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Dragon Notes

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  UNDER CONSTRUCTION

Normal Incidence [Diel]


When normally incident on a boundary separating two dissipationless (\(σ_1\),\(σ_2\)) dielectric media with respective intrinsic impedances \(η_1,η_2\), the incident EM-wave described in phasor form by \[{\overrightarrow{\boldsymbol{E}}}_i\left(z\right)=\widehat{\boldsymbol{x}}E_{i0}e^{-j{\beta }_1z},\ \ \ {\overrightarrow{\boldsymbol{H}}}_i\left(z\right)=\widehat{\boldsymbol{y}}\frac{E_{i0}}{{\eta }_1}e^{-j{\beta }_1z}\] is partially reflected off the boundary as a wave described by \[{\overrightarrow{\boldsymbol{E}}}_r\left(z\right)=\widehat{\boldsymbol{x}}E_{r0}e^{+j{\beta }_1z},\ {\overrightarrow{\boldsymbol{H}}}_r\left(z\right)=\widehat{\boldsymbol{y}}\frac{E_{r0}}{{\eta }_1}e^{+j{\beta }_1z},\] and partially transmitted into
[#2]
as \[{\overrightarrow{\boldsymbol{E}}}_t\left(z\right)=\widehat{\boldsymbol{x}}E_{t0}e^{-j{\beta }_2z},\ {\overrightarrow{\boldsymbol{H}}}_t\left(z\right)=\widehat{\boldsymbol{y}}\frac{E_{t0}}{{\eta }_2}e^{-j{\beta }_2z}.\] The two waves in
[#1]
sum to \[{\overrightarrow{\boldsymbol{E}}}_1\left(z\right)=\widehat{\boldsymbol{x}}E_{i0}\left[\tau e^{-j{\beta }_1z}+\mathrm{\Gamma } \left(j2{\mathrm{sin} \left({\beta }_1z\right)} \right)\ \right],\] which is composed of a traveling and a standing wave with amplitudes \(\tau E_{i0}\) and \(2 \Gamma E_{i0}\), respectively. Due to the traveling component, \(E_{\text{1}}\) does not have zeros at fixed distances from the interface, but does have locations of maximum and minimum values: \[\underline{{\eta }_2>{\eta}_1}\text{: } \boxed{z_{\text{max}}=-n\frac{{\lambda }_1}{2},\ n=0,1,2,\dots }\ \boxed{z_{\text{min}}=-\left(2n+1\right)\frac{{\lambda }_1}{4},\ n=0,1,2,\dots }\] \[\underline{{\eta }_2<{\eta }_1}\text{: } \boxed{z_{\text{min}}=-n\frac{{\lambda }_1}{2},\ n=0,1,2,\dots }\ \boxed{z_{\text{max}}=-\left(2n+1\right)\frac{{\lambda }_1}{4},\ n=0,1,2,\dots }\] In a dissipationless medium, \(\Gamma\) is real, and there is no phase shift; the \(H\)-field then will have maxima where the \(E\)-field has minima, and minima at its maxima. The \(H\)-field, and the \(E\)-field rewritten in the same form, is: \[{\overrightarrow{\boldsymbol{E}}}_1\left(z\right)=\widehat{\boldsymbol{x}}E_{i0}e^{-j{\beta }_1z}\left(1+\mathrm{\Gamma }e^{j2{\beta }_1z}\right),\ \ \ {\overrightarrow{\boldsymbol{H}}}_1\left(z\right)=\widehat{\boldsymbol{x}}\frac{E_{i0}}{{\eta }_1}e^{-j{\beta }_1z}\left(1-\mathrm{\Gamma }e^{j2{\beta }_1z}\right).\] The reflection and transmission coefficients for the waves above are \[\boxed{\mathrm{\Gamma }=\frac{{\eta }_2-{\eta }_1}{{\eta }_2+{\eta }_1}}\ \boxed{\tau =\frac{2{\eta }_2}{{\eta }_2+{\eta }_1}},\] and the reflected and transmitted field amplitudes are correspondingly \[E_{r0}=\mathrm{\Gamma }E_{i0},\ E_{t0}=\tau E_{i0}.\] The time-average power transmitted into the medium is \[\boxed{\overrightarrow{\mathcal{P}}_{\text{av}}\left(z\right)=\widehat{\boldsymbol{z}}\frac{1}{2}{\left| E_t \right|}^2\mathcal{R}e\left[\frac{1}{{\eta }_2}\right]} .\]

\(n_1=1,\text{ } n_2=2, \text{ }\)

\(\beta_1 =1, \text{ }\beta_2 =3 \text{ }\)






Dragon Notes,   Est. 2018     About

By OverLordGoldDragon