Optical Modeling with FDTD: Practical Applications with MEMs and Confocal Microscopy A Thesis Presented by Christopher Jason Foster to The Department of Electrical and Computer Engineering in partial fulfillment of the requirements for the degree of Master of Science in Electrical Engineering in the field of Electromagnetics and Optics Northeastern University Boston, Massachusetts April 2004
107
Embed
Optical Modeling with FDTD: Practical Applications with ......Optical Modeling with FDTD: Practical Applications with MEMs and Confocal Microscopy A Thesis Presented by Christopher
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
Optical Modeling with FDTD: Practical Applications with MEMs and Confocal Microscopy
A Thesis Presented
by
Christopher Jason Foster
to
The Department of Electrical and Computer Engineering
in partial fulfillment of the requirements for the degree of
Master of Science
in
Electrical Engineering
in the field of
Electromagnetics and Optics
Northeastern University Boston, Massachusetts
April 2004
ii
NORTHEASTERN UNIVERSITY
Graduate School of Engineering
Thesis Title: Optical Modeling with FDTD: Practical Applications with MEMs and Confocal Microscopy
Author: Christopher Jason Foster Department: Electrical and Computer Engineering Approved for Thesis Requirement of the Master of Science Degree ____________________________________________________ __________________ Thesis Advisor Date ____________________________________________________ __________________ Thesis Reader Date ____________________________________________________ __________________ Thesis Reader Date ____________________________________________________ __________________ Department Chair Date Graduate School Notified of Acceptance: ____________________________________________________ __________________ Director of Graduate School Date
iii
NORTHEASTERN UNIVERSITY
Graduate School of Engineering
Thesis Title: Optical Modeling with FDTD: Practical Applications with MEMs and Confocal Microscopy
Author: Christopher Jason Foster Department: Electrical and Computer Engineering Approved for Thesis Requirement of the Master of Science Degree ____________________________________________________ __________________ Thesis Advisor Date ____________________________________________________ __________________ Thesis Reader Date ____________________________________________________ __________________ Thesis Reader Date ____________________________________________________ __________________ Department Chair Date Graduate School Notified of Acceptance: ____________________________________________________ __________________ Director of Graduate School Date Copy Deposited in Library: ____________________________________________________ __________________ Reference Librarian Date
iv
ABSTRACT:
A two dimensional FDTD model is used to analyze the reflected power and detected field for two distinct optical systems. This model includes the application of several types of electromagnetic stimulation, makes use of absorbing boundaries, and allows for the specification of arbitrary geometry index of refraction and dielectric loss media parameters. The first analyzed system, a MEMS cantilever device, is used as a mechanically stimulated, optically interrogated Fabry-Perot interferometer. It is analyzed using a multiple beam interference (MBI) algorithm whose results are compared to the results of FDTD analysis. The resulting comparison to the MBI method is favorable and shows that etch holes for device release have little impact on the relative amount of reflected light. Specifically, the reflected power is reduced by approximately 12%.
The second system, a confocal microscope, is discussed in various detection modes analytically including monostatic and quasi-static point detection using coherent and incoherent detectors. These closed form approaches to the problem are used to simplify the FDTD modeling of the system and provide a means of comparison. An FDTD simulation of light propagating through the perturbations in index of refraction caused by the epidermal and dermal layers of human skin are conducted to compare with current research. Further development of the skin model is necessary to achieve agreement with actual observed behavior for existing quasi-static confocal point detectors. A discussion of the systemic problem of detected signal dropout is discussed along with a commentary on the relatively small change in normalized power observed by modeling.
v
For Erin
vi
Table of Contents Introduction...............................................................................................................1 Background................................................................................................................3 Case Definition and Requirements................................................................3 Analytic Methods...........................................................................................7 A Conceptual Overview of the FDTD Method.............................................14 Development of the Computational Model...............................................................18 The FDTD Equations.....................................................................................18 Absorbing Boundary Conditions....................................................................24 Excitation Functions.......................................................................................29 Analysis of Time Domain Data and Test Cases.............................................33 MEMS Vibrometer.....................................................................................................42 Quasi-Monostatic Confocal Coherent Modeling........................................................50 Conclusions / Discussion............................................................................................59 References...................................................................................................................68 Code Appendix............................................................................................................70
1
I. INTRODUCTION
The analysis of optical systems containing complex geometries composed of
turbid media with multiple indicies of refraction is a nontrivial problem to approach
analytically. To establish this in the cases which are described, common analytic
approaches and their limitations must be explored. Optical systems that are considered
herein are monostatic or quasi-monostatic systems where some variable in the focal or
target plane will affect the resulting field at the receiver. Specifically, the optical systems
analyzed use coherent light sources which are considered ideal in their assumed profiles.
