ECE 6341 Spring 2016 Prof. David R. Jackson ECE Dept. Notes 29

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

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

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

High-Frequency Scattering by Cylinder PEC cylinder Assume

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

High-Frequency Scattering by Cylinder (cont.) or We next calculate the radiation from this current.

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

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

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

High-Frequency Scattering by Cylinder (cont.) or Hence Integrating over the lit region, we have, For the integral

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

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

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

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

Geometrical Optics (cont.) Specular point Proof

High-Frequency Scattering by Cylinder (cont.) Note that there is always a stationary-phase point, for all observation angles (except  = 0). Then Note: The stationary-phase method will fail for  =  / 2.

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

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

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

High-Frequency Scattering by Cylinder (cont.) Simplify using or Then we have

High-Frequency Scattering by Cylinder (cont.) Recall: Then Therefore, we have or Simplifying, we have

Final high-frequency radiation pattern of cylinder High-Frequency Scattering by Cylinder (cont.) Final high-frequency radiation pattern of cylinder (scattered field) Then

High-Frequency Scattering by Cylinder (cont.) Solution for  = 0 Then In this case we have: Recall: Hence:

High-Frequency Scattering by Cylinder (cont.) In the backscattered direction ( = ): Then Echo width (monostatic RCS): Note: E0 = 1 in our case.

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