6 无界媒质中的均匀平面波

6.1 理想介质中的均匀平面波

6.1.1 理想介质中的波动方程

在无源的线性、均匀、各向同性的无限大理想介质中，

• 对于一般场，齐次波动方程为

  • ${\displaystyle {\mathbf{\nabla }}^2\mathit{E}(\mathit{r},t)-\mu \epsilon \frac{{\partial }^2\mathit{E}(\mathit{r},t)}{\partial t^2}=0}$$\d\grad^2 \bm E(\bm r, t) - \mu \ve \pdv[2]{\bm E(\bm r, t)}{t} = 0$.

  • ${\displaystyle {\mathbf{\nabla }}^2\mathit{H}(\mathit{r},t)-\mu \epsilon \frac{{\partial }^2\mathit{H}(\mathit{r},t)}{\partial t^2}=0}$$\d\grad^2 \bm H(\bm r, t) - \mu \ve \pdv[2]{\bm H(\bm r, t)}{t} = 0$.

• 对于时谐场，亥姆霍兹方程为

  • ${\mathbf{\nabla }}^2\dot{\mathit{E}}(\mathit{r})+k^2\dot{\mathit{E}}(\mathit{r})=\dot{\mathbf{0}}$$\grad^2 \dot{\bm E}(\bm r) + k^2 \dot{\bm E}(\bm r) = \dot{\bm 0}$.

  • ${\mathbf{\nabla }}^2\dot{\mathit{E}}(\mathit{r})+k^2\dot{\mathit{E}}(\mathit{r})=\dot{\mathbf{0}}$$\grad^2 \dot{\bm E}(\bm r) + k^2 \dot{\bm E}(\bm r) = \dot{\bm 0}$.

• 概念说明

  • 波数：$k=\omega \sqrt{\mu \epsilon }={\displaystyle \frac{\omega }{v}}$$k = \omega \sqrt{\mu \ve} = \dfrac{\omega}{v}$.

  • 波速：$v={\displaystyle \frac{1}{\sqrt{\mu \epsilon }}}={\displaystyle \frac{c}{\sqrt{{\mu }_{\text{r}}{\epsilon }_{\text{r}}}}}$$v = \dfrac{1}{\sqrt{\mu \ve}} = \dfrac{c}{\sqrt{\mu_\text r \ve_\text r}}$.

6.1.2 理想介质中均匀平面波

• 考虑沿 $z$$z$ 轴方向传播的均匀平面波

  • ${\displaystyle \frac{{\partial }^2\mathit{E}(z,t)}{\partial z^2}=\mu \epsilon \frac{{\partial }^2\mathit{E}(z,t)}{\partial t^2}={\displaystyle \frac{1}{v^2}}\frac{{\partial }^2\mathit{E}(z,t)}{\partial t^2}}$$\d\pdv[2]{\bm E(z, t)}{z} = \mu\ve \pdv[2]{\bm E(z, t)}{t} = \dfrac{1}{v^2} \pdv[2]{\bm E(z, t)}{t}$.

  • ${\displaystyle \frac{{\partial }^2\mathit{H}(z,t)}{\partial z^2}=\mu \epsilon \frac{{\partial }^2\mathit{H}(z,t)}{\partial t^2}={\displaystyle \frac{1}{v^2}}\frac{{\partial }^2\mathit{H}(z,t)}{\partial t^2}}$$\d\pdv[2]{\bm H(z, t)}{z} = \mu\ve \pdv[2]{\bm H(z, t)}{t} = \dfrac{1}{v^2} \pdv[2]{\bm H(z, t)}{t}$.

• 麦克斯韦第一与第二方程的约束

  • $E_z(z,t)=H_z(z,t)=0$$E_z(z, t) = H_z(z, t) = 0$.

  • ${\displaystyle \frac{\partial E_x(z,t)}{\partial z}=-\mu \frac{\partial H_y(z,t)}{\partial t}}$$\dpdv{E_x(z, t)}{z} = -\mu \pdv{H_y(z, t)}{t}$.

  • $-{\displaystyle \frac{\partial H_y(z,t)}{\partial z}=\epsilon \frac{\partial E_x(z,t)}{\partial t}}$$-\dpdv{H_y(z, t)}{z} = \ve \pdv{E_x(z, t)}{t}$.

• 齐次一维标量波动方程的标准形式

  • ${\displaystyle \frac{{\partial }^2{E}_x(z,t)}{\partial z^2}={\displaystyle \frac{1}{v^2}}\frac{{\partial }^2{E}_x(z,t)}{\partial t^2}}$$\d\pdv[2]{E_x(z, t)}{z} = \dfrac{1}{v^2} \pdv[2]{E_x(z, t)}{t}$.

  • ${\displaystyle \frac{{\partial }^2{H}_y(z,t)}{\partial z^2}={\displaystyle \frac{1}{v^2}}\frac{{\partial }^2{H}_y(z,t)}{\partial t^2}}$$\d\pdv[2]{H_y(z, t)}{z} = \dfrac{1}{v^2} \pdv[2]{H_y(z, t)}{t}$.

• 上述方程的解

  • $E_x(z,t)=f_1(t-z/v)+f_2(t+z/v)$$E_x(z, t) = f_1(t - z / v) + f_2(t + z / v)$.

    • 入射波 $E_x^{+}(z,t)=f_1(t-z/v)$$E_x^+(z, t) = f_1(t - z / v)$ 沿 $+{\mathit{a}}_z$$+\bm a_z$ 方向传播.

    • 反射波 $E_x^{-}(z,t)=f_2(t+z/v)$$E_x^-(z, t) = f_2(t + z / v)$ 沿 $-{\mathit{a}}_z$$-\bm a_z$ 方向传播.

  • $H_y(z,t)=H_y^{+}(z,t)+H_y^{-}(z,t)$$H_y(z, t) = H_y^+(z, t) + H_y^-(z, t)$.

    • $H_y^{+}(z,t)={\displaystyle \frac{1}{\mu v}}{E}_x^{+}(z,t)$$H_y^+(z, t) = \dfrac{1}{\mu v} E_x^+(z, t)$.

    • $H_y^{-}(z,t)=-{\displaystyle \frac{1}{\mu v}}{E}_x^{-}(z,t)$$H_y^-(z, t) = -\dfrac{1}{\mu v} E_x^-(z, t)$.

• 本征波阻抗

  • $\eta :={\displaystyle \frac{E_x^{+}(z,t)}{H_y^{+}(z,t)}}=\sqrt{{\displaystyle \frac{\mu }{\epsilon }}}$$\eta := \dfrac{E_x^+(z, t)}{H_y^+(z, t)} = \sqrt{\dfrac{\mu}{\ve}}$.

  • ${\eta }_0=\sqrt{{\displaystyle \frac{{\mu }_0}{{\epsilon }_0}}}\approx 120\pi \approx 377\mathrm{\Omega }$$\eta_0 = \sqrt{\dfrac{\mu_0}{\ve_0}} \approx 120 \pi \approx 377\ \Omega$.

6.1.3 时谐均匀平面波瞬时值

• 时谐电磁场

  • ${\displaystyle \frac{{\mathrm{d}}^2{\dot{E}}_x(z)}{\mathrm{d}{z}^2}=-{\omega }^2\mu \epsilon {\dot{E}}_x(z)={\gamma }^2{\dot{E}}_x(z)}$$\d\dv[2]{\dot{ E}_x(z)}{z} = -\omega^2 \mu \ve \dot{E}_x(z) = \gamma^2 \dot{E}_x(z)$.

  • ${\displaystyle \frac{{\mathrm{d}}^2{\dot{H}}_y(z)}{\mathrm{d}{z}^2}=-{\omega }^2\mu \epsilon {\dot{H}}_y(z)={\gamma }^2{\dot{H}}_y(z)}$$\d\dv[2]{\dot{H}_y(z)}{z} = -\omega^2 \mu \ve \dot{H}_y(z) = \gamma^2 \dot{H}_y(z)$.

• 符号说明

  • 传播常数：$\gamma :=\sqrt{-k^2}=\sqrt{-{\omega }^2\mu \epsilon }\equiv \mathrm{j}\beta$$\gamma := \sqrt{-k^2} = \sqrt{-\omega^2 \mu \ve} \equiv \j\beta$

  • 相位常数：$\beta :=\omega \sqrt{\mu \epsilon }=k$$\beta := \omega \sqrt{\mu \ve} = k$，单位为 $\mathrm{rad}/\mathrm{m}$$\rm{rad/m}$.