Therefore, our definition of an optical system will be limited to those systems using
visible light to make a monostatic detection of that light which is received at the
source/detector after having passed through the system of complex geometries and
refractive indicies.
The impetus behind this project is its beginning as a simple 2D FDTD simulation
program. Through further development the hope was to obtain a more robust model
capable of answering research questions posed by members of the Optical Science
Laboratory at Northeastern University about confocal microscopy. While still in the
development and testing stages of the more robust FDTD code, we were approached by
Patricia Nieva whose doctoral dissertation concerned optical modeling of an almost one
dimensional optical layout MEMS device that she needed to model and wanted
confirmation of her analytic results. This provided the first real test of the model and was
used to further generalize its application, particularly in the way fields were analyzed
after simulation to compute power reflectance measurements. Following our combined
efforts in publishing this work, the FDTD model was again improved and used to model
2
confocal microscopy. In this way, it is hoped that the code will serve to aid the
development of optical systems in the Optical Science Laboratory (OSL) at Northeastern
University and simplify and analyze design through accurate simulation.
Further, in order to be a verifiable model, the analytic validation of results of the
model are extremely important. In all cases where it is possible great lengths are taken to
ensure that closed form methods of analysis match closely with their computational
counterparts. This careful analysis of results and comparison makes the tool
immeasurably more valuable as a tool for research for it requires a secure understanding
of the underlying optical effects being modeled. It is for these reasons that this work was
deemed appropriate to fulfill a need of the OSL, and carried out as a series of simulations
that build the versatility of the model.
The modeling conducted of Patricia Nieva’s MEMS cantilever was an evaluation
of the necessity for the detailed FDTD method for use in accounting the contributions of
various nontrivial geometries in the form of etch holes for device release. Only through
completing the model could the contributions of these elements be quantified, and
determined insignificant for her application. In all, only a power reflection change of
12% was observed, which though measurable did not alter the form of the result as a
function of beam displacement to necessitate the inclusion of the FDTD model into her
solution.
The intent of the confocal microscope modeling was to provide an answer to the
questionable results achieved in the lab for a particular type of incoherent quasi-
monostatic point detector that was experiencing signal dropouts at depths into human
skin tissue greater than that of the epidermal dermal junction. The objective was to
3
determine whether this behavior could be modeled simply by creating that junction and
the layers above and below it in computational space and finding the coherent (for
mathematical simplification purposes that will become apparent later) detected field.
Unfortunately this method of simulation did not generate the “dropouts” associated with
the real instrument so more skin features must be added to the model.
II. BACKGROUND
A) Case Definition and Requirements
Two optical systems will be analyzed. The first is one whose construction is
accomplished using a MEMS (microelectromechanical systems) fabrication process.
This process creates very small structures to tolerances of less than 1µm. Studying the
process by which MEMS devices are constructed gives a knowledge of the materials used
and their spatial distributions. The knowledge of materials and their properties would
include refractive index which is then used to compute the refractive and reflective
natures of the material composites. Additionally, geometric anomalies rising from the
requirements and limitations of fabrication techniques also can contribute to optical
analysis complexity.
The basic technology of MEMS fabrication is similar to that of IC (integrated
circuit) fabrication. The differences arise in size tolerances, material used in each
processing step, and their packaging and connections. Like IC fabrication, a MEMS
fabrication will have several steps of material layer deposition interleaved with
photoresist patterning and etch steps. The result is a sandwich of many layers of different
materials where certain layers have been patterned to form the desired structure. This
solid sandwich structure is then “released” by way removing a sacrificial SiO2 layer
4
which causes the once-solid structure to become a standing mechanical structure that can
actuate or sense in some meaningful way. Typical means of release are acid wash or acid
vapor exposure [19].
The particular device examined in this work is a singly fixed cantilever structure.
The spacing between the cantilever and substrate and their associated reflectivities acts as
a Fabry-Perot interferometer. Based on external vibration that spacing will change.
Patricia Nieva’s mechanical model of external vibration’s relation to cantilever spacing
change corresponds in a change of the system’s interferometric response. Combining
optical and mechanical models yields a means of measuring external vibration from
interferometric response. The second system to be analyzed is a quasi-monostatic
confocal microscope. Confocal microscopy is a means of obtaining an image from a
specific depth into a target through the focusing of incident light and spatial filtering of
the detected light. In this way, scatter outside the focus is strongly rejected making it
possible to resolve cellular sized objects in all three directions. This is inherently a point
detector that must be scanned over a surface to obtain a 2-D picture of the target at the
desired depth. The following illustration shows the basic premise of the confocal
microscopy point detection scheme. Notice that the focusing objective will send the light
to the focus only if there is no scattering before it reaches that point. By placing index of
refraction perturbations between the objective lens and the focus, the incident beam is
altered and the light may not go through the focal point or may not focus sharply. In such
a case the light may then be blocked by the spatial filter at the detector as is illustrated by
the red dashed path. If the light passes through the medium undeviated however, and
5
reaches the focus, it will be able to pass through the spatial filter at the detector because
the spatial filter is the conjugate point to the focus.
