2/11/2016
ECE 5322 21st Century Electromagnetics
Instructor:
Office:
Phone:
E‐Mail:
Dr. Raymond C. Rumpf
A‐337
(915) 747‐6958
[email protected]
Lecture #5
Coupled-Mode Theory
Lecture 5
1
Lecture Outline
•
•
•
•
•
•
•
Lecture 5
Electromagnetic modes
Coupled-mode theory
Codirectional coupling
Contradirectional coupling
Non-directional coupling
Phase matching with gratings
Mode-matching vs. coupled-wave models
Slide 2
1
2/11/2016
Electromagnetic
Modes
What are modes?
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.
Lecture 5
Slide 4
2
2/11/2016
Modes in a Waveguide
Lecture 5
Slide 5
Waves in Free Space
Poincare Sphere
Lecture 5
Slide 6
3
2/11/2016
Resonant Modes
Lecture 5
Slide 7
Coupled-Mode
Theory
4
2/11/2016
Modes in Two Waveguides
Triangle Waveguide


E1  E0,1  x, y  e  j 1z


H1  H 0,1  x, y  e  j 1z
Square Waveguide


E2  E0,2  x, y  e  j 2 z


H 2  H 0,2  x, y  e  j 2 z
Lecture 5
Slide 9
Supermodes
When two waveguides are in close proximity, they become coupled.
The pair forms “supermodes.”
Coupled Waveguides
Lecture 5
Coupled Waveguides
Slide 10
5
2/11/2016
Visualization of Coupled-Modes
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 complicated
to analyze, but involve the same
concepts.
z
Launch
Lecture 5
Slide 11
Perturbation Analysis
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.



E  A  z  E1  B  z  E2



H  A  z  H1  B  z  H 2
A  z   amplitude of 1st mode
B  z   amplitude of 2nd mode
When two waveguides are in close proximity, they become coupled
and exchange power as a function of z.
Lecture 5
Slide 12
6
2/11/2016
Assumed Solution in
Perturbation Analysis
We start with the following solution.
Ignoring magnetic response


  E   j0 H


  H  j 0 r E



E  A  z  E1  B  z  E2



H  A  z  H1  B  z  H 2
We substitute these into Maxwell’s curl equations to obtain
 dA
 dB
 zˆ  E2
0
zˆ  E1
dz
dz
 dA

 dB

 j 0   r  1  AE1  zˆ  H 2
 j 0   r   2  BE2  0
zˆ  H1
dz
dz








To do this, we made use of the following vector identity



 dA

  AE  A  E  A  E  A  E 
zˆ  E
dz
 


Lecture 5
Slide 13
Derivation of the Generalized
Coupled-Mode Equations
We have the following equations enforcing Maxwell’s equations.


dB
  zˆ  E 
0
 zˆ  E  dA
dz
dz
1

 j
 zˆ  H  dA
dz
1
2

0

 r  1  AE1   zˆ  H 2 

dB
 j 0   r   2  BE2  0
dz
Eq. (1)
Eq. (2)
We derive the generalized coupled-mode equations by substituting
the above expressions into the following integral equations.
 

 
 

 
Lecture 5


 E1*   Eq. 2   H1*   Eq. 1  dxdy  0


 E2*   Eq. 2   H 2*   Eq. 1  dxdy  0
Slide 14
7
2/11/2016
Generalized Coupled-Mode
Equations
After LOTS of algebra, we get (i.e. it is easily shown that… )
dA
dB  j  2  1  z
 j   z
 c12
 j 1 A  j12 Be  2 1   0
e
dz
dz
dB
dA  j  2  1  z
 j   z
 c21
 j  2 B  j 21 Ae  2 1   0
e
dz
dz
These are called the generalized coupled-mode equations. These are
solved to describe the coupling between the two waveguides.
Mode Coupling Coefficient
Butt Coupling Coefficient
 
