BPM anglais
Transcript of BPM anglais
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components. Boundary conditions are also implemented to reduce the effect of the
simulation space limits on the solution, and so create a practical algorithm.
First a brief look is taken at the Silicon on Insulator, ASOC style structures
analysed in this work. Then the principles of operation of the FD algorithms is
summarised.
3.2 Waveguide Structures and their Simulation using FD methods
This section introduces the ASOC-style structures simulated using FD
methods in this work, and then summarises the principles of the FD methods
introduced in this chapter.
3.2.1 Types of Waveguide Structure
Waveguides are structures that guide light to form components in Opto-
Electronic Integrated Circuits (OEICs). Some of the types of waveguide that have
been developed for optoelectronics are displayed in Figure 3.1. This work primarily
considers rib waveguide structures, but buried waveguides are also considered in
Chapter 7 and Effective Index (EI) approximations (introduced later in this section)
of 3D structures to a 2D slab waveguide are used as a useful approximation.
Figure 3.1 : Types of dielectric waveguide: (a) Rib Waveguide, (b) Buried Waveguide, (c) slab
waveguide. Approximate field profiles for the fundamental mode are superimposed.
(a) (b) (c)
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3.2.2 Modes in a dielectric waveguide
A mode is a characteristic pattern of light that will propagate in a z-invariant
optical structure (where z is the direction of propagation). Superimposed on Figure
3.1 are the approximate fundamental mode patterns supported by such structures.
Waveguides are often designed to be single-moded, however these and other
structures often support higher order modes, i.e. fields with more than one peak.
Each mode has a characteristic field pattern and propagation properties, i.e. the
propagation constant (or Effective Index), (or neff ), and attenuation characteristics,
measured as the imaginary part of the propagation constant, or as an attenuation in
decibels per centimetre (dB/cm). Structures that support multiple modes are studied
in detail in Chapters 6 and 7.
3.2.3 The large area- SoI / ASOC waveguide
Silicon is a well developed material for electronic applications, with well
established manufacturing processes Together with promising optical properties,
such as a material loss of less than 0.5dB/cm at optical wavelengths of 1.3 to l.55m,
encompassing the normal operating wavelength-range of long distance optical
communications, it proves an interesting material in the development of monolithic
opto-electronic circuits [3.6]. Several proposals have been made as to how the silicon
waveguide may be best realised, but the Silicon on Insulator method of fabrication,
suggested in [3.7], is the most promising, since it allows true monolithic integration
with electronic circuits. The dimensions of the waveguide, designed in [3.7], are very
small, causing problems with fibre coupling and exhibiting fairly high fundamental
loss of around 5dB/cm. However, in [3.8], single-mode silicon on insulator
waveguides are designed and fabricated with dimensions of several micrometers,
allowing efficient fibre coupling, and low fundamental mode loss of less than
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0.5dB/cm in waveguides of rib width greater than 3m. It is on this type of structure
that ASOC technology [3.9], and the waveguides studied in this work, are based.
Figure 3.2 shows the basic waveguide structure that is studied in detail in this work.
The dimensions and refractive indices are an approximation to a known ASOC
waveguide used in Arrayed Waveguide Grating (AWG) structures. The ASOC
structure is designed to be single-moded.
Figure 3.2: The structure studied in this work
At several points throughout this work it proves beneficial to use a semi-
analytical 2D slab approximation of the 3D structure shown in Figure 3.2, called the
Effective Index approximation [3.10], introduced next.
3.2.4 The Effective Index Approximation
The Effective Index approximation [3.10] is used in this work to generate a
2D slab structure with properties similar to the 3D structure in Figure 3.1. The
Effective Index approximation is created by splitting the 3D structure into a series of
vertical slabs, Figure 3.3(a), and using a slab-modesolver on each vertical slab. The
modal index ( neff ) found for each slab, calculating the fundamental scalar slab mode
Figure 3.3(b), is subsequently used to construct a 2D slab structure, Figure 3.3(c).
The Effective Index approximation is widely accepted to be accurate enough for fast
0.5m
2.5m
4.5m
4m n1 = 3.48, Si
n2 = 1.45, SiO 2
n3 = 3.48, Si
n0 = 1.00, Air
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The FD mode solver and FD-BPM sample both the structure and field in
analysis. Sampling is most commonly performed at regular spatial intervals, but in
some cases the sampling interval is changed dynamically, allowing denser sampling
where required (i.e. as implemented in [3.12]). Throughout this work, regular
sampling is used.
Figure 3.4 shows how the Cartesian FD-BPM solver propagates, in the 2D
case, (a), and the 3D case, (b). From a known set of present points, forming a line
in 2D implementations Figure 3.4(a) or a plane in 3D implementations Figure 3.4(b),
the field at the next set of points is calculated. Knowledge of the structure at all
points is required. This process is repeated for each new set of known field points,
and so the simulation propagates in the z-direction.
(a) (b)
Figure 3.4 : (a) The 2D BPM algorithm calculates the next field from the present field. (b)
The 3D algorithm propagates a plane, rather than a line, of sample points along the z-
propagation direction.
The Mode solver determines the characteristic light patterns, or modes, that
will propagate in a structure that is invariant in the z-direction. Examples of 2D and
3D fundamental modes and the respective structures they propagate in are shown in
Figure 3.5 and Figure 3.6. The mode solver can be set up to find the complex
TransverseDirection (x)
PropagationDirection (z)
NextSample Points
PresentSample points
n1
n2
y
x z
n1
n2n3
n4
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propagation constants (phase velocity and attenuation coefficient), , as well as the
mode-patterns in the structure.
Transverse Position (x), m
Figure 3.5 : The normalised field magnitude of the fundamental mode of a 2D slab waveguide.
Calculated using an FD mode Solver. = 1.528m
Figure 3.6 : Cross-sectional scalar field magnitude pattern of the Fundamental Mode supported
by the illustrated z-invariant waveguide. Calculated using an FD Mode Solver. Contours at 10%
field intervals. = 1.528m
The next section derives the basic FD-BPM algorithm. Following sections in
this chapter derive an FD Mode Solver algorithm, and then implement both solvers
n1 = 1.0
n2 = 3.48n3 = 1.45
n4 = 3.48
n1 = 3.46804 n 2 = 3.47513 n1 = 3.46804
N o r m a l
i s e d
E l e c t r i c
F i e l d S t r e n g
t h
Index
Discontinuity
0
1
0 5 10 14.5 24.5
T r a n s v e r s e
P o s
i t i o n
( S a m p l e s
) y -
d i r e c t
i o n .
y = 0 . 0
5 m
Transverse Position (Samples) x-direction. x = 0.05m
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into practical computer algorithms, including the application of Boundary Conditions
(BCs), necessary to cope with the edges of the simulation space.
3.3 Derivation of the FD-BPM Algorithm
3.3.1 Maxwells Equations
The Finite Difference Beam Propagation Method (FD-BPM) is based on
Maxwells Equations [3.3]. In a source free region, and assuming a periodic time
variation, e j t , they are:
H j E = (3.1a)
E j H = (3.1b)
0= E (3.1c)
0= H (3.1d)
where r 0= and r 0= , and the vector quantity E (V/m) is the electric field
vector and H (A/m) is the magnetic field vector and a periodic time variation e j t has
been assumed. The quantities and define the electromagnetic properties of the
medium, and are the dielectric constant and the magnetic permeability of the
medium, respectively. 2120 10854.8= Fm x is the dielectric constant in a vacuum
and 270 104= Hm x the magnetic permeability in a vacuum. r and r are the
relative permittivity and permeability of the material. In all cases during this work
r =1 since only non-magnetic materials are considered. When analysing the optical
properties of a material, it is convenient to work with its refractive index, n , which is
defined as r r n = .
