Intensity and Wavefront Curvature
The following describes the algorithm used to compute the ray intensity when Intensity computation is set to Compute intensity or Compute intensity and power in the physics interface Intensity Computation section. In this algorithm, each ray is treated as a propagating wavefront subtending a small solid angle. The algorithm is only valid when the rays propagate in homogeneous media. The following auxiliary dependent variables are defined for each ray:
In 3D:
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The initial ray intensity I0, which is reinitialized at material discontinuities and walls.
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The integral A of the attenuation coefficient along the ray path.
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Two initial principal radii of curvature, r1,i and r2,i.
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Two principal radii of curvature, r1 and r2, which represent the maximum and minimum radii of curvature of the intersection of the wavefront with an arbitrary plane.
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Three components of a unit vector e1 in the direction corresponding to the first principal radius of curvature. This information is used to reinitialize the principal radii of curvature at curved boundaries.
In 2D:
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The initial ray intensity I0.
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The integral A of the attenuation coefficient along the ray path.
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One initial principal radius of curvature, r1,i.
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One radius of curvature r1 of the wavefront. All wavefronts are assumed to be converging or diverging cylindrical waves, so it is not necessary to define a second radius of curvature.
A different algorithm is used to compute the intensity of each ray if the Intensity computation is instead set to Compute intensity in graded media or Compute intensity and power in graded media. This algorithm accounts for the effect of graded media on the ray intensity, but it is also slower and less accurate for homogeneous media. The following auxiliary dependent variables are defined for each ray:
In 3D:
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The intensity help variable Γ.
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The integral A of the attenuation coefficient along the ray path.
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Two principal curvature calculation help variables α1 and α2 and the rotation angle ϕ, which indicates the orientation of the principal curvature directions. An additional help variable is used internally to detect poles in the local coordinate system definition and to redefine the local coordinate system accordingly.
In 2D:
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The intensity help variable Γ.
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The integral A of the attenuation coefficient along the ray path.
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One principal curvature calculation help variable α1.
For the options Compute Power, Compute intensity and power, and Compute intensity and power in graded media an additional auxiliary dependent variable is defined to indicate the total power transmitted by each ray. Like intensity, power is affected by absorbing media and by reflection or refraction at boundaries. Unlike intensity, power does not change due to the focusing or divergence of a ray bundle.
Principal Radii of Curvature
In 3D models, each ray is treated as a wavefront for which two principal radii of curvature, r1 and r2, are defined. In addition, the values of the two principal radii of curvature are stored as r1,i and r2,i whenever the ray reaches a boundary.
Within domains, the equations
are solved. Negative radii of curvature indicate that the wavefront is expanding as the ray propagates, while positive radii of curvature indicate that the wavefront is converging. A continuous locus of points at which either of the principal radii of curvature equals zero is called a caustic surface. The unit vector e1 is defined so that r1 is the radius of curvature of the intersection of the wavefront with the plane tangent to e1 and the wave vector k. Similarly, if a vector e2 is defined so that
Then r2 is the radius of curvature of the intersection of the wavefront with a plane tangent to e2 and k. The components of e1 are stored as auxiliary dependent variables for each ray. The components of e2 can then be derived from e1 and k at any time.
The principal radii of curvature are reinitialized at material discontinuities, and the orientation of the unit vectors e1 and e2 may change. If the unit vector in the direction of the incident ray ni is not parallel to the surface normal ns, then it is possible to define a unique tangent plane, called the plane of incidence, that contains the incident, reflected, and refracted rays. The unit vector normal to this plane, u0, is defined as
If the ray is normal to the surface, then the incident ray is parallel to the refracted ray and antiparallel to the reflected ray, and u0 can be any arbitrary unit vector orthogonal to ni. In addition to the unit vector normal to the plane of incidence, the following unit vectors tangent to the incident wavefront, refracted wavefront, reflected wavefront, and surface normal are defined:
where the subscripts i, t, and r denote the incident, refracted, and reflected wavefronts, respectively. For a wavefront propagating in a direction n, with principal curvatures k1 = 1/r1 and k2 = 1/r2 defined for directions e1 and e2, respectively, the principal curvatures in two other orthogonal directions e1' and e2' (both orthogonal to n) are
(8-1)
where θ is the angle of rotation about n which transforms the vectors e1 and e2 to e1' and e2', respectively. Because k1' and k2' are not principal curvatures, it is necessary to include off-diagonal elements of the wavefront curvature tensor equal to k12'.
The following algorithm is used to reinitialize the principal radii of curvature of the wavefront and their orientations. The reinitialization of the curvature variables follows the method of Stavroudis in Ref. 2.
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Given ni and ns, compute unit vectors in the directions of the reflected and refracted rays, nr and nt.
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Compute the vectors u0, ui, ut, ur, and us.
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Compute the angle of rotation θ(i) needed to transform the local coordinate system with axes parallel to e1, e2, and ni to a local coordinate system with axes parallel to u0, ui, and ni. (Note: the superscript (i) is used to distinguish this coordinate system rotation angle from the angle of incidence of the ray, often denoted θi.)
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Transform the principal curvatures to the local coordinate system defined by u0, ui, and ni by using Equation 8-1, substituting θ(i) for θ.
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Given the two principal curvatures of the surface, k1,s and k2,s with directions e1,s and e2,s, transform the curvature variables to a local coordinate system with axes parallel to u0, us, and ns. Let the new curvature variables be denoted by k1,s', k2,s', and k12,s'.
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The curvatures of the wavefront and the surface have now been defined in coordinate systems that share the axis u0 and only differ by a rotation by the angle of incidence θi about u0. Defining the variables η and γ as in Material Discontinuity Theory, compute the curvature variables of the refracted ray in a coordinate system defined by u0, ut, and nt using the equations
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Obtain the principal curvatures of the refracted ray by rotating the coordinate system defined by u0, ut, and nt by an angle θ(t) about nt. The angle θ(t) is defined as
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Reinitialize the principal curvatures of the refracted ray:
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Invert the principal curvatures to obtain the principal radii of curvature of the refracted ray. Rotate u0 about nt by the angle θ(t) to obtain the reinitialized principal curvature direction e1,t.
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If a reflected ray is released, compute the curvature variables of the reflected ray in a coordinate system defined by u0, ur, and nr using the equations
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Obtain the principal curvatures of the reflected ray by rotating the coordinate system defined by u0, ur, and nr by an angle θ(r) about nr. The angle θ(r) is defined as
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Initialize the principal curvatures of the reflected ray:
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Invert the principal curvatures to obtain the principal radii of curvature of the reflected ray. Rotate u0 about nr by the angle θ(r) to obtain the initialized principal curvature direction e1,r.
In 2D components, only one principal radius of curvature is computed; each ray is treated as a cylindrical wave. In addition, no auxiliary degrees of freedom for the principal curvature direction are required because the out-of-plane direction can always be treated as one of the axes of the local coordinate system that defines the orientation of the wavefront.
In 2D axisymmetric model components, radii of curvature are computed for the in-plane direction (that is, the rz-plane) as well as the out-of-plane or azimuthal directions. During ray-boundary interactions, these radii of curvature are reinitialized as if the ray interacted with a 3D surface of revolution, which may have finite radii of curvature in both the in-plane and out-of-plane directions.
Intensity Calculation
The value of the ray intensity is stored as the auxiliary dependent variable I0 when a ray is released. At any point along the ray’s trajectory, the intensity is equal to
In 2D, the second principal radius of curvature r2 is treated as an arbitrarily large distance that remains constant for each ray.