here - Faculty of Science - University of Waterloo

Evolution Operators and Boundary Conditions
for Propagation and Reflection Methods
David Yevick
Department of Physics
University of Waterloo
Faculty of Science - Department of
Physics 5/3/2009
D. Yevick - Evolution Operators and
Boundary Conditions
1

Collaborators
• Frank Schmidt (ZIB)
• Tilmann Friese (ZIB, University of Waterloo]
• Hatem El-Refaei, Nortel Networks
• Ian Betty, Nortel Networks
• Chunlin Yu, Queen’s University
Faculty of Science - Department of
Physics 5/3/2009
D. Yevick - Evolution Operators and
Boundary Conditions
2

Outline
• Fundamental Equations
• Non-Local Boundary Conditions
• Improving Accuracy in Fast Reflection
Calculations
Faculty of Science - Department of
Physics 5/3/2009
D. Yevick - Evolution Operators and
Boundary Conditions
3

Part I - Fundamental Equations
Faculty of Science - Department of
Physics 5/3/2009
D. Yevick - Evolution Operators and
Boundary Conditions
4

Scalar Wave Equation
• Scalar, Monochromatic Electric Field
⎛ ∂
⎜
⎝ ∂z
Defining
Y
0
2
⎛ ∂
⎜
⎝ ∂z
2
2
=
2
+
k
+
0
∂
2
∂x
k
2
n
1 ∂
2 2
n ∂y
2
0
0
0
n
∂
+
∂y
= n
2
0
2
2
( X
2
0
2
+
k
reference
, X
and N =
+ Y
0
2
0
n
2
⎞
( r )
⎟ Ε( x,
y,
z)
= 0
⎠
0
2
1 ∂
= ,
2 2 2
k0
n ∂x
0
2
n ( r )
−1,
we have
2
n
0
⎞
+ N)
⎟ E( x,
y,
z)
= 0
⎠
Faculty of Science - Department of
Physics 5/3/2009
D. Yevick - Evolution Operators and
Boundary Conditions
5

Forward Solution
• Define H = X
0
+ Y0
+ N . For forwardtravelling
waves ( time-dependence
i t
)
E( x,
y,
z
+ ∆z)
=
e
δ∆z 1+H
E( x
y
z
⎛
⎜
⎝
∂
∂z
+
e ω , , )
⎞
ik0 n0
1+
H ⎟ E( x,
y,
z)
=
⎠
0
• We then have with
δ = −ik 0 n 0
Faculty of Science - Department of
Physics 5/3/2009
D. Yevick - Evolution Operators and
Boundary Conditions
6

Modal Analysis
• Modal Decomposition
E( x,
y,
z)
= ∑ a
m
m
E
m
( x,
y,
z)
with
⎡ ∂
⎢
⎣∂x
2
2
+
∂
2
∂y
2
+
k
2
0
n
2
( x,
⎤
y,
z)
⎥ E
⎦
m
( x,
y,
z)
=
β ( x,
2
m
y,
z) E
m
( x,
y,
z)
• Approximate Forward Solution
E(
∑
−iβ
m ( x,
y,
z ) ∆z
x,
y,
z + ∆z)
= e E
m
( x,
y,
z)
m
Faculty of Science - Department of
Physics 5/3/2009
D. Yevick - Evolution Operators and
Boundary Conditions
7

Fresnel Approximation
• Fresnel Approximation
1
+
H ≈ 1+
H
2
• Slowly-Varying Envelope
E(
x,
y,
z)
= E( x,
y,
z)
e
−δ z
⎛
⎜
⎝
∂
∂z
δ
⎞
+ ( X
0
+ Y0
+ N)
⎟E(
x,
y,
z)
=
2
⎠
0
Faculty of Science - Department of
Physics 5/3/2009
D. Yevick - Evolution Operators and
Boundary Conditions
8

