* Local **spherical interface separating two transparent media of different refractive indices**.
* **Interface separating two transparent media of different refractive indices**.
* can be **foud in nature** : examples : when we look through the flat and quiet surface of a lake*, or when we look at something inside a fish ball aquarium.
* can be **found in nature** :<br>
Examples :<br>
\- a **plane refracting surface** : the *flat and quiet surface of a lake*.
\- a **spherical refracting surface** : a *fish ball aquarium*.
Fig. 1. The spherical refracting interface of a fish ball aquarium.
* **appears in the design and modeling of other optical elements** : examples:<br>
\- a glass window pane is the combinaison of two parallel plane refracting interfaces (air/glass, then glass/air) that are rotationaly symmetrical around a same axis.
* **appears in the design and modeling of other optical elements** :<br>
Examples :<br>
\- 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>
\- 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).
!!!!
!!!! <br>
!!!! 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)
!!!!
!!!! <br>
!!!! 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)<br>
!!!! (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).
!!!!
!!!! <br>
!!!! 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 :<br>
!!!!
!!!! * First case , refracting surface such as :<br>
!!!! \- refracting index of the medium of the incident light : $n_{inc} = n_{water} = 1.33$<br>
!!!! \- refracting index of the medium of the emergent light : $n_{eme} = n_{air} = 1$<br>
!!!!
!!!! *¨ Second case , refracting surface such as :<br>
!!!! \- refracting index of the medium of the incident light : $n_{inc} = n_{air} = 1$<br>
!!!! \- refracting index of the medium of the emergent light : $n_{eme} = n_{water} = 1.33$.
!!!!
#### Non stigmatism of spherical refracting surfaces
<br>
Fig. 3. : In each point of the spherical refracting surface, the Snell-Descartes relation applies.
<br>
Fig. 4. : A spherical refracting surface is not stigmatic: The rays (or their extensions) coming
from an object point generally do not converge towards an image point.
<br>
Fig. 5a. : If we limit the opening of the spherical refracting surface so that only the rays
meeting the surface near the vertex are refracted through the surface.
<br>
Fig. 5b. : and if the object points remain close 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.
#### Gauss conditions / paraxial approximation and quasi-stigmatism
When spherical refracting surfaces are used under the following conditions, named **Gauss conditions** :<br>
\- All *incident rays lie close to the optical axis*<br>
\- The *angles of incidence and refraction are small*<br>
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 $`alpha <or \approx10^\circ`$,thefollowingapproximationscanbemade:<br>
$`sin(\alpha) \approx tan (\alpha) \approx \alpha`$, and $`cos(\alpha) \approx 1`$.
*Geometrical optics limited to Gaussian conditions* is called *Gaussian optics* or *paraxial optics*.
#### 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 :<br>
\- **$`n-{inc} : refractive index of the medium of the incident light**.<br>
\- **$`n-{eme} : refractive index of the medium of the emergent light**.
* 1 arrow : indicates the *direction of light propagation*
#### Thin spherical refracting surface
We call **thin spherical refracting surface** a spherical refracting surface *used in the Gauss conditions*.
