David CHen Waveguide Cavity

8/3/2019 David CHen Waveguide Cavity

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Field and WaveField and Wave ElectromagneticsElectromagnetics (II)(II)

CHCH--1010

Assistant Professor: Yi-Pai Huang

Department of Photonicsand

Display Institute


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Outlines

Waveguide and Resonator10-2 General Wave Behaviors along Uniform Guiding Structures

- Transverse Electromagnetic Waves

- Transverse Magnetic Waves

- Transverse Electric Waves


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Transmission line (1) Guiding TEM wave

(2) Higher frequency, higher resistance

When frequency increase to micro-waves, the loss in the

transmission line were to serious to be suffered.

Therefore, the new wave guiding structure have to be used.

For simply classified, the wave guide can be separated into:

(1) Parallel plate(2) Rectangular

(3) Circular

(4) Dielectric

f)(R

The wave guide has following two characteristics:

(1) A hollow waveguide cannot support TEM mode.

(2) To transmit TE or TM mode, the frequency must be higher than

cutoff frequency.

Introductions


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The wave propagate at +z direction, with propagation constant

The wave equation can be written as:

Homogeneous Helmholtzs equation:

The Laplacian operator:

With rectangular waveguide:

General Wave Behaviors along Uniform Guiding Structures


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Utilize B.C. to determine 6 equations

E and H are dependent, from curve equation in source free field, can get:

To find out and

The can be in terms of and



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For example : 10-9a and 10-10b can determine and

The wave behavior in a waveguide can be determined:

Where



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For the TEM wave in the wave guide,

=0 =0

=0

Propagation (phase) velocity of TEM wave

Wave impedance

Formula for a TEM wave in +z direction

Transverse Electromagnetic Waves


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Discussion:

1. Same expression of the propagation constant of a uniform plane wave

2. The phase velocity and the wave impedance for TEM waves areindependent of the frequency of the waves

3. TEM wave cannot exist in a single-conductor hollow (or dielectric-filled)waveguide of any shape.

ani

E

H

Conductor

A. B and H must form a close loop in a transverse plane.B. From Amperes circuital law, it should has conduction and

displacement current.C. TEM wave doesnt has Ez-component, therefore, no any longitudinal

current in the single-conductor hollow (or dielectric-filled) waveguide .

Transverse Electromagnetic Waves


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TM wave

With convenience, the equations can be re-write as:

Where:

Transverse Magnetic Waves


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and

Consider the wave impedance

Here Because

How to find ?

can be real or imaginary, and =0 is the critical point

Cutoff-frequency

The value of for a particular mode in a

waveguide depends on the eigenvalue of this mode



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(a)

(b)

For the frequency higher or lower than the cutoff-frequency, will be discussed :



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and

Consider the wave impedance

Here Because

How to find ?

can be real or imaginary, and =0 is the critical point

Cutoff-frequency

The value of for a particular mode in a

waveguide depends on the eigenvalue of this mode



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(a)

For the frequency higher or lower than the cutoff-frequency, will be discussed :

Transverse Magnetic Waves Cutoff Frequency

PropagationConstant

Phase

Constant

Wavelength inwaveguide

Wavelength in uniformdielectric space

Wavelength ofcutoff frequency


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In free space (air)

(1) TM mode phase velocity always faster than the light speed in the medium

(2) TM mode group velocity always slower than the light speed in the medium

(3) Depends on frequency dispersive transmission systems

(4) Propagation velocity (velocity of energy transport) = group velocity

Phasevelocity

Group

velocity

Wave impedance of TM mode

The wave impedance of TM mode with a lossless dielectric filled waveguide is alwaysless than the intrinsic impedance of the dielectric medium.



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(b)

The incident frequency < cutoff frequency Becomes evanescent waveWaveguide is a kind of high-pass filter

Normalized wave impedances for

propagating TM waves

Normalizedwaveimpedance



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TE mode and

Transverse Electric Waves


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/ZTE

/ZTM

(1) Similar to the TM mode, but ZTE

always larger than the intrinsic

impedance of the dielectric material(2) The frequency lower than cut-offfrequency cannot be existed

Transverse Electric Waves Cutoff Frequency


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(a) Determine Z and of TE and TM mode

(b) Determine Z and of TE and TM mode

(c) For any frequency, determine Z and of TEM mode

(a)

Example 10-1


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For TEM wave, theres no cutoff frequency. And Ex, Hx, Ey, Hy

Exsit when

Therefore, Z and are dependent with frequency, and same asuniform plane wave in infinite dielectric material

The wave is evanescent wave, not propagation wave

no significant of

(b)

(c)

Example 10-1


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TEM mode:

TM and TE mode:

Discussion:(1) Slope of P and O is the phase velocity

(2) Slope of point P is the group velocity

(3) For TE and TM mode

(4) When frequency >> cutoff frequency

(5) dependent on the eigenvalue

- Diagram (Phase and Group Velocity)


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Please plot the diagram of and f

2

c

22 fyxCircle equation

h0f

0ff c No attenuation

Attenuation depends on the eigenvalue

Example 10-2


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Outlines

Waveguide and Resonator

10-3 Parallel-Plate Waveguide

- TM waves

- TE waves

- Energy-Transport Velocity

- Attenuation in Parallel-Plate Waveguides


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(a)

(b)

(c)

(d)

Parallel-plate waveguide

For B.C.

