rigorous stigmatism
Rigorous stigmatism
Considering an optical system, the stigmatism between two points A and Ai is realized when all rays from the object A are focusing on its image Ai after propagation through the optical system.
The stigmatism can be approached. It is for instance the case for most of the optical systems working in Gauss conditions (see "paraxial parameters" and "paraxial conjugation" tutorials).
The stigmatism can be rigorous for some special optical systems and particular object positions. According to the Fermat principle, when rigorous stigmatism occures, the optical distance from the object to its image is constant whatever the rays considered.
The most common configurations for rigorous stigmatism are detailed in the next sections of this tutorial. they include the following cases : spherical dioptres working for Weiesrtrass points as object and image, ellipsoidal mirror working for its focal points as object and image, parabolic mirror used for focusing a collimated beam.
There is one optical component for which the stigmatism is rigorous whatever the object position : it is the flat mirror. Indeed, according to Snell-Descartes law (see tutorial on "refraction-reflexion on a dioptre"), the conjugate is the symmetric of the object with respect to the plane mirror. This is illustrated on the nearby ray tracing. Therefore, flat mirrors are aberration free. They are currently used for instance to make imaging systems more compact by folding them.
Also, all systems made of a single surface ( dioptres or mirrors) are rigorously stigmatic for all points of the surface. Indeed, each point of the surface is obviously rigorously conjugated with itself.
Weierstrass points of an elliptic mirror
In the 2D space, an ellipse (E) is a curve defined by the set of points M satisfying the following equation: 
where F and F' are two carecteristic points called the focal points and a is the half length of the large axis of the ellipse. Optically speaking, if the ellipse is a mirror, whatever a ray coming from F, once reflected by a point M of the ellipse it passes through F'. Indeed, as he optical distance from F to F' is constant, the Fermat principle is verified. F and F' are also called the Weierstrass points of the elliptic mirror.
The equation of the ellipse in a coordinate system centered in the middle of the focal points is : 
. As defined above, a is the length of the large axis. b is the length of the small axis. If c = |FF'| / 2 is the half distance between the two focal points F and F', the link between a and b is given by the formula below :
.
The rigorous stigmatism is illustrated in the leftmost ray tracing where the object is the focal point the nearest from the mirror and the image is the other one. Note that the rays can make a large angle with the ellipse axis (larger than 90 degrees). For this reason, elliptical mirrors with a large aperture are often part of illumination systems enabling to capt a lot of rays. For instance, elliptic mirrors are largely used in optically pumped lasers (like Nd:Yag lasers) for imaging pumping light source into the gain medium (see "laser" tutorials). Note also that any change in the object position induces aberration preventing from rigorous stigmatism as shown in the rightmost ray tracing.
Rigourous focus of a collimated beam reflected by a parabolic mirror
In the 2D space, a parabola (P) is a curve defined by the set of points M satisfying the following equation : 
where F is the focal point of the parabola, (D) is a straight line perpendicular to the parabola axis called directrix and P is the intersection with (D) of the straight line parallel to the axis and passing by M. Optically speaking, if the parabola is a mirror, whatever a ray parallel to the parabola axis, once reflected by the parabola it passes through F. In other words, an on axis object at infinity (from which rays are all parallel to the parabola axis) is rigorously conjugated with F. Indeed, (D) can be taken as a reference line for the optical distance from an object at infinity. As he optical distance from (D) to F is constant, the Fermat principle is verified.
The apex of the parabola is at half distance between the directrix and the focal point : 
. The equation of the parabola in a coordinate system centered on its vertex is : 
. R is the radius of the circle tangent to the parabola at the vertex. R/2 is the focal length of the parabola.
The rigourous stigmatism between an on axis object at infinity and the focal point is illustrated in the left side ray tracing. Once again, this rigorous stigatism disappears for off-axis objects as illustrated on the right side ray tracing. Parabolic mirrors are largely used in optics to focus rays coming from far away objects on a detector as for instance in astronomy or free space telecom. They can also be used in illumination devices to produce collimated beam from a small source located at the focal point.
Weierstrass points of a spherical refracting surface
For a spherical dioptre, two singular points A and Ai on the optical axis are rigorously stigmatic : they are called the Weiestrass points (or Young points). According to the Fermat principle, they are calculated by imposing the optical path for rays coming from A and ending on the optical axis after refraction to be constant whatever the incident ray direction. This leads to the following formulas:
.
R is the radius of the dioptre and C is its center. n0 and n1 are the refraction indexes respectively before and after the dioptre.
CA and CAi are the algebric distances respectively from C to A and from C to Ai.
Therefore, for a convex dioptre (R negative) both Weierstrass points are on the left side of the center. This means that the object is real and the image is virtual. For a concave dioptre (R positive), object and image are on both sides of the dioptre. Depending on the refraction indexes, either the object or the image is virtual.
The rigorous stimatism of a dioptre is illustrated with the ray tracing on the left. In this case, R is negative. The object is the most distant from the dioptre ( n1 larger than n0).
Spherical dioptres working for Weierstrass points are for instance used in high aperture imaging systems (like microscope lenses) in order to reduce the aperture and correct more easily further aberrations.
References
"Optique Fondements et applications" - 2004 - author : José-Philippe Perez.
"Optique géométrique Imagerie et instruments" - 2007 - author : Bernard Balland.
"Optique géométrique paraxiale" - Institut doptique théorique et appliquée - 1985 - author : Michel Cagnet.
"Formation des images Aberrations" - Institut doptique théorique et appliquée - 1985 - author : Michel Cagnet.