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---title: 'The thin lens'media_order: 'Const_lens_conv_point_AapresO.gif,lens-convergent-N2-en.jpeg,Const_lens_conv_point_AentreFO.gif,lens-convergent-N2-es.jpeg,lens-convergent-N2-fr.jpeg,Const_lens_conv_point_AavantF.gif,lens-divergent-N2-es.jpeg,lens-divergent-N2-fr.jpeg,lens-divergent-N2-en.jpeg,diverging-thin-lens-representation.jpeg,converging-thin-lens-representation.jpeg'---
### What is a lens ?
#### Objective
* initial : to **focuse or disperse the light**.* ultimate : to **realize images**, alone or as part of optical instruments.
#### Physical principle
* **uses the refractive phenomenon**, described by the Snell-Descartes' law.
#### Constitution
* Piece of **glass, quartz, plastic** (for visible and near infrared and UV).* **Rotationally symmetrical**.* **2 polished surfaces** perpendicular to its axis of symmetry, **either or both curved** (and most often spherical).
<!--image to build : a thin lens-->
#### Interest in optics : thin lenses
* **Thin lens** : *thickness << diameter** Thins lens : **most important simple optical element** that is *used alone or combined in serie in most optical instruments* : magnifying glasses, microscopes, tele and macro objectives, camera, refracting telescopes.
<!--image to build N1 ou N2 : a composition :
upper medium : a unic thin lensupper part towards utilization of a unique lens : magnigfying glass and eyeglasseslower medium : small serie of centered naked lenseslower part toward utilization of combined lenses : macroscope, camera (apparatus and objective of a cellular), refracting telescope, teleopbjective-->
### Modeling a thin lens surrounded by air, gaz or vaccum.
#### Why modeling ?
* To **understand, calculate and predict images** of objects given by thin lenses
<!--picture when we see the object, the lens and the image-->
##### Why surrounded by air, gaz or vaccum?
* **In most optical instruments**, lenses are *surrounding by air*.* **air, gaz and vaccum** have refractive index values in the range "$1.000\pm0.001$, and can be approximated by *$n_{air}=n_{gaz}=n_{vaccum}=1$*<br>$\Longrightarrow$ same optical behavior in air, gaz and vacuum.
#### Types and characterization of thin lenses
**Convergent** = **converging** = **convexe** = **positive** lenses
* Characterized by :<br>\- **Focal lenght** (usually in cm) always >0 *+* adjective "**converging**"<br> or<br>\- Its **image focal length** $f'$ (in *algebraic value*, usually in cm), that is *positive $f'>0$*.<br> or<br>\- Its **vergence** $V$ (in ophtalmology) that is *positive $V>0$*,<br>with $V (\delta)=\dfrac{1}{f'(m)}$ ($f'$ being expresssed in m "meter" and $V$ in $\delta$ "dioptre", so $\delta=m^{-1}$).<br> **Divergent** = **diverging** = **concave ** = **negative** lenses
* Characterized by :<br>\- **Focal lenght** (usually in cm) always >0 *+* adjective "**diverging**"<br> or<br>\- Its **image focal length** $f'$ (in *algebraic value*, usually in cm), that is *negative $f'<0$*.<br> or<br>\- Its **vergence** $V$ (in ophtalmology) that is *negative $V<0$*,<br>with $V (\delta)=\dfrac{1}{f'(m)}$ ($f'$ being expresssed in m "meter" and $V$ in $\delta$ "dioptre", so $\delta=m^{-1}$).<br>
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#### What physical framework, model and technics ?
* _Framework : Geometrical Optics = Light rays optics $\longrightarrow$ foothills stage_.
* _Model : paraxial model = gaussian model $\longrightarrow$ foothills stage_.
* Model splits in *two different technics (but equivalent)* :<br> **graphical modeling** AND **analytical modeling**
* *Differences between model predictions and experimental observations* : ** optical aberrations** (_under control, minimized and negligeable in optical instruments_).-->
### Analytical modeling
(_for thin lens surrounded by air, gaz or vaccum_)
##### Thin lens equation
**$\dfrac{1}{\overline{OA'}}-\dfrac{1}{\overline{OA}}=V=-\dfrac{1}{\overline{OF}}=\dfrac{1}{\overline{OF'}}$**
##### Transverse magnification expression
**$M_{T-thinlens}=\dfrac{\overline{OA'}}{\overline{OA}}$**
### Graphical modeling
#### Thin lens representation
* **optical axis** = *revolution axis* of the lens, positively *oriented* in the direction of propagation of the light (_from the object towards the lens_).
* **thins lens representation** :<br><br>\- *line segment*, perpendicular to optical axis, centered on the axis with symbolic *indication of the lens shape* at its extremities (_convexe or concave_).<br><br>\- **S = C = O** : vertex S = nodal point C = center O of the thin lens $\Longrightarrow$ is used point O.<br><br>\- *point O*, intersection of the line segment with optical axis.<br><br>\- *object focal point F* and *image focal point F'*, positioned on the optical axis symmetrically with respect to the point O ($f=-f'$) at algebraic distances $\overline{OF}=f$ and $\overline{OF'}=f'$.<br><br> \- *object focal plane (P)* and *image focal plane (P')*, planes perpendicular to the optical axis at respectively points $F$ and $F'$.
<br>_Converging thin lens representation : $\overline{OF}<0$ , $\overline{OF'}>0$ and $|\overline{OF}|=|\overline{OF'}|$_
<br> _Divverging thin lens representation : $\overline{OF}>0$ , $\overline{OF'}<0$ and $|\overline{OF}|=|\overline{OF'}|$_ #### Determining conjugate points :
##### Converging thin lens
**Towards geogebra animations** :<br>\- Graphical construction<br>[Click here for geogebra animation](https://www.geogebra.org/material/iframe/id/zqwazusz)<br>\- Graphical construction and light pencils <br>[Click here for geogebra animation](https://www.geogebra.org/material/iframe/id/wkrw5qgm)<br>\- Graphical construction and transverse magnification<br>[Click here for geogebra animation](https://www.geogebra.org/material/iframe/id/xwbwedft)<br>
* **Point source located between ∞ et F**
* **Point source located between F et O**
* **Virtual object point** (will be seen at level foothills, to remove from here).
##### Diverging thin lens
(to be implemented)
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