r\left(x,z\right)=-\widehat{\boldsymbol{y}}\frac{E_{r0}}{{\eta }_1}e^{-j{\beta }_1\left(x{\mathrm{sin} \left({\theta }_r\right)}-z{\mathrm{cos} \left({\theta }_r\right)}\right)};$ ${\overrightarrow{\boldsymbol{E}}}_t\left(x,z\right)=\left[\widehat{\boldsymbol{x}}{\mathrm{cos} \left({\theta }_t\right)}-\widehat{\boldsymbol{z}}{\mathrm{sin} \left({\theta }_t\right)}\right]E_{t0}e^{-j{\beta }_2\left(x{\mathrm{sin} \left({\theta }_t\right)}+z{\mathrm{cos} \left({\theta }_t\right)}\right)},$ $\ {\overrightarrow{\boldsymbol{H}}}_t\left(x,z\right)=\widehat{\boldsymbol{y}}\frac{E_{t0}}{{\eta }_2}e^{-j{\beta }_2\left(x{\mathrm{sin} \left({\theta }_t\right)}+z{\mathrm{cos} \left({\theta }_t\right)}\right)}.$ The wave coefficients are then $\boxed{{\mathrm{\Gamma }}_{\mathrm{\parallel }}=\frac{{\eta }_2{\mathrm{cos} \left({\theta }_t\right)}-{\eta }_1{\mathrm{cos} \left({\theta }_i\right)}}{{\eta }_2{\mathrm{cos} \left({\theta }_t\right)}+{\eta }_1{\mathrm{cos} \left({\theta }_i\right)}}}\ \boxed{{\tau }_{\parallel }=\frac{2{\eta }_2{\mathrm{cos} \left({\theta }_i\right)}}{{\eta }_2{\mathrm{cos} \left({\theta }_i\right)}+{\eta }_1{\mathrm{cos} \left({\theta }_t\right)}}}$
All Polarization

Any incident wave’s reflected and transmitted wave angles are then related to the incident wave’s via Snell’s laws: $\boxed{{\theta }_i={\theta }_r}\ \boxed{\frac{{\mathrm{sin} \left({\theta }_t\right)}}{{\mathrm{sin} \left({\theta }_i\right)}}=\frac{{\beta }_1}{{\beta }_2}=\frac{n_1}{n_2}}$