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Prof. David R. Jackson ECE Dept.

ECE 6341. Spring 2014. Prof. David R. Jackson ECE Dept. Notes 26. EM Plane-Wave Transformation. z. y. x. Note: The incident field will be represented using both A r and F r. EM Plane-Wave Transformation. E r directly corresponds to A r H r directly corresponds to F r.

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Prof. David R. Jackson ECE Dept.

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  1. ECE 6341 Spring 2014 Prof. David R. Jackson ECE Dept. Notes 26

  2. EM Plane-Wave Transformation z y x Note: The incident field will be represented using both Ar and Fr.

  3. EM Plane-Wave Transformation Er directly corresponds to Ar Hrdirectly corresponds to Fr Any other field component will involve both ArandFr (see the next slide).

  4. EM Plane-Wave Transformation (cont.)

  5. EM Plane-Wave Transformation (cont.) Not in the form of a spherical-wave expansion! We need to put this in a form so that is matches with what we get from the Debye potential representation.

  6. EM Plane-Wave Transformation (cont.) Try this:

  7. EM Plane-Wave Transformation (cont.) Now use the “integration formula” Harrington notation (Schaum’s outline Eq. (26.2)) Note: The (-1)m term is added to agree with the Harrington notation. Hence Eq. (E.16) in Harrington or

  8. EM Plane-Wave Transformation (cont.) Thus we have Hence

  9. EM Plane-Wave Transformation (cont.) Next, use so Now let Goal: solve for cn

  10. EM Plane-Wave Transformation (cont.) Compare these two equations for the incident radial field component: Er that is known for plane wave We need to put these in the same form so we can solve for cn. Er from Debye potential

  11. EM Plane-Wave Transformation (cont.) We now need to evaluate To evaluate this, use

  12. EM Plane-Wave Transformation (cont.) Hence To simplify this, use the spherical Bessel Eq: or Hence:

  13. EM Plane-Wave Transformation (cont.) Therefore: Hence:

  14. EM Plane-Wave Transformation (cont.) or Er from Debye potential Compare with the known expansion for : Er that is known for plane wave We see that

  15. EM Plane-Wave Transformation (cont.) Hence we have

  16. EM Plane-Wave Transformation (cont.) Hence or Hence

  17. EM Plane-Wave Transformation (cont.) Similarly, to find Fr for the incident plane wave, we use: Note: Hence we have

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