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title: 'Spherical refracting surface : overview' |
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media_order: dioptre-1.gif |
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published: false |
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lessons: |
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- slug: simple-optical-elements |
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- order: 1 |
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--- |
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!!!! *LESSON UNDER CONSTRUCTION :* <br> |
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!!!! 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. <br> |
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!!!! Working document intended only for the pedagogical team. |
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<!--MetaData : ... --> |
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Very imperfect courses, to be re-designed from the beginning |
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---------------------- |
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### What is a refracting interface ? |
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#### Refracting surface : physical description |
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* **Interface separating two transparent media of different refractive indices**. |
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* can be **found in nature** :<br> |
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Examples :<br> |
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\- a **plane refracting surface** : the *flat and quiet surface of a lake*. |
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\- a **spherical refracting surface** : a *fish ball aquarium*. |
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<br> |
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Fig. 1. The spherical refracting interface of a fish ball aquarium. |
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* **appears in the design and modeling of other optical elements** :<br> |
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Examples :<br> |
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\- 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.<br> |
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\- 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. |
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#### Refracting interface versus refracting surface |
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!!!! *DIFFICULT POINT* : One plane or spherical refracting interface has two different optical behaviors for image formation, |
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is characterized by two different sets of parameters, depending of the direction of the light propagation. |
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!!!! |
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!!!!Consider a plane interface (a thick window whose thickness and effect can be neglected) separating air and water, |
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and two twins (Thompson and Thomson) at equal distances on both sides of the interfaces (Fig. 2a). |
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!!!! |
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!!!! <br> |
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!!!! Fig. 2a : The situation is not symmetrical. |
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!!!! |
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!!!! * When Thompson (in air) looks at Thomson (in water), the light propagates from Thomson to Thompson’s eyes. |
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The fact is that Thompson sees the image of his brother closer than the real position of his brother (Fig. 2b) |
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!!!! |
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!!!! <br> |
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!!!! Fig. 2b. Thompson sees his brother closer than his real position in water. |
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!!!! |
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!!!! * In the opposite situation, when Thomson (in water) looks at Thompson (in air), |
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the light propagates from Thompson to Thomson’s eyes. |
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And the fact is that Thomson sees the image of his brother farther away than his real position (Fig. 2c)<br> |
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!!!! (Strictly speaking, the eye of a fish should be considered in this situation, eyes well adapted to the vision in water, |
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and in direct contact with water. If not, we should consider that the Thompson’s dive mask is filled with water, |
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to have Thomson’s eyes in contact with water and not add another water/air refracting interface |
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(that of the dive mask) on the path of the light). |
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!!!! |
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!!!! <br> |
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!!!! Fig. 2c. Thomson sees in brother farther than his real position in air. |
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!!!! |
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!!!! All this can be predicted and calculated, but this example shows that this air/water plane |
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refracting interface corresponds to two different plane refracting surfaces :<br> |
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!!!! |
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!!!! * First case , refracting surface such as :<br> |
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!!!! \- refracting index of the medium of the incident light : $n_{inc} = n_{water} = 1.33$<br> |
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!!!! \- refracting index of the medium of the emergent light : $n_{eme} = n_{air} = 1$<br> |
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!!!! |
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!!!! *¨ Second case , refracting surface such as :<br> |
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!!!! \- refracting index of the medium of the incident light : $n_{inc} = n_{air} = 1$<br> |
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!!!! \- refracting index of the medium of the emergent light : $n_{eme} = n_{water} = 1.33$. |
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!!!! |
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#### Difference in terminology between Spanish, French and English |
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!!!! *BE CAREFUL* :<br> |
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!!!! 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". |
