All right, let’s tackle this one. To make this a little more straightforward, I’m going to assume you’re bouncing a laser beam down a hall of mirrors. In this case, your question is really how many bounces will the laser beam make before enough light is lost that no more bounces are possible.
Some of the answer will depend on how bright your light is to start with – if you lose 60% of a faint light, you’ll lose sight of that light much faster than 60% of a bright laser light. The other thing we need to be careful with is how much of the light needs to remain for us to count it as still going. We could push down to a single photon, but the human eye definitely does not take a single photon and tell the brain that there was a light. Even the best detectors on some of the biggest optical telescopes don’t have a guarantee of detecting every single photon – at best they capture around 95% of the total light, but that capture fraction drops at the very blue and very red ends to something more like a 50% capture rate.
But we can do the math anyways, making some assumptions along the way. If the mirrored hallway is made of your run-of-the-mill mirror material (basically what you have in your bathrooms), these mirrors reflect around 90% of the visible light that hits them. A second bounce will take 10% of the light that remains, and so on and so on. By the time you’ve bounced off of an 8th mirror, you’ve already lost more than half of the light that you started with. To drop down to 1% of the light you had at the start, you only need 45 mirrors.
So, let’s try the math with a laser pointer. According to this site, a 5 milli-Watt green laser pointer (which is the typical strength for a presentation style laser pointer) has a luminance of 5 x 10^12 candela per square meter. If that’s the brightness of the laser before it hits the first mirror, we should expect that by the time we hit the 9th mirror, the laser light is half as bright. This is the case, but half of 10^12 is still a pretty big number.
In fact, the human eye switches from color vision to night vision around a brightness of 0.001 candela per square meter. We’re going to have to bounce this light off of a bunch more mirrors before we drop the laser light down that faint. In fact, you’ll need 344 mirrors before the light is so faint that you’d have to have night-adjusted vision to see it.
To have that laser spot drop below even your night vision? 409 mirrors. After that 409th bounce, enough of the laser light would have been absorbed into each of those mirrors that nothing is left, as far as your eye is concerned, anyways.
Now, obviously, I’ve done some conservative math here – you can do better if you have a very high end mirror, or if you had a high powered laser, or had attached a camera more sensitive than your eye at the end of your hall of mirrors. In any of those cases you’d be able to tell that some light has still made it through after the 409th mirror. But inevitably, with an infinitely long hall of mirrors, at some point you will run out of photons, as some fraction of them will absorb into the mirror with every bounce.
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