Results for the May 16 flare

Table 1 shows the data. +45, etc. indicates the direction of the arm of the diffraction pattern; +45 indicates 45 degrees up from right, -45 indicates 45 degrees down from right, etc.

Table 1: Intensities of each peak in the diffraction pattern: May 16, 1999

order# ($m$)	+45	+135	-135	-45	average	st. dev.
1	20578.4	20235	27072	25751.2	23409	3511
2	20779.8	18306	20992	-	20026	1493
3	18612.2	17100	13721.6	13161.4	15649	2632
4	11919.6	11960	13790	11586.4	12314	998
5	6980.4	7954	6177.6	6901.2	7003	729
6	3697.4	3650	2705	3908.8	3490	535
7	1016.4	1352	1584	1028.5	1245	274
8	-	171.5	-	91.2	131	57
9	0	0	0	0	0	-
10	345.6	399	262.5	324	333	57
11	699.6	907.2	815	489.6	728	180
12	985.8	1440	1170	980	1144	216
13	984	1505	1320	817.6	1157	312
14	608	837	745.2	562.5	688	125
15	-	-	800	365.7	583	307
16	-	-	0	0	0	-
17	-	-	0	0	0	-
18	-	-	0	0	0	-
19	-	-	0	0	0	-
20	-	-	253	255	254	1.4
21	-	-	279	245.7	262	34
22	-	-	204	202.1	203	2.3
23	-	-	369.6	-	369.6	-

The value of $b/a$: Analyzing the data, we see that the intensity of the peaks drops to zero slightly before the 9th order (because the 8th order is much dimmer than the 10th order), and right between the 17th and 18th orders. So the intensity vanishes close to every 9 orders. We conclude that $b/a$ must be close to an even fraction of 9, such as $\frac{7}{9}$, $\frac{5}{9}$, etc. The transmission of the grating is 82%, and since $b$ is the width of the opening and $a$ is the distance between openings, $(b/a)^{2} = .82$, or $b/a \approx 0.905$. The closest fraction of 9 to 0.9 is $\frac{8}{9}$ = 0.889. Since the graph actually dips slightly more often than every 9 orders, $b/a$ is probably slightly smaller than $\frac{8}{9}$. The value that best fits the zeroes of the orders is 0.885.

Fig. 5 shows that the theoretical curve fits the data very well. This would lead us to believe that we have used the correct theoretical representation of the diffraction pattern.

Figure 5: The dotted line represents (where $b/a$ = 0.885)

Using Equation (4), we can take any order peak and plug the intensity in as $I(m)$, and divide by the modulating function to find $I(0)$. We used the first seven orders to calculate an average extrapolated value for $I(0)$ (see Table 2).

Table 2: Predicted intensity of zeroth order

order#	Average $I(m)$	Predicted $I(0)$ (by equation (4))
1	23409	$1.45\times10^{6}$
2	20026	$1.42\times10^{6}$
3	15649	$1.39\times10^{6}$
4	12314	$1.55\times10^{6}$
5	7003	$1.43\times10^{6}$
6	3490	$1.42\times10^{6}$
7	1245	$1.43\times10^{6}$

We obtain an average result of $I(0) = 1.45 \times 10^{6}$. The uncertainty of the mean is $6.10 \times 10^{4}$, about 4.2% of the calculated value.

We must keep in mind that the zeroth order does not give the actual intensity of the flare, only the largest portion of it. Much of the light is diffracted off into the diffraction pattern, so to find the actual intensity of the flare, we have to add together all the orders. This gives us $1.45 \times 10^{6} + 3.56 \times 10^{5}$ (we add the zeroth order to the total of the intensities of all the peaks; but since orders past $m=15$ are cut off in the +45 and +135 arms, we take the average of each order and multiply by 4 to be the sum of the intensities in that order), which is $1.81 \times 10^{6}$.

Significantly, the total intensity of the entire May 16 diffraction pattern through 23 orders is $19.7\pm 0.32\%$ of the intensity of the flare. The error is derived from the standard deviations of each order. This means that 19.7% of all the light coming through the filter gets diffracted away from its original angle, even when the diffraction pattern is too dim to be visible. Also, this indicates that much of the light is being lost from the focused position and being spread out over the entire image. This is slightly less than the theoretical value of 20.0% diffracted away, but the theoretical value is within the inherent error of the number 19.7%.

Now we examine how the intensity range of the camera has been increased. Table 3 compares the intensity of the highest pixel, $I_{max}$, to the total intensity, $I_{tot}$, at each order for the May 16th event.

Table 3: Comparing $I_{max}$ and $I_{tot}$, at each order

order#	Average $I_{max}$	Average $I_{tot}$	$I_{tot}/I_{max}$
1	2610	23409	8.97
2	2127	20026	9.42
3	1515	15649	10.30
4	970	12314	12.70
5	524	7003	13.37
6	228	3490	15.30
7	69	1245	18.11

The graph in Fig. 6 shows that we can estimate that $I_{tot}$ for the zeroth order to be 8.7 times greater than $I_{max}$. We conclude that the highest pixel value is $\frac{ 1.45 \times 10^{6} }{ 8.7 }$, or $1.67 \times 10^{5}$. This means that our intensity range has been effectively increased by a factor of 41, from 1 : 4096 to 1 : $1.67 \times 10^{5}$. Furthermore, we can expect that we can still use this method to determine intensities of even brighter flares; even if the diffraction pattern itself saturates, we can use higher orders to determine all of the orders, and the total intensity of the flare.

Figure 6: The ratio $I_{tot}/I_{max}$ as a function of order number $m$; we extrapolate backwards to find the ratio for the 0th order.