Lab #4 - Gaussian Beams ECE/PH 482/582 Goals: 1. Verify that the transverse beam intensity for a laser operating in the fundamental or TEMoo mode follows a Gaussian profile. 2. Calculate and experimentally verify the beam waist and divergence angle for a typical HeNe laser using a commercial beam profile instrument, the Photon, Inc. Beam Scan. 3. Build a beam-expanding telescope, expand the HeNe beam to a larger diameter, and measure its expansion. 4. Verify the focusing of a Gaussian beam and how it varies with incident beam waist. 5. Examine and compare the beam shapes of the HeNe laser and a collimated visible diode laser and understand why they are different. Reading: Kuhn, Laser Engineering, Chapter 4: sections 4.3 – 4.4 Other Resources: Two classic papers on the subject – both are posted on the class web site under “Links”. Kogelnik and Li, Laser Beams and Resonators, Proc. IEEE, vol. 54, no.10, pp. 13121329, Oct.1966 Kogelnik, On the Propagation of Gaussian Beams of Light Through Lenslike Media Including those with a Loss or Gain Variation, Applied Optics, Vol. 4, no. 12, pp. 15621569, Dec.1965 Yariv, Quantum Electronics, 3rd Edition, (Wiley and Sons), Section 6.6 Java applet - www.iap.uni-bonn.de/lehre/ss01_laserphysics/GaussianBeam.html Preparatory Questions: Consider a HeNe laser (λo = 632.8 nm) with a beam waist wo = 0.5 mm: 1. Assume the beam waist is located at the output mirror of the laser. Calculate the size of the beam waist w(z) after the beam travels across the laboratory. (Assume the lab is ~5 meters long.) 2. Now assume you focus the beam down into a spot 0.1 mm in diameter. Calculate the size of the beam waist after the new Gaussian beam travels across the laboratory. (Assume the lab is ~5 meters long.) 3. Now, assume you build a small Keplerian telescope and expand the beam to 5 mm in diameter. Calculate the size of the beam waist after this new Gaussian beam travels across the laboratory. (Assume the lab is ~5 meters long.) 4. In the lab, you will use a short focal length and a longer focal length lens to make a Keplerian telescope to expand the beam diameter by 5x. Sketch such a telescope design and specify the longer f.l. lens focal length required if the short f.l. is 1 inch. 5. The equation for the radial variation of intensity for a TEMoo Gaussian beam is given below in Eqn. (1). Rewrite this equation so that the x-axis is “r2” and it gives a linear relationship in r2 from r2 = 0 to ∞. [Note: This would “prove” that the Gaussian relationship holds.] Laboratory Experiment: The transverse intensity profile of the TEMoo mode is a Gaussian distribution and is given by 2 r 2 I (r ) I 0e ( z )2 (1) where Io is the peak beam intensity at r = 0, r is the transverse beam radius, n is the refractive index of the medium (usually air so n = 1), and w(z) is the spot size at a distance z. The spot size represents the beam radius at which the intensity has decreased to 1/e-2 times its peak value. The peak value I 0 occurs at r = 0. Note that as z approaches infinity, w(z) varies as z, and hence I ( r ) varies as 1/z2. In the far field then, the beam acts as an expanding spherical wave with a finite angular extent. There is another parameter appearing in the above equations, which is of particular interest. The beam waist at a distance z is given by: 2 z z R w( z ) w0 1 (2) Where zR is the “Rayleigh range” or “confocal parameter” and nw0 2 zR 0 (3) The minimum beam radius is wo. It is a function of the cavity length of the laser d, the radii of curvature of the mirrors R1 and R2, and the wavelength of the light λo. 0 2 n d ( R1 d )( R2 d )( R1 R2 d ) ( ) 0 ( R1 R2 2d )2 (4) The last major beam property we are interested in is the far field divergence angle. This is given by 1 / 2 0 n 0 (5) where 1/2 is the half angle shown in the Figure 4.3 in the text (Kuhn) as θa. Using the “1/2” subscript reminds us that it is only the half-angle of divergence and not the full angle. The divergence angle can be used to estimate the spot size of a focused beam. Procedure: 1. Make measurements of the HeNe transverse beam intensity at a fixed value of z of ~ 1 m. This is done by carefully aligning the center of the HeNe laser beam reflecting off the rotating mirror to pass directly across the pinhole in front of the Si PIN detector. ω (rev/s) HeNe Laser r PIN Detector Scan speed across pinhole is 2(2π r ω). Pinhole [Why the “2” in front? Derive this.] 2. Plot the oscilloscope trace and use the oscilloscope cursors to measure x at several points on the trace. These measurements will be used to calculate the radius r of the beam. Measure and record your distance of z. 3. Now shine the laser beam against a far wall and calculate the far-field divergence angle. [Note: Measure the beam "diameter" of the visible spot you can easily see. This is a good approximation of 2w.] Use Eqn. (2) in the far-field limit to estimate the minimum beam waist of the laser wo. From wo calculate zR. Were you really measuring in the “far-field,” ie. z >> zR ? 