Beam profile/radius

The laser beam profile at the end of the medium can be direct evidence of the profile flattening. Figure 8.6b and c shows the laser profiles at the gas jet positions z = 0 and z = —18 mm. At z = 0, the laser profile was severely distorted and weakened after the propagation. On the other hand, at z = —18 mm, the laser beam formed a flattened profile with a radius of 60 pm, which closely matched the 3-D calculation. It is thus clear that a proper selection of the gas jet position is critical for the profile flattening and self-guiding of laser pulses. 

Filaments were visible in the beam profile at the exit of the fog as soon as the transmitted pulse energy was greater than 25 mJ (45 GW), corresponding to 15% transmission (extinction coefficient 0.2 m-1). Hence, filamentation can be transmitted in a fog over a distance comparable with the visibility. In this configuration the particle density was 8.6 x 104 cm 3, so that the mean free path for a 100 pm filament was 0.5 mm Each single filament hits on average 2000 droplets per meter propagation. However, since the droplet radius is about 100 times smaller than the filament diameter, the filaments them-... 

For the indication of absolute laser fluence values, e.g., for ablation, modification, melting etc. of a material, the determination of the spot size is crucial. Fluence values allow the comparison of laser treatment with different types of lasers showing various spatial beam characteristics like a square (e.g., excimer laser) or a Gaussian beam profile (e.g., Tksapphire laser). In the following, the determination of the Gaussian beam radius is described. 

For a Gaussian beam profile the calculation is more difficult, and the solution can only be obtained numerically. The result for a Gaussian beam with the radius w (Sect. 5.3) is [4.34]... 

Consider a source which emits a collimated beam of radius r. The beam can be focused onto the endface of the fiber at Q on the axis by introducing a thin lens of radius equal to the beam radius and focal length/, as shown in Fig. 4-6(a). All of the light from the source excites only bound rays provided the angle 0 subtended by the lens at Q does not exceed the maximum angle of incidence 0 i(O) defined by Eq. (4-6). If for convenience we assume a step profile and set n(0) = /ijo, then 0m(0) = sin ( c<>/ o) c - setting = /tan in the... 

Conversely, a Gaussian beam with a = 0.75r has nearly the same diffraction intensity pattern as the uniform beam of radius r. The situation is illustrated in Fig. 10-1. Clearly A (u) is also approximately Gaussian for various smoothed-out beam profiles intermediate between the uniform and Gaussian cases, e.g. the profiles given by Eq. (15-9), so the results and conclusions for the Gaussian beam have wide applicability. 

For the FEA calculation, precise information about the laser energy (i.e. the laser power distribution) represents one of the important boundary conditions. Since the intensity varies with the laser beam radius and the transparait paHner absorbs part of this intensity, the lasCT beam profile at the joining area has to be drtermined. For this, Ae lasa- beam profile was measured under the defined expainaital welding conditions with a special detecting camoa. Resulting beam profiles are shown in Fig. 3 which are used for the FEA simulation. 

MEIS, by contrast, is frequently used to achieve depth profiling information with close to monolayer resolution. The energy of the incident ion beam is in the order of 100 keV. At such energies, the shadow cone radius is relatively small and incident ions are able to channel hundreds of nanometres into the bulk of a crystalline lattice. It is possible, with careful sample alignment, to selectively illuminate a given number of surface layers (see below), in which case one may achieve layer by layer compositional information as a function of depth. 

The input beam was a spatially filtered beam from one of the laser sources described below. The spatial profile of the input beam usually showed a greater than 96% correlation with a gaussian profile. The pulse width was 2.5 nanoseconds. The experiments at low energies were conducted at 10 Hz. For incident fluences above about 10 mJ/cm, the repetition rate was reduced to 0.5 Hz. This removed any effect of persistent thermal effects. The laser intensity was controlled by wave plate/polarizer combination. The input beam was focused onto the sample using approximately f/45 optics. The focal spot size (beam radius) was measured to be 2.6 0.1 pm at 550 nm. 

Two simple cases can be identified from last equation. The first case is trivial and it occurs when the radius of the circle is zero, d=0, and F(u,v) is a constant. The diffraction free beam corresponds with a plane wave. The second case occurs when the radius is different of zero and F(u,v) is a constant. For this case we have that the profile of the beam is... 

A non-contact laser surface profilometer which allows for precise characterization of surface profile has been purchased from Rodenstock Precision Optics. Profiles of microscopic surface terrain over spans of up to 60mm are achievable at a scan rate of 30 rnm/min. ITie laser beam allows surface height resolution on the order of one micron. In initial tests with the system, both longitudinal and radial profile scanning were successfully performed on cylindrical tensile test specimens. Radial scanning is possible because the footprint of the focused solid state laser (0.8 pm) is small compared to the radius of curvature of the specimen. 

At the center of the resonator z = 0-> o = 0 R = oc. The radius R becomes infinite. At the beam waist the constant phase surface becomes a plane z = 0. This is illustrated by Fig. 5.10, which depicts the phase fronts and intensity profiles of the fundamental mode at different locations inside a confocal resonator. 

The fundamental modes have a Gaussian profile. For r = w z) the intensity decreases to 1/e of its maximum value io = C on the axis r = 0). The value r = w z) is called the beam radius or mode radius. The smallest beam radius wo within the confocal resonator is the beam waist, which is located at the center z = 0. From (5.31) we obtain with d = R... 

Gaussian laser beam, with one quadrant of the beam cut away to reveal the radial profile of power density [9]. Here r is the radius of the coaxial circle, w is the effective radius of the beam, pr is the power flowing through the circle, Pc is the power density at the center of the laser beam, and s is the base of the natural logarithms (e = 2.71828...). 

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