Telescope Reviews: Illumination, Brightness, Exit Pupil

Hi all,

This is my first post on these forums. I've been reading these forums for a while and I want to thank you all for making such an incredible resource. I've learned so much reading these forums, particularly from EdZ.

I'm posting because I did some analysis of EdZ's light meter data which is posted a few threads up. I was motivated by the fact that there are multiple factors determining brightness so it can be difficult to sort out which factors are important and which are not. For example, AFov was argued above as being important, but it is difficult to decide from the data if it actually is important because variations in exit pupil size, for example, may be interfering.

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Attached are four scatter plots that help sort out the relevant factors:

1) Each point in each plot represents one of 36 binoculars. The y-axis in all four plots is the light meter reading. The x-axis in plot #1 is the exit pupil area in mm squared ( = pi * (diameter/2)^2 ). Disregarding the colors for the moment, we see there is a positive relationship between brightness and large exit pupils, but there is a large amount of deviation from the linear least squares fit (black line), particularly at small exit pupil's. Now looking at the colors, which show the AFov, we see that all the binoculars that deviate significantly above the line have fields of view of greater than 57 degrees. Similarly, the number of binoculars with small fields of view (< 58) under the line outnumber the binoculars with large fields under the line (although there is still quite a bit of "noise" at small exit pupils). (Some dots are unfortunately hidden by others.)

2) In the second plot the x-axis is now the product of exit pupil area and Afov area (steradians). The AFov in steradians is given by 2*pi*(1-cos(A/2)), where A is the traditional AFov specification given in degrees. The linear fit improves significantly although there are one or two big outliers (Zen Ray ZEN ED2 8x43 & Oberwerk 8x42). If one just looks at the porro prism binoculars (blue dots) then one gets the impression that the relationship is quite strong. For all binoculars the correlation increases from 0.67 in plot #1 to 0.83 in plot #2 (see top right corner). In summary, when we also include the effect of exit pupil, we see that AFov is adding important information about the light meter measurement.

We could have also performed multi-linear regression with two predictors (Exit pupil (EP) and AFov) instead of single-variable regression with just their product. We choose the product because it provides a slightly better fit to the data. The product also makes physical sense. For example, consider the hypothetical situation of a fully illuminated exit pupil. In this case the total illumination is the sum of exit-pupil-sized light beams added up over all directions in the field. This integral is simply the exit pupil times the AFov area in steradians.

(Note: the AFov given in steradians is hardly different that the AFov given in square degrees (= pi*(A/2)^2 ) for the range of A present in this sample. Only large A cause the two to differ significantly.)

3) It is quite obvious the roof prisms tend to be below the line in plot #2 and vice versa for the porro prisms. Hence in plot #3 I created a new binary predictor variable which is one if the binocular is a roof prism and zero otherwise. Using both EP * AFov (* == multiplication) and this "roof prism variable" (called RP from now on) in multi-linear regression leads to an improved fit. The correlation is now 0.92. The x-axis is now a combination of EP*AFov and RP scaled by the constants determined from the linear regression. The coefficients in the linear fit are given below the x-axis. The black line is the line y = x.

4) The next best predictor after EP*AFov and RP is another sort of "derived" predictor I came up with which is EP*AFov*(the fraction of AFOV fully illuminated). Hence, the x-axis in Plot #4 is the three predictors scaled by the coefficients from the linear regression. This new predictor, however, only adds a small amount of increased skill. For the analysis since plot #2, I've been using a technique called cross validation, whereby I remove a binocular from the analysis and perform the fit on the remaining binoculars. Removing a binocular from the analysis allows me to test the fit from the other binoculars on this "independent" data point. I give each binocular a chance to be removed from the analysis. This cross validation analysis suggests that the three predictors added here have real skill in predicting independent data, but no other predictors add additional skill beyond the three given here. (Note: I've also been systematically adding the best predictor at each step from plot #2 to plot #4.) Other predictors I tried include EP, AFov and Illumination each by itself. This doesn't necessarily mean that EP, for example, doesn't have any skill. It just means that it doesn't add any additional information beyond the predictors I did use. Also there could be a function of the three variables that I did not consider here that is better than the different functions I tried here. (see http://en.wikipedia.org/wiki/Cross-validation_(statistics) for more info on cross validation)

The coefficients in front of the predictors change slightly from plot #3 to plot #4. The coefficient in front of the RP (plot #4) means that on average a roof prism leads to a 148 reduction in the light meter reading compared to an equivalent porro. I'm not sure what to make of this because one would think that the illumination measurement would take into account the difference between roof and porro prisms. This might be because I haven't come up with the best way to utilize EdZ's illumination measurement data. Basically the problem hinges on transferring EdZ's measurement to an estimate of illumination as a function of off axis angle for the entire AFov. With this info, you then basically sum up the exit pupil size times the fraction of illumination over the field of view (using spherical coordinates):

Brightness = integral_from_0_to_A/2_of { 2*pi*E*I(a)*sin(a) } da ,

where a is the angle from the on-axis direction, E is the exit pupil area, I(a) is the illumination as a function of a. The sin(a) is there because we are in spherical coordinates. The above assumes everything is symmetric in the azimuthal direction, but if one has the illumination everywhere then you can easily generalize the above integral. I believe EdZ has more illumination data besides the percent fully illuminated that I think will help constrain I(a) significantly. I just haven't taken the time yet to do this yet.

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So in summary it appears that both exit pupil and AFov of view are important. They both enter into the relationships described in plots #2 - #4 in the same way (i.e. the brightness is proportional to Exit Pupil * AFov ). Exit pupil seems more important only because the range of exit pupil areas varies by a factor of 3.2 while the range of AFov varies by a factor of 2.4 across the sample of binoculars measured by EdZ. Another contributing factor to the difficulty of seeing the AFov relationship is the fact that very large exit pupil binoculars almost always have small AFov. Hence an entire class of small AFov binoculars are also exceedingly bright not dim.

Another point to remember is that for the brightness of the IMAGE instead of the brightness of the entire FIELD, exit pupil matters while AFov does not.

The role of AFov in these measurements can be seen by considering the Fujinon FMT-SX 10x50. The Fujinon is the purple dot in plot #1 which is way above the "expected" brightness suggested by the linear fit to all binoculars. When we consider the fact that the Fujinon as a large AFov (plots #2 - #4), however, the Fujinon is not drastically brighter than the other binoculars. For example, in plot #2 the Fujinon (which is the dot closest to the 1000 light meter reading) is only about 60 to 70 units above the linear fit to all the other binoculars.

One disturbing thing about the linear fits is that the intercept is significantly larger than zero. For example, the intercept for the fit in plot #3 is 427. This says that as the exit pupil or Afov go to zero there is still 427 units of brightness (for a porro prism), which is obviously wrong. The intercept gets close to zero if I raise the EP*Afov to a power approaching 1/2 or square root and then perform the linear regression (the goodness of fit stays more or less the same, too!). But this doesn't make sense to me because the brightness should scale with the area of the exit pupil or AFov not the linear dimension.

That's all my thoughts for now.

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My thanks go to EdZ for sharing his measurements with us. There is nothing on the Internet or anywhere else that compares with his measurements and reviews. Thanks!

-Dave