section_path: Construction of simple lenses/Lensmaker's equation/Derivation
section_name: Derivation
paragraph_text: Derivation
  
  The spherical thin lens equation in paraxial approximation is derived here with respect to the right figure. The 1st spherical lens surface (which meets the optical axis at V 1 {\textstyle V_{1}} as its vertex) images an on-axis object point O to the virtual image I', which can be described by the following equation, n 1 u + n 2 v ′ = n 2 − n 1 R 1. {\displaystyle {\frac {n_{1}}{u}}+{\frac {n_{2}}{v'}}={\frac {n_{2}-n_{1}}{R_{1}}}.} For the imaging by second lens surface, by taking the above sign convention, u ′ = − v ′ + d {\textstyle u'=-v'+d} and n 2 − v ′ + d + n 1 v = n 1 − n 2 R 2. {\displaystyle {\frac {n_{2}}{-v'+d}}+{\frac {n_{1}}{v}}={\frac {n_{1}-n_{2}}{R_{2}}}.} Adding these two equations yields,
  
  n 1 u + n 1 v = (n 2 − n 1) (1 R 1 − 1 R 2) + n 2 d (v ′ − d) v ′. {\displaystyle {\frac {n_{1}}{u}}+{\frac {n_{1}}{v}}=(n_{2}-n_{1})({\frac {1}{R_{1}}}-{\frac {1}{R_{2}}})+{\frac {n_{2}d}{(v'-d)v'}}.}
  
  For the thin lens approximation where d → 0 {\displaystyle d\rightarrow 0}, the 2nd term of the RHS (Right Hand Side) is gone, so n 1 u + n 1 v = (n 2 − n 1) (1 R 1 − 1 R 2). {\displaystyle {\frac {n_{1}}{u}}+{\frac {n_{1}}{v}}=(n_{2}-n_{1})({\frac {1}{R_{1}}}-{\frac {1}{R_{2}}}).} The focal length f {\displaystyle f} of the thin lens is found by limiting u → − ∞ {\displaystyle u\rightarrow -\infty }, n 1 f = (n 2 − n 1) (1 R 1 − 1 R 2) → 1 f = (n 2 n 1 − 1) (1 R 1 − 1 R 2). {\displaystyle {\frac {n_{1}}{f}}=(n_{2}-n_{1})({\frac {1}{R_{1}}}-{\frac {1}{R_{2}}})\rightarrow {\frac {1}{f}}=({\frac {n_{2}}{n_{1}}}-1)({\frac {1}{R_{1}}}-{\frac {1}{R_{2}}}).} So, the Gaussian thin lens equation is
  
  1 u + 1 v = 1 f. {\displaystyle {\frac {1}{u}}+{\frac {1}{v}}={\frac {1}{f}}.}
  
  For the thin lens in air or vacuum where n 1 = 1 {\textstyle n_{1}=1} can be assumed, f {\textstyle f} becomes
  
  1 f = (n − 1) (1 R 1 − 1 R 2) {\displaystyle {\frac {1}{f}}=(n-1)({\frac {1}{R_{1}}}-{\frac {1}{R_{2}}})}
  
  where the subscript of 2 in n 2 {\textstyle n_{2}} is dropped.