# 菲涅耳衍射

${\displaystyle F\ {\stackrel {def}{=}}\ {\frac {a^{2}}{L\lambda }}}$

## 菲涅耳衍射

${\displaystyle \psi (x,y,z)=-\ {\frac {i}{\lambda }}\int _{\mathbb {S} }\psi (x',y',0){\frac {e^{ikR}}{R}}K(\chi )\ \mathrm {d} x'\mathrm {d} y'}$

${\displaystyle K(\chi )={\frac {1}{2}}(1+\cos \chi )}$

### 菲涅耳近似

${\displaystyle \rho ={\sqrt {(x-x')^{2}+(y-y')^{2}}}}$

${\displaystyle (x',y',0)}$ ${\displaystyle (x,y,z)}$  之間的距離 ${\displaystyle R}$  可以以泰勒級數表示為

{\displaystyle {\begin{aligned}R&={\sqrt {(x-x')^{2}+(y-y')^{2}+z^{2}}}={\sqrt {\rho ^{2}+z^{2}}}\\&=z{\sqrt {1+{\frac {\rho ^{2}}{z^{2}}}}}\\&=z\left[1+{\frac {\rho ^{2}}{2z^{2}}}-{\frac {1}{8}}\left({\frac {\rho ^{2}}{z^{2}}}\right)^{2}+\cdots \right]\\&=z+{\frac {\rho ^{2}}{2z}}-{\frac {\rho ^{4}}{8z^{3}}}+\cdots \\\end{aligned}}}

${\displaystyle {\frac {k\rho ^{4}}{8z^{3}}}\ll 2\pi }$

${\displaystyle {\frac {\rho ^{4}}{8z^{3}\lambda }}\ll 1}$

${\displaystyle {\frac {[(x-x')^{2}+(y-y')^{2}]^{2}}{8z^{3}\lambda }}\ll 1}$

${\displaystyle R\approx z+{\frac {\rho ^{2}}{2z}}=z+{\frac {(x-x')^{2}+(y-y')^{2}}{2z}}}$

${\displaystyle z\gg \left({\frac {\rho ^{4}}{8\lambda }}\right)^{1/3}=\left[{\frac {0.002^{4}}{8\cdot 500\cdot 10^{-9}}}\right]^{1/3}\approx 0.016[m]}$

### 菲涅耳衍射積分式

 菲涅耳數 ${\displaystyle F=a^{2}/L\lambda }$ 菲涅耳衍射區域：${\displaystyle F\geq 1}$ 夫朗和斐繞射區域：${\displaystyle F\ll 1}$ ${\displaystyle a}$  － 孔徑或狹縫的尺寸${\displaystyle \lambda }$  － 波長${\displaystyle L}$  － 離開孔徑或狹縫的距離

${\displaystyle \psi (x,y,z)=-\ {\frac {ie^{ikz}}{\lambda z}}\int _{\mathbb {S} }\psi (x',y',0)e^{ik[(x-x')^{2}+(y-y')^{2}]/2z}\ \mathrm {d} x'\mathrm {d} y'}$

### 圓孔衍射

${\displaystyle \psi (0,0,z)=-\ {\frac {ie^{ikz}\psi _{0}}{\lambda z}}\int _{\mathbb {S} }e^{ik(x'^{2}+y'^{2})/2z}\ \mathrm {d} x'\mathrm {d} y'}$

{\displaystyle {\begin{aligned}\psi (0,0,z)&=-\ {\frac {ie^{ikz}\psi _{0}}{\lambda z}}\int _{0}^{a}e^{ik\rho '^{2}/2z}\ \rho '\mathrm {d} \rho '\\&=-\psi _{0}e^{ikz}(e^{ika^{2}/2z}-1)\\\end{aligned}}}

${\displaystyle I(z)=\psi ^{*}\psi /2=\psi _{0}^{\ 2}\ 2\sin ^{2}(ka^{2}/4z)=I_{0}\sin ^{2}(ka^{2}/4z)}$