Figure 1: A simple confocal system where the scattered field originating at the focus is received and out of focus rays are rejected.
Only when a photon scatters from a point other than the focus and then scatters
again such that its final trajectory brings it through the spatial filter at the detector will a
received photon be one that did not come from the focus. This multiply scattered photon
is unlikely because of the two spatial filters in the system. First, the incident light is
restricted to a diffraction limited spherical wave converging upon the focus, and then it is
also spatially filtered at the detector. It will be shown that these two effects are
multiplicative in nature, which greatly increases the likelihood that the detected photons
are actually scattering from the focus, and therefore improves greatly the SNR (signal to
6
noise ratio) [1,2,3]. This is especially useful when the effect of scattering in the media
between the focus and the objective is great.
In the specific case that is analyzed, we consider human tissue, specifically skin,
to be the medium upon which the light is incident. In the model, the skin will include the
epidermal and dermal layer boundaries, and simple elliptical model cells randomly
positioned. The geometric variation is random, but with mean sizes taken from skin
images [4,5,20].
In both of these applications a result is achieved that practically useful, but its
analysis unattainable in closed form. For the case of the MEMS device, the system tested
is designed for use as a vibrometer that can function at high temperatures. Its intended
use is for jet engine vibration measurement [21]. By optically interrogating the device,
the need for temperature sensitive connective equipment is eliminated and the entire
system becomes more robust. In the case of the confocal microscope, the
characterization of subsurface abnormalities in the skin of human patients can be used for
surgical guidance in the treatment of skin cancers [4]. By modeling the skin in this way a
device capable of this detection can be more fully understood. Computational modeling
becomes necessary in both of these cases because of the geometric abnormalities
involved. Human tissue takes on a limited randomness in its shape and distribution at the
cellular level, and the etch holes used for acid release in the MEMS present an equally
difficult geometric distribution of refractive index to model analytically.
The required performance of the optical model used in both of these cases is
dictated by the information we wish to obtain. In the case of the MEMS model we
require the power reflection coefficient for the system, and for the confocal case we
7
require the field at the focal plane. These two results are each obtainable by a model
which can calculate the field everywhere in the computational domain, since that result
can be used to obtain the power reflection coefficient analytically and the field at the
focal plane directly. Therefore we can set a requirement on the model to obtain all of
these field values.
In order to accommodate the analysis required for both of these cases, an optical
system model with sufficient detail to account for the complexities of the geometry and
varying index must be constructed. This model must be able to calculate field values at
points and planes of interest in the system. Further it must be flexible enough to allow a
set of realistic coherent sources to be chosen as a stimulus for the system. Finally, it must
allow the specification of spatial variation in the index of refraction on the spatial scale of
that of the true system. There are several optical modeling systems that would allow for
these capabilities. However, additional specific case considerations and features limited
the choice of model further. The system did not suit the diffusive optics model for highly
scattering media since there was not a large enough attenuation of field in the material,
nor did it have regular enough geometry or great enough incident light bandwidth to
make a frequency domain analysis appropriate.
B) Analytic Methods
The MEMS case can be analytically modeled by the matrix method of multiple
beam interference [22]. This method of modeling uses a one dimensional approximation
of a multiple layer system of index of refraction to calculate a composite reflection
coefficient. Such a model, because of its one dimensional nature, can only specify the
distance that light travels through each medium with a single distance parameter,
8
disallowing the specification of any source other than plane waves, and any geometry
other than infinite dielectric sheets of material of some defined depths. However, it is not
limited in the specific type of index that is used, and a complex index can account for
attenuation. Further, an arbitrary number of layers can be specified to represent a
complex composite of materials as in the MEMS case. The specification of this method
is completely defined by the following equation
0 0
1 1 1i r t
t
U U M Un n n
+ = − , (1.1)
where n0 is the index of refraction of the material before the dielectric layers begin, nt is
the is the index of refraction after they cease and the U values are the fields incident,
reflected and transmitted respectively. Throughout this text U is used for electric field
except in those places where specific vector components are necessary. This allows the
variable E to be used for Irradiance. The matrix M is given by
cos( ) sin( )
sin( ) cos( )
iA B ks ksM n
C D in ks ks
− = = −
. (1.2)
In this matrix, n is the index of that a given layer, i is the imaginary number, k is 2π/λ and
s is the thickness of the layer. These matrices can then be cascaded to represent a
multilayer equivalent system matrix. To extract the reflection coefficient, we use
0 0
0 0
t t
t t
An Bn n C DnAn Bn n C Dn
ρ + − −=
+ + + . (1.3)
This is the field reflection coefficient which we square to obtain the power reflection
coefficient, the value which is sought in our MEMS model solution. The obvious
shortcoming of this method is the lack of multidimensionality. For any etch holes to be
9
taken into account in the multiple beam interference method of analysis it will be
required that at least a two-dimensional geometry be allowed.
