Identify a compatible pair of ball lenses for coupling light from one optical fiber into another using the numerical aperture of each fiber, the ball lens material, and the ball lens diameter. Identical ball lenses are used when coupling light from one fiber optic to another fiber optic of similar NA, but different ball lenses may be needed when coupling light between fibers of different NA. The ball lens output NA must be less than or equal to the NA of the second fiber in order to maximize throughput.
d: 
EFL_{1} (mm): 
BFL_{1} (mm): 
EFL_{2} (mm): 
BFL_{2} (mm): 
Ball Lens Output NA: 
$$ \text{NA} = \frac{2 \, d \left(n  1 \right)}{n \, D} $$

$$ d = \frac{n \, D}{4 \left(n  1 \right) \sqrt{ \frac{ \left( \frac{1}{\text{NA}} \right)^2  1 }{4} } } $$

$$ \text{EFL} = \frac{n \, D}{4 \left( n  1 \right)} $$

$$ \text{BFL} = \text{EFL}  \frac{D}{2} $$

NA_{1}  Represents numerical aperture of the first fiber 
NA_{2}  Represents numerical aperture of the second fiber 
EFL_{1}  Represents the effective focal length of the first ball lens 
EFL_{2}  Represents the effective focal length of the second ball lens 
BFL_{1}  Represents the back focal length of the first ball lens 
BFL_{2}  Represents the back focal length of the second ball lens 
n_{1}  Represents the refractive index of the first ball lens 
n_{2}  Represents the refractive index of the second ball lens 
D_{1}  Represents the diameter of the first ball lens 
D_{2}  Represents the diameter of the second ball lens 
d  Diameter of the beam between the ball lenses 