K. Wood

# Waves and Optics Lab

#### Spring 2017

 Course Information Expectations Lab Manuals and Notes

### Lab Manuals & Notes

# Date Experiment (lab manuals linked)
1 Jan. 31, Feb. 1 Resonance
The main objective here is to measure the energy damping rate γ in the two different ways outlined in the manual: directly and via a frequency sweep. Analyzing the amplitude oscillations in both the time domain and the frequency domain will give us a handle on γ. We can then compare the results against one another, and they should be consistent.

Sidenote: The setup reminds me of the detection technique of Atomic Force Microscopy (AFM). I spent a summer of my undergrad modeling AFM for a simulation and wrote a little report outlining some of the basics - check it out if you're interested.
2 Feb. 7, 8 Coupled Oscillators
In this lab we're going to model a system of carts (masses) connected by springs and sliding on a ~frictionless track as a system of coupled harmonic oscillators. By driving this system at various frequencies and recording the resulting amplitudes of steady state motion (aka performing a frequency sweep) we will deduce the frequencies of the normal modes. There should be one normal mode per cart (mass). Can you qualitatively describe the motion of all normal modes for 2 and 3 carts?

We will do this for a 2 cart system and a 3 cart system. Think about how you can analyze these frequency spectra to compare with theoretical predictions (Hint: see the discussion in the manual about the ratio of normal mode frequecies.)
3 Feb. 14, 15 Speed of Sound in Solids
Again, we will measure the same observable two different ways and compare the results (are you sensing a pattern?). This time the observable will be the speed of sound through a solid tube.

Both techniques make use of piezoelectric transducers - so be sure you understand what these things do, and how we're going to use them. First, we measure the speed of sound directly by sending a pressure wave down the tube and measuring how long it takes to arrive at the end. We will also set up standing waves inside the tubes to deduce the speed of these (sound) waves.
4 Feb. 21, 22 Wave Propogation in Periodic Structures
Our main objective is to measure the resonant frequency of our system and compare to the expected value of ω0 = (LC)-1/2 given the reported L & C values of our circuit. We will also study the dependency of phase velocity on driving frequency ω.

So how do we measure ω0? The manual motivates the relation
ω = 2ω0sin(ka/2) ≈ (ka)ω0
for ka<<1. Then if we plot ω vs. ka, the slope in the linear regime (ka<<1) will give ω0. We control ω, but how can we measure ka? Simple - we measure how many LC units our wave needs to traverse to achieve a phase shift of 2π , call it N' (in units of a). Then
λ = N'a ⇒ ka = 2π/N'.

Now we won't always observe a full 2π phase shift, in which case we will need to extrapolate our data. In doing so, there will be some uncertainty in N' associated with the uncertainty of the slope in your linear fit. Schematically: 5 Feb. 28, March 1 Polarization of Light
This lab is very straight forward. Our setup will consist of a source of randomly polarized light (a candescent bulb will do) that will traverse a system of various polarizers and quarter wave plates before reaching a detector that will measure the transmitted intensity of the light.

Begin with two polarizers between the source and detector and measure the transmitted intesity as a function of relative angle between the two polarizers. We can use Jones matrices to predict the behavior, and you should plot the result (the continuous function) I/I0 = cos2θ. After measuring I/I0 as a function of θ, you should put your data point on top of the theoretical curve. If Nature agrees with the theory (and we make careful measurements and properly quantify the uncertainty), our data points should overlap with the curve within their uncertainties.

Next repeat this excercise with the two polarizers oriented in such a way to produce a minimum (θ = π/2) and place a third polarizer in between. Change the relative angle between the first and middle polarizer. The results might surprise you. Think about it!

Finally, replace the middle middle polarizer with a quarter wave plate and follow the instructions in the manual. [Hint: think about the difference between elliptical polarized light and circular polarized light.]

6 March 7, 8 Michelson Interferometer
After getting a feel for how sentive an instrument the Michelson interferometer is, we will use it to measure the wavelength of the laser light. You should compare your result with the value given on the laser. Next, we will use this measurement (and others) to determine the index of refraction for air, given by

n=1+(λpatm/2L)(m/Δpgauge)

Here L is distance the laser travels in the gauge (where we control the pressure pgauge). By measuring the number of fringe shifts as a function of Δpgauge we can determine (m/Δpgauge) and thus measure the index of refraction of air.

BE MINDFUL OF UNCERTAINTIES AND PROPOGATE ALL ERRORS!
7 March 21, 22 Fabry-Perot Interferometer
Once you've set up your Fabry-Perot interferometer you should observe the characteristic "bulls-eye" pattern on your viewing screen (or wall). We're going to characterize the Fabry-Perot by measuring the finesse of the instrument. Once again, we will do this using two different methods and compare our results.

First, we will extract the finesse by measureing the transmitted intensity as a function of angular displacement of the diffraction pattern. Use a photodetector with an aperature covering the lens that is sufficiently small to provide the necessary spacial resolution (small compared to the fringe width) yet big enough to let in enough light for our detectors to work. The FWHM of the first peak in the diffraction pattern is related to the finesse via
F = 16/(FWHM)2.
As described in the manual, we can also determine the finesse by measuring the reflectance of FB mirrors.
F = 4R/(1-R)2.
Do this and compare with your result from the diffraction pattern.

NOTE: Last week I mentioned that the exciting result from LIGO came from what is essentially a Michelson interferometer. LIGO actually uses Fabry-Perot interferometers at the end of its Michelson interferometer legs to increase the optical pathlength and increase their sensitivity by a factor of 1000!
8 March 28, 29 Diffraction
We're going to be looking at a variety of diffraction patterns, but we'll focus on measuring the intensity profile of single slit and double slit diffraction patterns. After measuring I(θ)/I0 I'd like for us to use the same data visualization technique as we did in the polarization lab. That is, I'd like for you to plot the continuous theoretical curve and put your discrete data points on top. This makes it very convenient to visually compare your data to the theory. I used some of Seth Messer's (Spring 2016) data to make the following example: (Although I really should have included a legend.) I did this in Mathematica - here's the notebook for your reference. You'll need to download it and open it in Mathematica for it to make sense... otherwise it's just a ridiculous looking text file. As always, let me know if you have questions!

In case some of you prefer python over Mathematica, I made the same plot using the (standard) matlibplot package in python. Check it out.
9 April 4, 5 Optical Instruments
We're eventually going to make a telescope using two lenses. Decide on which two lenses you'd like to use and measure the focal lengths using (i) the direct method (ii) Bessel's Method (iii) Abbé's Method.

Next, position these lenses correctly to make a telescope. Be sure to check your beam's collimation to verify you do in fact have a telescope (beam width should be constant/focus at infinity). The magnitude of the magnification "should" be given by the ratio of the two focal lengths. Lets check this. We measured the focal lengths so we can predict what the magnification "should" be (be sure to propogate errors). Now lets measure the magnification directly. Send a laser beam through your telescope and measure the beam width (≡ FWHM). The ratio of this with the beam width of the laser without going through the telescope should give you the same magnification that you calculated using the focal lengths.
10 April 11, 12 Gaussian Beam Optics
11 April 18, 19 Laser Speckle