# Physics307L:People/Allen/Lightspeed

## Introduction

The speed of light is a fundamental constant and is the maximum speed found to be attainable in this universe. It is therefore somewhat amazing that we can measure this speed to such a degree of accuracy with such a simple experiment. The speed of light is calculated by measuring the differences in the time interval between when a signal is transmitted and when it is received a short distance away, as the distance between the emitter and the receiver is changed by regular increments.

A low intensity light signal is sent by a light-emitting diode at one end of a 3-meter cardboard cylinder and received by a photomultiplier tube at the other end. The time of flight is measured by a device that converts the time interval, between the signal emission and its reception, to a voltage level that can be displayed and measured by an oscilloscope.

I took measurements at 10-50 cm distance intervals over 0-300 cm, in three different trials, attenuating the light signal to keep the intensity constant at the photomultiplier tube (PMT) among the measurements for each trial. As the distance is decreased, the intensity of the signal at the receiver, the PMT, is greater, resulting in a higher amplitude output from the PMT to the time-to-amplitude converter (TAC).

The TAC is triggered when a pulse strength passes a certain threshold, which is reached more quickly by a higher amplitude pulse. This results in a decrease in the recorded time intervals with respect to the actual time intervals as the emitter, a light emitting diode (LED), and the PMT are moved closer together ~ a phenomenon known as time walk. To compensate for this, there are two polarized lenses placed between the LED and the PMT which can be rotated with respect to each other, to adjust the light intensity before each time interval measurement and keep the amplitude of the output signal from the PMT constant, as the distance from the LED changes.

The data was plotted and fitted to a linear regression, whose inverse slope is an estimate of the speed of light. The error of the slope was calculated by dividing the variability of the residuals of the time measurements by the spread in the values of the distance. Standard error of the slope = √[Σ(yi-yi’)ˆ2/(n-1)]/√[Σ(xi-<xi>)ˆ2] where <x> is the average distance, yi is the time measurement at xi, and yi’ is the expected time measurement due to the linear best fit line. The percent of this error was then applied to the inverse slope to determine its respective error. The inverse slopes and their errors were averaged to derive the final calculation of the speed of light.

In averaging the resulting inverses to the slope of the linear best fit line from each trial, my measurements of time intervals with respect to distance resulted in a calculation of the speed of light to be 3.003 x 10^8 m/s with an uncertainty of +/- 0.1232 x 10^8 m/s (4%), in excellent accord with the accepted value of 2.9979 x 10^8 m/s, to within 0.2%. σy/√Σ(xi-<x>)ˆ2

## Equipment

The equipment is as follows:

• Tektronix Oscilloscope (Model TDS 1002)
• Bertan Power Supply (Model 215, 3000V, 5mADC)
• Canberra Delay Module (Model 2058)
• Ortec TAC/SCA Module (Model 567)
• Harshaw NIM Bin (Model NQ-75)
• Harrison Laboratories Power Supply (Model 6207A, 160V, 0.2A)
• Photomultiplier Tube (PMT)
• LED circuit
• BNC Cables

These are from David Weiss' notes from when we worked together two years ago. There is now a longer cable from the PMT to the TAC, so I did not need to use the delay module in collecting my data this time.

< User:David J Weiss | Notebook

## Safety

~ The PMT is connected to a high voltage power source set to near 2400V. Be sure all cables are insulated and not worn.

~ Do not handle the 200V power source to the LED when the power is on, as has exposed contacts in the back and two exposed wires in the front with worn insulation.

~ Do not expose the PMT to ambient light when the high voltage power source is on, or it will be destroyed by the high current passing through it due to the photomultiplier.

## Set-Up

The PMT is set inside one end of a 3-4 meter long cardboard cylinder, with a polarizing lens attached to the receiving end, such that this assembly can be rotated with respect to the cylinder. An LED, mounted on a long rod consisting of a series of meter sticks, is located at the other end of the cylinder so it can be moved by measured intervals of distance with respect to the PMT. The LED also has a polarizing lens attached to it between it and the PMT. By rotating the PMT to adjust the angle of alignment of the two polarizing lenses, the intensity of the light pulse received by the PMT from the LED can be attenuated, and kept nearly constant as the distance is changed.