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3.3.2 The Wave Equation
The total electromagnetic field that is supported by a waveguide can be
expressed in terms of only the electric or magnetic field components to produce wave
equations. In this section the derivation that models the electric, rather than magnetic
field is considered. In this case, the magnetic field is removed from the derivation by
taking the curl of equation (3.1a) and substituting in (3.1b), i.e.
E
H j E
2
)(
==
(3.2)
For convenience, we define 0022
0 =k and 2022 k nk = where k is known as the
wave-number. Note that the speed of light in free space,00
1
=c . In free space,
2
0 =k .
With these definitions, (3.2) becomes
E k E 2= (3.3)
To simplify the LHS of the equation, the vector identity (3.4) is used:
)(2 A A A += (3.4)
so that (3.3) becomes:
E k n E E 2022 )( =+ (3.5)
Eq. (3.5) is the electric field vector wave equation. A wave equation expressed in
terms of the magnetic field can be derived similarly.
3.3.3 SCALAR approximation
The scalar approximation is defined such that neither the gradient nor the
magnitude of the field changes across a dielectric boundary. The vector field, E , is
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replaced by the scalar field In (3.1(c)), is assumed to not be a function of (x,y,z),
such that 0= and (3.5) becomes:
020
22 =+ k n (3.6)
This scalar analysis is considered sufficiently accurate for the initial analysis of a
particular waveguide structure, with no polarisation information. However for more
detailed and accurate analyses, a polarised and/or vector analysis of the component is
required, as described in the following sections.
3.3.4 Full Vector formulation and Polarised Approximations
Full vector and polarised analyses provide a more detailed analysis of an
optical structure. Taking (3.1c), the equation is split into transverse and propagation
direction components (3.7).
0=+=
+
+
=
z t t
z y x
E z
E
E z
E y
E x
E
(3.7)
(in a sourceless region). In FD-BPM, the variation of is assumed locally invariant
in the propagation direction (i.e. the structure changes slowly). Under the assumption
(3.8) can be assumed true, all assumptions that will subsequently allow the z-
derivative to be eliminated later on.
z t t E
z E
=
(3.8)
The wave equation, (3.5) is split, and the transverse components considered, i.e.
0)(22 =+ E x
E k E x x (3.9a)
and
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0)(22 =+ E y
E k E y y (3.9b)
where
z
E
y
E
x
E E z y x
+
+
= .The z-derivative in E is removed using (3.8) so
that
t t y x E
y
E
x E
E
+
= (3.10)
Combining (3.10) and (3.9 a, b) and expanding gives
011
2
22
2
2
2
2
2
2
=
+
+
+
+
+
y
E
x x
E
x y
E
x x
E E k
z
E
y
E
x
E y x y x x
x x x
(3.11a)
and
011
2
22
2
2
2
2
2
2
=
+
+
+
+
+
x E
y y
E
y y
E
x E
y E k
z
E
y
E
x
E x y y x y
y y y
(3.11b)
which can be written in matrix form for clarity, in (3.11c)
=
++
+
+
+
+
0
0
11
1
1
22
2
2
2
22
2
2
2
y
x
rmCouplingTe
rmCouplingTe
E
E
M
k z y y x x y x y
y x y x
L
k z y x x
4 4 4 4 34 4 4 4 214 4 4 34 4 4 21
4 4 4 84 4 4 764 4 4 4 84 4 4 4 76
(3.11c)
These coupled equations could now be developed into a full-vector FD-BPM
algorithm.
Considering (3.11c) in conjunction with a typical dielectric waveguide (e.g.
that shown in Figure 3.6) it is seen that for coupling to occur the symmetry of the
waveguide (and the modes it supports) must be broken. In polarisation rotation
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grid is usually, but not exclusively, uniform [3.12]. The Crank Nicholson method is
based on the discrete approximations of the derivatives in (3.12). The following
sections describe its implementation.
3.4.2 The Slowly Varying Envelope Approximation (SVEA)
The Slowly Varying Envelope Approximation (SVEA) assumes that, since
the simulation follows the propagation of light in the structure, the optical field can
be defined in terms of its envelope and rapid phase components, i.e. (in the TE case):
z j x x
be z y x z y x E = ),,(),,( (3.14)
where x is the envelope of the electric field, E x and b becomes the background
index of the BPM simulation. The SVEA allows larger steps in the propagation (z)
direction to be taken, i.e. of the same order of magnitude as the wavelength. Without
the SVEA the propagation step-length would typically be limited to a maximum of
around 1/10 th the wavelength, as with FD-TD methods [3.15]. The first and second z-
derivatives of (3.14) are as follows:
( ) z j x xb z j x bb e z je z
+=
(3.15a)
( ) z j xb xb x z j x bb e z j z e z
=
2
2
2
2
2
2 (3.15b)
In all orthogonal co-ordinate systems phase is assumed invariant with the transverse
parts. We can re-write (3.12) in full, dividing through with the fast phase term, as
(3.16):
021 22
2
2
2
2
=
+
+
+
xk z j
z y x x
(3.16)
This is the full TE BPM algorithm. The TM version is similarly derived. Note that
there is a second derivative to be solved in the x,y and z directions. In practical
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terms, direct calculation of the second derivative in the z-direction is problematic,
since BPM based on three z field points (required to calculate the second z-
derivative) is normally unstable [3.16]. Normally a Pad approximation is used, as
summarised next.
3.4.3 Paraxial Approximation (Pad order 0)
The simplest and fastest method is to assume that 022
z
, which is accurate
for when the envelope of the electric field changes slowly in the z-direction. In this
case, (3.16) simplifies to (3.17):
021 22
2
2
=
+
+
xk z j
y x x
(3.17)
This proves accurate for cases where the waveguides are weakly guiding, or where
the chosen of the simulation is very close to the actual propagation constant of the
propagating mode [3.1]. This approximation can becomes inaccurate in the case of structures guiding light at a large angle to the assumed propagation direction (wide
angle situations). In some cases, a higher order approximation of 22
z
is required, as
summarised in the next section.
3.4.4 Higher Pad Orders (Wide Angle BPM)
It is possible to improve the tolerance of the solver to wide angle (i.e. non-
paraxial) propagation by improving the approximation to 22
z
[3.17]. This allows
more accurate calculation of the propagation of light at large angles from the
assumed simulation propagation direction. However, this comes at a cost to increased
memory use and reduced simulation speed and consequently wide angle methods are
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Wide angle BPM significantly increases the number of calculations, and therefore
the calculation time per propagation step. In this work a different method using
structure related co-ordinate systems reduces the requirement for wide-angle BPM,
which is consequently not studied in detail. Now the BPM equations have been
defined, in the next section we implement the paraxial equations into a practical
algorithm.
3.4.5 Calculation of the Transverse & z - Derivatives
The BPM algorithm is based on the discrete sampling of the structure and
field, and calculating the derivatives in the chosen BPM equation. In this case the
polarised paraxial version is implemented (3.17).
Figure 3.7 : Sampling points used to calculate the x & y transverse derivatives.
