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?