The Interference Pattern plot can be used to visualize the fringes resulting from the interference of two or more rays at a surface. The interference pattern is only valid when it is plotted over an area with a length scale that is much smaller than the principal radius of curvature of any incident wavefront. This is a consequence of the treatment of each ray as a wavefront that subtends a small solid angle. Furthermore, this plot type requires calculation of the ray intensity and instantaneous phase.
Given the phase Ψ0 of a ray at the point
(qx, qy) where the ray intersects a plane, the phase
Ψ at a nearby point
(qx + Δx, qy + Δy) is computed as follows. Without loss of generality assume that the
x- and
y-axes are oriented so that they are parallel to the projections of the principal curvature directions onto the cut plane. For a spherical wavefront, the orientations of these axes may be determined arbitrarily as long as they are orthogonal. Let
ni,x and
ni,y denote the
x- and
y- components, respectively, of the normalized ray direction vector. Let
r1 and
r2 denote the principal radii of curvature in the directions parallel to the
x- and
y-axes, respectively. For an incident ray with wave vector magnitude
k, the phase at
(qx + Δx, qy) is
Where Δr1 is the change in the principal radius of curvature of the wavefront when moving between the two points.
The rationale for discarding higher-order terms is that they all involve division by higher powers of r1 and
r2. Therefore this series approximation is only valid when the perturbations in the
x and
y directions are much smaller than the principal radii of curvature. To assume otherwise would mean that the incident wavefronts subtend very large solid angles.
