From 3b32bb588d995e4ff05825eb8e8a19b135b00dcd Mon Sep 17 00:00:00 2001 From: Claude Meny Date: Sat, 20 Mar 2021 11:11:29 +0100 Subject: [PATCH] Delete cheatsheet.en.md --- .../cheatsheet.en.md | 259 ------------------ 1 file changed, 259 deletions(-) delete mode 100644 01.curriculum/01.physics-chemistry-biology/02.Niv2/04.optics/04.use-of-basic-optical-elements/01.plane-refracting-surface/02.plane-refracting-surface-overview/cheatsheet.en.md diff --git a/01.curriculum/01.physics-chemistry-biology/02.Niv2/04.optics/04.use-of-basic-optical-elements/01.plane-refracting-surface/02.plane-refracting-surface-overview/cheatsheet.en.md b/01.curriculum/01.physics-chemistry-biology/02.Niv2/04.optics/04.use-of-basic-optical-elements/01.plane-refracting-surface/02.plane-refracting-surface-overview/cheatsheet.en.md deleted file mode 100644 index d3733759a..000000000 --- a/01.curriculum/01.physics-chemistry-biology/02.Niv2/04.optics/04.use-of-basic-optical-elements/01.plane-refracting-surface/02.plane-refracting-surface-overview/cheatsheet.en.md +++ /dev/null @@ -1,259 +0,0 @@ ---- -title: 'Spherical refracting surface : overview' -media_order: dioptre-1.gif -published: false -visible: false ---- - -### What is a refracting interface ? - -#### Refracting surface : physical description - -* **Interface separating two transparent media of different refractive indices**. - -* can be **found in nature** :
-Examples :
-\- a **plane refracting surface** : the *flat and quiet surface of a lake*. -\- a **spherical refracting surface** : a *fish ball aquarium*. - -![](spherical-refracting-surface-example-1.jpg)
-Fig. 1. The spherical refracting interface of a fish ball aquarium. - -* **appears in the design and modeling of other optical elements** :
-Examples :
-\- a **glass window pane** is the combinaison of *two parallel plane refracting interfaces* (air/glass, then glass/air), separated by the thickness of the pane.
-\- a **lens** is composed of *two consecutive curved (often spherical) refracting interfaces* (air/glass, then glass/air) that are rotational symmetrical around a same axis. - -#### Refracting interface versus refracting surface - -!!!! *DIFFICULT POINT* : One plane or spherical refracting interface has two different optical behaviors for image formation, -is characterized by two different sets of parameters, depending of the direction of the light propagation. -!!!! -!!!!Consider a plane interface (a thick window whose thickness and effect can be neglected) separating air and water, -and two twins (Thompson and Thomson) at equal distances on both sides of the interfaces (Fig. 2a). -!!!! -!!!! ![](plane-refracting-surface-1.jpg)
-!!!! Fig. 2a : The situation is not symmetrical. -!!!! -!!!! * When Thompson (in air) looks at Thomson (in water), the light propagates from Thomson to Thompson’s eyes. -The fact is that Thompson sees the image of his brother closer than the real position of his brother (Fig. 2b) -!!!! -!!!! ![](plane-refracting-surface-2.jpg)
-!!!! Fig. 2b. Thompson sees his brother closer than his real position in water. -!!!! -!!!! * In the opposite situation, when Thomson (in water) looks at Thompson (in air), -the light propagates from Thompson to Thomson’s eyes. -And the fact is that Thomson sees the image of his brother farther away than his real position (Fig. 2c)
-!!!! (Strictly speaking, the eye of a fish should be considered in this situation, eyes well adapted to the vision in water, -and in direct contact with water. If not, we should consider that the Thompson’s dive mask is filled with water, -to have Thomson’s eyes in contact with water and not add another water/air refracting interface -(that of the dive mask) on the path of the light). -!!!! -!!!! ![](plane-refracting-surface-3.jpg)