* 
  r   r ,q  E p  Eq dxdy
 
 pq 
 0 
 
c pq 
 




zˆ  E *p  H p  E p  H *p dxdy
 

 




zˆ  E *p  H q  Eq  H *p dxdy
 
 
 





zˆ  E *p  H p  E p  H *p dxdy
 

 
Change in Propagation Constant
 
 
 0     r   r ,q  E *p  E p dxdy
 
p   
* 

*
  zˆ   E p  H p  E p  H p  dxdy
 
p, q  1 or 2
Lecture 5
Slide 15
Mode Coupling Coefficient, pq
The mode coupling coefficient is calculated according to
 
 pq 
 0 
 
 
 
  r ,q  E *p  Eq dxdy
 




zˆ  E *p  H p  E p  H *p dxdy
 
 
r

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.
Lecture 5
Slide 16
8
2/11/2016
Butt Coupling Coefficient, cpq
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
 
c pq 




zˆ  E *p  H q  Eq  H *p dxdy
 
 
 





zˆ  E *p  H p  E p  H *p dxdy
 

 
Butt coupling
Lecture 5
Slide 17
Change in Propagation Constant, p
When the qth waveguide is brought into proximity to pth waveguide,
the propagation constant in the pth waveguide changes by p.
 
p 
 0 
 
 
r
 
  r , q  E *p  E p dxdy
 




zˆ  E *p  H p  E p  H *p 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.
Lecture 5
Slide 18
9
2/11/2016
Mode-Coupling Vs. Butt Coupling
Butt Coupling
This is an “end‐fire” mechanism and occurs because parts of the mode from one waveguide match the mode from the second.
Mode Coupling
This is a “leakage” mechanism and occurs due to the propagation behavior of the supermode.
Lecture 5
19
Normalized Power in EigenModes
The total power in waveguide p is
Pp 
 

*
1
ˆ
E

H
p
p  zdxdy
 
2 


We see that the denominator in the prior equations is 4Pp.
Without loss of generality, we normalize the power in the eigenmodes according to
 
4 Pp 




zˆ  E *p  H p  E p  H *p dxdy  1
 

 
After normalizing the power, it is then easily shown that…
c21  c
*
12
Lecture 5
p  

*
q
Slide 20
10
2/11/2016
Power in Supermode
The power in the supermode is
P
 
 *
1
ˆ
E
   H  zdxdy
2 


After some algebra, this becomes
P

1 2
2
A  B  A* Bc12 e j 2 z  AB*c12* e j 2 z 


4
 2  1
2
Total power
Difference in propagation constants of the two waveguides.
Lecture 5
Slide 21
Consequences of Conservation
of Power
For waveguides without loss or gain,
dP
0
dz
So, we now differentiate our equation for total power to get
*
jA* B  21
 12  2 c12  e  j 2 z  jAB*  21  12*  2 c12*  e j 2 z  0
For this to be satisfied independent of z, we must have
 21  12*  2 c12*
Note, we only have when:
 21  12*
• 1=2 (identical waveguides)  =0, or
• Waveguides are sufficiently separated so that c12*  0
If the waveguides are very close or are very different, the 2 c12* term cannot be ignored.
Lecture 5
Slide 22
11
2/11/2016
Revised Coupled Mode
Equations
Our coupled-mode equations can now be written as
dA
  j a Be j 2 z  j a A
dz
dB
  j b Ae j 2 z  j b B
dz
a 
b 
a 
b 
12  c12  2
1  c12
2
 21  c12* 1
1  c12
2
 21c21  1
1  c12
2
12 c12*   2
1  c12
2
Lecture 5
Slide 23
Simplified Coupled Mode
Equations
Assuming cpq=p=0 (i.e. modes in the individual waveguides
unperturbed), the coupled-mode equations are written as
dA
  j12 Be  j  2  1  z
dz
dB
  j 21 Ae  j  2  1  z
dz
These are the equations that most analyses use.
Lecture 5
Slide 24
12
2/11/2016
Codirectional
Coupling
Picture of Codirectional Coupling
Exit
Launch
Lecture 5
Slide 26
13
2/11/2016
General Coupled-Mode Solution
In codirectional coupling, both modes are propagating in the same
direction and usually with similar propagation constants.
1  0 and  2  0
*
Reciprocity requires that 12   21 . Most often, pq is real so
  12   21
The general solution to the simplified coupled-mode equations are
A  z    a1e j z  a2 e  j z  e  j z
B  z   b1e
j z
 b2 e
 j z
 e
 j z
Lecture 5
Initial conditions…
a1  a2  A  0 
b1  b2  B  0 
Slide 27
Solution with Boundary
Conditions
The final solution for A(z) and B(z) are then



j
j


sin  z   A  0  
sin  z  B  0   e  j z
A  z    cos  z  



 





j
 j

B  z    sin  z  A  0   cos  z  
sin  z   B  0   e j z



 

