1

Prof. David R. JacksonECE Dept.

Spring 2014

Notes 6

ECE 6341

2

Leaky Modes

f = fs

v

u

TM1 Mode

ISW

SW

f > fs We will examine the solutions asthe frequency is lowered.

Splitting point

1tan

r

v u u

R

20 1( ) 1R k h n

3

Leaky Modes (cont.)

a) f > fc SW+ISW

v

u

ISW

SW

TM1 Mode

The TM1 surface wave is above cutoff. There is also an improper TM1 SW mode.

Note: There is also a TM0 mode, but this is not shown

4

Leaky Modes (cont.)

a) f > fc SW+ISW

Im kz

k1

SW

Re kz

k0

ISW

The red arrows indicate the direction of movement as the frequency is lowered.

2 2 1/20

2 20

( )z

z

v h k k

k k

0zk k

0zk k 0zk k

v

u

ISW

SW

TM1 Mode

5

b) f = fc

Leaky Modes (cont.)

v

u

TM1 Mode

The TM1 surface wave is now at cutoff. There is also an improper SW mode.

6

b) f = fc

Leaky Modes (cont.)

Im kz

k1

Re kz

k0

v

u

TM1 Mode

7

c) f < fc 2 ISWs

Leaky Modes (cont.)

v

u

The TM1 surface wave is now an improper SW, so there are two improper SW modes.

TM1 Mode

8

c) f < fc 2 ISWs

Im kz k1

Re kz

k0

Leaky Modes (cont.)

The two improper SW modes approach each other.

v

u

TM1 Mode

9

d) f = fs

Leaky Modes (cont.)

v

u

The two improper SW modes now coalesce.

TM1 Mode

10

d) f = fs

Leaky Modes (cont.)

Im kz k1

Re kz

k0

Splitting point

v

u

TM1 Mode

11

e) f < fs

Leaky Modes (cont.)v

u

The graphical solution fails! (It cannot show us complex leaky-wave modal solutions.)

The wavenumber kz becomes complex (and hence so do u and v).

12

Leaky Modes (cont.)

Im kz

k1

e) f < fs 2 LWs

Re kz

k0

LW

LW

This solution is rejected as completely non-physical since it grows with distance z.

The growing solution is the complex conjugate of the decaying one (for a lossless slab).

z z zk j

13

Leaky Modes (cont.)

Proof of conjugate property (lossless slab)

Take conjugate of both sides:

112 2 2

2 21 211

2 2 20

( )tan ( )

( )

zr z

z

k kk k h

k k

112 *2 2

2 *21 211

*2 2 20

( )tan ( )

( )

zr z

z

k kk k h

k k

Hence, the conjugate is a valid solution.

TRE: TMx Mode

14

Leaky Modes (cont.)

a) f > fc

Im kz

k1

SW

Re kz

k0

ISW

b) f = fc

Im kz

k1

Re kz

k0c) f < fc

Im kz

k1

Re kz

k0

splitting point

d) f = fs

Im kz

k1

Re kz

k0

LW

LW

e) f < fs

Here we see a summary of the

frequency behavior for a typical surface-wave

mode (e.g., TM1).

TM0: Always remains a proper physical SW mode.

TE1: Goes from proper physical SW to nonphysical ISW; remains nonphysical ISW down to zero frequency.

Exceptions:

Im kz

k1

Re kz

k0

15

Leaky Modes (cont.)

A leaky mode is a mode that has a complex wavenumber (even for a lossless structure). It loses energy as it propagates due to radiation.

z

Power flow

x

z z zk j

ˆ ˆˆ ˆRe Re x z x zk xk zk x z

0

0tan /z x

16

Leaky Modes (cont.)

One interesting aspect: The fields of the leaky mode must be improper (exponentially increasing).

Proof:1

2 2 20 0( )x zk k k

2 2 20 0x zk k k

2 220x x z zj k j

2 2 2 2 202 2x x x x z z z zj k j

x x z z Taking the imaginary part of both sides:

Notes:z > 0 (propagation in +z direction)z > 0 (propagation in +z direction)

x > 0 (outward radiation)

17

Leaky Modes (cont.)For a leaky wave excited by a source, the exponential growth will only persist out to a “shadow boundary” once a source is considered.

This is justified later in the course by an asymptotic analysis: In the source problem, the LW pole is only captured when the observation point lies within

the leakage region (region of exponential growth).

0

z

Power flow

Leaky mode

x

Region of exponential

growth

Region of weak fields

z z zk j Source

A hypothetical source launches a leaky wave going in one direction.

18

A requirement for a leaky mode to be strongly “physical” is that the wavenumber must lie within the physical region (z = Re kz < k0) where is wave is a fast wave*.

Leaky Modes (cont.)

A leaky-mode is considered to be “physical” if we can measure a significant contribution from it along the interface (0 = 90o) .

(Basic reason: The LW pole is not captured in the complex plane in the source problem if the LW is a slow wave.)

* This is justified by asymptotic analysis, given later.

19

f) f < fp Physical LW

Physical leaky wave region (Re kz < k0)

Leaky Modes (cont.)

Im kz

k1

Re kz

k0 LW

f = fp

PhysicalNon-physical

Note: The physical region is also the fast-wave region.

20

If the leaky mode is within the physical (fast-wave) region, a wedge-shaped radiation region will exist.

Leaky Modes (cont.)

This is illustrated on the next two slides.

21

0sinz

2 2 2 2 2 20x z x zk k k

0 0sinz k Hence

Leaky Modes (cont.)

