Markus Testorf, Jürgen Jahns
Imaging with lenses is the most important operations in free space
optics. The off-axis
mode of propagation in planar-optical systems causes a special situation,
first because
aberrations are introduced, and second because of geometrical
constraints, both of which
depend on the propagation angle 
(Fig. 1). The maximum number
of data channels that
can be transmitted can be expressed in terms of the space bandwidth
product SBP of the
planar-optical imaging system. Here we consider the influence of
geometrical constraints
and aberrations on the SBP with the propagation angle as a
variable parameter.
Fig. 2 shows the unfolded version of Fig. 1. To concentrate on the
constraints due to
planar optics, we consider the one-dimensional case. From 
and 
it can be shown, that the maximum number of resolvable points
in x is given as [3]
which shows a maximum value for 
.
To obtain the full space bandwidth product has to be
multiplied with
the number of points in y.
Fig. 1: Planar optical imaging system
Fig. 2: Unfolded off axis system
Now, we consider wave aberrations. We restrict our discussion to astigmatism and defocusing in one dimension. From a Taylor expansion of the point spread function of free space we calculate the phase of the aberrations
[3]:
From the term linear in 
we again estimate the number of points in x. If we only tolerate wave
aberrations
smaller than 
we get
For 
, 
and 
,
a value 
is found, which is about 15 times less than the value obtained from Eq.
(2). Thus
our results demonstrate the significance of wave aberrations to
planar-optical imaging systems.
[1] J. Jahns and J. S. Walker, Opt. Comm. 76(5,6), 313 (1990).
[2] M. Testorf et al., "Free space propagation in planar optical systems", this report.
[3] M. Testorf, J. Jahns, "Imaging in planar optics: system design for oblique diffraction angles", OSA topical meeting on Diffractive Optics & Micro-optics, Boston, April 29 - May 3, 1996.