Figure 3.7 shows the 5-points used to calculate the x and y derivatives at each point
(u,v). Calculation of the field derivatives in this way causes the solver to be most
accurate when dielectric boundaries fall exactly half-way between sample points
[3.18]. Where one or more of the points falls outside the sample area, it will be
assumed for the time being that the field and refractive index value at that point is
n(u-1,v)(u-1,v)
n(u,v)(u,v) n(u+1,v)
(u+1,v)
n(u,v-1)(u,v-1)
n1 n2
n(u,v+1)
(u,v+1)
x
y
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zero. This creates an artificial reflective boundary around the edges of the simulation
which could affect the accuracy of the simulations. This is addressed later in Section
3.7.
Since for now we are dealing with the TE polarisation (principal electric field
E x), the y-direction in this case is a straight-forward field derivative, since the field
and its gradient are continuous across dielectric boundaries. In the x-direction the
field is discontinuous across boundaries, and so the variation of the local refractive
index has to be taken into account in the calculation. So, for a point (u,v), calculating
the y-direction differential
22
2 )1,(),(2)1,(),( y
vuvuvu y
vu x x x x
++
(3.24a)
For the x-direction
( )
( )2
22
22
22
22
)),(),1((),1(),1(),(),(2
)),(),1((),(),(),1(),1(2
),(1
xvunvun
vuvunvuvun
vunvunvuvunvuvun
vu x x
x x
x x
x
+
++++
(3.25a)
To calculate the TM polarisation the dx and dy derivatives are effectively swapped
round, so the y-direction experiences a discontinuity across boundaries. I.e. for the y-
direction
( )
( )
2
22
22
22
22
)),()1,((
)1,()1,(),(),(2
)),()1,((
),(),()1,()1,(2
),(1
yvunvun
vuvunvuvun
vunvun
vuvunvuvun
vu y y
y y
y y
y
+
++++
(3.24b)
For the x-direction
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22
2 )1,(),(2)1,(),( vuvuvuvu y y y y
++
(3.25b)
Finally, when calculating the scalar field, the field is assumed continuous in
magnitude and gradient across all boundaries, so
22
2 )1,(),(2)1,(),( y
vuvuvu y
vu
++
(3.24c)
and
22
2 )1,(),(2)1,(),( x
vuvuvuvu
++
(3.25c)
In the case of 2D FD-BPM, the y-derivative does not exist, so only the x-derivative is
considered.
Shown above is the most basic method of deriving the required difference
equations. The consequence of calculating the second order differential in this way is
that dielectric boundaries are simulated with a zero-order error [3.2]. This is
addressed in Chapter 5 where Improved FD methods are implemented.
Now the transverse derivatives can be calculated, estimation of the z-step will
complete the practical BPM algorithm. Since the calculation of the z-differential is
most accurate exactly half-way between the sample points, the transverse (x,y)
second order differentials should be calculated at the same point. A simulation
weighting, , is introduced here, which allows the simulation to be adjusted for
stability purposes. The choice ( )5.0= sets the point of calculation half-way
between z-samples and is most accurate. (3.17) becomes:
( )
( )
+
++
=
+
+
+
2
222
2
222
12
1)1()1(
2
y x xk
z j
y x xk
z j
z xb
z xb
z z xb
z z xb
(3.26)
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It is noted that x is in fact an array of all points in the transverse plane, and (3.26) is
more clearly written in matrix form. In the algorithm based on (3.26) the transverse
sample points are scanned column by column, as in Figure 3.8.
Figure 3.8 : Scanning the grid of sample points.
The result is a sparse matrix problem, the matrices consisting of 5 diagonal non-zero
lines, (3.27). Each line will exhibit a regular pattern of zeros, representing a point
outside of the simulation area (addressed further in Section 3.7). Since simulations
are likely to be wider than taller in terms of sample points, vertical, rather than
horizontal scanning reduces the width of the diagonal band. This has memory
benefits, as will be covered later in the chapter. The matrix can be stored as five
vectors, one for each diagonal line, optimising the algorithms use of computer
memory. In 2D the matrix problem is reduced to a tri-diagonal problem. This has the
advantage of being solvable directly, so iterative solvers are not required and it is
much faster.
In the next section we derive the equivalent matrix problem for the mode
solver. Then we examine how the matrix problems are solved.
x
y
N - 1
0
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=
+
+
+
+
+
+
++++
+
++++
+++
z n
z
z n
z n
z n
z n
z y
z y
z y
z y
z y
z z z z
z z z
z z n
z z
z z n
z z n
z z n
z z n
z z
y
z z
y
z z
y
z z
y
z z y
z z z z z z z z
z z z z z z
C T L
B
BC T L
R
R BC T
R BC
C T L
B
BC T L
R
R BC T
R BC
1
0
111
2
1
1111
000
1
0
111
2
1
1111
000
..
....
....
...
....
....
..
....
....
...
....
....
(3.27)
( ) ( ) ( )
+
+
+
=
+ )(2
)(222
222
2
222
2
222
m ym
m
m ym
mbm
b z m nn x
nnn x
n y
k z
jC
2
1 y
BT z m z
m == ( )
)(2
222
2
m ym
ym z m nn x
n L
+=
( )
)(2
222
2
m ym
ym z m nn x
n R
+=
+
+
( ) ( ) ( ) ( )
+
+
=
+
+
)(2
)(22
12
222
2
222
2
222
m ym
m
m ym
mbm
b z z m nn x
nnn x
n y
k z
jC
( ) 21
1 y
BT z z m z z
m == ++
( ) ( ))(
21 222
2
m ym
ym z z m nn x
n L
+=
+ ( ) ( ))(
21 222
2
m ym
ym z z m nn x
n R
+=
+
++
3.5 The FD Mode Solver
3.5.1 Introduction
In the last section we derived the FD-BPM algorithm, which will be used to
simulate an arbitrary field profile propagating in an arbitrary structure. An FD mode
solver is developed here to examine the modes that are able to propagate in the
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structure in question. In Chapters 6 and 7 of this thesis the mode-solver will prove
essential in the analysis of complex coupled waveguide structures.
Different types of FD mode-solver exist. The imaginary distance method of
mode solving [3.19], which is similarly derived to the BPM algorithm, allows
efficient solving of structures when combined with other methods, such as the
Alternating Direction Implicit method [3.20]. The Imaginary Distance method works
well provided the structure is not heavily multi-moded and the lowest-order modes
are the ones of interest, since this type of solver will naturally converge to the lowest
order mode. In the investigation of the large and complex structures, such as the
AWG, that potentially support large numbers of high-order modes (see Chapter 7), a
mode solver is required where it is possible to control convergence to any mode
supported by the structure, hence in this case the imaginary distance solver is not
suitable.
Another method common in the modal solution of complex structures utilises
the Shifted Inverse Power Method (SIPM) (as used in [3.21]), which solves
eigenvector problems, converging to the eigenvector with the closest corresponding
eigenvalue to a background value set at runtime. Consequently this method allows
the direct solution of any supported mode of the structure. With some manipulation
of the wave equation, (3.12), an eigenvalue problem can be created where the modal
propagation constant ( ) is the eigenvalue, and the corresponding field the
eigenvector. Convergence of the system relies on the accuracy with which the matrix
problem is solved. The creation of an Eigenvalue problem suitable for the SIPM is
detailed in the next section.