In contrast to the matrix method used to obtain the reflected power from a
dielectric stack, the analytic methods employed to find the fields at a focal plane for any
optical system (including the confocal microscope) are somewhat complicated. The
standard method employed is generally spatial transform by using the Fresnel-Kirchoff
Integral formula for free space, or assuming one has the appropriate Green’s Function for
the intervening space, using that Green’s function. The formulation for this method is
given as
( ') ( , ') ( ) rSource
U r G r r U r dA= ∫ , (1.4)
where the function G is given by the Kirchoff approximation [11] as
2 2( ) ( )[ ]22
4
t tx x y yikz ikzike e
zπ
− + −
(1.5)
in the far-field approximation. This specialized Green’s function will only hold for free
space, and only for z>>λ. Therefore where the source is known but the region over
which the source must propagate is not free space, this approximation will not hold.
Specifically, for arbitrary geometries as exist in models of human skin, it would be
impossible to find an analytic expression for a Green’s Function without prior knowledge
of the fields at both locations and using an inverse method to obtain the function. Other
methods of solution that can be applied to this case include surface equivalent scattering
methods [11] or computational methods [6]. However, using the former also requires
more knowledge of fields than are given in the excitation condition.
10
One of the previous attempts at creating a different method of expressing the
effect of Green’s Functions with confocal microscopes was discussed by DiMarzio and
Lindberg [2,3]. This method uses the back-propagated local oscillator and a coherent
detection scheme in such a way that ideally, the signal received at the detector is simply
the signal at the focal plane. This makes use of the reciprocity condition and assumes
that the medium is linear. Since these conditions should hold in a simple model such as
the one proposed herein for human tissue, a coherent detection scheme could be used to
analytically determine the signal at the receiver, but only given that the focal plane signal
was obtained. To more explicitly detail the method of coherent confocal detection that is
required, an analysis of the system and its signals is necessary. The system is illustrated
below.
Figure 2: A confocal detection system which makes use of a beam splitter and collimated beams.
The signal is transmitted from the source, through the beam splitter and objective lens to
the target. Scattered light from the target is then sent back through the objective so that it
11
is recollimated, sent through the beam splitter again, and finally focused on the detector
through the detector objective. An alternate path is from the source, through the beam
splitter, off the mirror and directly through the detector objective and to the detector.
These two signals are summed at the detector and the signal is squared to obtain the
power. Writing these signals mathematically, some careful attention to notation is
required. At the transmitter a subscript ‘t’ will be used, the target will have no subscript,
and the detector will have a subscript ‘r’. The field at the source will be used throughout
to express fields at these locations by the use of the Green’s Function transforms. The
vector r will express all three spatial coordinates and its subscript will show its location.
In this way, we can write the field at the target as
( ) ( ) ( , )t t t t tTransmitter
U r U r g r r dA= ∫∫ , (1.6)
where the Green’s Function, g, above represents one trip through the scattering media
from the transmitter to the target plane. The field that is subsequently scattered from the
target to the detector plane is represented by
( ) ( ) ( , ) ( , )scat r t r t rTarget
U r U r g r r sp dVθ θ= ∫∫∫ , (1.7)
where Uscat is the field scattered from the target, and is given at the target by
( ) ( ) ( , )scat t t rU r U r sp θ θ= , (1.8)
and s and p are the field equivalent scattering parameters to the power scattering
parameters σ and ρ, as defined by
2 * *2
2 22
| (0) | ( ) ( )( ) | ( ) | U ss p pE r U rr
θ θ= = , (1.9)
where E is irradiance and *x is the complex conjugate of x .
