Numerical Aperture vs. F-Number

Numerical Aperture and F-Number(or focal ratio) are unit-less numbers. They describe the angle extended by the diameter of a lens. These numbers characterize the resolving power, depth of field of a lens, as well as how much light it could collect.

1. Numerical Aperture

The numerical aperture of a lens is defined as

$NA = nsin\theta$

where $n$ is the refraction of index of the mediumi in which the lens is placed (1.0 for air, 1.3 for water, 1.52 for typical immersion oil). $\theta$ is the half cone extended by the lens.

2. F-Number
The F-number (or focal ratio) of a lens is defined as

$N = \frac{f}{D} $

where $f$  and $D$ are focal length and diameter of the lens respectively. As can be seen, the definition of NA associates with the index of the refraction of the medium in which the lens is working, while F-number does not. Therefore, numerical aperture is commonly used in scientific scenario, i.e. microscopic objective lens, which may work in different medium. F-number is more widely used in general photography, which more likely takes place in the air.

3.  Numerical Aperture and F-Number

Noticed that

$tan\theta=\frac{D/2}{f}=\frac{D}{2f}$ 

Under paraxial approximation, we have

$NA = nsin\theta=n\left(arctan(\frac{D}{2f})\right)\approx \frac{nD}{2f} = \frac{n}{2f/D} = \frac{n}{2N} $

For the special case, where the lens is placed in the air, last equation can be simplified into 

 $NA = \frac{1}{2N}$

4. Resolving Power

The fundamental resolution limit of an optical system is due to diffraction. In a diffraction limited system, the smallest angular separation two objects can have before they blur together is given by Rayleigh criterion as:

$sin\theta=1.22\frac{\lambda}{D}$

where $\lambda$ is the wavelength. The separation of the two objects in image (plane) can be expressed as

$r = 1.22\frac{\lambda f}{D}= 1.22\lambda N = 1.22\frac{\lambda}{2NA}$

This also gives the radius of smallest spot size that can be distinguish in a diffraction limited system

For a given wavelength, the ability of an imaging system in resolving details is limited by the diameter of the aperture(lens). The larger the aperture is, the smaller the diffraction spot size is, thus finer the detail can be distinguished in the image plane.

5. Depth of Field

For high numerical apertures lens, depth of field is determined primarily by wave optics, while for lower numerical apertures, the geometrical optical circle of confusion

$d_{total} = d_{wave}+d_{geom}$

           $=\frac{\lambda \cdot n}{NA^{2}} + \frac{n}{M \cdot NA}r$

where $M$ is the magnification factor of the lens.