Различия
Показаны различия между двумя версиями страницы.
| Предыдущая версия справа и слева Предыдущая версия Следующая версия | Предыдущая версия | ||
| optics:res2.3 [2020/02/07 18:42] – root | optics:res2.3 [2020/02/10 08:26] (текущий) – root | ||
|---|---|---|---|
| Строка 6: | Строка 6: | ||
| рассмотрим образ | рассмотрим образ | ||
| - | $\left[\frac{\partial\mathbf{B}}{\partial t}\right]_{\omega}=\frac{1}{2\pi}\intop_{-\infty}^{\infty}\frac{\partial\mathbf{B}}{\partial t}e^{i\omega t}dt=\frac{1}{2\pi}\intop_{-\infty}^{\infty}\frac{\partial\left(\mathbf{B}e^{i\omega t}\right)}{\partial t}-i\omega\mathbf{B}e^{i\omega t}dt=$ | + | $$ |
| - | + | \left[\frac{\partial\mathbf{B}}{\partial t}\right]_{\omega}=\frac{1}{2\pi}\intop_{-\infty}^{\infty}\frac{\partial\mathbf{B}}{\partial t}e^{i\omega t}dt=\frac{1}{2\pi}\intop_{-\infty}^{\infty}\frac{\partial\left(\mathbf{B}e^{i\omega t}\right)}{\partial t}-i\omega\mathbf{B}e^{i\omega t}dt= | |
| - | $\frac{1}{2\pi}\left.\left(\mathbf{B}e^{i\omega t}\right)\right|_{-\infty}^{\infty}-\frac{i\omega}{2\pi}\intop_{-\infty}^{\infty}\mathbf{B}e^{i\omega t}dt=0-i\omega\mathbf{B}_{\omega}=-i\omega\mathbf{B}_{\omega}$ | + | $$ |
| - | + | $$ | |
| - | аналогично для пространственных компонент: | + | \frac{1}{2\pi}\left.\left(\mathbf{B}e^{i\omega t}\right)\right|_{-\infty}^{\infty}-\frac{i\omega}{2\pi}\intop_{-\infty}^{\infty}\mathbf{B}e^{i\omega t}dt=0-i\omega\mathbf{B}_{\omega}=-i\omega\mathbf{B}_{\omega} |
| - | + | . | |
| - | $\left[div\mathbf{B}\right]_{\mathbf{k}}=\left[\frac{\partial B_{x}}{\partial x}+\frac{\partial B_{y}}{\partial y}+\frac{\partial B_{z}}{\partial z}\right]_{\mathbf{k}}$ | + | $$ |
| - | + | Аналогично | |
| - | рассмотрим | + | $$ |
| - | + | \left[\text{div} \mathbf{B}\right]_{\mathbf{k}}=\left[\frac{\partial B_{x}}{\partial x}+\frac{\partial B_{y}}{\partial y}+\frac{\partial B_{z}}{\partial z}\right]_{\mathbf{k}}, | |
| - | $\left[\frac{\partial B_{x}}{\partial x}\right]_{\mathbf{k}}=\frac{1}{\left(2\pi\right)^{3}}\intop_{-\infty}^{\infty}\frac{\partial B_{x}}{\partial x}e^{-ik_{x}x}\, | + | $$ |
| - | + | но для начала | |
| - | $\frac{1}{\left(2\pi\right)^{3}}\intop_{-\infty}^{\infty}\left(\frac{\partial\left(B_{x}e^{-ik_{x}x}\right)}{\partial x}+ik_{x}B_{x}e^{-ik_{x}x}\right)\, | + | $$ |
| - | + | \left[\frac{\partial B_{x}}{\partial x}\right]_{\mathbf{k}}=\frac{1}{\left(2\pi\right)^{3}} | |
