---
title: 'Spherical refracting surface  : overview'
media_order: dioptre-1.gif
---

### Spherical refracting surface modeling.

#### Description 

![](dioptre-1.gif)

with :
* arrow : indicates direction of light propagation.
* $`n_{ini}`$ : refractive index of the initial medium.
* $`n_{fin}`$ : refractive index of the final medium.
* $`\overline{SC}`$ : algebraic distance between vertex S and center C of curvature on optical axis.








!!!! *BE CAREFUL* :<br>
!!!! In the same way as we use in English the single word "mirror" to qualify a "reflecting surface", in French is use the single word "dioptre" to qualify a "refracting surface".
!!!! The term "dioptre" in English is a unit of mesure of the vergence of an optical system. In French, the same unit of measure is named "dioptrie".
!!!! So keep in mind the following scheme :
!!!!
!!!! refracting surface : *EN : refracting surface* , *ES : superficie refractiva* , *FR : dioptre*.<br>
!!!! _A crystal ball forms a spherical refracting surface : un "dioptre sphérique" in French._
!!!!
!!!! unit of measure : *EN : dioptre* , *ES : dioptría* , *FR : dioptrie*.<br>
!!!! _My corrective lens for both eyes are 4 dioptres : "4 dioptries" in French._


#### Spherical refracting surface.

#### Analytical study

A **spherical refracting surface** in analytical paraxial optics is defined by *three quantities* :
* **$`n_{ini}`$** : *refractive index of the initial medium* (the medium on the side on the incident light).
* **$`n_{fin}`$** : *refractive index of the final medium* (the medium on the side on the emerging light, after crossing the refracting surface).
* **$`\overline{SC}`$** : the *algebraic distance between the __vertex S__* (sometimes called "pole", is the centre of the aperture) *and the __center of curvature C__* of the refracting surface.

! *USEFUL* : The whole analytic study below also applies to a plane refracting surface. We just need to remark that a plane surface is a spherical surface whose radius of curvature tends towards infinity.

##### Spherical refracting surface equation

**spherical refracting surface equation** = **"conjuction equation" for a spherical refracting surface**

**$`\dfrac{n_{fin}}{\overline{SA_{ima}}}-\dfrac{n_{ini}}{\overline{SA_{obj}}}=\dfrac{n_{fin}-n_{ini}}{\overline{SC}}`$**

##### Transverse magnification expression

2. I use the **"transverse magnification equation" for a spherical refracting surface**, to calculate the *__algebraic value__ of the transverse magnification* **$`\overline{M_T}`$**, then to derive the *__algebraic length__* **$`\overline{A_{ima}B_{ima}}`$** of the segment $`[A_{ima}B_{ima}]`$, that is the algebraic distance of the point image $`B_{ima}`$ from its orthogonal projection $`A_{ima}`$ on the optical axis.

By *definition :* **$`\overline{M_T}=\dfrac{\overline{A_{ima}B_{ima}}}{\overline{A_{obj}B_{obj}}}`$**.
Its *expression for spherical refracting surface :* **$`\overline{M_T}=\dfrac{n_{ini}\cdot\overline{SA_{ima}}}{n_{fin}\cdot\overline{SA_{obj}}}`$**.

I know $`\overline{SA_{obj}}$, $n_{ini}$ and $n_{fin}$, I have previously calculated $`\overline{SA_{ima}}$, so I can calculate $`\overline{M_T}`$ and deduced $`\overline{A_{ima}B_{ima}}`$


! *USEFUL* : The conjuction equation and the transverse magnification equation for a plane refracting surface are obtained by rewriting these equations for a spherical refracting surface in the limit when $`|\overline{SC}|\longrightarrow\infty`$.<br> Then we get *for a plane refracting surface :*
!
! * *conjuction equation :*&nbsp;&nbsp; $`\dfrac{n_{fin}}{\overline{SA_{ima}}}-\dfrac{n_{ini}}{\overline{SA_{obj}}}=0`$.
!
! * *transverse magnification equation :*&nbsp;&nbsp; $`\dfrac{n_{ini}\cdot\overline{SA_{ima}}}{n_{fin}\cdot\overline{SA_{obj}}}`$ &nbsp;&nbsp; (unchanged).
!
! This generalizes and completes the knowledge you get about plane refracting surfaces seen in your pedagogical paths in plain and hills.


#### Graphical study