12
Finally, the Green’s Function that represents the propagation from the target to the
receiver is ( , )rg r r . The signal ( )scat rU r is one of the two signals transmitted to the
detector, the other of which is the local oscillator reflected from the mirror directly to the
detector. This second signal is given by
( ) ( ) ( , )LO r t t FS t r tTransmitter
U r U r g r r dA= ∫∫ , (1.10)
where gFS is the free-space Green’s Function given in the description of the Fresnel-
Kirchoff integral above. The detected field is then
( ) ( ) ( )det r LO r scat rU r U r U r= + , (1.11)
and the detected power (with normalized wave impedance) is given by
2( ) ( ) ( ) ( )| |Opt r Opt r r scat r LO r rReceiver Receiver
P r E r dA U r U r dA= = +∫∫ ∫∫ . (1.12)
This is the total optical power that is received at the detector, and can be expanded into
three integrand terms, two of which represent the so-called self-power which is
proportional to the scattered field squared or the local oscillator squared. The other term
we will refer to as the cross-term and involves the product of the scattered field and the
local oscillator field. This terms is also called the coherent power term and can raise the
signal to noise ratio (SNR) of the scattered field by multiplying it by a potentially much
higher power local oscillator [1] shown as
. . (1.13)
The result of isolating only this one mix term in the expression is that it provides a
simplifying result to the expression for the Green’s Function. Expressing the scattered
*( ) ( )Mix scat r LO r rReceiver
P U r U r dA= ∫∫
13
field in terms of the source field in the above expression we can make that simplification:
*( ) ( ) ( , ) ( , ) ( ) ( , ) [ ] Mix r LO r r t r t t t t rReceiver Target Transmitter
P r U r g r r sp U r g r r dA dV dAθ θ= ∫∫ ∫∫∫ ∫∫ , (1.14)
where the integration over the transmitter simply results in the field Ut(r). Assuming that
the integration over receiver and target is reversible, and reciprocity exists such that
*( , ) ( , )r rg r r g r r= , (1.15)
then the local oscillator is spatially transformed from the receiver to the target by the new
Green’s Function *( , )rg r r . This is referred to as the back-propagated local oscillator
(BPLO). The function Ut(r) remains untransformed such that the final result is
*( ) ( ) ( ) ( , )Mix d LO t s t rTarget
P r U r U r dVµ ρ θ θ= ∫∫∫ . (1.16)
Considering this result, we can see that using a coherent detection scheme it is possible to
completely eliminate the need to specify the Green’s Function explicitly for the field after
the total trip through the optical system if one has the field at the focal plane (target) only.
Therefore, the field at the detector can be found from the field at the target. Further, in a
true monostatic system the BPLO will have the same form as the field at the target, and
the final form will simply take the value of the field at the target squared.
Removing the local oscillator from the system however creates a radical change in
the final result. Of course, there will no longer be a coherent detection scheme, and of
the power terms there will be only the one self term of the detected field squared.
Unfortunately however, no similar simplification as was accomplished in the coherent
method is possible. Though the Green’s Function can be made implicit, instead of having
a back-propagated local oscillator which is ideally equal to that of the field at the target,
14
the signal that is back-propagated is the detected signal itself. This is written
mathematically
*( ) ( ) ( ) ( , )Opt d scat t s t rTarget
P r U r U r dVµ ρ θ θ= ∫∫∫ . (1.17)
This result is in fact more difficult to obtain than actually finding the scattered field at the
receiver plane. In order to obtain ( )scatU r , ( )scat rU r must first be found, where
( ) 2
scat rU r is in fact the final signal required. Therefore, in order to calculate the
detected field incoherent detector, there is no simplification that eliminates the need to
find the effect of the field at the target propagating to the detector.
The above discussion of analytic methods of approach therefore leaves much
undetermined. In order to account for the effect of etch release holes in the MEMS
vibrometer device we must reject a one-dimensional analytic model. Further, in order to
predict the detected field in a confocal microscope that uses a coherent detection scheme
we need a means of finding the focal plane field distribution. The incoherent detection
scheme presents even more stringent demands requiring that any model that hopes to
determine the field at the detector must in fact model the entire optical system following
the field from the transmitter, on a round trip through the target and to the detector plane.
These demands are only met by computational methods, specifically, considering the fact
that all sources are coherent, they are best modeled by a time domain solution to
Maxwell’s equations, or the finite difference time domain (FDTD) method.
C) A Conceptual Overview of the FDTD Method
The FDTD method is an extension of Maxwell’s equations which is easily
adaptable to a computer system for solution. Noting the definition of the derivative by a
15
ratio of limits where the limit represents an infinitesimal difference in some value, the
finite difference method approximates this by substituting limits for extremely small
values. Therefore, in defining a finite difference method, one of the most important
factors is the chosen minimum unit for the parameter used. In the case of Maxwell’s
Equations, the full definition in three spatial dimensions and one time dimension would
require the discretization of all four of these measures by some minimum unit. It follows
that the chosen discretization unit will determine the maximum amount of error that can
be accrued by a successive series of these approximations. Further, in the case of the
approximation of Maxwell’s equations specifically, the time domain method will have a
limit based on wavelength of light used. In general, a limit of 0.1λmin is an accepted
minimum distance value that produces stable results [5,10].