-!!!! Fig. 2c. Thomson sees in brother farther than his real position in air. -!!!! -!!!! All this can be predicted and calculated, but this example shows that this air/water plane -refracting interface corresponds to two different plane refracting surfaces :
-!!!! -!!!! * First case , refracting surface such as :
-!!!! \- refracting index of the medium of the incident light : $n_{inc} = n_{water} = 1.33$
-!!!! \- refracting index of the medium of the emergent light : $n_{eme} = n_{air} = 1$
-!!!! -!!!! *¨ Second case , refracting surface such as :
-!!!! \- refracting index of the medium of the incident light : $n_{inc} = n_{air} = 1$
-!!!! \- refracting index of the medium of the emergent light : $n_{eme} = n_{water} = 1.33$. -!!!! - -#### Difference in terminology between Spanish, French and English - -!!!! *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 measure 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._ - - -#### Non stigmatism of spherical refracting surfaces - -Ray tracing study of a **spherical refracting surface** : - - -* **At each impact point** of the rays upon the spherical refracting surface, the **Snell-Descartes relation applies**. - -![](dioptre-spherique-snell-law.png)
- -* A spherical refracting surface is **not stigmatic** : The *rays (or their extensions)* originating *from a same object point* and that emerge from the surfac egenerally *do not converge towards an image point*. - -![](dioptre-spherique-non-stigmatique-1.png)
- -* **If we limit the aperture** of the spherical refracting surface so that only the rays -meeting the surface near the vertex are refracted through the surface. - -![](dioptre-spherique-non-stigmatique-2.png)
- -* **and if** the object points remain close enough to the optical axis, so that the **angles of -incidence and refraction remain small**, then for each object point an image point can be almost -defined, and therefore the spherical refracting surface becomes *quasi-stigmatic*. - -![](dioptre-spherique-gauss-conditions.png)
- - -#### Gauss conditions / paraxial approximation and quasi-stigmatism - -When spherical refracting surfaces are used under the following conditions, named **Gauss conditions** :
-\- The *angles of incidence and refraction are small*
-(the rays are slightly inclined on the optical axis, and intercept the spherical surface in the vicinity of its vertex),
-then *the spherical refracting surfaces* can be considered *quasi-stigmatic*, and therefore they can be used to build optical images. - -Mathematically, when an angle $`\alpha`$ is small $`\alpha < or \approx 10 ^\circ`$, the following approximations can be made :
-$`sin(\alpha) \approx tan (\alpha) \approx \alpha`$, and $`cos(\alpha) \approx 1`$. - -*Geometrical optics limited to Gaussian conditions* is called *Gaussian optics* or *paraxial optics*. - - -#### Thin spherical refracting surface - -We call **thin spherical refracting surface** a spherical refracting surface *used in the Gauss conditions*. - - -### How is modeled a spherical refracting surface in paraxial optics ? - - -#### Characterization of a spherical refracting surface - -* 2 distincts points : **vextex S** and **center of curvature C** on the optical axis, -which defines $`\overline{SC}`$ : algebraic distance between vertex S and center C of curvature on optical axis. - -* 2 refractive index values :
-\- **$`n_{inc}`$ : refractive index of the medium of the incident light**.
-\- **$`n_{eme}`$ : refractive index of the medium of the emergent light**. - -* 1 arrow : indicates the *direction of light propagation* - -![](dioptre-1.gif) - - -#### Analytical study - - -* **Thin spherical refracting surface equation** = **conjuction equation** for a spherical refracting surface