Where

Parallel-Plate Waveguide TM mode


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Where and

Re-call Ch4 : B.C. of Cartesian Coordinate



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Propagate

Evanescent

Propagation constant

Criticalpoint

Cutofffrequency

For different eigenvalue : has different TM mode (eigenmode)

.

.

.

Candetermine

If - Dominate mode

TEM mode is the dominate mode of parallel-plate waveguide

Only the transverse components



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(a) The instantaneous field expression for TM1 mode in parallel-plate

)zt(je

1n

Where

At yz plane, E has y and z components, at time t, the line equation:

t=0

(b)Sketch the E and M lines in the yz plane

Example 10-3


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Where

From

E-field line:

H-field line:

E-field:(1) Repeat with 2

(2) Inverse with

H-field:(1) Cos at y-direction(2) Sin at z-direction

Example 10-3


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TM1: Propagate in z-direction:

Propagate in+z with phase constant-y with phase constant /b

Propagate in+z with phase constant+y with phase constant /b

Example 10-4


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Compare with CH8-7,(1) x z, and z -y

(2)

CH8-7


P ll l Pl W id TM d


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of

TM1

mode propagate when:

Where

Same as the general form


P ll l Pl t W id TE d


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and

(1) Same cutoff-frequency of TEn and TMn mode in parallel-plate waveguide

(2) When n=0, Hy=0 and Ex=0 No TE0 mode in parallel-plate waveguide

Parallel-Plate Waveguide TE mode

E l 10 5


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(a) The instantaneous field expression for TE1 mode in parallel-plate

(b)Sketch the E and M lines in the yz plane

E-field line:

H-field line:

Example 10-5

Example 10 5


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TE1 mode in parallel-plate

TM1 mode in parallel-plate

Example 10-5

Energy transport Velocity


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Energy-transport Velocity

Group velocity loses its significance because the low frequencycomponents may be below cutoff

Group velocity = Propagation velocity (velocity of energy transport)

= The energy propagates along a waveguide

Time-average propagate power

Time-average stored energy per unit guide length

Example 10 6


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Time-average propagate power

Determine the energy-transport velocity of the TMn mode in a lossless

parallel-plate waveguide

Poynting vector:

where

2

b

)yb

n

(d2

)yb

n2cos(21

n

b

dy)yb

n

(cos

n

0

b

0

2

and

Example 10-6

Example 10-6


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2

m

2

e H2

1

w,E2

1

w

Time-average stored energy per unit guide length

Groupvelocity

Re-call

Example 10-6

Attenuation in Parallel-Plate Waveguides


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Attenuation constant in a waveguide:

: Loses in the dielectric medium

: Ohmic power loss in the imperfectly conducting walls

where

(1) In the TEM mode:

''j'jc

is Independent to f

is proportional to(a)

(b)




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(2) In the TM mode:

thus(a) calculation:

Re-call Cutoff-frequency




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(b) calculation:

Attenuation in Parallel Plate Waveguides



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(2) In the TE mode:

(a) calculation:

(b) calculation:




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TM mode: TE mode:


Outlines


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Outlines

Plane Electromagnetic Waves

10-4 Rectangular Waveguides

- TM Waves

- TE Waves

- Attenuation in Rectangular Waveguide

- Discontinuity of rect. waveguide

Rectangular Waveguides TM Mode


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where

g g



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From TM mode waveguide equations:

g g



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Where the propagation constant:

For TM modes in rectangular waveguides, neither m or n can be zero. WHY???

TM11

is the minimum cutoff frequency of TM modes in rectangular waveguides.

g g

Example 10-7


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where

(a)

p

Example 10-7


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xy-plane

yz-plane

Perpendicular to each other

Rectangular Waveguides TE Mode


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Rectangular Waveguides TE Mode


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For TE mode, m or n can be zero, but not both.

If a>b, than m=1, n=0 will be the dominate mode, due to the lowest fc

Example 10-8


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(a)

Example 10-8


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(b-1) x-y plane

(b-2) y-z plane

Example 10-8


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(b-3) x-z plane

Example 10-8


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(c) Surface current

Example 10-9


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1.

2.

Outlines


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10-5 Circular Waveguides

- Bessels differential equation and Bessels functions

- TM Waves in Circular Waveguides

- TE Waves in Circular Waveguides


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Attenuation in Rectangular Waveguide


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(b) calculation:

Derive is too complex.

