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12.temporary_ins/65.geometrical-optics/50.simple-elements/20.mirror/20.overview/cheatsheet.en.md

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+--
+title: 'new course  : overview'
+published: true
+routable: true
+visible: false
+lessons:
+   slug: simple-optical-elements
+   order: 2
+---
+
+!!!! *LESSON  UNDER  CONSTRUCTION :* <br>
+!!!! 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>
+!!!! Working document intended only for the pedagogical team.
+
+<!--MetaData : ... -->
+
+---------------------
+
+### The mirror
+
+#### What is a mirror ?
+
+##### Objective
+
+* initial : to **reflect** light, to **focus or disperse light**.
+* Ultimate : to **realize images**, alone or as part of optical instruments.
+
+##### Physical principle
+
+* uses the **phenomenon of reflection**, described by the law of reflection.
+
+##### Constitution
+
+*  Usually plane or curved (spherical for the most simple to realize,
+parabolic or elliptical) **surface, highly polished** so that its surface 
+state deviates from its theoretical form of less than $`\lambda/10`$ at each point
+of its surface ($`\lambda`$ being the wavelength in vacuum of the light to be reflected).
+To increase the reflectivity of the mirror (percentage of reflected light 
+intensity per the total of the incident light intensity), the surface is
+**most often metallized**.
+
+##### Interest in optics
+
+* **One of the most importante simple optical component** that is used *alone or combined in a series in most optical instruments* :
+some telephotos, reflecting telescopes.
+
+#### Why to study plane and spherical mirrors?
+
+* **Plane and spherical mirrors** are the *most technically easy to realize*, 
+so they are the *most common and cheap*.
+* In paraxial optics, the optical properties of a **plane mirror** are those 
+of a *spherical mirror whose radius of curvature tends towards infinity*.
+Plane mirror, concave and convex spherical mirror
+
+![](plane-concave-convex-mirrors.png)<br>
+Fig. 1. a) plane    b) concave    c) convex   mirrors
+
+#### Are plane and spherical mirrors stigmatic?
+
+##### Perfect stigmatism of the plane mirror
+
+* A plane mirror is **perfectly stigmatic**.
+* Object and image are symmetrical on both side of the surface of the plane mirror.<br>
+$`\Longrightarrow`$ A real object gives a virtual image.<br> 
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; nbsp;&nbsp;A virtual object gives a real image.
+
+##### Non stigmatism of the spherical mirror
+
+* In each point of the spherical mirror, the law of reflection applies.
+* A spherical mirror is not stigmatic: The rays (or their extensions) coming from an object point generally do not converge towards an image point (see Fig. 2.)
+* A spherical mirrors with a limited aperture (see the angle $`\alpha`$ (rad) which is reduced on Fig. 3. and 4.) and used so that 
+angles of incidence remain small (see Fig. 4.) become quasi-stigmatic.
+
+![](spherical-mirror-rays-stigmatism-1000-1.jpg)<br>
+Fig. 2. Non stigmatism of a convexe mirror.
+
+![](spherical-mirror-rays-stigmatism-1000-2.jpg)<br>
+Fig. 3. But when we limit the aperture of the mirror
+
+![](spherical-mirror-rays-stigmatism-1000-3.jpg)<br>
+Fig. 4 . and limit the conditions of use to small angles of incidence, then a image point can almost be defined : the mirror becomes
+quasi-stigmatic.
+
+
+##### Gauss conditions / paraxial approximation and quasi-stigmatism
+
+* When spherical mirrors are used under the following conditions, named **Gauss conditions** :<br>
+\- The *angles of incidence are small*<br>
+(the rays are slightly inclined on the optical axis, and intercept the spherical mirror in the 
+vicinity of its vertex),<br>
+then the spherical mirrors can be considered *quasi- stigmatic*, and therefore they 
+*can be used to build optical images*.
+
+* Mathematically, when an angle $`i`$ is small ($`i < or \approx 10 ^\circ`$), the following 
+approximations can be made :<br>
+$`sin(i) \approx  tan (i) \approx i`$ (rad), et $`cos(i) \approx 1`$.
+
+* Geometrical optics limited to Gaussian conditions is called **Gaussian optical** or **paraxial optics**.
