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ECE 5322 21st Century Electromagnetics
Instructor:Office:Phone:E‐Mail:
Dr. Raymond C. RumpfA‐337(915) 747‐[email protected]
Coupled-Mode Theory
Lecture #5
Lecture 5 1
Lecture Outline
• Electromagnetic modes
• Coupled-mode theory
• Codirectional coupling
• Contradirectional coupling
• Non-directional coupling
• Phase matching with gratings
• Mode-matching vs. coupled-wave models
Lecture 5 Slide 2
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Electromagnetic Modes
What are modes?
Lecture 5 Slide 4
Modes can mean many different things depending on the context it is being used.
• Different discrete eigen‐modes in a waveguide• Different polarizations• Different directions• Etc.
Generalized Definition:
An electromagnetic mode is electromagnetic power that exists independent and different from other electromagnetic power.
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Modes in a Waveguide
Lecture 5 Slide 5
Waves in Free Space
Lecture 5 Slide 6
Poincare Sphere
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Modes in Two Waveguides
Lecture 5 Slide 9
Triangle Waveguide Square Waveguide
1
1
1 0,1
1 0,1
,
,
j z
j z
E E x y e
H H x y e
2
2
2 0,2
2 0,2
,
,
j z
j z
E E x y e
H H x y e
Supermodes
Lecture 5 Slide 10
Coupled WaveguidesCoupled Waveguides
When two waveguides are in close proximity, they become coupled.
The pair forms “supermodes.”
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Visualization of Coupled-Modes
Lecture 5 Slide 11
When two waveguides are in close proximity, they become coupled and exchange power as a function of z.
Very often, this leads to a periodic exchange of power between the waveguides.
Waveguide arrays are more complicatedto analyze, but involve the same concepts.
z
Launch
Perturbation Analysis
Lecture 5 Slide 12
1 2
1 2
E A z E B z E
H A z H B z H
Assumption – To simplify the analysis, it will be assumed that the supermodes can be represented as a weighted sum of the individual guided modes. This implies that the modes do not change at all with the introduction of the second guide. In reality, the modes are deformed slightly, but are still coupled.
When two waveguides are in close proximity, they become coupled and exchange power as a function of z.
st
nd
amplitude of 1 mode
amplitude of 2 mode
A z
B z
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Assumed Solution in Perturbation Analysis
Lecture 5 Slide 13
1 2
1 2
E A z E B z E
H A z H B z H
We start with the following solution.
We substitute these into Maxwell’s curl equations to obtain
1 2
1 0 1 1 2 0 2 2
ˆ ˆ 0
ˆ ˆ 0r r
dA dBz E z E
dz dzdA dB
z H j AE z H j BEdz dz
To do this, we made use of the following vector identity
ˆdA
AE A E A E A E z Edz
0
0 r
E j H
H j E
Ignoring magnetic response
Derivation of the Generalized Coupled-Mode Equations
Lecture 5 Slide 14
We have the following equations enforcing Maxwell’s equations.
We derive the generalized coupled-mode equations by substituting the above expressions into the following integral equations.
1 2
1 0 1 1 2 0 2 2
ˆ ˆ 0
ˆ ˆ 0r r
dA dBz E z E
dz dzdA dB
z H j AE z H j BEdz dz
Eq. (1)
Eq. (2)
* *1 1Eq. 2 Eq. 1 0E H dxdy
* *2 2Eq. 2 Eq. 1 0E H dxdy
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Generalized Coupled-Mode Equations
Lecture 5 Slide 15
2 1 2 1
2 1 2 1
12 1 12
21 2 21
0
0
j z j z
j z j z
dA dBc e j A j Be
dz dzdB dA
c e j B j Aedz dz
After LOTS of algebra, we get (i.e. it is easily shown that… )
These are called the generalized coupled-mode equations. These are solved to describe the coupling between the two waveguides.
