 # Dragon Notes UNDER CONSTRUCTION
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## 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
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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
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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 $\boxed{\overrightarrow{\mathcal{P}}_{\text{av}}\left(z\right)=\widehat{\boldsymbol{z}}\frac{1}{2}{\left| E_t \right|}^2\mathcal{R}e\left[\frac{1}{{\eta }_2}\right]} .$

$$n_1=1,\text{ } n_2=2, \text{ }$$

$$\beta_1 =1, \text{ }\beta_2 =3 \text{ }$$