Notes 14

ECE 5317-6351 Microwave Engineering

Fall 2011

Surface Waves

Prof. David R. JacksonDept. of ECE

Fall 2019

1

Adapted from notes by Prof. Jeffery T. Williams

Substrate(ground plane below)

Microstrip line

Surface-wave field

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:Surface waves can also be used as a 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.

3

,r rε µ

x

zh

1 0 r rk k µ ε=

Lossless slab

Ground plane

Dielectric Slab

TMx & TEx modes: Note:These modes may also be classified as TMz and TEz.

4

,r rε µPlane wave

x

z

HExTM

,r rε µPlane wave

x

z

E HxTE

Surface Wave

The internal angle is larger than the critical angle, so there is exponential decay in the air region.

( )1 1 0 1 0sin sinz ck k k k kθ θ= > =

The surface wave is a “slow wave”:

1 cθ θ>

( )1/22 2 2 20 0 0 0x z z xk k k j k k jα= − = − − = −Hence

5

,r rε µ1θ

Exponential decay

Plane wave

x

z

0z pk k v c> ⇒ <

TMx SolutionAssume TMx:

( ) ( ), zjk zx xE x z e e x−=

2 22

2 2 2 0x x xx

E E E k Ex y z

∂ ∂ ∂+ + + =

∂ ∂ ∂

( )2

2 22 0x

z xE k k Ex

∂+ − =

∂

6

( ) ( ) ( )2

2 22 0x

z x

d e xk k e x

dx+ − =

so

( ) ( ) ( )2

2 22 0x

z x

d e xk k e x

dx+ − =

Denote ( )1/22 2 2 20 0 0 0x z z xk k k j k k jα= − = − − = −

2 21 1x zk k k= −

( ) ( )2

202 0x

x x

d e xe x

dxα− =

( ) ( )2

212 0x

x x

d e xk e x

dx+ =x h≤

x h≥

TMx Solution (cont.)

Then we have:

7

0 1k k k= or

x h≤

x h≥

1 1cos( )zjk zx xE e k x−=

00

xz xjk zxE Ae e α−−=

Applying boundary conditions at the ground plane, we have:

TMx Solution (cont.)

0xEx

∂=

∂

This follows since

Note: on PEC

8

,r rε µPlane wave

x

z

HExTM

( )

0

0yx z

xz z

EEE E

x y zE jk Ex

∇ ⋅ =∂∂ ∂

⇒ + + =∂ ∂ ∂

∂⇒ = − −

∂

Boundary Conditions

BC 1) 0 1 @x xD D x h= =

0 1x r xE Eε=

0 1x xE Ex x

∂ ∂=

∂ ∂

0 1 @z zE E x h= =BC 2)

Recall:

⇒

⇒

9

,r rε µPlane wave

x

z

HExTM

xz z

E jk Ex

∂=

∂

00

1 1cos( )

xz

z

xjk zx

jk zx x

E Ae e

E e k x

α−−

−

=

=

Boundary Conditions (cont.)

These two BC equations yield:

0

0

1

0 1 1

cos( )

( ) sin( )

x

x

hr x

hx x x

Ae k h

Ae k k h

α

α

ε

α

−

−

=

− = −

Divide second by first:

0 1 11 ( ) tan( )x x x

r

k k hαε

− = −

0 1 1tan( )x r x xk k hα ε =

or

10

BC 1)

BC 2)

00

1 1cos( )

xz

z

xjk zx

jk zx x

E Ae e

E e k x

α−−

−

=

=

0 1x r xE Eε=

0 1x xE Ex x

∂ ∂=

∂ ∂

Final Result: TMx

This may be written as:

2 2 2 2 2 20 1 1tanr z z zk k k k k k hε − = − −

This is a transcendental equation for the unknown wavenumber kz.

11

Final Result: TEx

Omitting the derivation, the final result for TEx modes is:

This is a transcendental equation for the unknown wavenumber kz.

( )2 2 2 2 2 20 1 1

1 cotz z zr

k k k k k k hµ

− = − − −

12

Graphical Solution for SW ModesConsider TMx:

Define:

0 1 1tan( )x r x xk k hα ε =

1

0

x

x

u k hv hα

≡≡

1 tanr

v u uε

=

or

Then

0 1 11 ( ) tan( )x x x

r

h k h k hαε

=

13

Graphical Solution (cont.)

Hence

2 21

2 20

z

z

u h k k

v h k k

= −

= −

2 2 2 21

2 2 2 20

( )( )

z

z

u h k kv h k k

= −

= −Add

We can develop another equation by relating u and v:

2 2 2 2 21 0

2 20 1

( )( ) ( 1)

u v h k kk h n

+ = −

= − 1 1 0/ r rn k k ε µ≡ =

where

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2 2 20 1( ) ( 1)R k h n≡ −

2 2 2u v R+ =

Define

Then

Graphical Solution (cont.)