  2  2
Note, when perturbation of the modes in the waveguide is minimal,   0 and   .
Lecture 5
Slide 28
14
2/11/2016
Typical Solution in Terms of
Power
In most cases, power is injected into only one waveguide.
A  0   A0
B 0  0
Our equations for A(z) and B(z) reduce to


j
A  z   A0  cos  z  
sin  z   e  j z



B  z    A0
j

sin  z  e j z
It is often more meaningful to write similar expressions in terms of
the power in each waveguide as a function of z.
A z 
Pa  z  
A0
A0
 1  F sin 2  z 
2
Maximum power‐coupling efficiency…
2
Bz
Pb  z  
2
 
1
F   
2

  1    
2
2
 F sin 2  z 
Lecture 5
Slide 29
Typical Response of
Codirectional Couplers
Maximums occur at

zm 
 2m  1 m  0,1, 2,...
2

 0  F 1

 2

3 2
2
Coupling Length
The length over which maximum power is transferred to the second waveguide is called the coupling length.
Lc 



2 2  2   2

 2  F  0.2

When 1=2 (i.e.  = 0),
Lc 
Lecture 5

2
 2

3 2
2
Slide 30
15
2/11/2016
Visualization of the Terms
sin
sin2
Lc
Lc

  
Lc 

z
2


4

2 Lc
Launch
Lecture 5
Slide 31
Contradirectional
Coupling
(Bragg Grating)
16
2/11/2016
Contradirectional Coupling
In contradirectional coupling, the coupled-modes are propagating in
opposite directions.
Let the second mode be the backward propagation mode.
1  0 and  2  0
*
Reciprocity requires that 12   21
.
Lecture 5
Slide 33
Conditions for Contradirectional
Coupling
Contradirectional coupling cannot occur by simply bringing two
waveguides in proximity. Typically a grating is used to couple the
counter propagating modes.
waveguide 1
grating
waveguide 2
12  z    G e
Lecture 5
j
2
z

The mode coupling coefficient is now a periodic function.
Slide 34
17
2/11/2016
Contradirectional Coupled-Mode
Equations
The coupled-mode equations are now written as

2 
 j   2  1 
z
dA
 
  j G Be 
dz
dB
  j G Ae
dz
2

 j   2  1 



z

12     G e
*
21
j
2
z

Lecture 5
Slide 35
Phase Matching Conditions
We introduce the following phase matching condition of the grating.

1   2 
2

2
We will have three cases
Case 1:    G
Case 2:    G
Case 3:    G
Lecture 5
Pass band. Forward output. Temporary and confined peak in reflected mode.
Band edge.
Stop band. Reflected output. Band of reflection.
Slide 36
18
2/11/2016
Case 1: ||>G (Pass Band)
The mode amplitudes are:
A  z   A0
B  z   A0
 cos    z  L    j sin    z  L   j z
e
 cos   L   j sin   L 
j G sin    z  L  
 cos   L   j sin   L 
   2   G2
e  j z
The normalized forward and backward power
Pf  z  
Pb  z  
A z 
2

2
A0
Bz
2
2
A0
 2   G2 sin 2    z  L  
 2   G2 sin 2   L 
 G2 sin 2    z  L  
 2
   G2 sin 2   L 
L is the distance over which the periodic perturbation exists.
Lecture 5
Slide 37
Case 2: ||=G (Band Edge)
The mode amplitudes are:
1  j  z  L  j z
A  z   A0
e
1  j L
B  z   A0
j G  z  L   j z
e
1  j L
The normalized forward and backward power
Pf  z  
Pb  z  
Lecture 5
A z 
A0

2
Bz
A0
2
2
2
1   G2  z  L 
1   G2 L2
 2  z  L
 G 2 2
1  G L
2
2
Slide 38
19
2/11/2016
Case 3: ||<G (Stop Band)
The mode amplitudes are:
A  z   A0
B  z   A0
 cosh   z  L    j sinh   z  L   j z
e
 cosh  L   j sinh  L 
j G sinh   z  L  
 cosh  L   j sinh  L 
e
   G2   2
 j z
The normalized forward and backward power
Pf  z  
Pb  z  
A z 
A0

