Q:

Missing factor of 1/2 in the definition of light intensity

The light field can be written as $E=A \cos\theta$ where $\theta=kz+wt$ is the phase of light wave. 
What we measure or see is the intensity which is the square of the light field
$$I=E^2=A^2 \cos^2\theta$$
Can anyone explain   why most of the textbooks directly show $\ I=E^2=A^2 \cos^2\theta=A^2$ with the reason the phase is changing very fast.
I am confused  why the fast change results in  $\ I=A^2$ rather than $\ I=(1/2)A^2$.

A:

You're correct in that the angular average $\langle \cos^2 \theta \rangle = 1/2$; the simplest way to see this is to take the average of the trigonometric identity $\cos^2 \theta + \sin^2 \theta = 1$, and use the fact that $\langle \cos^2 \theta \rangle = \langle \sin^2 \theta \rangle$. Thus, if your textbook defined the irradiance as $I\equiv \langle \mathbf E^2(\theta)\rangle$, where the electric field $\mathbf E(\theta) = \mathbf E_0 \cos \theta$, then the correct equation is:
$$ I= \frac12 \mathbf E_0^2 $$
However, as Sofia mentioned in the comments, it is also common to represent electromagnetic fields in a complex notation $\mathbf{\tilde E}(\theta) = \mathbf {\tilde E}_0 e^{i\theta}$ to simplify calculations, where the physical field is understood to be the real part $\mathbf{E(\theta)} = \text{Re}\big\{\mathbf{\tilde{E}}(\theta)\big\}$. In this notation, the irradiance is usually defined as $I \equiv \langle \mathbf{\tilde E}^*\!\!(\theta)\cdot\mathbf{\tilde E}(\theta) \rangle$, in which case you get:
$$ I = \left|\mathbf{\tilde E}_0\right|^2$$

