EE421 Gaussian Waves
1. The Gaussian output of a helium neon laser is used to illuminate a target. a) If the laser has an output beam diameter of 1 cm. What is the maximum distance that the beam can travel before the diameter has expanded to 10 times the waist? b) If the laser has an output beam diameter of 10 cm. What is the maximum distance that the beam can travel before the diameter has expanded to 10 times the waist?
2. A TEM
0,0
Gaussian wave is traveling through space with a wave length of .5 um. The wave enters a thin lens with a focal length of .5 meters. When the wave enters the lens it has a curvature R
1
= 1 meter and a spot size w
1
(z) = 10 cm. What is the curvature and spot size of the wave just as it exits the lens?
.
3. Given the end pumped 4-level laser operating at 5000 angstroms. The rod has an ion density of 2 x 10
19
ions/cm
3
. The rod has a length of 15 cm and a diameter of 1 cm.

A) What should be the reflectance (0% or 100%) of the beam splitter at the pump wavelength?
B) What should be the reflectance (0% or 100%) of the beam splitter at a
C) wavelength of 5000 angstroms?
Where in the cavity would the largest spot size be?
D) Where in the cavity would the smallest spot size be?
E) What is the maximum theoretical energy that can be extracted from the rod as coherent radiation?
G) Assume the atomic line width is 1000 megahertz and calculate the number of longitudinal modes that can be expected in the output. For simplicity, assume the laser material has an index of refraction equal to one.
H) Find the system matrix and determine the stability of the laser cavity.
Again, for simplicity, assume the laser material has an index of refraction equal to one.

4. A single mode fiber cable is used to carry control signals for Miller’s new truck. The core index n1 = 1.460, cladding index n2 = 1.450, and outside index n
0
= 1.000. The signal wavelength

= 6000 angstroms. a) Find the maximum core diameter for this cable. b) Find the maximum beam entrance cone angle 2
 m
. c) What is the minimum skip distance for a signal traveling in this fiber?
5. Given a ruby laser with a rod 12 cm. long and 1 cm. in diameter having a chromium ion density of 1.50 x 10
19
ions/cm
3
.
a) What is the theoretical maximum energy that can be extracted from this
laser in the form of coherent radiation?
b) If the slope efficiency is 1.2% and the (threshold lamp input) / (rod volume)
is 300 J/cm
3
, find:
1) The lamp input at threshold.
2) The required lamp input for a 12 joule output.

Gaussian waves:
An optical cavity has one plane output coupler mirror and one curved feedback mirror with a radius of 20 meters. The mirror spacing is 1/2 meter. Assume the cavity is stable and is resonating with a wavelength of 6000 Angstroms. Find the minimum beam diameter in the cavity. Also, find the mode volume for the TEM
0,0
mode for this cavity.
Hint: Use the ABCD law for gaussian waves starting at the assumed waist. Remember the wave curvature and size repeat at any Z plus a round trip.