1. EM Waveguiding Overview Waveguide may refer to any structure that conveys electromagnetic waves between its endpoints Most common meaning is a hollow metal pipe used to carry radio waves May be used to transport radiation of a single frequency Transverse Electric (TE) modes have E ┴ k g (propagation wavevector) Transverse Magnetic (TM) modes have B ┴ k g Transverse Electric-Magnetic modes (TEM) have E, B ┴ k g A cutoff frequency exists, below which no radiation propagates

2. EM Waveguiding Electromagnetic wave reflection by perfect conductor E ┴ can be finite just outside conducting surface E || vanishes just outside and inside conducting surface ii rr EIEI ERER z y E I|| EI┴EI┴ ER┴ER┴ E R|| z y EI┴EI┴ ER┴ER┴ - - - - - - - D ┴ 2 =  o  E ┴ 2 D ┴ 1 =  o E ┴ 1 D ┴ 1 = D ┴ 2 z y E oI + E oR = 0 E || 1 = E || 2 E I|| E R|| E I|| E R|| E I|| E R|| E oT = 0

3. EM Waveguiding Electromagnetic wave propagation between conducting plates Boundary conditions B ┴ 1 = B ┴ 2 E || 1 = E || 2 (1,2 inside, outside here) E || must vanish just outside conducting surface since E = 0 inside E ┴ may be finite just outside since induced surface charges allow E = 0 inside (TM modes only) B ┴ = 0 at surface since B 1 = 0 Two parallel plates, TE mode b E1E1 E2E2 k1k1 k2k2 y x z b 

4. EM Waveguiding Fields in vacuum E 1 = e x E o e i (  t - k 1.r) k 1 = -e y k sin  + e z k cos  k 1.r = - ky sin  + kz cos  E 2 = -e x E o e i (  t - k 2.r) k 2 = +e y k sin  + e z k cos  k 2.r = + ky sin  + kz cos 

5. EM Waveguiding Fields E 1 = e x E o e i (  t - k 1.r) k 1 = -e y k sin  + e z k cos  k 1.r = - ky sin  + kz cos  E 2 = -e x E o e i (  t - k 2.r) k 2 = +e y k sin  + e z k cos  k 2.r = + ky sin  + kz cos  ExEx y sin(n  y/b)

6. EM Waveguiding

7. bb  ’’ kgkg 1 propagating mode 2 modes vacuum propagation

8. EM Waveguiding x y x y

9. Electric components of TE n guided fields viewed along x (plan view) n = 1 n = 2 n = 3 n = 4 Magnetic components of TE n guided fields viewed along x (plan view) z y z y

10. EM Waveguiding Rectangular waveguides Boundary conditions B ┴ 1 = B ┴ 2 E || 1 = E || 2 E || must vanish just outside conducting surface since E = 0 inside E ┴ may be finite just outside since induced surface charges allow E = 0 inside B ┴ = 0 at surface Infinite, rectangular conduit b y x z a

11. EM Waveguiding TE mn modes in rectangular waveguides TE n modes for two infinite plates are also solutions for the rectangular guide E field vanishes on xz plane plates as before, but not on the yz plane plates Charges are induced on the yz plates such that E = 0 inside the conductors Let E x = C f(x) sin(n  y/b) e i(  t - k g z) In free space .E = 0 and E z = 0 for a TE mn mode and ∂E z /∂z = 0 Hence ∂E x /∂x = -∂E y /∂y f(x) = -n  / b cos(m  x/a) satisfies this condition By integration E x = -C n  / b cos(m  x/a) sin(n  y/b) e i(  t - k g z) E y = C m  / a sin(m  x/a) cos(n  y/b) e i(  t - k g z) E z = 0

12. EM Waveguiding

14. Electric components of TE mn guided fields viewed along k g m = 0 n = 1 m = 1 n = 1 m = 2 n = 2 m = 3 n = 1 Magnetic components of TE mn guided fields viewed along k g x y x y

15. EM Waveguiding Comparison of fields in TE and TM modes www.opamp-electronics.com/tutorials/waveguides_2_14_08.htm

16. EM Waveguiding