# 干涉 (物理学)

## 干涉的条件

 波的叠加 波 1 波 2 相长干涉 相消干涉

## 两列波的干涉

### 基础理论

${\displaystyle I=\left\langle \mathbf {S} \right\rangle ={\frac {c}{4\pi }}{\sqrt {\frac {\epsilon }{\mu }}}\left\langle \mathbf {E} ^{2}\right\rangle \,}$

${\displaystyle \mathbf {E} (\mathbf {r} ,t)={\frac {1}{2}}\left[\mathbf {A} (\mathbf {r} )e^{-i\omega t}+\mathbf {A} ^{*}(\mathbf {r} )e^{i\omega t}\right]\,}$

${\displaystyle \mathbf {E} ^{2}={\frac {1}{4}}\left[\mathbf {A} ^{2}e^{-2i\omega t}+\mathbf {A} ^{*2}e^{2i\omega t}+2\mathbf {A} \cdot \mathbf {A} ^{*}\right]\,}$

${\displaystyle I=\left\langle \mathbf {E} ^{2}\right\rangle ={\frac {1}{2}}\mathbf {A} \cdot \mathbf {A} ^{*}={\frac {1}{2}}\left(a_{1}^{2}+a_{2}^{2}+a_{3}^{2}\right)\,}$

${\displaystyle \mathbf {E} =\mathbf {E} _{1}+\mathbf {E} _{2}\,}$

${\displaystyle I=\left\langle \mathbf {E} ^{2}\right\rangle =\left\langle \mathbf {E} _{1}^{2}\right\rangle +\left\langle \mathbf {E} _{2}^{2}\right\rangle +2\left\langle \mathbf {E} _{1}\cdot \mathbf {E} _{2}\right\rangle \,}$

{\displaystyle {\begin{aligned}\mathbf {E} _{1}\cdot \mathbf {E} _{2}&={\frac {1}{4}}\left[\mathbf {A} e^{-i\omega t}+\mathbf {A} ^{*}e^{i\omega t}\right]\left[\mathbf {B} e^{-i\omega t}+\mathbf {B} ^{*}e^{i\omega t}\right]\\&={\frac {1}{4}}\left(\mathbf {A} \cdot \mathbf {B} e^{-2i\omega t}+\mathbf {A} ^{*}\cdot \mathbf {B} ^{*}e^{2i\omega t}+\mathbf {A} \cdot \mathbf {B} ^{*}+\mathbf {A} ^{*}\cdot \mathbf {B} \right)\end{aligned}}\,}

${\displaystyle 2\left\langle \mathbf {E} _{1}\cdot \mathbf {E} _{2}\right\rangle ={\frac {1}{2}}\left(\mathbf {A} \cdot \mathbf {B} ^{*}+\mathbf {A} ^{*}\cdot \mathbf {B} \right)\,}$

${\displaystyle \mathbf {A} =\sum _{i=1}^{3}a_{i}e^{i\phi _{i}}\mathbf {e} _{i}\qquad i=1,2,3\,}$

${\displaystyle \mathbf {B} =\sum _{i=1}^{3}b_{i}e^{i\psi _{i}}\mathbf {e} _{i}\qquad i=1,2,3\,}$

${\displaystyle \delta =\phi _{1}-\psi _{1}=\phi _{2}-\psi _{2}=\phi _{3}-\psi _{3}={\frac {2\pi }{\lambda }}\Delta L\,}$

${\displaystyle 2\left\langle \mathbf {E} _{1}\cdot \mathbf {E} _{2}\right\rangle =(a_{1}b_{1}+a_{2}b_{2}+a_{3}b_{3})\cos \delta =(a_{1}b_{1}+a_{2}b_{2}+a_{3}b_{3})\cos {\frac {2\pi }{\lambda }}\Delta L\,}$

${\displaystyle a_{2}=b_{2}=a_{3}=b_{3}=0\,}$

{\displaystyle {\begin{aligned}I&={\frac {1}{2}}a_{1}^{2}+{\frac {1}{2}}b_{1}^{2}+a_{1}b_{1}\cos \delta \\&=I_{1}+I_{2}+2{\sqrt {I_{1}I_{2}}}\cos \delta \end{aligned}}}

${\displaystyle I=4I_{0}\cos ^{2}{\frac {\delta }{2}}\,}$ ，此时对应的极大值为${\displaystyle 4I_{0}\,}$ ，极小值为0。

${\displaystyle {\mathcal {V}}={\frac {I_{max}-I_{min}}{I_{max}+I_{min}}}\,}$ ，即可见度的范围为0到1之间。

