色散方程、透镜
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色散方程
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$$q_e E=m_e \frac{d^2 x}{d t^2}+m_e \gamma \frac{dx}{dt}+m_e \omega_0^2 x\\
\quad\\
\small令E=E_0 e^{i \omega t},x=x_0 e^{i (\omega t-\alpha)}\\
\quad\\
\small可得x_0 e^{i (\omega t-\alpha)}=\frac{q_e}{m_e}\frac{E_0 e^{i \omega t}}{\omega_0^2-\omega^2+i\gamma\omega}\\
\quad\\
\small滞后相位\alpha=\arctan(\frac{\gamma\omega}{\omega_0^2-\omega^2})\\
\quad\\
\small振幅x_0=\frac{q_e}{m_e}\frac{E_0}{\sqrt{(\omega_0^2-\omega^2)^2+\gamma^2\omega^2}}\\
\quad\\
x=\frac{q_e}{m_e}\frac{E}{\omega_0^2-\omega^2+i\gamma\omega}\\
\quad\\
Nq_ex=\frac{Nq_e^2}{m_e}\frac{E}{\omega_0^2-\omega^2+i\gamma\omega}\\
\quad\\
\small各向同性材料中\\
\quad\\
P=E_总(\varepsilon_r-1)\varepsilon_0=\frac{Nq_e^2}{m_e}\frac{E_{空腔}}{\omega_0^2-\omega^2+i\gamma\omega}\\
\quad\\
\small稀疏介质中E_{空腔}=E_总\\
\quad\\
n^2=1+\frac{Nq_e^2}{\varepsilon_0m_e}\frac{1}{\omega_0^2-\omega^2+i\gamma\omega}\\
\quad\\
\small稠密介质中E_{空腔}-\frac{(\varepsilon_r-1)E_总}{3}=E_总,即E_{空腔}=\frac{\varepsilon_r+2}{3}E_总\\
\quad\\
\frac{n^2-1}{n^2+2}=\frac{Nq_e^2}{3\varepsilon_0m_e}\frac{1}{\omega_0^2-\omega^2+i\gamma\omega}\\
\quad\\
\small当有多个特征频率时\\
\quad\\
n^2=1+\frac{Nq_e^2}{\varepsilon_0m_e}\sum\limits_k\frac{f_k}{\omega_{0k}^2-\omega^2+i\gamma\omega}\\
\quad\\
\frac{n^2-1}{n^2+2}=\frac{Nq_e^2}{3\varepsilon_0m_e}\sum\limits_k\frac{f_k}{\omega_{0k}^2-\omega^2+i\gamma\omega}
$$
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成像曲线
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$$透镜\\
\quad\\
\small等光程原理\\
\quad\\
n_1l_1+n_2l_2=\Delta\\
\quad\\
\small令L=\frac{\Delta}{n_2},l_0=L-l_2\\
\quad\\
\small则有n_1l_1-n_2l_0=0\\
\quad\\
\small即\frac{l_1}{l_0}=\frac{n_2}{n_1}=e\\
\quad\\
\small为双曲线方程$$
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球面透镜成像
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$$\small等光程原理\\
\quad\\
\Delta=n_ul_u+n_vl_v=n_uu+n_vv\\
\quad\\
\small余弦定理\\
\quad\\
l_u=[R^2+(u+R)^2-2R(u+R)\cos\varphi]^{\frac{1}{2}}\\
\quad\\
l_u=[R^2+(v-R)^2+2R(v-R)\cos\varphi]^{\frac{1}{2}}\\
\quad\\
\frac{d\Delta}{d\varphi}=0\\
\quad\\
\frac{n_u(u+R)}{l_u}-\frac{n_v(v-R)}{l_v}=0\\
\quad\\
\frac{n_u}{l_u}+\frac{n_v}{l_v}=\frac{1}{R}(\frac{vn_v}{l_v}-\frac{un_u}{l_u})\\
\quad\\
