Editing Lens (section)

==== Derivation ====
[[File:A Diagram for a Spherical Lens Equation with Paraxial Rays, 2024-08-27.png|thumb|A Diagram for a Spherical Lens Equation with Paraxial Rays.]]
The spherical thin lens equation in [[paraxial approximation]] is derived here with respect to the right figure.<ref name="Hecht-2017b" /> The 1st spherical lens surface (which meets the optical axis at <math display="inline">\ V_1\ </math> as its vertex) images an on-axis object point ''O'' to the virtual image ''I'', which can be described by the following equation,<math display="block">\ \frac{\ n_1\ }{\ u\ } + \frac{\ n_2\ }{\ v'\ } = \frac{\ n_2 - n_1\ }{\ R_1\ } ~.</math> For the imaging by second lens surface, by taking the above sign convention, <math display="inline">\ u' = - v' + d\ </math> and <math display="block">\ \frac{ n_2 }{\ -v' + d\ } + \frac{\ n_1\ }{\ v\ } = \frac{\ n_1 - n_2\ }{\ R_2\ } ~.</math> Adding these two equations yields <math display="block">\ \frac{\ n_1\ }{ u } + \frac{\ n_1\ }{ v } = \left( n_2 - n_1 \right) \left( \frac{ 1 }{\ R_1\ } - \frac{ 1 }{\ R_2\ } \right) + \frac{\ n_2\ d\ }{\ \left(\ v' - d\ \right)\ v'\ } ~.</math> For the thin lens approximation where <math>\ d \rightarrow 0\ ,</math> the 2nd term of the RHS (Right Hand Side) is gone, so

<math display="block">\ \frac{\ n_1\ }{ u } + \frac{\ n_1\ }{ v } = \left( n_2 - n_1 \right) \left( \frac{ 1 }{\ R_1\ } - \frac{ 1 }{\ R_2\ } \right) ~.</math>

The focal length <math>\ f\ </math> of the thin lens is found by limiting <math>\ u \rightarrow - \infty\ ,</math>

<math display="block">\ \frac{\ n_1\ }{\ f\ } = \left( n_2 - n_1 \right) \left( \frac{ 1 }{\ R_1\ } - \frac{ 1 }{\ R_2\ } \right) \rightarrow \frac{ 1 }{\ f\ } = \left( \frac{\ n_2\ }{\ n_1\ } - 1 \right) \left( \frac{ 1 }{\ R_1\ } - \frac{ 1 }{\ R_2\ } \right) ~.</math>

So, the Gaussian thin lens equation is

<math display="block">\ \frac{ 1 }{\ u\ } + \frac{ 1 }{\ v\ } = \frac{ 1 }{\ f\ } ~.</math>

For the thin lens in air or vacuum where <math display="inline">\ n_1 = 1\ </math> can be assumed, <math display="inline">\ f\ </math> becomes

<math display="block">\ \frac{ 1 }{\ f\ } = \left( n - 1 \right)\left(\frac{ 1 }{\ R_1\ } - \frac{ 1 }{\ R_2\ } \right)\ </math>

where the subscript of 2 in <math display="inline">\ n_2\ </math> is dropped.