| - | $\frac{1}{\left(2\pi\right)^{3}}\left.\left(B_{x}e^{-ik_{x}x}\right)\right|_{-\infty}^{\infty}+\frac{1}{\left(2\pi\right)^{3}}\intop_{-\infty}^{\infty}\left(\frac{\partial\left(B_{x}e^{-ik_{x}x}\right)}{\partial x}+ik_{x}B_{x}e^{-ik_{x}x}\right)\, | + | \intop_{-\infty}^{\infty} |
| - | + | \frac{\partial B_{x}}{\partial x}e^{-ik_{x}x}\, | |
| - | $0+ik_{x}\left(B_{x}\right)_{\mathbf{k}}=ik_{x}\left(B_{x}\right)_{\mathbf{k}}$ | + | $$ |
| + | $$ | ||
| + | \frac{1}{\left(2\pi\right)^{3}}\intop_{-\infty}^{\infty}\intop_{-\infty}^{\infty} | ||
| + | \intop_{-\infty}^{\infty}\left(\frac{\partial\left(B_{x}e^{-ik_{x}x}\right)}{\partial x}+ik_{x}B_{x}e^{-ik_{x}x}\right)\, | ||
| + | $$ | ||
| + | $$ | ||
| + | \frac{1}{\left(2\pi\right)^{3}}\intop_{-\infty}^{\infty} \intop_{-\infty}^{\infty} | ||
| + | \left.\left(B_{x}e^{-ik_{x}x}\right)\right|_{-\infty}^{\infty} | ||
| + | \, | ||
| + | $$ | ||
| + | $$ | ||
| + | \frac{1}{\left(2\pi\right)^{3}} | ||
| + | \intop_{-\infty}^{\infty}\intop_{-\infty}^{\infty}\intop_{-\infty}^{\infty} | ||
| + | \left(ik_{x}B_{x}e^{-ik_{x}x}\right)\, | ||
| + | $$ | ||
| + | $$ | ||
| + | 0 \ +ik_{x}\left(B_{x}\right)_{\mathbf{k}}=ik_{x}\left(B_{x}\right)_{\mathbf{k}} | ||
| + | $$ | ||
| + | первое слагаемое зануляется т.к. поле на больших расстояния обращается в ноль. | ||
| Складывая, | Складывая, | ||
| - | + | $$ | |
| - | $\left[div\mathbf{B}\right]_{\mathbf{k}}=\left(\nabla\cdot\mathbf{B}\right)_{\mathbf{k}}=\left[\frac{\partial B_{x}}{\partial x}+\frac{\partial B_{y}}{\partial y}+\frac{\partial B_{z}}{\partial z}\right]_{\mathbf{k}}=i\left(\mathbf{k}\cdot\mathbf{B}_{\mathbf{k}}\right)$ | + | \left[\text{div}\mathbf{B}\right]_{\mathbf{k}}=\left(\nabla\cdot\mathbf{B}\right)_{\mathbf{k}}=\left[\frac{\partial B_{x}}{\partial x}+\frac{\partial B_{y}}{\partial y}+\frac{\partial B_{z}}{\partial z}\right]_{\mathbf{k}}=i\left(\mathbf{k}\cdot\mathbf{B}_{\mathbf{k}}\right) |
| + | . | ||
| + | $$ | ||
| совершенно аналогично из покопонентного вычисления | совершенно аналогично из покопонентного вычисления | ||
| - | $\left[rot\mathbf{B}\right]_{\mathbf{k}}=\left[\nabla\times\mathbf{B}\right]_{\mathbf{k}}=i\left[\mathbf{k}\times\mathbf{B}_{\mathbf{k}}\right]$ | + | $$ |
| - | + | \left[\text{rot}\mathbf{B}\right]_{\mathbf{k}}=\left[\nabla\times\mathbf{B}\right]_{\mathbf{k}}=i\left[\mathbf{k}\times\mathbf{B}_{\mathbf{k}}\right] | |
| - | \textemdash{} | + | . |
| - | + | $$ | |
| - | $\left[\nabla\times\mathbf{E}\right]_{\omega}=\frac{1}{2\pi}\intop_{-\infty}^{\infty}\left[\nabla\times\mathbf{E}\right]e^{i\omega t}dt=\nabla\times\frac{1}{2\pi}\intop_{-\infty}^{\infty}\mathbf{E}e^{i\omega t}dt=\left[\nabla\times\mathbf{E}_{\omega}\right]$ | + | |
