Prof. David R. Jackson ECE Dept. Spring 2014 Notes 29 ECE

High-Frequency Scattering by Cylinder Assume PEC cylinder 2

Physical Optics Physical Optics Approximation Lit Dark Dark region (not seen by incident plane wave) Normal 3

Physical Optics (cont.) Physical Optics Approximation Lit Dark Lit region: Normal Locally, the reflection acts like plane-wave reflection from a flat surface. 4

Physical Optics Approximation Lit Dark Lit region: Dark region: Physical Optics (cont.) 5

Physical Optics Approximation High-Frequency Scattering by Cylinder (cont.) PEC lit Lit region: 6

or High-Frequency Scattering by Cylinder (cont.) 7

Scattered field: Consider a z -directed line source at the origin: High-Frequency Scattering by Cylinder (cont.) 8

For current at, include phase shift terms Far field: Next, consider the line source to be located at ( x ´, y ´ ): High-Frequency Scattering by Cylinder (cont.) 9 Hence

or Hence, letting or Hence High-Frequency Scattering by Cylinder (cont.) 10

For the integral Hence Integrating, High-Frequency Scattering by Cylinder (cont.) 11 or

For the integral Hence This may be written as where Hence, we can identify High-Frequency Scattering by Cylinder (cont.) Compare with 12

or SPP (No SPP. Assume   2  n ) Find the stationary-phase point (SPP): High-Frequency Scattering by Cylinder (cont.) AB 13

(b) since choose We require the restriction that Hence, choose n = -1 : From the previous slide, High-Frequency Scattering by Cylinder (cont.) Also, 14

Geometrical Optics Specular point The specular point of reflection is the point at which the ray reflects off and travels to the observation point. We can show that Observation point Specular point 15

Geometrical Optics (cont.) Specular point Proof 16

Then High-Frequency Scattering by Cylinder (cont.) Note that there is always a stationary-phase point, for all observation angles (except  = 0 ). 17

At SPP: Next, calculate the g function at the stationary-phase point: High-Frequency Scattering by Cylinder (cont.) 18

At SPP: Next, calculate the second derivative of the g function: Note: High-Frequency Scattering by Cylinder (cont.) 19

Hence the integral is Hence High-Frequency Scattering by Cylinder (cont.) 20 Recall:

or High-Frequency Scattering by Cylinder (cont.) 21 Use Then we have

Then or Therefore High-Frequency Scattering by Cylinder (cont.) Recall 22

Then Radiation pattern of cylinder (scattered field) High-Frequency Scattering by Cylinder (cont.) 23

Then High-Frequency Scattering by Cylinder (cont.) 24 In the backscattered direction (  =  ): Echo width (monostatic RCS): Note: E 0 = 1 in our case.

Then High-Frequency Scattering by Cylinder (cont.) 25 Hence (circumference of lit region)