Gaussian content as a laser beam quality parameter

We propose the Gaussian content (GC) as an optional quality parameter for the characterization of laser beams. It is defined as the overlap integral of a given field with an optimally defined Gaussian. The definition is especially suited for applications where coherence properties are targeted. Mathematical definitions and basic calculation procedures are given along with results for basic beam profiles. The coherent combination of an array of laser beams and the optimal coupling between a diode laser and a single-mode fiber are elaborated as application examples. The measurement of the GC and its conservation upon propagation are experimentally confirmed.     

Relaxation oscillations in a laser with a Gaussian mirror

We present an analysis of the relaxation oscillations in a laser with a Gaussian mirror by taking into account the three-dimensional spatial field distribution of the laser modes and the spatial hole burning effect. In particular, we discuss the influence of the Gaussian mirror peak reflectivity and a Gaussian parameter on the damping rate and frequency of the relaxation oscillation for two different laser structures, i.e., with a classically unstable resonator and a classically stable resonator.     

Propagation and focusing of Gaussian beams generated by Gaussian mirror resonators

The propagation and focusing properties of a class of Gaussian beams generated by optical resonators with Gaussian reflectivity mirrors are investigated. Attention is concentrated on the following two beams in this class: (a) the annular Gaussian beam (the Gaussian doughnut mode) and (b) the flat-topped Gaussian beam. A class of flat-topped Gaussian beams is introduced. All analysis is limited to a coherent superposition scheme of the lowest-order Gaussian modes (TEM00) that have different parameters.     

Beam-divergence control of excimers with plane-parallel Gaussian cavities

Plane-parallel cavities with Gaussian-reflectivity-profile mirrors as full reflectors were applied to a XeCl laser, and the near- and the far-field characteristics of the laser radiation were analyzed. It is shown that excimer lasers fitted with plane-parallel Gaussian cavities deliver laser radiation with a beam-quality factor M(2) that is more than 50% smaller than that of laser beams delivered by conventional plane-parallel cavities. The effect of the Gaussian mirror spot size on M(2) was also investigated, and it is shown that the narrowing of the Gaussian mirror spot size reduces the beam-quality-factor value.     

Apodizing holographic gratings for the modal control of semiconductor lasers

We first present the fabrication technique of apodizing holographic gratings. Gratings with a spatially variable reflectivity profile were obtained by the interference of two Gaussian beams on a glass plate covered with a photoresist. When the exposure time was short enough to avoid saturation of the photoresist, gratings with a quasi-Gaussian reflectivity profile for the beam reflected in the -1 order were produced; the reflectivity at the center could be as high as 71%, and the half-width of the reflectivity profile at the e(-1) position could be as small as 180 mum. Apodizing gratings were used as the end mirror of the external cavity of a broad-area semiconductor laser. Single longitudinal- and lateral-mode operation was observed over the full range of allowed injection currents.     

Characterizing output beams for lasers that use high-magnification unstable resonators.

Laser beams generated from high-magnification on-axis unstable resonators by use of hard-edged optics typically have a doughnut-shaped distribution in the near field (i.e., a flat-top profile with a hole in the middle for an axially coupled beam). We derive analytical expressions describing this distribution by using the flattened Gaussian beams concept. The superposition of two flattened Gaussian beams whose flatness and steepness of edges are controlled by defined parameters (i.e., the beam width and the order) is used to analyze the output beam intensity along the propagation axis. Finally, experimental measurements of beam propagation from a copper-vapor laser fitted with a high-magnification unstable resonator show excellent agreement with theoretical predictions.     

Focusing Properties of the Gaussian Beam Generated by Optical Resonators with Gaussian Reflectivity Mirrors

Abstract

This paper presents a parametric study concerning the focusing properties of the light beam generated by optical resonators with Gaussian reflectivity mirrors. The diffraction integral has been evaluated and analytical expressions are derived for intensity patterns near the focus in systems of different Fresnel numbers. Universal curves have been plotted against dimensionless parameters to show the variations in the intensity patterns due to the change in focusing geometry and the incident beam intensity profile. Modification to the well known diffraction formula are considered in order to extend the conventional Gaussian formalism to the case of Gaussian mirror resonators.     

