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