How to Calculate Relative Uncertainties for Laser Beam Waist and Divergence?

[says]

Homework Statement

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.

Homework Equations

half angle of divergence = 2*λ / π*s₀
s²=s₀²+Φ²z²
s₀: beam waist
z: distance of beam from source
Φ: angle of divergence

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.

 

[anathema]

Thank you for sharing your experimental data and approach. Calculating relative error for the beam waist and angle of divergence can help to determine the accuracy of your measurements. To calculate relative error, you will need to compare your experimental values to a known or expected value.

For the beam waist, you can compare your calculated value of 8.2E-1 mm to the manufacturer's specified value. Let's say the manufacturer's value is 8.0E-1 mm. To calculate the relative error, you would use the formula: (experimental value - expected value)/expected value * 100%. In this case, the relative error would be (8.2E-1 - 8.0E-1)/8.0E-1 * 100% = 2.5%.

For the angle of divergence, you can use a similar approach. Let's say the manufacturer's specified angle of divergence is 1.0E-3 rad. To calculate the relative error, you would use the formula: (experimental value - expected value)/expected value * 100%. In this case, the relative error would be (9.6E-4 - 1.0E-3)/1.0E-3 * 100% = -4%.

It is important to note that negative relative error indicates that your experimental value is lower than the expected value, while positive relative error indicates that your experimental value is higher than the expected value. Ideally, you want your relative error to be as close to 0% as possible, indicating a high level of accuracy in your measurements.

I hope this helps to guide you in calculating the relative error for your experimental data. If you have any further questions, please do not hesitate to ask.