This post introduces some of the concepts that are important when analyzing the data measured from the Holometer. The last post introduced the concept of an interferometer. In the Holometer, the photodiodes measure the intensity of the laser at the dark port. As described in the previous post, if everything was perfectly aligned and nothing moved, the dark port would be completely dark. However, in reality, everything is moving a bit and the intensity of laser at the dark port varies. The photodiode produces a voltage corresponding to the intensity of the laser which is sampled by a computer-based measurement system. (I’ll write much more about the computer-based measurement system later since it is the one part of this experiment with which I have expertise.) If this voltage is plotted over time, it looks like noise. That is, there is no discernible pattern to the voltages. Here is an example (not from a photodiode, but a simulated signal):
It is very difficult to see repeating patterns in a signal when there are either multiple patterns present or there is significant random noise. In order to see these repeating patterns we travel to a parallel universe of sorts. In this parallel universe we observe the signal in the frequency domain (amplitude as a function of frequency) instead of in the time domain (amplitude as a function of time). This is done with a Fourier transform. I’m not going to focus on how to calculate Fourier transforms of signals, but I do want to describe Fourier transforms conceptually.
Let’s start with a sine wave with a frequency of 4 Hz, an amplitude of 1 V, and a phase of 0 degrees. Frequency is how many cycles of the sine wave are in every second; amplitude is the difference in volts from the equilibrium position (0 V in this case) to the maximum voltage; and phase is where the sine wave starts at time 0 s. I’m not going to worry about the phase at all. It looks like this in the time-domain (graph of amplitude in Volts vs. time in seconds):
It could also be described in the frequency domain (amplitude in Vrms vs. time in seconds):
The amplitude of the signal in the frequency domain is zero everywhere except at 4 Hz, where it is 0.7 V (Vrms value of an 1-V amplitude sine wave). (For those of you wondering, I’m not going to discuss the phase information in the frequency domain.)
What if our signal is the combination of two sine waves? Let’s add a second sine wave with a frequency of 10 Hz, an amplitude of 1.5 V, and a phase of 0 degrees. The signal still looks like a repeating pattern, but is more complicated:
However, the signal is easily described in the frequency domain:
The amplitude of the signal in the frequency domain is zero everywhere except at 4 Hz and 10 Hz.
The idea behind a frequency-domain representation of a signal is that any signal can be represented as a combination of sine waves of various frequencies, amplitudes, and phases. Here’s an example of a signal that is a combination of five different sine waves of various frequency and amplitudes:
and here are the five frequencies clearly shown in the frequency domain:
Good luck determining that the time-domain graph was comprised of five sine waves of frequency 4 Hz, 10 Hz, 13 Hz, 15 Hz, and 18 Hz.
Let’s go back and look at the original signal in the frequency domain:
It is now evident that what appeared to be random noise is composed of a 60 Hz signal. The signal to noise ratio for this example was worse than 1/50. That is, the amplitude of the noise was more than 50 times greater than that of the 60 Hz signal. This is the power of analyzing signals in the frequency domain. When performing this analysis, the Fourier transform is calculated multiple times and the results are averaged together. This reduces the influence of the random noise and makes the repeating signal more apparent. There are other techniques besides averaging to improve the measurement such as windowing, but we aren’t going to worry about that for now.
There are many things that contribute to the noise measured by the photodiode in the interferometer. A truck driving by, seismic activity, vibrations due to a fan, 60 Hz AC power, and the digitizer itself are all possibilities. In addition, the holographic noise presented in the first post in this series is also present. If the two interferometers which make up the Holometer are isolated from each other, than much of their noise will be uncorrelated, that is, not related to each other. Some sources of noise would be correlated, such as seismic activity and a truck driving near the experiment. However, these sources of noise occur at much lower frequencies that the holographic noise. If the two interferometers overlap in their space-time (I’ll explain this more later), than the holographic noise should be correlated. We can remove uncorrelated signals between two signals by calculating the cross power spectrum. As an example, suppose I have two signals:
We can filter out uncorrelated signals by calculating the cross-power spectrum for the two signals (again, just focus on the concept and not how the calculation is done). The graph of the resulting cross-power spectrum for these two signals is:
Whoa! There is a 60 Hz correlated signal between the two signals. This signal was lost in the noise when looking at the individual signals in the frequency domain because it was so much weaker than the other components (unless you look really carefully). However, by minimizing the uncorrelated frequencies between the two signals, the 60 Hz correlated signal becomes more apparent. (The uncorrelated 80-Hz and 110-Hz signals are not completely eliminated, but they are minimized.)
The Holometer experiment uses these techniques to search for the correlated holographic noise signal between two adjacent interferometers.
I have the opportunity to meet with Craig Hogan this week and ask him some of my questions which should help me more clearly explain the ideas of the holographic principle, in general, and the Holometer experiment, in particular. I hope to have clear answers for questions such as:
- Why is the interferometer 40-m long?
- Given the incredible small scale of the Planck length, how to the two adjacent interferometers “share” the same space-time?
- What is the connection between the holographic information principle and the Planck length?
- Why is the holographic noise expected to be around 4 MHz?
- What’s up with this?
What questions do you have?
This post is one in a series about The Holometer experiment and my work at Fermilab in the Summer of 2011:
- Holometer: Holographic Noise
- Holometer: Interferometer
- Holometer: Spectral Analysis
- Holometer: Transverse Jitter
- Holometer: Correlated Interferometers
- Holometer: Computer-Based Measurements