数值孔径

普通光学中的数值孔径概念

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

数值孔径与焦比的关系

$\ N = f/D$

$\mathrm{NA_i} = n \sin \theta = n \sin \arctan \frac{D}{2f} \approx n \frac {D}{2f}$

$N \approx \frac{1}{2\;\mathrm{NA_i}}$

工作焦比（有效焦比）

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

$\frac{1}{2 \mathrm{NA_o}} = \frac{m-1}{m}\, N$

激光物理中的数值孔径概念

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

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）：

$\mathrm{NA}\simeq \frac{\lambda_0}{\pi w_0},$

光纤光学中的数值孔径概念

$n \sin \theta_\max = \sqrt{n_1^2 - n_2^2}$

$n\sin\theta_\mathrm{max} = n_1\sin\theta_r\$

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

$\frac{n}{n_{1}}\sin\theta_\mathrm{max} = \cos\theta_{c}$

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

$n \sin \theta_\mathrm{max} = \sqrt{n_1^2 - n_2^2}$

$\mathrm{NA} = \sqrt{n_1^2 - n_2^2}$

参考文献

1. ^ "High-def Disc Update: Where things stand with HD DVD and Blu-ray" by Steve Kindig, Crutchfield Advisor. Accessed 2008-01-18.
2. ^ 2.0 2.1 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.