Lab #1: Photon counting with a PM tube statistics of light James. R. Graham, UC Berkeley

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1 Lab #: Photon counting with a PM tube statistics of light James. R. Graham, UC Berkeley Your report is due on 2008, September 6 at 5:59:59PM PDT. NO EXTENSIONS! Overview. Schedule This is a two-week lab 9/2 & 9/9. Your lab report is due on 9/6. For show-and-tell on 9/9 you should have progressed at least to step 7 (see below)..2 Goals Explore the fundamental physical limitations on the detection of light. Investigate how precisely brightness can be specified, and what determines that precision. Ultimately, what are the statistical properties of photons?.3 Reading assignments (available on line) Unix tutorial IDL tutorial TeX tutorial Taylor Chapters & 2.4 Key steps You will execute four primary steps in this lab:. Collect data from the photometer experiment and use IDL to make a plot of the counts per sample versus time. 2. Plot histograms of sets of samples from the photometer. 3. Compute the mean and standard deviation for your samples and investigate the variability of the count rate. 4. Compare the observed histograms with the theoretical Poisson probability distribution function.

2 2 Getting started: collecting data from the photomultiplier tube This lab activity is based on a light detector that can record the arrival of individual photons. The detector, known as a photomultiplier PMT, is very sensitive, and must be treated with care. Before you modify anything associated with the PMT make sure that you have read and understood the PMT web page: Do not expose the PMT to direct room light! Do not expose to light so that the count rate is > MHz. To figure the count rate multiply the counts per sample by the sample rate.e.g., 000 counts per sample when the sample rate is khz corresponds to a,000,000 counts per second or MHz. The first step is to acquire a time sequence of digitized data and save the data as a file. Log into one of the sun workstations. Make a Unix sub-directory (using mkdir) for your project and type the following at the Unix prompt: % echo counter nsamples=00 rate=000 fname=c dat sendpc (Note: % indicates the Unix prompt do not type an extra %). This cryptic command takes 00 samples of data from the photo-multiplier tube at a rate of 000 Hz, and puts the data in a file called c dat. A cartoon of the counting process is shown in Figure. Δt time Sample n Sample n+ Sample n+2 Figure : A cartoon of the samples generated by the counter. The bin width is constant, and for an interval Δt = /rate photon counts (smiley faces) are accumulated. In this example the sequence of counts is 3,, 2, 4. Only four samples from a longer sequence are illustrated. You should get a message indicating success. Since you asked for 00 samples, this file should contain 00 lines. Figure out for this example how long is the counter active for? What would the answer be if you had chosen: nsamples=000 rate=00? 2

3 A rate of 000 Hz means that for each sample the computer accumulates counts from the PMT for /000 of a second ( ms). Typically, at this rate you should get few counts per sample. Play around with the sample rate and convince yourself that the length of the sample is inversely proportional to the sample rate. If the rate is significantly higher or lower than this then ask an instructor for help. The maximum sample rate is 5000 Hz. Note that not all sample rates are supported by the hardware. If you ask for a counter rate that is not supported then the counter program will choose the closest one available. 3 Reading data The next challenge is to read in the data file into IDL. View your file by asking the computer to print out the contents, e.g., % cat c dat Notice that the file comprises a single column of numbers. Each number corresponds to the number of counts in each sample. There is a handy IDL program that reads a file into computer memory and labels it with an IDL variable name, in this example it is called "mydata": readcol, 'c dat', mydata Notice that the file name must be enclosed in single quotes. IDL names between single quotes are called strings. 4 Plotting data Once you have the data in an IDL variable you can plot it using the IDL command plot, mydata To make a useful graph you need to plot your data as a function of time. Label the x- (i.e., time) and y-axes (counts). Make sure that your labels include units. A plot title is also helpful and can be used to denote the sample rate. Invoke IDL help by typing a question mark at the IDL command line. Look up "PLOT" in the index and figure out how to add a title and axes to your plot. 3

4 Some useful on-line resources are To make your life easier we have provided a handy IDL function that calls that sendpc program for you so that you do not have to do so much typing. Here s an example: t = savephotons(nsamples=00, rate=00) Data saved in file: photon txt Note that savephotons makes up the file name for you if you do not specify it. This program is what is known as a function in IDL, i.e. a routine with arguments (the values of nsamples and rate) and a return value. In this case the return value, t, lets you know if the program worked. print,t The IDL code for savephotons can be found in /home/global/ay2/idl/pc/savephotons.pro So a set of IDL instruction to collect some data and plot them would be: t = savephotons(nsamples=00, rate=00) Data saved in file: photon txt readcol,'photon txt',data % READCOL: Format keyword not supplied - All columns assumed floating point % READCOL: 00 valid lines read plot,data 5 Statistics making a histogram First, be sure to review and understand what a histogram (see Taylor) Calculate the mean and standard deviation of the count rate for a set of data collected using savephotons. The IDL function total will turn out to be very useful for computing statistical quantities. Be sure to look up total in the IDL help. Bin the data and make a histogram of the counts. Binning means sorting the data into unique categories, and counting the number of occurrences of those categories. Use my example program in the statistics hand out to help you solve this programming problem. 4