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3.5.2 Creating the Eigenvalue Problem, (TE)
The starting point is the TE version of the wave equation, (3.12). As before, TM,
Scalar and Magnetic Field versions of the solver are similarly derived. In this case
the aim is to create an eigenvector problem, suitable for the SIPM, where the vector
is a field (mode) contained in the structure, and the eigenvalue the propagation
constant of that mode. Let us start with (3.12), re-written in full, i.e.:
01 2
2
2
2
2
=
+
+
+
x E k z y x x
(3.12)
The solution will come in the form of the following function, i.e. an envelope and a
complex phase term
z j x e y x F E
= ),( (3.28)
where is the propagation constant of the mode described by F(x,y). The structure is
invariant and therefore the envelope of the mode is z-invariant, so
),(
),(
22
2
y x F e z E
y x F e j z
E
z j x
z j x
=
=(3.29 a,b)
Removing the fast phase term to consider the envelope, and substituting this into
(3.12) we can create an eigenvector problem, i.e.
x xE E k
y x x22
2
21
=
+
+
(3.30)
When implemented as the BPM solver in section (3.3.4) the left-hand side becomes a
band-diagonal sparse matrix problem similar to the matrix in (3.27), or in the case of
a 2D slab (Figure 3.3), a tri-diagonal matrix problem . This eigenvector problem can
be solved using the Shifted Inverse Power Method [3.21], the implementation of
which is described next.
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3.5.3 The Shifted Inverse Power Method
In general, A Matrix M has m eigenvalues and eigenvectors. The eigenvalue
i and the eigenvector v i are a solution of M such that :
Mv i = i vi (3.31a)
Note that when considering 3.30,
+
+
= 22
21k
y x xM
, E x = v i , i = 2.
Considering (3.31a), it can be said that:
(M- i)v i = 0 (3.31b)
Consider an arbitrary vector, V. V will consist of a weighted sum of the eigenvectors
of M, i.e.:
=
=m
iiivC V
1
(3.32)
with co-efficients, C. Consider the sequence of solutions defined by the equation
( ) nn V V M = +1 (3.33a)
where is an input approximation to an eigenvalue. I.e. when considering (3.30)
= 2m, where 2m is the background propagation constant of the modesolver. (3.33a)
expanded is (3.33b)
( ) im
i
nii
m
i
ni vC vC M
==
+ =11
1 (3.33b)
Due to the fundamental relationship in (3.31a) it can be said that
( ) im
i
n
iii
m
i
n
i vC vC ==+
= 111
(3.34)
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and since eigenvectors are independent, from (3.34)
( ) niini C C =+ 1 (3.35a)
so
( ) =+
i
nin
iC
C 1 (3.35b)
i.e. the amplitude of the eigensolution closest to will be amplified the most. After
sufficient iterations, V will have become a quasi-eigenvector, such that:
( )
+
in
n
V V 11
(3.36)
i.e. the vector is amplified by( ) i
1each iteration. From this we can recover the
eigenvalue for the dominant eigenvector now stored in V .
A problem occurs when two adjacent eigenvalues are very close in value. In
this case convergence rate depends on:
1
1
i
j
(3.37)
where i and j are the two eigenvalues, j the farthest eigenvalue from . Overall
this means that the more similar in distance each eigensolution is to the input guess,
the slower the convergence. When considering (3.30), the consequence is that slow
convergence would be expected when solving heavily multi-moded structures with
very close s, as is found later in Chapter 7. The SIPM creates a matrix problem,
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(3.33a), similar to the matrix problem we need to solve for the BPM algorithm
(3.27). The next sections look at solving these matrix problems.
3.6 Solution of the 3D FD BPM and Mode Solver Matrix Problems
3.6.1 Introduction
In the previous sections, similar matrix problems were created for both the
BPM algorithm, (3.27), and FD Mode Solver, (3.33a). How these problems are
solved will determine the efficiency of the solvers, in terms of speed and memory
consumption, and ultimately the size and complexity of optical structure that can be
simulated. Since the aim is to simulate accurately as large a structure as possible, this
is an important area for optimisation. Methods by which the matrix problem may be
solved, and their relative merits, as found during the course of the PhD, are
summarised in Table 3.1.
The chosen solution was the Bi-Conjugate Gradient Method [3.22], [3.23]. In
the course of the work undertaken, the basic Bi-Conjugate method (the algorithm
detailed in [3.22]) was originally implemented, but it was found necessary to upgrade
to the improved convergence, more stable, BCGStab(l) version (with the order, l =
2), the algorithm of which is described in [3.23]. This was necessary because
convergence became unreliable with the basic Bi-conjugate Gradient Method when
Perfectly Matched Layers (see section 3.7) were introduced at the edges of the
simulation window. The basic Bi-conjugate Gradient Method and Incomplete LU
Decomposition is summarised in the following sections. The more advanced
BiCGStab(l) works on the same principle as the basic BiCG and has the additional
advantage of requiring half the memory of the original method, due to requiring the
storage of only one LU matrix, rather than two, explained later in the chapter.
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Method Advantages Disadvantages Ref.
Direct solution, through
Calculation of Inverse of
Matrix
Exact Solution (depending
on machine accuracy)
Prohibitive in memory use
(inverse is not a sparse
matrix). N 3 process
[3.22]
Band Matrix Solver,
(Numerical Recipes
Algorithm)[],
incorporating LU
decomposition
Exact Solution (depending
on machine accuracy)
Full LU decomposition
high memory use, N 3
process.
[3.22]
Iterative Bi-Conjugate
Gradient Method, basic
preconditioner (numerical
recipes)
Significantly reduced
memory use
Convergence issues,
accuracy dependent on
convergence and time
allowed for simulation
[3.22]
Iterative Bi-Conjugate
Gradient Method,
Incomplete LU
decomposition (ILUD) as
preconditioner
(Numerical Recipes)
A balance between memory
usage and convergence can
be reached. (through
altering the LU
decomposition level)
Convergence issues still
exist due to linear nature of
searching the solution
space. Memory use is
inefficient, since the ILUD
of the matrix and its
transpose is required.
[3.22]
Improved Bi-Conjugate
Gradient Method
(BiCGStab(l)), with ILUD
Only the ILUD of the
matrix is required,
effectively halving memory
use. More reliable
convergence achieved
through changing the order
of convergence, (l).
Increased time required for
iteration (x l). Although
faster, more reliable
convergence compensates
for this.
[3.23]
Table 3.1 : Relative merits of matrix problem solution methods
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3.6.2 Bi-Conjugate Gradient Method
The Bi-Conjugate Gradient Method (BiCGM), [3.22], iteratively solves equations of
the form:
b x A =. (3.38)
Typically, the matrix, A, is sparse and non-symmetric and the size of the problem is
very large. With each iteration step, the (initially guessed) vector xk is corrected by a
true residual ( k k Axbr = ) and a shadow residual ( k r ~ ) found through a multiplication
involving ( AT ), the transpose matrix. The residuals are forced to be orthogonal to the
shadow residuals. In this way the BiCGM forces convergence to the solution. This
method initially appears ideal, using fast, direct, multiplications to achieve
convergence. However in practical cases, the method requires additional help to
encourage convergence, in the form of preconditioning, examined below.
3.6.3 Addressing Convergence Issues Through Preconditioning
The BiCGM converges most effectively when and b are similar, i.e. the matrix
A is nearly the identity matrix. Conversely, the BiCGM does not cope well with
operations where x and b are very different. With the mode-solver this is especially
true when the initial propagation constant entered into the solver is very close to an
actual root, hence amplification is large (3.36). This is also an issue with the BPM
algorithm when the step size is large and the field has the potential to change
significantly. To improve convergence, the problem is preconditioned, i.e. we write
( ) ( )b P x P A .. = (3.39)
where P is the preconditioner matrix. P is chosen to be an approximation to the
inverse of A . The level of preconditioning determines the speed at which the
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problem will converge (in some cases, if it converges at all). Different levels of
preconditioning include:
a) P is the inverse of the leading diagonal of A .
b) P is some approximation of 1
A
(a) is a simple case of (b). In some cases, (a) is sufficient to allow reliable
convergence, and requires the least work mathematically to implement and use.
