1. 1 Electromagnetic Sensing for Space-borne Imaging Lecture 6 Antenna Lore, Transmission and Reception Patterns, Phased Arrays, Sparse Aperture Theorem

2. 2 Gain Function and Radiating Pattern Antenna Boresight  

3. 3 Gain Function and Radiating Pattern

4. 4

5. 5 Transmitted power I A Antenna Feed Calculating the gain function from what we’ve learned so far

6. 6

7. 7 Calculating the gain function Example: Circular aperture of diameter D

8. 8 Intensity pattern (or point source image) from a circular aperture The intensity pattern depends only on the radial distance of the look angle from the point source direction. Almost all the energy is contained in the central Airy disk within x < , or for look angles such that:   / D

9. 9 Intensity pattern (or point source image) from a circular aperture Intensity distribution as would be seen without saturation. Only the central maximum is visible Intensity distribution with over 1200% saturation, so that the secondary fringes are visible.

10. 10 Receiving Pattern

11. 11 Reciprocity Measurement signal APAP Transmitted power AQAQ

12. 12 “The old RF guys never died. They just phased array” - Anon. I 44 Time delay 11 33 22 55 Gain

13. 13 Phased Array – Complex Amplitude on the Ground Plane

14. 14 Phased Array – Complex Amplitude on the Ground Plane

15. 15 Phased Array – Complex Amplitude on the Ground Plane

16. 16 Phased Array – Gain Function

17. 17 Simple 1-D Array Example

18. 18 Simple 1-D Example: N = 20 Beam width is inversely proportional to aperture separation Max gain function is independent of separation

19. 19 Simple 1-D Example:  =0.1, N = 20, 40, 80 Maximum gain function is proportional to number of apertures

20. 20 Simple 1-D Example:  greater than 1, N = 20, For large separations. There are side-beams at multiples of 1/  radians on either side of the central beam  = 1  = 2

21. 21 Power density on the ground - 1-D Case Early in the history of this technology, people thought that if they increased the transmitter separations and thereby decreased the beam width, they would increase the power density of the central beam. This is false. In the 1-D case, we see that:  Doubling the separation does decrease the beam width by 2.  But the max value of G (which is proportional to power density) is independent of separation.  Therefore the total power in the main beam is smaller by a factor of 2.  Total overall power remains the same. Power lost from the main beam reappears in side-beams Analogous results hold for a general 2-D array

22. 22 The Sparse (or Thinned) Aperture Theorem In the 2-D case:  Increasing the separation by a factor R decreases the beam width by the same factor.  But the max value of G (which is proportional to power density) is independent of separation.  Therefore the total power in the main beam is smaller by a factor of 1/R 2.  Total overall power remains the same. Power lost from the main beam reappears in the side-beams These observations form the “Sparse Aperture Theorem”, aka “Sparse Aperture Curse” * * Robert L. Forward, “Roundtrip Interstellar Travel Using Laser Pushed Lightsails, J. Spacecraft and Rockets, Vol. 21, No. 2, Mar.-Apr. 1984, p.190.