PURDUE UNIVERSITY DEPARTMENT OF PHYSICS PMT EFFECTIVE RADIUS AND UNIFORMITY TESTING AUTHORS: MIHAI CARA, RUDY GILMORE, JOHN P. FINLEY January 14, 2002
ABSTRACT... 3 1. NOTATION... 3 2. RAW DATA FORMAT... 3 3. METHODOLOGY AND DATA ANALYSIS... 4 Introdution... 4 Effetive Radius... 5 Computation of the Effetive Radius... 6 Bakground Subtration... 8 Stability of the Method... 9 Error Estimation... 9 Uniformity Testing... 9 4. RESULTS... 10 5. INDIVIDUAL TUBES DATA... 11 Hamamatsu UA0041... 11 Hamamatsu UA0041 (rotated by 90 )... 12 Hamamatsu UA0042... 13 Hamamatsu UA0044 (λ = 420nm)... 14 Hamamatsu UA0044 (λ = 550nm)... 15 Hamamatsu UA0044 (λ = 650nm)... 16 Photonis 25216... 17 Photonis 25832... 18 Photonis 25892... 19 Eletron Tubes 122... 20 Eletron Tubes 125... 21 2
PMT EFFECTIVE RADIUS AND UNIFORMITY TESTING ABSTRACT As a part of the VERITAS projet we have tested several photomultiplier tubes from different manufaturers for surfae uniformity of their sensitivity and effetive diameter. This report presents the results of our tests and the methods implied in data analysis. The Hamamatsu 41 tube is used to illustrate the methods used in data proessing. The data aquisition program was written C++ and the analysis was done using Mathematia 4. 1. NOTATION For this report we have tested 3 photomultiplier tubes (PMT) from Hamamatsu (model: R7056ASSY), 3 from Photonis (model: XP2900/02), and 2 from Eletron Tubes (model: P25VN-02) (hereafter ET). In order to distinguish tubes from the same ompany we append the serial number of the tube to the name of the orresponding ompany, for example: Hamamatsu 41 (S/N: UA0041), Photonis 25216, Eletron Tubes (ET) 122 and so on. Sometimes, we will further abbreviate these names, for example, to HUA41 (Hamamatsu R7056ASSY tube with S/N UA0041), Ph25216 (Photonis XP2900/02 tube with S/N 25216) or ET122 (Eletron Tubes P25VN-02 tube with S/N 122). 2. RAW DATA FORMAT The 2D data (anode urrent) was taken by sanning the surfae of a PMT along both the X and Y axes with a step of about 0.5mm (exat value: 0.4953mm). During this san the photomultiplier s surfae was illuminated by an optial fiber of diameter 0.6 mm that was oupled to a broadband light soure. The wavelength used in Fig. 1 Density profile for Hamamatsu 41 3
these measurements was λ = 420nm. The sanned area was a box of side 34mm resulting in a 2D pixel struture of 70 by 70. The tubes were plaed in the enter of the sanned area. In Fig. 1 a density plot of the raw data for Hamamatsu 41 (an interpolation of 1 st order was used) is displayed. This setup allowed us to obtain a quantitative measure of the effetive radius and non-uniformity of the tubes sensitivity. In addition, orientation of dynodes is irrelevant in our measurements (see setion 4). A new method of data analysis has been developed whih, ompared to the methods reported at the summer VERITAS meeting in Leeds, does not require that the PMT be plaed exatly in the enter of the sanned region. The new method (presented in the next setion) automatially finds the enter of the PMT. The density plot adjusted to the new enter position is presented in Fig. 1b (oordinates of the enter of the PMT determined from Fig. 1b are: x = 0.606504mm and y = 0.421581mm ). 3. METHODOLOGY AND DATA ANALYSIS Introdution The most important harateristis studied were effetive radius of the PMT, radial profile and deviation from uniformity of the surfae sensitivity. We have onsidered several proedures that an be used to define this effetive radius. The first proedure is based on the analysis of the radial profile of the sensitivity surfae of the PMT. To build the radial profile of a tube we integrated the signal over irular rings of width 0.25mm for radii varying from 1mm to about 15mm. The integrals were normalized over the integration domain. From the outset this method was found to have the following defets: if the integration is done without interpolating the experimental data (in pratie we average pixels whih lie within a speifi ring) then we introdue errors owing to the disrete nature of the data. On the other hand, the interpolation of data ontaining Fig. 2 Radial profile for Hamamatsu 41 experimental errors has its own problems. The effetive radius was then defined as the radius for whih the intensity of the signal is ½ of the average signal inside a irle of some onvenient radius (but smaller than the radius