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---title: 'Spherical refracting surface : overview'media_order: dioptre-1.gif---
### Spherical refracting surface modeling.
#### Description
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.
Consider a *point object* **$`B_{obj}`$** whose orthogonal projection on the optical axis gives the *point object* **$`A_{obj}`$**. If the point object is located on the optical axis, then $`B_{obj}=A_{obj}`$ and we will use to named it point object $`A_{obj}`$. The point object $`B_{obj}`$ can be **real** *as well as* **virtual**.
The **calculation of the position** of the *point image* **$`B_{ima}`$**, *conjugated point of the point object $`B_{obj}`$* by the refracting surface, is carried out in **two steps** :
1. I use the **spherical refracting surface equation** (known too as the **"conjuction equation" for a spherical refracting surface**) to calculate the *position of the point* **$`A_{ima}`$**, $`A_{ima}`$ being the *orthogonal projection on the optical axis of the point image* $`B_{ima}`$.
**$`\dfrac{n_{fin}}{\overline{SA_{ima}}}-\dfrac{n_{ini}}{\overline{SA_{obj}}}=\dfrac{n_{fin}-n_{ini}}{\overline{SC}}`$**
To perform this I *need to know the __algebraic distance__* **$`\overline{SA_{obj}}`$**, and the *calculation of the __algebraic distance__* **$`\overline{SA_{ima}}`$** along the optical axis *gives me the position of $`A_{ima}`$*.<!--conjugación-->
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 :* $`\dfrac{n_{fin}}{\overline{SA_{ima}}}-\dfrac{n_{ini}}{\overline{SA_{obj}}}=0`$.!! * *transverse magnification equation :* $`\dfrac{n_{ini}\cdot\overline{SA_{ima}}}{n_{fin}\cdot\overline{SA_{obj}}}`$ (unchanged).!! This generalizes and completes the knowledge you get about plane refracting surfaces seen in your pedagogical paths in plain and hills.
#### Graphical study
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