Wide-Angle Approximations
• Taylor Series Expansion
1 1 2 1 3 5 4
1+
H ≈ 1+
H − H + H − H + O(
H
2 8 16 128
• Padé [2,0] approximant:
1+
H
≈ 1+
H
2
H
−
8
• Padé [1,1] approximant
1+
H
1+
3H
/ 4
≈
1+
H / 4
1
= 1+
H −
2
2
1
8
H
2
+
1
32
H
3
−
1
128
H
4
5
)
+ O(
H
5
)
Faculty of Science - Department of
Physics 5/3/2009
D. Yevick - Evolution Operators and
Boundary Conditions
9

Square-Root Operator Recursion
• Recursion Relation
1+
H −1
=
=
=
( 1+
H −1)
1+
2 +
H
H + 1
H
⎛
⎜
⎝
1+
1+
( 1+
H −1)
H
H
+ 1⎞
⎟
+ 1
⎠
f
• Thus if
1 1 ) ( − + = H x
we have
f ( x)
= x /(2 + f ( x))
Faculty of Science - Department of
Physics 5/3/2009
D. Yevick - Evolution Operators and
Boundary Conditions
10

Continued Fraction Expansion
• Iterating the recursion relation yields
1+
H
−1
=
2
• Note that we have employed f ( x)
= 0
to terminate the fraction, yielding a real
expression.
+
2
+
H
H
...
2
H
H
+
2
Faculty of Science - Department of
Physics 5/3/2009
D. Yevick - Evolution Operators and
Boundary Conditions
11

Padé Representations
• The Padé approximant can be factored as
1+
H
≈
s
∏
⎡
⎢1+
sin
⎢
⎢
⎢
1+
cos
⎣
r=
1 2
⎛ rπ
⎞
⎜ ⎟H
⎝ 2s
+ 1⎠
⎛ rπ
⎞
⎜ ⎟H
⎝ 2s
+ 1⎠
• In a partial fraction representation
2
⎤
⎥
⎥
⎥
⎥
⎦
1+
H
Faculty of Science - Department of
Physics 5/3/2009
= 1+
s
∑
⎡ 2
⎢
⎢
2s
+
⎢
⎢
1+
cos
⎣
sin
1
r=
1 2
⎛
⎜
⎝
⎛ rπ
⎞
⎜ ⎟H
⎝ 2s
+ 1⎠
rπ
⎞
⎟H
2s
+ 1⎠
D. Yevick - Evolution Operators and
Boundary Conditions
2
⎤
⎥
⎥
⎥
⎥
⎦
12

Finite Difference Method
• Applying a [1,1] Padé approximant yields the
Crank-Nicholson procedure
E(
z
+ ∆z)
=
e
H
δ
2
E(
z)
=
e
δ
( X
2
0
+ Y
0
+ N )
E(
z)
≈
⎛
⎜
⎝
1
1
+
−
δ
δ
H
H
/
/
4
4
⎞
⎟E(
z)
⎠
+
O(
δ
3
)
Faculty of Science - Department of
Physics 5/3/2009
D. Yevick - Evolution Operators and
Boundary Conditions
13

Discrete Representation
• On a one-dimensional transverse grid
{ } x i
E(
x
where
D
2
x
i
E
1−
, z + ∆z)
=
1+
i
=
E
i+
1
− 2E
∆z
i
2
i∆z
4k0n
i∆z
4k
n
+
and for any operator O,
0
E
0
0
i−1
1
O
( k
( k
2
0
2
0
( n
( n
2
2
( x
( x
) − n
) − n
represents O
i
i
2
0
2
0
−1
) +
) +
.
D
D
2
x
2
x
)
)
E(
x
i
, z)
Faculty of Science - Department of
Physics 5/3/2009
D. Yevick - Evolution Operators and
Boundary Conditions
14

Part II - Nonlocal Boundary
Conditions
Faculty of Science - Department of
Physics 5/3/2009
D. Yevick - Evolution Operators and
Boundary Conditions
15

