Adolf W. Lohmann

Speckle masking in astronomy: triple correlation theory and applications.

Due to the turbulent atmosphere the resolution of conventional astrophotography is limited to ∼1 sec of arc. However, the speckle-masking method can yield diffraction-limited resolution, i.e., 0.03 sec of arc with a 3.6-m telescope. Speckle masking yields true images of general astronomical objects. No point source is required in the isoplanatic field of the object. We present the theory of speckle masking; it makes use of triple correlations and their Fourier counterparts, the bispectra. We show algorithms for the recovery of the object from genuine astronomical bispectra data.

Making an array illuminator based on the Talbot effect

Binary phase elements in photoresist have been implemented which transform a uniform beam of light into an array of output light spots by means of the fractional Talbot effect. Arrays of more than 30 x 30 light spots with varying spot shapes have been achieved with compression ratios of up to 1:9.

The wigner distribution function and its optical production

Abstract An optical signal (image etc.) can be described by its complex amplitude u (x, y), or by its spatial frequency spectrum. Both descriptions are complte and also equivalent, because one can be derived from the other by a Fourier transformation. Neither the complex amplitude nor the spatial frequency spectrum is suitable for answering a question like “what is the spatial frequency in a certain part of the image?”. Here the term “local spectrum” is adequate. A rigorous definition of the “local spectrum” can be based on the Wigner distribution function. We developed optical methods for producing this “local spectrum” and we applied these methods to the investigation of sound patterns.

Space–bandwidth product of optical signals and systems

The space–bandwidth product (SW) is fundamental for judging the performance of an optical system. Often the SW of a system is defined only as a pure number that counts the degrees of freedom of the system. We claim that a quasi-geometrical representation of the SW in the Wigner domain is more useful. We also represent the input signal as a SW in the Wigner domain. For perfect signal processing it is necessary that the system SW fully embrace the signal SW.

Relationships between the Radon–Wigner and fractional Fourier transforms

Two recently described transforms are shown to be related. The Radon–Wigner transform is the squared modulus of the fractional Fourier transform. This new theorem may serve to translate signal and image processing results between different signal representations. Some consequences regarding moments are presented, including a new fractional-Fourier-transform uncertainty relation. Implications for processing are suggested.

Phase and amplitude recovery from bispectra

The bispectrum is the Fourier transform of the triple correlation, sometimes also referred to as triple product integral. We are concerned here with the bispectrum of the autotriple correlation. Bispectrum analysis can be used to solve phase problems in signal processing, since the knowledge of the bispectrum of a signal usually allows one to reconstruct both amplitude and phase of the Fourier transform signal. We present mathematical proof based on the theory of analytic functions and discuss the restrictions involved. A recursive algorithm is outlined for the reconstruction from sampled data. In addition, possibilities for noise reduction by averaging redundant information will be described. Examples are included for 1-D signals.

Fractional Hilbert transform

We have generalized the Hilbert transform by defining the fractional Hilbert transform (FHT) operation. In the first stage, two different approaches for defining the FHT are suggested. One is based on modifying only the spatial filter, and the other proposes using the fractional Fourier plane for filtering. In the second stage, the two definitions are combined into a fractional Hilbert transform, which is characterized by two parameters. Computer simulations are presented.

Graded-index fibers, Wigner-distribution functions, and the fractional Fourier transform

Two definitions of a fractional Fourier transform have been proposed previously. One is based on the propagation of a wave field through a graded-index medium, and the other is based on rotating a functions Wigner distribution. It is shown that both definitions are equivalent. An important result of this equivalency is that the Wigner distribution of a wave field rotates as the wave field propagates through a quadratic graded-index medium. The relation with ray-optics phase space is discussed.

What classical optics can do for the digital optical computer

Recently, quite strong nonlinear optical configurations such as MQW have been invented. As a consequence, optical logic components with reasonable parameters are now feasible, but that is not enough to justify the development of a digital optical computer. The natural parallelism of optical instruments provides the impetus for developing a highly parallel digital optical computer. The optical technology is not so far behind the electronics technology as one might suspect. We show how varieties of computer subsystems can be implemented relatively easily by classical optical hardware.

Accuracy of phase shifting interferometry

The accuracy of phase shifting interferometers is impaired by mechanical drifts and vibrations, intensity variations, nonlinearities of the photoelectric detection device, and, most seriously, by inaccuracies of the reference phase shifter. The phase shifting procedure enables the detection of most of the errors listed above by a special Lissajous display technique described here. Furthermore, it is possible to correct phase shifter inaccuracies by using an iterative process relying solely on the interference pattern itself and the Fourier sums used in phase shifting interferometry.