diff --git a/12.temporary_ins/65.geometrical-optics/50.simple-elements/10.dioptre/10.main/textbook.en.md b/12.temporary_ins/65.geometrical-optics/50.simple-elements/10.dioptre/10.main/textbook.en.md new file mode 100644 index 000000000..afbdf28b9 --- /dev/null +++ b/12.temporary_ins/65.geometrical-optics/50.simple-elements/10.dioptre/10.main/textbook.en.md @@ -0,0 +1,80 @@ +--- +title: 'Refracting surface ' +published: false +routable: false +visible: false +--- + +!!!! *LESSON UNDER CONSTRUCTION :*
+!!!! Published but invisible: does not appear in the tree structure of the m3p2.com site. This course is *under construction*, it is *not approved by the pedagogical team* at this stage.
+!!!! Working document intended only for the pedagogical team. + + + +------------- + +### Spherical refracting surface in paraxial approximation. + +#### Refracting surface. + +A **refracting surface** is a *polished surface between two media with different refractive indexes*. + +!!!! *BE CAREFUL* :
+!!!! 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 mesaure is named "dioptrie". +!!!! So keep in mind the following scheme : +!!!! +!!!! refracting surface : *EN : refracting surface* , *ES : superficie refractiva* , *FR : dioptre*.
+!!!! _A crystal ball forms a spherical refracting surface : un "dioptre sphérique" in French._ +!!!! +!!!! unit of measure : *EN : dioptre* , *ES : dioptría* , *FR : dioptrie*.
+!!!! _My corrective lens for both eyes are 4 dioptres : "4 dioptries" in French._ + + +#### Spherical refracting surface. + + + +#### Analytical study of the position and shape of an image. + +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 $`A_{ima}`$*. + + +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 $`|\overline{SC}|\longrightarrow\infty`$.
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 of the position and shape of an image. + + +