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ECE 5317-6351 Microwave Engineering Fall Fall2011 2011 Prof. David R. Jackson Dept. of ECE Notes 12 Surface Waves 1 Grounded Dielectric Slab Discontinuities on planar transmission lines such as microstrip will radiate surface-wave fields. Microstrip line Substrate (ground plane below) Surface-wave field It is important to understand these waves. Note: Sometimes layers are also used as a desired propagation mechanism for microwave and millimeter-wave frequencies. (The physics is similar to that of a fiber-optic guide.) 2 Grounded Dielectric Slab x h r , r z Goal: Determine the modes of propagation and their wavenumbers. Assumption: There is no variation of the fields in the y direction, and propagation is along the z direction. 3 Dielectric Slab TMx & TEx modes: Note: These modes may also be classified as TMz and TEz. x TMx H Plane wave E r , r z x TEx E H Plane wave r , r z 4 Surface Wave Exponential decay x 1 Plane wave r , r z 1 c The internal angle is greater than the critical angle, so there is exponential decay in the air region. kz k1 sin1 k0 The surface wave is a “slow wave”. Hence kx0 k k 2 0 2 1/2 z j k z2 k02 j x 0 5 TMx Solution Assume TMx Ex e jkz z ex x 2 Ex Ex 2 Ex 2 k Ex 0 2 2 2 x y z 2 Ex 2 2 k k z Ex 0 2 x 6 TMx Solution (cont.) 2ex x 2 2 k k ex x 0 z 2 x k x 0 k02 k z2 1/2 Denote j k z2 k02 j x 0 k x1 k12 k z2 Then we have xh 2ex x 2 k x 0 ex x 0 2 x xh 2ex x 2 k x1ex x 0 2 x 7 TMx Solution (cont.) Applying boundary conditions at the ground plane, we have: xh Ex1 e xh Ex 0 Ae jkz z e x 0 x jkz z Note: cos(kx1x) En Ex 0 0 n x This follows since D v 0 x h Et 0 r , r z 8 Boundary Conditions BC 1) Dx0 Dx1 @x h Ex 0 r Ex1 BC 2) Ez 0 Ez1 @ x h E x 0 E x1 x x Note: Ex Ey Ez Ex E 0 0 jk z Ez x y z x 9 Boundary Conditions (cont.) These two BC equations yield: Ae x 0h r cos(k x1h) x 0 Ae x 0h (k x1 ) sin(k x1h) Divide second by first: x 0 1 r (k x1 ) tan(k x1h) or x0 r kx1 tan(kx1h) 10 Final Result: TMx This may be written as: r kz 2 k02 k12 kz 2 tan k12 kz 2 h This is a transcendental equation for the unknown wavenumber kz. 11 Final Result: TEx Omitting the derivation, the final result for TEx modes is: k z 2 k0 2 1 r k12 k z 2 cot k12 k z 2 h This is a transcendental equation for the unknown wavenumber kz. 12 Graphical Solution for SW Modes x0 r kx1 tan(kx1h) Consider TMx: x0h or Let 1 r (k x1h) tan(k x1h) u k x1h v x0h Then v 1 r u tan u 13 Graphical Solution (cont.) We can develop another equation by relating u and v: u h k12 k z 2 v h k z 2 k0 2 Hence u 2 h 2 (k12 k z 2 ) v h ( k z k0 ) 2 2 u v h (k k0 ) 2 2 2 2 1 2 (k0 h) (n 1) 2 2 1 2 2 Add where n1 k1 / k0 r r 14 Graphical Solution (cont.) Define R2 (k0h)2 (n12 1) R ( k0 h) n 1 2 1 Note: R is proportional to frequency. Then u v R 2 2 2 15 Graphical Solution (cont.) Summary for TMx Case v 1 r u tan u u v R 2 2 2 u k x1h h k12 k z2 v x 0 h h k z2 k02 16 Graphical Solution (cont.) v v TM0 1 r u tan u R p /2 p 3p /2 u u 2 v2 R2 R (k0 h) n12 1 17 Graphical Solution (cont.) v Graph for a Higher Frequency TM0 TM1 R p /2 p 3 p /2 u Improper SW (v < 0) (We reject this solution.) 