• 波动方程的复数解

  • ${\dot{E}}_x(z)={\dot{E}}_{x0}^{+}{\mathrm{e}}^{-\gamma z}+{\dot{E}}_{x0}^{-}{\mathrm{e}}^{\gamma z}={\dot{E}}_{x0}^{+}{\mathrm{e}}^{-\mathrm{j}\beta z}+{\dot{E}}_{x0}^{-}{\mathrm{e}}^{\mathrm{j}\beta z}$$\dot E_x(z) = \dot E_{x0}^+ \e^{-\gamma z} + \dot E_{x0}^- \e^{\gamma z} = \dot E_{x0}^+ \e^{-\j\beta z} + \dot E_{x0}^- \e^{\j\beta z}$.

  • ${\dot{H}}_y(z)={\dot{H}}_{y0}^{+}{\mathrm{e}}^{-\gamma z}+{\dot{H}}_{y0}^{-}{\mathrm{e}}^{\gamma z}={\displaystyle \frac{1}{\eta }}({\dot{E}}_{x0}^{+}{\mathrm{e}}^{-\mathrm{j}\beta z}-{\dot{E}}_{x0}^{-}{\mathrm{e}}^{\mathrm{j}\beta z})$$\dot H_y(z) = \dot H_{y0}^+ \e^{-\gamma z} + \dot H_{y0}^- \e^{\gamma z} = \dfrac{1}{\eta} \pqty{ \dot E_{x0}^+ \e^{-\j \beta z} - \dot E_{x0}^- \e^{\j\beta z} }$.

• 给定初值 ${\dot{E}}_{x0}^{+}=E_{x0}^{+}\mathrm{\angle }{\phi }_{\mathrm{e}}$$\dot E_{x0}^+ = E_{x0}^+ \angle\varphi_\e$，则瞬时值为

  • ${\mathit{E}}_x^{+}(z,t)={\mathit{a}}_x\sqrt{2}{E}_{x0}^{+}\cos (\omega t-\beta z+{\phi }_{\mathrm{e}})$$\bm E_x^+(z, t) = \bm a_x \sqrt 2 E_{x0}^+ \cos(\omega t - \beta z + \varphi_\e)$.

  • ${\mathit{H}}_y^{+}(z,t)={\mathit{a}}_y{\displaystyle \frac{\sqrt{2}{E}_{x0}^{+}}{\eta }}\cos (\omega t-\beta z+{\phi }_{\mathrm{e}})$$\bm H_y^+(z, t) = \bm a_y \dfrac{\sqrt 2 E_{x0}^+}{\eta} \cos(\omega t - \beta z + \varphi_\e)$.

  • ${\mathit{S}}^{+}(z,t)={\mathit{a}}_z{\displaystyle \frac{2{E_{x0}^{+}}^2}{\eta }}{\cos }^2(\omega t-\beta z+{\phi }_{\mathrm{e}})$$\bm S^+(z, t) = \bm a_z \dfrac{2 {E_{x0}^+}^2}{\eta} \cos[2] (\omega t - \beta z + \varphi_\e)$.

• 均匀平面波的说明

  • 坡应廷矢量的角频率为电磁场的两倍.

  • 电磁场量与坡应廷矢量的振幅为常数.

  • 电磁场量的相位相等，${\phi }_{\text{p}}=\omega t-\beta z+{\phi }_{\mathrm{e}}$$\varphi_\text p = \omega t - \beta z + \varphi_\e$.

6.1.4 时谐均匀平面波的概念

• 定义

  • 周期：$T\equiv {\displaystyle \frac{1}{f}}:={\displaystyle \frac{2\pi }{\omega }}$$T \equiv \dfrac{1}{f} := \dfrac{2\pi}{\omega}$.

  • 传播常数：$\gamma :=\sqrt{-{\omega }^2\mu \epsilon }$$\gamma := \sqrt{-\omega^2 \mu \ve}$.

• 波

  • 波速：$v:={\displaystyle \frac{1}{\sqrt{\mu \epsilon }}}\equiv {\displaystyle \frac{c}{\sqrt{{\mu }_{\text{r}}{\epsilon }_{\text{r}}}}}$$v := \dfrac{1}{\sqrt{\mu \ve}} \equiv \dfrac{c}{\sqrt{\mu_\text r \ve_\text r}}$.

  • 波数：$k:=\omega \sqrt{\mu \epsilon }={\displaystyle \frac{\omega }{v}}$$k := \omega\sqrt{\mu \ve} = \dfrac{\omega}{v}$.

  • 波长：$\lambda :=vT={\displaystyle \frac{2\pi }{k}}$$\lambda := v T = \dfrac{2\pi}{k}$.

• 相

  • 相位：${\phi }_{\text{p}}:=\omega t-\beta z+{\phi }_{\mathrm{e}}$$\varphi_\text p := \omega t - \beta z + \varphi_\e$.

  • 相数：$\beta :=\mathrm{Im}\gamma =\omega \sqrt{\mu \epsilon }$$\beta := \Im\gamma = \omega\sqrt{\mu\ve}$.

  • 相速：$v_{\text{p}}:={\displaystyle {\frac{\mathrm{d}z}{\mathrm{d}t}\phantom{\int }|}_{{\phi }_{\text{p}}=\text{常}\text{数}}={\displaystyle \frac{\omega }{\beta }}={\displaystyle \frac{1}{\sqrt{\mu \epsilon }}}}$$v_\text p := \d\eval{\dv{z}{t}}_{\varphi_\text p = 常数} = \dfrac{\omega}{\beta} = \dfrac{1}{\sqrt{\mu\ve}}$.

• 理想介质中

  • $\beta =k={\displaystyle \frac{2\pi }{\lambda }}={\displaystyle \frac{\omega }{v_{\text{p}}}}$$\beta = k = \dfrac{2\pi}{\lambda} = \dfrac{\omega}{v_\text p}$.

  • $v=v_{\text{p}}={\displaystyle \frac{1}{\sqrt{\mu \epsilon }}}={\displaystyle \frac{\omega }{\beta }}$$v = v_\text p = \dfrac{1}{\sqrt{\mu\ve}} = \dfrac{\omega}{\beta}$.

6.1.5 均匀平面波的传播特性

• 均匀平面波的分量

  • 电磁场沿平面波传播方向无分量，称为横电磁波（TEM 波）

  • 横向分量中，两组相互垂直的电磁场分量分别组成独立分量波组.

  • 每组分量波可存在入射波与反射波，其速度均为 $v={\displaystyle \frac{1}{\sqrt{\mu \epsilon }}}$$v = \dfrac{1}{\sqrt{\mu\ve}}$.

• 电场与磁场的关系

  • 平面波的性质

    • $\dot{\mathit{H}}={\displaystyle \frac{1}{\eta }}{\mathit{a}}_{\text{n}}\times \dot{\mathit{E}}$$\dot{\bm H} = \dfrac{1}{\eta} \bm a_\text n \cp \dot{\bm E}$.

    • $\dot{\mathit{E}}=\eta \dot{\mathit{H}}\times {\mathit{a}}_{\text{n}}$$\dot{\bm E} = \eta \dot{\bm H} \cp \bm a_\text n$.

  • 麦克斯韦方程

    • $\dot{\mathit{H}}=-{\displaystyle \frac{1}{\mathrm{j}\omega \mu }}\mathbf{\nabla }\times \dot{\mathit{E}}$$\dot{\bm H} = -\dfrac{1}{\j\omega\mu} \curl \dot{\bm E}$.

    • $\dot{\mathit{E}}={\displaystyle \frac{1}{\mathrm{j}\omega \epsilon }}\mathbf{\nabla }\times \dot{\mathit{H}}$$\dot{\bm E} = \dfrac{1}{\j\omega \ve} \curl \dot{\bm H}$.

• 理想媒质（无损耗媒质）

  • 本征波阻抗 $\eta =\sqrt{{\displaystyle \frac{\mu }{\epsilon }}}$$\eta = \sqrt{\dfrac{\mu}{\ve}}$ 为实数.

  • 电场与磁场的相位相同.

• 电磁波的能量

  • 电磁场能量密度相等

    • 电场：$w_{\mathrm{e}}^{+}(z,t)={\displaystyle \frac{\epsilon }{2}}{E}_x^{+}(z,t{)}^2={\displaystyle \frac{\sqrt{\mu \epsilon }}{2}}{E}_x^{+}{H}_x^{+}(z,t)$$w_\e^+(z, t) = \dfrac{\ve}{2} E_x^+(z, t)^2 = \dfrac{\sqrt{\mu\ve}}{2} E_x^+ H_x^+(z, t)$.