-**$`\dfrac{n_{eme}}{\overline{SA_{ima}}}-\dfrac{n_{inc}}{\overline{SA_{obj}}}=\dfrac{n_{eme}-n_{inc}}{\overline{SC}}`$**   (equ.1) - -* **Transverse magnification expression**

- **$`\overline{M_T}=\dfrac{n_{inc}\cdot\overline{SA_{ima}}}{n_{eme}\cdot\overline{SA_{obj}}}`$** -   (equ.2)

-You know $`\overline{SA_{obj}}`$, $`n_{inc}`$ and $`n_{eme}`$, you have previously calculated $`\overline{SA_{ima}}`$, so you 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 :*   $`\dfrac{n_{fin}}{\overline{SA_{ima}}}-\dfrac{n_{ini}}{\overline{SA_{obj}}}=0`$    (equ.3) -! -! * *transverse magnification equation :*   $`\dfrac{n_{ini}\cdot\overline{SA_{ima}}}{n_{fin}\cdot\overline{SA_{obj}}}`$ -   (equ.2, unchanged)

-! but (equ.3) gives $`\dfrac{\overline{SA_{ima}}}{\overline{SA_{obj}}}=\dfrac{n_{inc}}{n_{eme}}`$.
-! Copy this result into (equ.2) leads to $`\overline{M_T}=+1`$. - - -#### Graphical study - -##### 1 - Determining object and image focal points - -Positions of object focal point F and image focal point F’ are easily obtained from the conjunction equation (equ. 1). - -* Image focal length $`\overline{OF'}`$ : $`\left(|\overline{OA_{obj}}|\rightarrow\infty\Rightarrow A_{ima}=F'\right)`$
-    (equ.1)$`\Longrightarrow\dfrac{n_{eme}}{\overline{SF'}}=\dfrac{n_{eme}-n_{inc}}{\overline{SC}}`$ -$`\Longrightarrow\overline{SF'}=\dfrac{n_{eme}\cdot\overline{SC}}{n_{eme}-n_{inc}}`$   (equ.4) - -* Object focal length $`\overline{OF}`$ : $`\left(|\overline{OA_{ima}}|\rightarrow\infty\Rightarrow A_{obj}=F\right)`$
-    (equ.1) $`\Longrightarrow-\dfrac{n_{inc}}{\overline{SF}}=\dfrac{n_{eme}-n_{inc}}{\overline{SC}}`$ -$`\Longrightarrow\overline{SF}=-\dfrac{n_{inc}\cdot\overline{SC}}{n_{eme}-n_{inc}}`$   (equ.5) - -!!!! *ADVISE* :
-!!!! Memory does not replace understanding. Do not memorise (equ.4) and (equ.5), but understand -!!!! the definitions of the object and image focal points, and know how to find these two equations -!!! from the conjuction equation for a spherical refracting surface. -!!!! - -! *NOTE 1* :
-! An optical element being convergent when the image focal point is real, -! so when $`\overline{OF}>0`$ (with optically axis positively oriented in the direction of the light propagation), -! you can deduce from (equ.4) that is spherical refracting surface is convergent if and only if its center -! of curvature C is in the mmedium of highest refractive index. -! - -##### 2 - Thin spherical refracting surface representation - -* **Optical axis = revolution axis** of the refracting surface, positively **oriented** in the direction of -propagation of the light. - -* Thin spherical refracting surface representation :

-\- **line segment**, perpendicular to the optical axis, centered on the axis with symbolic -**indication of the direction of curvature** of the surface at its extremities.

-\- **vertex S**, that locates the refracting surface on the optical axis.

-\- **nodal point C = center of curvature**.

-\- **object focal point F and image focal point F’**. - -! *NOTE 2*
-! The direction of the curvature does not presume the convergent or divergent character -! of the diopter. It also depends on the refractive index values on each side of the spherical -! refracting surface. look at what happens to the incident ray parallel to the optical axis -in Figures 3 and 4, and 5 and 6 below, and review NOTE 1. -! - -#### Examples of graphical situations, with analytical results to train - -!!!! *IMPORTANT* :
-!!!! Even for only one of the following figures, the real or virtual character of the -!!!! image may depend on the position of the object. This paragraph is only for you -!!!! to understand how to determine the 3 rays that determine the image. It is -!!!! important not to memorize these figures, which would be limiting, misleading -!!!! and without interest. -!!!! -!!!! All the useful numerical values are given for each figure, making it possible -!!!! also to check that you master the analytical study of each presented case. -!!!! - - - -* with **real objects** - -![](thin-spherical-surface-1.png)
-Fig. 4. - -![](thin-spherical-surface-2.png)
-Fig. 5. - -![](thin-spherical-surface-3.png)
-Fig. 6. - -![](thin-spherical-surface-4.png)
-Fig. 7. - -* with **virtual objects** - -![](thin-spherical-surface-5.png)
-Fig. 8. - -![](thin-spherical-surface-6.png)
-Fig. 9. - -![](thin-spherical-surface-7.png)
-Fig. 10. - -![](thin-spherical-surface-8.png)
-Fig. 11. -