Hi I'm a computer science graduate and I am working on a computer program to find out the radar cross section of an object in 3d space. I would be really grateful if some one can help me out in some physics problems here. The idea is basically to simulate a plane wave as a dense grid of parallel rays and follow each ray as they travel in the medium, how they get reflected etc. If the power density of the incident plane wave is Pi, then the body will intercept a power equal to RCS * Pi. This power it will scatter uniformly in all possible direction as in the far field, the body is as good as a point isotropic source. So, the scattered power density at a distance r is given as:

Ps = (RCS * Pi)/4 * PI * r^2 (Spherical spreading of the rays reflected of the surface)

thus,

RCS = 4 * Pi * r^2 *(Ps / Pi)

in far field, r-> infinity

The formula can also be written in terms of electric field.

RCS = 4 * pi * r^2 * |Es|^2 / |Ei|^2

where Es is scattered electric field as seen at a distance r from the target and ei is the incident electric field.

So basically what I want to do is:

1. See where each incident ray intersects the object. Based on this calculate the electric field at the point of intersection. To find the total electric field which is incident, do the same for every ray and take a summation.

2. To calculate the Es or scattered electric field, I will follow each reflected ray(specular reflection is only considered) or scattered ray and see where it intersects a bounding sphere of radius r. At that point calculate the electric field. Do this for all the scattered rays and take the summation to find the total electric field scattered. If a huge number of rays are taken and the mesh is really fine, I believe one can get results very close to an isotropic type of radiation.

3. Take the square of magnitude of electric field vectors and substitute it in the RCS formula.


I was reading a thesis and the formula given for the electric field along a ray traveling through the space was:

E(s) = E(0) * exp(-jks) -------( 1 )

where

E(s) is electrical field at a distance s(along the direction of propogation of the wave) from the reference point (say F0). E(0) is electrical field at reference point F0. k is the wave number 2 * PI / wavelength. exp(-jks) represents the phase variation of electric field along the ray.

There was a formula given for the ray which has been reflected off a
surface as well. It was like this:

E(s) = Eincident(Q) * R * exp(-jks) / s ----------- ( 2 )

where Q is the point at which reflection took place and Eincident(Q)
is the electrical field incident at the point Q. R is the reflection
matrix. According to thesis the reflection matrix is given as:

-1 0
0 1

My problem with these formulas are as follows:

1. Shouldn't E(s) and E(0) be 3d vectors ? In my program the user should specify the source electric field magnitude and i have already figured out the direction of electric field vector given the direction vector for the plane wave. So shouldn't that be sufficient or some more information is required.

2. What is the point of having a complex phase ? If E(0) was defined to be a 3d vector, then E(s) becomes a complex vector ( a vector whose components are complex numbers) due to multiplication with exp(-jks). Or it could be that E(0) is a complex vector, so that would make E(s) also a complex vector. The reflection matrix makes perfect sense to me because at the plane of incidence, the direction of the horizontal component of the electric field gets reveresed after the reflection whereas the direction of the vertical component remains the same.

Some one please help me out.

Well, bad luck.

A radar cross section has very little to do with the "optical" surface of the object. For instance, a half-wave dipole can be very thin, but has an important cross-section. There, the intersection with rays, even dense ones, gives a very false result. Bad method hence. This is of real concern, because low frequency radars developed against stealth planes use this effect which is difficult to counter.

Also, the object doesn't at all radiate back as a point source would do. This is half of the trick by stealth planes. And anyway, there can't be a spherical source in electromagnetics - at least with linearly polarized fields. But you may hope that each point of the object radiates like a short dipole.

Very short basics in http://en.wikipedia.org/wiki/Radar_cross_section
Only 10 pages in chapter 9 on "Antennas" by Kraus
but if you aren't already good at electromagnetism, you have no chance to model the phenomena by some optical analogy, no will you get the knowledge within any reasonable time.

Your only chance is to be associated with an EM expert.

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The cross-section you describe if for a single frequency, but radars avoid this case. Over-The-Horizon radars cover a wide spectrum, as do impulse radars developed against stealth planes. And even more classical radars spread their spectrum enough to have changed the phase and frequency within the propagation time that corresponds to a plane's size.

Also, the RCS is defined when the radar receiver is at the same location as the transmitter, but Over-The-Horizon radars carefully avoid this. Transmitter on one island, receiver on another. Nice toys: each covers 10% of the Earth's surface.

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Where they intersect: look at rendering engines in 3D games.

Electric field: bad luck if the materials are unknown. And anyway, you must use multiple reflections, since for instance the perpendicular surfaces of a tail are very important in the RCS. Even the engine inlets with the blades are very important! And this requires a very detailed description of the engine.

Specular reflection only: no! And anyway, this is incompatible with a spherical source - or even a dipole. The very idea of summing many reflections is to take only diffuse reflection; their combination may become specular with some forms and materials.

Before reading a thesis, you may try to read textbooks about electromagnetics, but I see no hope in any reasonable time.

E is a 3D vector that depends on time and position. At a single frequency, you may write the cos(2piFt) and sin(2piFt) part of each of its 3D components as the real and imaginary part of a complex number - this is the principal use of complex numbers. Find a book about electrical engineering that describes it. exp(-jks) accounts for the propagation time, during which the phase goes on rotating. Book about waves.

Reflection: don't expect any coefficient near to 1 on a military airplane, even an older one. And againt a metal, the vertical electric field would be reversed as well.

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You know you're tackling a really difficult task? My impression is that you don't have the knowledge for it and can't acquire it in time - and probably those who suggested the project don't have this knowledge neither.

Can you associate with an E-M specialist who would program all the computations abouts fields?

Or can you choose another project?

The Wikipedia entry contains gross mistakes from the beginning on.

It states the cross-section to be dimensionless. Very bad impression right on.

And anyway, introducing a geometric surface is the worst way to define a radar cross-section.

In short: don't read this entry.

Another difficulty is the amount of computation needed.

Engine inlets are a heavy contributor to RCS, and in a ray analysis, you need to compute a huge number of interactions. This may well be too much for a PC, for instance.

By the way, computing a steady state at a single frequency isn't necessarily the best way! It needs to solve a huge matrix (the interactions of all surface elements) which isn't very sparse.

Computing the time response to a Dirac pulse should be faster. It needs to compute over many instants, but at least surface elements don't interact then (as they're separated by the propagation time), and you get the RCS for all frequencies at once. What you need is store the field for each surface element and at many instants.

I definitely prefer this way.

Even more complicated than I thought.

Computing all surface currents, potentials etc looks faster than all volume fields - and should give the same answer if the engine exhaust still is negligible in stealth planes, but it needs to compute near-fields as well, and distinguish whether currents run at the fore or aft surface etc. Also, materials are dispersive, especially in stealth airplanes.

This is true for both an impulse response computation and a fixed frequency one.

Our special prize goes to the cockpit, which is partially reflective on a civilian plane but tries to absorb waves in a stealth one, and for the plane's radar antenna.

Getting a not-too-false result needs an extremely detailed model of the plane.

By the way, expect all military planes to have RCS-minimizing measures. They appeared well over 20 years before the F-117, though the F-117 put much emphasis on them.