Lab III Laser Beam Properties ECE 476 I. Purpose The goals of this experiment are to understand the irradiance profile of a Gaussian laser beam and to learn how to test for temporal coherence using a simple interferometer. II. Background Gaussian Beam: beam waist and divergence The He-Ne laser in the lab oscillates in the 00TEM mode. The beam produced by this oscillation is a Gaussian beam, meaning that the wave fronts of constant phase are spherical and thus cause the beam to diverge. The beam diameter, 2w, increases as the beam propagates down the z axis. The narrowest portion of the beam is called the beam waist and it has a diameter 2w0, as shown in Figure 1. Figure 1: Gaussian Beam. For a Gaussian beam the beam diameter changes as a function of distance z, where z is measured as the distance from the beam waist 2w0. Also a Gaussian beam has both a cylindrical and a spherical character which comes naturally out of the solution to the wave equation in cylindrical coordinates (see lecture 3 Fall 2008). Concentrating on its cylindrical character, the radius of the cylinder w(z) at a distance z away from the beam waist is given by: 22001)(⎟⎟⎠⎞⎜⎜⎝⎛+=wzwzwπλ Equation 1 where l is the wavelength of light. Considering the spherical character of the beam, the half divergence angle q (as shown in Figure 1) of the beam is given by 0wπλθ= Equation 2Note that to calculate the radius w(z) and the divergence q of a Gaussian laser beam, we need to know the minimum waist radius w0. However, the beam waist typically occurs inside the laser’s casing, so we can not measure it. By using Eq. 1, the minimum spot size parameter ow can be expressed in terms of two measured beam radii w(1z ) and w(2z ) at z locations 1z and 2z outside the laser. This expression is )()(122221220zwzwzzw−−=πλ Equation 3 Remember that z is measured from the beam waist. The location of the beam waist w0 is approximately in the center of the laser’s casing. Using Equation 2, the half divergence angle is therefore given by 21221222)()(zzzwzw−−=θ Equation 4 Another way to estimate the divergence of the beam is by a geometrical approach, as shown in Figure 2. Figure 2: Divergence angle geometry. Using two measured beam radii, w(z1) and w(z2), the divergence angle can be approximated by ⎟⎟⎠⎞⎜⎜⎝⎛−−=−12121)()(tanzzzwzwθ Equation 5Gaussian Beam: measuring the beam radius One method to measure the diameter of a laser beam is to measure the power of the beam as a sharp edge is moved to block the beam as shown in Figure 3. Figure 3: Theoretical setup of Gaussian beam experiment. The cross-sectional irradiance distribution for a TEM00 wave is graphed in Figure 4. Figure 4: Irradiance profile of Gaussian beam. Mathematically, the irradiance distribution I(x,y) can be expressed as Equation 6 where the Cartesian coordinates x and y are measured from the beam center and are perpendicular to the direction of propagation z. The power transmitted past the sharp edge as the edge is inserted to block the beam is given by Equation 7 where a is the depth of the knife-edge in the beam, P0 is the power measured by the detector without the knife edge in the beam, and erfc( ) is the complementary error function. A sample plot of the expected relative power plot versus knife-edge position is shown in Figure 5.Figure 5: Relative power distribution as a function of knife-edge position. Plotted in Figure 4 are the two positions -w and w which occur at 90% and 10% power. Therefore the beam diameter 2w can be determined by noting the positions of the 10% and 90% relative power points. The choice to use 10% and 90% to determine the beam diameter is arbitrary; the thing that really matters is that the percentages are consistent for all beam diameter measurements. Coherence The light from a laser is in phase across the diameter of the laser beam. This is called spatial coherence. Another type of coherence found in lasers is temporal coherence where the light emitted at one time has a definite phase relationship with light emitted at a different time. Perfect temporal coherence can be thought of as a perfect sine wave of light wavefronts being emitted by the laser. In real lasers there may occasionally be a jump or discontinuity in the sine wave of the emitted light. One measure of the emitted laser light coherence is over what period of time the emitted light is acting as a perfect sine wave. One way to check the temporal coherence is to build a Michelson interferometer which allows light to be split into two beams, sent along different paths (of different travel times), and then recombined to see if the light is temporally coherent. A Michelson interferometer set up is shown in Figure 6. The light from the laser is split into two beams with a beam splitter, sent to two mirrors, and then recombined at the beam splitter. The two beams are then expanded with a lens and viewed on a screen. The light on the screen will show a series of fringes, which correspond to constructive and destructive interference of the two beam. If fringes can be seen, then the light in the two beams is coherent with respect to each other. If the distance from the beam splitter to the mirrors is different for the two paths then temporal coherence exists.Figure 6: Theoretical setup of interferometer. III. Procedure Part A: Gaussian Beam Set-up 0. Look up “Gaussian Beam” in the index of Kasap. Add a reference to page 6. 1. Mount your Class II He-Ne laser. 2. Get out your optical power meter and mount the photo detector at least 50 cm from the laser. 3. Get out your 1-dimensional translational stage and mount your six-inch metal ruler to it. This will be used as the knife-edge from the theoretical setup shown in Figure 3. You may find a cross post adapter useful. 4. Mount the knife-edge between the laser and the photo detector. The knife-edge should be about 20 cm from the front of the laser. Experiment 1. Measure the actual distance between the front of the laser and the knife edge. Measure the power of the laser beam versus knife-edge position and develop a plot as shown in Figure 5. Take measurements at intervals of 0.05 mm (every 5 ticks on the micrometer adjustment knob) as the knife edge is inserted into the beam and as the knife edge is removed from the beam. Plot your data for both inserting and removing the knife-edge.If your