### 波前分割干涉（分波前干涉[8]）

#### 杨氏双缝

${\displaystyle L_{1}={\sqrt {d^{2}+y^{2}+(x-{\frac {a}{2}})^{2}}}\,}$
${\displaystyle L_{2}={\sqrt {d^{2}+y^{2}+(x+{\frac {a}{2}})^{2}}}\,}$

${\displaystyle \Delta s=a\sin \alpha ^{\prime }\approx a{\frac {x}{d}}\,}$

#### 菲涅耳双棱镜

${\displaystyle d=2a\tan \beta \approx 2a\beta =2a\alpha (n-1)\,}$

#### 迈克耳孙测星干涉仪

${\displaystyle \Delta x={\frac {\lambda f}{d}}\,}$

${\displaystyle {\mathcal {V}}={\frac {2J_{1}(u)}{u}}\,}$

${\displaystyle D=1.22{\frac {\lambda }{2\alpha }}\,}$

### 振幅分割干涉（分振幅干涉）

#### 等倾干涉

${\displaystyle \Delta L=n_{2}({\overline {AB}}+{\overline {BC}})-n_{1}{\overline {AN}}\,}$

${\displaystyle {\overline {AB}}={\overline {BC}}={\frac {d}{\cos \theta ^{\prime }}}\,}$
${\displaystyle {\overline {AN}}={\overline {AC}}\sin \theta \,}$

${\displaystyle \delta ={\frac {4\pi }{\lambda }}n_{2}d\cos \theta ^{\prime }\pm \pi \,}$

#### 等厚干涉

${\displaystyle \Delta L=n_{1}({\overline {SB}}+{\overline {DP}}-{\overline {SA}}-{\overline {AP}})+n_{2}({\overline {BC}}+{\overline {CD}})\,}$

${\displaystyle \Delta L=2n_{2}d\cos \theta ^{\prime }\,}$

${\displaystyle 2n_{2}d{\overline {\cos \theta ^{\prime }}}\pm {\frac {\lambda }{2}}=m\lambda \,}$

${\displaystyle d={\frac {m\lambda }{2}}\quad m=0,1,2,...\,}$ ，即对於相邻明条纹，在该点的厚度差为${\displaystyle {\frac {\lambda }{2}}\,}$ ；若表面厚度绝对均匀，则在表面上无干涉条纹。

${\displaystyle 2nd\pm {\frac {\lambda }{2}}=m\lambda \,}$

${\displaystyle 2d\pm {\frac {\lambda }{2}}=m\lambda \,}$ ，其中m为整数时是亮条纹，m为半整数时是暗条纹。其干涉条纹是一组同心圆，并且中心为零级暗纹。

${\displaystyle r^{2}=R^{2}-(R-d)^{2}=2Rd-d^{2}\approx 2Rd\,}$

### 相干性

#### 时间相干性

{\displaystyle {\begin{aligned}f(\nu )&=f_{0}\int _{-{\frac {\Delta \tau }{2}}}^{\frac {\Delta \tau }{2}}e^{2i\pi (\nu -\nu _{0})t}\,dt\\&=f_{0}\Delta \tau \left[{\frac {\sin {\pi (\nu -\nu _{0})\Delta \tau }}{\pi (\nu -\nu _{0})\Delta \tau }}\right]\end{aligned}}}

#### 空间相干性

${\displaystyle {\mathcal {L}}\sim {\frac {R}{b}}\lambda \,}$

## 多光束干涉

### 平行平面板的多光束干涉

${\displaystyle r_{1}A,\quad t_{1}t_{2}r_{2}Ae^{i\delta },\quad t_{1}t_{2}r_{2}^{3}Ae^{2i\delta },...\quad t_{1}t_{2}r_{2}^{(2p-3)}Ae^{i(p-1)\delta },\quad ...\,}$

${\displaystyle t_{1}t_{2}A,\quad t_{1}t_{2}r_{2}^{2}Ae^{i\delta },\quad t_{1}t_{2}r_{2}^{4}Ae^{2i\delta },...\quad t_{1}t_{2}r_{2}^{(2p-2)}Ae^{i(p-1)\delta },\quad ...\,}$

${\displaystyle A_{r}={\frac {r[1-(r^{2}+t_{1}t_{2})e^{i\delta }]}{1-r^{2}e^{i\delta }}}A\,}$

${\displaystyle A_{r}={\frac {{\sqrt {R}}(1-e^{i\delta })}{1-Re^{i\delta }}}A\,}$

${\displaystyle I_{r}={\frac {4R\sin ^{2}{\frac {\delta }{2}}}{(1-R)^{2}+4R\sin ^{2}{\frac {\delta }{2}}}}I\,}$

