Markus Testorf, Jürgen Jahns, Nikolay A. Khilo+, Andrey M. Goncharenko+
In planar-optical systems the light signal travels along a zigzag path under an angle
relative to the surface normal. We want to understand the effects of the off-axis light
propagation on image formation and related phenomena. For our analysis, we use the
concept of the transfer function. For small diffraction angles it is shown that off-axis
propagation adds a term to the parabolic transfer function which can be interpreted as an
aberration.
The transfer function of free space obtained as a solution of Helmholtz' equation is given
as [1]
where 
and 
are the spatial frequencies in x
and y direction, respectively. In planar optics propagation is centered around a
frequency 
,
where 
is the angle
that the folded optical axis forms with the surface normal of the substrate.
Therefore, it is convenient to make the substitution
We consider small angle diffraction, which means 
, for all contributing
frequencies. The transfer function of free space can be approximated by a
truncated Taylor series
Our result contains three terms. The phase linear in 
corresponds to a lateral
shift 
in space. The second term contains the parabolic approximation, which contains
different scaling factors for the and coordinate. The last expression corresponds to
aberrations which depend on the deflection angle . That means, for large angles
the
parabolic approximation may not be sufficient to describe small angle diffraction.
If we calculate the Fourier transformation of the aberrations we obtain the
corresponding point spread function as shown in Fig. 1. The amplitude
distribution computed from the parabolic approximation has to be convoluted
by a coma like distribution.
Fig. 1: Gray scale picture of the point spread function of the aberrations
+ Dr. Khilo and Prof. Goncharenko are with the Belarussian Academy of Sciences.
References:[1] J. W. Goodman, Introduction to Fourier optics, (MacGraw Hill, 1968), chap. 3.
[2] M. Testorf, J. Jahns, N. A. Khilo and A. M. Goncharenko, "Off-axis Talbot effect and array generation in planar optics," OSA topical meeting on Diffractive Optics & Micro-optics, Boston, April 29 - May 3, 1996.