Experimental observation of interfering Bessel beams.

We experimentally verified the interference resulting of the superposition of two Bessel beams propagating in free space and showed for first time the self imaging effect using nondiffracting beams. Our results are supported by numerical simulations and possible applications are discussed.

Talbot interferometer with simultaneous dark and bright fields.

We describe an optical setup for implementing a Talbot interferometer that uses two mutually incoherent sets of self-images to simultaneously produce interference bands due to dark and bright fields. A spatial filter can be used to visualize these fields without interference fringes. Experimental results are included.

Apodization of annular apertures: Strehl ratio.

We show that for any rotationally symmetric apodizer on the full aperture, there is a family of apodizers on the annular aperture with the same functional Strehl ratio vs defocus (W(20)) and vs spherical aberration (W(40)). However, in the latter case, the coefficients W(20) and W(40) are reduced by the factors (1 - epsilon(2)) and (1 - epsilon(2))(2) respectively, where is the central obscuration ratio. We indicate that the best focal plane is shifted from W(20) = - W(40) to W(20) = -(1 + epsilon(2)) W(40). These general results allow us to design and to compare novel apodizers on annular apertures which reduce the influence of W(20) and W(40). The Strehl ratios of a novel family of apodizers are discussed to illustrate our general results.

Arbitrarily high focal depth with a quasioptimum real and positive transmittance apodizer.

Focal depth is assessed by the average value of the square modulus of the slope associated with the complex amplitude along the optical axis. Then, the calculus of variations is used for identifying the optimum apodizer, characterized by a Strehl ratio vs defocus with high focal depth, for a specified light throughput. We show that a certain Lorentzian profile is a quasioptimum solution for the above requirements. This apodizer has real and positive transmittance, and it can be modified to achieve arbitrarily high focal depth. A closed formula relates focal depth to light throughput.

Spatial filters for replicating images.

The conditions for self-imaging, longitudinal periodicity are relaxed to demand image replication only along the optical axis. Lateral periodicity in the square of the radius, in the spatial-frequency domain, is necessary and sufficient for image replication along the optical axis. This formulation allows one to employ any zone plate as a spatial filter that creates self-replicating images of a given object, under either coherent or incoherent illumination.

Optical implementation of the fractional Hilbert transform for two-dimensional objects.

The classical Hilbert transform can be implemented optically as a spatial-filtering process, whereby half the Fourier spectrum is pi-phase shifted. Recently the Hilbert transform was generalized. The generalized version, called the fractional Hilbert transform, is quite easy to implement optically if the input is one dimensional. Here we show how to implement the fractional Hilbert transform for two-dimensional inputs. Hence the new transform is now suitable for image processing.