Bernd Eppich

Determination of the ten second order intensity moments

Abstract The general astigmatic beam can be characterized by its ten second order moments in first order approach. Most intensity moments, except the beam twist, can be determined by measuring the intensity in a reasonable number of positions around the waist of the beam. The beam twist is determined by applying a rotated cylindrical lens. The ten intensity moments of two kinds of astigmatic beam were determined: a simple astigmatic TEM 8,0 Hermite–Gaussian beam and a twisted beam generated from the TEM 8,0 mode. The experimental results were compared with the theoretical calculations and demonstrate that the ten second order moments of a beam can be determined in a rather simple way.

Inside-pumped Nd:YAG tube laser.

A novel type of solid-state laser with a tubular active element made of Nd:YAG is presented. Using the tube geometry, one can achieve higher efficiency and higher output power per laser crystal than by using a rod or slab geometry. Our tube laser is pumped from the inside by four flash lamps. The slope efficiency of 9% and the total efficiency of 7.5% at as much as 400 W of output power are, to our knowledge, the highest values reported for flash-lamp-pumped Nd:YAG lasers. At 15 kW, 75% of the maximum available pumping power, the output power is 1000 W. Thermal lensing is five times lower than for an equivalent rod laser.

Measurement of the Wigner distribution function based on the inverse Radon transformation

The Wigner Distribution Function (WDF) has been well known in quantum mechanics since the thirties. Applied to paraxial optics it can be considered as a phase-space representation of quasi-monochromatic, partially coherent beams. For a two-dimensional beam (one transverse dimension) it depends on one spatial and one angular coordinate. Due to some properties of the WDF, it may be considered as an intensity distribution of geometrical rays, depending on position and direction, although this analogy is limited. Once the WDF of a laser beam in one transverse plane is known, the intensity distribution behind any optical system can be derived from it. The measurement of the WDF cannot be connected with a single intensity measurement as can be shown easily. Several suggestions for measurement procedures involving different kinds of apertures have been made. But all of them suffer from the influences of those apertures. Here we present a method which uses only a single focusing lens and several intensity measurements. It is based on a known mathematical procedure called the inverse radon transformation. An experimental result of applying this method to a laser beam emerged by an unstable resonator is presented, too.

Twist of coherent fields and beam quality

An optical system is presented which transforms an arbitrary astigmatic beam into one of circular symmetry with twist. This holds for coherent and partially coherent fields. Experiments were performed with Gauss-Hermitian modes TEMno, transforming them into ring-like structures of Gauss- Laguerre type modes. The theoretical results concerning intensity structure and beam propagation factor were confirmed.

Measuring laser beam parameters: phase and spatial coherence using the Wigner function

Detailed laser beam characterization is essential for the proper choice of lasers source according to the respective application as well as for the optimization of optical systems. Since most lasers generate partially coherent beams, intensity and phase distributions are not enough to describe them. In addition the knowledge of the coherence distribution is necessary. So far different setups have been used to measure phase and coherence distribution with limited accuracy. Here we demonstrate a new measurement procedure which is based on the retrieval of the Wigner distribution, from which all relevant information can be derived. The setup is very simple and the results seems to be fairly accurate.

Generation of partially coherent fields with twist

It is shown, theoretically and experimentally that any astigmatic beam with M2>1, coherent or partially coherent, can be transformed by a first order system into a beam with second intensity moments of circular symmetry and twist. Experiments were performed with coherent Gauss- Hermite beams of high order and partially coherent diode arrays. Furthermore it is demonstrated that the bema parameter product is not constant in ideal ABCD-system.

Definition, meaning, and measurement of coherence parameters

It is a well-known rule of thumb that the intensity profiles of a beam are smoother andless structured the less the transverse coherence of the beam is. In this paper it will be shown that the global degree of coherence as well as the lateral coherence length is related to the width of the Fourier power spectra ofthe profiles the beam takes on under propagation and hencecan be considered as a quantitative measure of the smoothness of the beam. A spin-off of these considerations is a simple non-interferometric measurement procedure for both, the global degree of coherence of a beam and the lateral coherence length.

Measurement of the four-dimensional Wigner distribution of paraxial light sources

The complete knowledge and description of light sources is a fundamental base of optical design. Partially coherent, paraxial light sources of homogeneous polarisation state are represented by the four-dimensional Wigner distribution function, which contains all information on amplitude, phase relations and spatial coherence. Hence, knowledge of the Wigner distribution enables the prediction of power density distributions after propagation through a wide range of optical systems by numerical simulation. A simple optical setup consisting of a spherical lens, a cylindrical lens, and a CCD camera can be used to experimentally retrieve the Wigner distribution function of a spatially confined light source by a tomographic reconstruction scheme. This paper briefly introduces the Wigner distribution and outline the reconstruction scheme. Measurements on partially coherent laser beams are presented including examples of successful predictions of power density distributions behind some optical systems.

Modal decomposition of partially coherent beams using the ambiguity function

Phase-space representations of optical beams such as the ambiguity function or the Wigner distribution function have recently gained considerable importance for the characterization of coherent and partially coherent beams. There is growing interest in beam properties such as the beam propagation factor and the coherence and phase information that can be extracted from these phase-space representations. A method is proposed to decompose a partially coherent beam into Hermite-Gaussian modes by using the ambiguity function. The modal weights and the possible phase relations of the Hermite-Gaussian modes are retrieved. The method can also be applied for the decomposition of the Wigner distribution function. Some examples are discussed in the scope of beam characterization.

Spatial coherence: comparison of interferometric and non-interferometric measurements

Coherence properties of real laser beams can be crucial for many applications, e.g. in lithographic processes or for Bragg grating writing. Knowledge of the coherence distribution together with the amplitude and phase distribution allows for a complete beam characterization, which enables the numerical simulation of beam propagation through virtually any relevant optical system. Classical measurement methods of coherence properties are Young’s double hole interferometry and Shear interferometry. But due to their interferometric nature experimental realization of both methods is quite difficult and obtainable accuracies are usually not satisfying. The reconstruction of the Wigner distribution from a couple of measured intensity distributions in the waist region of a beam provides a simple measurement setup delivering fairly accurate results. This is demonstrated by an experimental comparison of these three methods.