.
From the parabolic approximation of free space propagation [2, 3] we calculate the
Talbot distance, which is different for the x and y:Markus Testorf, Jürgen Jahns, Nikolay A. Khilo+, Andrey M. Goncharenko+
Wave fields with a lateral periodicity (in x and y) also exhibit a longitudinal periodicity along the z-axis. This so-called "self-imaging" or Talbot effect [1] has been investigated for on-axis light propagation. We are interested in using it for interconnection purposes in connection with planar optics. For this we need to analyze this phenomenon for off-axis light propagation.
We consider the configuration as shown in Fig. 1: A plane wave propagates off the
optical axis z and is diffracted by a grating. Then, the first self-image of the grating
occurs after one Talbot distance .
From the parabolic approximation of free space propagation [2, 3] we calculate the
Talbot distance, which is different for the x and y:
with m integer. Indices x and y denote the lateral coordinates and d is the grating period.
Fig. 1: The off-axis Talbot effect
Experimentally, we determined the Talbot distance 
to verify Eq. (1). For our experiments we used a
linear Ronchi grating of period d=50mm. For angles between 
and 
we found a good match between theory and experiment
with deviations of less than 1 %. The intensity profile of the self-image was
recorded and compared to numerical simulations (Fig. 2), where wave aberrations
arising from off-axis propagation [2, 3] where taken into account. The correspondence
between measurement and simulation indicates the significance of these aberrations
for a design of planar-optical systems.
Fig. 2: Experiment and simulation of the intesity pattern in the first Talbot plane. Dashed lines correspond to the profile of the Ronchi grating
+ Dr. Khilo and Prof. Goncharenko are with the Belarussian Academy of Sciences.
References:[1] K. Patorski, in Progress in Optics, Vol. XXVII (ed. E. Wolf, 1989), pp. 1-108.
[2] M. Testorf et al., "Free space propagation in planar optical systems", this report.
[3] M. Testorf, J. Jahns, N. A. Khilo and A. M. Goncharenko, "Talbot effect for oblique angle of light propagation", Opt. Comm. (accepted).