5 Does the histogram plot that that you have made really reflect that data that you collected? Carefully compare the list of counts in the data file and the plot. Do a reality check on a data set where you can rapidly compute the histogram with pencil and paper. Once you can plot histograms with confidence, repeat the experiment, say, six times and compare the results of your experiments. Does the histogram change? Do you always get the same mean count rate? Become adept at inspecting the histogram plot and guessing what the mean and standard deviations are. Calculate the mean and standard deviation of the six count rates you just measured. Be sure to use a unique file name for each sequence of data, because you may want to go back and look at your data files again. If you do not change the name the file will be overwritten. To examine the data in IDL it will be helpful to understand how to use "for" loops. A quick and sophisticated way to approach repetitive tasks involves writing the entire sequence of data acquisition in IDL using for loops. The simplest IDL FOR loop can be executed at the command line. Try this: for i=0,4 do print,i Suppose we want to take a sequence of data sets where the rate increased by steps of a factor of two, then the following would be useful: for i=0,4 do print,2^i This is perhaps the most important programming lesson here. Always test your program on a problem where you know the answer. For example test your histogram program with x =[0,0,0,,,] and x=[0,,2,3,4,5,6]. 5

6 If you need multple instructions within the loop, then you need to write a little program. Create a new file with emacs called test.pro ; Testing savephotons with a for loop. Each time savephotons is called ; a unique file name is used. Written by JRG 8/25/2008 ; mynsamp = 00 ; Define the number of samples myrate = 000 ; Define the sample rate (in Hz) nrep = 4 ; Define the number of repitions for i=0,nrep- do begin ; construct the file name myfilename = 'test_' + $ string(i,format='(i03)') + '_' + $ string(mynsamp,format='(i03)') + '_' + $ string(myrate,format='(i04)') + '.dat' ; get the data t = savephotons(nsamp=mynsamp, rate = myrate, file = myfilename) ; print out some useful information print,'i= ',i,' file = ',myfilename,' status = ',t print,' ' endfor end Note that the semicolon is a comment character everything after a ; is ignored. To execute this program in IDL, all you need to do is type.run test % Compiled module: $MAIN$. Data saved in file: test_000_00_000.dat i= 0 file = test_000_00_000.dat status = Data saved in file: test_00_00_000.dat i= file = test_00_00_000.dat status = Data saved in file: test_002_00_000.dat i= 2 file = test_002_00_000.dat status = Data saved in file: test_003_00_000.dat i= 3 file = test_003_00_000.dat status = Notice that the file name is printed out, and each time that savephotons is invoked the new file name is used. 6

7 If you are plotting multiple sequences of data investigate what happens when you set the IDL plotting variable!p.multi = [0,2,3] What does this do? To get back to normal plotting set!p.multi=0 Now repeat with a bigger sample of data, e.g., increase the number of samples by a factor of 4, but keep the sample rate the same as before. Again, take six sets of data and repeat the exercise of calculating the mean and standard deviation for each longer set of data. Calculate the means and standard deviations of the six count rates you just measured. What do you notice about the mean count rate and the standard deviation for these sequences? 6 Mean and standard deviation Let's try and get to the root of these variations. First, let s explore the relation between the mean number of counts and the standard deviation. Take a sequence of data with increasingly long (i.e. slow) sample times. For this part of the experiment you need to know that the shortest sample is 200 µs (5000 Hz) and that the duration of the sample can only increase in 00 µs increments. Thus the allowable rates are 0000 Hz/n, where is an integer n = 2, 3, 4 A handy aspect of savephotons is that you can call it with the argument dt, which is the sample duration in ms, so that t = savephotons(nsamples=00, dt=0.2) t = savephotons(nsamples=00, dt=0.3) t = savephotons(nsamples=00, dt=0.4) are all valid data requests. Calculate the mean count for each sequence and also the standard deviation. Suppose xbar and s are the means and standard deviations, use the command plot, xbar, s^2 to make a plot of the mean versus the variance (the standard deviation squared). Now over plot a line representing x=y. What does this tell you about the relation between mean and variance for counting (Poisson) statistics? 7