2
Bz
A0
2
2
2
 2   G2 sinh 2   z  L  
 2   G2 sinh 2  L 
 G2 sinh 2   z  L  
 2
   G2 sinh 2  L 
Lecture 5
Slide 39
Typical Bragg Response
RESPONSE OF A BRAGG GRATING
100%
Pass Band
  G
Pass Band
  G
Transmittance (T)
R
Stop Band
  G
T
Reflectance (R)
B  0
A0
2
A L
A0
2
2
2
Frequency (k0)
k0 neff 


 G
B  2neff 
k0 neff 


 G
Bragg wavelength
Lecture 5
Slide 40
20
2/11/2016
Non-Directional
Coupling
Non-Directional Coupling
It turns out that we can couple waves travelling in different directions. This is called non‐directional coupling.
 

K   k1  k2



k2
Grating vectors in opposite directions describe the same grating.

k1
Wave 1
Lecture 5

k2
Wave 2

K

k1

K
Grating that would couple wave 1 and wave 2
42
21
2/11/2016
Phase Matching
with Gratings
Generalized Framework
How do we couple two completely different modes so they can exchange power? Ordinarily, this will not happen.

1
Lecture 5

2
Slide 44
22
2/11/2016
Phase Matching
We can couple any two modes using a grating.


1
2

K
The phase matching condition to couple energy between two modes is


1

 


K   1   2

2

K
Lecture 5
Slide 45
Grating Coupler Regimes
Short period gratings
Bragg gratings
Contradirectional coupling
“Medium” period gratings
Non‐directional coupling
Long period gratings
Codirectional coupling
Lecture 5
Slide 46
23
2/11/2016
Mode-Matching
Vs.
Coupled-Wave
Frameworks to Model
Propagation
Both mode-matching and coupled-mode frameworks view
devices as consisting of a series of segments that are
uniform in the z-direction.
Lecture 5
Slide 48
24
2/11/2016
Mode-Matching Framework (1 of 3)
Mode matching views the field in a segment as being the
sum of a set of orthogonal basis functions (eigen-modes).
E  x
f1  x 
=
f2  x 
+
f3  x 
+
f4  x 
+
fm  x 
+
E  x    am f m  x 
m
Lecture 5
Slide 49
Mode-Matching Framework (2 of 3)
The modes within a segment accumulate phase differently
as they propagate, but they do not interact and they
propagate independently.
 j z
E  x, z    am f m  x  e m

m
complete description
of the m th eigen-mode
E  x, z 
f1  x  e j 1z
f 2  x  e  j2 z
+
f 3  x  e  j 3 z
=
+
f 4  x  e j 4 z
+
Lecture 5
Slide 50
25
2/11/2016
Mode-Matching Framework (3 of 3)
At an interface, the power redistributes itself among the
eigen-modes in the next segment.
boundary conditions
+ +
+
+
=
=
+
+
+
+
E1  x, z0   E2  x, z0 
a
f
1, m 1, m
m
 x  e j
1,m z0
  a2,m f 2,m  x  e
 j  2,m z0
m
Lecture 5
Slide 51
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
26
2/11/2016
Coupled-Wave Framework (1 of 3)
Coupled-wave views the field in a segment as being the
sum of a set of plane wave basis functions.
E  x
e
 jk x ,1 x
=
+
E  x    am e
+
+
+
 jk x ,m x
m
Lecture 5
Slide 53
Coupled-Wave Framework (2 of 3)
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.
 jk x
E  x, z    am  z  e
x ,m
m
Lecture 5
Slide 54
27
2/11/2016
Coupled-Wave Framework (3 of 3)
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
+
+
+
+
=
=
+
+
+
+
E1  x, z0   E2  x, z0 
 a  ze
1, m
m
 jk x ,m x
  a2,m  z  e
 jk x ,m x
m
Lecture 5
Slide 55
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
28
2/11/2016
How Do We Reconcile These
Two Theories?
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. Lecture 5
57
29