Light Emitting Diodes Used as Avalanche Photodiodes: Errors, Statistics, Probability Distributions
This experiment is intended to demonstrate certain properties of random errors, statistical analysis,
and the probability distributions--reinforcing the theory you studied in Physics 362 and 204. Along the
way, we will re-learn about how to effectively use op-amps in a simple circuit.
Experiment 1A: Measurement Errors
Random errors may be studied in any measurement that can be repeated many times, provided that
the variations between readings are greater than the smallest observable unit. The difficulty is to find a
simple measurement that can be made quickly and in which the errors are large compared with the
graduations on the measuring instrument.
To generate our data, we are going to measure the number of photons detected in a particular time
interval by a simple photodetector. The photodetector we will use is a Light Emitting Diode (LED) that
would normally be forward biased to produce light. Here, we will reverse bias the LED so that the
absorption of a photon will cause a momentary pulse of current to flow through the LED - this is called
the 'Geiger' mode because the detector now behaves similar to a Geiger-Mueller tube. By detecting the
current pulse, we can count the number of photons hitting the LED/detector.
The voltage pulse produced is fairly small, so we will construct a circuit to increase the amplitude
of the pulse to about 5V. This will let us use a LabQuest interface to collect our data easily and quickly.
For this experiment, a good lab notebook should contain detailed descriptions and explanations
of (including references):
Semiconductors, p-n junctions, LEDs, photodiodes, and avalanche
photodiodes.1-3 Basically, I expect that you will be able to explain the physics behind the behavior we will
observe, you’ll want that information when you write a formal paper on this topic.
Detection circuit:
Figure 1 shows the basic circuit we will use to detect the signal from the LED. It uses a reversebiased LED as the detector, and a comparator to allow the voltage on the back side of the LED to be read
out without impacting the operation of the LED, much.
+5V
LM311
+15V
1 - 20 k
Vout
Vbias
100 900 k
+5V
10k
1
Figure 1: The circuit used to
detect single photons incident
on a reversed-biased LED
operating in the Geiger mode.
A 311 comparator is used to
condition the signal so that it
meets TTL signal level
requirements. The circuit is
discussed further in the body
of this handout. NOTE:
Figure 1 is not a sufficient
diagram for a laboratory
notebook, as it does not
describe the actual circuit that
will be created in the
experiment - more details and
actual pictures are required.
The LM311 Comparator is really just an op-amp, but without the feedback from the output to one
of the inputs. Use the Golden Rules to explain how this part of the circuit will allow us to turn our
initially small voltage pulse into a pulse with an amplitude of 5V.
Recall the two Golden Rules of an ideal op-amp: 4
1)
The output changes to make the voltage of the two inputs equal to each other.
2)
No current flows in or out of the inputs.
What is the purpose of the resistor that sits between the LED and ground?
Procedure:
Construct and test your single-photon detector circuit. Ideally, you want a circuit that gives zero
detection events ('counts') when absolutely no light is falling on it, and detects lots of photons (all of
them?) when light is shining on the circuit. Making this work well will entail adjustments of Vbias and
the second potentiometer, among other things. The use of a good oscilloscope to examine the pulses
produced by the op-amp and the comparator is highly recommended. You should verify (i.e. do tests that
provide actual evidence, like a graph) that the detector is light sensitive. All of this work needs to be well
documented in your lab notebook.
Once everything is working, choose a time interval and measure the number of counts per interval
over many intervals. Correct the data for the background signal, if one exists. Use Chauvenet's criteria, as
appropriate, to remove data points.5,6
(I)
For the entire valid data set, calculate the mean, variance 2, standard deviation , and the
σ
standard deviation of the mean, σ x 
. An explanation of what σ x means, and what it is
N
used for is a required part of the lab.
(II)
Calculate the average absolute deviation d' = x  x , for the whole valid set of readings. If the
2
σ . (This relationship should be derived.)
π
Calculate d' for your data set and see if it agrees with this expected relationship.
Make a histogram of the data, with error bars. Does it seem like a Gaussian distribution?
Report your mean value and its uncertainty with the correct number of significant digits.
distribution is Gaussian, we should find that d'
(III)
(IV)
References (by no means is this an exhaustive list!)
