What is the formula for the intensity of light, and how are amplitude, frequency and number of photons considered?

Question

One formula for light intensity is$$
I = frac{nfh}{At}
,,$$where:

$n$ is the number of photons;

$h$ is Planck's constant;

$f$ is the frequency;

$A$ is the incident area;

$t$ is time.

Another formula describes intensity as a function of the magnitude of electric field squared:
$$Ileft(tright) propto left|Eleft(tright)right|^2$$
$$I=left|Sright|=frac{left|Eright|^2}{Z_0}$$
How do I reconcile these two formulas?

zhutchens1 · Accepted Answer

Classical/Wave Model

An electromagnetic wave is composed of an oscillating electric and magnetic fields, which are orthogonal. Our field equations might be described by
$$mathbf{E}(x,t) = {E_0}sinleft(kx-omega tright)mathbf{hat x}$$
and
$$mathbf{B}(x,t) = {B_0}sinleft(kx-omega tright)mathbf{hat y}.$$
Here the frequency is given by $f = frac{2pi}{omega}$ and the wavelength by $lambda = frac{2pi}{k}$. The amplitues are given by $E_0$ and $B_0$. These equations form a plane wave which has a total intensity, at any point in time, as given by the Poynting vector
$$ mathbf{S} = frac{1}{mu_0}left(mathbf{E} times mathbf{B}right). $$
The time-average of the Poynting vector turns out to be
$$ I(t) = left< mathbf{S}(t) right> = frac{1}{2cmu_0} E_0^2.$$

This is the equation you mention. There are no photons to be counted in this paradigm, for photons are waves and not particles by classical electrodynamics theory.

Particle/Quantum Model

In the high-energy limit, photons act more like particles than waves.

The intensity is defined as power per unit area, and power is defined as energy per unit time. Thus:
$$I = frac{P}{A} = frac{E}{Delta t} frac{1}{A}.$$
The energy of a photon is $E = hf$, so the total intensity for $n$ photons is
$$I = n cdot frac{hf}{ADelta t}. $$
In this model, photons are only counted, and not seen as waves. Thus there is no amplitude to be considered.

What is the formula for the intensity of light, and how are amplitude, frequency and number of photons considered?

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