# Homework Help: Relative Uncertainties

1. Apr 13, 2017

### says

1. The problem statement, all variables and given/known data
How can I calculate the relative error of a laser beam waist and angle of divergence using this experimental data:
I have measured the increase in laser beam diameter as the distance from the laser increases. With this data I plotted a graph of distance from the laser beam2 vs. laser beam diameter2.

2. Relevant equations
half angle of divergence = 2*λ / π*s0
s2=s022z2
s0: beam waist
z: distance of beam from source
Φ: angle of divergence

3. The attempt at a solution
I fitted a LINEST trendline in excel to the graph of laser beam2 vs. laser beam diameter2, which equals y = 4.7E-7x + 3.4E-1. I also got absolute uncertainties in both of these values, being 5.5E-9 for the 4.7E-7, and 6.24E-3 for the 3.4E-1.

I know that from the trendline 3.4E-1 = beam waist2. Taking the √3.4E-1 gives me the beam waist = 5.8E-1. Converting this beam waist to 1/e2 gives a beam waist = 8.2E-1 mm

If I use the equation: half angle of divergence = 2*λ / π*beam waist = 4.8E-4. Multiplying by 2 and the full angle of divergence = 9.6E-4 rad.

beam waist = 8.2E-1 mm and angle of divergence = 9.6E-4 rad are consistent with the manufacturers specs, but I'm not sure how to calculate the relative uncertainties of each.

Any guidance would be much appreciated.

2. Apr 19, 2017

### PF_Help_Bot

Thanks for the thread! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post? The more details the better.