About the first-order electric field correlation function.

[rocdoc]

For a fixed point in space, the first-order electric field correlation function may be given as (Possibly incorrectly, see my "second" post to this thread)!
$$\langle\vec E^*(t)\vec E(t+\tau)\rangle = {\frac {1} {T}} \int_T\vec E^*(t)\vec E(t+\tau)dt~~~~~(1)$$
Where T is a very large time and * denotes the complex conjugate.$$~$$
Now, the electric field is a vector quantity, so in cartesian coordinates
$$\vec E(t)=(~E_x(t),~E_y(t),~E_z(t)~)$$
So, my question is, in cartesian coordinates, should the integrand in the right-hand side of EQ(1) mean
$$ E^*_x(t) E_x(t+\tau)+~E^*_y(t)E_y(t+\tau)+E^*_z(t)E_z(t+\tau)$$

 

[rocdoc]

For a fixed point in space, the first-order electric field correlation function may be defined as, see Loudon 1, pg. 83
$$\langle E^*(t) E(t+\tau)\rangle = {\frac {1} {T}} \int\limits_T E^*(t) E(t+\tau)~dt~~~~(3.4)$$
Should E be just one component of the electric field?
Reference
1. R. Loudon, The quantum theory of light, 2nd Ed, Oxford University Press,1983.

 

[Andy Resnick]

For a fixed point in space, the first-order electric field correlation function may be defined as, see Loudon 1, pg. 83
$$\langle E^*(t) E(t+\tau)\rangle = {\frac {1} {T}} \int\limits_T E^*(t) E(t+\tau)~dt~~~~(3.4)$$
Should E be just one component of the electric field?
Reference
1. R. Loudon, The quantum theory of light, 2nd Ed, Oxford University Press,1983.

Usually, these correlation functions are first worked for scalar fields. Going over to vector fields, the correlation function is typically written in terms of a matrix: a 2x2 cross-spectral density matrix (or mutual coherence matrix) with matrix elements <E*_i( r1, t) E_j(r2,t+τ)> is used in optics when polarization is included. I suppose you could go to a full 3x3 matrix if you wanted to...