final exam – written part - École Polytechnique de Montréal

QUESTION 5 :
Propagation in a graded index lens
A graded index lens with parallel end-faces has a refractive index with radial dependence
n 2 (r) n 0
2
ar 2
where r is the distance from the central axis of the lens in a cylindrical coordinate system and n 0 is the index on
the central axis.
Using the ray equation of geometrical optics
d
ds n d r
ds
n
answer the following questions :
1. Calculate a function r(z) describing the path of a ray incident at a distance r 0 from the center of the input
face and perpendicular to this face. Use the paraxial approximation and eliminate negligible terms in
your calculation if needed.
2. What is the minimal thickness of the lens for an incident plane wave on the input face to be focused to a
point on the output face?
3. What is the transfer matrix of this lens?
4. What is the transfer matrix of four of these lenses placed in sequence and in contact? Comment.
Examen général de synthèse - Partie écrite
Vendredi 16 juin 2006
Page 7 sur 11
Département de génie physique