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| new course : overview | objeto-fisico-imagen-virtual-haz-3-rayos-bb.jpg,objeto-fisico-imagen-virtual-pantalla-bb.jpg,objeto-imagen-posicion-tamano-bb.jpg,objeto-imagen-bb.jpg,objeto-imagen-3-rayos-bb.jpg,plano-imagen-real-convergencia-bb.jpg,plano-imagen-real-convergencia-pantalla-bb.jpg,plano-objeto-fisico-bb.jpg,vision-object-b.jpg,imagerie-rays-optics.gif,vision-object-image-2.jpg,vision-image.jpg,axe_opt.gif,plano-objeto-plan-image-bb.jpg,objeto-imagen-haz-3-rayos-bb.jpg,objeto-fisico-imagen-virtual-haz-3-rayons-pantalla-bb.jpg,plano-objeto-plano-imagen-real-pantalla-bb.jpg,sym_rev_2.gif,imagerie-rayons-lens-1.jpg,imagerie-rayons.jpg,imagerie-rays-optics.jpg |
What is optical imaging ?
Objective
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Optical $\Longrightarrow$ visible range + near infrared + near UV
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to realize optical images of physical objects by the use of simple optical elements which can be combined in optical systems to form optical instruments.
Physical object
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Physical object : large (compared to $\lambda_{optical}$) *volume* of matter (liquid or solid) whose *external surface breaks down in a huge number of microscopic surfaces*.
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Physical object point = point source :
- microscopic surface part of the overall surface of the physical object.
- emits or diffuses light in all direction outside the volume. That means in equivalent ways : emits spherical waves (wave optics), emits light rays (rays optics), emits photons (photons optics) that diverge from the object point. -
Point source pencil of light = bundle of rays : part of the light emitted by a point source that intercept an optical system or pass through a limiting aperture.
- naked eyes = direct vision of an object : pencils from all visible point sources of the object intercept the pupil of my eyes.
Direct vision : the pencils of each visible point souces intercept the iris of my eyes
Optical image
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Object seen from a specific angle of view, whose principal direction named line of sight is (when oriented positively in direction of the eye) the optical axis of the imaging system.
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Part of the light that diverge from any point source of the seen object, has to converge back in a new location in space named image point.
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Image : set of all the image points.
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form consistency between initial object and its image, but shape distortions may appear.
Image vision : an optical imager (rectangle) has modified the incident pencils. Only pencils from image points enter my eyes. I don't see anymore the initial extended object.
Optical imager and basis physical principles.
** Imager** :
- intercept part of the light emitted or diffused by the physical object.
- modify the pencils of light from each object point to converge them back into a new location in space.
** Optical imager** :
- create a real three dimensional image of the extended object surface oriented towards the imager.
- use refraction and/or reflexion phenomena.
- Imagers can be : individual thin simple optical elements or centered optical systems
By use of refraction and/or reflexion phenomena, an optical imager modifies all incident pencils to converge towards image points.
Thin simple optical element
- often has a symetry of revolution about an axis.
simple optical element : refracting or reflecting element, rotationaly symmetrical around an axis
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Thin $\Longrightarrow$ diameter $\gg$ thickness or depth)
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Simple : surfaces of simple optical element are plane or spherical
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Thin optical elements studied are :
- plane or thin curved refracting surfaces.
- plane or thin curved mirrors.
- thin lenses.
Centered optical systems
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Combination of thin simple optical elements centered on a common axis that becomes the optical axis of the system (when positively oriented in direction of the incident light on the system).
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Interest : can be characterized as a whole.
optical system : combination of thin simple optical elements, centered on a same optical axis
What physical framework to describe optical imaging ?
From idealization to physical and usefull reality
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Point : mathematical concept of vanishing volume.
has a location in space, but no extension, no orientation. -
Image point physical meaning : the pencil emerging from the imager focuses on a so small volume that its extension can be neglected.
- extension of the volume can not be resolved naked eye vision.
- surface illumated in the sensor plane is below the size of a pixel. -
perfectly stigmatic optical system : gives one image point for each object point (don't exist).
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quasi stigmatic optical system : under certain conditions of use a set of optical elements is quasi-stigmatic and so becomes an imager.
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Optical imager = quasi-stigmatic optical element or system used to give images.
Framework of light rays optics.
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We use the concept of light rays, coming from light rays optics
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light rays optics = geometrical optics
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A light pencil that diverges from a point source, is modified by an imager and converges back towards a image point.
equivalent to
- All rays emerging from a point source are deviated by the imager and cross back on the conjugated image point.
$\Longrightarrow$ knowledge of only two different rays from a point source through the imager is sufficient to determine image position. -
For any object point of any imager, trajectories of 3 specific rays will be specified.
Light rays optics : 3 specific ray are specified (2 are sufficient) to locate the image point of any object point (in this figure, the thin imager is a thin lens)
How is modeled and characterized a thin simple optical element in light rays optics?
Thin simple optical element
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Thin simple optical element = optical element whose thickness can be neglected in front of diameter $\Longrightarrow$ represented by a plane (Elt).