    • 磁场：$w_{\text{m}}^{+}(z,t)={\displaystyle \frac{\mu }{2}}{H}_x^{+}(z,t{)}^2={\displaystyle \frac{\sqrt{\mu \epsilon }}{2}}{E}_x^{+}{H}_x^{+}(z,t)$$w_\text m^+(z, t) = \dfrac{\mu}{2} H_x^+(z, t)^2 = \dfrac{\sqrt{\mu\ve}}{2} E_x^+ H_x^+(z, t)$.

    • 于是：$w_{\mathrm{e}}^{+}(z,t)=w_{\text{m}}^{+}(z,t),w_{\mathrm{e}}^{-}(z,t)=w_{\text{m}}^{-}(z,t)$$w_\e^+(z, t) = w_\text m^+(z, t), \quad w_\e^-(z, t) = w_\text m^-(z, t)$.

  • 能量流动密度与均值

    • ${\tilde{\mathit{S}}}^{+}=\dot{\mathit{E}}(z)\times {\dot{\mathit{H}}}^{\ast }(z)={\displaystyle \frac{{E_{x0}^{+}}^2}{\eta }}{\mathit{a}}_z={\displaystyle \frac{{E_{x0\text{m}}^{+}}^2}{2\eta }}{\mathit{a}}_z$$\wt{\bm S}^+ = \dot{\bm E}(z) \cp \dot{\bm H}^*(z) = \dfrac{{E_{x0}^+}^2}{\eta} \bm a_z = \dfrac{{E_{x0\text m}^+}^2}{2\eta} \bm a_z$.

    • ${\mathit{S}}_{\mathrm{avg}}^{+}(z)=\mathrm{Re}{\tilde{\mathit{S}}}^{+}=E_{x0}^{+}{H}_{x0}^{+}{\mathit{a}}_z$$\bm S_\avg^+(z) = \Re\wt{\bm S}^+ = E_{x0}^+ H_{x0}^+ \bm a_z$.

    • ${\mathit{S}}_{\mathrm{avg}}^{-}(z)=\mathrm{Re}{\tilde{\mathit{S}}}^{-}=-E_{x0}^{-}{H}_{x0}^{-}{\mathit{a}}_z$$\bm S_\avg^-(z) = \Re\wt{\bm S}^- = -E_{x0}^- H_{x0}^- \bm a_z$.

6.2 平面波的极化

6.2.1 平面波极化

• 对于沿 z 轴方向传播的均匀平面波，

  • $E_x(z,t)=E_{x\text{m}}\cos (\omega t-\beta z+{\phi }_x)$$E_x(z, t) = E_{x \text m} \cos(\omega t - \beta z + \varphi_x)$.

  • $E_y(z,t)=E_{y\text{m}}\cos (\omega t-\beta z+{\phi }_y)$$E_y(z, t) = E_{y \text m} \cos(\omega t - \beta z + \varphi_y)$.

  • $\mathit{E}={\mathit{a}}_x{E}_x(z,t)+{\mathit{a}}_y{E}_y(z,t)$$\bm E = \bm a_x E_x(z, t) + \bm a_y E_y(z, t)$.

• 波的极化（偏振）

  • 定义：空间中给定电场矢量的末端点随时间变化的轨迹.

  • 分类：线极化、圆极化、椭圆极化.

6.2.2 直线极化波

• 当 ${\phi }_x={\phi }_y$$\varphi_x = \varphi_y$ 时，

  • ${\displaystyle \frac{E_x}{E_y}}={\displaystyle \frac{E_{x\text{m}}}{E_{y\text{m}}}}$$\dfrac{E_x}{E_y} = \dfrac{E_{x \text m}}{E_{y \text m}}$.

  • $\phi :=\arctan {\displaystyle \frac{E_y}{E_x}}=\arctan {\displaystyle \frac{E_{y\text{m}}}{E_{x\text{m}}}}$$\varphi := \arctan\dfrac{E_y}{E_x} = \arctan\dfrac{E_{y \text m}}{E_{x \text m}}$.

• 当 $|{\phi }_x-{\phi }_y|=\pi$$\vqty{\varphi_x - \varphi_y} = \pi$ 时，

  • ${\displaystyle \frac{E_x}{E_y}}=-{\displaystyle \frac{E_{x\text{m}}}{E_{y\text{m}}}}$$\dfrac{E_x}{E_y} = -\dfrac{E_{x \text m}}{E_{y \text m}}$.

  • $\phi :=\arctan {\displaystyle \frac{E_y}{E_x}}=-\arctan {\displaystyle \frac{E_{y\text{m}}}{E_{x\text{m}}}}$$\varphi := \arctan\dfrac{E_y}{E_x} = -\arctan\dfrac{E_{y \text m}}{E_{x \text m}}$.

• 特殊情况

  • $\phi =0$$\varphi = 0$：$x$$x$ 轴取向的线性极化波.

  • $\phi ={\displaystyle \frac{\pi }{2}}$$\varphi = \dfrac{\pi}{2}$：$y$$y$ 轴取向的线性极化波.

6.2.3 圆形极化波

当 $E_{x\text{m}}=E_{y\text{m}}=E_{\text{m}}$$E_{x \text m} = E_{y \text m} = E_\text m$ 且 $|{\phi }_x-{\phi }_y|={\displaystyle \frac{\pi }{2}}$$\vqty{\varphi_x - \varphi_y} = \dfrac{\pi}{2}$ 时，$\left|\mathit{E}\right|=E_{\text{m}}$$\vqty{\bm E} = E_\text m$.

• 当 ${\phi }_x-{\phi }_y={\displaystyle \frac{\pi }{2}}$$\varphi_x - \varphi_y = \dfrac{\pi}{2}$ 时，

  • $\phi =\omega t-kz+{\phi }_x$$\varphi = \omega t - k z + \varphi_x$.

  • 此时称为右旋圆极化.

• 当 ${\phi }_y-{\phi }_x={\displaystyle \frac{\pi }{2}}$$\varphi_y - \varphi_x = \dfrac{\pi}{2}$ 时，

  • $\phi =-(\omega t-kz+{\phi }_x)$$\varphi = -(\omega t - k z + \varphi_x)$.

  • 此时称为左旋圆极化.

6.2.4 椭圆极化波

消去 $t$$t$，记 $\theta ={\phi }_x-{\phi }_y$$\theta = \varphi_x - \varphi_y$，得

1
Cos[a + b]^2 - 2 Cos[a + b] Cos[a + c] Cos[b - c] + Cos[a + c]^2 // Simplify

• 直线极化

  • 当 $\theta =0$$\theta = 0$ 时，为一三象限的直线极化.

  • 当 $\theta =\pi$$\theta = \pi$ 时，为二四象限的直线极化.

• 圆形极化（$E_{x\text{m}}=E_{y\text{m}}=E_{\text{m}}$$E_{x \text m} = E_{y \text m} = E_\text m$）

  • 当 $\theta ={\displaystyle \frac{\pi }{2}}$$\theta = \dfrac{\pi}{2}$ 时，为右旋圆极化.

  • 当 $\theta =-{\displaystyle \frac{\pi }{2}}$$\theta = -\dfrac{\pi}{2}$ 时，为左旋圆极化.

• 椭圆极化

  • $\tan 2\phi ={\displaystyle \frac{2E_{x\text{m}}{E}_{y\text{m}}\cos \theta }{E_{x\text{m}}^2-E_{y\text{m}}^2}}$$\tan2\varphi = \dfrac{2 E_{x \text m} E_{y \text m} \cos\theta}{E_{x \text m}^2 - E_{y \text m}^2}$.

  • 当 $\theta =\pm {\displaystyle \frac{\pi }{2}},E_{x\text{m}}\ne E_{y\text{m}}$$\theta = \pm \dfrac{\pi}{2}, E_{x \text m} \ne E_{y \text m}$ 时，为正椭圆极化.

  • 固定 $z$$z$，增加 $t$$t$，

    • 当 $\theta >0$$\theta >0$ 时，为右旋椭圆极化.

    • 当 $\theta <0$$\theta < 0$ 时，为左旋椭圆极化.

  • 固定 $t$$t$，增加 $z$$z$，

    • 当 $\theta >0$$\theta >0$ 时，为左旋椭圆极化.

    • 当 $\theta <0$$\theta < 0$ 时，为右旋椭圆极化.

