Notes 12 ECE Microwave Engineering Fall Surface Waves Prof. David R. Jackson Dept. of ECE Fall 2011

Grounded Dielectric Slab Discontinuities on planar transmission lines such as microstrip will radiate surface-wave fields. Substrate (ground plane below) Microstrip line 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 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. x z h 3

Dielectric Slab TM x & TE x modes: x z H E TM x z E H TE x x Note: These modes may also be classified as TM z and TE z. 4 Plane wave

Surface Wave The internal angle is greater than the critical angle, so there is exponential decay in the air region. The surface wave is a “slow wave”. z x Exponential decay Hence 5 Plane wave

TM x Solution Assume TM x 6

Denote TM x Solution (cont.) Then we have 7

Applying boundary conditions at the ground plane, we have: TM x Solution (cont.) This follows since Note: 8

Boundary Conditions BC 1) BC 2) Note: 9

Boundary Conditions (cont.) These two BC equations yield: Divide second by first: or 10

Final Result: TM x This may be written as: This is a transcendental equation for the unknown wavenumber k z. 11

Final Result: TE x Omitting the derivation, the final result for TE x modes is: This is a transcendental equation for the unknown wavenumber k z. 12

Graphical Solution for SW Modes Consider TM x : Let or Then 13

Graphical Solution (cont.) Hence Add We can develop another equation by relating u and v: where 14

Define Then Graphical Solution (cont.) Note: R is proportional to frequency. 15

Summary for TM x Case Graphical Solution (cont.) 16

TM 0 R  2  3  v u Graphical Solution (cont.) 17

Graphical Solution (cont.) Graph for a Higher Frequency TM 0 R  v u TM 1 Improper SW ( v < 0)  / 2 3  / 2 (We reject this solution.) 18

Proper vs. Improper If v < 0 : “improper SW” (fields increase in x direction) If v > 0 : “proper SW” (fields decrease in x direction) Cutoff frequency: TM 1 mode: Cutoff frequency: The transition between a proper and improper mode. Note: This definition is different from that for a closed waveguide structure (where k z = 0 at the cutoff frequency.) 19

TM x Cutoff Frequency u R  v For other TM n modes: TM 1 : 20

TM 0 Mode The TM 0 mode has no cutoff frequency (it can propagate at any frequency: TM n1n1 f TM 1 k z / k 0 21 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).

TM 0 Mode After making some approximations to the transcendental equation, valid for low frequency, we have the following approximate result for the TM 0 mode: 22

TE x Modes R  v u   TE 1 TE 2 Hence 23

TE x Modes (cont.) No TE 0 mode ( f c = 0 ). The lowest TE x mode is the TE 1 mode. TE 1 cut-off frequency at ( R    2  In general, we have TE n : The TE 1 mode will start to propagate when the substrate thickness is roughly 1/4 of a dielectric wavelength. 24

TE x Modes (cont.) 25 Here we examine the radiation efficiency e r of a small electric dipole placed on top of the substrate (which could mode a microstrip antenna). TM 0 SW

26 Dielectric Rod a z This serves as a model for a single-mode fiber-optic cable. The physics is similar to that of the TM 0 surface wave on a layer.

Fiber-Optic Guide (cont.) Two types of fiber-optic guides: 1) Single-mode fiber 2) Multi-mode fiber This fiber carries a single mode (HE 11 ). 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). 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

28 Dominant mode (lowest cutoff frequency): HE 11 (f c = 0) The field shape is somewhat similar to the TE 11 waveguide mode. Dielectric Rod (cont.) Note: The notation HE means that the mode is hybrid, and has both E z and H z, although H z is stronger. (For an EH mode, E z would be stronger.) The physical properties of the fields are similar to those of the TM 0 surface wave on a slab (For example, at low frequency the field is loosely bound to the rod.) The dominant mode is a hybrid mode (it has both E z and H z ). Single-mode fiber

Dielectric Rod (cont.) What they look like in practice: Single-mode fiber

Fiber-Optic Guide (cont.) Higher index core region 30 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”).