BPM anglais

8/8/2019 BPM anglais

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35

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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36

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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37

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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39

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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40

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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41

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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43

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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44

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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46

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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47

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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49

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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50

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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52

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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54

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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55

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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56

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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57

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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58

(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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59

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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61

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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62

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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63

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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64

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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65

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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66

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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77

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


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


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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Oxide Silicon-on-Insulator Structures, Optics Letters, 1988, Vol. 13, pp. 175

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