Objective
• To simulate on a finite, discrete
computational grid the field radiated from
a local source into a homogeneous semiinfinite
medium.
Faculty of Science - Department of
Physics 5/3/2009
D. Yevick - Evolution Operators and
Boundary Conditions
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Electrorefraction Modulator
Faculty of Science - Department of
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D. Yevick - Evolution Operators and
Boundary Conditions
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Standard Boundary Conditions
Faculty of Science - Department of
Physics 5/3/2009
D. Yevick - Evolution Operators and
Boundary Conditions
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Improved Boundary Conditions
Faculty of Science - Department of
Physics 5/3/2009
D. Yevick - Evolution Operators and
Boundary Conditions
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Boundary Layers
– The approximate propagation operators
introduced above are unitary. To remove the
outward propagating electric field at the
boundary we can introduce absorbing or
impedance-matched boundary layers.
z
x 1
x
N
Faculty of Science - Department of
Physics 5/3/2009
D. Yevick - Evolution Operators and
Boundary Conditions
20

Transparent Boundaries
E E N + 1
• Set 0 and to be consistent with
purely outgoing waves at the boundary.
– Local Boundary Conditions: E , 0
E N +1 are
computed from Eat the last propagation step.
– Nonlocal Boundary Conditions: E0
, E N +1 are
obtained from previous values of .
E
Faculty of Science - Department of
Physics 5/3/2009
D. Yevick - Evolution Operators and
Boundary Conditions
21

Impedance-Matched Layer
• For a non-equidistant grid, ∆ X
i
= (1 − bi
) ∆X
the governing equation in a homogeneous
refractive index layer near the boundary is
⎛
⎜−
2ik
⎝
• For continuous
• Thus, if
E
k
∂
d
⎞
)
⎟E(
x,
y,
z)
⎠
2
2 2 2
0
n0
+ + k0
( n − n
2
b 0
=
∂z
dx
x, z
b → ia
i i
x
, k
( x,
z)
z
, no spurious effects.
, we have
∝
e
ik
x
(1+
ia
i
) x+
ik
z
z
0
Faculty of Science - Department of
Physics 5/3/2009
D. Yevick - Evolution Operators and
Boundary Conditions
22

Impedance-Matched Layer
Attenuation
=
e
∑
−2k
∆x
a −2k0n
x
l
l
=
e
b
∆xsinθ
∑
l
a
l
n b
n(r
Z = L a
)
n b
θ
/ tanθ
x
L a
z
Faculty of Science - Department of
Physics 5/3/2009
D. Yevick - Evolution Operators and
Boundary Conditions
23

Approximate and Exact Results
Faculty of Science - Department of
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D. Yevick - Evolution Operators and
Boundary Conditions
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Continuous Nonlocal Boundary
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D. Yevick - Evolution Operators and
Boundary Conditions
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Continuous Nonlocal Boundary
Faculty of Science - Department of
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D. Yevick - Evolution Operators and
Boundary Conditions
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Gaussian Beam - Continuous N.L.
Faculty of Science - Department of
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D. Yevick - Evolution Operators and
Boundary Conditions
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Remaining Power - Continuous
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D. Yevick - Evolution Operators and
Boundary Conditions
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Exact Nonlocal Boundary
Faculty of Science - Department of
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D. Yevick - Evolution Operators and
Boundary Conditions
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Remaining Power - Discrete
Faculty of Science - Department of
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D. Yevick - Evolution Operators and
Boundary Conditions
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Padé [1,1] Boundary Conditions
• [1,1] Padé Approximation
−1+
1+
H ≈
• Claerbout’s Equation
H / 2
1+
H / 4
⎡⎛ 1 H ⎞ ∂ H ⎤
⎢⎜
+ ⎟ + δ ( , , )
4 2
⎥ E x y z
⎣⎝
⎠ ∂z
⎦
=
• Boundary Condition Equation ( n b
= n 0 )
⎛
⎜1
+
⎝
X
4
0
⎞⎡1−
s ⎤
⎟⎢
E(
z
z ⎥
⎠⎣
∆ ⎦
+ ∆z)
X
= − δ
4
0
(1 +
0
s)
E(
z
+ ∆z)
Faculty of Science - Department of
Physics 5/3/2009
D. Yevick - Evolution Operators and
Boundary Conditions
31