| - | + | ||
| - | \textemdash{} | + | |
| - | + | ||
| - | $rot\mathbf{E}=-\frac{1}{c}\frac{\partial\mathbf{B}}{\partial t}$ | + | |
| - | $\mapsto$ $\left[\nabla\times\mathbf{E}_{\omega}\right]=\frac{i\omega}{c}\mathbf{B}_{\omega}$ | + | |
| - | + | ||
| - | $rot\mathbf{H}=\frac{4\pi}{c}\mathbf{j}+\frac{1}{c}\frac{\partial\mathbf{D}}{\partial t}$ | + | |
| - | $\mapsto$ $\left[\nabla\times\mathbf{H}_{\omega}\right]=\frac{4\pi}{c}\mathbf{j}_{\omega}+\frac{i\omega}{c}\mathbf{D}_{\omega}$ | + | |
| - | + | ||
| - | $div\mathbf{D}=4\pi\rho$ $\mapsto$ $\left(\nabla\cdot\mathbf{D}_{\omega}\right)=4\pi\rho_{\omega}$ | + | |
| - | + | ||
| - | $div\mathbf{B}=0$ $\mapsto$ $\left(\nabla\cdot\mathbf{B}_{\omega}\right)=0$ | + | |
| - | + | ||
| - | \textemdash{} | + | |
| - | + | ||
| - | $rot\mathbf{E}=-\frac{1}{c}\frac{\partial\mathbf{B}}{\partial t}$ | + | |
| - | $\mapsto$ $i\left[\mathbf{k}\times\mathbf{E}_{\mathbf{k}}\right]=-\frac{1}{c}\frac{\partial\mathbf{B_{\mathbf{k}}}}{\partial t}$ | + | |
| - | $rot\mathbf{H}=\frac{4\pi}{c}\mathbf{j}+\frac{1}{c}\frac{\partial\mathbf{D}}{\partial t}$ | + | $$ |
| - | $\mapsto$ $i\left[\mathbf{k}\times\mathbf{H}_{\mathbf{k}}\right]=\frac{4\pi}{c}\mathbf{j}_{\mathbf{k}}+\frac{1}{c}\frac{\partial\mathbf{D}_{\mathbf{k}}}{\partial t}$ | + | \left[\nabla\times\mathbf{E}\right]_{\omega}=\frac{1}{2\pi}\intop_{-\infty}^{\infty}\left[\nabla\times\mathbf{E}\right]e^{i\omega t}dt=\nabla\times\frac{1}{2\pi}\intop_{-\infty}^{\infty}\mathbf{E}e^{i\omega t}dt=\left[\nabla\times\mathbf{E}_{\omega}\right] |
| + | .$$ | ||
| - | $div\mathbf{D}=4\pi\rho$ $\mapsto$ $i\left(\mathbf{k}\cdot\mathbf{D}_{\mathbf{k}}\right)=4\pi\rho_{\mathbf{k}}$ | + | Тогда окончательно запишем разложение по монохроматическим волнам: |
| - | $div\mathbf{B}=0$ $\mapsto$ $\left(\mathbf{k}\cdot\mathbf{B}_{\mathbf{k}}\right)=0$ | + | $$\text{rot}\mathbf{E}=-\frac{1}{c}\frac{\partial\mathbf{B}}{\partial t} \mapsto \left[\nabla\times\mathbf{E}_{\omega}\right]=\frac{i\omega}{c}\mathbf{B}_{\omega} |
| + | ,$$ | ||
| + | $$ | ||
| + | \text{rot}\mathbf{H}=\frac{4\pi}{c}\mathbf{j}+\frac{1}{c}\frac{\partial\mathbf{D}}{\partial t} \mapsto | ||
| + | \left[\nabla\times\mathbf{H}_{\omega}\right]=\frac{4\pi}{c}\mathbf{j}_{\omega}+\frac{i\omega}{c}\mathbf{D}_{\omega} | ||
| + | ,$$ | ||
| - | \textemdash{} | + | $$\text{div}\mathbf{D}=4\pi\rho \mapsto \left(\nabla\cdot\mathbf{D}_{\omega}\right)=4\pi\rho_{\omega} |
| + | ,$$ | ||
| + | $$ | ||