*0 ,
* *ˆ
r r q p q
pq
p p p p
E E dxdy
z E H E H dxdy
* *
* *
ˆ
ˆ
p p
pq
p p
q
p
q
p
z E H E H dxdy
c
z E H E H dxdy
*0 ,
* *ˆ
r r q p p
p
p p p p
E E dxdy
z E H E H dxdy
Mode Coupling Coefficient Butt Coupling Coefficient Change in Propagation Constant
, 1 or 2p q
Mode Coupling Coefficient, pq
Lecture 5 Slide 16
*0 ,
* *ˆ
r r q p q
pq
p p p p
E E dxdy
z E H E H dxdy
The mode coupling coefficient is calculated according to
This parameter quantifies how efficiently power “leaks” from waveguide p to waveguide q due to the behavior of the supermode.
r is the dielectric function containing both waveguides. r,q is dielectric function with only waveguide q.
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Butt Coupling Coefficient, cpq
Lecture 5 Slide 17
* *
* *
ˆ
ˆ
p p
pq
p p
q
p
q
p
z E H E H dxdy
c
z E H E H dxdy
The coefficient cpq quantifies the excitation efficiency from one waveguide to the other. It is called the butt coupling coefficient and is calculated according to
Butt coupling
Change in Propagation Constant, p
Lecture 5 Slide 18
When the qth waveguide is brought into proximity to pth waveguide, the propagation constant in the pth waveguide changes by p.
*0 ,
* *ˆ
r r q p p
p
p p p p
E E dxdy
z E H E H dxdy
We expect p to be largest when the waveguides are the closest and the fields are perturbed more strongly affecting the propagation constant.
Many analyses just assume = 0.
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Mode-Coupling Vs. Butt Coupling
Lecture 5 19
Butt CouplingThis is an “end‐fire” mechanism and occurs because parts of the mode from one waveguide match the mode from the second.
Mode CouplingThis is a “leakage” mechanism and occurs due to the propagation behavior of the supermode.
Normalized Power in Eigen-Modes
Lecture 5 Slide 20
The total power in waveguide p is
*1ˆ
2p p pP E H zdxdy
We see that the denominator in the prior equations is 4Pp.
Without loss of generality, we normalize the power in the eigen-modes according to
* *ˆ4 1p p p p pP z E H E H dxdy
After normalizing the power, it is then easily shown that… *
21 12c c *p q
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Power in Supermode
Lecture 5 Slide 21
The power in the supermode is
*1ˆ
2P E H zdxdy
After some algebra, this becomes
2 2 * 2 * * 212 12
1
4j z j zP A B A Bc e AB c e
2 1
2
Total power
Difference in propagation constants of the two waveguides.
Consequences of Conservation of Power
Lecture 5 Slide 22
For waveguides without loss or gain,
0dP
dz
So, we now differentiate our equation for total power to get
* * 2 * * * 221 12 12 21 12 122 2 0j z j zjA B c e jAB c e
For this to be satisfied independent of z, we must have
* *21 12 122 c Note, we only have when:
• 1=2 (identical waveguides) =0, or• Waveguides are sufficiently separated so that
*21 12
*12 0c
If the waveguides are very close or are very different, the term cannot be ignored.*
122 c
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Revised Coupled Mode Equations
Lecture 5 Slide 23
2
2
j za a
j zb b
dAj Be j A
dzdB
j Ae j Bdz
Our coupled-mode equations can now be written as
12 12 22
12
*21 12 1
2
12
21 21 12
12
*12 12 2
2
12
1
1
1
1
a
b
a
b
c
c
c
c
c
c
c
c
Simplified Coupled Mode Equations
Lecture 5 Slide 24
2 1
2 1
12
21
j z
j z
dAj Be
dzdB
j Aedz
Assuming cpq=p=0 (i.e. modes in the individual waveguides unperturbed), the coupled-mode equations are written as
These are the equations that most analyses use.
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General Coupled-Mode Solution
Lecture 5 Slide 27
1 2
1 2
j z j z j z
j z j z j z
A z a e a e e
B z b e b e e
In codirectional coupling, both modes are propagating in the same direction and usually with similar propagation constants.
The general solution to the simplified coupled-mode equations are
1 20 and 0
Reciprocity requires that . Most often, pq is real so*
12 21
12 21
1 2
1 2
0
0
a a A
b b B
Initial conditions…
Solution with Boundary Conditions
Lecture 5 Slide 28
cos sin 0 sin 0
sin 0 cos sin 0
j z
j z
j jA z z z A z B e
j jB z z A z z B e
The final solution for A(z) and B(z) are then
2 2
Note, when perturbation of the modes in the waveguide is minimal, 0 and .