• 極大值：當 ${\displaystyle z={\frac {a^{2}}{2n\lambda }},\qquad n=1,2,3,\dots }$
• 極小值：當 ${\displaystyle z={\frac {a^{2}}{(2n-1)\lambda }},\qquad n=1,2,3,\dots }$

${\displaystyle Z_{F}={\frac {0.001^{2}}{500\cdot 10^{-9}}}\approx 2[m]}$

[6]

${\displaystyle I=\left(V_{0}-\cos \left({\frac {u^{2}+v^{2}}{2u}}\right)\right)^{2}+\left(V_{1}-\sin \left({\frac {u^{2}+v^{2}}{2u}}\right)\right)^{2}}$

${\displaystyle V_{m}=\sum _{n=0}^{\infty }*((-1)^{n}*({\frac {v}{u}})^{2*n+m}*J_{2n+m}(v))}$

${\displaystyle J_{2n+m}(v)}$  为 第一类${\displaystyle 2n+m}$ 贝塞尔函数

### 圆盘衍射

${\displaystyle I=I_{0}*\lambda ^{2}/4}$

### 单缝衍射

${\displaystyle I=(Cp(Y)-Cq(Y))^{2}+(Sp(Y)-Sq(Y))^{2}}$

${\displaystyle Cp(Y):=\int _{0}^{p}(\cos((1/2)*\pi *t^{2})\,dt}$

${\displaystyle Cq(Y)=\int _{0}^{q}(\cos((1/2)*\pi *t^{2})\,dt}$ ;

Sp,Sq 为正弦菲涅耳积分：

${\displaystyle Sp(Y):=\int _{0}^{p}(\sin((1/2)*\pi *t^{2})\,dt}$

${\displaystyle Sq(Y)=\int _{0}^{q}(\sin((1/2)*\pi *t^{2})\,dt}$ ;

### 直边衍射

${\displaystyle I=(Cp(Y)+0.5)^{2}+(Sp(Y)+0.5))^{2}}$

${\displaystyle Cq(Y)=\int _{0}^{q}(\cos((1/2)*\pi *t^{2})\,dt}$ ;

Sp 为正弦菲涅耳积分：

${\displaystyle Sp(Y):=\int _{0}^{p}(\sin((1/2)*\pi *t^{2})\,dt}$

## 進階理論

### 卷積

${\displaystyle h(x,y,z)=-\ {\frac {ie^{ikz}}{\lambda z}}e^{i{\frac {k}{2z}}(x^{2}+y^{2})}}$

${\displaystyle \psi (x,y,z)=\iint \limits _{-\infty }^{\ \ \ \infty }\psi (x',y',0)h(x-x',y-y',z)\ \mathrm {d} x'\mathrm {d} y'}$

${\displaystyle \psi _{z}(x,y)=\iint \limits _{-\infty }^{\ \ \ \infty }\psi _{0}(x',y')h_{z}(x-x',y-y')\ \mathrm {d} x'\mathrm {d} y'}$

${\displaystyle \psi _{z}(x,y)=\psi _{0}(x,y)*h_{z}(x,y)}$

${\displaystyle {\mathcal {F}}\{\psi _{z}(x,y)\}={\mathcal {F}}\{\psi _{0}(x,y)*h(x,y)\}={\mathcal {F}}\{\psi _{0}(x,y)\}\cdot {\mathcal {F}}\{h_{z}(x,y)\}}$

${\displaystyle G(X,Y)={\mathcal {L}}\{f(x,y)\}}$

${\displaystyle G(X,Y)={\mathcal {L}}\left\{\iint \limits _{-\infty }^{\ \ \ \infty }f(x',y')\delta (x-x')\delta (y-y')\ \mathrm {d} x'\mathrm {d} y'\right\}}$

${\displaystyle f(x',y')}$  視為函數 ${\displaystyle \delta (x-x')\delta (y-y')}$  權重係數，應用線性系統的性質，可以將積分式寫為

${\displaystyle G(X,Y)=\iint \limits _{-\infty }^{\ \ \ \infty }f(x',y'){\mathcal {L}}\{\delta (x-x')\delta (y-y')\}\ \mathrm {d} x'\mathrm {d} y'}$