Other limitations and considerations that must be accounted for when selecting
the minimum distance for discretization as well. In addition to failing to fall below a
maximum discretization error by choosing too large a δ, one can create a problem of
computational complexity by choosing one too small. This is a more platform dependent
decision considering that any choice below 0.1λmin should give stable results. The gains
in accuracy are observable at smaller discretizations, but at the cost of increasing the
computational complexity by a factor proportional to the ratio of discretization sizes
raised to the power of the number of spatial and time dimensions. For example, a three
dimensional (in space) model with δ =λmin/20 would require approximately 24=16 times
more calculations to be performed than the tent wavelength case. Further, in addition to
processing time, storage requirements will also increase. In order to reduce the required
computer resources, one must make great effort to ensure that the stored data required for
16
calculations does not exceed the amount of available RAM. Unfortunately, with a
memory allocation of 8 bytes for each data point (using double precision float data types)
and 6 fields in the full 3D model, even storing one complete time step can be extremely
costly. In contrast, the matrix method trial was run in considerably less than one minute,
generating the data trend. The total time for all computational measurements was in
excess of 2 computer days. The systems used to make these calculations were Pentium 4
based PCs without additional fast processing or unusually large fast memory access. The
standard RAM used in each system varied in quantity, but in all cases was less than 1GB.
The fact that this particular finite difference approximation is that of the time
domain expressions for Maxwell’s equations in differential form completely specifies the
solutions that we will derive in the following. The derivation of the set of discretized
equations used for FDTD was first proposed by Yee in 1966 [15]. This formulation is
commonly associated with the so-called Yee cell, a graphical aid to understanding what it
is that these equations actually solve. The two main equations in the 2D TM mode are:
1 12 2
1 1 , 1 ,, ,2 2
1 12 2
1 1 1, ,, ,2 2
( )
( )
j k j kj k j k
j k j kj k j k
n n n ntx x
n n n nty y
H H E E
H H E E
µδ
µδ
++ +
++ +
+ − ∆
+ − ∆
= − −
= + − (1.18)
1 1 1 12 2 2 21 1 1 12 2 2 2
1, , , , , ,[ ]n n n nn n t
j k j k y y x xj k j k j k j kE E H H H Hεδ+ + + ++ ∆+ − + −= + − − + (1.19)
are actually the discretized form of Ampere’s and Faraday’s Laws in differential form.
By stepping through all spatial indicies for every time step these two equations allow
solutions for all field values at all locations and times. The indicies j and k are spatial
locations while the index n is that of time. An index into space is multiplied by the
minimum space step described above as a maximum of 0.1λmin. This is referred to in the
equation as the value δ. The derivation of these equations will be shown later, however,
17
it is now sufficient to say that using Ampere’s law to find H values, and Faraday’s law
to find E values iteratively, these equations simulate the propagation of EM waves in
time and space.
In the equations for FDTD above, Maxwell’s equations are discretized so that an
iterative solution can simulate the propagation of an EM wave. In order to start this
propagation we need only specify the electrical parameters of the space in which we will
simulate. This is done on the same level as that of the spatial discretization itself. A
reference to a matrix which stores the parameters for each spatial increment is used to
execute this in code. Then each time step is able to reference the physical geometry of
the simulated system, and accurately simulate its reaction to impinging EM waves.
Normally the values stored relate to the wave impedance, and if there is loss in the model,
some indication of the conductivity of the material.
Finally, halting the propagation of the waves at mesh truncations can be more
complicated than it might otherwise seem. Any immediate truncation of the grid of wave
impedances would create a perceived change in material parameters and some reflection
would occur. In order to prevent the reflection of power from this imaginary boundary,
many schemes for artificially attenuating the impinging waves on the edge of the mesh
(or lattice termination) have been proposed [8,12,13,14,23]. Therefore, before one can
make a full solution to the problem of modeling a system in some surrounding space, one
must choose such a boundary condition to eliminate the unwanted reflections due to
lattice termination.
In the past, this method of computational modeling has been used for a
variety of systems. Since it is not restricted to a specific wavelength, the same technique
18
can be used for RF modeling as optical modeling. As a result, everything from antenna
arrays and stealth aircraft radar signatures to cell phone interaction with human tissue has
been analyzed using this method [5,6,7]. Additionally, there are codes commercially
available for companies wishing to obtain design criteria for their products that may rely
on or have sensitivity to impinging electromagnetic waves [24].
