From 83f1ae197c6dc1ba719153ce52d2cd17771475ec Mon Sep 17 00:00:00 2001 From: Claude Meny Date: Sat, 20 Mar 2021 11:11:16 +0100 Subject: [PATCH] Add new file --- .../10.dioptre/20.overview/cheatsheet.en.md | 272 ++++++++++++++++++ 1 file changed, 272 insertions(+) create mode 100644 12.temporary_ins/65.geometrical-optics/50.simple-elements/10.dioptre/20.overview/cheatsheet.en.md diff --git a/12.temporary_ins/65.geometrical-optics/50.simple-elements/10.dioptre/20.overview/cheatsheet.en.md b/12.temporary_ins/65.geometrical-optics/50.simple-elements/10.dioptre/20.overview/cheatsheet.en.md new file mode 100644 index 000000000..6192074d3 --- /dev/null +++ b/12.temporary_ins/65.geometrical-optics/50.simple-elements/10.dioptre/20.overview/cheatsheet.en.md @@ -0,0 +1,272 @@ +--- +title: 'Spherical refracting surface : overview' +media_order: dioptre-1.gif +published: false +visible: false +lessons: + - slug: simple-optical-elements + - order: 1 +--- + +!!!! *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. + + + +Very imperfect courses, to be re-designed from the beginning + +---------------------- + +### 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. +