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

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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

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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

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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

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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

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TM x Solution Assume TM x 6

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

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

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Boundary Conditions BC 1) BC 2) Note: 9

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

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

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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

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

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

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

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

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TM 0 R 2 3 v u Graphical Solution (cont.) 17

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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

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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

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

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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).

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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

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TE x Modes R v u TE 1 TE 2 Hence 23

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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

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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

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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.

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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

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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

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

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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”).

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