 # Dragon Notes UNDER CONSTRUCTION
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## Oblique Incidence [PEC]

When obliquely incident on a PEC boundary, the incident EM-wave propagating in the direction ${\widehat{\boldsymbol{n}}}_i\boldsymbol{=}\widehat{\boldsymbol{x}}{\mathrm{sin} \left({\theta }_i\right)}+\widehat{\boldsymbol{z}}{\mathrm{cos} \left({\theta }_i\right)\ }$ will have corresponding $$E$$- and $$H$$- fields described by ${\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)},\ \$ $\ {\overrightarrow{\boldsymbol{H}}}_i\left(x,z\right)=\widehat{\boldsymbol{y}}\frac{E_{i0}}{{\eta }_1}\left(-\widehat{\boldsymbol{x}}{\mathrm{cos} \left({\theta }_i\right)\ }+\widehat{\boldsymbol{z}}\sin{\left(\theta_i \right)}\right)e^{-j{\beta_1 }\left(x{\mathrm{sin} \left({\theta }_i\right)}+z{\mathrm{cos} \left({\theta }_i\right)}\right)}.$ The reflected wave will then propagate in the direction ${\widehat{\boldsymbol{n}}}_r\boldsymbol{=}\widehat{\boldsymbol{x}}{\mathrm{sin} \left({\theta }_i\right)}-\widehat{\boldsymbol{z}}{\mathrm{cos} \left({\theta }_i\right)}.$ and have corresponding $$E$$- and $$H$$- fields described by ${\overrightarrow{\boldsymbol{E}}}_r\left(x,z\right)=\widehat{\boldsymbol{y}}E_{r0}e^{-j{\beta_1 }\left(x{\mathrm{sin} \left({\theta }_i\right)}-z{\mathrm{cos} \left({\theta }_i\right)}\right)},\ \$ $\ {\overrightarrow{\boldsymbol{H}}}_i\left(x,z\right)=\widehat{\boldsymbol{y}}\frac{E_{i0}}{{\eta }_1}\left(-\widehat{\boldsymbol{x}}{\mathrm{cos} \left({\theta }_i\right)}-\widehat{\boldsymbol{z}}\sin{\left(\theta_i \right)}\right)e^{-j{\beta_1 }\left(x{\mathrm{sin} \left({\theta }_i\right)}-z{\mathrm{cos} \left({\theta }_i\right)}\right)}.$ Assuming light is incident over the entire boundary, then at each point $$(x,z)$$ outside the boundary the total $$E$$- and $$H$$- fields will be the vector-sums of incident and (other) reflected waves – yielding: ${\overrightarrow{\boldsymbol{E}}}_1\left(x,z\right)=-\widehat{\boldsymbol{y}}2E_{i0}{\mathrm{sin} \left({\beta }_1z{\mathrm{cos} \left({\theta }_i\right)}\right)e^{-j{\beta }_1x{\mathrm{sin} \left({\theta }_i\right)}}},\ \ \$ ${\overrightarrow{\boldsymbol{H}}}_1\left(z,t\right)=-\widehat{\boldsymbol{y}}2\frac{E_{i0}}{{\eta }_1}\left[\widehat{\boldsymbol{x}}{\mathrm{cos} \left({\theta }_i\right)}{\mathrm{cos} ({\beta }_1z{\mathrm{cos} \left({\theta }_i\right)}}e^{-j{\beta }_1x{\mathrm{sin} \left({\theta }_i\right)}}+\widehat{\boldsymbol{z}}j{\mathrm{sin} \left({\theta }_i\right)}{\mathrm{sin} \left({\beta }_1z{\mathrm{cos} \left({\theta }_i\right)}\right)}e^{-j{\beta }_1x{\mathrm{sin} \left({\theta }_i\right)}}\right].$
The above expressions convey the following properties:
• : In the direction ($$z$$-direction) normal to the boundary, $$E_{1y}$$ and $$H_{1x}$$ maintain standing-wave patterns according to $$\sin{⁡\left(β_{1z}z\right)}$$ and $$\cos{⁡\left(β_{1z}z\right)}$$, respectively. $$[\beta_{1z} = \beta_1 \cos{\left( \theta_i \right)}]$$
• : No average power is propagated in the $$z$$-direction. ($$E_{1y}$$ and $$H_{1x}$$ are $$90^o$$ out of phase)
• : In the direction ($$x$$-direction) parallel to the boundary, $$E_{1y}$$ and $$H_{1x}$$ are in both time and space phase and propagate with a phase velocity and a wavelength$${}^1$$
• $u_{1x}=\frac{u_1}{{\mathrm{sin} \left({\theta }_i\right)\ }},\ {\lambda }_{1x}=\frac{{\lambda }_1}{{\mathrm{sin} \left({\theta }_i\right)}}$
• : The propagating wave in the $$x$$-direction is a nonuniform plane wave because its amplitude varies with $$z$$.
• : A conducting plate could be inserted at$${}^2$$
• $z=-\frac{m{\lambda }_1}{2{\mathrm{cos} \left({\theta }_i\right)\ }},\ m=1,\ 2,\ 3,\ \dots$
• without changing the field pattern that exists between the plate and the boundary ($$z=0$$). A transverse electric (TE) wave ($$E_{1x}=0$$) would bounce back and forth between the conducting planes and propagate in the $$x$$-direction – rendering the plat$$E$$-boundary configuration effectively a parallel-plate waveguide. (For $$∥$$ polariz., a transverse magnetic wave would propagate instead)

• 1 - $$u_{1x}=\omega/\beta_{1x}=\omega/[\beta_1 \sin{\left(\theta_1 \right)}]$$; $$[λ_{1x}=2π/β_{1x}]$$.     2 - $${\overrightarrow{\boldsymbol{E}}}_1 = 0$$ for all $$x$$ when $$\sin{\left(\beta_1 z \cos{\left(\theta_i \right)}\right)} = 0$$ 