### 傅立葉變換

${\displaystyle K_{x}\ {\stackrel {def}{=}}\ kx/z}$
${\displaystyle K_{y}\ {\stackrel {def}{=}}\ ky/z}$

${\displaystyle (x-x')^{2}=x^{2}+x'^{2}-2xx'}$
${\displaystyle (y-y')^{2}=y^{2}+y'^{2}-2yy'}$

${\displaystyle G(K_{x},K_{y})\ {\stackrel {def}{=}}\ {\mathcal {F}}\left\{g(x',y')\right\}\ {\stackrel {def}{=}}\ \iint \limits _{-\infty }^{\ \ \ \infty }g(x',y')e^{-i(K_{x}x'+K_{y}y')}\ \mathrm {d} x'\mathrm {d} y'}$

${\displaystyle g(x',y')=\psi _{0}(x',y')e^{ik(x'^{2}+y'^{2})/2z}}$

{\displaystyle {\begin{aligned}\psi _{z}(x,y)&=-\ {\frac {ie^{ikz}}{\lambda z}}e^{ik(x^{2}+y^{2})/2z}\ {\mathcal {F}}\{\psi _{0}(x',y')e^{ik(x'^{2}+y'^{2})/2z}\}\\&=-\ {\frac {ie^{ikz}}{\lambda z}}e^{ik(x^{2}+y^{2})/2z}\ {\mathcal {F}}\{g(x',y')\}\\&=-\ {\frac {ie^{ikz}}{\lambda z}}e^{ik(x^{2}+y^{2})/2z}\ G(K_{x},K_{y})\\&=h_{z}(x,y)\ G(K_{x},K_{y})\\\end{aligned}}}

## 參考文獻

1. ^ 楊伯溫，《光電工程實驗》，第75頁。
2. ^ M. Born & E. Wolf, Principles of Optics, 1999, Cambridge University Press, Cambridge
3. ^ 實際而言，在先前一個步驟裏做了一個近似，即假定 ${\displaystyle e^{ikr}/r}$  是真實波，但這不是向量亥姆霍茲方程式的解答，而是純量亥姆霍茲方程式的解答。請參閱條目純量波近似（scalar wave approximation）。
4. ^ Gillen, Glen; Guha, Shekhar. Modeling and Propagation of Near-Field Diffraction Patterns: A More Complete Approach. American Journal of Physics. 2004, 72 (8): 1195–1201. ISSN 0002-9505. （原始内容存档于2015-12-20）.
5. ^ Bekefi, George; Barrett, Alan, Electromagnetic Vibrations, Waves and Radiations, The MIT Press: pp. 563–567, 1977, ISBN 978-0262520478
6. ^ Hone-Ene Hwang and Gwo-Huei Yang, Study and improvement of near‐field diffraction limits of circular aperture imaging systems, Journal of the Chinese Institute of Engineers, Vol. 25, No. 3, pp. 335-340 (2002)
7. ^ Karl Dieter Möller, Optics 2nd edtion p136
8. ^ Karl Dieter Möller, Optics, 2nd Ed p174
9. ^ Karl Dieter Möller, Optics, 2nd Ed p. 174
10. ^ M. Born & E. Wolf, Principles of Optics, Fresnel diffraction at a straight edge p493
11. ^ 叶玉堂、肖峻、饶建珍等编著 《光学教程》 第二版 251页, ISBN 9787302114611
12. ^ Hecht, Eugene, Optics 4th, United States of America: Addison Wesley: pp. 529–532, 2002, ISBN 0-8053-8566-5 （英语）
13. ^ http://www.ils.uec.ac.jp/~dima/PhysRevLett_94_013203.pdf页面存档备份，存于互联网档案馆） H. Oberst, D. Kouznetsov, K. Shimizu, J. Fujita, F. Shimizu. Fresnel diffraction mirror for atomic wave, Physical Review Letters, 94, 013203 (2005).
• Goodman, Joseph W. Introduction to Fourier optics. New York: McGraw-Hill. 1996. ISBN 0-07-024254-2.