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!!!! 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". |
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!!!! So keep in mind the following scheme : |
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!!!! |
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!!!! refracting surface : *EN : refracting surface* , *ES : superficie refractiva* , *FR : dioptre*.<br> |
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!!!! _A crystal ball forms a spherical refracting surface : un "dioptre sphérique" in French._ |
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!!!! |
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!!!! unit of measure : *EN : dioptre* , *ES : dioptría* , *FR : dioptrie*.<br> |
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!!!! _My corrective lens for both eyes are 4 dioptres : "4 dioptries" in French._ |
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#### Non stigmatism of spherical refracting surfaces |
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Ray tracing study of a **spherical refracting surface** : |
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<!--à supprimer |
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[Click here for geogebra animation](https://www.geogebra.org/material/iframe/id/x4hxqekd)<br> |
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--> |
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* **At each impact point** of the rays upon the spherical refracting surface, the **Snell-Descartes relation applies**. |
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<br> |
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* 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*. |
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<br> |
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* **If we limit the aperture** of the spherical refracting surface so that only the rays |
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meeting the surface near the vertex are refracted through the surface. |
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<br> |
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* **and if** the object points remain close enough to the optical axis, so that the **angles of |
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incidence and refraction remain small**, then for each object point an image point can be almost |
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defined, and therefore the spherical refracting surface becomes *quasi-stigmatic*. |
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<br> |
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#### Gauss conditions / paraxial approximation and quasi-stigmatism |
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When spherical refracting surfaces are used under the following conditions, named **Gauss conditions** :<br> |
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\- The *angles of incidence and refraction are small*<br> |
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(the rays are slightly inclined on the optical axis, and intercept the spherical surface in the vicinity of its vertex),<br> |
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then *the spherical refracting surfaces* can be considered *quasi-stigmatic*, and therefore they can be used to build optical images. |
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Mathematically, when an angle $`\alpha`$ is small $`\alpha < or \approx 10 ^\circ`$, the following approximations can be made :<br> |
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$`sin(\alpha) \approx tan (\alpha) \approx \alpha`$, and $`cos(\alpha) \approx 1`$. |
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*Geometrical optics limited to Gaussian conditions* is called *Gaussian optics* or *paraxial optics*. |
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#### Thin spherical refracting surface |
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We call **thin spherical refracting surface** a spherical refracting surface *used in the Gauss conditions*. |
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### How is modeled a spherical refracting surface in paraxial optics ? |
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#### Characterization of a spherical refracting surface |
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* 2 distincts points : **vextex S** and **center of curvature C** on the optical axis, |
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which defines $`\overline{SC}`$ : algebraic distance between vertex S and center C of curvature on optical axis. |
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* 2 refractive index values :<br> |
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\- **$`n_{inc}`$ : refractive index of the medium of the incident light**.<br> |
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\- **$`n_{eme}`$ : refractive index of the medium of the emergent light**. |
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* 1 arrow : indicates the *direction of light propagation* |
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#### Analytical study |
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* **Thin spherical refracting surface equation** = **conjuction equation** for a spherical refracting surface<br><br> |
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**$`\dfrac{n_{eme}}{\overline{SA_{ima}}}-\dfrac{n_{inc}}{\overline{SA_{obj}}}=\dfrac{n_{eme}-n_{inc}}{\overline{SC}}`$** (equ.1) |
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* **Transverse magnification expression**<br><br> |
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**$`\overline{M_T}=\dfrac{n_{inc}\cdot\overline{SA_{ima}}}{n_{eme}\cdot\overline{SA_{obj}}}`$** |
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(equ.2)<br><br> |
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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 $`\overline{A_{ima}B_{ima}}`$. |
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! *USEFUL* : The conjuction equation and the transverse magnification equation for a plane refracting |
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!surface are obtained by rewriting these equations for a spherical refracting surface in the limit when |
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! |
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! $`|\overline{SC}|\longrightarrow\infty`$.<br> |
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! Then we get *for a plane refracting surface :* |
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! |
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! * *conjuction equation :* $`\dfrac{n_{fin}}{\overline{SA_{ima}}}-\dfrac{n_{ini}}{\overline{SA_{obj}}}=0`$ (equ.3) |
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! |
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! * *transverse magnification equation :* $`\dfrac{n_{ini}\cdot\overline{SA_{ima}}}{n_{fin}\cdot\overline{SA_{obj}}}`$ |
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(equ.2, unchanged)<br><br> |
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! but (equ.3) gives $`\dfrac{\overline{SA_{ima}}}{\overline{SA_{obj}}}=\dfrac{n_{inc}}{n_{eme}}`$.<br> |
