Arc Length Limitations, Arc lengths effect on light into triplet Options

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First off let me explain that while I'm an engineer with a few more years of experience than I care to admit (yes, I'm old) I don't have much expereince in optics. But I was wondering how we could evaluate the effect of arc length on the amount of light getting into the triplet. I've had to make some simplifying assumptions so this may not be entirely valid but I think that I've discovered some useful concepts that might help others in their lamp selection and PJ design.

Arc length has been mentioned here in LL but rarely quantified, this characteristic is only sometimes included in bulb datasheets. It is certainly logical to beleive that shorter arc lengths being closer to point light sources, should work with the fresnels and spherical reflectors that assume point sources better than a long arc or filiment. Light not coming from the focus point should get though the fresnel but is it useful to the image? Points near the arc will probably still be useful but the effectiveness would be expected to drop off the further away we get from the focal point.

So I started playing with ideas in the RAYTRACE program. Below is my test case, in which I assume a 37mm arc (the length in the bulb I planned to use). Vertical black lines marked 18.55 are the two halves of the arc. The two vertical pink lines in the middle are the 220mm fresnel on the left and 330 on the right, the horizontal dashed lines show the focal points and are dimensioned 220 and 330 respectively. The 50mm vertical line on the right is the rear lens of the triplet at 320mm from the 330mm fresnel as specified in the LL plans. The red lines (rays) represent the light coming from one end of the arc, green the middle and blue the other end of the arc. Each of these 3 points has rays going to the center, outside and half way point of the fresnel to show the range of angles involved. I expected it to be difficult to keep track of where the light ended up after the two fresnels, but as can be seen the rays converge to points on the other side of the optical axis. So light starting at 18.5mm above the optical axis on the 220mm side, converges to a point 28mm below the optical axis on the 330mm side at 330mm. So the ratio of the fresnels 330/220 = 1.5 means that a point starting at 18.5 x -1.5 = -28 (it doesn't come out exact because I didn't align the model perfectly, but you get the idea).

 arc37.gif ( 10.26K ) Number of downloads: 54


So given our 220/330 fresnel pairing the effective arc length will get multiplied by 1.5. I assumed a 50mm triplet side as I don't know the exact dimensions. 28 + 28 = 56mm so not all the light will be captured by this triplet. So atleast some of the light form the 37mm arc will not get into the triplet and FOV will force us to get light into an even smaller area.