However, for complex and large problems, as created by simulation of large
structures with the FD solvers, an improved approximation is required. Where
finding an inverse matrix directly would be prohibitive in terms of time and memory,
LU decomposition, [3.22], can be used. An optimised form of Incomplete LU
decomposition is used in this case, to reduce processing and memory use to an
acceptable level whilst still achieving convergence. This is described in the next
section.
3.6.4 Incomplete LU Decomposition
LU decomposition, detailed in [3.22], splits a matrix, A , into two triangular
matrices, (3.34).
4 4 4 84 4 4 764 4 4 4 84 4 4 4 764 4 4 4 84 4 4 4 76 AU L
aaaa
aaaa
aaaa
aaaa
=
33323130
23222120
13121110
03020100
33
2322
131211
03020100
33323130
222120
1110
00
000
00
0
0
00
000
(3.40)
A full LU decomposition would allow us to solve the equation b x A =. through the
linear set
y xU
b y L==
.(3.41 a,b)
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where y is an intermediate vector. The solutions can be calculated directly, due to
the matrices being triangular. Thus LU decomposition allows the solution of a matrix
problem without requiring the calculation of an inverse matrix. However, the size of
the matrices typically created in FD studies of dielectric waveguide problems
prohibits the storage of a whole, non-sparse, matrix. Hence partial LU decomposition
is used here to generate an approximate solution of the problem to encourage the
convergence of the BiCGM. This is detailed next.
3.6.5 Incomplete LU Decomposition for the BiCGM
As explained in 3.6.3, applying some preconditioning to the BiCGM will
improve convergence, but an accurate LU decomposition is not necessary. The LU
decomposition as it stands requires approximately 1/3 N 3 calculations, as
programmed from [3.22], and will require enough memory to store an NxN matrix,
where N is the number of sample points in the problem. Consequently as the size of
the problem is increased, this method will quickly become too slow and bloated to be
useable.
Since in this case only an approximation of an inverse is required, the
computer memory required by LU decomposition can be reduced by ignoring less
significant values of the matrix, defined in this case as values below a certain
threshold in magnitude (referred to in later experiments as LU lim), known as
Incomplete LU Decomposition. When used as a preconditioner to the BiCGM, the
threshold that determines a significant matrix element can be lowered, improving the
estimate to allow faster convergence, or raised to reduce the memory requirement.
Where incomplete LU decomposition reduces the memory requirements, it
does not significantly reduce the number of calculations required to perform the
operation. However, when performing an LU decomposition on a band-matrix
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problem (as we have in this case), the LU matrices (3.40) will be band matrices of
the same width. Taking this into account drastically reduces the number of elements
that we need to consider, and consequently the time taken to perform the
decomposition. The algorithm developed is optimised with this in mind.
Now all the components of the basic BPM and Mode Solver algorithms have
been created, we turn our attention to how the boundaries of the numerical work
space affect the simulation, and implement methods by which the boundaries are
made insignificant to simulation accuracy.
3.7 Implementation of boundary conditions
3.7.1 Introduction
To create a practical solver we also have to consider the effects of the
simulation boundaries on the simulation accuracy. Basic BPM and Mode Solver
boundary conditions set the field just outside the simulation area to zero (Section
3.5), simulating a perfectly conducting metal box. This section explores methods by
which the energy arriving at the boundaries can be absorbed or otherwise removed to
avoid reflected light interfering with the simulation. Two popular methods are
considered in this work, the Transparent Boundary Condition [3.24] and the Perfectly
Matched Layer [3.25]. The two methods and their implementations are summarised
in the following sections.
3.7.2 Transparent Boundary Conditions (TBCs)
Transparent Boundary Conditions (TBCs) [3.24], implemented in a BPM
algorithm (Section 3.3), are based on the assumption that the radiation field behaves
as a complex exponential near the boundary. The field outside the sample area is
predicted using this assumption, making the boundary transparent and allowing
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energy to leave the simulation area. The reflection co-efficient of such a method is
quoted as being in the region of 3 x 10 -8 under ideal conditions, [3.24], and as such
is considered for use in this solver. Heavily radiative and/or wide angle simulations,
however, can reduce the effectiveness of this method.
Consider the condition where one of the points used to calculate the
transverse derivatives in the FD method falls outside of the solution space. Currently
this point is set to zero field, which in effect simulates a perfectly conducting
boundary. The Transparent Boundary Condition estimates the unknown point outside
of the simulation area by examining the points next to it (Figure 3.9). If the points
indicate an incoming wave, the outside point is set to zero.
Figure 3.9 : Field points at simulation boundary.
Taking the scalar case as an example we estimate E(u+1,v) equation through (3.42),
x jevu E
vu E vu E
vu E =+
= ),1(
),(),(
),1((3.42)
i.e.
x jvu E
vu E
= ),(),1(
ln
If real( )>0, then outgoing wave.
If real( )
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3.7.3 Perfectly Matched Layers (PMLs)
One explanation of the behaviour of Berengers Perfectly Matched Layers
(PMLs)[3.25] is that the real spatial co-ordinate system is mapped into a complex
plane in regions at the edge of the simulation space, i.e. in Figure 3.10. This causes
the radiation entering this region to attenuate.
Figure 3.10 : Mapping the real co-ordinate system into complex space
Unlike some other methods that simulate absorbing materials at the edge of
the simulation space (briefly noted in [3.26]), where the transition into the absorbing
area is gradual and as smooth as possible so as to avoid reflections, the PMLs can
employ a reasonably steep transition into a high absorbing region without the risk of
reflections, resulting in an efficient and reliable method. Compared to TBCs, which
have literally no calculation overhead, PMLs require additional simulation space, so
will exhibit some additional overhead. It is normally possible to keep this at a
reasonable level, however, by keeping the PML width small and the absorption as
high as possible [3.26]. PMLs are implementable in both the BPM [3.27] and Mode
Solvers derived in Chapter 2.
PMLs are applied by making the transverse component of the co-ordinate
system complex near the edges of the simulation, i.e. (3.43a)
= 0 )](1[)( d jp P (3.43a)
ComplexMapping
Reflective boundaryof simulation, incomplex space.
Real SpaceAxis
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where P is the transverse co-ordinate in the PML, is the distance into the PML, (in
m, starting at zero at the inside edge of the PML). w is the width of the PML and
)( p an equation describing the PML profile, normally set so that the gradient of the
PML is low near 0= to avoid reflections occurring at the inner PML boundary.
From (3.43a)
)](1[)(
jp P =
(3.43b)
In this work )( p was set to:
c
w p0
2
2
2)( = (3.44)
where is set at runtime and which sets the attenuation strength of the PML. (3.44)
was used to smoothly grade in the PML, reducing the potential for reflections at the
intersection between absorbing and non-absorbing regions. Consider a plane wave of
form )exp()( isx x = entering and travelling through the PML. Map s:->p(1-j ) so
that in p space px jpx x j jp eee =)1( and hence the wave is attenuated, its
amplitude will decrease as ))(exp( d p s . The wave will reach the end of thesimulation space at w= and be reflected. The reflected wave will continue to be
attenuated, since the sign of s has been reversed. The overall reflection co-efficient of
the region is (3.45), assuming complete reflection at the simulation boundary.