The LED is connected to the 200V power supply and then to the start input of the TAC. The PMT is connected to the high voltage power supply and then connected to the stop input of the TAC with a T-joint that allows the signal to continue on directly to the oscilloscope Ch 1 as well. Both connections to the TAC are with BNC cables of similar lengths, that carry pulses when the light flash is transmitted and when it is received. If there is not a large enough time interval between the start and stop inputs to meet the time requirements of the TAC, the stop input needs to be put through a delay module before passing on to the TAC. Since the cables from the TAC and the LED were of similar lengths, I left all switches on the delay module to the delay out side.

The TAC output is a signal with a voltage level proportional to the start-stop time interval. The output from the TAC is connected to the oscilloscope to be measured and displayed. The output from the PMT is also directly connected to the oscilloscope to measure the intensity of the light received by it. This intensity can then be kept nearly constant by rotating the PMT to change the alignment of the lenses.

I set the polarizer alignment to have the highest possible amplitude signal from the PMT at the furthest distance, then changed the alignment to maintain the intensity of light as the LED was moved closer to the PMT. I checked the amplitude of the PMT output signal via the oscilloscope reading before the TAC measurement at each distance. Without this adjustment, the TAC measurements taken moving in from further to closer would give a smaller change in time and moving out would give a larger change in time due to the time walk. This would result in a calculation of the speed of light to be much less due to the distortion in the relative time intervals.

## Day 1:

Dr. Koch suggested that I observe the signals from the LED over an interval of time that showed a diminished repeat of the signal on the oscilloscope. This appeared as two or three “echoes” of the initial signal, as if the signal was being reflected back from an interface at some position along the coaxial BNC cable acting as a wave-guide.

This day, I took only three measurements to clarify the procedure, starting at the furthest distance; the TAC signal amplitudes were 4.96 V at 300 cm, 4.92V at 290 cm, and 4.88 V at 280 cm. The cursor was set first on Ch 2, Type as Voltage, with Cursor 1 at the level line and Cursor 2 at the voltage of the signal pulse, which gave a Delta reading of 520 mV from the PMT, after adjusting the polarizing lenses. Then the cursor was set on Ch 1 in a similar manner to read the voltage from the TAC signal which represented the time interval.

Settings:

Amplitude from PMT ~ 520 mV first day, 512 mV Trial 1, 672 mV Trials 2 & 3 Voltage to LED ~ 190 V Voltage to PMT ~ 2400 V TAC multiplier set at 1x, range ~ 100 ns TAC Start and Stop switches both set at anti, Output set to out Delay Module switches all set to the delay out side Oscilloscope Ch 1 from TAC ~ 1.00 V Oscilloscope Ch 2 from PMT ~ 200 mV Cursor set on Ch 2, Type as Voltage, with cursor 1 at the level line and cursor 2 at the minimum voltage, which gave a Delta reading of 520 mV Trigger set on Ch 1, Average, 64x, to stabilize the display

## Day 2:

On the second day, the set-up, settings and procedures were as for the first day, except for the PMT output, which was slightly lower, at 512 mV. I started with 10 cm intervals between measurements, but noticed the voltage output from the PMT did not change more than the error in measurement due to the fluctuations in the Oscilloscope display. The measurement intervals for the voltage readings from the TAC were 0.04 V, and the display would fluctuate +/- 0.08 V. After five measurements, I increased the interval to 25 cm. The second and third sets of measurements were taken at 50 cm intervals, first from 250 cm to 0 cm, and then from 0 cm to 250 cm, with a PMT setting of 672 mV. The higher setting helped to stabilize the oscilloscope display. I would record an average measurement based on my sense of both the frequency of the voltage level and the length of time the display would stay at that level.

