Prof. David R. Jackson ECE Dept. Spring 2014 Notes 27 ECE

Scattering by Sphere z y x PEC sphere PW 2

Scattering by Sphere (cont.) where 3

Scattering by Sphere (cont.) Let Examine the field components to determine the boundary conditions. where 4

Scattering by Sphere (cont.) For example, 5

Scattering by Sphere (cont.) Hence 6

Scattering by Sphere (cont.) † Gustav Adolf Feodor Wilhelm Ludwig Mie (Sept. 29, 1869 – Feb. 13, 1957) was a German physicist. He was the first to publish the solution to scattering by a dielectric sphere. The exact solution to scattering by spherical particles is called the Mie † series solution (first published in 1908). 7 He received a doctorate degree in mathematics at the age of 22.

Backscattered Field From symmetry, 8 z y x PEC scattered incident

Choose this one or This backscattered field can be represented in different ways, depending on the angle  that is chosen. z y x PEC y 9 Backscattered Field (cont.)

From the TE r / TM r table: 10 so

Let Keep the dominant term in the series, which is the n = 1 term See the formulas for d n and e n to verify that lower terms are more dominant. 11 Low-Frequency Approximation

12 Low-Frequency Approximation (cont.)

Next, evaluate the Legendre functions: so Hence 13 Low-Frequency Approximation (cont.) Harrington notation

Hence, or 14 Low-Frequency Approximation (cont.)

Low-Frequency Backscattered Field Hence, at Recall that Examine e 1 : so 15

Next, examine the Bessel function terms as x = ka  0 : 16 Low-Frequency Backscattered Field (cont.)

Also 17 Low-Frequency Backscattered Field (cont.)

Hence or 18 Low-Frequency Backscattered Field (cont.)

so or 19 Low-Frequency Backscattered Field (cont.) Hence

Next, consider the TM r term: with 20 Low-Frequency Backscattered Field (cont.) Therefore

21 Low-Frequency Backscattered Field (cont.) We need to evaluate As a shortcut, compare with what we just did for e 1 :

Hence 22 Low-Frequency Backscattered Field (cont.) From the previous approximations for e 1, we note that Therefore

We then have The total backscattered field is then: 23 Low-Frequency Backscattered Field (cont.) Recall:

or This is the final backscattered field from a small sphere. 24 Low-Frequency Backscattered Field (cont.)

Echo Area (Radar Cross Section) Physical interpretation: This is the radiated power of an isotropic radiating “black body” disk that absorbs all incident power. x y z Perfectly absorbing disk 25 Monostatic RCS

Echo Area (Radar Cross Section) (cont.) Simplifying, we have 26

Echo Area (Radar Cross Section) (cont.) For electrically small particles, there is a much stronger scattering from the particles as the size increases. 27 Scattering from small dielectric or metallic particles is called “Rayleigh scattering.” Final Result

Echo Area (Radar Cross Section) (cont.) 28 Rayleigh scattering (low frequency) High-frequency (geometrical optics) limit Resonances

Echo Area (Radar Cross Section) (cont.) 29 Result for Dielectric Sphere Extra factor added

Dipole Moment of Particle 30 Dipole Approximation The sphere acts as a small radiating dipole in the x direction (the direction of he incident electric field), with a dipole moment that is given by

Rayleigh Scattering in the Sky 31 The change of sky color at sunset (red nearest the sun, blue furthest away) is caused by Rayleigh scattering from atmospheric gas particles which are much smaller than the wavelengths of visible light. The grey/white color of the clouds is caused by Mie scattering by water droplets which are of a comparable size to the wavelengths of visible light. John Strutt (Lord Rayleigh), “On the transmission of light through an atmosphere containing small particles in suspension, and on the origin of the blue of the sky,” Philosophical Magazine, series 5, vol. 47, pp , 1899.

32 Rayleigh scattering is more evident after sunset. This picture was taken about one hour after sunset at 500m altitude, looking at the horizon where the sun had set. Rayleigh Scattering in the Sky (cont.)

33 Scattered blue light is polarized. The picture on the right is shot through a polarizing filter which removes light that is linearly polarized in a specific direction Note: When looking up at the sky at sunset or sunrise, the light will be polarized in a north-south direction. (There is little radiated field from the dipole moment that is vertically polarized; therefore the horizontal (north-south) polarization dominates.) Rayleigh Scattering in the Sky (cont.)