Far-field intensity distribution, M2 factor, and propagation of flattened Gaussian beams

Explicit expressions for the M(2) factor and field distribution of flattened Gaussian beams that propagate through an unapertured paraxial ABCD system are derived and illustrated with numerical examples. We show that the far-field expression of flattened Gaussian beams can be readily obtained by use of our results.     

Frequency-dependent graded reflectivity mirror: characterization and laser implementation

We present a theoretical and experimental investigation of a new family of variable reflectivity mirrors, based on frustrated total internal reflection and interference effects. These mirrors are sensitive to frequency in the sense that their transverse reflectivity distribution changes significantly as a function of the frequency of light. The mirror reported here shows total power reflectivity changes of 50% within 8.0 GHz. The mirror was tested as the output coupler of an unstable resonator in a Nd:YAG laser working in the free-running regime. This configuration was compared with the standard stable multimode resonator configuration. The beam divergence was reduced by more than 1 order of magnitude and the output power was reduced by only 10%. The laser resonator mode competition that is due to the frequency-dependent mirror reflectivity distribution is discussed.     

General propagation equation of flattened Gaussian beams

The closed-form propagation equation of flattened Gaussian beams passing through a paraxial optical ABCD system, in which the linear gain and absorption media are included, is derived, and its general applicable advantage is illustrated with numerical examples.     

Complex Gaussian functions expansion method applied to truncated Gaussian beams

An analytical expression for the beam propagation factor (M ² factor) of truncated Gaussian beams was derived by using the complex Gaussian functions expansion method. The reasonability of the approximation of complex Gaussian functions expansion method is studied, and a comparison of this method with the generalised truncated second-order moments method and the asymptotic analysis method is also made. In general, an easy analytical expression for the M ² factor of truncated laser beams can be derived by using the complex Gaussian functions expansion method. The M ² factor obtained by using the complex Gaussian functions expansion method is more consistent with that in practice than that obtained by using two other methods. The analytical results obtained by using the complex Gaussian functions expansion method can be reduced to that for the non-truncated case when the truncation parameter is sufficiently large. Therefore, the complex Gaussian functions expansion method is a suitable approximation method for studying the M ² factor of truncated laser beams.     

Temperature distributions and thermal deformations of mirror substrates in laser resonators

For finite-thickness media with convective surface losses, the three-dimensional temperature distributions and thermal deformations of mirror substrates in laser resonators that are due to absorption of laser light with a Gaussian power-density profile are calculated by use of the well-known Green's function methods. Some expressions and theoretical profiles of the temperature distributions and thermal deformations as functions of the radius and the thickness of a mirror substrate are obtained. The results of the calculations show that the rise in temperature is closely related to the absorption coefficient of the medium as well as to the convective heat-transfer coefficient, that the initial thermal deformations of mirror surfaces increase quickly at the beginning of laser heating and that then the thermal deformations are insensitive to laser heating times. Meanwhile, thermal deformations of a silicon mirror are experimentally demonstrated by use of CO(2) laser irradiation. The experimental trends of thermal deformations are in agreement with the theoretical profiles.     

Inspection of commercial optical devices for data storage using a three Gaussian beam microscope interferometer

Recently, an interferometric profilometer based on the heterodyning of three Gaussian beams has been reported. This microscope interferometer, called a three Gaussian beam interferometer, has been used to profile high quality optical surfaces that exhibit constant reflectivity with high vertical resolution and lateral resolution near lambda. We report the use of this interferometer to measure the profiles of two commercially available optical surfaces for data storage, namely, the compact disk (CD-R) and the digital versatile disk (DVD-R). We include experimental results from a one-dimensional radial scan of these devices without data marks. The measurements are taken by placing the devices with the polycarbonate surface facing the probe beam of the interferometer. This microscope interferometer is unique when compared with other optical measuring instruments because it uses narrowband detection, filters out undesirable noisy signals, and because the amplitude of the output voltage signal is basically proportional to the local vertical height of the surface under test, thus detecting with high sensitivity. We show that the resulting profiles, measured with this interferometer across the polycarbonate layer, provide valuable information about the track profiles, making this interferometer a suitable tool for quality control of surface storage devices.     

