# 菲涅耳衍射

## 维基百科，自由的百科全书

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

## 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)}$ 之间的距离 ${\displaystyle R}$ 可以以泰勒级数表示为

{\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))}

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

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

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

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

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

### 菲涅耳衍射积分式

 菲涅耳数 ${\displaystyle F=a^{2}/L\lambda }$ 菲涅耳衍射区域：${\displaystyle F\geq 1}$ 夫琅禾费衍射区域：${\displaystyle F\ll 1}$ ${\displaystyle a}$ － 孔径或狭缝的尺寸 ${\displaystyle \lambda }$ － 波长 ${\displaystyle L}$ － 离开孔径或狭缝的距离

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

### 圆孔衍射

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

{\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))}

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

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

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

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

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

${\displaystyle J_{2n+m}(v)}$ 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：

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

## 进阶理论

### 卷积

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

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

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

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

${\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\))$

${\displaystyle f(x',y')}$ 视为函数 ${\displaystyle \delta (x-x')\delta (y-y')}$ 权重系数，应用线性系统的性质，可以将积分式写为

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

### 傅里叶变换

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

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

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

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

{\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))}

## 参考文献

1. ^ M. Born & E. Wolf, Principles of Optics, 1999, Cambridge University Press, Cambridge
2. ^ 实际而言，在先前一个步骤里做了一个近似，即假定 ${\displaystyle e^{ikr}/r}$ 是真实波，但这不是矢量亥姆霍兹方程的解答，而是标量亥姆霍兹方程的解答。请参阅条目标量波近似（scalar wave approximation）。
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11. ^ Hecht, Eugene, Optics 4th, United States of America: Addison Wesley: pp. 529–532, 2002, ISBN 0-8053-8566-5 （英语）
12. ^ http://www.ils.uec.ac.jp/~dima/PhysRevLett_94_013203.pdf 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.