Padé [2,0] Boundary Conditions
• [2,2] Padé Equation
⎛1−
s ⎞
⎜ ⎟E
⎝ ∆z
⎠
2
⎛ X
0
X ⎞
0 ⎛ 1+
s ⎞
+ 1(
x)
= −δ
⎜1
⎟
+ − ⎜ ⎟E
j 1(
x)
2 8
⎝
⎠⎝
2 ⎠
j +
• Laplace transform this equation with
respect to x in the exterior region.
• Requiring that no poles are present in the
right-hand plane of the transform yields the
desired boundary condition.
Faculty of Science - Department of
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D. Yevick - Evolution Operators and
Boundary Conditions
32

[2,2] Boundary Condition Results
Faculty of Science - Department of
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D. Yevick - Evolution Operators and
Boundary Conditions
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Padé [N,N] Boundary Conditions
• For the [N,N] case,
g
g
g
(1)
i
(2)
i
( k −1)
i
sE
⎛ ′ 2
1 a ⎞
1
( x)
⎜ − ∂
x
= ⎟E
⎜
2
1−
a1∂
⎟
x
⎝ ⎠
⎛ ′ 2
1 a ⎞
2 x
( x)
⎜ − ∂
= ⎟g
⎜
2
1−
a ⎟
2∂
x
⎝ ⎠
i−1
(1)
i
⎛ ′ 2
1 a ⎞
1
( x)
⎜ −
k −
∂
x
= ⎟g
⎜
2
1−
ak
−1∂
⎟
x
⎝ ⎠
⎛ ′ 2
1 a ⎞
k x ( k
Ei
( x)
⎜ − ∂
= ⎟g
2 i
⎜ 1−
a ⎟
k∂
x
⎝ ⎠
i
( x)
= E
1(
x)
, where
( x)
( x)
( k −2)
i
−1)
( x)
( x)
i−
Faculty of Science - Department of
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D. Yevick - Evolution Operators and
Boundary Conditions
34

General Boundary Conditions (2)
• Introducing a vector
g
g i
(x) with
( j)
, j
( x)
= g
i
( x),
j = 1k
−1,
g
i,
k
( x)
Ei
( x)
i
=
yields
( E
2
+ A ∂ ) ( x)
x g i
with boundary conditions
g
B
g
g
= 0
B
i, +
=
+ i,
+
,
i,
−
=
+ i,
−
g
Faculty of Science - Department of
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D. Yevick - Evolution Operators and
Boundary Conditions
35

General Boundary Conditions (3)
• After Laplace transforming, this yields
(
2
E + p A) g ( p)
= A(
pg
+ g
)
or, defining C
2 = −A
−1 E ,
• Problem: Construct
of ( I + C) −1
have
ˆ
i
i,0
i,
0
(
2 2
p I − C ) gˆ
( p)
= pg
,0
+ g
, 0
i
i
C
p Rp
> 0 j
i
such that all poles
Faculty of Science - Department of
Physics 5/3/2009
D. Yevick - Evolution Operators and
Boundary Conditions
36

[N,N] Boundary Condition Results
Faculty of Science - Department of
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D. Yevick - Evolution Operators and
Boundary Conditions
37

Part III - Improving Accuracy in
Fast Reflection Calculations
Faculty of Science - Department of
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D. Yevick - Evolution Operators and
Boundary Conditions
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Facet Reflection Coefficient
E y
∂
E y
• Matching and at the boundary gives
∂z
Ψ
y
= Ψ
+
o
e
−ik
o
n
ol
L
l
z
+ Ψ
−
o
e
ik
o
n
ol
L
l
z
A
B
( 1)
E k +
yr
[ R]
TE
=
=
1 ( k
L )(
)
B
LA
E
yr
(1 −
2
E
E
yr
yi
=
n
n
oA
oA
L
L
A
A
−
−
+
E
n
n
yi
oB
oB
)
L
L
, or
B
B
Faculty of Science - Department of
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D. Yevick - Evolution Operators and
Boundary Conditions
39

