18-01-2013, 03:37 PM
Simulation and Optimization of Photonic Crystals Using the Multiple Multipole Program
[attachment=47562]
INTRODUCTION AND OVERVIEW
The multiple multipole program (MMP) is a powerful semianalytical technique
that was developed during the last 25 years. Because of its accuracy and reliability
it is especially well suited for numerical model optimization. This is highly
191
important for the design of structures within photonic crystals because currently
almost no useful design rules are available for devices embedded in photonic
crystals.
For the efficient analysis of photonic crystals several techniques were added
to the original MMP codes. These include:
1. The modeling of periodic structures using the concepts of fictitious
boundaries and periodic boundary conditions
2. Novel eigenvalue solvers based on fictitious excitations and weighted
residual minimization techniques
3. So-called connections, a unique macro feature of MMP that allows one
to accurately handle waveguide discontinuities without any absorbing
boundaries
4. Eigenvalue estimation techniques and parameter estimation techniques
for a drastic reduction of the computation time of frequency domain
methods
These techniques also are valuable in conjunction with other numerical methods.
Beside the MMP features mentioned above, we present a promising alternative
to the well-known supercell approach for the analysis of waveguides in photonic
crystals. Our new approach is considerably faster than the well-known
supercell approach, and it is more realistic at the same time. It allows us to compute
not only the propagation constants of all modes but also the radiation losses
of the waveguide modes in photonic crystals of finite size.
Finally, we embed the MMP approximation of each mode of a waveguide into
a connection that can be used exactly as an analytic expansion that describes the
electromagnetic field of the mode everywhere. This combination of MMP with the
mode matching technique allows us to model waveguide discontinuities efficiently
and accurately. Furthermore, it avoids all problems associated with reflected waves
at the output ports caused by absorbing boundary conditions. As a consequence,
we can drastically reduce the model size and computation time. We then address
the problem of fabrication tolerances, which has a strong impact on the quality of
photonic crystal devices. With an extensive sensitivity analysis we explore the
impact of the individual cell geometry (for example, radii and locations of the rods
of a 2D photonic crystal) on device characteristics such as reflection and transmission
coefficients. We finally take advantage of the sensitivity analysis in order
to improve the device characteristics. The entire procedure is demonstrated using
two relatively simple but interesting examples, the 90° bend and the filtering
T-junction. We demonstrate that a 90° bend can be optimized so that almost zero
reflection is obtained over a very broad frequency range that covers almost the entire
bandgap. Furthermore, we outline a design procedure for filtering T-junctions that
is based on the analysis of waveguide modes while avoiding time-consuming modeling
optimization of the entire device. In both examples, we apply our sensitivity
analysis for the optimization of these structures.
192 Electromagnetic Theory and Applications for Photonic Crystals
INTRODUCTION TO PHOTONIC CRYSTAL SIMULATION
In 1987 Yablonovich [1] proposed photonic crystals (PhCs) as an optical equivalent
to semiconductors (i.e., in PhCs photons play essentially the role that electrons
play in semiconductors). Although PhCs are rarely observed in nature, nanotechnology
allows us to fabricate both 2D and 3D PhCs [2–4]. Pure PhCs are strictly
periodic structures that exhibit photonic bandgaps [3,4] equivalent to the bandgap
in semiconductors. Therefore, the bandgap computation is closely related to the
computation of periodic structures, such as gratings. Since the waves in perfect
PhCs are solutions of an eigenvalue problem, codes for computing band diagrams
that describe a perfect PhC must be able to efficiently handle both periodic structures
and eigenvalue problems.
Since no photon can penetrate a PhC within the bandgap, a PhC of finite size
totally reflects light within a limited frequency range that corresponds to the bandgap.
