# 基尔霍夫衍射公式本文重定向自 菲涅耳－基尔霍夫衍射公式

## 惠更斯-菲涅耳原理

• 它是波扰的传播
• 它具有时空周期性，能够相干叠加

${\displaystyle \psi (\mathbf {r} )=\oint _{\mathbb {S} }\,\mathrm {d} \psi (\mathbf {r} ,\mathbf {r} ')}$

• 它应当正比于面元素的面积：
${\displaystyle \mathrm {d} \psi (\mathbf {r} ,\mathbf {r} ')\propto \mathrm {d} S'}$
• 它应当正比于次波源的复振幅：
${\displaystyle \mathrm {d} \psi (\mathbf {r} ,\mathbf {r} ')\propto \psi (\mathbf {r} ')}$
• 次波源发射出的次波应是球面波，其中${\displaystyle k}$波数
${\displaystyle \mathrm {d} \psi (\mathbf {r} ,\mathbf {r} ')\propto {\frac {e^{ikR}}{R}}}$；其中，${\displaystyle \mathbf {R} =\mathbf {r} -\mathbf {r} '}$是从点Q到点P的位移矢量。
• 次波源发射出的次波是各向异性的。假设${\displaystyle {\hat {\mathbf {n} }}}$是与面元素矢量${\displaystyle \mathrm {d} \mathbf {S} '}$同方向的单位矢量，${\displaystyle \chi }$${\displaystyle {\hat {\mathbf {n} }}}$${\displaystyle {\hat {\mathbf {R} }}}$之间的夹角，则倾斜因子${\displaystyle K(\chi )}$${\displaystyle \mathrm {d} \psi (\mathbf {r} ,\mathbf {r} ')}$的关系为
${\displaystyle \mathrm {d} \psi (\mathbf {r} ,\mathbf {r} ')\propto K(\chi )}$

${\displaystyle \psi (\mathbf {r} )=c\oint _{\mathbb {S} }\,\psi (\mathbf {r} ')K(\chi ){\frac {e^{ikR}}{R}}\,\mathrm {d} S'}$

## 菲涅耳-基尔霍夫衍射公式

${\displaystyle \psi (\mathbf {r} )=-\ {\frac {i\psi _{0}}{2\lambda }}\oint _{\mathbb {S} }\left({\frac {e^{ik(r'+R)}}{r'R}}\right)[\cos \alpha +\cos \chi ]\,\mathrm {d} S'}$

## 严格导引

${\displaystyle \psi (\mathbf {r} )={\frac {1}{4\pi }}\oint _{\mathbb {S} }\left[\psi (\mathbf {r} ')\nabla '\left({\frac {e^{ikR}}{R}}\right)-\left({\frac {e^{ikR}}{R}}\right)\nabla '\psi (\mathbf {r} ')\right]\cdot \,\mathrm {d} \mathbf {S} '}$

${\displaystyle \psi (\mathbf {r} )={\frac {1}{4\pi }}\oint _{\mathbb {S} }\left[\psi (\mathbf {r} '){\frac {\partial }{\partial n'}}\left({\frac {e^{ikR}}{R}}\right)-\left({\frac {e^{ikR}}{R}}\right){\frac {\partial \psi (\mathbf {r} ')}{\partial n'}}\right]\,\mathrm {d} S'}$

• 点波源与孔隙之间的距离${\displaystyle r'}$超大于波长${\displaystyle \lambda =2\pi /k}$
• ${\displaystyle R}$超大于波长${\displaystyle \lambda }$

### 点波源

${\displaystyle \psi (\mathbf {r} ')=\psi _{0}{\frac {e^{ikr'}}{r'}}}$

${\displaystyle \psi (\mathbf {r} )={\frac {\psi _{0}}{4\pi }}\oint _{\mathbb {S} }\left[\left({\frac {e^{ikr'}}{r'}}\right)\nabla '\left({\frac {e^{ikR}}{R}}\right)-\left({\frac {e^{ikR}}{R}}\right)\nabla '\left({\frac {e^{ikr'}}{r'}}\right)\right]\cdot \,{\hat {\mathbf {n} }}\mathrm {d} S'}$

${\displaystyle \nabla '\left({\frac {e^{ikR}}{R}}\right)=-\left({\frac {e^{ikR}}{R}}\right)\left(ik-\ {\frac {1}{R}}\right){\hat {\mathbf {R} }}}$
${\displaystyle \nabla '\left({\frac {e^{ikr'}}{r'}}\right)=\left({\frac {e^{ikr'}}{r'}}\right)\left(ik-\ {\frac {1}{r'}}\right){\hat {\mathbf {r} '}}}$