6.2.5 合成与分解

6.3 导电媒质中的均匀平面波

6.3.1 导电媒质中的波动方程

在均匀、线性、各向同性、无局外电源的导电媒质中，

• 复数形式的麦克斯韦方程组

  • 麦克斯韦第一方程

    • $\mathbf{\nabla }\times \dot{\mathit{H}}=\dot{\mathit{J}}+\mathrm{j}\omega \dot{\mathit{D}}=\mathrm{j}\omega \epsilon (1-\mathrm{j}{\displaystyle \frac{\sigma }{\omega \epsilon }})\dot{\mathit{E}}=\mathrm{j}\omega {\epsilon }_{\text{c}}\dot{\mathit{E}}$$\curl \dot{\bm H} = \dot{\bm J} + \j\omega \dot{\bm D} = \j\omega \ve \pqty{ 1 - \j \dfrac{\sigma}{\omega \ve} } \dot{\bm E} = \j\omega \ve_\text c \dot{\bm E}$.

    • 复介电常数：${\epsilon }_{\text{c}}=\epsilon (1-\mathrm{j}{\displaystyle \frac{\sigma }{\omega \epsilon }})=\left|{\epsilon }_{\text{c}}\right|{\mathrm{e}}^{\mathrm{j}{\delta }_{\text{c}}}$$\ve_\text c = \ve \pqty{1 - \j\dfrac{\sigma}{\omega \ve}} = \vqty{\ve_\text c} \e^{\j \delta_\text c}$.

    • 损耗角正切：$\tan {\delta }_{\text{c}}={\displaystyle \frac{\sigma }{\omega \epsilon }}$$\tan\delta_\text c = \dfrac{\sigma}{\omega\ve}$.

  • 麦克斯韦第二方程：$\mathbf{\nabla }\times \dot{\mathit{E}}=-\mathrm{j}\omega \mu \dot{\mathit{H}}$$\curl \dot{\bm E} = -\j\omega \mu \dot{\bm H}$.

  • 麦克斯韦第三方程：$\mathbf{\nabla }\mathbf{\cdot }\dot{\mathit{H}}=0$$\div \dot{\bm H} = 0$.

  • 麦克斯韦第四方程：$\mathbf{\nabla }\mathbf{\cdot }\dot{\mathit{E}}=0$$\div \dot{\bm E} = 0$.

• 复数波动方程

  • ${\mathbf{\nabla }}^2\dot{\mathit{E}}(\mathit{r})+{\omega }^2\mu {\epsilon }_{\text{c}}\dot{\mathit{E}}(\mathit{r})=0$$\grad^2 \dot{\bm E}(\bm r) + \omega^2 \mu \ve_\text c \dot{\bm E}(\bm r) = 0$.

  • ${\mathbf{\nabla }}^2\dot{\mathit{H}}(\mathit{r})+{\omega }^2\mu {\epsilon }_{\text{c}}\dot{\mathit{H}}(\mathit{r})=0$$\grad^2 \dot{\bm H}(\bm r) + \omega^2 \mu \ve_\text c \dot{\bm H}(\bm r) = 0$.

  • $k^2={\omega }^2\mu {\epsilon }_{\text{c}}={\omega }^2\mu \epsilon (1-\mathrm{j}{\displaystyle \frac{\sigma }{\omega \epsilon }})$$k^2 = \omega^2 \mu \ve_\text c = \omega^2 \mu \ve \pqty{1 - \j\dfrac{\sigma}{\omega\ve}}$.

6.3.2 时谐均匀平面波瞬时值

• 考虑电磁波沿 $z$$z$ 轴方向传播，

  • ${\displaystyle \frac{{\mathrm{d}}^2{\dot{E}}_x(z)}{\mathrm{d}{z}^2}=-{\omega }^2\mu {\epsilon }_{\text{c}}{\dot{E}}_x(z)={\gamma }^2{\dot{E}}_x(z)}$$\d\dv[2]{\dot{E}_x(z)}{z} = -\omega^2 \mu \ve_\text c \dot{E}_x(z) = \gamma^2 \dot{E}_x(z)$.

  • ${\displaystyle \frac{{\mathrm{d}}^2{\dot{H}}_y(z)}{\mathrm{d}{z}^2}=-{\omega }^2\mu {\epsilon }_{\text{c}}{\dot{H}}_y(z)={\gamma }^2{\dot{H}}_y(z)}$$\d\dv[2]{\dot{H}_y(z)}{z} = -\omega^2 \mu \ve_\text c \dot{H}_y(z) = \gamma^2 \dot{H}_y(z)$.

• 复数 传播常数

  • 传播常数：$\gamma =\sqrt{-k^2}=\alpha +\mathrm{j}\beta$$\gamma = \sqrt{-k^2} = \alpha + \j\beta$.

  • 衰减常数：$\alpha =\omega \sqrt{{\displaystyle \frac{\mu \epsilon }{2}}(\sqrt{1+{\displaystyle \frac{{\sigma }^2}{{\omega }^2{\epsilon }^2}}}-1)}$$\alpha = \omega \sqrt{ \dfrac{\mu\ve}{2} \pqty{ \sqrt{1 + \dfrac{\sigma^2}{\omega^2 \ve^2}} - 1 } }$，单位为 $1\mathrm{Np}=20\lg \mathrm{e}\mathrm{dB}$$1\ \rm{Np} = 20 \lg\e\ \rm{dB}$.

  • 相位常数：$\beta =\omega \sqrt{{\displaystyle \frac{\mu \epsilon }{2}}(\sqrt{1+{\displaystyle \frac{{\sigma }^2}{{\omega }^2{\epsilon }^2}}}+1)}>\alpha$$\beta = \omega \sqrt{ \dfrac{\mu\ve}{2} \pqty{ \sqrt{1 + \dfrac{\sigma^2}{\omega^2 \ve^2}} + 1 } } > \alpha$，单位为 $\mathrm{rad}/\mathrm{m}$$\rm{rad / m}$.

• 波动方程的复数解

  • ${\dot{E}}_x(z)={\dot{E}}_{x0}^{+}{\mathrm{e}}^{-\gamma z}+{\dot{E}}_{x0}^{-}{\mathrm{e}}^{\gamma z}={\dot{E}}_{x0}^{+}{\mathrm{e}}^{-\alpha z}{\mathrm{e}}^{-\mathrm{j}\beta z}+{\dot{E}}_{x0}^{-}{\mathrm{e}}^{\alpha z}{\mathrm{e}}^{\mathrm{j}\beta z}$$\dot E_x(z) = \dot E_{x0}^+ \e^{-\gamma z} + \dot E_{x0}^- \e^{\gamma z} = \dot E_{x0}^+ \e^{-\alpha z} \e^{-\j\beta z} + \dot E_{x0}^- \e^{\alpha z} \e^{\j\beta z}$.

  • ${\dot{H}}_y(z)={\dot{H}}_{y0}^{+}{\mathrm{e}}^{-\gamma z}+{\dot{H}}_{y0}^{-}{\mathrm{e}}^{\gamma z}={\displaystyle \frac{1}{\eta }}({\dot{E}}_{x0}^{+}{\mathrm{e}}^{-\alpha z}{\mathrm{e}}^{-\mathrm{j}\beta z}-{\dot{E}}_{x0}^{-}{\mathrm{e}}^{\alpha z}{\mathrm{e}}^{\mathrm{j}\beta z})$$\dot H_y(z) = \dot H_{y0}^+ \e^{-\gamma z} + \dot H_{y0}^- \e^{\gamma z} = \dfrac{1}{\eta} \pqty{ \dot E_{x0}^+ \e^{-\alpha z} \e^{-\j \beta z} - \dot E_{x0}^- \e^{\alpha z} \e^{\j\beta z} }$.

• 复数 本征波阻抗

  • ${\eta }_{\text{c}}=\sqrt{{\displaystyle \frac{\mu }{{\epsilon }_{\text{c}}}}}={\displaystyle \frac{\eta }{\sqrt{1-\mathrm{j}\frac{\sigma }{\omega \epsilon }}}}=\left|{\eta }_{\text{c}}\right|\mathrm{\angle }\varphi$$\eta_\text c = \sqrt{\dfrac{\mu}{\ve_\text c}} = \dfrac{\eta}{\sqrt{ 1 - \j \cfrac{\sigma}{\omega \ve} }} = \vqty{\eta_\text c} \angle\phi$.

  • ${\eta }_{\text{c}}={\displaystyle \frac{{\dot{E}}_x^{+}(z)}{{\dot{H}}_y^{+}(z)}}={\displaystyle \frac{{\dot{E}}_x^{-}(z)}{-{\dot{H}}_y^{-}(z)}}$$\eta_\text c = \dfrac{\dot{E}_x^+(z)}{\dot{H}_y^+(z)} = \dfrac{\dot{E}_x^-(z)}{-\dot{H}_y^-(z)}$.