Although this is an interesting effect, only the introduction of defects or cell
modifications — corresponding to the doping of semiconductors — explains the
high attention that is currently paid to PhCs in integrated optics. In the design of
finite PhCs with modified cells, one has much more freedom than in the doping of
semiconductors because nanotechnology allows one easily to modify the geometry
of any particular cell of a PhC. For example, when the initial PhC consists of a
regular array of circular rods, one can modify the radius and location and radius
(or even shape) of each rod. Furthermore, tunable materials [5,6] allow one also to
modify the material properties within some range. When identical cell modifications
are introduced along a line in a PhC, one obtains simple waveguides. We will
show that the characteristic properties of these waveguides may be varied in a wide
range. The composition of different waveguides of finite length can then be used
for more advanced structures such as filters. Although PhC waveguides are similar
to standard waveguides, the proper computation of PhC waveguides is more
demanding because the simple cylindrical symmetry of standard waveguides is
replaced by periodic symmetry. As a consequence, the computation of PhC waveguide
modes requires the proper handling of periodic symmetry in addition to
the eigenvalue problem that defines the modes. Since these ingredients also are
required for the computation of band diagrams, essentially the same codes may be
used for both the computation of the band diagrams of perfect PhCs and for the
computation of PhC waveguide modes.
A more severe problem in the PhC waveguide computation results because the
symmetry perpendicular to the direction of the waveguide is broken by the defect
or cell modification that generates the waveguide. As a consequence, the PhC waveguide
model is periodic along the waveguide but not periodic in the transverse
direction. Thus, the theoretical waveguide model is infinite and must be either truncated
or modified in such a way that the numerical effort remains finite. In the
following, we will present two different techniques for solving this problem.
Once waveguides have been introduced in a PhC, it is natural also to introduce
waveguide discontinuities. Already the most simple waveguide discontinuities (i.e.,
waveguide bends) make the PhC concept very attractive for integrated optics for the
Simulation and Optimization of Photonic Crystals 193
following reason. Conventional waveguide bends in integrated optics either exhibit
significant radiation losses or are large in size, which is the most important reason
for the large extent of traditional optical chips. Within PhCs one can easily obtain
sharp waveguide bends without any radiation loss and even with zero reflection coefficients
[7]. In the following we will analyze and optimize a sharp 90° bend in a PhC.
From the numerical point of view, the analysis of waveguide discontinuities
is most demanding because one typically must model structures that are large
compared to the wavelength and consist of many cells. First of all, the symmetry
of the initial crystal is completely broken. Furthermore, the model must be appropriately
truncated. Finally, a high accuracy is required because small variations
of some model parameters may have unexpectedly big effects. This especially
holds when structures are being optimized. We will show that optimization may
render some parts of the PhC very critical or sensitive, which causes problems
for the fabrication and may be attractive at the same time when one is interested
in tuning or switching. Finally, the critical parts are attractive for the design of
sensors.
Because no design rules are currently known for the design of PhC waveguide
devices, the only way to develop useful structures without overly stressing intuition
is to combine numerical simulation with optimization. When this is done, it is most
important to know that optimizers are heavily disturbed by inaccuracies of the
forward solver that is used to analyze the PhC structure. We therefore prefer accurate
numerical techniques that are close to analytic solutions but flexible enough for
computing all kind of PhC structures. In the following, we outline the basics of the
MMP, which may be considered a semianalytic method. We show how MMP can
handle all important problems associated with PhC simulations, namely periodic
symmetries, eigenvalue problems, and waveguide discontinuity problems. Finally,
we use MMP for the detailed analysis of PhC structures and for the optimization of
such structures.
For reasons of simplicity, we focus on 2D PhCs. Beside much higher memory
requirements, much longer computation time, and difficulties in the graphic
representation, the handling of 3D PhC models provides no essential new problems
(i.e., the fundamental procedures remain the same for 3D).
BASICS OF THE MULTIPLE MULTIPOLE PROGRAM
Two-dimensional PhCs consist of infinite 2D arrays of either dielectric rods in
free space (or some other material) or of holes (air or some other material) within
photonic bandgap, the dielectric contrast (i.e., the ratio of the higher refraction
index divided by the lower refraction index) must be sufficiently big. Typically,
this ratio is near 3. For a numerical method, it does not matter whether the refractive
index in the rod is bigger or smaller than in the surrounding medium, that is,
if one has an array of rods or of holes. In the following, we will focus on the rod
type PhCs.
