A biconvex lens has the following general geometry
Where there's an index of refraction $n_1$ outside of the lens and $n_2$ for the lens material.
In addition we will give the radii of curvature as $R_1$ and $R_2$ for the input and output sides of the lens respectively.
# Ball lens
A biconvex lens where $R_1=R_2$ is referred to as a [ball lens](Ball%20lens.md) if the lens itself takes on a spherical shape.
# Paraxial approximation
Consider an optical setup under the [paraxial approximation](Paraxial%20approximation.md).
## Thin lens model
We may model the lens as an infinitely thin plane if $R_1$ and $R_2$ are large relative to the thickness of the lens - however, the radii of curvature at each [optical Interface](Optical%20Interface.md) and the [Index of refraction](Index%20of%20refraction.md) of refraction of the lens material still remain as intrinsic qualities, thus the [focal length](Optics%20(Index).md#focal%20length) is dependent on $R_1$ and $R_2.$ Taking the $d=0$ limit of the [thick lens model](Biconvex%20lens.md#Thick%20lens%20model) gives the general [thin lens ABCD matrix](Thin%20lens.md#ABCD%20matrix) where
$f=\frac{n_1}{n_2-n_1}\frac{R_1R_2}{R_1+R_2}.$
## Thick lens model
Under the Paraxial approximation we model a [thick](Thick%20lens.md#paraxial%20approximation) biconvex lens as a pair of [spherical interfaces](Optical%20Interface.md#Spherical%20Interface) where the first interface (at the input) is _convex_ and the second interface (at the output) is _concave_ in the direction of propagation as shown in the diagram below
### ABCD matrix
By treating each boundary as a [spherical interface](Optical%20Interface.md#Spherical%20Interface), we use the derived [ABCD matrices](Optical%20Interface.md#ABCD%20matrix) for the _convex_ configuration at the input and the _concave_ configuration at the output with the [free space](Free%20space%20light%20propagation.md#ABCD%20matrix) sandwiched between them according to the rules for [cascading transfer matrices](Ray%20transfer%20matrix.md#Cascading%20Transfer%20Matrices) yielding
$M=\begin{pmatrix}
1 & 0\\
\frac{1}{R_2}\bigg(1-\frac{n_2}{n_1}\bigg) & \frac{n_2}{n_1}
\end{pmatrix}\begin{pmatrix}1 & d \\ 0 & 1\end{pmatrix}\begin{pmatrix}
1 & 0\\
-\frac{1}{R_1}\bigg(1-\frac{n_1}{n_2}\bigg) & \frac{n_1}{n_2}
\end{pmatrix}$
$=\begin{pmatrix}1-\frac{d}{R_1}\bigg(1-\frac{n_1}{n_2}\bigg)& d\frac{n_1}{n_2}\\
\frac{n_1-n_2}{n_1n_2R_1R_2}\bigg[(R_1+R_2)n_2+(n_1-n_2)d\bigg] & 1+\frac{d}{R_2}\bigg(\frac{n_1}{n_2}-1\bigg)\end{pmatrix}$
One way of at least hinting at the validity of this result is to take the $d=0$, returning a matrix with the same form as that of the [thin lens model](Biconvex%20lens.md#Thin%20lens%20model).
#Electromagnetism/Optics
#ExperimentalTools/ExperimentalToolsinOptics