Radar cross section

Radar cross section (RCS) is a description of how an object reflects an incident electromagnetic wave. For an arbitrary object, the RCS is highly dependent on the radar wavelength and incident direction of the radio wave. The usual definition or RCS differs by a factor of 4 π from the standard physics definition of differential cross section at 180 degrees. Bistatic radar cross section is defined similarly for other angles.

The RCS is integral to the development of radar stealth technology, particularly in applications involving aircraft and ballistic missiles. RCS data for current military aircraft are almost all highly classified.

Measurement

Measurement of a target's RCS is performed at a radar reflectivity range or scattering range. The first type of range is an outdoor range where the target is positioned on the pylon some distance down-range from the transmitters. Such a range eliminates the need for placing radar absorbers behind the target, however multi-path effects due to the ground must be mitigated.

An anechoic chamber is also commonly used. In such a room, the target is placed on a rotating pillar in the center, and the walls, floors and ceiling are covered by stacks of radar absorbing material. These absorbers prevent corruption of the measurement due to reflections. A compact range is an anechoic chamber with a reflector to simulate far field conditions.

Calculation

Quantitatively, the RCS is an effective surface area that intercepts the incident wave and that scatters the energy isotropically in space. For the RCS,

\sigma

is defined in three-dimensions as

\sigma = 4 \pi R^{2} \frac{P_{s}}{P_{i}}

Where $\sigma$ is the RCS, $P_{i}$ is the incident power density measured at the target, and $P_{s}$ is the scattered power density seen at a distance $R$ away from the target.

In electromagnetic analysis this is also commonly written as

\sigma = 4 \pi R^{2} \frac{|E_{s}|^{2}}{|E_{i}|^{2}}

where $E_{i}$ and $E_{s}$ are the incident and scattered electric field intensities, respectively.

In the design phase, it is often desirable to employ a computer to predict what the RCS will look like before fabricating an actual object. Many iterations of this prediction process can be performed in a short time at low cost, whereas use of a measurement range is often time-consuming, expensive and error-prone. The linearity of Maxwell's equations makes RCS relatively straightforward to calculate with a variety of analytic and numerical methods, but changing levels of military interest and the need for secrecy have made the field challenging, none the less.

The field of solving Maxwell's equations through numerical algorithms is called computational electromagnetics, and many effective analysis methods have been applied to the RCS prediction problem. RCS prediction software are often run on large supercomputers and employ high-resolution CAD models of real radar targets. High frequency approximations such as geometric optics, Physical Optics, the geometric theory of diffraction, the uniform theory of diffraction and the physical theory of diffraction are used when the wavelength is much shorter than the target feature size.

Statistical models include chi-square, rice, and the log-normal target models. These models are used to predict likely values of the RCS given an average value, and are useful when running radar Monte Carlo simulations.

Purely numerical methods such as the boundary element method (method of moments), finite difference time domain method (FDTD) and finite element methods are limited by computer performance to longer wavelengths or smaller features.

Though, for simple cases, the wavelength ranges of these two types of method overlap considerably, for difficult shapes and materials or very high accuracy they are combined in various sorts of hybrid methods.

Reduction

RCS reduction is chiefly important in stealth technology. With smaller RCS, aircraft and other military vehicles may better evade radar detection, whether it be from land-based installations or other vehicles.

Purpose Shaping

Purpose shaping is an RCS reduction technique in which the shape of the target’s reflecting surfaces is designed such that they reflect energy away from the source. The aim is usually to create a “cone-of-silence” about the aircraft’s direction of flight. Note that due to the energy reflection, this RCS reduction method is defeated by using Passive (multistatic) radars.

Purpose-shaping techniques can be seen in the design of surface faceting on the F-117A Nighthawk stealth fighter. This aircraft, designed in the late 1970s though only revealed to the public in 1988, uses a multitude of flat surfaces to reflect incident radar energy away from the source. Yue suggests that limited available computing power for the design phase kept the number of surfaces to a minimum. The B-2 Spirit stealth bomber benefited from increased computing power, enabling its contoured shapes and further reduction in RCS. The F-22 Raptor and F-35 Joint Strike Fighter continue the trend in purpose shaping and promise to have even smaller monostatic RCS.

Active Cancellation

In active cancellation techniques, the target aircraft generates a radar signal equal in intensity but opposite in phase to the predicted reflection of an incident radar signal (similarly to noise canceling ear phones). This creates destructive interference between the reflected and generated signals, resulting in reduced RCS. To incorporate active cancellation techniques, the precise characteristics of the waveform and angle of arrival of the illuminating radar signal must be known, since they define the nature of generated energy required for cancellation. Except against simple or low frequency radar systems, the implementation of active cancellation techniques is extremely difficult due to the complex processing requirements and the difficulty of predicting the exact nature of the reflected radar signal over a broad aspect of an aircraft, missile or other target.

