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If we use an integrating sphere to couple a single LED to the Ring of Fire, we can expect of order 1-10 photons to strike each pixel of an optical detector per second. The exposure time needed to fill the detector to its full-well capacity would be many thousands of seconds. Both (Richmond) and (Lampton & Sholl) reach this same conclusion.
I think it a very good sign that two independent estimates for the photon flux at the focal plane reach roughly the same value when both assume the same optical setup (including an integrating sphere).
One can increase the photon flux on the detector a bit by using a bundle of fibers to carry light from an integrating sphere to the Ring of Fire. However, to increase the flux by orders of magnitude -- which would make the exposure times reasonable in practice -- one must find a way to couple the LEDs to the optical fiber more directly than a very simple integrating sphere.
Suppose that the SNAP telescope uses LEDs to project light onto the focal plane, both to check the sensitivity of the detectors, and to act as a source of high-frequency flatfielding. How long will a typical exposure have to be in order to fill the detector pixels to their full-well capacity? How many LEDs will we need?
I tried to make a very rough calculation to estimate the answers to these questions in a back-of-the-envelope manner. Please treat these numbers as little more than "sanity checks."
I consider only the simple case of an LED emitting light at 600 nm, since the University of Indiana group has used this sort of LED in their tests.
The basic calculation goes like this:
After our SNAP teleconference on Aug 18, 2005, I learned that the values quoted by the U of Indiana group were for power leaving the LED itself, NOT leaving the integrating sphere in their setup. That makes a very large difference in the computations.
So, I will revise my calculations in the following manner. I'll take the number of photons emitted by the LED to be the value gleaned from the U of Indiana presentations.
N(LED) = 7.6E14 photons/sec (at 100% duty, 3V, 9mA)Next, I'll assume that the final SNAP design does contain an integrating sphere to couple light from (a number of) input LEDs to a single fiber; one can scale the final values by the number of output fibers leaving the sphere. The geometric dilution Dg from an integrating sphere varies over a very wide range, depending on the radius of the sphere Rs and the radius of the output fiber Rf. I will consider two cases which may bracket realistic cases:
- optimistic: Rs = 5 cm, Rf = 500 microns. This yields a dilution of Dg = 10E-4.
- pessimistic: Rs = 10 cm, Rf = 50 microns. This yields a dilution of Dg = 2.5E-7.
I came up with two estimates for each value (except for the CCD pixel size): an "optimistic" one, and a "pessimistic" one.
optimistic pessimistic -------------------------------------------------------------------- N(LED) 7.55E15 photons/sec 7.55E14 photons/sec Dg 1E-4 2.5E-7 fiber_trans 0.99 0.97 RoF_trans 0.95 0.729 RoF_area 0.255 sq.m. 0.510 sq.m. pixel_area 1.1E-10 sq.m. 1.1E-10 sq.m. ---------------------------------------------------------------------
Using these values, I then calculate the photon flux reaching each pixel (number of photons per second per pixel) via the formula:
pixel_area photon_flux = N(LED) * Dg * fiber_trans * RoF_trans * -------------- RoF_area
The results are
optimistic pessimistic -------------------------------------------------------------------- photons photons photon_flux 30 ----------- 0.03 ----------- sec * pixel sec * pixel time to reach 100,000 photons 3,300 seconds 3.3E6 seconds per pixel --------------------------------------------------------------------
The most important factor by far here is the means by which light leaving the LED is coupled to the optical fiber. If one could shine the LED directly into the fiber, without the intermediate step of an integrating sphere, then the photon flux at the focal plane would be orders of magnitude greater.
One can gain factors of a few (but not orders of magnitude) by making a bundle of output fibers which together move light from an integrating sphere to the Ring of Fire.
Mike Lampton and Mike Sholl went through a similar series of calculations months earlier.
The exact sequence of computations and approximations is a bit different, but the ideas are the same. Lampton and Sholl consider several cases: using an integrating sphere or directly coupling of LED to fiber.
Please read their calculations for details. I'll provide here only a very quick summary of a few of their final results.