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In electromagnetic theory, we define electromagnetic waves as two fluctuating electric and magnetic field which travel in a direction. We have a property called Poynting vector which is the power per unit area and it's written: $$\vec{S} = \frac{1}{\mu_0} \vec{E} \times \vec{B} $$ With some calculation we can write: $S = \frac{1}{2 \mu_0 c} E_0^2$ which here $S$ is the average power.

But we know that the energy of photons is $I=N(hf)$ where N is number of photons per unit area per time. So when we change the frequency, the energy and intensity will change.

But it seems power flux calculated from Poynting vector is frequency independent. Where is the problem? I don't really know the relation between these two approaches (wave and particle)

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They are independent because $N$ is not fixed.

The intensity of a single photon depends on the frequency (you know, $E=h\nu$). However, a wave is usually a bunch of many photons. The total intensity is the sum of energies per unit area and unit time.

Imagine you want to achive a certain level $I_0$.

For a red light, each photon has "little" energy, so you'll need a lot of photons to get $I_0$

However, with blue light, which is much more energetic, you'll need fewer photons to achive $I_0$

So, in sum, the intensity of a wave is not frequency dependent because you still have the numebr of photons, $N$, as a degree of freedom.

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There is no contradiction. You miss the point that $N$ (the number of photons per unit area per time) is not necessarily constant when one changes frequency.

It is true that the energy of a single photon is dependent on frequency, but this does not mean that frequency and power are actually dependent.

They are two independent quantities.

A source of high frequency just emits less photons per unit area and time than a source of low freqeuncy at given intensity. That is all.

The interesting thing is that the noise level (i.e. the fluctuation of the instantaneous intensity of the light beam (also called "shot noise")) at given light intensity is higher for higher frequency light. This is due to the fact that the rate of photons is less and hence the relative statistical variation in the photon flux is stronger.

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