lecture 17 -- beam propagation methodemlab.utep.edu/ee5390cem/lecture 17 -- beam propagation...
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Lecture 17 Slide 1
EE 5337
Computational Electromagnetics (CEM)
Lecture #17
Beam Propagation Method These notes may contain copyrighted material obtained under fair use rules. Distribution of these materials is strictly prohibited
InstructorDr. Raymond Rumpf(915) 747‐[email protected]
Outline
• Overview
• Formulation of 2D finite‐difference beam propagation method (FD‐BPM)
• Implementation of 2D FD‐BPM
• Formulation of 3D FD‐BPM
• Alternative formulations of BPM
– FFT‐BPM
– Wide Angle FD‐BPM
– Bi‐Directional BPM
Lecture 17 Slide 2
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Lecture 17 Slide 3
Overview
Lecture 17 Slide 4
Geometry of BPM
z
x
longitudinal direction
tran
sverse direction
BPM is primarily a “forward” propagating algorithm where the dominant direction of propagation is longitudinal.
The grid is computed and interpreted as it is in FDFD. The algorithm and implementation looks more like the method of lines than it does FDFD.
BPM is implemented on a grid. There are no layers.
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Lecture 17 Slide 5
Example Simulation of a Coupled‐Line Filter
This animation is NOT of the wave propagating through the device. Instead, it is the sequence of how the solution is calculated.
Lecture 17 Slide 6
Formulation of 2D Finite‐Difference
Beam Propagation Method
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Lecture 17 Slide 7
Starting Point
We start with Maxwell’s equations in the following form.
yzxx x
x zyy y
y xzz z
EEH
y z
E EH
z xE E
Hx y
yzxx x
x zyy y
y xzz z
HHE
y z
H HE
z x
H HE
x y
Recall that we have normalized the grid according to
0x k x 0y k y 0z k z
Recall that the material properties potentially incorporate a PML at the x and y axis boundaries (propagation along z).
y xxx xx yy yy zz zz x y
x y
s ss s
s s
y xxx xx yy yy zz zz x y
x y
s ss s
s s
Lecture 17 Slide 8
Reduction to Two Dimensions
Assuming the device is uniform along the y direction,
zE
y
yxx x
x zyy y
y x
EH
z
E EH
z xE E
x y
zz zH
zH
y
y
xx x
x zyy y
y x
HE
z
H HE
z x
H H
x y
zz zE
0y
and Maxwell’s equations reduce to
yxx x
x zyy y
yzz z
EH
zE E
Hz x
EH
x
yxx x
x zyy y
yzz z
HE
z
H HE
z x
HE
x
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Lecture 17 Slide 9
Two Distinct Modes
We see that Maxwell’s equations have decoupled into two distinct modes.
E Mode
x zyy y
yxx x
yzz z
H HE
z xE
HzE
Hx
H Mode
x zyy y
yxx x
yzz z
E EH
z x
HE
z
HE
x
Lecture 17 Slide 10
Slowly Varying Envelope Approximation
Assuming the field is not changing rapidly, we can write the field as
eff eff, , , ,jn z jn zE x z x z e H x z x z e
Substituting these solutions into our two sets of equations yields
E Mode
eff eff eff
eff eff
eff eff
, , ,
, ,
, ,
jn z jn z jn zx z yy y
jn z jn zy xx x
jn z jn zy zz z
x z e x z e x z ez x
x z e x z ez
x z e x z ex
H Mode
eff eff eff
eff eff
eff eff
, , ,
, ,
, ,
jn z jn z jn zx z yy y
jn z jn zy xx x
jn z jn zy zz z
x z e x z e x z ez x
x z e x z ez
x z e x z ex
eff
eff
x zx yy y
yy xx x
yzz z
jnz x
jnz
x
eff
eff
x zx yy y
yy xx x
yzz z
jnz x
jnz
x
Not a good approximation to make for metamaterials and photonic crystals!
Better for lenses, waveguides, etc.
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Lecture 17 Slide 11
Matrix Form of Differential Equations
Each of these equations is written once for every point in the grid. This large set of equations can be written in matrix form as
E Mode H Mode
eff
eff
x zx yy y
yy xx x
yzz z
jnz x
jnz
x
eff
eff
hxx x z yy y
yy xx x
ex y zz z
djn
dzd
jndz
hh D h ε e
ee μ h
D e μ h
eff
eff
x zx yy y
yy xx x
yzz z
jnz x
jnz
x
eff
eff
exx x z yy y
yy xx x
hx y zz z
djn
dzd
jndz
ee D e μ h
hh ε e
D h ε e
Lecture 17 Slide 12
Wave Equation for E‐Mode
We can reduce the set of three equations to a single equation. This is the matrix wave equation.
eff
eff
Hx yy y
yy xx x
Ex y zz
xz
z
x
djn
dzd
jndz
D ε e
ee μ h
D e μ
h h
h
h
eff
21 2
eff eff2
11 1eff eff
2 0
H Ezz xx yy y
y y H Exx x zz x y xx
yxx y xx y
yy y
djn
dz
d dj n n
d djn jn
dz d
dz d
z
z
μ DD εe e
e eμ D μ D e μ ε I
μ I e
e
μ I e
1
1effxx
zz x
y
Ez
x
y
jnz
μ D
μ I
h
e
e
h
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Lecture 17 Slide 13
Small Angle Approximation (1 of 2)
When this is the case, we can make the “small angle approximation.”