[attachment=47562]
INTRODUCTION AND OVERVIEW
The multiple multipole program (MMP) is a powerful semianalytical technique
that was developed during the last 25 years. Because of its accuracy and reliability
it is especially well suited for numerical model optimization. This is highly
191
important for the design of structures within photonic crystals because currently
almost no useful design rules are available for devices embedded in photonic
crystals.
For the efficient analysis of photonic crystals several techniques were added
to the original MMP codes. These include:
1. The modeling of periodic structures using the concepts of fictitious
boundaries and periodic boundary conditions
2. Novel eigenvalue solvers based on fictitious excitations and weighted
residual minimization techniques
3. So-called connections, a unique macro feature of MMP that allows one
to accurately handle waveguide discontinuities without any absorbing
boundaries
4. Eigenvalue estimation techniques and parameter estimation techniques
for a drastic reduction of the computation time of frequency domain
methods
These techniques also are valuable in conjunction with other numerical methods.
Beside the MMP features mentioned above, we present a promising alternative
to the well-known supercell approach for the analysis of waveguides in photonic
crystals. Our new approach is considerably faster than the well-known
supercell approach, and it is more realistic at the same time. It allows us to compute
not only the propagation constants of all modes but also the radiation losses
of the waveguide modes in photonic crystals of finite size.
Finally, we embed the MMP approximation of each mode of a waveguide into
a connection that can be used exactly as an analytic expansion that describes the
electromagnetic field of the mode everywhere. This combination of MMP with the
mode matching technique allows us to model waveguide discontinuities efficiently
and accurately. Furthermore, it avoids all problems associated with reflected waves
at the output ports caused by absorbing boundary conditions. As a consequence,
we can drastically reduce the model size and computation time. We then address
the problem of fabrication tolerances, which has a strong impact on the quality of
photonic crystal devices. With an extensive sensitivity analysis we explore the
impact of the individual cell geometry (for example, radii and locations of the rods
of a 2D photonic crystal) on device characteristics such as reflection and transmission
coefficients. We finally take advantage of the sensitivity analysis in order
to improve the device characteristics. The entire procedure is demonstrated using
two relatively simple but interesting examples, the 90° bend and the filtering
T-junction. We demonstrate that a 90° bend can be optimized so that almost zero
reflection is obtained over a very broad frequency range that covers almost the entire
bandgap. Furthermore, we outline a design procedure for filtering T-junctions that
is based on the analysis of waveguide modes while avoiding time-consuming modeling
optimization of the entire device. In both examples, we apply our sensitivity
analysis for the optimization of these structures.
192 Electromagnetic Theory and Applications for Photonic Crystals
INTRODUCTION TO PHOTONIC CRYSTAL SIMULATION
In 1987 Yablonovich [1] proposed photonic crystals (PhCs) as an optical equivalent
to semiconductors (i.e., in PhCs photons play essentially the role that electrons
play in semiconductors). Although PhCs are rarely observed in nature, nanotechnology
allows us to fabricate both 2D and 3D PhCs [2–4]. Pure PhCs are strictly
periodic structures that exhibit photonic bandgaps [3,4] equivalent to the bandgap
in semiconductors. Therefore, the bandgap computation is closely related to the
computation of periodic structures, such as gratings. Since the waves in perfect
PhCs are solutions of an eigenvalue problem, codes for computing band diagrams
that describe a perfect PhC must be able to efficiently handle both periodic structures
and eigenvalue problems.
Since no photon can penetrate a PhC within the bandgap, a PhC of finite size
totally reflects light within a limited frequency range that corresponds to the bandgap.
