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Dragon Notes

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  UNDER CONSTRUCTION

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:
  • [1]: 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. \([\beta_{1z} = \beta_1 \cos{\left( \theta_i \right)}]\)
  • [2]: No average power is propagated in the \(z\)-direction. (\(E_{1y}\) and \(H_{1x}\) are \(90^o\) out of phase)
  • [3]: 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\({}^1\)
  • \[u_{1x}=\frac{u_1}{{\mathrm{sin} \left({\theta }_i\right)\ }},\ {\lambda }_{1x}=\frac{{\lambda }_1}{{\mathrm{sin} \left({\theta }_i\right)}}\]
  • [4]: The propagating wave in the \(x\)-direction is a nonuniform plane wave because its amplitude varies with \(z\).
  • [5]: 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 \(\sin{\left(\beta_1 z \cos{\left(\theta_i \right)}\right)} = 0\)
Waves


Dragon Notes,   Est. 2018     About

By OverLordGoldDragon