1. Monopole antenna Lorentz lemma Monopole antenna exciting the rectangular WG Horn antenna Green’s formula and free space Green’s function Huygens’ principle derived from Lorentz’s lemma Vector-potential and Lorentz‘ gauge Radiated power, radiation resistance and resonance of linear dipole Conclusions Elements of electromagnetic field theory and guided waves

2. Monopole antenna in a waveguide and Lorentz’s lemma l Image (- I A ) l VAVA IAIA JsJs Image J s Apply the so-called Lorentz’s lemma IAIA The wall can be extended to the whole plane, and the image principle can be applied to the equivalent structure

3. Lorentz lemma application z z S0S0 Closed surface S z=∞z=∞ -I A 2l Here the integrands =0 y IAIA

4. Derivations Let the arbitrary EM field = transverse part of the TE 10 field, backward wave TEmn(y) TE10(y) TEmn(x)TE10(x)

5. Result Choosing E arb,H arb as transverse-backward TE NM we may find all C NM +. All C N>1,M>0 <<C 10 at frequencies f c10 <f<f c11 l opt = wg /4 E arb. (z = 0)E arb. (z = -2l )

6. Horn antenna R at the horn aperture <<1 Matching to free space – smooth variation of Z along the horn Radiation - Radiating aperture

7. Vector-potential again Lienard- Wiechert

8. Lorentz’s gauge If A=A z (R,   E   (R>>d, )=j   A z ( R,   d R  Lorentz’s gauge =0=0 Lorentz’s gauge for the far zone:

9. Green’s function from Lorentz lemma S0S0 S=S 0 +S 1 +S 2 +S ∞ S1S1 S2S2 S∞S∞ r (x,y,z) l<< V Il=1 r’ (x’,y’,z’) x y z 0 E rad. = H rad. = 0 E rad =H rad. =0 Imagined osc. dipole

10. Huygens’ principle in Green’s interpretation P s =j  0 H t and M s = E t – effective electric and magnetic surface polarizations d p = j  0 H t dS – electric dipole, d m = E t dS – magnetic dipole Huygens: Surface with EM field = el.+mag. surface polarization d p +d m – Huygens’ radiating element. Radiates only forward! Green’s result keeps valid for arbitrary surfaces S 0 -- Huygen’s far-zone field dmdm dpdp EdpEdp E dm EdpEdp

11. Radiation pattern Z’Z’ ImIm h R R0R0  R R0R0 Current distr. z  .

12. Radiated power and radiation resonance R rad. ≠ R in. R0R0  ImIm Apply the Lorentz gauge for the far zone E  =j   A z

13. Concluding remarks One may conditionally split the course onto 3 parts: Part 1 - EM fields, their excitation and interaction with charges and currents, their properties in free space and in different media Part 2 –Waveguides, transmission lines, and resonators, properties of guided and resonant modes Part 3 – Radiation of EM waves by oscillating currents to free space, to waveguides and through apertures