2
2
y yd d
dz dz
e e
also called the Fresnel approximation
In many cases, the envelope of the electromagnetic field does not change rapidly as it propagates.
Rapid Variation Slow Variation
Lecture 17 Slide 14
Small Angle Approximation (2 of 2)
Given the small angle approximation2
2
y yd d
dz dz
e e
We can drop the term from our wave equation.2
2
y
z
e
2
2
yd
dze
1 2eff eff
1 2eff eff
1 2eff
eff
2 0
2 0
2
y H Exx x zz x y xx yy y
y H Exx x zz x y xx yy y
y H Exx x zz x xx yy y
dj n n
dz
dj n n
dz
d jn
dz n
eμ D μ D e μ ε I e
eμ D μ D e μ ε I e
eμ D μ D μ ε I e
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Lecture 17 Slide 15
Different Finite‐Difference Solutions
How do we approximate the z‐derivative with a finite‐difference?
First, we write our matrix wave equation more compactly as
1 2eff
eff
2
y H Ey xx x zz x xx yy
d jn
dz n
e
Ae A μ D μ D μ ε I
1
eff
11
1eff
1 11
eff
Forward Euler2
Backward Euler2
Crank-Nicolson2 2
i iy y i
i y
i iy y i
i y
i i i iy y i y i y
j
z n
j
z n
j
z n
e eA e
e eA e
e e A e A e
This is a standard finite‐difference evaluated at i+0.5
This term is interpolated at i+0.5.
Computationally simpler, but solution is nonlinear and only first‐order accurate at best. Can be unstable.
The Crank‐Nicolson method is second‐order accurate and unconditionally stable.
Lecture 17 Slide 16
Forward Step Equation
1 11
eff
1
11
eff eff
2 2
4 4
i i i iy y i y i y
i iy i i y
j
z n
j z j z
n n
e e A e A e
e I A I A e
Solving our new equation for leads to1iye
We now have a way of calculating the field in a following slice based only the field in the previous slice.
Backward waves are ignored in this formulation.
Note: We need to make a good guess for the value of neff.
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Lecture 17 Slide 17
Implementationof 2D FD‐BPM
Lecture 17 Slide 18
Grid Schemes for BPM
Periodic Structures Finite Structures
PBCP
BC
No BC Needed!
No BC Needed!
No BC Needed!
No BC Needed!
Dirich
letBCD
irichletBC
PMLP
ML
Spacer
Spacer
No spacer needed!
No spacer needed!No spacerneeded!
No spacerneeded!
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Lecture 17 Slide 19
The Effective Refractive Index, neffRecall the slowly varying envelop approximation.
eff eff, , , ,jn z jn zE x z x z e H x z x z e
The BPM does not calculate neff. We must tell BPM what is neff.
How do we know neff without modeling the device?
We have to calculate it or estimate it.
Techniques:
1. For plane waves and beams, calculate the average refractive index in the cross section of your grid to estimate the longitudinal wave vector.
2. For waveguide problems, calculate the effective index of your guided mode rigorously and use that in BPM.
Compute field in next plane
1 1i i i i e P e
Compute source at first plane
Ey(:,1) = ?;
Compute matrix derivative operators
and e hx xD D
Build deviceon grid
Calculate optimized grid
Lecture 17 Slide 20
Block Diagram of BPM
Define simulation parameters
Compute propagator P
1
1 1eff eff
12eff
4 4i i i i
i i i ih ei xx x zz x xx yy
j z j z
n n
n
P I A I A
A μ D μ D μ ε I
Post processDone?
Loop over z
BPM can handle modes, beams, plane waves, etc., very easily.
Extract slice & diagonalize. and i iμ ε
yes
no
Chooseneff
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Lecture 17 Slide 21
Formulation of 3D Finite‐Difference
Beam Propagation Method
Lecture 17 Slide 22
Starting Point
We have the same starting point as with 2D FD‐BPM.
yzxx x
x zyy y
y xzz z
EEH
y z
E EH
z xE E
Hx y
yzxx x
x zyy y
y xzz z
HHE
y z
H HE
z x
H HE
x y
Recall that we have normalized the grid according to
0x k x 0y k y 0z k z
Recall that the material properties potentially incorporate a PML at the x and y axis boundaries (propagation along z).
y xxx xx yy yy zz zz x y
x y
s ss s
s s
y xxx xx yy yy zz zz x y
x y
s ss s
s s
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Lecture 17 Slide 23
Slowly Varying Envelope Approximation
Assuming the field is not changing rapidly, we can write the field as
eff eff, , , ,jn z jn zE x z x z e H x z x z e
Maxwell’s equations become
Not a good approximation to make for metamaterials and photonic crystals!
