ENE 451 Homework 4 No due date 1. For a symmetric Gaussian beam with = 1 μm, w0 = 3 μm, n = 1, calculate w and R for z-z0? = 0, 10, 20, 50, 100, 200, 500, and 1000 μm. Calculate the power density in W/cm2 along the beam axis for these values of z-z0, assuming that the total power is 1 mW. 2. Consider the use of a single lens with f = 2 cm as a beam expander, in conjunction with a 5 cm diameter, f/1.6 lens used to produce a small spot Assume that the beam incident on the 2-cm focal length lens is collimated, with w0 = 1 mm, and that we use the e-4 criterion with the f/1.6 lens. (a) How far apart should the lenses be? (b) Where is the image plane for the f/1.6 lens? (c) How large is the focal spot formed by the f/1.6 lens ( = 0.8 μm, n = 1)? (d) How large would the spot be if the input to the f/1.6 lens were collimated? 3. A collimated Gaussian beam 5 mm in diameter to e-2 relative power density from a Nd: YAG laser ( = 1.06 μm) is incident on an f/3.0 lens 2 cm in diameter. Total power of the beam is 5 mW. (a) What are the diameters of the beam and the power density in W/cm2 at the center of the beam in the focal plane of the f/3.0 lens? (b) An f/1.5 lens 1 mm in diameter is placed a distance d to the right of the f/3.0 lens. For what range of values of d will the e-2 power points be included within the aperture of the smaller lens? (c) What values of d within the range of values determined in (b) will give the smallest spot in the image plane of the f/1.5 lens? (d) For the value of d determined in (c), where is the image plane of the f/1.5 lens located relative to it? (e) What are the diameter of the beam and the power density in W/cm2 at the center of the beam in the image plane of the f/1.5 lens?