{\displaystyle {\begin{aligned}A_{t}&={\frac {t_{1}t_{2}}{1-r_{2}^{2}e^{i\delta }}}A\\&={\frac {T}{1-Re^{i\delta }}}A\\I_{t}&={\frac {T}{(1-R)^{2}+4R\sin ^{2}{\frac {\delta }{2}}}}I\end{aligned}}}

${\displaystyle 2nd\cos \theta _{2}=m\lambda \,}$

${\displaystyle {\frac {I_{r}}{I}}={\frac {F\sin ^{2}{\frac {\delta }{2}}}{1+F\sin ^{2}{\frac {\delta }{2}}}}\,}$
${\displaystyle {\frac {I_{t}}{I}}={\frac {1}{1+F\sin ^{2}{\frac {\delta }{2}}}}\,}$

${\displaystyle \Delta \lambda \approx {\frac {\lambda _{0}^{2}}{2nd\cos \theta _{2}}}}$

${\displaystyle {\mathcal {F}}={\frac {\Delta \lambda }{\delta \lambda }}={\frac {\pi }{2\arcsin(1/{\sqrt {F}})}}}$ .

${\displaystyle {\mathcal {F}}\approx {\frac {\pi {\sqrt {F}}}{2}}={\frac {\pi R^{1/2}}{1-R}}}$

### 法布里－珀罗干涉仪

${\displaystyle m_{0}={\frac {2nd}{\lambda }}\,}$

${\displaystyle \theta _{p}={\sqrt {\frac {n\lambda }{d}}}{\sqrt {p-1+e}}\,}$

${\displaystyle D_{p}^{2}=(2f\theta _{p})^{2}={\frac {4n\lambda f^{2}}{d}}(p-1+e)\,}$

## 量子干涉

1905年至1917年间，爱因斯坦通过马克斯·普朗克能量量子化假设和对光电效应的解释，在《关于光的产生和转化的一个试探性的观点》、《论我们关于辐射的本性和组成的观点的发展》、《论辐射的量子理论》等论文中提出电磁波的能量由不连续的能量子组成，这些能量子被称为光量子光子）。[註 1][14]:xiii[15]因此，电磁辐射必须同时具有波动性和粒子性两种自然属性，这被称作波粒二象性。自罗伯特·密立根於1916年完成了光电效应的一系列实验，以及阿瑟·康普顿於1923年观察到了X射线被自由电子的散射，并於1926年测定了光子的动量，物理学界都逐渐接受了电磁波也具有粒子性的这一事实[16]:67-68, 161

${\displaystyle |\psi \rangle =(|\psi _{1}\rangle +|\psi _{2}\rangle )/{\sqrt {2}}\,}$

${\displaystyle |S(\theta )\rangle =(|\psi _{1}\rangle +e^{i\theta }|\psi _{2}\rangle )/{\sqrt {2}}\,}$

${\displaystyle p(\theta )=|\langle S(\theta )|\psi _{1}\rangle |^{2}=(1+\cos(\theta ))/2\,}$

## 註釋

1. ^ 這三篇論文的英文標題分別為《On a Heuristic Point of View Concerning the Production and Transformation of Light》、《On the Development of Our Views Concerning the Nature and Constitution of Radiation》、《On the Quantum Mechanics of Radiation》。
2. ^

尽管在理论上可以在双缝干涉中每次从相干光源只发射一个光子，根据波函数的统计诠释，经过长时间的积累在屏上将得到经典的干涉条纹；然而在当前的技术下，製備单光子态还十分困难——即使是采用作为相干光源，多个光子仍然会彼此非常接近地进入光检测器，这是光子作为玻色子的一种量子效应，稱為光子群聚[6]:253。实际操作中相对可行的办法是产生光子对，从而可以作为产生单光子态的一个近似，此时在一个光子对中第二个光子的频率和传播方向都和第一个光子相关，从而可被看作是单光子的福柯态英语Fock state[6]:254

另一种更常见的方法是利用非线性光学中的自發参量下转换，用晶体中的单个紫外光子作为泵浦光，其通过非线性效应产生一个信号光子和一个闲频光子，这两个光子的波长都近似为泵浦光子的波长的2倍，偏振方向都和泵浦光子互相垂直；通过采用双折射晶体可以实现泵浦光和下转换光的相位匹配，从而使输出光强得到最大[21]。产生的两个下转换光子都携带了泵浦光子的相位信息，从而处於一个纠缠态，对信号光子的任何测量都会影响到闲频光子的量子态，反之亦然[22]

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