Good for lenses, waveguides, etc.
eff
eff
yzy xx x
x zx yy y
y xzz z
jny z
jnz x
x y
eff
eff
yzy xx x
x zx yy y
y xzz z
jny z
jnz x
x y
yzxx x
x zyy y
y xzz z
EEH
y z
E EH
z xE E
Hx y
yzxx x
x zyy y
y xzz z
HHE
y z
H HE
z x
H HE
x y
Lecture 17 Slide 24
Matrix Form of Differential Equations
Each of these equations is written once for every point in the grid. This large set of equations can be written in matrix form as
eff
eff
yey z y xx x
exx x z yy y
e ex y y x zz z
djn
dzd
jndz
eD e e μ h
ee D e μ h
D e D e μ h
eff
eff
yhy z y xx x
hxx x z yy y
h hx y y x zz z
djn
dzd
jndz
hD h h ε e
hh D h ε e
D h D h ε e
eff
eff
yzy xx x
x zx yy y
y xzz z
jny z
jnz x
x y
eff
eff
yzy xx x
x zx yy y
y xzz z
jny z
jnz x
x y
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Lecture 17 Slide 25
Eliminate Longitudinal Components
We solve the third equation in each set for the longitudinal components ez and hz.
1 e ez zz x y y x
h μ D e D e 1 h h
z zz x y y x
e ε D h D h
We now substitute these expressions into the remaining equations.
1eff
1eff
ye h hy zz x y y x y xx x
e h hxx x zz x y y x yy y
djn
dzd
jndz
eD ε D h D h e μ h
ee D ε D h D h μ h
1eff
1eff
yh e ey zz x y y x y xx x
h e exx x zz x y y x yy y
djn
dzd
jndz
hD μ D e D e h ε e
hh D μ D e D e ε e
Lecture 17 Slide 26
Rearrange Terms
We rearrange or remaining finite‐difference equations and collect the common terms.
1 1eff
1 1eff
e h e hxx x zz y x yy x zz x y
y e h e hy xx y zz y x y zz x y
djn
dzd
jndz
ee D ε D h μ D ε D h
ee μ D ε D h D ε D h
1 1eff
1 1eff
h e h exx zz y x yy x zz x y x
y h e h exx y zz y x y zz x y y
djn
dzd
jndz
hD μ D e ε D μ D e h
hε D μ D e D μ D e h
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Lecture 17 Slide 27
Block Matrix Form
We can now cast these four matrix equations into two block matrix equations.
1 1eff
1 1eff
e h e hxx x zz y x yy x zz x y
y e h e hy xx y zz y x y zz x y
djn
dzd
jndz
ee D ε D h μ D ε D h
ee μ D ε D h D ε D h
1 1eff
1 1eff
h e h exx zz y x yy x zz x y x
y h e h exx y zz y x y zz x y y
djn
dzd
jndz
hD μ D e ε D μ D e h
hε D μ D e D μ D e h
1 1
eff 1 1
e h e hx zz y yy x zz xx x x
e h e hy y y xx y zz y y zz x
djn
dz
D ε D μ D ε De e hP P
e e h μ D ε D D ε D
1 1
eff 1 1
h e h ex zz y yy x zz xx x x
h e h ey y y xx y zz y y zz x
djn
dz
D μ D ε D μ Dh h eQ Q
h h e ε D μ D D μ D
Lecture 17 Slide 28
Matrix Wave Equation
We derive the matrix wave equation by combining the two block matrix equations. First, we solve the first block matrix wave equation for the magnetic field term.
1eff
x x x
y y y
djn
dz
h e eP
h e e
Second, we substitute this into the second block matrix equation.