Photobleaching of a solid photochromic medium by a Gaussian laser beam

Equations that describe the photobleaching of photochromic layers illuminated by Gaussian laser beams are given. The photochromic samples are made of thionine, triethanolamine, and polyvinyl alcohol and follow simple kinetics law. Spatial absorbance and time-dependent power transmittance are well described by the developed equations. The model is used to calculate the Gaussian dimension of helium-neon laser beams from kinetics data.     

Laser diode facet modal reflectivity measurements

A simple and accurate method for measuring the front facet modal reflectivity of a Fabry-Perot laser diode is presented. In this method, optical feedback from an external mirror of known reflectivity, R(ext), is used to alter the laser diode threshold current. The effect of the external mirror and front facet reflectivities on the threshold current then allows for a measurement of the front facet modal reflectivity of the laser diode and is theoretically and experimentally studied. This method was used to measure a facet reflectivity of R(2) = 0.0151(+0.0018/-0.0032) [R(2) = 0.00592(+0.00085/-0.00123)] for a commercially antireflection-coated facet of a laser diode with a center wavelength of 795 nm (935 nm). The results of the reflectivity measurements based on the threshold current as a function of the external mirror reflectivity are compared with the results of the reflectivity measurements based on modulation depth of the optical spectrum [IEEE J. Quantum Electron. QE-19, 493 (1983)].     

The pointing stability of flattened Gaussian beams

Based on the moment characterization of the beam pointing stability proposed by Morin et al., the pointing stability of flattened Gaussian beams (FGBs) is studied. The analytical expression for the misalignment factor of FGBs is derived and a comparison with the approximate result is made. The effect of relative transverse and angular shifts on the misalignment factor is analysed. The validity of the exact and approximate expressions is discussed with illustrative numerical examples. The pointing stability of Gaussian beams and truncated plane waves is also studied, and treated as the spatial cases of FGBs.     

Elegant Laguerre-Gauss beams as a new tool for describing axisymmetric flattened Gaussian beams.

A new superposition scheme for representing flattened Gaussian (FG) beams is proposed. Such a representation, unlike the original proposed by Gori [Opt. Commun. 107, 335 (1994)], is based on an expansion in terms of the so-called elegant Laguerre-Gaussian beams. This new representation allows us to obtain the closed-form expression of a FG beam of any order propagating through a paraxial ABCD optical system by means of a simple recurrence rule, which turns out to be particularly stable even when it is applied to FG beams of very high orders (> 10(4)).     

Unstable Cr:LiSAF laser resonator with a variable reflectivity output coupler

The performance of a flash-lamp-pumped Cr:LiSAF unstable laser resonator utilizing a fourth-order super-Gaussian variable reflectivity mirror as an output coupler is described. The super-Gaussian mirror results in a smooth, flattop transverse beam profile in the near field that is advantageous for nonlinear frequency-conversion applications. Long-pulse and Q-switched operation of the Cr:LiSAF unstable laser resonator are described and compared with stable resonator operation. We obtained tunable ultraviolet radiation extending from 267 to 290 nm by frequency mixing theQ-switched Cr:LiSAF laser output with lithium triborate and beta-barium borate nonlinear crystals.     

Fractional Fourier transform of a higher-order cosh–Gaussian beam

A higher-order cosh–Gaussian beam is an appropriate model to describe the flattened laser beam. The fractional Fourier transform (FRFT) is applied to treat the propagation of higher-order cosh–Gaussian beams. An analytical expression for a higher-order cosh–Gaussian beam passing through a FRFT system has been derived. By using the derived expression, the properties of a higher-order cosh–Gaussian beam in the FRFT plane are graphically illustrated with numerical examples.