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Data handling Assignment 2 This second part of the assignment looks at four more experimental situations. The first scenario considers the data from two scientists, you are asked to analyse the data, compare and combine their results, then propagate this through an expression. In the second scenario you are given a set of data on which to perform the relevant analysis and then to propagate the uncertainties through the given expression. The third scenario asks you to calculate the numerical aperture of an optical fibre, you are given a series of data from which you are asked to calculate the angular divergence and provide an uncertainty. Finally, in the fourth, you are presented data and must make a best fit using a standard function to describe the observation. Use a spreadsheet, programming language etc. of your choice for the analysis. Present your findings in a document including any figures, plots etc. that you wish. Please ensure you discuss how you processed the data. Present your calculated data with appropriate format for a published result. a) Alice and Bob carry out some more measurements this time they measure the wavelength of red laser light from a helium–neon laser. They take 10 repeat readings each. Their results are shown below, all values are in nanometres. Wavelength/nm Alice Bob 629.99 631.8 634.13 639.7 633.92 627.9 629.98 633.7 632.38 635.8 633.41 638.6 633.97 632.1 631.45 640.8 629.87 631.6 633.8 634.8 Perform data analysis on each set of data and compare the two sets of results. To improve the precision of the experiment, combine their findings and comment on the result. Compare the results with the accepted value and comment on the accuracy of each of their, and the combined, measurements. b) Alice measures the radius of a single mode optical fibre (a) and its numerical aperture (NA) over a number of days. From each set of measurements she calculates a standard error (SE) from the set of data, the results of which are shown below. She performs all the measurements at a wavelength (λ) of (1550 ± 0.001) nm. Using the data she wants to calculate the V number (including the uncertainty) of her optical fibre. The V number (V) is given by the relationship: 𝑉 = 2𝜋𝑎 𝜆 𝑁𝐴 Where λ is the wavelength (in vacuum), a the radius of the fibre core and NA is the numerical aperture. a/µm SE a/µm NA SE NA 5.1 0.1 0.114 0.006 4.9 0.2 0.12 0.009 5.2 0.1 0.112 0.008 0.101 0.01 0.118 0.009 Combine the data to give the most precise result, comment on the precision of the individual values compared to the combined result. Using the combined data calculate the V number and associated error using the expression above. c) Alice makes some improvements to her NA measurement system for single mode optical fibres used in the previous question. She has measured the beam diameter (d) at a number of positions (z) from the end facet of the fibre, see table below. Calculate and plot the angular divergence (θ) and provide an uncertainty, i.e. use points in pairs. z/mm d/mm 10 2.5 20 4.57 30 7.22 40 9.42 50 11.65 60 14.25 70 16.59 80 18.65 90 21.12 100 23.46 Schematic of the beam divergence from an optical fibre Finally propagate your calculated value through the expression 𝑁𝐴 = 𝑛 sin 𝜃 and calculate the NA of the optical fibre. The measurement was performed in air and so refractive index (n) can be assumed to be 1. d) Bob is investigating a new laser medium and wants to determine the fluorescence lifetime of the active ions. He sets up a photodiode to measure the decay of the sample’s emission at the end of an excitation pulse. The signal is amplified and captured on a digital oscilloscope after averaging over 64 waveforms. A text file to the captured data points can be obtained from the Blackboard Data Analysis tab (Assignment_2_d_data.csv). Assuming the fluorescence decay is a single exponential, with a small offset coming from the amplifier circuit, make a best fit to the data to find the lifetime of the material. Utilise the method of minimising the  2 determined from the best fit formula and data. Plot the residuals and the data, along with the best fit function, and determine the lifetime value for Bob. Note the digitisation error coming from the oscilloscope as the signal decays to negligible levels, assume that this is the error for every data point measured in Bob’s setup

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