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Typical Solution in Terms of Power
Lecture 5 Slide 29
0 0cos sin sinj z j zj jA z A z z e B z A z e
In most cases, power is injected into only one waveguide.
00 0 0A A B
Our equations for A(z) and B(z) reduce to
It is often more meaningful to write similar expressions in terms of the power in each waveguide as a function of z.
2
22
0
2
22
0
1 sin
sin
a
b
A zP z F z
A
B zP z F z
A
2
2
1
1F
Maximum power‐coupling efficiency…
Typical Response of Codirectional Couplers
Lecture 5 Slide 30
2 3 2 2
2 3 2 2
0 1F
2 0.2F
Maximums occur at
2 1 0,1,2,...2mz m m
Coupling LengthThe length over which maximum power is transferred to the second waveguide is called the coupling length.
When 1=2 (i.e. = 0),
2 22 2cL
2cL
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Visualization of the Terms
Lecture 5 Slide 31
z
Launch
cL
sin
sin2
cL
2
4
2
c
c
L
L
ContradirectionalCoupling
(Bragg Grating)
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Contradirectional Coupling
Lecture 5 Slide 33
In contradirectional coupling, the coupled-modes are propagating in opposite directions.
Let the second mode be the backward propagation mode.
1 20 and 0
Reciprocity requires that .*12 21
Conditions for ContradirectionalCoupling
Lecture 5 Slide 34
Contradirectional coupling cannot occur by simply bringing two waveguides in proximity. Typically a grating is used to couple the counter propagating modes.
waveguide 1
waveguide 2
grating
2
12
j z
Gz e
The mode coupling coefficient is now a periodic function.
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Contradirectional Coupled-Mode Equations
Lecture 5 Slide 35
The coupled-mode equations are now written as
2 1
2 1
2
2
j z
G
j z
G
dAj Be
dz
dBj Ae
dz
2*
12 21
j z
Ge
Phase Matching Conditions
Lecture 5 Slide 36
1 2
2
2
We introduce the following phase matching condition of the grating.
We will have three cases
Case 1:
Case 2:
Case 3:
G
G
G
Band edge.
Pass band. Forward output. Temporary and confined peak in reflected mode.
Stop band. Reflected output. Band of reflection.
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Case 1: ||>G (Pass Band)
Lecture 5 Slide 37
0
0
cos sin
cos sin
sin
cos sin
j z
G j z
z L j z LA z A e
L j L
j z LB z A e
L j L
The mode amplitudes are:
The normalized forward and backward power
2 2G
2 2 2 2
2 2 2 2
0
2 2 2
2 2 2 2
0
sin
sin
sin
sin
Gf
G
Gb
G
A z z LP z
LA
B z z LP z
LA
L is the distance over which the periodic perturbation exists.
Case 2: ||=G (Band Edge)
Lecture 5 Slide 38
0
0
1
1
1
j z
G j z
j z LA z A e
j L
j z LB z A e
j L
The mode amplitudes are:
The normalized forward and backward power
2 22
2 2 2
0
2 22
2 2 2
0
1
1
1
Gf
G
Gb
G
A z z LP z
LA
B z z LP z
LA
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Case 3: ||<G (Stop Band)
Lecture 5 Slide 39
0
0
cosh sinh
cosh sinh
sinh
cosh sinh
j z
G j z
z L j z LA z A e
L j L
j z LB z A e
L j L
2 2G
The mode amplitudes are:
The normalized forward and backward power
2 2 2 2
2 2 2 2
0
2 2 2
2 2 2 2
0
sinh
sinh
sinh
sinh
Gf
G
Gb
G
A z z LP z
LA
B z z LP z
LA
Typical Bragg Response
Lecture 5 Slide 40
100%
RESPONSE OF A BRAGG GRATING
Stop Band
Pass Band Pass Band
G G
G
0 eff Gk n
Frequency (k0)
0 eff Gk n
Reflectance (R)
Transmittance (T)
2
2
0
2
0
2
0
A LT
BR
A
A
eff2B n Bragg wavelength
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Non-Directional Coupling
Non-Directional Coupling
Lecture 5 42
It turns out that we can couple waves travelling in different directions. This is called non‐directional coupling.