Only discuss , which is the most important mode of Rect. Waveguide.

00

0

0

0



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where



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Both TE10 and TM11 has minimum, both has broad minimum

If a fixed, b increase can reduce attenuation, but reduce available band width of TE10

Available band: frequency over which TE10 is the only possible propagating mode Choose b/a=1/2 is the usual way to optimize.

TE10 mode always has less attenuation constant than TM11 mode

Example 10-10


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Example 10-10


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Discontinuity of rect. waveguide


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,

Capactive iris Inductive iris

Circular Waveguide


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Circular Waveguide Bessels Differential Equation


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Circular Waveguide Bessel Function


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except

Circular Waveguide Bessel Function


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except

Circular Waveguide Neumann Function


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0

Circular Waveguide TM mode


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(1)

(2)

Compensated with consider the time variable (t)

Now we have E 0, how to find other components?

Circular Waveguide TM mode


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Re-call: in Cartesian coordinate

where

Now we have Ez , how to find other components?

where

Now: in Cylindrical coordinate

Circular Waveguide TM01 mode (Eigenvalue)


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Circular Waveguide TM01 mode


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where

r=a

Circular Waveguide TE mode


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)y,rx(

Circular Waveguide TE11 mode


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Example 10-12


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What is Eigenvalue ?

Question : What does Eigenvalue mean?


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World

Country Taiwan, Japan

University NCTU, NTHU,

Department EO, EE,.

Student

Courses available courses

Score

Waveguide

Material ,

Structure Rect., Circ

Size x=a, y=b, .

Eigenvalue

Frequency available f

Value

g

You are the Eigenvalue which represent an

unique property in the world (waveguide)

Outlines


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10-6 Dielectric Waveguides- TM Waves along a Dielectric Slab

- TE Waves along a Dielectric Slab

- Dielectric Fiber

TM Waves along a Dielectric Slab


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(1)

(2)

(3)

In the space (2)

How about the equation in the space (1) and (3) ?



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Re-call:

In TIR condition, it has surface wave


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Now, we got , how about ?



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Combine with



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y=0

y=d/2

y=-d/2

From dispersion relation



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TE Waves along a Dielectric Slab


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TE Waves along a Dielectric Slab


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Example 10-13


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Example 10-14


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(a)

(b)

Example 10-14


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Utilize

(c)

0

Dielectric Fiber


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In the waveguide

Out of the waveguide

Dielectric Fiber


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Step-index fiber

Dielectric Fiber - Applications


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Multi-mode fiber

- High coupling efficiency

Single-mode fiber- Low dispersion

- Low loss with distortion

Graded-index fiber

- High coupling efficiency

- Low dispersion

- Self-adjustment


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Outlines


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10-7 Cavity Resonators- Rectangular Cavity Resonators

- Quality Factor

- Circular Cavity Resonators

- Application of Resonators

Dimension of the cavity are a, b, and d.

TM and TE mode in a resonator is not unique

Introduction of Resonator


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TM and TE mode in a resonator is not unique

Because x or y or z are free to be chosen as the direction of propagation

z-axis are chosen as the reference direction of propagation

Has conducting walls at z=0 and z=d and setup standing waves.

Three-symbol(mnp) were used to designate a TM or TE in a cavity resonator

Rect. Cavity Resonator TM Mode


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Re-call CH-8:

In Rect. Waveguide:



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Consider the B.C.: =0 at z=0 and z=d

Re-call : Plan wave reflection of perfect conductor

How about Ez(x,y,z) and other components?

Re-call : Waveguide TM mode and



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Rect. Cavity Resonator TE Mode


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(1)

(2)

(3)


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How to couple energy from a waveguide to a cavity ?

- Generate a hole or iris at an appropriate location

- Transmits a desired mode from the waveguide to be excited by the cavity resonant

How to Use Cavity Resonator?


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Example 10-15


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m and n0,p can be zero

Either m and n =0, (not both)

p0


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Step1: find non-zero components

Quality Factor


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Step2: find W, PL, and f

(b)

(a) Resonant frequency:

( )

(b)

Quality Factor


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(c)

=

=

=


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Example 10-16


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Circular Cavity Resonator


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where

Circular Cavity Resonator


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Example 10-17


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Applications


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1967 (General Conference on Weights and Measures) Cs1339,192,631,7701

250 m/s F3 A F4-133B F49,192,631,770 Hz http://www.hle.com.tw/index1.asp

Applications


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Resonant

(DWDM)

(VCSEL)

DWDM

http://www.itrc.org.tw/Bulletin/News/micrograting.php

http://www.itrc.org.tw/Bulletin/News/microinterferometer.php

Applications


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Documents

David CHen Waveguide Cavity