Although this is an interesting effect, only the introduction of defects or cell
modifications — corresponding to the doping of semiconductors — explains the
high attention that is currently paid to PhCs in integrated optics. In the design of
finite PhCs with modified cells, one has much more freedom than in the doping of
semiconductors because nanotechnology allows one easily to modify the geometry
of any particular cell of a PhC. For example, when the initial PhC consists of a
regular array of circular rods, one can modify the radius and location and radius
(or even shape) of each rod. Furthermore, tunable materials [5,6] allow one also to
modify the material properties within some range. When identical cell modifications
are introduced along a line in a PhC, one obtains simple waveguides. We will
show that the characteristic properties of these waveguides may be varied in a wide
range. The composition of different waveguides of finite length can then be used
for more advanced structures such as filters. Although PhC waveguides are similar
to standard waveguides, the proper computation of PhC waveguides is more
demanding because the simple cylindrical symmetry of standard waveguides is
replaced by periodic symmetry. As a consequence, the computation of PhC waveguide
modes requires the proper handling of periodic symmetry in addition to
the eigenvalue problem that defines the modes. Since these ingredients also are
required for the computation of band diagrams, essentially the same codes may be
used for both the computation of the band diagrams of perfect PhCs and for the
computation of PhC waveguide modes.
A more severe problem in the PhC waveguide computation results because the
symmetry perpendicular to the direction of the waveguide is broken by the defect
or cell modification that generates the waveguide. As a consequence, the PhC waveguide
model is periodic along the waveguide but not periodic in the transverse
direction. Thus, the theoretical waveguide model is infinite and must be either truncated
or modified in such a way that the numerical effort remains finite. In the
following, we will present two different techniques for solving this problem.
Once waveguides have been introduced in a PhC, it is natural also to introduce
waveguide discontinuities. Already the most simple waveguide discontinuities (i.e.,
waveguide bends) make the PhC concept very attractive for integrated optics for the
Simulation and Optimization of Photonic Crystals 193
following reason. Conventional waveguide bends in integrated optics either exhibit
significant radiation losses or are large in size, which is the most important reason
for the large extent of traditional optical chips. Within PhCs one can easily obtain
sharp waveguide bends without any radiation loss and even with zero reflection coefficients
[7]. In the following we will analyze and optimize a sharp 90° bend in a PhC.
From the numerical point of view, the analysis of waveguide discontinuities
is most demanding because one typically must model structures that are large
compared to the wavelength and consist of many cells. First of all, the symmetry
of the initial crystal is completely broken. Furthermore, the model must be appropriately
truncated. Finally, a high accuracy is required because small variations
of some model parameters may have unexpectedly big effects. This especially
holds when structures are being optimized. We will show that optimization may
render some parts of the PhC very critical or sensitive, which causes problems
for the fabrication and may be attractive at the same time when one is interested
in tuning or switching. Finally, the critical parts are attractive for the design of
sensors.
Because no design rules are currently known for the design of PhC waveguide
devices, the only way to develop useful structures without overly stressing intuition
is to combine numerical simulation with optimization. When this is done, it is most
important to know that optimizers are heavily disturbed by inaccuracies of the
forward solver that is used to analyze the PhC structure. We therefore prefer accurate
numerical techniques that are close to analytic solutions but flexible enough for
computing all kind of PhC structures. In the following, we outline the basics of the
MMP, which may be considered a semianalytic method. We show how MMP can
handle all important problems associated with PhC simulations, namely periodic
symmetries, eigenvalue problems, and waveguide discontinuity problems. Finally,
we use MMP for the detailed analysis of PhC structures and for the optimization of
such structures.
For reasons of simplicity, we focus on 2D PhCs. Beside much higher memory
requirements, much longer computation time, and difficulties in the graphic
representation, the handling of 3D PhC models provides no essential new problems
(i.e., the fundamental procedures remain the same for 3D).
BASICS OF THE MULTIPLE MULTIPOLE PROGRAM
Two-dimensional PhCs consist of infinite 2D arrays of either dielectric rods in
free space (or some other material) or of holes (air or some other material) within
photonic bandgap, the dielectric contrast (i.e., the ratio of the higher refraction
index divided by the lower refraction index) must be sufficiently big. Typically,
this ratio is near 3. For a numerical method, it does not matter whether the refractive
index in the rod is bigger or smaller than in the surrounding medium, that is,
if one has an array of rods or of holes. In the following, we will focus on the rod
type PhCs.