orresponding to the roll- off in the sensitivity). However, this introdues an ambiguity as to the orret hoie of radius. In Fig. 2 radial profiles for Hamamatsu 41 are presented: red dots orrespond to disrete averaging while green dots to the ase when data are interpolated before integration. As one an observe the turn over obtained by the two methods, whih is of major importane in determining the effetive radius, differ enough to produe quite different values for the effetive radius. Another approah was to define the effetive radius as the radius of a ylinder having the same volume as that enlosed by the data with a height equal to the aforementioned average over some onveniently hosen irle. The motivation for this method omes from the fat that we expet a perfet PMT to have an intensity surfae in the shape of a ylinder. This method of determining the effetive radius, however, has problems similar to the previous approah. 4
Effetive Radius The last approah disussed in the previous setion, instead, suggested to us the use of another surfae as ideal for the PMT sensitivity surfae the surfae obtained by rotating the a funtion zx ( ) : = { 1 tanh[ bx ( r)] }around an axis parallel to the z-axis and passing through a point with 2 x, y : oordinates ( ) a z( ρ): = { 1 tanh[ b( ρ r )]}, (1) 2 where or 2 ( ) ( ) 2 ρ ( xy, ): = x x + y y, (2) { ( ) ( ) } 2 2 a zxy (, ) : = 1 tanh[ b( x x + y y r)]. (3) 2 The two additional parameters x, ywere introdued in order to take into aount the possibility that a PMT is not plaed exatly in the enter of the sanned region. They allow us to find the position of the enter of a PMT in an automated way. In Fig. 3 we plotted the funtion (1) using the following parameters: a= 1, b= 2, r = 12.5, x = y = 0. Fig. 4 shows a 3D view of the funtion (3) with the same parameters. This funtion has the following useful properties: it is a ontinuous funtion; it is almost flat for a large range of values of ρ (for example, for the funtion in Fig. 3, ( 10) 0.999955 z = a derease of 0.0045% from 1 and ( ) about 0.25% from its magnitude in the enter and this derease strongly depends on the steepness of the funtion given by parameter b); and z goes to zero as ρ. The parameter a is equal to the amplitude of the funtion in the enter of the oordinate system while parameter b desribes the slope of the funtion. The parameter of interest, r, is z ρ = r = a. One the value of ρ for whih ( ) 1 2 an observe a onnetion between this property of the parameter r and the previous definition of the effetive radius (see the beginning of this setion). We exploit the above desribed properties of funtion (1) and of the parameter r to define the effetive radius of the sensitivity of a PMT: z 11 = 0.997527 a derease of Fig. 3 Ideal radial profile 5
The effetive radius of a PMT is defined as the value of the radius ρ for whih the funtion z( ρ ) dereases by ½ from its value in the origin (enter). For funtion (1) or (3) it is given by the value of the parameter r. The two additional parameters x, ywere introdued in order to take into aount the possibility when a PMT is not plaed exatly in the enter of the sanned region. They allow us to find the position of the enter of a PMT in an automated way. We used these two parameters to enter the density plot in Fig. 1b. Computation of the Effetive Radius So far, we introdued the ideal surfae of sensitivity for a PMT and defined the effetive radius of a PMT. Now, we have to speify a method for omputing the parameters of the ideal surfae of sensitivity. Then, the parameter r will give us the desired effetive radius of the PMT. We found that we an aomplish this by using the Fig. 4 3D view of the funtion z(x,y) least squares method to fit the ideal surfae to the experimental data. In pratie, the least squares method will try to average the experimental data and represent them with the ideal surfae. NOTE 1: Our purpose here is to replae the experimental surfae of sensitivity with an equivalent ideal surfae a simple enough surfae (see, for example, Fig. 4) that allows us to give a lear and meaningful definition of the effetive radius. We do not want to find the best approximation to experimental data; for this purpose the use of some large set of basis funtions would be more advisable than the use of funtion (3). NOTE 2: The bakground should be eliminated from the signal before applying the least