## Data Analysis

To calculate the measured speed of light from this data, it is necessary to convert the voltage measurements from the TAC to time intervals. Since the time interval range for the TAC was 100 ns and the voltage interval range was 10 volts, the conversion factor is 10 ns/V. This resulted in the measurements to be seconds x 10^-8, rather than in nanoseconds.

Using LINEST to calculate the linear regression line, I found the slope for time vs. distance, with a given error to the slope and error to the time measurements for the first trial. The value of the error to the slope as a percentage of the slope was then calculated. This value was multiplied by the value of the inverse slope to find the error of the inverse slope. As the error in the time carried more weight in determining the slope, I did not simply invert the data entries into my chart.

With my charting and linear fit of the first set of data, I found a slope for time vs. distance of 3.362 ns/m with an error on the time of +/- 0.505 ns and an error on the slope of +/- 0.1407 ns/m. This translates into an inverse slope of 2.974 x 10^8 m/s +/- 0.1245 x 10^8 m/s (4%), which represents the measurement of the speed of light. This is within 0.8% of the accepted value of 2.99792458 x 10^8 m/s (Wikipedia), which falls within the error of my measurements.

With the charting of my second and third sets of data, I found the slopes for each to be the same, resulting in a slope for time vs. distance of 3.314 ns/m with an error on the time of +/- 0.2815 ns and an error on the slope of +/- 0.1346 ns/m. This translates into an inverse slope of 3.017 x 10^8 m/s +/- 0.1225 x 10^8 m/s (4%), within 0.6% of the accepted value, which is again within the error of my measurements

I could not average the measurements at respective distances among the three trials, as they were taken at different light intensities at the PMT between the first and the second two trials. However, in averaging the three resulting slopes, the resulting calculation of the speed of light (using further decimal places in the calculations) is 3.003 x 10^8 m/s with an error of +/- 0.1232 x 10^8 m/s. This is now within 0.2% of the accepted value of the speed of light.

The uncertainty to the slope was calculated through Linest for the first trial and by hand for the second two trials using the equation,

```  Standard error of the slope = σy/√Σ(xi-<x>)ˆ2
= √[Σ(yi-yi’)ˆ2/(n-1)]/√[Σ(xi-<xi>)ˆ2]
where <x> is the average distance, yi is the time measurement at xi,
and yi’ is the expected time measurement due to the linear best fit line.
```

This represents dividing the variability of the residuals of the time measurements by the spread in the values of the distances*. The error was then calculated as a percentage of the slope and that percentage was multiplied by the inverse slope to find the error to the inverse slope, i.e. the error to the calculation of the speed of light.

## Concerns and Sources of Error

It seems that the main source of error is due to the measurements of the output voltage from the TAC that represented the time interval for the passage of light. The readings and display on the oscilloscope varied constantly, even when averaged over 64 pulses. The readings from the PMT signal voltage also varied constantly. Thus all recorded measurements were based on approximations with regard to maintaining the intensity of the light received by the PMT, and estimates with regard to determining the time interval for the passage of light. My recorded measurements were determined by my sense of both the frequency of the voltage level and the length of time the display would stay at that level.

With more time, it would be interesting to measure the time between the echoes in the signal from the LED and compare them to the length of the BNC cable to determine whether they could be due to a reflection along the cable from the interface of the connection to the oscilloscope. This might then be a way to measure the speed of the signal through the cables. I would also like to explore other ways to allow for time walk in the calculations to apply to this phenomenon in situations where one cannot adjust the cause. In such a situation, one could check for influences away from the linear by creating a residual plot of the differences of measured time from the expected times due to the linear regression line. A pattern in this plot away from random error distribution can reveal a systematic non-linear source of error.

## References

• Dr. Gold’s Physics 307L Lab Manual
• Watkins, Ann E., Statistics in Action, Key Curriculum Press, CA, 2004. p.638.
• Wikipedia
• David Weiss’s Lab Manual