# 数值孔径

## 普通光学中的数值孔径概念

${\displaystyle \mathrm {NA} =n\sin \theta \;}$

### 数值孔径与焦比的关系

${\displaystyle \ N=f/D}$

${\displaystyle \mathrm {NA_{i}} =n\sin \theta =n\sin \arctan {\frac {D}{2f}}\approx n{\frac {D}{2f}}}$

${\displaystyle N\approx {\frac {1}{2\;\mathrm {NA_{i}} }}}$

#### 工作焦比（有效焦比）

${\displaystyle {\frac {1}{2\mathrm {NA_{i}} }}=N_{\mathrm {w} }=(1-m)\,N}$

${\displaystyle {\frac {1}{2\mathrm {NA_{o}} }}={\frac {m-1}{m}}\,N}$

## 激光物理中的数值孔径概念

${\displaystyle \mathrm {NA} =n\sin \theta \;}$

θ的定义则与之前所述不同。激光光束的并不是一个因受到光阑限制而产生的锐利圆锥，而是一个辐照度随着离光束中心距离而逐渐降低的高斯光束。针对这种情况，激光物理学家们选择用光束的发散程度来定义θ，也就是θ由光的传播方向，以及辐照度降低到波前总辐照度1/e2时距光束中轴的距离决定。对于高斯激光束，其数值孔径与激光最小束斑尺寸有关（其数值孔径表示激光的发散程度，激光发散程度与激光最小光束直径有关）：

${\displaystyle \mathrm {NA} \simeq {\frac {\lambda _{0}}{\pi w_{0}}},}$

## 光纤光学中的数值孔径概念

${\displaystyle n\sin \theta _{\max }={\sqrt {n_{1}^{2}-n_{2}^{2}}}}$

${\displaystyle n\sin \theta _{\mathrm {max} }=n_{1}\sin \theta _{r}\ }$

${\displaystyle \sin \theta _{r}=\sin \left({90^{\circ }}-\theta _{c}\right)=\cos \theta _{c}\ }$

${\displaystyle {\frac {n}{n_{1}}}\sin \theta _{\mathrm {max} }=\cos \theta _{c}}$

${\displaystyle {\frac {n^{2}}{n_{1}^{2}}}\sin ^{2}\theta _{\mathrm {max} }=\cos ^{2}\theta _{c}=1-\sin ^{2}\theta _{c}=1-{\frac {n_{2}^{2}}{n_{1}^{2}}}}$

${\displaystyle n\sin \theta _{\mathrm {max} }={\sqrt {n_{1}^{2}-n_{2}^{2}}}}$

${\displaystyle \mathrm {NA} ={\sqrt {n_{1}^{2}-n_{2}^{2}}}}$

## 参考文献

1. ^ "High-def Disc Update: Where things stand with HD DVD and Blu-ray"页面存档备份，存于互联网档案馆） by Steve Kindig, Crutchfield Advisor. Accessed 2008-01-18.
2. Greivenkamp, John E. Field Guide to Geometrical Optics. SPIE Field Guides vol. FG01. SPIE. 2004. ISBN 0-8194-5294-7. p. 29.
3. ^ Rudolf Kingslake. Lenses in photography: the practical guide to optics for photographers. Case-Hoyt, for Garden City Books. 1951: 97–98.
4. ^ Angelo V Arecchi, Tahar Messadi, and R. John Koshel. Field Guide to Illumination. SPIE. 2007: 48. ISBN 978-0-8194-6768-3.