8 7 The Poisson distribution Plot a histogram for one of your sequences with a small count rate, e.g, 2-4 counts per sample and lots of samples, e.g., 000. Calculate the mean count rate and compare the resultant histogram with the theoretical Poisson probability distribution, P( x,µ ) = µ x exp ( µ x! ), where µ is the mean. Use IDL's OPLOT function to compare your data and the prediction. Think about that! How do you compare a histogram and a theoretical distribution!? The Poisson distribution gives a probability. You have measured counts. Explain how to choose the correct scaling factor (or normalization) to compare the measured and theoretical distributions. Does the Poisson distribution provide a good description of the data? Now arrange so that the counts per sample is increased (be careful that the count rate does not exceed MHz). Aim for several hundred counts per sample. Plot the histogram again. What has happened to the shape of the histogram? Calculate the mean and standard deviation and over-plot the corresponding Gaussian probability distribution, P( x,µ,σ ) = 2π σ exp x µ 2 σ 2. Is a Gaussian curve a good approximation to the Poisson distribution? Under what conditions is the Gaussian probability distribution a good approximation? Some notes regarding Poisson fluctuations and blackbody radiation are included in the Appendix. 8 Standard deviation of the mean The more events you count the more accurately you can measure the number of counts per sample (i.e., the count rate). To illustrate the effect take ten sets of data with a given number of samples, say 6. Choose a fixed sample rate, say, khz. For each of these ten sets calculate the mean. Due to statistical variations the ten means will be different, so also calculate the mean of the means (MOM) and the standard deviation of the means (SDOM). The MOM is the best estimate of the counts per sample and the SDOM is a measure of how precisely we know the average counts per sample. How does the SDOM vary with the number of samples in the individual sequences? Intuition suggests that if we have more samples in each of our ten measurements the SDOM will be smaller. To quantify this effect repeat with 2, 4, 8, 6, 32, 64, 28, 256, 52, 024, 2048 etc. 8

9 samples per sequence. Don't vary the sample rate, and be sure that no one changes the LED brightness while you are collecting your data! For each sample size consider the ten data sets and calculate the mean of the means (MOM) and the standard deviation of the means (SDOM). Plot the MOM and the SDOM as a function of the number of samples. Describe how the MOM and SDOM vary as the sample size increases. Based on your knowledge of Poisson statistics and error propagation predict the SDOM given the measured mean count per sample and the sample size. Use the IDL OPLOT function to compare your prediction with the data. If I want to improve the accuracy of a measurement of the mean by a factor of two, by what factor do I need to increase the number of samples? How accurate is your best estimate of the count rate, i.e., how accurate is the MOM? Is it possible to construct a light source for the photometer experiment that would not show variations in the count rate? Write up your lab report describing each of the above exercises. Show your results by including IDL plots in your report. 9

10 9 Appendix: Photon noise Why do photons exhibit Poisson fluctuations? Is the detection of photons really governed by the Poisson probability distribution? Consider the number of photons, n, in a cavity temperature T, and frequency v. As photons are emitted and absorbed by the walls the number of photons in the cavity will change. Boltzmann s law gives the probability that a state with energy E = nhv occurs, where n may take on any value 0,, 2, 3, P(n) = U exp ( nhν kt ), where h is Planck s constant, and k is Boltzmann s constant. The normalization U is found from the condition that the probability must sum to unity The average energy at frequency v is E = ( ) U = exp nhν k T. = U = n =0 EP(n) U n =0 n =0 ( ) nhν exp nhν k T hν exp hν k T ( ) We can use this result to compute the energy density of photons. Photons are spin zero particles, and therefore unlike half-integer spin particles like electrons can congregate in a cell of phase space. The number of phase cells per unit volume for photons with momenta p to p + dp is dn = 2 4πp2 dp = 8πν 2 dν, h 3 c 3 where the factor of two accounts for the two spin polarization states. Thus, the energy density of blackbody radiation at frequency v to v + dv is which is the famous Planck formula. ρ ν dν = E dn = 8πν 2 c 3 hν exp( hν k T ) dν,. 0

11 The mean number of photons is n = np(n) = U n =0 exp( hν k T ), and Hence, the variance n 2 - n 2 is n 2 = n 2 P(n) = U n =0 ( ) + ( ) exp hν k T [ exp hν k T ]. 2 ( ) ( ) exp hν k T σ 2 = [ exp hν k T ], 2 or by substituting for n we see that the variance is proportional to n σ 2 = n exp( hν k T), with a correction factor. Often the correction factor is very close to unity. For example, Figure 2 shows the correction factor an incandescent lamp (3300 K) at wavelength between 0. and 00 µm. In the UV and visible we have to a very good approximation σ 2 = n, i.e., the fluctuations are Poisson. However, at infrared and radio wavelengths the Poisson approximation is not useful. Figure 2: The Boson correction factor /[-exp(hv/kt)] versus wavelength for a temperature typical of a tungsten filament lamp. At UV and visible wavelengths the correction factor is close to unity and blackbody radiation shows Poisson fluctuations.