#### Rayleigh- versus Mie- Scattering

Rayleigh-
Region
Mie- Region
optical
region
A
B

Figure 1: Rayleigh- and Mie- Scattering, and optical region

Rayleigh-
Region
Mie- Region
optical
region
A
B

Figure 1: Rayleigh- and Mie- Scattering, and optical region

#### Rayleigh- versus Mie- Scattering

The figure 1 shows the different regions applicable to computing the radar cross section of a sphere. The optical region (“far field” counterpart) rules apply when (2π·r/λ);>10. In this region, the radar cross section of a sphere is independent of frequency. The reflecting area of the sphere is here equal to the area of a circle with the radius of the sphere

σ = π·r2

(1)

The radar cross section equation breaks down primarily due to creeping waves in the area where 2π · r. The largest positive perturbation of the radar cross section (point A) would be 4 times higher than the radar cross section computed using the optical region formula. Just slightly a minimum occurs (point B) and the actual radar cross section would be 0.26 times the value calculated by using the optical region formula. This area is known as the “Mie” or “resonance region”.

If we used a one-meter diameter sphere, the perturbations would occur at 95 MHz, so any frequency above 950 MHz would give predicted results.

The size of the spherical reflection area is smaller than the wavelength in the area of the “Rayleigh-Scattering”. The radar cross section is calculated the formula

σ = π·r2 · 7,11 · (2π·r/ λ)4

(2)

here. The “Rayleigh-Scattering” is a typical application case for weather radar.

Approximately the lower L-Band still takes the Mie scattering into account at air defense and air traffic control radar sets. There are predominantly optical conditions at frequencies above 1 GHz.

##### Qualitative derivation

Figure 2: Time delay of the circulating wave to the directly reflected wave.

Figure 2: Time delay of the creeping wave to the directly reflected wave.

The energy components present in the interference are on the one hand the energy directly reflected in the center of the sphere, which, however, is subject to a phase jump of 180° during the reflection. The second part results from a creeping wave, which is generated by a continuous diffraction at the surface of the sphere. This creeping wave has to take a detour depending on the diameter of the sphere. Both components overlap in phase in the local maxima of the diagram in Figure 1 and in phase opposition in the local minima.

If it is assumed for simplification that the circulating wave runs directly on the surface of the sphere, the detour can be calculated according to Fig. 2 from the sum of the diameter and half the circumference of the circle (spherical section). Thus, the first minimum occurs at the earliest when the detour is equal to half the wavelength and the phase shift due to the time delay of the detour (like the phase jump in reflection) is also 180°. All further local maxima and minima occur at a size of the detour equal to the even as well as odd multiple of the half wavelength.

Since there is a small distance between the surface of the sphere and the path of the circulating wave, the approximation 2πr can be used instead of (2+π)r.

For example, the classic Russian VHF radars operated on frequencies between 145 and 175 MHz, which corresponds to a wavelength of 1.7 to 2.1 meters. For the geometric dimensions of a fighter (about 2.5 to 4 m circumference of the fuselage), this corresponds to a position in the diagram shown about the second maximum (above the letter B).