II. DEVELOPMENT OF THE COMPUTATIONAL MODEL
A) The FDTD Equations
The theoretical development of the computational model used to predict fields in
the cases considered herein is FDTD. This well established model is a means of
rewriting Maxwell’s Equations such that they can be expressed as discrete space and time
additions, subtractions, multiplications and divisions. These operations are easily
programmable at the high level language desired for implementation. The chosen
implementation language is MATLAB for its ease of use. Some discussion is appropriate
for the derivation of this simply realizable iterative solution to Maxwell’s equations
however, since the entirety of the success of the FDTD algorithm relies upon its ease of
use in as an iterative algorithm by a general computing platform.
For the purposes of this study, several assumptions are made concerning the
model that is used for simulation. The media are assumed to have the potential for field
loss by conductivity. Electrical parameters including the wave impedance and
conductivity will be assumed constant over frequency, which though it deviates from the
true nature of media will be sufficient given that all excitations are of a narrow band of
frequencies. This property of dispersion may actually play a significant role in the way
that broadband light interacts with the media in question, however, this study is limited in
19
that it will only cover the fields due to coherent source stimulation and since the effect of
second harmonics and other nonlinear effects are disregarded in our model as well, the
frequency of all concerned excitations and should fall within a designated narrow range.
Given this, a set of parameters optical parameters drawn from that frequency range
should serve to give accurate results despite this overall simplification of the model’s
treatment of dispersive loss.
The choice to neglect dispersion in the parameters that define the optical media in
question and the choice to include the non-ideal dielectric loss parameter of conductivity
shape the specific formulation of the two time-domain differential form curl equations we
will use in the derivation of their discrete counterparts. Faraday and Ampere’s Laws are
given in their differential forms as:
BEt
∂∇× = −
∂ (2.1)
DH Jt
∂∇× = +
∂ , (2.2)
where the auxiliary equations,
D E
B H
ε
µ
=
=, (2.3)
specify the relation of flux densities to field intensities by the physical parameters of the
medium. Here, µ=µ0 since there will be little contribution to wave propagation from
magnetic effects of the material at optical frequencies. In contrast, ε will be a space-
varying, frequency independent constant that will be expressed by proportionality to the
dielectric constant of free space by the index of refraction n:
20nε ε= . (2.4)
20
In cases where a frequency domain expression of these quantities are given, complex
values of both n and ε are allowable, with the imaginary part of the complex number
indicating the attenuation of field in the material. Since this is a time domain expression
however, we will use the conductivity σ, which gives rise to the so called conduction
current J in Ampere’s Law. It is helpful to derive σ from the imaginary part of the index
of refraction n, since a time domain solution is rarely discussed and complex indicies of
refraction are normally preferred in the field of optics [5,16]. The expression for
conductivity arises from n when one accounts for the frequency dependent loss expressed
in terms of the permittivity as
0riσε ε εω
= − . (2.5)
Solving using (2.4) and (2.5) the conductivity can be written as
02Re( ) Im( )n nσ ωε= , (2.6)
which, when combined with the expression from the conduction current,
J Eσ= , (2.7)
can be applied to the computational model as a means to implement the reduction of the
electric field as it propagates through space.
For the considered cases of the MEMS and confocal microscopy simulations, the
problems can be limited to a 2D analysis. There are losses in accuracy in this
approximation because there can be transverse plane variations that can not be expressed
with only one transverse direction. A better approximation is achieved in general in the
MEMS case because it does not have variations in the transverse plane that are larger
than the beam diameter. However the en-face model of a skin cell is inherently round
21
and larger than the beam diameter out of focus. Further, over that second planar
dimension abnormalities such as moles, hairs and sweat glands can create great variations
in the behavior of the confocal system that would not be accounted for by a simple 2D
model. Finally, a loss of the ability to express polarization states other than TM means
that the loss of polarization is not possible. In optical systems where this is important to
polarized optical components or the Fresnel reflection coefficients vary greatly depending
on the polarization state of the source due to extreme angles of incidence this would be a
decisive factor in determining the results. Since we are not adding large unique features
like moles, sweat glands and hair follicles to the skin model for confocal and Fresnel
reflection coefficients are small and at small angles these effects are negligible.
Therefore we may simplify the field components used in the computational model to
restrict E to a single direction z resulting in fields propagating in the x - y plane. H
will have components in x and y . By choosing a 2-D model, we make the assumption
that the change of all fields in z is zero, and the resulting field is called TM polarized.
The Yee cell shown below illustrates graphically what the centered time centered
space (CTCS) discretization method creates. The E-field component created by a spatial
change in H that is described by Ampére’s law is located centrally between the H-field
components which create it. Further, the reverse is true of the H components whose E
contributors are located equidistant from its location as described in Faraday’s law. One
can also, from this graphical representation of the fields, see how the difference in the H
fields surrounding a given E component create a time change in that component. The
fact that we measure the rotation of H fields strength about E to generate a time change
in E (Ampére’s Law) explains where the term curl originates.
