# 夫琅禾费衍射

## 夫琅禾费近似

$U(x,y) = \frac{e^{i k z} e^{\frac{ik}{2z} (x^2 + y^2)}}{i \lambda z} \iint_{-\infty}^{\infty} \,u(x',y') e^{-i \frac{2\pi}{\lambda z}(x' x + y' y)}dx'\,dy'$[3]

## 形式

### 解釋

 符合下式就會產生菲涅耳衍射： $F = \frac{a^2}{L\lambda} \ge 1$ 符合下式就會產生夫琅禾费衍射： $F = \frac{a^2}{L\lambda} \ll 1$ $a$ － 圓孔半徑或狹縫寬度， $\lambda$ － 波長， $L$ － 離圓孔的矩離

### 振幅透射率

$\psi_0 (x,y,z,t) = \ e^{i (k z - \omega t)}.$

$\psi_{\mathrm{rad}}(\theta,\phi,r) \, \propto \, \frac{e^{ikr}}{r}\int_{-\infty}^{\infty} \int_{-\infty}^{\infty} \psi_0(x,y) \, T(x,y) \, e^{i k \sin \theta (x \cos \phi + y \sin \phi)} \, dx \,dy .$

$\ e^{i k \sin \theta (x \cos \phi + y \sin \phi)}.$

$\psi_{\mathrm{rad}}(\theta) \propto \int_{-\infty}^{\infty} \psi_0(x)\, T(x) \, e^{i (k\theta) x} \, dx,$

$\int |\psi_\mathrm{rad}(\theta,\phi,r)|^2 r^2 \sin\theta\,d\theta\,d\phi = \int |\psi_0\, T(x,y)|^2\, dx\,dy.$

## 例子

### 狹縫衍射

$\psi_{rad}(\theta) \propto \mathrm{sinc}\left(\frac{\pi a \theta}{\lambda}\right),$

### 高斯剖面

$\psi(\theta) = \exp\left(\frac{-k^2\theta^2}{4a}\right).$

$\psi(\theta) = \exp\left(-\frac{\pi^2W^2}{2 \lambda^2\ln 2}\theta^2\right),$

## 參考資料

### 註釋

1. ^ Hecht, E. (1987), p396 -- Definition of Fraunhofer diffraction and explanation of forms.
2. ^ 2.0 2.1 Hecht, E. (1987), p397 -- diagram and explanation of Fraunhofer diffraction with reference to an opaque shield w/ aperture.
3. ^ Goodman, Joseph. Introduction to Fourier Optics. Englewood, Co: Roberts & Company. 2005. ISBN 0-9747077-2-4.
4. ^ 4.0 4.1 Hecht, E. (1987), p396 - description of the Fraunhofer diffraction through an aperture; details the main equations for the identification of Fresnel and Fraunhofer diffraction.
5. ^ 菲涅耳與夫琅禾费間的一般計算例子：$F_\mathrm{fraunhofer} = \frac{a^2}{L \lambda}= \frac{3^2}{6\times 6} = 0.25$
$F_\mathrm{Fresnel} = \frac {3^2}{2\times 2} = 2.25$

### 文獻

• Hecht, E. Optics, 2nd edition. Addison Wesley. 1987. ISBN 0-201-11611-1.
• Jenkins, F., White, H. Fundamentals of Optics, 4th edition. McGraw-Hill INC. 1976. ISBN 0-07-032330-5.