4. Use the Photon Inc. "Beam Scan" to record the transverse profile and beam diameter of the HeNe laser with z equal to the distance chosen for the part1 measurement. Look at the “signal out” trace from the Beam Scan on the oscilloscope, measure the full width at the 1/e2 power points, and verify that this width agrees with the meter reading of the 13.5% diameter. [The Beam Scan scans a slit across the beam at a linear rate of ______ cm/s based on the diameter of the slit cylinder and its rate of rotation.] 5. Rotate the Beam Scan head by 90o to measure the HeNe beam parameters in the perpendicular direction. Are they they same? Is the beam round? 6. Once you have verified that the Beam Scan is giving the correct value of 2w use mirrors to measure the HeNe laser beam diameter for 5 different values of z ranging from close to the laser to across the room. If the beam gets too large for the Beam Scan (~7 mm diameter), just measure with a ruler. Using these values, you will plot w(z) later. 7. Now place the 10 cm focal length lens in front of the laser to transform to a new Gaussian beam and re-measure the beam diameter as a function of z for the focused beam. What is the minimum spot size you measure at the focus of the lens? What is the new divergence angle of the beam? Which slit (Aperture #1 or #2) did you use on the Beam Scan? Why? 8. Now use the two lenses – one short f.l. and one longer f.l. – to construct a beam expanding telescope to expand the HeNe laser beam to ~5 mm diameter leaving the longer f.l. lens … and collimated. Measure the far-field (across the lab) diameter of the beam. 9. Finally use the Beam Scan to measure the “collimated” beam dimensions of the diode laser module in the lab. Is this beam round like the HeNe laser? What are the beam dimensions you measure in the two perpendicular directions? Measure the far-field divergence angles in both axial directions. Questions to Answer: 1. Make a plot of the ln(I/Io) vs. r2 for the HeNe laser beam a distance of 1 m from the laser which you measured using the rotating mirror. 2. The PIN detector outputs a signal voltage. Is this signal voltage proportional to the optical beam’s electric field (V/cm) or to its intensity (W/cm2)? Explain your answer clearly. [This is a critical point for all these measurements.] 3. Using your plot in part 1, verify that the beam profile is indeed Gaussian and determine the “spot size” w(z). For the remaining values of z, you can just read off the 1/e2 power point from the Beam Scan once you have used a scope to check this value from the Beam Scan unit. 4. Knowing w(z), calculate the minimum beam waist wo and beam divergence angle θfull for the laser without a lens. Repeat for the measurements taken using the 10 cm f.l. lens and for the expanded beam. Sketch all three cases showing where the minimum beam waist is located. 5. Why must the pinhole used on the detector be much smaller than the beam diameter? 6. Describe the beam from the collimated diode laser giving its waist dimension(s) and divergence angles. 7. Comment on anything you observed that didn't quite fit the "conventional theory". Conclusion: After this lab you should have a good understanding of how Gaussian beams behave and how to calculate and measure important beam parameters. A solid knowledge of Gaussian beams is very important when working with lasers and optical systems that use lasers, such as fiber optic communications, free space communications, LIDAR, spectroscopy, etc. Many times a laser must be focused to do machining or to be input into an optical fiber or nonlinear crystal, so it is important to be able to calculate what the spot size and the Rayleigh range are for the focused beam. Revised Oct 2009 tkp

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# Gaussian Beams