A macroscopic theory of interference and diffraction of light from finite sources II. Fields with a spectral range of arbitrary width

The results of part I of this investigation are generalized to stationary fields with a spectral range of arbitrary width. For this purpose it is found necessary to introduce in place of the mutual intensity function of Zernike a more general correlation function $\hat{\Gamma}$(x$_{1}$, x$_{2}$, $\tau $) = $\langle \hat{V}$(x$_{1}$, t + $\tau $) $\hat{V}^{\ast}$(x$_{2}$, t)$\rangle $, which expresses the correlation between disturbances at any two given points P$_{1}$(x$_{1}$), P$_{2}$(x$_{2}$) in the field, the disturbance at P$_{1}$ being considered at a time $\tau $ later than at P$_{2}$. It is shown that $\hat{\Gamma}$ is an observable quantity. Expressions for $\hat{\Gamma}$ in terms of functions which specify the source and the transmission properties of the medium are derived. Further, it is shown that in vacuo the correlation function obeys rigorously the two wave equations $\nabla _{s}^{2}\hat{\Gamma}$ = $\frac{1}{c^{2}}\frac{\partial ^{2}\hat{\Gamma}}{\partial \tau ^{2}}$ (s = 1, 2), where $\nabla _{s}^{2}$ is the Laplacian operator with respect to the co-ordinates (x$_{s}$, y$_{s}$, z$_{s}$) of P$_{s}$(x$_{s}$). Using this result, a formula is obtained which expresses rigorously the correlation between disturbances at P$_{1}$ and P$_{2}$ in terms of the values of the correlation and of its derivatives at all pairs of points on an arbitrary closed surface which surrounds P$_{1}$ and P$_{2}$. A special case of this formula (P$_{2}$ = P$_{1}$, $\tau $ = 0) represents a rigorous formulation of the generalized Huygens principle, involving observable quantities only.