1
S.M. Sze, Physics of Semiconductor Devices, 2nd Ed. (Wiley, New York, 1981).
Michael Levinshtein, Juha Kostamovaara, and Sergey Vainshtein, Breakdown phenomena in semiconductors and
semiconductor devices, (World Scientific, New Jersey, 2005).
3
Earl Hergert and Slawomir Piatek, "Understanding key parameters of silicon photomultipliers," Laser Focus
World, November 2014, pp 45-49. (www.laserfocusworld.com)
4
Paul Horowitz and Winfield Hill, The Art of Electronics,. (Cambridge University Press, 1989).
5
Philip R. Bevington, Data Reduction and Error Analysis for the Physical Sciences. (McGraw-Hill, New York,
1969)
6
John P.G. Shepherd, Data Analysis. (UWRF Textbook Library, ~1995)
2
2
Experiment 1B: Statistics of Random Events
1. Introduction
In this experiment we will investigate the statistics of random events. Rutherford and his collaborators
demonstrated the randomness of radioactivity by showing that their experiments on radioactive decay were
in agreement with the mathematical theory of random events. 1 In the present experiment, we will perform
essentially the same experiments on the light detected by our Single Photon Avalanche Diode.
2. Experimental
You should review the theory of the binomial, Poisson, Gaussian (normal), and Exponential
distributions2,3 and then carry out the experiments below. Remember to use error bars on the plots you
make and that any time you have an exponential function, it is best to plot it on a ln-linear plot.
Neat tabulation of results is important in this experiment and you are advised to work out a suitable
scheme of tabulation. This is worth thinking about even if you are not writing down the data by hand.
Truly massive data sets are rarely ever printed out, but documenting their electronic location is then
absolutely crucial! It bears repeating that all graphs ought to have error bars, and all fits should be
weighted fits…..
Part A- Counts per interval (allocate ONE lab period)
In this part you should measure the distribution of numbers of counts per time interval (not counts per
second!) for 3 different intervals. Choose a source and a source-detector distance, then vary the time
interval so that the mean number of counts per interval are approximately 1, 10 and 100 for the 3 sets,
respectively. Make a total of at least 100 observations for each set. Before going for a really massive
number of observations, explore the limitations of your LabQuest/computer combination. Compare
histograms of your observations with Poisson and Gaussian distributions. (Use the mean and  from your
data to create these theoretical ditributions. Use the 2 test to make a numerical estimate of the goodness of
fit between theory and experiment. Doing fits of your data to these functions is extra.)
For all measured distributions calculate the mean observed counting rate and the variance. Compare
the variance with the mean. Finally, correct the mean rates (and their uncertainties) with the backround
rate.
Part B-Time distribution of counts.
In this part you need to measure the time between pulses from the detector. Originally Geiger and
Marsden did this by hand, observing scintillator flashes and using a stop watch! Electronic timing is the
only feasible way to do this for count rates of higher than 1 per second and that is why we are using the
LabQuest interface. One way to do this is to tell LoggerPro that you have a Photogate sensor (using the
Photogate timing calibration). Then you can create a calculated column using one of the special Photogate
functions (see LoggerPro's Help menu under 'Photogate timing modes') to calculate the time between the
leading edges of the pulses. The data can then be transferred into Origin and/or Excel for analysis.
Determine the mean count rate from your data. Was any data lost due to the finite speed of your
circuit or the LabQuest timer? Compare your data to the expected theoretical distribution, and fit the data
to an appropriate function to get a second determination of the mean count rate. Compare and discuss the
different mean count rates.
Investigate any "dead-time" effects of your detection circuit.
There are a huge number of items that you can examine as extras in this lab, just be sure to discuss
your plans with the instructor before you begin.
References
1
E. Rutherford, Radioactive Substances and their Radiation. (Cambridge University Press, 1913).
Philip R. Bevington, Data Reduction and Error Analysis for the Physical Sciences. (McGraw-Hill, New York,
1969)
3
John P.G. Shepherd, Data Analysis. (UWRF Textbook Library, ~1995)
2
3