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Thin simple optical elements* are **rotationally symetrical around an axis** :
$\Longrightarrow$ same optical behavior in all plane containing the axis of symetry.
$\Longrightarrow$ *object point and conjugate image point belong to a same plane that contains the axis of symetry.
$\Longrightarrow$ working in the **sectional view** corresponding to that plane is **sufficient**. -
Representation on a sectional view containing the optical axis: straight line [Elt] perpendicular to the optical axis.
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Location with its vertex S : intersection of [Elt] with the optical axis.
Optical behavior
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- At each object point B corresponds to a unic image point B'.
- At each image point B 'corresponds a unic object point B :
$\Longrightarrow$ B and B' are conjugate points.
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All object point (A ; B, C, ...) of an object plane (PO) perpendicular to the optical axis have conjugated points images (A' ; B', C', ...) in a same image plane (PI) perpendicular to the optical axis :
$\Longrightarrow$ (PO) and (PI) are conjugate planes.
Coordinates to locate object and image points
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Each point (object and image) is projected perpendicularly on the optical axis :
- point source B $\Longrightarrow$ point A on the optical axis.
- conjugate image point B' $\Longrightarrow$ point A' on the optical axis. -
Distance of a point from thin imager : algebraic distance between imager vertex and point projection.
- distance of point source B from imager [Elt] : $\overline{SA}$
- distance of conjugate image point B' from imager [Elt] : $\overline{SA'}$ -
Distance of a point from optical axis : algebraic distance between point projection and point itself.
- distance of point source B from axis [Elt] : $\overline{AB}$
- distance of conjugate image point B' from axis : $\overline{A'B'}$
Characterization of a thin simple optical element
*4 points ** * located on the optical axis that characterize optical behavoir : S , C, F and F
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S : vertex of the thin imager : indicates its position in space, and on the optical axis.
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C : nodal point : by definition all rays (or its extension) that pass through nodal point C has unchanged direction when leaving the thin optical element. Position characterizes by its algebraic distance from vertex S : $\overline{SC}$.
The nodel point is a center (whose exact physical meaning depends of the type of thin simple optical element) -
F' : image focal point = second focal point = image focus : incident rays (or their extensions) parallel to the optical axis (or their extensions), after leaving the thin imager, pass through F'.
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F : object focal point = first focal point = object focus : incident rays (or their extensions) passing through F leave (or their extensions) the imager parallel to the optical axis.
which define 3 important planes, perpendicular to the optical axis
- (Elt) : representation of the thin imager interface, contains S : incident rays (or their extension) change of direction when passing through (Elt).
- (P') : image focal plane = second focal plane , contains F' : all incident rays parallel to each other originated to a unic point at infinity leave (Elt) to converge (or their extension) on a unic image point B' located in (P'). Location of B' in (P') is the intersection of the ray passing through C with (P').
- (P) : object focal plane = first focal plane , contains F : all incident rays originated from a unic point source B leave (Elt) parallel to each other, given an image point B' located at infinity, in direction of the ray passing through C.
and 2 important algebraic distances
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$f'=\overline{SF'}$ : algebraic distance from thin imager (Elt) to image focal plane (P) :
$f'=\overline{SF'}$ $>0 \Longleftrightarrow$ converging thin imager.
$f'=\overline{SF'}$ $<0 \Longleftrightarrow$ diverging thin imager. -
$f=\overline{SF}$ : algebraic distance from thin imager (Elt) to object focal plane (P).
Important significations of focal planes
Image focal plane :
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Physical sense :
object B at infinity $\Longleftrightarrow |\overline{SB}| \ll |f|$. -
Object at infinity (P) $\Longleftrightarrow$ object in (P'), can be viewed by naked eye whether convergent or divergent optical element.
example : direct vision of the universe through a telescope (telescope is not a thin imager, but same image focal plane defintion)
- Object at infinity (P) $\Longleftrightarrow$ object in (P'), to be captured by an image sensor if convergent optical element.
example : picture taken with a telephoto lens (telephoto lens is not a thin imager, but same image focal plane defintion)
Object focal plane (P)
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Physical sense :
image B' at infinity $\Longleftrightarrow |\overline{SB'}| \ll |f'|$. -
Object in (P) $\Longleftrightarrow$ image at infinity
(examples : Object can be the lightbulb of a lighthouse or a headlight, the film in a film projector)
How to determine the image given by a thin simple optical element ?
Graphical study
Two different scales
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Object and image : tranverse sizes $|\overline{AB}|$, $|\overline{A'B'};;\ll$ distances from optical element along optical axis $|\overline{SA}|$ , $|\overline{SA'}|$.
$\Longrightarrow$ dimensions perpendicular to optical axis $\ll$ dimensions along optical axis -
So Accurate graphical study $\Longrightarrow$ greatly magnify scale perpendicular to optical axis.
Determining of conjugate points
From given point (object or image) 3 specific light rays can be drawn ( only 2 required), whose intersection gives the conjugate point :
- [Ray1] : Incident rays (or their extensions) passing through the object focal point F leave (or their extensions) the imager parallel to the optical axis.