Reflection Coefficients
Air
.
.
.
.
.
.
n 2
n 1
2d
nco
n cl
Waveguide Geometry
Faculty of Science - Department of
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D. Yevick - Evolution Operators and
Boundary Conditions
40

Standard Operator Results
Faculty of Science - Department of
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D. Yevick - Evolution Operators and
Boundary Conditions
41

Calculated Reflection Error
• Since the Padé approximation for L
has poles in the evanescent spectral
region, uncontrollable errors can develop.
• One method to resolve this - Generate an
approximant with complex coefficients by
selecting an imaginary termination
condition for the continued fraction
representation of 1+ H .
Faculty of Science - Department of
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D. Yevick - Evolution Operators and
Boundary Conditions
42

Complex Padé Reflection
0.45
0.41
Complex Pade
Real Pade
Reflection Power
0.37
0.33
0.29
0.25
0 2 4 6 8 10 12 14 16 18 20
Pade Order
Faculty of Science - Department of
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D. Yevick - Evolution Operators and
Boundary Conditions
43

Rotated Padé Approximants
• A second method: Write
1+
H
=
e
1+
[(1 +
x)
e
iα / 2
−iα
−1]
and perform a Padé expansion in the
variable
y
=
− α
( 1+
x)
e
i −1
Faculty of Science - Department of
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D. Yevick - Evolution Operators and
Boundary Conditions
44

Rotated Padé Reflection
0.4
0.38
Rotation Angle 0
Rotation Angle 30
Rotation Angle 60
Rotation Angle 90
Reflection Power
0.36
0.34
0.32
0.3
0 2 4 6 8 10 12 14 16 18 20
Pade Order
Faculty of Science - Department of
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D. Yevick - Evolution Operators and
Boundary Conditions
45

Refractive Index Discretization
+
Ψ in
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D. Yevick - Evolution Operators and
Boundary Conditions
46

Transition, Propagation Operator
+
Ψin
j
−
Ψin
j
+
Ψout
j + 1
−
Ψout
j + 1
P
m
=
⎛e
⎜
⎝0
− jk
o
n
om
L
e
m
jk
z
o
n
om
L
0
m
z
⎟ ⎞
⎠
T
j
=
⎛
⎜1+
1 ⎜
⎜
2 ⎜
⎜1−
⎝
n
n
n
o
n
o
o
j+
1
o
j
j
j+
1
L
L
−1
j+
1
−1
j+
1
L
L
j
j
1−
1+
n
n
n
o
n
o
o
j+
1
o
j
j
j+
1
L
L
−1
j+
1
−1
j+
1
L
L
j
j
⎞
⎟
⎟
⎟
⎟
⎟
⎠
Faculty of Science - Department of
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D. Yevick - Evolution Operators and
Boundary Conditions
47

Distributed Feedback
Normalized power
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0
0.800 0.801 0.802 0.803 0.804 0.805 0.806 0.807
Wavelength ( µ m)
Reflectivity using rotated [1/1] Padé
Reflectivity using rotated [3/3] Padé
Reflectivity using rotated [5/5] Padé
Coupled wave theory [14]
Total power using [1/1] Padé
Total power using [3/3] Padé
Total power using [5/5] Padé
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D. Yevick - Evolution Operators and
Boundary Conditions
48

Conclusions
• Procedures now exist for constructing
exact, nonlocal boundary conditions for
wide-classes of two-dimensional
parabolic partial differential equations.
• Modified Padé operators can be employed
to increase the accuracy of reflection
calculations at abrupt interfaces.
Faculty of Science - Department of
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D. Yevick - Evolution Operators and
Boundary Conditions
49