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! Copy this result into (equ.2) leads to $`\overline{M_T}=+1`$. |
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#### Graphical study |
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##### 1 - Determining object and image focal points |
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Positions of object focal point F and image focal point F’ are easily obtained from the conjunction equation (equ. 1). |
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* Image focal length $`\overline{OF'}`$ : $`\left(|\overline{OA_{obj}}|\rightarrow\infty\Rightarrow A_{ima}=F'\right)`$<br> |
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(equ.1)$`\Longrightarrow\dfrac{n_{eme}}{\overline{SF'}}=\dfrac{n_{eme}-n_{inc}}{\overline{SC}}`$ |
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$`\Longrightarrow\overline{SF'}=\dfrac{n_{eme}\cdot\overline{SC}}{n_{eme}-n_{inc}}`$ (equ.4) |
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* Object focal length $`\overline{OF}`$ : $`\left(|\overline{OA_{ima}}|\rightarrow\infty\Rightarrow A_{obj}=F\right)`$<br> |
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(equ.1) $`\Longrightarrow-\dfrac{n_{inc}}{\overline{SF}}=\dfrac{n_{eme}-n_{inc}}{\overline{SC}}`$ |
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$`\Longrightarrow\overline{SF}=-\dfrac{n_{inc}\cdot\overline{SC}}{n_{eme}-n_{inc}}`$ (equ.5) |
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!!!! *ADVISE* :<br> |
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!!!! Memory does not replace understanding. Do not memorise (equ.4) and (equ.5), but understand |
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!!!! the definitions of the object and image focal points, and know how to find these two equations |
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!!! from the conjuction equation for a spherical refracting surface. |
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!!!! |
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! *NOTE 1* :<br> |
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! An optical element being convergent when the image focal point is real, |
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! so when $`\overline{OF}>0`$ (with optically axis positively oriented in the direction of the light propagation), |
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! you can deduce from (equ.4) that is spherical refracting surface is convergent if and only if its center |
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! of curvature C is in the mmedium of highest refractive index. |
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! |
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##### 2 - Thin spherical refracting surface representation |
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* **Optical axis = revolution axis** of the refracting surface, positively **oriented** in the direction of |
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propagation of the light. |
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* Thin spherical refracting surface representation :<br><br> |
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\- **line segment**, perpendicular to the optical axis, centered on the axis with symbolic |
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**indication of the direction of curvature** of the surface at its extremities.<br><br> |
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\- **vertex S**, that locates the refracting surface on the optical axis.<br><br> |
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\- **nodal point C = center of curvature**.<br><br> |
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\- **object focal point F and image focal point F’**. |
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! *NOTE 2*<br> |
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! The direction of the curvature does not presume the convergent or divergent character |
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! of the diopter. It also depends on the refractive index values on each side of the spherical |
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! refracting surface. look at what happens to the incident ray parallel to the optical axis |
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in Figures 3 and 4, and 5 and 6 below, and review NOTE 1. |
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! |
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#### Examples of graphical situations, with analytical results to train |
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!!!! *IMPORTANT* :<br> |
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!!!! Even for only one of the following figures, the real or virtual character of the |
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!!!! image may depend on the position of the object. This paragraph is only for you |
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!!!! to understand how to determine the 3 rays that determine the image. It is |
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!!!! important not to memorize these figures, which would be limiting, misleading |
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!!!! and without interest. |
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!!!! |
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!!!! All the useful numerical values are given for each figure, making it possible |
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!!!! also to check that you master the analytical study of each presented case. |
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!!!! |
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<!--a supprimer |
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[Click here for geogebra animation](https://www.geogebra.org/material/iframe/id/gvkqgrpe)<br> |
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--> |
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* with **real objects** |
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<br> |
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Fig. 4. |
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<br> |
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Fig. 5. |
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<br> |
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Fig. 6. |
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<br> |
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Fig. 7. |
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* with **virtual objects** |
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<br> |
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Fig. 8. |
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<br> |
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Fig. 9. |
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<br> |
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Fig. 10. |
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<br> |
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Fig. 11. |
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