+ 
+#### The thin spherical mirror (paraxial optics)
+
+* We call **thin spherical mirror** a *spherical mirror used in the Gauss conditions*.
+
+##### Analytical study (in paraxial optics)
+
+* **Spherical mirror equation** = **conjuction equation** for a spherical mirror :<br><br>
+$`\dfrac{1}{\overline{SA_{ima}}}+\dfrac{1}{\overline{SA_{obj}}}=\dfrac{2}{\overline{SC}}`$&nbsp;&nbsp;(equ.1)
+
+* **Transverse magnification expression** :<br><br>
+$`\overline{M_T}=-\dfrac{\overline{SA_{ima}}}{\overline{SA_{obj}}}`$&nbsp;&nbsp;(equ.2)
+
+You know $`\overline{SA_{obj}}`$ , calculate $`\overline{SA_{ima}}`$ using (equ. 1) 
+then $`\overline{M_T}`$ with (equ.2), and deduce $`\overline{A_{ima}B_{ima}}`$.
+
+! *USEFUL 1* :<br>
+! The conjunction equation and the transverse magnification equation for a plane mirror 
+! are obtained by rewriting these two equations for a spherical mirror in the limit when
+! $`|\overline{SC}|\longrightarrow\infty`$.
+! Then we get for a plane mirror : $`\overline{SA_{ima}}= - \overline{SA_{obj}}`$ and $`\overline{M_T}=+1`$.
+
+! *USEFUL 2* :<br>
+! *You can find* the conjunction and the transverse magnification *equations for a plane or spherical mirror as well as for a plane refracting surface directly from 
+! those of the spherical refracting surface*, with the following assumptions :<br>
+! - to go from refracting surface to mirror : $`n_{eme}=-n_{inc}`$<br>
+! (to memorize : medium of incidence=medium of emergence, therefor same speed of light, but direction
+! of propagation reverses after reflection on the mirror)<br>
+! - to go from spherical to plane : $`|\overline{SC}|\longrightarrow\infty`$.
+! Then we get for a plane mirror :<br>
+! $`\overline{SA_{ima}}= - \overline{SA_{obj}}`$ and $`\overline{M_T}=+1`$
+
+##### 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 $`\overline{OF'}`$ : $`\left(|\overline{OA_{obj}}|\rightarrow\infty\Rightarrow A_{ima}=F'\right)`$<br><br>
+(equ.1) $`\Longrightarrow\dfrac{1}{\overline{SF'}}=\dfrac{2}{\overline{SC}}\Longrightarrow\overline{SF'}=\dfrac{\overline{SC}}{2}`$
+
+* Object focal length $`\overline{OF}`$ : $`\left(|\overline{OA_{ima}}|\rightarrow\infty\Rightarrow A_{obj}=F\right)`$<br><br>
+(equ.2) $`\Longrightarrow\dfrac{1}{\overline{SF}}=\dfrac{2}{\overline{SC}}\Longrightarrow\overline{SF}=\dfrac{\overline{SC}}{2}`$
+
+*2 - Thin spherical mirror representation*
+
+* **Optical axis = revolution axis** of the mirror, positively **oriented** in the direction of propagation of the incident light.
+
+* Thin spherical mirror equation :<br><br>
+\-**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, and *dark or hatched area on the non-reflective 
+side* of the mirror.<br><br>
+\-**vertex S**, that indicates the position of the mirror along the optical axis;<br><br>
+\-**nodal point C = center of curvature**.<br><br>
+\-**object focal point F** and **image focal point F’**.
+
+##### Examples of graphical situations, with analytical results to train
+
+<!--a supprimer
+[Click here for geogebra animation](https://www.geogebra.org/m/jwgy9q7z)
+-->
+
+* with **real objects** 
+
+![](Thin-spherical-mirror-InfAC-1000.jpg)<br>
+Fig. 5.  Concave mirror with object between infinity and C
+
+![](Thin-spherical-mirror-CAF-1000.jpg)<br>
+Fig. 6.  Concave mirror with object between C and F/F’
+
+![](Thin-spherical-mirror-FAS-1000.jpg)<br>
+Fig. 7.  Concave mirror with object between F/F’ and S
+
+![](Thin-spherical-mirror-InfAS-1000.jpg)<br>
+Fig. 8.  Convex mirror
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