section_path: Construction of simple lenses/For a spherical surface
section_name: For a spherical surface
paragraph_text: For a spherical surface
  
  For a single refraction for a circular boundary, the relation between object and its image in the paraxial approximation is given by
  
  n 1 u + n 2 v = n 2 − n 1 R {\displaystyle {\frac {n_{1}}{u}}+{\frac {n_{2}}{v}}={\frac {n_{2}-n_{1}}{R}}}
  
  where R is the radius of the spherical surface, n is the refractive index of the material of the surface, n is the refractive index of medium (the medium other than the spherical surface material), u {\textstyle u} is the on-axis (on the optical axis) object distance from the line perpendicular to the axis toward the refraction point on the surface (which height is h), and v {\textstyle v} is the on-axis image distance from the line. Due to paraxial approximation where the line of h is close to the vertex of the spherical surface meeting the optical axis on the left, u {\textstyle u} and v {\textstyle v} are also considered distances with respect to the vertex.
  
  Moving v {\textstyle v} toward the right infinity leads to the first or object focal length f 0 {\textstyle f_{0}} for the spherical surface. Similarly, u {\textstyle u} toward the left infinity leads to the second or image focal length f i {\displaystyle f_{i}}.
  
  f 0 = n 1 n 2 − n 1 R, f i = n 2 n 2 − n 1 R {\displaystyle {\begin{aligned}f_{0}&={\frac {n_{1}}{n_{2}-n_{1}}}R,\\f_{i}&={\frac {n_{2}}{n_{2}-n_{1}}}R\end{aligned}}}
  
  Applying this equation on the two spherical surfaces of a lens and approximating the lens thickness to zero (so a thin lens) leads to the lensmaker's formula.