${\displaystyle \nabla '\left({\frac {e^{ikR}}{R}}\right)=-ik\left({\frac {e^{ikR}}{R}}\right){\hat {\mathbf {R} }}}$
${\displaystyle \nabla '\left({\frac {e^{ikr'}}{r'}}\right)=ik\left({\frac {e^{ikr'}}{r'}}\right){\hat {\mathbf {r} '}}}$

{\displaystyle {\begin{aligned}\psi (\mathbf {r} )&=-\ {\frac {i\psi _{0}}{2\lambda }}\oint _{\mathbb {S} }\left({\frac {e^{ik(r'+R)}}{r'R}}\right)({\hat {\mathbf {r} '}}\cdot {\hat {\mathbf {n} }}+{\hat {\mathbf {R} }}\cdot {\hat {\mathbf {n} }})\,\mathrm {d} S'\\&=-\ {\frac {i\psi _{0}}{2\lambda }}\oint _{\mathbb {S} }\left({\frac {e^{ik(r'+R)}}{r'R}}\right)(\cos \alpha +\cos \chi )\,\mathrm {d} S'\\\end{aligned}}}

### 倾斜因子

${\displaystyle \cos \alpha =1}$

${\displaystyle \psi (\mathbf {r} )=-\ {\frac {i\psi (\mathbf {r} ')}{\lambda }}\oint _{\mathbb {S} }\left({\frac {e^{ikR}}{R}}\right)K(\chi )\,\mathrm {d} S'}$

### 惠更斯－菲涅耳原理

${\displaystyle -\ {\frac {i\psi (\mathbf {r} ')}{\lambda }}\left({\frac {e^{ikR}}{R}}\right)K(\chi )}$

${\displaystyle \psi (\mathbf {r} )=\psi _{0}{\frac {e^{ikr}}{r}}}$

${\displaystyle \psi _{0}{\frac {e^{ikr}}{r}}=-\ {\frac {i}{\lambda }}\oint _{\mathbb {S} _{1}}\left({\frac {\psi _{0}e^{ikr'}}{r'}}\right)\left({\frac {e^{ikR}}{R}}\right)K(\chi )\,\mathrm {d} S'}$

${\displaystyle R\approx r-r'\cos(\theta )}$
${\displaystyle R^{2}\approx r^{2}-2rr'\cos(\theta )}$
${\displaystyle K(\chi )={\frac {1+\cos(\chi )}{2}}\approx {\frac {1+\cos(\theta )}{2}}}$

{\displaystyle {\begin{aligned}\psi (\mathbf {r} )&\approx -\ {\frac {i\psi _{0}}{2\lambda }}{\frac {e^{ik(r'+r)}}{r'r}}\int _{0}^{\pi }e^{-ikr'\cos(\theta )}[1+\cos(\theta )]2\pi r'^{2}\sin(\theta )\mathrm {d} \theta \\&\approx -\ {\frac {ik\psi _{0}r'e^{ik(r'+r)}}{2r}}\int _{0}^{\pi }e^{-ikr'\cos(\theta )}[1+\cos(\theta )]\sin(\theta )\mathrm {d} \theta \\&\approx -\ {\frac {ik\psi _{0}r'e^{ik(r'+r)}}{2r}}\ {\frac {2[\sin(kr')+i\cos(kr')]}{kr'}}\\&\approx \psi _{0}{\frac {e^{ikr}}{r}}\\\end{aligned}}}

### 有限尺寸波源

${\displaystyle \psi (\mathbf {r} )={\frac {1}{4\pi }}\oint _{\mathbb {S} }\left[\psi (\mathbf {r} '){\frac {\partial }{\partial n'}}\left({\frac {e^{ikR}}{R}}\right)-\left({\frac {e^{ikR}}{R}}\right){\frac {\partial \psi (\mathbf {r} ')}{\partial n'}}\right]\,\mathrm {d} S'}$

${\displaystyle \psi (\mathbf {r} )=-\ {\frac {1}{4\pi }}\oint _{\mathbb {S} }\left({\frac {e^{ikR}}{R}}\right)\left[ik\psi (\mathbf {r} ')\cos {({\hat {\mathbf {R} }},{\hat {\mathbf {n} }}})+{\frac {\partial \psi (\mathbf {r} ')}{\partial n'}}\right]\,\mathrm {d} S'}$

## 参考文献

1. ^ M. Born and E. Wolf, Principles of Optics, 1999, Cambridge University Press, Cambridge
2. ^ RS Longhurst, Gemoetrical and Physical Optics, 1969, Longmans, London
3. Hecht, Eugene. Optics 4th. United States of America: Addison Wesley. 2002: pp. 510–512. ISBN 0-8053-8566-5 （英语）.
4. ^ G. Kirchhoff, Ann. d. Physik. 1883, 2, 18, p663
5. ^ Goodman, Joseph. Introduction to Fourier Optics 3rd. Roberts and Company Publishers. 2004: pp. 35. ISBN 978-0974707723.