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

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

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

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

5. High-Frequency Scattering by Cylinder
PEC cylinder
Assume

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

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

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

9. 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

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

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

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

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

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

15. 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

16. Geometrical Optics (cont.)
Specular point
Proof

17. 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.

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

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

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

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

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

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

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

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

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