= d p s R
w
0)(2exp (3.45)
Implementation of the PML in BPM is straightforward. For example, the 2D
Scalar Paraxial BPM implementation when inside the PML is shown in (3.46):
0)(11
2)(11
)(11 22
=
+
k z p j x jp x jp (3.46)
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3.7.4 Summary
The methods introduced here are used to minimise reflections from the
simulation boundaries, and improve the accuracy of the solvers. In the following
sections the Solvers are assembled and tested for effectiveness, evaluating the
boundary methods introduced above. First we summarise the variables of the FD-
BPM and Mode Solver algorithms.
3.8 Implementation and Operation of the FD Solvers
3.8.1 Introduction
The previous sections describe the individual components of the FD solvers
algorithms. In this section we examine how the components of the solvers are
assembled to form practical tools. These tools are then used later in this chapter to
determine basic guidelines of operation, and in the following chapters the solvers are
developed in various ways so that they can be successfully applied to the Arrayed
Waveguide structures in question in Chapters 6 and 7.
3.8.2 Assembling the Solvers
In total, four solvers have been developed for this investigation, i.e. an FD
based mode solver and a FD-BPM solver for 2D and 3D simulations. The flow charts
detailing the operation of the programmed solvers are shown in Figure 3.11. The
orange boxes in the flow chart indicate the parts of the program that take significant
computer time. It should be noted that the LU decomposition, applicable to the 3D
solvers only, takes up the most amount of time per iteration. The LU decomposition
has to be performed each time the structure or input propagation constant of the
simulation changes. For the FD-BPM solver this means that continuously variant
structures are much slower to simulate than invariant ones. For the mode solver, LU
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decomposition has to be repeated if the input propagation constant, m, changes, i.e.
when performing a second run to improve the accuracy of the result (see later in this
Section). Since, in the case of 2D based solvers tri-diagonal matrices are generated, a
fast, direct, tri-diagonal solver can be used [3.22], eliminating the requirement for the
BiCG method and LU decomposition. Figure 3.12 shows the flow diagram for a
single BiCG iteration, in the case of the mode solver (a) and the FD-BPM algorithm
(b) for a 3D structure. Figure 3.13 shows the flow diagram of an iteration for the
mode solver (a) and the FD-BPM algorithm (b) for a 2D structure.
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Figure 3.11: Flow chart of Mode Solver (left) and BPM algorithm (right) operation. In
implementation, the orange boxes take significant computer time to execute (darker = longer). *
Required for solvers of 3D structures only
START
END END
Read Input Parameters
Read Input Field
Read Input Structure
Set up Matrices
Set up Boundary Conditions
Incomplete LU Decomposition *
Set up Structure
Read Input Parameters
Read Input Field
Read Input Structure
Set up Matrices
Set up Boundary Conditions
Incomplete LU Decomposition *
Set up Structure
Perform M.S. Iteration (a)
Has converged?
Reached LimitOf Passes?
Update
Perform Propagation Step (b)
Is SimulationFinished?
Does StructureChange?
NO
YESYES
YES
YES NO
NO NO
START
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Figure 3.12 : MS Iteration (a) and BPM Propagation Step (b) for a 3D struture, showing
preconditioning and BiCG iterations.
Figure 3.13 : MS Iteration (a) and BPM Propagation Step (b) for 2D slab waveguide. In this case
a fast, direct, tri-diagonal solver is used. Matrix problems 3.33a and 3.27 are tri-diagonal when
the y- transverse direction does not exist.
Solve (3.33a) using direct tri-diagonal solver [REF]
START
END
START
END
Solve RHS directly (3.27)
(a) (b)
Solve LHS (3.27) using directtri-diagonal solver [REF]
BiCG Iteration (3.33a)
START
Has BiCGconverged?
END
BiCG Iteration LHS (3.27)
START
Has BiCGconverged?
END
Post-Condition with ILUM
Precondition with ILUM
Solve RHS directly (3.27)
Post-Condition with ILUM
(a) (b)
Precondition with ILUM
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3.8.3 Input Parameters to the Solvers
The assembled FD solvers have several variables, potentially affecting the
accuracy and the convergence of the output simulation. These variables are
summarised in Table 3.2. The ILU Threshold (LU lim) and BiCGM Threshold
(BCG lim), affect the convergence of the BiCGM. The ILU Threshold is set so that the
BiCGM conducts only a few (normally less than 10) iterations before converging to
ensure accuracy is retained [3.22]. The BiCGM Threshold (BCG lim), which sets the
maximum average difference of each value of the matrix, compared to the previous
iteration, is set so that it converges sufficiently to ensure accurate propagation
constant, , and field outputs. The Convergence Threshold ( lim) affects how many
Mode Solver iterations are performed and sets the accuracy to which is found.
PML widths ( wl,r,t,b ) and strengths ( l,r,t,b ) affect absorption of the outgoing wave, and
simulation weighting, , affects stability of the Crank Nicholson FD-BPM solver, but
is set to 0.5 throughout this work since no stability problems were encountered.All variables discussed in so far affect convergence to a solution. In the
majority of cases these are found through trial and error. A table at the beginning of
each experiment in this work shows the values that are used for these variables for
the particular experiment. Other variables are set according to the type of simulation
required. Experiments later on in this chapter show the effect of the other variables
(Input m,b , x, y and z) on the simulations. As well as the input variables, two
other inputs are required, discussed next.
a) Input Field
For the mode solver, this field can be arbitrary, provided the field contains elements
of all possible eigenvectors for convergence to any mode. For the BPM algorithm the
input field is the field to be launched into the structure.
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b) Structure
For both solvers this is ultimately an array of refractive indices, representing the
structure to be modelled. For the BPM algorithm this array will z-dependent. The
amount of space that is left either side of the waveguide is important, and should be
set to avoid the boundary conditions affecting the simulation accuracy when the
calculated field is too close to the boundary. In all experiments in this work, ample
space has been left, to ensure that accuracy is not affected.
Electric / Magnetic Field Field used as the basis of the formulation, i.e. that is being calculated
TE / TM / Scalar Polarisation of field (or scalar approximation)
Wavelength, Simulated Free-Space Wavelength light
Input m or b Estimate of the propagation constant of the simulation
x,y Transverse sampling interval
z Propagation step size
Simulation weighting, affects stability and accuracy
ILU Threshold (LU lim ) Magnitude below which LU elements are discarded
BiCGM Threshold
(BCG lim )
Threshold to test whether BiCGM has converged. (Measured as
average deviation from previous iteration)
Convergence
Threshold, lim
Threshold to test whether has converged.
(Measured as deviation from previous iteration)
PML Width (w l,r,t,b ) Affects simulation time and PML effectiveness
PML Strength ( l,r,t,b ) Affects absorption of incoming waves / reflection off PML boundary
Table 3.2 : Simulation parameters. GREEN = BPM solver only, YELLOW = BiCG parameters
for 3D FD-BPM and 2D MS, ORANGE = Mode Solver only, BLUE = PML Parameters
The Solvers have now been assembled and the parameters defined. Now we
look at the known characteristics and operation of the solver.
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3.8.4 Solver Characteristics and Problems
Here we summarise known characteristics of the FD Solvers that cause
inaccuracies or breakdown in simulation:
1. Simulation is accurate only when boundaries are exactly half-way between sample
points. This can severely limit the choice of grid we can make especially in the
simulation of complex and continuously changing structures. However this limitation
is successfully addressed through Improved FD in Chapter 5.
2. Staircase error in BPM is caused through approximating the structure onto a
discrete grid, an example shown in Figure 3.14. The structure becomes distorted
causing propagation errors and scattering of light, especially when using coarse
sample densities. In this work, this problem is addressed through the combination of
Structure Related Co-ordinates in Chapter 4, and Improved FD in Chapter 5.