1 1eff eff eff
22
eff eff eff2
2
2
x x x x x
y y y y y
x x x x x
y y y y y
x
d d djn jn jn
dz dz dz
d d djn jn n
dz dz dz
d
dz
e e e e eP P Q
e e e e e
e e e e ePQ
e e e e e
e
e 2eff eff2 x x
y y y
dj n n
dz
e eI PQ
e e
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Lecture 17 Slide 29
Small Angle Approximation
Assuming the fields to not diverge rapidly, then the variation in the field longitudinally will be slow. This means
2
2
x x
y y
d d
dz dz
e e
e e
This means we can drop the term from our wave equation.2
2
x
yz
e
e
also called the Fresnel approximation
2
2
x
y
d
dz
e
e
2eff eff
2eff
eff
2
1
2
x x
y y
x x
y y
dj n n
dz
dn
dz j n
e eI PQ
e e
e eI PQ
e e
Lecture 17 Slide 30
Explicit Finite‐Difference Approximation
We explicitly approximate the z‐derivative with a finite‐difference.
eff
1 1 1
eff
1
2
1
2 2i i i i i i
d
dz j n
z j n
eAe
e e A e A e
This is a standard finite‐difference evaluated at i+0.5
First, we write our matrix wave equation more compactly as
2eff
eff
1
2x
y
dn
dz j n
eeAe e A I PQ
e
, 2eff ,
,
x ii i i i i
y i
n
ee A I PQ
e
This term is interpolated at i+0.5.
Note: This is called the Crank‐Nicolson scheme because it is a central finite‐difference.
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Lecture 17 Slide 31
Forward Step Equation
1 1 1
eff
1
1 1eff eff
1
2 2
4 4
i i i i i i
i i i i
z j n
z z
j n j n
e e A e A e
e I A I A e
Solving our new equation for leads to1ie
We now have a way of calculating the field in a following slice based only the field in the previous slice.
Backward waves are ignored in this formulation.
Note: We need to make a good guess for the value of neff.
Lecture 17 Slide 32
Alternative Formulationsof BPM
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Lecture 17 Slide 33
FFT‐BPM
The FFT based BPM was the first BPM. It was essentially replaced by FD‐BPM because FFT‐BPM has the following disadvantages:
• Simulations were slow (FFTs are computationally intensive)• Discretization in the transverse dimension must be uniform • No transparent boundary condition could be used• Very small discretization widths were not feasible• Polarization cannot be treated• Inaccurate for high contrast devices• Propagation step must be small
one layer
propagate plane waves
propagate plane waves
Introduce material phase in real‐space.
Algorithm to Propagate One Layer1. FFT the fields to calculate plane
wave spectrum.2. Add phase for one half of layer
to plane waves according to their longitudinal wave vector.
3. Inverse FFT at mid‐point to reconstruct the real‐space field.
4. Introduce the phase due to the materials in the layer.
5. Repeat steps 1 to 3 for the second half of the layer.
M. D. Feit, J. A. Fleck, Jr., “Light propagation in graded-index optical fibers,” Applied Optics, Vol. 17, No. 24, December 1978.
Lecture 17 Slide 34
Wide Angle FD‐BPM – Recurrence Formula
The wave equation before the small angle approximation was made was:
2
2eff eff2
12 0y y y
xx xx yy yzz
j n nz z x x
This can be written more compactly as2
2eff eff2
12 0 y y
y xx xx yyzz
j n A A nz z x x
We can rearrange this differential equation to derive a recurrence formula for the derivatives.
eff
eff
21
12
y y
A
j n
zj n z
eff
1eff
21
12
m
m
Aj n
zj n z
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Lecture 17 Slide 35
Wide Angle FD‐BPM – Padé Approximant Operators
We initialize the recurrence with
1
0z
0th Order (Small Angle)
eff
0 eff
1eff
21 2
12
Aj n A
jz n
j n z
1st Order (Wide Angle)
eff eff
12eff0eff
2 21 11
42
A Aj n n
jAznj n z
2nd Order (Wide Angle)2
3eff eff eff
22eff1eff
2 2 81 11
22
A A Aj n n n
jAznj n z
The numerator and denominator are polynomials of the operator A of orders N and D respectively.
This leads to the Padé Approximate operators denoted as Padé(N, D)
Lecture 17 Slide 36
Wide Angle FD‐BPM – Implementation
Previously we solved by setting . This
was the small angle approximation.
For wide angle BPM, we instead solve where N(A)
and D(A) are the polynomials of A.
This equation is usually implemented with finite‐differences using
the “multi‐step” method.
2
eff22 0y y
yj n Az z
2
20y
z
yy
N Aj
z D A
G. Ronald Hadley, “Wide-angle beam propagation using Padé approximant operators,” Optics Lett., Vol. 17, No. 20, October 1992.
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Lecture 17 Slide 37
Bi‐Directional BPM
The beam propagation method inherently propagates waves in only the forward direction.
It is possible to modify the method so as to account for backward scattered waves.
This is accomplished in a manner similar to how we derived scattering matrices.
By the time BPM is modified to be bidirectional and wide‐angle, it approaches being a rigorous method. The implementation, however, is tedious. At this point, use the method of lines which is fully rigorous and has a simpler implementation.
Hatem El-Refaei, David Yevick, and Ian Betty, “Stable and Noniterative Bidirectional Beam Propagation Method,” IEEE Photonics Technol. Lett, Vol. 12, No. 4, April 2000.