1. High Frequency Techniques in Electromagnetics Ayhan Altıntaş Bilkent University, Dept. of Electrical Engineering, Ankara, Turkey E-mail: altintas@ee.bilkent.edu.tr Ayhan Altıntaş Bilkent University, Dept. of Electrical Engineering, Ankara, Turkey E-mail: altintas@ee.bilkent.edu.tr

2. Outline  Ray-based Techniques  Geometrical Optics (GO)  Geometrical Theory of Diffraction (GTD-UTD)  Integral-based Techniques  Physical Optics (PO)  Physical Theory of Diffraction (PTD)  Equivalent Edge Currents (EEC)

3. Scattering Problem J J: induced surface current E scat (  ) E inc (  ) PEC Scatterer E = E inc (  ) + E scat (  ) Total Field Radiated by J Determine E or E scat !

4. Geometrical Optics PROPERTIES Abides power conservation in the ray tubes Phase factor is introduced along rays (local plane waves) Polarization is preserved in ray- fixed coordinates Can be derived from Maxwell’s Equations DIFFICULTY Not valid in caustics s Astigmatic Ray Tube 0 s Line Caustics are two caustic distances

5. Geometrical Optics Properties:  Conceptually simple  Localized scattering  Requires only tracing of incident and reflected rays  Pinpoints flash points Reflected rays Incident rays Shadow Region Scatterer Shadow boundary Disadvantages:  Requires finding of reflection point on the surface  Predicts null field in shadow regions  Predicts discontinuous field along shadow boundaries

6. Geometrical Optics Geometrical Optics for reflection Source Image s QrQr Wavefront S’ Caustic distance for reflected rays Radius of curvature of the surface at Q r Note that in 2-D there is only one caustic distance

7. Geometrical Optics Example – A strip

8. Half Plane Fields

9. Geometrical Theory of Diffraction (GTD) Incident ray Q1Q1 Q2Q2 Diffracted rays Surface diffraction Diffracted rays Incident ray Edge diffraction s Observation direction Shadow boundary Ray Theory Solves some of GO difficulties

10. GTD Calculation GTD Formulation: Properties:  Conceptionally simple  Local phenomena  Tracing of diffracted rays  Pinpoints flash points  Predicts non-zero field in shadow regions  A higher order approximation than GO in terms of frequency  Uniform versions yield smooth and continuous fields at and around shadow boundaries (transition regions) Disadvantages:  Requires searching for diffraction points on the edge  Requires finding of attachment and launching points and geodesics on the surface  Fails at caustics where many diffracted rays merge Properties:  Conceptionally simple  Local phenomena  Tracing of diffracted rays  Pinpoints flash points  Predicts non-zero field in shadow regions  A higher order approximation than GO in terms of frequency  Uniform versions yield smooth and continuous fields at and around shadow boundaries (transition regions) Disadvantages:  Requires searching for diffraction points on the edge  Requires finding of attachment and launching points and geodesics on the surface  Fails at caustics where many diffracted rays merge

11. e Edge s´ o´o´ s Keller cone Plane of Diffraction E i  o´ Ei´Ei´ EdEd EdoEdo oo Incident ray Diffracted ray 3-D Edge Diffraction Keller Cone becomes a disk in 2-D problems

12. Edge Diffraction Coefficients Note there is only one caustic distance Where is the other one?

13. Keller’s Diffraction Coefficients (GTD) Keller´s edge diffraction coefficients Not valid when Non-uniform

14. Numerical Result – GTD

15. Numerical Result - UTD In the Uniform Geometrical Theory of Diffraction (UTD) D s,h contain Fresnel integrals to make them valid in transition regions. (Invented at Ohio State University by Kouyoumjian and Pathak Uniform Asypmtotic Theory(UAT) is similar to UTD but uses Keller diffraction and modifies reflected field, not very suitable for numerical work.(Invented at U.of Illinois) In the Uniform Geometrical Theory of Diffraction (UTD) D s,h contain Fresnel integrals to make them valid in transition regions. (Invented at Ohio State University by Kouyoumjian and Pathak Uniform Asypmtotic Theory(UAT) is similar to UTD but uses Keller diffraction and modifies reflected field, not very suitable for numerical work.(Invented at U.of Illinois)

16. GTD-UTD Example – A Disk

17. Backscattering from a square plate z y x  a a e inc h inc Diffracted Ray Caustics

18. Flat Plate Modeling Scattered field for RCS has many Caustics Ray based techniques fail at caustics

19. Physical Optics approximation is the GO based surface current. Properties: Simple No need to search for flash points Stationary phase evaluation of the radiation integral yields reflected field, so PO asymptotically reduces to GO Stays bounded in the caustics Suited well for the RCS of targets build up with flat polygonal plates Disadvantages : Surface integral required Reciprocity is not satisfied Not accurate away from specular reflection

20. Physical Theory of Diffraction We do not know J fw ! How do we calculate the second integral? Use High frequency asymptotic approximation to E ! Incident Plane Wave Half plane

21. Physical Theory of Diffraction Note that singularities of and cancel so is valid in transition regions

22. PTD Equivalent Edge Currents (EEC) PTD - EEC Derived from the integration of fringe wave currents on a half plane. Then use asymptotic methods to convert the 2-D surface integral into a 1-D line integral. Surface Integral: Line Integral:

23. PTD Coefficients coefficients depending on angles of the geometry Various approaches exist to determine these coefficients, most useful ones are by Mitzner (ILDC) and Ando.

24. RCS of a Flat Plate z y x a e inc h inc

25. Disk Example – Revisited

26. Disk - Cross Polar Radiation

27. HF work of A. Altintas

28. HF Work of A. Altintas

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34. GTD Equivalent Edge Currents (EEC) Advantages:  Finite fields at or around caustics.  Field prediction even when there is no GO/GTD ray reaching the observation (corner diffraction).  Spatial variations of the incident field are inherently included. Problems:  Not valid in the transition regions of shadow boundaries.  Derived heuristically. GTD - EEC Replaces the edge with non-uniform electric and magnetic line sources.