squares proedure. The proedure we used for this is desribed in the next setion. NOTE 3: Mathematia s NonlinearFit pakage ontains two funtions whih an be used to fit funtion (3) to the experimental data: NonlinearFit and NonlinearRegress. Both of them aept as an option the weights of the experimental data points. For this report we did not use this option. The ontinuous blue line in Fig. 2 is the radial profile of the ideal surfae of sensitivity for the Hamamatsu 41 tube omputed using the least squares method. We found the following value for the parameter r: 13.32mm. It is interesting to ompare this value with the values obtained using three methods from the preliminary report presented at Leeds (two of these methods were outlined in the introdution to this setion; the third method is a variation of the seond method): 13.35mm (method 1 ½ of the average over a irle of radius of 10mm); 13.39mm (method 2 radius of an equivalent ylinder with the volume equal to the volume enlosed by the experimental intensity surfae and height omputed as in method 1) and 12.75mm (method 3 a variation of method 2; it uses a different (adaptive) way to ompute the height of the ylinder; we found that this method tends to grossly underestimate the effetive radius. We an see that the new value of the effetive radius is lose to the values obtained using methods 1 and 2 from the preliminary report. 6
For some PMTs we have found that the turn over of the ideal profile did not math very well the turn over of the disrete radial profile. We explain this by the way these two profiles are built: the disrete one, unlike the ideal radial profile, does not take into aount global properties of the experimental data it is built by averaging data over narrow rings. Atually, the slope of the disrete profile depends on the width of the integration rings. In order to have a more intuitive representation of how well the ideal surfae fits the experimental data we built several plots in whih we overlapped ideal and experimental surfaes. Fig. 5 shows the experimental sensitivity surfae of Hamamatsu 41 (plotted without mesh), the ideal sensitivity surfae omputed using the least squares method (plotted with mesh) and their overlap from different view points. It is interesting to see how this looks in ross-setion. For this purpose, in Fig. 6 we present uts through the X-Z plane of the overlapping surfaes. From the frontal view one an observe how the ideal surfae averages the experimental data. Fig. 5 Approximation of the experimental data by an ideal surfae for Hamamatsu 41 7
Fig. 6 Cuts of the overlapped surfaes through X-Y plane for Hamamatsu 41 Bakground Subtration The bakground signal in our measurements mainly omes from multiple refletions of the light emerging from the optial fiber from the walls of the dark box as well as eletronis noise. This bakground signal is small in omparison with the useful signal (as one an see from Fig. 1, Fig. 5, or Fig. 6). Nevertheless, it is very advisable to eliminate this bakground. The most simple and elegant solution is to introdue an additional parameter into the equation for the ideal sensitivity surfae (3) to ount for the bakground rather than subtrat it. We onsidered the following funtion: { ( ) ( ) } 2 2 a zxy (, ): = 1 tanh[ b( x x + y y r)] + z0. (4) 2 Unfortunately, the introdution of the parameter z 0 made the numerial algorithm unstable. Instead, from Fig. 1 we observe that the peripheral pixels of the sanned region bear only the bakground signal plus some statistial noise. In order to eliminate the statistial noise we ompute the average value of the signal orresponding to the peripheral pixels (for a 70 70 grid we have 276 peripheral pixels): n n 1 1 bakground : = z + z + z + z. 4 1 ( n ) ( 1, i n, i) ( i,1 i, n) (5) i= 1 i= 2 8