RAM

The third RCS reduction technique for aircraft and missiles is the use of radar absorbing material (RAM) either in the original construction or as an addition to highly reflective surfaces. There are at least three types of RAM: resonant, non-resonant magnetic and non-resonant large volume. Resonant but somewhat “lossy” materials are applied to the reflecting surfaces of the target. The thickness of the material corresponds to one-quarter wavelength of the expected illuminating radar-wave. The incident radar energy is reflected from the outside and inside surfaces of the RAM to create a destructive interference pattern. This results in the cancellation of the reflected energy. Non-resonant magnetic RAM uses ferrite particles suspended in epoxy or paint to reduce the reflectivity of the surface to incident radar waves. Because the non-resonant RAM dissipates incident radar energy over a larger surface area, it usually results in an increase in surface temperature, thus reducing RCS at the expense of an increase in infrared signature. A major advantage of non-resonant RAM is that it can be effective over a broad range of frequencies, whereas resonant RAM is limited to a narrow range of design frequencies. Large volume RAM is usually resistive carbon loading added to fiberglass hexagonal cell aircraft structures or other non-conducting components. Fins of resistive materials can also be added. Thin resistive sheets spaced by foam or aerogel may be suitable for space craft.

Thin coatings made of only dielectrics and conductors have very limited absorbing bandwidth, so magnetic materials are used when weight and cost permit, either in resonant RAM or as non-resonant RAM.

Optimization methods

Thin non-resonant or broad resonance coatings can be modeled with a Leontovich impedance boundary condition. This is the ratio of the tangential electric field to the tangential magnetic field on the surface, and ignores fields propagating along the surface within the coating. This is particularly convenient when using boundary element method calculations. The surface impedance can be calculated and tested separately. For an isotropic surface the ideal surface impedance is equal to the 377 Ohm impedance of free space. For non-isotropic (anisotropic) coatings, the optimal coating depends on the shape of the target and the radar direction, but duality, the symmetry of Maxwell's equations between the electric and magnetic fields, tells one that optimal coatings have η₀ × η₁ = 377² Ω², where η₀ and η₁ are perpendicular components of the anisotropic surface impedance, aligned with edges and/or the radar direction. A perfect electric conductor has more back scatter from a leading edge for the linear polarization with the electric field parallel to the edge and more from a trailing edge with the electric field perpendicular to the edge, so the high surface impedance should be parallel to leading edges and perpendicular to trailing edges, for the greatest radar threat direction, with some sort of smooth transition between.

To calculate the radar cross section of such a stealth body, one would typically do one dimensional reflection calculations to calculate the surface impedance, then two dimensional numerical calculations to calculate the diffraction coefficients of edges and small three dimensional calculations to calculate the diffraction coefficients of corners and points. The cross section can then be calculated, using the diffraction coefficients, with the physical theory of diffraction or other high frequency method, combined with Physical Optics to include the contributions from illuminated smooth surfaces and Fock calculations to calculate creeping waves circling around any smooth shadowed parts.

Optimization is in the reverse order. First one does high frequency calculations to optimize the shape and find the most important features, then small calculations to find the best surface impedances in the problem areas, then reflection calculations to design coatings. One should avoid large numerical calculations that run too slowly for numerical optimization or distract workers from the physics, even when massive computing power is available.

References

Shaeffer, Tuley and Knott. Radar Cross Section. SciTech Publishing, 2004. ISBN 1891121251.
Harrington, Roger F. Time-Harmonic Electromagnetic Fields. McGraw-Hill, Inc., 1961. ISBN 070267456.
Balanis, Constantine A. Advanced Engineering Electromagnetics. Wiley, 1989. ISBN 0471621943.
“A Hybrid Method Based on Reciprocity for the Computation of Diffraction by Trailing Edges”David R. Ingham, IEEE Trans. Antennas Propagat., 43 No. 11, November 1995, pp. 1173–82.
“Revised Integration Methods in a Galerkin BoR Procedure” David R. Ingham, Applied Computational Electromagnetics Society (ACES ) Journal 10 No. 2, July, 1995, pp. 5–16.
“A Hybrid Approach to Trailing Edges and Trailing Ends” David R. Ingham, proceedings of the ACES Symposium, 1993, Monterey.
“Time-Domain Extrapolation to the Far Field Based on FDTD Calculations” Kane Yee, David Ingham and Kurt Shlager, IEEE Trans. Antennas Propagat., 39 No. 3, March 1991, pp.410–413.
“Numerical Calculation of Edge Diffraction, using Reciprocity” David Ingham, Proc. Int. Conf. Antennas Propagat., IV, May 1990, Dallas, pp.1574–1577.
“Time-Domain Extrapolation to the Far Field Based on FDTD Calculations”Kane Yee, David Ingham and Kurt Shlager, invited paper, Proc. URSI Conf., 1989, San José .

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