- [Ray2] : Incident rays (or their extensions) parallel to the optical axis pass (or their extensions), after leaving the thin imager, through the image focal point F'.
- [Ray3] : Incident ray (or its extension) passing through point C has unchanged direction when leaving the thin imager.
Consequences from focal points definitions
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Incident rays parallel to each other intersect on a same point in the image focal plane (P').
(location : intersection of (P') with [Ray3]) -
Emergent rays parallel to each other diverge from a same point source located in the object focal plane (P).
(location : intersection of (P) with [Ray3])
Determining the emergent ray corresponding to any incident ray
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deflection of a ray by a thin optical element : local interaction the point of impact (refraction or reflection) between incident ray and thin optical element $\Longrightarrow$ independent of the distance of the point source.
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so consider point source B to infinity $\Longrightarrow$ image point B' would be in the image focal plane (P').
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ray from B (to infinity) passing through point C has unchanged direction $\Longrightarrow$ location of B' in (P').
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Emerging ray (or its extension) = straight line between impact point and B'.
Determining the incident ray corresponding to any emergent ray
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deflection of a ray by a thin optical element : local interaction the point of impact (refraction or reflection) between emergent ray and thin optical element $\Longrightarrow$ independent of the distance of the image point.
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so consider image point B' to infinity $\Longrightarrow$ point source B would be in the object focal plane (P).
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emerging ray towards B (to infinity) passing through point C has unchanged direction $\Longrightarrow$ location of B in (P).
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incident ray (or its extension) = straight line between impact point and B.
Analytical determining of conjugate points
Distance of the conjugate point from thin optical element
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deduced from the thin optical element equation
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thin optical element equation = thin optical element formula
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will be given for :
- thin spherical mirror equation
- thin spherical refracting surface equation
- thin lens equation
(will be demonstrated in level foothills)
Distance of the conjugate point from optical axis
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deduced from the transverse magnification $M_T$.
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transverse magnification = lateral magnification
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Definition : $M_T=\dfrac{\overline{A'B'}}{\overline{A'B'}}$
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$M_T>0$ $\Longleftrightarrow$ erect image
$M_T<0$ $\Longleftrightarrow$ inverted image -
Expression : depends on type of simple optical element :
will be given for :
- thin spherical mirror equation
- thin spherical refracting surface equation
- thin lens equation -
algebraic value of $M_T$ : is a function of $\overline{AB}$ and $\overline{A'B'}$
$\Longrightarrow$ depends on conjugate points locations
$\Longrightarrow$ does not characterized the optical element itself.
How to characterize the action of an imager ?
Characterization of an extended object
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Extended object [AB] or [B$_1$B$_2$], perpendicularly to the optical axis :
- characterized by the algebraic transverse size (distance between its extremities) : $\overline{AB}$ or $\overline{B_1B_2}$
- or characterized by the apparent angle (in algebraic value or not) substended by the object at nodal point of the eye (direct vision) or nodal point of the thin imager : $\alpha$ or $\overline{\alpha}$ -
Extended object [A$_1$A$_2$] along the optical axis :
characterized by the algebraic longitudinal size (distance between its extremities) : $\overline{A_1A_2}$
Characterization of its conjugate extended image
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Extended image [A'B'] or [B'$_1$B'$_2$], perpendicularly to the optical axis :
- characterized by its algebraic size (distance between its extremities) : $\overline{A'B'}$ or $\overline{B_1'B_2'}$
- or characterized by the apparent angle (in algebraic value or not) substended by the object at nodal point of the eye (direct vision) or nodal point of the thin imager : $\alpha$ or $\overline{\alpha}$ -
Extended image [A'$_1$A'$_2$] along the optical axis :
characterized by the algebraic size (distance between its extremities) : $\overline{A'_1A'_2}$
Characterization of the imager action
The imager gives an image of an object. The characterization of imager action depends on how the object and the image are characterized.
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Object and image both characterized by their algebraic transverse sizes :
$\Longrightarrow$ imager action characterized by the transverse magnification $M_T$
$M_T=\dfrac{image:size}{object:size}=\dfrac{\overline{A'B'}}{\overline{AB}}$ -
Object and image both characterized their apparent angles :
$\Longrightarrow$ imager action characterized by the apparent magnification $M_A$
$M_A=\dfrac{image:apparent:angle}{object:apparent:angle}=\dfrac{\overline{\alpha'}}{\overline{\alpha}}$
or $M_A=\pm\dfrac{\alpha'}{\alpha}$, with sign + when erect image, sign - when inverted image.
apparent magnification = angular magnification
an apparent angle depends on distance from nodal point $\Longrightarrow$ more accurate definitions of apparent angles will be necessary (see chapter "optical instruments") -
Object and image both characterized by their algebraic longitudinal sizes :
$\Longrightarrow$ imager action characterized by the longitudinal magnification $M_L$
$M_L=\dfrac{image:size}{object:size}=\dfrac{\overline{A'_1A_2'}}{\overline{A_1A_2}}$