Kirchhoff physical optics

Kirchhoff replaces the true boundary field with a locally reflected incident field on the illuminated surface and suppresses the shadowed contribution. Numerical integration can be accurate at any chosen frequency while the physical-optics approximation remains inaccurate outside its validity regime. Do not equate an “exact integral” with exact wave scattering.

Reflection and amplitude

The normal-incidence pressure reflection factor used for a fluid interface is

\[R_c = \frac{gh-1}{gh+1}, \qquad g=\rho_{\mathrm{int}}/\rho_{\mathrm{ext}}, \quad h=c_{\mathrm{int}}/c_{\mathrm{ext}}.\]

Rigid and pressure-release boundaries use +1 and -1. A fluid-shell sphere uses a frequency- and thickness-dependent two-interface reflection approximation. There is no general implemented elastic-material Kirchhoff reflection law. This is the physical-optics approximation used for swimbladdered fish targets (Foote, 1985).

For a sphere, the finite-frequency expression used by the solver is

\[f_s = R_c a\left[\frac{1}{2}e^{-2ika} -\frac{i(e^{-2ika}-1)}{4ka}\right].\]

kirchhoff(Sphere(a), Rigid(), k)

Its leading magnitude tends to abs(R_c) * a / 2 at high frequency. The correction term is part of the physical-optics surface integral, not a replacement for the exact modal series at small ka.

The time convention is $\exp(-i\omega t)$, with incident pressure $\exp(ik\mathbf d\cdot\mathbf x)$ and outgoing scattered pressure $f\exp(ikr)/r$. For outward normal $\mathbf n$, the source-facing surface has $\mathbf n\cdot\mathbf d<0$. The monostatic physical-optics integral is

$

f=\frac{ikRc}{2\pi}\int{\mathbf n\cdot\mathbf d<0} (\mathbf n\cdot\mathbf d)\exp(2ik\mathbf d\cdot\mathbf x)\,dS. $

Translation by $\mathbf t$ multiplies the amplitude by $\exp(2ik\mathbf d\cdot\mathbf t)$. Target-strength agreement alone does not establish phase accuracy; coherent combinations require complex-amplitude checks as well.

Geometry-specific formulations

Straight cylinders use a Bessel-series lateral contribution plus an illuminated end-cap contribution. This differs from finite-cylinder modal calculations, which omit the caps. Spheroids use adaptive integration over polar and azimuthal coordinates, with illumination intervals determined geometrically. Bent cylinders use nested integration over the curved surface.

The quadrature tolerance controls integration error, not physical model error. The geometry tutorial runs a bent-cylinder example. The frequency tutorial provides the sphere workflow to adapt for comparisons.

kirchhoff(surface::Mesh, boundary, k) evaluates the same backscatter surface integral directly over a supplied full-3D mesh (mesh(path), mesh(generate), mesh(nodes, triangles), or mesh(body; method=:full)), retaining its actual shape, normals and endcaps rather than a closed-form reduction. It reproduces kirchhoff(Sphere/Spheroid/Cylinder, ...) to mesh-discretization accuracy when fed a matching generated mesh. Illumination follows the local outward normal against the incident direction, not a hidden-surface check, so a concave surface (e.g. a real swimbladder) can misclassify a self-shadowed region as illuminated. Check concave geometry with a full-wave solver.

surface = mesh(path; units = :mm)
kirchhoff(surface, Rigid(), k; incidence_angle = pi / 2)

Validity envelope

Against the exact modal series, a rigid or pressure-release sphere agrees to within 0.1 dB for ka ≳ 12 and disagrees by 10 dB or more for ka ≲ 0.5; the error is not monotonic in between, since Kirchhoff smooths over the modal series' own peaks and nulls. A rigid spheroid at broadside incidence agrees to within 1 dB for kb ≳ 5 (b the minor semi-axis), largely independent of aspect ratio from 1.5 to 8. At fixed kb = 10, error grows from 0.3 dB at broadside to 1.4 dB at end-on. Jech et al. (2015) report a similar pattern (agreement near broadside/high ka, larger deviations off-broadside) across independently implemented models. Check any case outside these ranges against modal or a converged numerical method.

These sampled trends are not uniform error bounds. For an orientation distribution, average scattering powers before converting to dB. Averaging target strengths gives a different result and can conceal large errors at individual angles. Concave self-shadowing and diffraction remain outside the local-normal Kirchhoff approximation.