Ray Termination Theory
The Ray Termination feature can annihilate rays when their intensity or power decreases below a specified threshold.
If the ray intensity is computed, then the threshold ray intensity Ith (SI unit: W/m2) can be specified. Then the termination criterion is
If the ray power is computed, then the threshold ray power Qth (SI unit: W) can be specified. Then the termination criterion is
It is also possible to specify termination criteria based on both intensity and power; in this case, the rays terminate if either criterion is satisfied.
Whenever possible, the Ray Termination feature attempts to stop rays at the exact instant when the ray intensity or power reaches a specified threshold value. A closed-form analytic solution for the exact stop time is available under the following circumstances:
•
The rays represent planar wavefronts in an absorbing medium, or
•
The rays represent spherical or ellipsoidal wavefronts in a non-absorbing medium.
If neither of these special conditions is met, the ray can only terminate if the termination criterion is met at the beginning of a time step taken by the solver; the ray intensity or power at the instant the ray is stopped may be less than the threshold.
Planar Wavefronts in an Absorbing Medium
In this special case, the principal wavefront radii of curvature are so large that the ray intensity can only change significantly in an attenuating medium. The exact time ts (SI unit: s) at which the ray intensity reaches the threshold value is
where
•
t0 (SI unit: s) is the previous time step, reflection time, or refraction time,
•
I (SI unit: W/m2) is the intensity at t0, and
•
Ith (SI unit: W/m2) is the threshold intensity.
If the termination criterion is based on the ray power Q (SI unit: W) instead of ray intensity, the analogous expression for the stop time is
The characteristic time for ray attenuation τ (SI unit: s) is
where
•
α (SI unit: 1/m) is the attenuation coefficient of the medium and
•
Vg (SI unit: m/s) is the group velocity magnitude.
Diverging Wavefronts in a Non-Absorbing Medium
In a non-absorbing medium in which the wavefront is non-planar, the ray can only terminate due to the intensity decreasing below the threshold, since the power does not change. The time ts (SI unit: s) at which the ray intensity reaches the threshold in a non-attenuating, homogeneous medium is
where
•
t0 (SI unit: s) is the previous time step, reflection time, or refraction time,
•
Vg (SI unit: m/s) is the group velocity magnitude,
•
r1 (SI unit: m) is the first principal radius of curvature of the wavefront at t0,
•
r2 (SI unit: m) is the second principal radius of curvature of the wavefront at t0,
•
I (SI unit: W/m2) is the intensity at t0, and
•
Ith (SI unit: W/m2) is the threshold intensity.
In 2D models, or in 3D models where the wavefronts are cylindrical (one extremely large radius of curvature and one finite radius of curvature), the stop time is instead
where r1 (SI unit: m) is the finite principal radius of curvature of the cylindrical wavefront.