Figure 3.14 : A continuous structure, grey, sampled onto a discrete grid, orange. The result is
shown in blue, a jagged structure that causes errors in calculated propagation, and scatters light
out of the structure.
3. Wide Angle Error, which is caused through the approximation of the second z-
derivative (Section 3.4.4). As the angle of propagation of the light moves away from
the propagation of the simulation, also shown in Figure 3.14, the calculation of the
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beam and the propagation constant becomes less accurate. An example from [3.28] is
shown in Figure 3.15, where the exact propagation constant is compared to the
calculated propagation constant as the angle of the structure simulated is changed,
and the Pad order is increased. In this work the paraxial solver is used throughout,
which in Figure 3.15, calculated in [3.28], appears to show reasonable agreement
with the exact value in the range of about +/- 20 degrees. The implementation of
Curved Co-ordinates in Chapter 5, allowing the simulation to follow light
propagation, should therefore eliminate the requirement for wide-angle propagation.
Figure 3.15 from [3.28] : Axial Propagation Constant as the angle of light propagation is
changed from the angle of simulation propagation, for paraxial and wide-angle schemes.
Next the behaviour of the solver when altering a few key variables will be
analysed, at the same time verifying its correct operation. We start with a brief look
at the effect on the solver of the different boundary conditions. For convenience most
simulations are conducted in 2D, with a quick evaluation of 3D made later in the
section.
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3.9 Initial Experimentation
3.9.1 The Effect of Boundary Conditions on BPM
This section briefly demonstrates the relative merits of three types of
boundary condition. The types of boundary condition under test are as follows :
a) Basic, perfectly conducting boundary conditions, (Section 3.4.5)
b) Transparent Boundary Conditions, (TBCs), (Section 3.7.2)
c) Perfectly Matched Layers, (PMLs), (Section 3.7.3)
The structure tested is a 2D slab, where 0 y and so the 2D versions if the FD
solvers can be used. Dimensions of the simulated structure are shown in Figure 3.16.
The input to the simulation is a square field, i.e. of magnitude 1 within the slab
waveguide and 0 outside, which is propagated for 500m. The mismatch between the
field input and the modes of the structure causes radiation from the waveguide,
allowing the effect of each type of boundary condition can be observed. Other
parameters are detailed in Table 3.3. PML parameters were chosen through some
initial experimentation and tested to ensure the PMLs did not affect simulation . It
was found that the PML strength, , could range between ~0.01 and ~1, when using a
width, w, of 5m (50 sample points of 0.1m separation), and exhibit favourable
absorption properties. A PML strength of > ~1 and the PMLs exhibited
instabilities, and light began to reflect off the inside edge of the PMLs. A PML
strength of < 0.01 and the attenuation of the reflections is not sufficient in the 50
sample space. In this case the PML parameters were set to w = 5.0m and = 0.05.
These values were found to optimally absorb the outgoing energy. In all simulations
in this work the PML strength was set to = 0.05. PML width varies.
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Figure 3.16 : The simulated 2D structure. (PML areas only simulated when the PML boundary
conditions are implemented).
Electric / Magnetic Field Electric
TE / TM / Scalar Scalar
Wavelength 1.55m
Input b/k 0 3.43
x 0.1
z 0.1
0.5
PML Width (W l,r ) 5.0m
PML Strength ( l,r ) 0.05
Table 3.3: 2D BPM Simulation Parameters. PML parameters only applicable with simulations
with implemented PMLs
Figure 3.17 a, b and c are field plots of the simulations with (a) zero-field, (b)
TBC and (c) PML boundary conditions respectively. In each part of the Figure the
radiation loss due to the mismatch of the excitation field and the waveguide modes is
clearly visible. It is also immediately obvious from Figure 3.17(a) that the reflections
shown from the zero-field boundary condition interfere significantly with the
simulation. The TBCs are largely effective at removing the outgoing field, Figure
3.17(b), but some reflections still occur. Figure 3.17(c) show that PMLs, however,
are effective at removing all reflections. The attenuation within the PMLs can clearly
be seen. An estimated 50% extra overhead is created by the unoptimised PMLs in
this case. However, this overhead can be reduced through optimisation of the PMLs.
Since some simulations in the investigations made in this work are likely to exhibit
neff = PML 3.46804 3.47513 3.46804 PML
width = 5.0m 8.0m 4.5m 8.0m 5.0m x
z
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high radiation, PMLs are used in all situations for consistency. Although not
explicitly demonstrated here, PMLs, used with the mode solver, were found to
attenuate outgoing radiation with similar efficiency.
0 1000 2000 3000 40000
50
100
150
Propagation (z) Axis, Samples
T r a n s v e r s e
( x ) A x i s ,
S a m p
l e s
0.8541 -- 1.4000.5211 -- 0.85410.3179 -- 0.52110.1939 -- 0.31790.1183 -- 0.19390.0722 -- 0.11830.0440 -- 0.07220.02687 -- 0.04400.01639 -- 0.026870.01000 -- 0.01639
0 1000 2000 3000 40000
50
100
150
Propagation (z) Axis, Samples
T r a n s v e r s e
( x ) A x
i s ,
S a m p
l e s
0.8541 -- 1.4000.5211 -- 0.85410.3179 -- 0.52110.1939 -- 0.31790.1183 -- 0.19390.0722 -- 0.11830.0440 -- 0.07220.02687 -- 0.04400.01639 -- 0.026870.01000 -- 0.01639
0 1000 2000 3000 40000
50
10 0
15 0
20 0
Propagation (z) Axis, Samples
T r a n s v e r s e
( x ) A x i s ,
S a m p
l e s
0.8541 -- 1.4000.5211 -- 0.85410.3179 -- 0.52110.1939 -- 0.31790.1183 -- 0.1939
0.0722 -- 0.11830.0440 -- 0.07220.02687 -- 0.04400.01639 -- 0.026870.01000 -- 0.01639
Figure 3.17 a,b,c : Field plots of propagation with a)Reflective Boundaries, b) Transparent
Boundary Conditions (TBCs) and c) Perfectly Matched Layers (PMLs) respectively
(a)
(b)
(c)
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3.9.2 Solver Error as a function of Transverse Step, x (Mode Solver)
This experiment tests the convergence of the calculated Modal Index ( neff =
/k 0) for the fundamental mode of a structure. The exact results from an analytical
slab waveguide mode solver were used to confirm the accuracy of the FD solvers.
The analytical solver also provided the exact background propagation constant b,
input into the simulations. The 2D FD Mode solver for slab waveguides was
executed using varying transverse sampling intervals and the error noted, measured
as the difference between the input and output Propagation Constant, after a
sufficient number of iterations had been performed to allow convergence. It was
found that two passes, (see Figure 3.11) were required to allow proper convergence
of the solver. The structure tested is shown in Figure 3.16. PMLs were used as
boundary conditions and the operating wavelength, = 1.528m. Other variables
used are summarised in Table 3.4.
The exact analytical Mode Solver found the modal index of the fundamentalmode to be n eff = 3.47330863, for the TE polarisation (field discontinuous across
boundaries) and n eff = 3.47331262 for the TM polarisation (field continuous across
boundaries). It should be noted that for this experiment the transverse step size ( x)
was chosen so that boundaries fell exactly half-way between sample points, since this
is when the solver is most accurate [3.18]. Figure 3.18 shows the output propagation
constant of the FD mode solver as a function of mesh size, x. Figure 3.19 compares
the solver accuracy with the analytical result as a function of x. Here, error is
defined as neff = ( ANALYTIC FD )/k 0. where ANALYTIC = neff(analytic) x k 0.