• 波动方程的瞬时值

  • ${\mathit{E}}_x^{+}(z,t)=\sqrt{2}{E}_{x0}^{+}{\mathrm{e}}^{-\alpha z}\cos (\omega t-\beta z+{\phi }_{\mathrm{e}}){\mathit{a}}_z$$\bm E_x^+(z, t) = \sqrt 2 E_{x0}^+ \e^{-\alpha z} \cos(\omega t - \beta z + \varphi_\e) \bm a_z$.

  • ${\mathit{H}}_y^{+}(z,t)=\sqrt{2}{\displaystyle \frac{E_{x0}^{+}}{\left|{\eta }_{\text{c}}\right|}}{\mathrm{e}}^{-\alpha z}\cos (\omega t-\beta z+{\phi }_{\text{e}}-\varphi ){\mathit{a}}_z$$\bm H_y^+(z, t) = \sqrt 2 \dfrac{E_{x0}^+}{\vqty{\eta_\text c}} \e^{-\alpha z} \cos(\omega t - \beta z + \varphi_\text e - \phi) \bm a_z$.

  • 电场相位超前磁场相位 $\varphi$$\phi$.

• 相速

  $$v_{\text{p}}={\displaystyle \frac{\omega }{\beta }}={\displaystyle \frac{v_{\text{p 无损}}}{\sqrt{\frac{1}{2}(\sqrt{1+{\left(\frac{\sigma }{\omega \epsilon }\right)}^2}+1)}}}.$$

• 波长

  $$\lambda ={\displaystyle \frac{2\pi }{\beta }}={\displaystyle \frac{{\lambda }_{\text{无}\text{损}}}{\sqrt{\frac{1}{2}(\sqrt{1+{\left(\frac{\sigma }{\omega \epsilon }\right)}^2}+1)}}}.$$

• 相速与 $\omega$$\omega$ 有关，不同频率的电磁波相速不同，从而发生色散，因此导电媒质是色散媒质.

6.3.3 导电媒质中平面波特性

• 均匀平面波的分量

  • 仍为横电磁波（TEM 波）.

  • 横向分量有两组独立分量波组.

  • 每组独立分量波存在入射波与反射波.

• 入射波与反射波

  • 波数 $k=\beta -\mathrm{j}\alpha$$k = \beta - \j\alpha$，传播常数 $\gamma =\alpha +\mathrm{j}\beta$$\gamma = \alpha + \j\beta$ 都是复数.

  • 波在传播的过程中按相速度 $\beta$$\beta$ 滞后，按 ${\mathrm{e}}^{-\alpha \left|z\right|}$$\e^{-\alpha \vqty{z}}$ 衰减.

  • 电导率 $\sigma$$\sigma$ 越大，衰减因子 $\alpha$$\alpha$ 越大，波长 $\lambda ={\displaystyle \frac{2\pi }{\beta }}$$\lambda = \dfrac{2\pi}{\beta}$ 越短，相速 $v_{\text{p}}={\displaystyle \frac{\omega }{\beta }}$$v_\text p = \dfrac{\omega}{\beta}$ 越慢.

• 电场与磁场的关系

  • 平面波的性质

    • $\dot{\mathit{H}}={\displaystyle \frac{1}{{\eta }_{\text{c}}}}{\mathit{a}}_{\text{n}}\times \dot{\mathit{E}}$$\dot{\bm H} = \dfrac{1}{\eta_\text c} \bm a_\text n \cp \dot{\bm E}$.

    • $\dot{\mathit{E}}={\eta }_{\text{c}}\dot{\mathit{H}}\times {\mathit{a}}_{\text{n}}$$\dot{\bm E} = \eta_\text c \dot{\bm H} \cp \bm a_\text n$.

  • 麦克斯韦方程

    • $\dot{\mathit{H}}=-{\displaystyle \frac{1}{\mathrm{j}\omega \mu }}\mathbf{\nabla }\times \dot{\mathit{E}}$$\dot{\bm H} = -\dfrac{1}{\j\omega\mu} \curl \dot{\bm E}$.

    • $\dot{\mathit{E}}={\displaystyle \frac{1}{\mathrm{j}\omega {\epsilon }_{\text{c}}}}\mathbf{\nabla }\times \dot{\mathit{H}}$$\dot{\bm E} = \dfrac{1}{\j\omega \ve_\text c} \curl \dot{\bm H}$.

• 导电媒质（有损耗媒质）

  • 本振波阻抗 ${\eta }_{\text{c}}={\displaystyle \frac{\sqrt{\mu /\epsilon }}{\sqrt{1-\mathrm{j}\frac{\sigma }{\omega \epsilon }}}}$$\eta_\text c = \dfrac{\sqrt{\mu/\ve}}{\sqrt{1 - \j\cfrac{\sigma}{\omega\ve}}}$ 为复数.

  • 电场相位超前磁场相位 $\varphi$$\phi$.

• 电磁波的能量

  • 电磁场能量密度不等

    • 电场：$w_{\mathrm{e}max}^{+}={\displaystyle \frac{\epsilon }{2}}{|E_x^{+}(z,t)|}^2$$w_{\e\max}^+ = \dfrac{\ve}{2} \vqty{E_x^+(z, t)}^2$.

    • 磁场：$w_{\text{m}max}^{+}={\displaystyle \frac{\mu }{2}}{|H_y^{+}(z,t)|}^2={\displaystyle \frac{\epsilon }{2}}{|E_x^{+}(z,t)|}^2\sqrt{1+{\left({\displaystyle \frac{\sigma }{\omega \epsilon }}\right)}^2}$$w_{\text m \max}^+ = \dfrac{\mu}{2} \vqty{H_y^+(z, t)}^2 = \dfrac{\ve}{2} \vqty{E_x^+(z, t)}^2 \sqrt{1 + \pqty{\dfrac{\sigma}{\omega\ve}}^2}$.

    • 于是：$w_{\mathrm{e}max}^{+}<w_{\text{m}max}^{+},w_{\mathrm{e}max}^{-}<w_{\text{m}max}^{-}$$w_{\e \max}^+ < w_{\text m \max}^+, \quad w_{\e \max}^- < w_{\text m \max}^-$.

  • 复数能流密度与均值

    • ${\tilde{\mathit{S}}}^{+}(z)=\dot{\mathit{E}}(z)\times {\dot{\mathit{H}}}^{\ast }(z)={\displaystyle \frac{{E_{x0}^{+}}^2}{\left|{\eta }_{\text{c}}\right|}}{\mathrm{e}}^{-2\alpha z}{\mathrm{e}}^{\mathrm{j}\varphi }{\mathit{a}}_z$$\wt{\bm S}^+(z) = \dot{\bm E}(z) \cp \dot{\bm H}^*(z) = \dfrac{{E_{x0}^+}^2}{\vqty{\eta_\text c}} \e^{-2\alpha z} \e^{\j\phi} \bm a_z$.

    • ${\mathit{S}}_{\mathrm{avg}}^{+}(z)=\mathrm{Re}{\tilde{\mathit{S}}}^{+}(z)={\displaystyle \frac{{E_{x0}^{+}}^2\cos \varphi }{\left|{\eta }_{\text{c}}\right|}}{\mathrm{e}}^{-2\alpha z}{\mathit{a}}_z$$\bm S_\avg^+(z) = \Re\wt{\bm S}^+(z) = \dfrac{{E_{x0}^+}^2 \cos\phi}{\vqty{\eta_\text c}} \e^{-2\alpha z} \bm a_z$.

    • ${\mathit{S}}_{\mathrm{avg}}^{-}(z)=\mathrm{Re}{\tilde{\mathit{S}}}^{-}(z)=-{\displaystyle \frac{{E_{x0}^{-}}^2\cos \varphi }{\left|{\eta }_{\text{c}}\right|}}{\mathrm{e}}^{-2\alpha z}{\mathit{a}}_z$$\bm S_\avg^-(z) = \Re\wt{\bm S}^-(z) = -\dfrac{{E_{x0}^-}^2 \cos\phi}{\vqty{\eta_\text c}} \e^{-2\alpha z} \bm a_z$.

6.3.4 低损耗媒质的近似计算

• 损耗角正切 $\tan {\delta }_{\text{c}}={\displaystyle \frac{\sigma }{\omega \epsilon }}={\displaystyle \frac{J_{\text{cm}}}{J_{\text{dm}}}}$$\tan\delta_\text c = \dfrac{\sigma}{\omega \ve} = \dfrac{J_\text{cm}}{J_\text{dm}}$.

  • 当 $\tan {\delta }_{\text{c}}\to 0$$\tan\delta_\text c \to 0$ 时，传导电流为零，是理想介质.

