A rough calculation of number of photons from LEDs reaching focal plane

Michael Richmond
Aug 18, 2005
revised later Aug 18, 2005: ver 2.0


Executive summary

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.

Richmond's estimate (Aug 2005)

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:

  1. Nsphere is the number of photons leaving integrating sphere. I based the value on a very rough integration by eye of the area under the curve in figure showing power output of LED600-03V, dated 15 July 2005; I believe that this was for an LED run at 100 percent duty cycle (could one of the U of Indiana folks confirm, please?); I also guessed a factor for the light lost in the monochronometer.

    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.

  2. fiber_trans is the fraction of light entering an optical fiber -- which carries the light from the integrating sphere to the Ring of Fire -- which comes out the far end. Based on properties of fibers in astronomical spectrographs, with length 2m and diameter 100 microns.
  3. RoF_trans is the fraction of light entering the Ring of Fire which leaves it; we assume some small fraction is absorbed by the rough walls of the Ring. Optimistic case: one bounce @ 95%. Pessimistic case: three bounces @ 90% each.
  4. RoF_area is the area of a circular region uniformly illuminated by the Ring of Fire; in other words, the area over which the light from the fiber is spread when it reaches the focal plane. Optimistic case is perfectly uniform illumination of exactly the area of the focal plane only. Pessimistic case is uniform illumination of an area twice as large.
  5. pixel_area is the area of one detector pixel.

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:

photon_flux  =  N(LED) * Dg * fiber_trans * RoF_trans *  --------------

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.

Lampton and Sholl's estimate (Feb 2005)

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.