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Convergence Threshold, lim 1x10 -10
PML Width (w l,r ) 5m
PML Strength ( l,r ) 0.05
Table 3.4 : FD-BPM Simulation Parameters
Figure 3.18 : mode solver output modal index ( b/k 0) as a function of sampling interval, ( x)
Simulation Accuracy with Delta-x
1.0E-10
1.0E-09
1.0E-08
1.0E-07
1.0E-06
1.0E-05
0.001 0.01 0.1
Delta-x (um)
E r r o r
i n E f f e c
t i v e
I n d e x
( n e
f f e r r o r )
Basic FD, TM, Electric Field
Basic FD, TE, E lectric Field
Figure 3.19 : Convergence of FD simulation result to analytic result, with sampling interval
(x).
Simulation Effective Index with Delta - x
3.473308
3.473309
3.47331
3.473311
3.473312
3.473313
3.473314
3.473315
0 0.02 0.04 0.06 0.08 0.1 Delta-x (um)
E f f e c
t i v e
I n d e x
( n e
f f )
Basic FD, TM, Electric Field
Basic FD, TE, Electric Field
TM
TE
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Convergence to an analytical solution confirms the correct operation of the
FD mode solver. The gradient in Figure 3.19 shows the error to be proportional to
x2, as expected with FD schemes [3.1]. Smaller sampling intervals improve
accuracy, but simultaneously increase simulation time and memory. This only really
becomes significant when simulating 3D structures, as in Section 3.9.4, since in 2D
only when sampling interval is set to below ~0.005m does the simulation last more
than a couple of seconds. The next simulation looks at how the input Propagation
Constant affects the accuracy of the BPM algorithm.
3.9.3 Input Propagation Constant & Propagation Step, Beam Propagation
Method
This subsection demonstrates how the background propagation constant ( b)
and propagation step affects the accuracy of simulation of the FD-BPM solver. The
true numerical solution, i.e. the mode profile and output propagation constant, MS ,
for the BPM simulation at a transverse sampling interval, x = 0.01m, is acquired
from the mode solver in the last experiment. The error was measured as a deviation
of the output propagation constant from the BPM solver, BPM , measured through the
phase difference of the output and input fields, from the output Propagation Constant
of the FD mode solver, MS , found in the last section (i.e. abs( MS - BPM ). The
background propagation constant, b, is purposely altered from the optimum (i.e. the
true numerical solution, MS ) to observe how the error is affected. The simulation is
of the TE polarisation, so the optimal modal index ( /k 0 = n eff ) of the simulation at x
= 0.01m is neff = 3.47331263 as calculated by the Mode Solver.
Figure 3.20 and Figure 3.21 show the calculated deviation of the BPM
propagation constant, i.e. abs( MS - BPM )/k 0, as a function of propagation step, Figure
3.20, and deviation of background modal index ( b /k 0), Figure 3.21. From Figure
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3.20 and Figure 3.21 it can be seen that solver accuracy is proportional to Oh (i.e.
first order), where h is the deviation of background index ( b /k 0). Similarly the
relationship between solver accuracy and z is O z . This highlights the importance
of choosing the correct propagation constant for the simulation. Also, in a multi-
moded structure, it is sensible to choose a small z to ensure accurate modelling of
all propagating modes.
Error with Propagation Step (delta-z)
1.E-09
1.E-08
1.E-07
1.E-06
1.E-050.1 1 10 100
Propagation Step (delta-z)
E r r o r
( d e
l t a - n e
f f )
3.473
3.4731
3.4732
3.4733
Figure 3.20 : Deviation of BPM /k 0 from MS /k 0 as a function of propagation step, with various
input effective indices (n eff = b/k 0). Deviation is of O z.
Error with Deviation of Input Effective Index
0.000000001
0.00000001
0.0000001
0.000001
0.000010.00001 0.0001 0.001
Deviation of Input Effective Index (delta - neff)
O u
t p u
t E f f e c
t i v e
I n d e x
E r r o r
( d e
l t a -
n e
f f )
delta-z = 1
delta-z = 0.1
delta-z = 10
Figure 3.21: Output Index Deviation ( BPM /k 0) as a function of deviation of background index
( b/k 0) from numerical optimal ( MS /k 0). Error is of O b .
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3D FD, Convergence with Delta-x
3.4725
3.47252
3.47254
3.47256
3.47258
3.4726
3.47262
3.47264
3.47266
3.47268
3.4727
0 0.02 0.04 0.06 0.08 0.1
Delta-x
E f f e c
t i v e
I n d e x
Figure 3.23 : Behaviour of Effective Index as a function of x, when calculating the fundamental
mode. Minimum solvable x (where boundaries fell exactly between sample points) was 0.04m
on a 2GB workstation.
Simulation Memory with Total Sample Points
1
10
100
1000
10000
100000
10000 100000 1000000Sample Points
M e m o r y ,
M B
x,y = 0.05
x,y = 0.04
x,y = 0.1
Figure 3.24 : Simulation Memory use with number of sample points.
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Simulation Time with Number of Sample Points
10
100
1000
10000
10000 100000 1000000
Number of Sample Points
S i m u
l a t i o n
T i m e ,
s
Figure 3.25 : Simulation Time with number of sample points
3D Simulation shows much greater error with transverse interval than the 2D
solver. Additionally it is difficult to tell whether the same transverse convergence
characteristics are shared with 2D simulation. Simulation memory use and time have
become significant factors, limiting the minimum transverse step to 0.04m before
the simulation could no longer run on a 2GB workstation. Observing Figure 3.24 and
Figure 3.25, both simulation memory and time appear to increase at roughly On1.5,
where n is the number of sample points. Simulation time and memory for a certain
accuracy is significantly reduced in Chapter 5, through use of Improved FD [3.29]
and Alternating Direction Implicit [3.30] techniques. The 3D solver shall be re-
examined then.
3.9.5 Conclusion
The 2D solver shows convergence with input parameters typical of FD
Solvers. The 3D solver shows much lower accuracy with transverse sample interval
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Method Benefit Chapter /
References
2D/3DFD Mode
Solver / BPM.
3D BPM allows more realistic simulations,allowing in-depth investigation into structure.
However, 3D simulations are inherently slow and
memory intensive (Figure 3.24, Figure 3.25).
Chapter 3[3.1]
[3.2]
Perfectly
Matched
Layers
High performance absorbing layers placed at the
edge of the simulation, to eliminate reflections
from the simulation edges.
Chapter 3
[3.27]
Curved Co-
ordinates
Allows the Simulation to follow the curve of the
structure, reducing simulation area, wide angle
errors and stair-casing errors
Chapter 4
Improved
Finite
Difference
Formulations
Improved approximation to index boundaries,
using transfer matrix to improve error from Oh 0 to
Oh 2 (h is the discretisation step), and allows
accurate simulation of boundaries not centred
between sample points, allowing coarser sampling
to be used for comparable accuracy.
Chapter 5
[3.29]
Alternating
Direction
Implicit
Method
Alternate Fully explicit/implicit steps in x/y
transverse directions. Eliminates requirement for
iterative matrix solving, so substantially reducing
memory use and time per simulation step. Allows
much larger 3D structures to be accurately
simulated.
Chapter 5
[3.30]
Table 3.6 : Methods by which the FD BPM and FD Mode Solvers are improved to allow
simulation of complex components.
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