  • 当 $\tan {\delta }_{\text{c}}\ll 1$$\tan\delta_\text c \ll 1$ 时，传导电流很小，是良介质（低损耗媒质）

  • 当 $\tan {\delta }_{\text{c}}\gg 1$$\tan\delta_\text c \gg 1$ 时，传导电流很大，是良导体（良导电媒质）

  • 当 $\tan {\delta }_{\text{c}}\to \mathrm{\infty }$$\tan\delta_\text c \to \infty$ 时，传导电流无穷，是理想导体.

• 低损耗媒质（$\sigma \ll \omega \epsilon$$\sigma \ll \omega \ve$）

  • 近似计算（1）

    • $\alpha \approx {\displaystyle \frac{\sigma }{2}}\sqrt{{\displaystyle \frac{\mu }{\epsilon }}}\approx {\displaystyle \frac{\sigma {\eta }_{\text{c}}}{2}}$$\alpha \approx \dfrac{\sigma}{2} \sqrt{\dfrac{\mu}{\ve}} \approx \dfrac{\sigma\, \eta_\text c}{2}$.

    • $\beta \approx \omega \sqrt{\mu \epsilon }(1+{\displaystyle \frac{{\sigma }^2}{8{\omega }^2{\epsilon }^2}})$$\beta \approx \omega\sqrt{\mu\ve} \pqty{1 + \dfrac{\sigma^2}{8\omega^2\ve^2}}$.

    • ${\eta }_{\text{c}}\approx \sqrt{{\displaystyle \frac{\mu }{\epsilon }}}(1+\mathrm{j}{\displaystyle \frac{\sigma }{2\omega \epsilon }})$$\eta_\text c \approx \sqrt{\dfrac{\mu}{\ve}} \pqty{1 + \j\dfrac{\sigma}{2\omega\ve}}$.

  • 近似计算（2）

    • $\alpha \approx {\displaystyle \frac{\sigma }{2}}\sqrt{{\displaystyle \frac{\mu }{\epsilon }}}\approx {\displaystyle \frac{\sigma {\eta }_{\text{c}}}{2}}$$\alpha \approx \dfrac{\sigma}{2} \sqrt{\dfrac{\mu}{\ve}} \approx \dfrac{\sigma\, \eta_\text c}{2}$.

    • $\beta \approx \omega \sqrt{\mu \epsilon }$$\beta \approx \omega\sqrt{\mu\ve}$.

    • ${\eta }_{\text{c}}\approx \eta =\sqrt{{\displaystyle \frac{\mu }{\epsilon }}}$$\eta_\text c \approx \eta = \sqrt{\dfrac{\mu}{\ve}}$.

  • 例子：云母、聚乙烯、聚苯乙烯等.

6.3.5 良导电媒质的表面阻抗

• 良导电媒质的近似计算（$\sigma \gg \omega \epsilon$$\sigma \gg \omega \ve$）

  • 近似计算

    • $\alpha \approx \beta \approx \sqrt{{\displaystyle \frac{\omega \mu \sigma }{2}}}=\sqrt{\pi f\mu \sigma }$$\alpha \approx \beta \approx \sqrt{\dfrac{\omega \mu \sigma}{2}} = \sqrt{\pi f \mu \sigma}$.

    • ${\eta }_{\text{c}}\approx \sqrt{\mathrm{j}{\displaystyle \frac{\mu \omega }{\sigma }}}=(1+\mathrm{j})\sqrt{{\displaystyle \frac{\pi f\mu }{\sigma }}}$$\eta_\text c \approx \sqrt{\j \dfrac{\mu\omega}{\sigma}} % = \sqrt{\dfrac{\omega\mu}{\sigma}} \angle{45^\circ} = (1 + \j) \sqrt{\dfrac{\pi f \mu}{\sigma}}$.

  • ${\eta }_{\text{c}}\equiv R_{\text{s}}+\mathrm{j}{X}_{\text{s}}$$\eta_\text c \equiv R_\text s + \j X_\text s$.

    • 表面电阻：$R_{\text{s}}\approx \sqrt{{\displaystyle \frac{\pi f\mu }{\sigma }}}$$R_\text s \approx \sqrt{\dfrac{\pi f \mu}{\sigma}}$.

    • 表面电抗：$X_{\text{s}}\approx \sqrt{{\displaystyle \frac{\pi f\mu }{\sigma }}}$$X_\text s \approx \sqrt{\dfrac{\pi f \mu}{\sigma}}$.

  • 性质说明

    • 电场超前磁场 ${45}^{\circ }$$45^\circ$.

    • 振幅按 ${\mathrm{e}}^{-\alpha z}$$\e^{-\alpha z}$ 衰减.

    • $\sigma ,\mu ,f$$\sigma, \mu, f$ 越大，衰减越快.

  • 趋肤效应

    • 高频电磁波进入良导体后，在数微米内基本衰减完毕.

    • 透入深度：衰减为表面值的 $e^{-1}$$e\inv$ 倍时经过的距离.

    • 透入深度：$\delta ={\displaystyle \frac{1}{\alpha }}\approx \sqrt{{\displaystyle \frac{2}{\omega \mu \sigma }}}={\displaystyle \frac{1}{\sqrt{\pi f\mu \sigma }}}$$\delta = \dfrac{1}{\alpha} \approx \sqrt{\dfrac{2}{\omega\mu\sigma}} = \dfrac{1}{\sqrt{\pi f \mu \sigma}}$.

    • 在经过距离 $4.6\delta$$4.6 \delta$ 时，电磁波振幅衰减为原来的 $1\mathrm{\%}$$1 \%$.

  • 静电屏蔽

    • 为实现静电屏蔽，一般取良导体的厚度为一个波长.

    • 即 $d=\lambda ={\displaystyle \frac{2\pi }{\beta }}=2\pi \delta ={\displaystyle \frac{2\pi }{\sqrt{\pi f\mu \sigma }}}$$d = \lambda = \dfrac{2\pi}{\beta} = 2\pi \delta = \dfrac{2\pi}{\sqrt{\pi f \mu \sigma}}$.

    • 经过一个波长后衰减 $54.5751\mathrm{dB}$$54.5751\rm{dB}$.

• 良导电媒质的表面阻抗

  • 导体内部（沿 $z$$z$ 方向传播）

    • 场强的振幅：${\dot{E}}_{x\text{m}}(z)={\dot{E}}_{x\text{m}}{\mathrm{e}}^{-\alpha z}{\mathrm{e}}^{-\mathrm{j}\beta z}$$\dot{E}_{x \text m}(z) = \dot{E}_{x \text m} \e^{-\alpha z} \e^{-\j\beta z}$.

    • 电流体密度：${\dot{J}}_{x\text{m}}(z)=\sigma {\dot{E}}_{x\text{m}}{\mathrm{e}}^{-\alpha z}{\mathrm{e}}^{-\mathrm{j}\beta z}$$\dot{J}_{x \text m}(z) = \sigma \dot{E}_{x \text m} \e^{-\alpha z} \e^{-\j \beta z}$.

    • 总传导电流：${\dot{I}}_{x\text{m}}={\displaystyle {\int }_0^{+\mathrm{\infty }}{\dot{J}}_{x\text{m}}(z)\mathrm{d}z={\displaystyle \frac{\sigma {\dot{E}}_{x\text{m}}}{\alpha +\mathrm{j}\beta }}}$$\dot{I}_{x \text m} = \intoi \dot{J}_{x \text m}(z) \dz = \dfrac{\sigma \dot{E}_{x \text m}}{\alpha + \j\beta}$.

  • 表面阻抗（$x$$x$ 方向单位长度，$y$$y$ 方向单位长度）

    • 表面阻抗：$Z_{\text{c}}={\displaystyle \frac{\alpha }{\sigma }}+\mathrm{j}{\displaystyle \frac{\beta }{\sigma }}={\eta }_{\text{c}}$$Z_\text c = \dfrac{\alpha}{\sigma} + \j\dfrac{\beta}{\sigma} = \eta_\text c$.

    • 表面电阻：$R_{\text{s}}={\displaystyle \frac{\alpha }{\sigma }}={\displaystyle \frac{1}{\sigma \delta }}\approx \sqrt{{\displaystyle \frac{\pi f\mu }{\sigma }}}$$R_\text s = \dfrac{\alpha}{\sigma} = \dfrac{1}{\sigma \delta} \approx \sqrt{\dfrac{\pi f \mu}{\sigma}}$.