18 Proper vs. Improper v x0h If v > 0 : “proper SW” (fields decrease in x direction) If v < 0 : “improper SW” (fields increase in x direction) Cutoff frequency: The transition between a proper and improper mode. Note: This definition is different from that for a closed waveguide structure (where kz = 0 at the cutoff frequency.) Cutoff frequency: TM1 mode: v0 u p 19 TMx Cutoff Frequency v R p TM1: k0 h n12 1 p p u h 0 1/ 2 n12 1 R For other TMn modes: TM n : h 0 n/2 n12 1 n 0,1, 2,... 20 TM0 Mode The TM0 mode has no cutoff frequency (it can propagate at any frequency: Note: The lower the frequency, the more loosely bound the field is in the air region (i.e., the slower it decays away from the interface). kz / k 0 x 0 k z2 k02 n1 TM0 TM1 1.0 fcTM1 f 21 TM0 Mode After making some approximations to the transcendental equation, valid for low frequency, we have the following approximate result for the TM0 mode: k h n2 1 0 1 k0 1 r2 2 TM 0 2 1/ 2 k0 h 1 22 TEx Modes v TE1 TE2 x0h 1 r kx1h cot kx1h R p/2 1/ r Hence p 3p/2 u v 1 r u cot u 23 TEx Modes (cont.) No TE0 mode ( fc= 0). The lowest TEx mode is the TE1 mode. TE1 cut-off frequency at (R p / 2 : k0 h h 0 n 1 2 1 p 2 1/ 4 n12 1 In general, we have TEn: h 0 2n 1 / 4 n12 1 n 1, 2,3,......... The TE1 mode will start to propagate when the substrate thickness is roughly 1/4 of a dielectric wavelength. 24 TEx Modes (cont.) Here we examine the radiation efficiency er of a small electric dipole placed on top of the substrate (which could mode a microstrip antenna). TM0 SW 100 EFFICIENCY (%) 80 r 2.2 60 r 10.8 40 exact 0 er CAD 20 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 hh// 00 Psp Psp Psw 0.1 25 Dielectric Rod z a r , r The physics is similar to that of the TM0 surface wave on a layer. This serves as a model for a single-mode fiber-optic cable. 26 Fiber-Optic Guide (cont.) Two types of fiber-optic guides: 1) Single-mode fiber This fiber carries a single mode (HE11). This requires the fiber diameter to be on the order of a wavelength. It has less loss, dispersion, and signal distortion. It is often used for long-distances (e.g., greater than 1 km). 2) Multi-mode fiber This fiber has a diameter that is large relative to a wavelength (e.g., 10 wavelengths). It operates on the principle of total internal reflection (critical-angle effect). It can handle more power than the single-mode fiber, but has more dispersion. 27 Dielectric Rod (cont.) Dominant mode (lowest cutoff frequency): HE11 (fc = 0) The dominant mode is a hybrid mode (it has both Ez and Hz). E Single-mode fiber Note: The notation HE means that the mode is hybrid, and has both Ez and Hz, although Hz is stronger. (For an EH mode, Ez would be stronger.) The field shape is somewhat similar to the TE11 waveguide mode. The physical properties of the fields are similar to those of the TM0 surface wave on a slab (For example, at low frequency the field is loosely bound to the rod.) 28 Dielectric Rod (cont.) http://en.wikipedia.org/wiki/Optical_fiber What they look like in practice: Single-mode fiber Fiber-Optic Guide (cont.) Higher index core region Multimode fiber A multimode fiber can be explained using geometrical optics and internal reflection. The “ray” of light is actually a superposition of many waveguide modes (hence the name “multimode”). http://en.wikipedia.org/wiki/Optical_fiber 30