We subtrat this value from the experimental data before trying to fit the ideal sensitivity surfae to the experimental data. Stability of the Method It is important that the method of omputing the effetive radius is stable to variations of the initial onditions. To hek that the method desribed in this report is stable to statistial flutuations we performed several numerial experiments. The two most important simulations performed onsist of: Adding pseudo-noise to the real signal (we used HUA41 data). We observed that an addition of statistial noise at a level of 10% of the useful signal led to variations of about 0.04% in the omputed value of the effetive radius from its original value (13.3197): r [13.3201, 13.3255]. A noise level of 20% led to variations in r of 0.16%: r [13.3133, 13.3349]. Varying the omputed value of the bakground level. Assuming that the bakground omputed using equation (5) is suseptible to flutuations we varied the omputed value of the bakground to observe the effets of suh variations on the resulting value of the effetive radius. We observed that if the bakground is not eliminated or its value is hanged by a fator of two the resulting error in r was less than 0.1%; for an error of 400% in omputing the bakground we got a variation of less than 0.2% of r, 0.5% for an error of 1000% and 2.4% for an error of 5000% (fator of 50) in omputing the bakground. These simulations tell us that both the methodology of omputing the effetive radius and the numerial algorithms used are very stable to statistial flutuations in the initial data. Error Estimation Mathematia s NonlinearRegress funtion returns the onfidene intervals of the omputed parameters. We will use this interval as a measure of the error (for nonlinear funtions omputation of onfidene regions is a diffiult problem; Mathematia gives only an estimation of these intervals) of the effetive radius (and of other parameters). In this report we will use a onfidene level of 99.7% to ompute onfidene intervals. For example, for HUA41 and a onfidene level of 99.7% we get the following onfidene interval: r [13.2828, 13.3567] with the average r = 13.3197 mm. In this ase we will write the result as r = ( 13.32 ± 0.04) mm. Uniformity Testing Unlike the effetive radius, it is more diffiult to give an absolute definition of the uniformity of the sensitivity of a PMT. Here we will try to find a measure of this uniformity whih an be used in omparing the uniformity of different PMTs. For a grid of size n mit seems natural to define the non-uniformity of a PMT as the average of the relative errors of data points: 1 z z( x, y ) ε = n m i, j i i 100%, (6) i, j z( xi, yi) 9
or, alternatively, for a model with 5 degrees of freedom: 2 zi, j z( xi, yi) 100%. i, j z xi yi 1 ε : = n m 5 (, ) (7) There is a problem with the above definitions: small values of zx ( i, yi) at the edge of the intensity surfae will make the orresponding experimental points (many onsisting only of statistial noise) to ontribute heavily to the value of the non-uniformity omputed with equations (6) and (7). We an orret this situation by replaing zx (, y ) at the denominator in equation (7) (or (6)) with a z( x, y )(see equation (3)). i i Thus, we define the non-uniformity of the sensitivity surfae of a PMT as 1 1 ( ) 2 1 ε : = zi, j z( xi, yi) 100% = EstimatedVariane 100%. (8) a n m 5 a i, j 4. RESULTS Here (see Table 1) we will present effetive radii and orresponding non-uniformities we have omputed for different PMTs. For the HUA41 tube we repeated the measurement with the tube rotated by 90. One an see that this didn t have (within experimental errors) any influene on the results. Also, for the HUA44 tube we measured the effetive radius and its non-uniformity for several wavelengths: 420nm, 550nm (green) and 650nm (red). For all these wavelengths its effetive radius is above 12.5 though for red light non-uniformity is somewhat greater. Manufaturer: Hamamatsu (R7056ASSY) Photonis (XP2900/02) Table 1: Effetive radii and non-uniformities of the studied PMTs Serial Number: Parameters of the ideal intensity surfae a, µa b, mm -1 r, mm (effetive radius) ε, % Passed UA0041 6.90±0.07 0.84±0.04 13.32±0.04 11.1 Yes UA0041 (90º) 6.90±0.06 0.85±0.04 13.33±0.04 10.9 Yes UA0042 7.34±0.07 1.12±0.08 13.47±0.04 13.3 Yes UA0044 (420nm) 5.81±0.05 1.09±0.06 13.51±0.03 11.3 Yes UA0044 (550nm) 7.69±0.07 1.12±0.07 13.60±0.03 12.4 Yes UA0044 (650nm) 9.6±0.1 1.4±0.2 13.69±0.05 19.9 Yes 25216 6.57±0.05 1.17±0.07 12.54±0.03 10.1 Yes 25832 9.45±0.07 1.17±0.06 12.32±0.03 9.2 No 25892 6.22±0.04 1.22±0.06 12.62±0.02 8.0 Yes Eletron Tubes 122 6.45±0.05 1.8±0.1 12.50±0.02 10.6 No (P25VN-02) 125 5.26±0.04 1.4±0.1 12.71±0.03 11.0 Yes 10
5. INDIVIDUAL TUBES DATA Hamamatsu UA0041 11
Hamamatsu UA0041 (rotated by 90 ) 12
Hamamatsu UA0042 13
Hamamatsu UA0044 (λ = 420nm) 14
Hamamatsu UA0044 (λ = 550nm) 15
Hamamatsu UA0044 (λ = 650nm) 16
Photonis 25216 17
Photonis 25832 18
Photonis 25892 19
Eletron Tubes 122 20
Eletron Tubes 125 21