    • 表面电抗：$X_{\text{s}}={\displaystyle \frac{\beta }{\sigma }}\approx {\displaystyle \frac{1}{\sigma \delta }}\approx \sqrt{{\displaystyle \frac{\pi f\mu }{\sigma }}}$$X_\text s = \dfrac{\beta}{\sigma} \approx \dfrac{1}{\sigma \delta} \approx \sqrt{\dfrac{\pi f \mu}{\sigma}}$.

  • 电磁平均功率

    • 等效表面电流密度：${\dot{J}}_{\text{sm}}(z)={\dot{I}}_{x\text{m}}={\displaystyle \frac{{\dot{E}}_{x\text{m}}(0^{+})}{{\eta }_{\text{c}}}}={\dot{H}}_{y\text{m}}(0^{+})$$\dot{J}_\text{sm}(z) = \dot{I}_{x \text m} = \dfrac{\dot{E}_{x \text m}(0^+)}{\eta_\text c} = \dot{H}_{y \text m}(0^+)$.

    • 平均功率：$p_0={\displaystyle \frac{1}{2}}{\dot{I}}_{x\text{m}}\mathbf{\cdot }{\dot{I}}_{x\text{m}}^{\ast }{R}_{\text{s}}={\displaystyle \frac{1}{2}}{J}_{\text{sm}}^2{R}_{\text{s}}$$p_0 = \dfrac{1}{2} \dot{I}_{x \text m} \vdot \dot{I}_{x \text m}^* R_\text s = \dfrac{1}{2} J_\text{sm}^2 R_\text s$.

    • 平均功率：${{\mathit{S}}_{\mathrm{avg}}(z)\phantom{\int }|}_{z=0}={\mathrm{Re}[{\displaystyle \frac{1}{2}}{\dot{\mathit{E}}}_{x\text{m}}(z)\times {\dot{H}}_{y\text{m}}^{\ast }(z)]\phantom{\int }|}_{z=0}={\displaystyle \frac{1}{2}}{H}_{y\text{m}}^2{R}_{\text{s}}{\mathit{a}}_z$$\eval{\bm S_\avg(z)}_{z = 0} = \eval{\Re\bqty{\dfrac{1}{2} \dot{\bm E}_{x \text m}(z) \cp \dot{H}_{y \text m}^*(z)}}_{z = 0} = \dfrac{1}{2} H_{y\text m}^2 R_\text s \bm a_z$.

    • 上述磁场可取导体表面内侧或外侧的总磁场.

6.4 均匀平面波对平面边界的垂直入射

6.4.1 垂直入射的反射波与折射波

• 记号约定：$\mathit{E},{\mathit{E}}^{\prime },{\mathit{E}}^{″}$$\bm E, \bm E', \bm E''$ 分别表示入射波、反射波和折射波.

• 电场与磁场

  • 媒质 1 中（入射波 + 反射波）

    • $ \dot{\bm E}_1(z) = \pqty{\dot{E}_{10} \e^{-\gamma_1 z} + \dot{E}'_{10} \e^{\gamma_1 z}} \bm a_x $.

    • $ \dot{\bm H}_1(z) = \dfrac{1}{\eta_1} \pqty{\dot{E}_{10} \e^{-\gamma_1 z} - \dot{E}'_{10} \e^{\gamma_1 z}} \bm a_y $.

  • 媒质 2 中（折射波）

    • ${\dot{\mathit{E}}}_2(z)={\dot{E}}_{20}^{″}{\mathrm{e}}^{-{\gamma }_{2}z}{\mathit{a}}_x$$\dot{\bm E}_2(z) = \dot{E}''_{20} \e^{-\gamma_2 z} \bm a_x$.

    • ${\dot{\mathit{H}}}_2(z)={\displaystyle \frac{1}{{\eta }_2}}{\dot{E}}_{20}^{″}{\mathrm{e}}^{-{\gamma }_{2}z}{\mathit{a}}_y$$\dot{\bm H}_2(z) = \dfrac{1}{\eta_2} \dot{E}''_{20} \e^{-\gamma_2 z} \bm a_y$.

• 边界条件

  • 联立 $E_{1\text{t}}=E_{2\text{t}},H_{1\text{t}}=H_{2\text{t}}$$E_{1\text t} = E_{2\text t}, H_{1 \text t} = H_{2 \text t}$.

  • 反射系数：$R:={\displaystyle \frac{{\dot{E}}_{10}^{\prime }}{{\dot{E}}_{10}}}={\displaystyle \frac{{\eta }_2-{\eta }_1}{{\eta }_2+{\eta }_1}}$$R := \dfrac{\dot{E}'_{10}}{\dot{E}_{10}} = \dfrac{\eta_2 - \eta_1}{\eta_2 + \eta_1}$.

  • 折射系数：$T:={\displaystyle \frac{{\dot{E}}_{10}^{″}}{{\dot{E}}_{10}}}={\displaystyle \frac{2{\eta }_2}{{\eta }_2+{\eta }_1}}=R+1$$T := \dfrac{\dot{E}''_{10}}{\dot{E}_{10}} = \dfrac{2\eta_2}{\eta_2 + \eta_1} = R + 1$.

6.4.2 理想介质与理想导体分界面

• 分界媒质

  • 媒质性质

    • 理想介质：${\sigma }_1=0,{\gamma }_1=\mathrm{j}{\beta }_1=\mathrm{j}\omega \sqrt{{\mu }_1{\epsilon }_1},{\eta }_1=\sqrt{{\mu }_1/{\epsilon }_1}$$\sigma_1 = 0, \gamma_1 = \j\beta_1 = \j\omega\sqrt{\mu_1\ve_1}, \eta_1 = \sqrt{\mu_1/\ve_1}$.

    • 理想导体：${\sigma }_2=\mathrm{\infty },{\dot{E}}_{20}^{″}={\dot{H}}_{20}^{″}=0$$\sigma_2 = \infty, \dot{E}’'_{20} = \dot{H}''_{20} = 0$，不存在电磁波.

  • 边界条件

    • $R=-1,T=0$$R = -1, T = 0$，即全反射现象.

    • 自由面电流：${\dot{\mathit{J}}}_{\text{s}}=\mathit{n}\times {\dot{\mathit{H}}}_1={\displaystyle \frac{2{\dot{E}}_{10}}{{\eta }_1}}{\mathit{a}}_x$$\dot{\bm J}_\text s = \bm n \cp \dot{\bm H}_1 = \dfrac{2\dot{E}_{10}}{\eta_1} \bm a_x$.

• 合成波

  • 频域

    • ${\dot{\mathit{E}}}_1(z)={\dot{E}}_{10}({\mathrm{e}}^{-\mathrm{j}{\beta }_{1}z}-{\mathrm{e}}^{\mathrm{j}{\beta }_{1}z}){\mathit{a}}_x=-2\mathrm{j}{\dot{E}}_{10}\sin \left({\beta }_{1}z\right){\mathit{a}}_x$$\dot{\bm E}_1(z) = \dot{E}_{10} \pqty{ \e^{-\j\beta_1 z} - \e^{\j\beta_1 z}} \bm a_x = -2\j \dot{E}_{10} \sin(\beta_1 z) \bm a_x$.

    • ${\dot{\mathit{H}}}_1(z)={\displaystyle \frac{{\dot{E}}_{10}}{{\eta }_1}}({\mathrm{e}}^{-\mathrm{j}{\beta }_{1}z}+{\mathrm{e}}^{\mathrm{j}{\beta }_{1}z}){\mathit{a}}_y=2{\displaystyle \frac{{\dot{E}}_{10}}{{\eta }_1}}\cos \left({\beta }_{1}z\right){\mathit{a}}_y$$\dot{\bm H}_1(z) = \dfrac{\dot{E}_{10}}{\eta_1} \pqty{\e^{-\j\beta_1 z} + \e^{\j\beta_1 z}} \bm a_y = 2 \dfrac{\dot{E}_{10}}{\eta_1} \cos(\beta_1 z) \bm a_y$.

  • 时域（初值 ${\dot{E}}_{10}=\sqrt{2}{E}_{10}\mathrm{\angle }{0}^{\circ }$$\dot{E}_{10} = \sqrt 2 E_{10} \angle 0^\circ$）

    • ${\mathit{E}}_1(z,t)=2\sqrt{2}{E}_{10}\sin \left({\beta }_{1}z\right)\sin \left(\omega t\right)$$\bm E_1(z, t) = 2\sqrt 2 E_{10} \sin(\beta_1 z) \sin(\omega t)$.

