Does brightness of light depend on number of quanta in the light? It makes sense assuming that the number of photons would affect the brightness of light.

  • $\begingroup$ I believe you are talking of luminous intensity. If so, see my answer below. $\endgroup$ – Wrichik Basu Jul 31 '17 at 14:13

Yes, it does.

It also depends on something which is not so clear in that formula, which is the square of the amplitude for each photon to get where it is going. So if we are looking at a detection screen, for example at the back of your eyes you have precisely such a screen that we call your retinas, we can parameterize it by two coordinates, call them $(x, y)$. Then a photon has an amplitude $\phi(x, y)$ to show up at any given place, and then the actual probability for the photon to arrive is $|\phi(x, y)|^2~dx~dy$ for it to be found in the little square $(x, x + dx) \times (y, y+dy).$

(If you're not familiar with the notation, this set is the set of all all points $(a, b)$ such that $x < a < x + dx$ and $y < b < y + dy$, and the $d\bullet$ expression stands for "a little difference in" and is just meant to say that $dy$ is a very small length in the $y$-direction.)

Now photons' amplitudes are complex numbers, which means that they are scaled rotation matrices -- they have a scale factor and a phase angle that they rotate through. For photons traveling over a distance $\ell$ at the speed of light $c$, the scale factor is just $1/\ell$ and the phase angle is $\nu~\ell/c$ where $\nu$ is the frequency. All of the wavy interference effects come from these phase angles interfering over many different trajectories, but the biggest effect on the "envelope" of brightness comes from this $1/\ell^2$ factor that comes after you square the amplitude. This says that if you're twice as far away from a source, it is one quarter as intense.

Finally, brightness depends on intrinsic response in the screen itself. The screen might be made out of detectors which respond to some frequencies more than others, with some curve $u(\nu)$ or so. For example your eye cannot see infrared or ultraviolet light, but snakes have dimples which can see infrared radiation and I think bees have receptors in the facets of their eyes for ultraviolet light. Furthermore green photons are a little more intense on your eyes than red or blue ones are; they look brighter to you.

Putting it all together, you have something which looks like:

brightness = (number of photons) * (probability for photon to go from source to detector) *
                (brightness response of detector per photon).

All three of these are important for how bright something looks to us or our detectors.


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