1. Current Distribution of a Printed Dipole with Arbitrary Length Embedded in Layered Uniaxial Anisotropic Dielectrics Benjamin D. Braaten* North Dakota State University, Fargo ND, USA David A. Rogers North Dakota State University, Fargo ND, USA Robert M. Nelson University of Wisconsin – Stout, Menomonie WI, USA North Dakota State University

2. Topics  Problem Definition  Spectral domain immittance functions  Spectral domain moment method  Results/Discussion  Conclusion North Dakota State University

3. Problem Definition Consider: North Dakota State University

4. The spectral domain immittance functions North Dakota State University Start with the following Hertz vector potentials: and Electric Hertz potential Magnetic Hertz potential

5. The spectral domain immittance functions North Dakota State University  Next, only the y-direction of the Hertz vector potential is needed. and  This is because the optical axis is in the y- direction and  this component satisfies the higher order TE and TM tangential boundary conditions.

6. The spectral domain immittance functions North Dakota State University Now define the following expression for the magnetic and electric field: where the Hertzian vector potentials are solutions to the following equations:

7. The spectral domain immittance functions North Dakota State University and

8. The spectral domain immittance functions North Dakota State University To simplify evaluating the previous expressions, we define the following Fourier transform: This results in the following relations:

9. The spectral domain immittance functions North Dakota State University This results in the following simplified expressions: where and

10. The spectral domain immittance functions North Dakota State University Similarly for and

11. The spectral domain immittance functions North Dakota State University Next, consider: Enforcing the B.C. with the previous expressions results in the following simple relation:

12. The Spectral Domain Moment Method North Dakota State University Next, the x-component of the electric field in each region can be written in the spatial domain as: Since is in the spatial domain, the two-dimensional inverse Fourier transform will need to be applied to

13. The Spectral Domain Moment Method North Dakota State University Proceeding in this manner results in the following expression: Next, the current in terms of the basis functions:

14. The Spectral Domain Moment Method North Dakota State University Defining weighting functions and rearranging gives: where and

15. The Spectral Domain Moment Method North Dakota State University Notice that the following expression is in the previous rearrangement: This is the 2D FT of the basis function. This can be useful if a basis function with an analytical FT is chosen.

16. The Spectral Domain Moment Method North Dakota State University Using this simplification results in the following expression: Notice Integration only on a single plane

17. The Spectral Domain Moment Method North Dakota State University For this work, PWS basis functions were used:

18. The Spectral Domain Moment Method North Dakota State University The alpha-beta plane of integration:

19. Numerical Results North Dakota State University Single Layer L =.5 λ 0 W =.0004 λ 0 d 1 =.1016 λ 0 1V source

20. Numerical Results North Dakota State University Single Layer Notice: Notice that the imaginary part can be individually modified (compare the solid lines with the dashed lines)

21. Numerical Results North Dakota State University Triple Layer L =.25 λ 0 W =.00083 λ 0 d 1 =.0026 λ 0 d 2 =.0026 λ 0 d 3 =.0026 λ 0 ε 1 = 3.25 (iso.) 1V source

22. Numerical Results North Dakota State University Triple Layer Notice: Notice that both the real and imaginary parts of the current change from the isotropic case when each permittivity component is modified.

23. Conclusion North Dakota State University  A summary on the spectral domain moment method has been presented.  A single printed dipole on a single anisotropic substrate has been investigated.  It was shown that with certain permittivity components, the imaginary part of the current could be modified while the real part of the current remained unchanged.  A single printed dipole in three anisotropic layers has been investigated.  It was shown that each component of the permittivity in the superstrate and substrate had an effect on both the real and imaginary part of the current.

24. Questions Thank you for listening North Dakota State University