| + | \text{div}\mathbf{B}=0 \mapsto \left(\nabla\cdot\mathbf{B}_{\omega}\right)=0. | ||
| + | $$ | ||
| - | $rot\mathbf{E}=-\frac{1}{c}\frac{\partial\mathbf{B}}{\partial t}$ | + | Разложение по плоским волнам: |
| - | $\mapsto$ $\left[\mathbf{k}\times\mathbf{E}_{\mathbf{k}\omega}\right]=\frac{\omega}{c}\mathbf{B}_{\mathbf{k}\omega}$ | + | |
| - | $rot\mathbf{H}=\frac{4\pi}{c}\mathbf{j}+\frac{1}{c}\frac{\partial\mathbf{D}}{\partial t}$ | + | $$ |
| - | $\mapsto$ $\left[\mathbf{k}\times\mathbf{H}_{\mathbf{k}\omega}\right]=-\frac{i4\pi}{c}\mathbf{j}_{\mathbf{k}\omega}+\frac{\omega}{c}\mathbf{D}_{\mathbf{k}\omega}$ | + | \text{rot}\mathbf{E}=-\frac{1}{c}\frac{\partial\mathbf{B}}{\partial t} |
| + | \mapsto | ||
| + | i\left[\mathbf{k}\times\mathbf{E}_{\mathbf{k}}\right]=-\frac{1}{c}\frac{\partial\mathbf{B_{\mathbf{k}}}}{\partial t} | ||
| + | ,$$ | ||
| + | $$ | ||
| + | \text{rot}\mathbf{H}=\frac{4\pi}{c}\mathbf{j}+\frac{1}{c}\frac{\partial\mathbf{D}}{\partial t} | ||
| + | \mapsto | ||
| + | i\left[\mathbf{k}\times\mathbf{H}_{\mathbf{k}}\right]=\frac{4\pi}{c}\mathbf{j}_{\mathbf{k}}+\frac{1}{c}\frac{\partial\mathbf{D}_{\mathbf{k}}}{\partial t} | ||
| + | ,$$ | ||
| + | $$ | ||
| + | \text{div}\mathbf{D}=4\pi\rho | ||
| + | \mapsto | ||
| + | i\left(\mathbf{k}\cdot\mathbf{D}_{\mathbf{k}}\right)=4\pi\rho_{\mathbf{k}} | ||
| + | ,$$ | ||
| + | $$ | ||
| + | \text{div}\mathbf{B}=0 \mapsto \left(\mathbf{k}\cdot\mathbf{B}_{\mathbf{k}}\right)=0 | ||
| + | . | ||
| + | $$ | ||
| - | $div\mathbf{D}=4\pi\rho$ $\mapsto$ $\left(\mathbf{k}\cdot\mathbf{D}_{\mathbf{k}\omega}\right)=-i4\pi\rho_{\mathbf{k}\omega}$ | + | Разложение по плоским и монохроматическим волнам: |
| - | $div\mathbf{B}=0$ $\mapsto$ $\left(\mathbf{k}\cdot\mathbf{B}_{\mathbf{k}\omega}\right)=0$ | + | $$ |
| + | \text{rot}\mathbf{E}=-\frac{1}{c}\frac{\partial\mathbf{B}}{\partial t} | ||
| + | \mapsto | ||
| + | \left[\mathbf{k}\times\mathbf{E}_{\mathbf{k}\omega}\right]=\frac{\omega}{c}\mathbf{B}_{\mathbf{k}\omega}, | ||
| + | $$\text{rot}\mathbf{H}=\frac{4\pi}{c}\mathbf{j}+\frac{1}{c}\frac{\partial\mathbf{D}}{\partial t} | ||
| + | \mapsto | ||
| + | \left[\mathbf{k}\times\mathbf{H}_{\mathbf{k}\omega}\right]=-\frac{i4\pi}{c}\mathbf{j}_{\mathbf{k}\omega}+\frac{\omega}{c}\mathbf{D}_{\mathbf{k}\omega}, | ||
| + | $$ | ||
| + | \text{div}\mathbf{D}=4\pi\rho \mapsto \left(\mathbf{k}\cdot\mathbf{D}_{\mathbf{k}\omega}\right)=-i4\pi\rho_{\mathbf{k}\omega}, | ||
| + | $$ | ||
| + | $$\text{div}\mathbf{B}=0 \mapsto \left(\mathbf{k}\cdot\mathbf{B}_{\mathbf{k}\omega}\right)=0 | ||
| + | .$$ | ||