# period detection for a series of photons

I have a dataset which records the arrival time for a series of photons.
I am using lomb-scargle method of scipy.signal to try to find some kind of period.

Since the arrival time is random,how to express the intensity of the signal?
If I bin the dataset and let the photon counts in every bins be the intensity , the dataset changes to a evenly spaced new dataset and information is lost.

lomb-scargle is good at finding period for unevenly spaced data.
Is it necessary to bin the data? If I bin the data, part of the data info will be lost(?) and I can use fft directly.

• The light intensity is usually defined as $\langle n \rangle^2$ which denotes the mean square of the number of photon arrivals in a certain time period $T=1/B_\mathrm{s}$ where $B_\mathrm{s}$ is the signal bandwidth. But I'm not sure whether this was your question... – Deve Dec 11 '13 at 16:05
• The arrival times of the photons are random. You mean I should bin the time,then let the photon numbers in every bin be the signal intensity? – questionhang Dec 12 '13 at 1:52
• The square thereof, yes. The outcome will of course depend on the length of the time interval you choose. As you said, the arrival time is random and light intensity is a function of photon count. So when counting the photons the information of their original arrival time is lost. But I don't see any other possibility. – Deve Dec 12 '13 at 7:51
• Q1 How much does it affect the period detection if the original arrival time is lost? It is a big problem? Q2 You mean I should use the photon number or the square of photon number? Is there any reference about your <n>**2 – questionhang Dec 12 '13 at 9:20
• A1 The choice of time period will be important. Does your dataset consist of arrival times each value representing the time at which one photon arrived? What quantity do you expect to change periodically? Light intensity? A2 I was wrong here. Light intensity is just $\langle n \rangle$, sorry for the confusion. – Deve Dec 12 '13 at 15:59

## 1 Answer

As discussed in the comments to your question the recorded data that consists of arrival times of photons must first be converted to photon counts $n$ for this is the light intensity. The result of this conversion will be a vector $n(k)$ (with discrete time $k$) that represents the evolution of light intensity over time and that can be analyzed in more detail with an FFT, for example. The Lomb-Scargle method won't help here as you'd like to analyse light intensity, not photon arrivals. It could be of use if your data was a sequence of unequally space light intensity values. But as I understand your question this is not the case.

In order to obtain $n(k)$ count the number of photon arrivals in every time interval $t_0 + kT$, where $t_0$ is some constant time offset (e.g. first arrival time) and $T$ is the length of the equidistant time intervals. $T$ determines the sampling frequency $f_\mathrm{s} = 1/T$ of $n(k)$ and consequently the range of any spectral analysis, that is $0 \ldots f_\mathrm{s}/2$. On the other hand $T$ should be sufficiently large to obtain a reliable photon count in each time interval.

To analyse the periodicity of $n(k)$ calculate its FFT and look for distinct peaks. Functions like periodogram also come handy sometimes.

• functions like periodogram? You mean in matlab? – questionhang Dec 13 '13 at 12:33
• For example, yes. There's no such function in Scipy? I'm not familiar with Scipy. – Deve Dec 13 '13 at 12:34
• In fact ,I have written a fft code to do that. [stackoverflow.com/questions/20074930/… But there are many fake peaks in the power spectrum.However I do not know how the python modules get the power. In this question, do you think there is a ready made function or formula to get the power? – questionhang Dec 13 '13 at 17:04
• The power of what? Light intensity is proportional to power already. Also, it's quite natural that your data has a DC offset als light intensity cannot be negative. Can you tell us what kind of signal you are measuring, what you get and what you'd expect? – Deve Dec 14 '13 at 10:54
• A light source emits photons randomly.The intensity, which defined by photon number in a time unit, may change as time elspses. I need to detect whether there are intensity periods. So I make a power spectral diagram. – questionhang Dec 14 '13 at 11:53