(assuming small attenuation)

ˆ ˆx zx z

Significant radiation requires z < k0.

0

z

Power flow

Leaky mode

x

z z zk j Source

22

10 0sin /z k

Leaky Modes (cont.)

0

z

Power flow

Leaky mode

x

z z zk j Source

As the mode approaches a slow wave (z k0), the leakage region shrinks to zero (0 90o).

23

Leaky Modes (cont.)

0 0sin0

j k d nnI I e

Phased-array analogy

z

I0

x

I1 In IN

d

0

0 0sinzk k

Note:A beam pointing at an angle in “visible space”

requires that kz < k0.

0 0sinz j k d njk z

z nde e

0 0sinn k d n

Equivalent phase constant:

24

Leaky Modes (cont.)

The angle 0 also forms the boundary between regions where the leaky-wave field increases and decreases with radial distance in cylindrical coordinates (proof omitted*).

0

z

Power flow

Leaky mode

x

Fields are decreasing

radially

z z zk j Source

Fields are increasing

radially

*Please see one of the homework problems.

25

The aperture field may strongly resemble the field of the leaky wave (creating a good leaky-wave antenna).

Leaky Modes (cont.)

Line source

Requirements:

1) The LW should be in the physical region.2) The amplitude of the LW should be strong.3) The attenuation constant of the LW should be small.

A non-physical LW usually does not contribute significantly to the aperture field (this is seen from asymptotic theory, discussed later).

Excitation problem:

26

Leaky Modes (cont.)

Summary of frequency regions:

a) f > fc physical SW (non-radiating, proper)

b) fs < f < fc non-physical ISW (non-radiating, improper)

c) fp < f < fs non-physical LW (radiating somewhat, improper)

d) f < fp physical LW (strong focused radiation, improper)

The frequency region fp < f < fc is called the “spectral-gap” region

(a term coined by Prof. A. A. Oliner).

The LW mode is usually considered to be nonphysical in the spectral-gap region.

27

Leaky Modes (cont.)

Im kz

k1

Re kz

k0

LW

f = fp

Physical Non-physical

ISW

f = fc

f = fs

Spectral-gap region

fp < f < fc

28

Field Radiated by Leaky Wave

1, 0, ,

2x zjk x jk z

y y z zE x z E k e e dk

0,LWzjk z

yE z e

22

0, 2LWz

y zLW

z z

kE k j

k k

For x > 0:

Then

Line sourcez

x

Aperture

1/22 20x zk k k

Note: The wavenumber kx is chosen to be either positive real or negative imaginary.

Assume:

TEx leaky wave

29

z / 0

x / 0

-10 100

0

10

0

3/ 0.02

2LWzk k j

5

x / 0

0

10

5

-10 0 10

Radiation occurs at 60o.

30

z / 0

x / 0

-10 100

0

10

0

3/ 0.002

2LWzk k j

5

x / 0

0

10

5

-10 0 10

Radiation occurs at 60o.

31

z / 0

x / 0

-10 100

0

10

0/ 1.5 0.02LWzk k j

5

x / 0

0

10

5

-10 0 10

The LW is nonphysical.

32

x / 0

Leaky-Wave Antenna

z / 0

x / 0

-10 100

0

10

5

x / 0

0

10

5

Near field

Far field

Line sourcez

x

Aperture

33

Leaky-Wave Antenna (cont.)

Far-Field Array Factor (AF)

0

0

sin

sin

0,

LWz

j k zy

j k zjk z

AF E z e dz

e e dz

22 2

0

2sin

LWz

LWz

kAF j

k k

x / 0

line sourcez

x

34

Leaky-Wave Antenna (cont.)

22 2

0

2sin

z z

z z

jAF j

k j

2 2 2 20

2sin 2

z z

z z z z

jAF

k j

1/22 2

2 22 2 2 20

2sin 2

z z

z z z z

AFk

A sharp beam occurs at 0 0sin zk

35

x / 0

Leaky-Wave Antenna (cont.)Two-layer (substrate/superstrate) structure excited by a line source.

2 1r r

1

2

/ 0.5

/ 0.25

b

t

1r2r

b

t

x

z

Far Field

Note: Here the two beams have merged to become a single beam at broadside.

36

x / 0

Implementation at millimeter-wave frequencies (62.2 GHz)

r1 = 1.0, r2 = 55, h = 2.41 mm, t = 0.484 mm, a = 3.73 0 (radius)

Leaky-Wave Antenna (cont.)

37

x / 0

Leaky-Wave Antenna (cont.)

(E-plane shown on one side, H-plane on the other side)

38

Leaky-Wave Antenna (cont.)W. W. Hansen, “Radiating electromagnetic waveguide,” Patent, 1940, U.S. Patent No. 2.402.622.

a

b

x

z

y

220z k

a

0z k Note:

•W. W. Hansen, “Radiating electromagnetic waveguide,” Patent, 1940, U.S. Patent No. 2.402.622.

y This is the first leaky-wave antenna invented.

Slotted waveguide

39

x / 0

Leaky-Wave Antenna (cont.)

The slotted waveguide illustrates in a simple way why the field is weak outside of the “leakage region.”

Top view

Source

Slot

Region of strong fields (leakage region)

Waveguidea TE10 mode

40

x / 0

Leaky-Wave Antenna (cont.)Another variation: Holey waveguide

220z k

a

a p

b

x

z

y

r

0p

y

41

x / 0

Leaky-Wave Antenna (cont.)

Another type of leaky-wave antenna: Surface-Integrated Waveguide (SIW)

h εr

via d s

p ls ws w

slot