Note:R is proportional to frequency.

20 1( ) 1R k h n= −

15

or

Summary for TMx Case

Graphical Solution (cont.)

1 tanr

v u uε

=

2 2 2u v R+ =

2 21 1

2 20 0

x z

x z

u k h h k k

v h h k kα

≡ = −

≡ = −

16

20 1( ) 1R k h n= −

Graphical Solution (cont.)

17Note: The TM0 mode exists at all frequencies (no cutoff frequency).

2 2 2u v R+ =

1 tanr

v u uε

=

20 1( ) 1R k h n= −

u

v

R

/ 2π π 3 / 2π

0TM

Graphical Solution (cont.)

Graph for a Higher Frequency

18

Improper SW (v < 0)

(We reject this solution.)

u

v

R

/ 2π π 3 / 2π

0TM

1TM

Proper vs. Improper

0xv hα=

If v < 0 : “improper SW” (fields increase in x direction)

If v > 0 : “proper SW” (fields decrease in x direction)

Cutoff frequency: TM1 mode:0v

u π==

Cutoff frequency: The transition between a proper and improper mode.

Note:The definition of cutoff frequency for this type of structure (an open structure) is

different from that for a closed waveguide structure (e.g., rectangular waveguide) (where kz = 0 at the cutoff frequency).

19

R π=

TM1 Cutoff Frequency

R π=

20 1 1k h n π− =

20 1

1/ 21

hnλ

=−

20 1

/ 2TM :1

0,1,2,...

nh n

nn

λ=

−

=

For other TMn modes:

TM1:

⇒

20

u

v

R/ 2π π

1TM

TM0 ModeThe TM0 mode has no cutoff frequency

(it can propagate at any frequency):

Note:The lower the frequency, the slower the field decays away from the

interface. As high frequency the wave decays very quickly since v → ∞.

( )22 20 0 0 0/ 1x z zk k k k kα = − = −

21

1TMcf

1 r rn µ ε=

0/zk k

f

1.0

0TM1TM

TM0 Mode (cont.)

After making some approximations to the transcendental equation, valid for low frequency, we have the following approximate result for the TM0mode (derivation omitted):

( ) ( )0

1/ 222 20 1

0 2

11TM

r

k h nkβ

ε

− ≈ +

0 1k h <<

22

Approximate CAD Formula

TEx Modes

( ) ( )0 1 11 cotx x x

r

h k h k hαµ

= −

1 cotr

v u uµ

= −

Hence

23

1

0

x

x

u k hv hα

≡≡

1/ rµ−

2 2 2u v R+ =

1 cotr

v u uµ

= −

u

v

R / 2π π 3 / 2π

1TE

2TE

TEx Modes (cont.)No TE0 mode ( fc= 0). The lowest TEx mode is the TE1 mode.

TE1 cut-off frequency at R = π / 2:

( ) 20 1 1

2k h n π

− =

20 1

1/ 41

hnλ

=−

In general, we have

TEn: ( )2

0 1

2 1 / 4

11,2,3,.........

nhn

n

λ−

=−

=

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 model a microstrip antenna, or a bend on a microstrip line).

spr

sp sw

Pe

P P≡

+

TM0 SW

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1h / λ0

0

20

40

60

80

100

EFFI

CIE

NC

Y (%

)

exactCAD

0/h λ

2.2rε =

10.8rε =

25Psp = power radiated into space Psw = power launched into surface wave

Dielectric Rod

a

z

,r rε µ

This serves as a model for a single-mode fiber-optic cable.

The physics is similar to that of the TM0 surface wave on a grounded

substrate.

26

Fiber-Optic Guide

Two types of fiber-optic guides:

1) Single-mode fiber

2) Multi-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 than multimode fiber. 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, and is less expensive, it but has more loss and dispersion.

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Dominant mode (lowest cutoff frequency): HE11 (fc = 0)

The field shape is somewhat similar to the TE11 circular waveguide mode.

Dielectric Rod: 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 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 more loosely bound to the rod.)

The dominant mode is a hybridmode (it has both Ez and Hz).

E

HE11 mode on single-mode fiber

28

2 20 0zk kρα = −

Single Mode Fiber (cont.)

What they look like in practice:Single-mode fiberhttp://en.wikipedia.org/wiki/Optical_fiber

29

Multimode Fiber

http://en.wikipedia.org/wiki/Optical_fiber

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”).

A laser bouncing down an acrylic rod, illustrating the total internal reflection of light in a multi-mode optical fiber

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