\small一阶近似\cos\varphi\approx 1\\
\quad\\
\frac{n_u}{u}+\frac{n_v}{v}=\frac{n_v-n_u}{R}\\
\quad\\
\small两侧相同介质透镜成像\\
\quad\\
\frac{n_1}{u_1}+\frac{n_2}{v_1}=\frac{n_2-n_1}{R_1}\\
\quad\\
\frac{n_2}{u_2}+\frac{n_1}{v_2}=\frac{n_1-n_2}{R_2}\\
\quad\\
\small由u_2=d-v_1可得\\
\quad\\
\frac{n_1}{u_1}+\frac{n_1}{v_2}=(n_2-n_1)(\frac{1}{R_1}-\frac{1}{R_2})+(\frac{n_2}{v_1-d}-\frac{n_2}{v_1})\\
\quad\\
\small令d\approx 0即可得到薄透镜成像公式\\
\quad\\
\frac{1}{u}+\frac{1}{v}=(\frac{n_2}{n_1}-1)(\frac{1}{R_1}-\frac{1}{R_2})=\frac{1}{f}
$$
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球面反射镜成像
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$$
l_u=\frac{R+(u-R)\cos\varphi}{\cos\theta}\\
\quad\\
l_v=\frac{R+(v-R)\cos\varphi}{\cos\theta}\\
\quad\\
\small角平分线定理\\
\quad\\
\frac{u-R}{l_u}=\frac{R-v}{l_v}\\
\quad\\
\frac{u-R}{R+(u-R)\cos\varphi}+\frac{v-R}{R+(v-R)\cos\varphi}=0\\
\quad\\
\small一阶近似\cos\varphi\approx 1\\
\quad\\
\frac{1}{u}+\frac{1}{v}=\frac{2}{R}\\
\quad\\
\small取凹面镜R符号为负\\
\quad\\
\frac{1}{u}+\frac{1}{v}=-\frac{2}{R}
$$
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色散棱镜
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$$\frac{\sin\theta_{i1}}{\sin\theta_{t1}}=n_{ti}\\
\quad\\
\frac{\sin\theta_{i2}}{\sin\theta_{t2}}=n_{it}\\
\quad\\
\small\begin{aligned}偏向角\delta &=\theta_{i1}-\theta_{t1}-\theta_{i2}+\theta_{t2}\\
\quad\\
&=\theta_{i1}+\theta_{t2}-\alpha\\
\quad\\
&=\theta_{i1}+\arcsin(n_{ti} \sin\theta_{i2})-\alpha\\
\quad\\
&=\theta_{i1}+\arcsin[n_{ti} \sin(\alpha-\theta_{t1})]-\alpha\\
\quad\\
&=\theta_{i1}+\arcsin[n_{ti} (\sin\alpha\cos\theta_{t1}-\cos\alpha\sin\theta_{t1})]-\alpha\\
\quad\\
&=\theta_{i1}+\arcsin[\sin\alpha\sqrt{n_{ti}^2-\sin^2\theta_{i1}}-\cos\alpha\sin\theta_{i1}]-\alpha\\
\end{aligned}
\quad\\
\small令\frac{d\delta}{d\theta_{t1}}=0有\\
\quad\\
\frac{\cos\theta_{t1}}{\sqrt{1-n_{ti}^2\sin^2\theta_{t1}}}-\frac{\cos(\alpha-\theta_{t1})}{\sqrt{1-n_{ti}^2\sin^2(\alpha-\theta_{t1})}}=0\\
\quad\\
\small\theta_{t1}=\frac{\alpha}{2}即对称透射时\delta取到极小值\\
\quad\\
\delta_m=2\arcsin(n_{ti}\sin \frac{\alpha}{2})-\alpha\\
\quad\\
n_{ti}=\frac{\sin\frac{\delta_m+\alpha}{2}}{\sin\frac{\alpha}{2}}
$$
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参考与引用来源
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赫克特光学第五版