    • ${\mathit{H}}_1(z,t)={\displaystyle \frac{2\sqrt{2}{E}_{10}}{{\eta }_1}}\cos \left({\beta }_{1}z\right)\cos \left(\omega t\right)$$\bm H_1(z, t) = \dfrac{2\sqrt 2 E_{10}}{\eta_1} \cos(\beta_1 z) \cos(\omega t)$.

• 说明

  • 合成波为纯驻波

    • 波节点：纯驻波的零点.

    • 波腹点：纯驻波的最值点.

  • 相位与振幅比较

    • 行波：相位随时间变化，振幅不变.

    • 驻波：振幅随时间变化，相位不变.

  • 能量的传输

    • 行波：可以传输能量.

    • 驻波：不能传输能量，只有电磁振荡.

6.4.3 理想介质与理想介质分界面

• 媒质 2

  • ${\dot{\mathit{E}}}_2(z)=T{E}_{10}{\mathrm{e}}^{-\mathrm{j}{\beta }_{2}z}{\mathit{a}}_x$$\dot{\bm E}_2(z) = T E_{10} \e^{-\j\beta_2 z} \bm a_x$.

  • ${\dot{\mathit{H}}}_2(z)={\displaystyle \frac{T{E}_{10}}{{\eta }_{\text{c}}}}{\mathrm{e}}^{-\mathrm{j}{\beta }_{2}z}{\mathit{a}}_y$$\dot{\bm H}_2(z) = \dfrac{T E_{10}}{\eta_\text c} \e^{-\j\beta_2 z} \bm a_y$.

• 媒质 1

  • 电场

    • ${\dot{\mathit{E}}}_1(z)=(E_{10}{\mathrm{e}}^{-\mathrm{j}{\beta }_{1}z}+R{E}_{10}{\mathrm{e}}^{\mathrm{j}{\beta }_{1}z}){\mathit{a}}_x$$\dot{\bm E}_1(z) = \pqty{ E_{10} \e^{-\j\beta_1 z} + R E_{10} \e^{\j\beta_1 z} } \bm a_x$，即入射波 + 反射波.

    • ${\dot{\mathit{E}}}_1(z)=E_{10}(T{\mathrm{e}}^{-\mathrm{j}{\beta }_{1}z}+\mathrm{j}2R\sin {\beta }_{1}z){\mathit{a}}_x$$\dot{\bm E}_1(z) = E_{10} \pqty{ T \e^{-\j\beta_1 z} + \j 2 R \sin \beta_1 z } \bm a_x$，即行波 + 驻波.

    • ${\dot{\mathit{E}}}_1(z)=E_{10}(1+R{\mathrm{e}}^{\mathrm{j}2{\beta }_{1}z}){\mathrm{e}}^{-\mathrm{j}{\beta }_{1}z}{\mathit{a}}_x$$\dot{\bm E}_1(z) = E_{10} \pqty{1 + R \e^{\j 2 \beta_1 z}} \e^{-\j\beta_1 z} \bm a_x$，即 $+{\mathit{a}}_z$$+\bm a_z$ 方向的平面波.

  • 磁场

    • ${\dot{\mathit{H}}}_1(z)={\displaystyle \frac{E_{10}}{{\eta }_1}}({\mathrm{e}}^{-\mathrm{j}{\beta }_{1}z}-R{\mathrm{e}}^{\mathrm{j}{\beta }_{1}z}){\mathit{a}}_y$$\dot{\bm H}_1(z) = \dfrac{E_{10}}{\eta_1} \pqty{ \e^{-\j\beta_1 z} - R \e^{\j\beta_1 z} } \bm a_y$，即入射波 + 反射波.

    • ${\dot{\mathit{H}}}_1(z)={\displaystyle \frac{E_{10}}{{\eta }_1}}(T{\mathrm{e}}^{-\mathrm{j}{\beta }_{1}z}-2R\cos {\beta }_{1}z){\mathit{a}}_y$$\dot{\bm H}_1(z) = \dfrac{E_{10}}{\eta_1} \pqty{T \e^{-\j\beta_1 z} - 2 R \cos\beta_1 z} \bm a_y$，即行波 + 驻波.

    • ${\dot{\mathit{H}}}_1(z)={\displaystyle \frac{E_{10}}{{\eta }_1}}(1-R{\mathrm{e}}^{\mathrm{j}2{\beta }_{1}z}){\mathrm{e}}^{-\mathrm{j}{\beta }_{1}z}{\mathit{a}}_y$$\dot{\bm H}_1(z) = \dfrac{E_{10}}{\eta_1} \pqty{1 - R \e^{\j 2 \beta_1 z}} \e^{-\j\beta_1 z} \bm a_y$，即 $+{\mathit{a}}_z$$+\bm a_z$ 方向的平面波.

  • 电磁场的振幅

    • $|{\dot{\mathit{E}}}_{1\text{m}}(z)|=\sqrt{2}{E}_{10}\sqrt{1+R^2+2R\cos \left(2{\beta }_{1}z\right)}$$\vqty{\dot{\bm E}_{1 \text m}(z)} = \sqrt 2 E_{10} \sqrt{1 + R^2 + 2 R \cos(2\beta_1 z)}$.

    • $|{\dot{\mathit{H}}}_{1\text{m}}(z)|={\displaystyle \frac{\sqrt{2}{E}_{10}}{{\eta }_1}}\sqrt{1+R^2-2R\cos \left(2{\beta }_{1}z\right)}$$\vqty{\dot{\bm H}_{1 \text m}(z)} = \dfrac{\sqrt 2 E_{10}}{\eta_1} \sqrt{1 + R^2 - 2 R \cos(2 \beta_1 z)}$.

  • 波幅点与波节点的分布

• 驻波系数（驻波比）

  • 驻波系数：$\rho :={\displaystyle \frac{{\left|\mathit{B}\right|}_{max}}{{\left|\mathit{B}\right|}_{min}}}={\displaystyle \frac{{\left|\mathit{H}\right|}_{max}}{{\left|\mathit{H}\right|}_{min}}}={\displaystyle \frac{1+\left|R\right|}{1-\left|R\right|}}$$\rho := \dfrac{\vqty{\bm B}_\max}{\vqty{\bm B}_\min} = \dfrac{\vqty{\bm H}_\max}{\vqty{\bm H}_\min} = \dfrac{1 + \vqty{R}}{1 - \vqty{R}}$.

    • 行波 $\rho =1$$\rho = 1$，纯驻波 $\rho =\mathrm{\infty }$$\rho = \infty$.

    • $\rho \ge 1$$\rho \ge 1$ 且越小越好.

  • 反射系数：$\left|R\right|={\displaystyle \frac{\rho -1}{\rho +1}}$$\vqty{R} = \dfrac{\rho - 1}{\rho + 1}$.

• 坡应廷矢量

  • ${\mathit{S}}_{\mathrm{avg}}=\mathrm{Re}[\dot{\mathit{E}}(z)\times {\dot{\mathit{H}}}^{\ast }(z)]={\displaystyle \frac{E_{10}^2}{{\eta }_1}}{\mathit{a}}_z$$\bm S_\avg = \Re\bqty{\dot{\bm E}(z) \cp \dot{\bm H}^*(z)} = \dfrac{E_{10}^2}{\eta_1} \bm a_z$.

  • ${\mathit{S}}_{\mathrm{avg}}^{\prime }=\mathrm{Re}[{\dot{\mathit{E}}}^{\prime }(z)\times {\dot{\mathit{H}}}^{\prime \ast }(z)]=-{\displaystyle \frac{R^2{E}_{10}^2}{{\eta }_1}}{\mathit{a}}_z$$\bm S'_\avg = \Re\bqty{\dot{\bm E}'(z) \cp \dot{\bm H}'^*(z)} = - \dfrac{R^2 E_{10}^2}{\eta_1} \bm a_z$.

  • ${\mathit{S}}_{\mathrm{avg}}^{″}=\mathrm{Re}[{\dot{\mathit{E}}}^{″}(z)\times {\dot{\mathit{H}}}^{″\ast }(z)]={\displaystyle \frac{T^2{E}_{10}^2}{{\eta }_2}}{\mathit{a}}_z$$\bm S''_\avg = \Re\bqty{\dot{\bm E}''(z) \cp \dot{\bm H}''^*(z)} = \dfrac{T^2 E_{10}^2}{\eta_2} \bm a_z$.

  • $\left|{\mathit{S}}_{\mathrm{avg}}\right|=\left|{\mathit{S}}_{\mathrm{avg}}^{\prime }\right|+\left|{\mathit{S}}_{\mathrm{avg}}^{″}\right|$$\vqty{\bm S_\avg} = \vqty{\bm S'_\avg} + \vqty{\bm S''_\avg}$.