  It is shown that the one-lens system in para-axial optics can serve as an
optical computer for contraction of Wigner's little groups and an analogue
computer which transforms analytically computations on a spherical surface to
those on a hyperbolic surface. It is shown possible to construct a set of
Lorentz transformations which leads to a two-by-two matrix whose expression is
the same as those in the para-axial lens optics. It is shown that the lens
focal condition corresponds to the contraction of the O(3)-like little group
for a massive particle to the E(2)-like little group for a massless particle,
and also to the contraction of the O(2,1)-like little group for a space-like
particle to the same E(2)-like little group. The lens-focusing transformations
presented in this paper allow us to continue analytically the spherical O(3)
world to the hyperbolic O(2,1) world, and vice versa.
