Optical and Fluorescence Formulae

 

Lens Makers Formula:
Lenses are characterized by their focal length. The focal length is given by the lens maker's formula:

where f is the focal length, n is the refractive index, and R1 and R2 are the radii of curvature of the two surfaces of the lens.  For a planar surface, R=¥.
 

Images and Ray Tracing:
In first order analysis, lens systems may be characterized by five variables: the focal length, f, the object to image distance, D, the lens to object distance, S1, the lens to image distance, S2, the magnification, M and the reduction, R.

These paraxial equations are first order relationships and as such are only approximations.  However, they are sufficiently accurate for the vast majority of situations.

Lens Pairs:
The following formulae may be used to determine the focal length of a pair of lenses:

where fc is the focal length of the combination, f1 is the focal length of the first lens, f2 is the focal length of the second lens and d is the distance between the principle planes of the two lenses.
 

Light Collection:
The light collection efficiency of a lens may be described by its f/# (pronounced "f" number) or Numerical Aperture (NA);

 

where f is the focal length, D is the diameter of the lens and ømax is the full angle of the cone of light rays that can pass through the system. 

For small ømax,

The solid angle that a lens collects is approximately:

The fraction of light that an optic collects is this solid angle divided by the total 4 steradians: 

Fluorophore "Brightness":
When comparing fluorophores, it is often useful to compare relative brightness:

where e is the Molar Absorptivity and QE is the Quantum Efficiency of the fluorophore.

Light Focusing:
Light from infinity, e.g. collimated light from a laser , may only be focused to a spot size with a finite radius, rs: 

 

Snell's Law: