Theory of refraction and reflection with partially coherent electromagnetic beams

Laws of refraction and reflection of light are governed by the classic Fresnel formulas. These formulas are not applicable to partially coherent light. We develop a general theory of refraction and reflection of electromagnetic beams of any state of coherence. We find that coherence properties of such beams change, in general, on refraction and on reflection at a planar interface.

Figure 1

The geometry relating to refraction and reflection of a monochromatic plane wave at an interface; h and v directions are chosen to be parallel and perpendicular to the plane of incidence.

Figure 2

The $(x_{\text{v}}^{\left(\mathbb{i}\right)},x_{\text{h}}^{\left(\mathbb{i}\right)},x_{\text{p}}^{\left(\mathbb{i}\right)})$ coordinate system.

Figure 3

The h and the p directions of the incident-, the refracted-, and the reflected-beam axis coordinate systems; the v direction for all three systems is the same, and points at a right angle into the plane of the figure.

Figure 4

The notation used for descriptions of the $(x_{{\text{v}}^{\prime }}^{\left(\mathbb{i}\right)},x_{{\text{h}}^{\prime }}^{\left(\mathbb{i}\right)},x_{{\text{p}}^{\prime }}^{\left(\mathbb{i}\right)})$ and the $(x^{\prime },y^{\prime },z^{\prime })$ coordinate systems.

Figure 5

Change of the coherence properties on refraction: the modulus $\left|\eta \right|$ of the degree of coherence of the incident (solid line) and of the transmitted (dotted line) beams are plotted as functions of $\rho =|{\mathbf{r}}^{\prime }-\mathbf{r}|$, at $\omega \approx 3.2\times {10}^{15}$ s${}^{-1}$, for values of the parameters ${\theta }_{\mathbb{i}}={40}^{\circ }$, $\delta =0.001$ m, $\sigma =0.01$ m, $A_{\text{v}}/A_{\text{h}}=1$, and $B_{\text{hv}}=9/16$.

Figure 6

Change of the coherence properties on reflection: the modulus $\left|\eta \right|$ of the degree of coherence of the incident (solid line) and of the reflected (dotted line) beams are plotted as functions of $\rho =|{\mathbf{r}}^{\prime }-\mathbf{r}|$, for the same choice of parameters as used in Fig. 5.

Figure 7

The geometry relating to the relationship between the $(x_{\text{v}}^{\left(\mathbb{i}\right)},x_{\text{h}}^{\left(\mathbb{i}\right)},x_{\text{p}}^{\left(\mathbb{i}\right)})$ and the $(x,y,z)$ coordinate systems.

Figure 8

The transformation from the $(x,y,z)$ to the $(x^{\prime },y^{\prime },z^{\prime })$ coordinate system.

Figure 9

The geometry showing that the vectors ${\kappa }^{\left(\mathbb{i}\right)}$, ${\kappa }^{\left(\mathbb{t}\right)}$, and $\mathbf{n}$ lie in the $y^{\prime }-z^{\prime }$ plane.