2k
1k
K
1 2k kK
Wave 1 Wave 2 Grating that would couple wave 1 and wave 2
1k
2k
K
Grating vectors in opposite directions describe the same grating.
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Phase Matching with Gratings
Generalized Framework
Lecture 5 Slide 44
How do we couple two completely different modes so they can exchange power? Ordinarily, this will not happen.
1
2
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Phase Matching
Lecture 5 Slide 45
We can couple any two modes using a grating.
1
2
K
The phase matching condition to couple energy between two modes is
1 2K 1
2
K
Grating Coupler Regimes
Lecture 5 Slide 46
Short period gratingsBragg gratingsContradirectional coupling
Long period gratingsCodirectional coupling
“Medium” period gratingsNon‐directional coupling
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Mode-Matching Vs.
Coupled-Wave
Frameworks to Model Propagation
Lecture 5 Slide 48
Both mode-matching and coupled-mode frameworks view devices as consisting of a series of segments that are uniform in the z-direction.
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Mode-Matching Framework (1 of 3)
Lecture 5 Slide 49
Mode matching views the field in a segment as being the sum of a set of orthogonal basis functions (eigen-modes).
+ + += +
m mm
E x a f x
E x 1f x 2f x 3f x 4f x mf x
Mode-Matching Framework (2 of 3)
Lecture 5 Slide 50
The modes within a segment accumulate phase differently as they propagate, but they do not interact and they propagate independently.
+
+
+
=
th
complete descriptionof the eigen-mode
, mj zm m
m
m
E x z a f x e
,E x z
11
j zf x e
22
j zf x e
33
j zf x e
44
j zf x e
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Mode-Matching Framework (3 of 3)
Lecture 5 Slide 51
At an interface, the power redistributes itself among the eigen-modes in the next segment.
+ + + =+
1, 0 2, 0
1 0 2 0
1, 1, 2, 2,
, ,
m mj z j zm m m m
m m
E x z E x z
a f x e a f x e
+ + += +
boundary conditions
Conclusions About Mode-Matching
• The mode-matching framework applies to more than waveguides– Metamaterials, gratings, electromagnetic band gap
materials, frequency selective surfaces, transmission lines, guided-mode resonance filters, photonic crystals, and more.
• Modes do not interact and they propagate independently with their own propagation constant.
• Power among the modes “scrambles” at an interface.
• The overall field is the sum of the eigen-modes
Lecture 5 Slide 52
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Coupled-Wave Framework (1 of 3)
Lecture 5 Slide 53
Coupled-wave views the field in a segment as being the sum of a set of plane wave basis functions.
+ + += +
,x mjk xm
m
E x a e
E x ,1xjk xe
Coupled-Wave Framework (2 of 3)
Lecture 5 Slide 54
The waves within a segment are coupled. So, in addition to accumulating phase as they propagate, they also interact by exchanging power (coupled). The mode coefficients are therefore a function of z.
,, x mjk xm
m
E x z a z e
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Coupled-Wave Framework (3 of 3)
Lecture 5 Slide 55
At an interface, the amplitudes of the plane waves on either side remain the same to enforce boundary conditions. This is because the same basis functions are being used on both sides..
+ + ++ =
boundary conditions
+ + += +
, ,
1 0 2 0
1, 2,
, ,
x m x mjk x jk xm m
m m
E x z E x z
a z e a z e
Conclusions about Coupled-Wave
• The coupled-mode framework applies to more than waveguides– Metamaterials, gratings, electromagnetic band gap materials,
frequency selective surfaces, transmission lines, guided-mode resonance filters, photonic crystals, and more.
• Modes can interact. In addition to accumulating phase as they propagate, modes can exchange power.
• Nothing interesting happens at an interface as the amplitudes of the modes remain constant across the interface (ignoring reflections)
• The overall field is the sum of the basis functions
Lecture 5 Slide 56
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How Do We Reconcile These Two Theories?
Lecture 5 57
Plane waves do not exist in inhomogeneous materials.
If we force them to exist, they exist in “sets” and the plane waves exchange energy as they propagate.
In this sense, we can think of modes as the set of plane waves that propagate independently of other sets of plane waves.
This transforms coupled‐mode framework to the mode‐matching frame work.