22
Figure 3: The Yee Cell. E and H are never collocated to allow curl to be defined by finite differences.
For a TM wave with 0z∂=
∂, we have propagation along the x-axis and the E-field only
in z . H is then only in x and y . Faraday’s law can then be broken into the
components of x and y as:
ˆ : z xy tx E Hµ∂ ∂∂ ∂= − (2.8)
ˆ : z yx ty E Hµ∂ ∂∂ ∂= , (2.9)
and Ampére’s Law is given in one component in z by
ˆ : y x z zx y tz H H E Eε σ∂ ∂ ∂∂ ∂ ∂− = + . (2.10)
Discretization is made on this form using the following relations:
; ;x j x y k y x y= ∆ = ∆ ∆ = ∆ = ∆ (2.11)
c tr ∆=
∆. (2.12)
23
Now, the FDTD is found by applying finite difference approximations to (2.8) – (2.10)
and substituting (2.11) and (2.12). We represent a CTCS approach for the H-field, and E-
field. This is well documented by Yee, and results in the evaluation of the above figure
in finite differences [15]. We adopt the notation for simplicity. (2.8) – (2.10) then
become:
1 12 2
1 1 , 1 ,, ,2 2
1 12 2
1 1 1, ,, ,2 2
( )
( )
j k j kj k j k
j k j kj k j k
n n n ntx x
n n n nty y
H H E E
H H E E
µδ
µδ
++ +
++ +
+ − ∆
+ − ∆
= − −
= + − (2.13)
1 1 1 12 2 2 21 1 1 12 2 2 2
1, , , , , ,[ ]n n n nn n t
j k j k y y x xj k j k j k j kE E H H H Hεδ+ + + ++ ∆+ − + −= + − − + (2.14)
where the relations 1cµε
= and µηε
= simplify the terms in 2.13-2.14 by:
;t r t rηµ η ε∆ ∆
= =∆ ∆ ,
(2.15)
resulting in the final form of the FDTD for 2-dimensions:
1 12 2
1 1 , 1 ,, ,2 2
( )j k j kj k j k
n n n nrx xH H E Eη ++ +
+ −= − − (2.16)
1 12 2
1 1 1, ,, ,2 2
( )j k j kj k j k
n n n nry yH H E Eη ++ +
+ −= + −
1 1 1 12 2 2 21 1 1 12 2 2 2
1, , , , , ,[ ]n n n nn n
j k j k y y x xj k j k j k j kE E r H H H Hη + + + +++ − + −= + − − +
. (2.17)
Therefore we can predict the next time value of the E and H fields by knowing only the
impedance of the propagation space, and the current time step.
The FDTD equations can be used iteratively, using only enough memory to store
two complete time frames of field values at any given time. The equations above each
generate the “next” time index which can then overwrite the current time index for the
next iteration in the time loop. In this way, the fields are continually updated, relying
24
only on the values generated in the last iteration. During each time iteration the entire
space mesh must have these equations applied. Further, since the discrete version of
Ampére’s law relies upon the values generated by Faraday’s Law, we must execute them
sequentially.
B) Absorbing Boundary Conditions
The method by which the lattice is terminated is the absorbing boundary condition
introduced by G. Mür [8]. This particular boundary condition is derived from separating
scattered from incident field in some discrete area at the edge of the mesh. The resulting
equations for implementation of this boundary need only be applied to the electric field
portion of the model and will differ depending on boundary edge on which they are
applied. The advantages to this method of absorbing boundary condition are its ease of
implementation and the low additional demand on additional computer resources.
Though there are a number of additional applications of updates to the electric fields at
the borders for every time step, no additional storage is required. The disadvantage is
that not all of the field is attenuated using this method that would normally be reflected
by the boundary. Specifically, only fields propagating perpendicular to the border over
which the field is attenuated will be affected. One component of the Mür implementation
is shown below. The equation for the electric field in the z direction is given at the four
boundaries of the discrete mesh terminations.
25
maxj x= : 1 1
, 1, 1, ,n n n nj k j k j k j k
c tE E E Ec t
δδ
+ +− −
∆ − = + − ∆ +, (2.18)
at maxk y= :
1 1, , 1 , 1 ,
n n n nj k j k j k j k
c tE E E Ec t
δδ
+ +− −
∆ − = + − ∆ +, (2.19)
at minj x= : 1 1
, 1, 1, ,n n n nj k j k j k j k
c tE E E Ec t
δδ
+ ++ +
∆ − = + − ∆ +, (2.20)
and at mink y= : 1 1
, , 1 , 1 ,n n n nj k j k j k j k
c tE E E Ec t
δδ
+ ++ +
∆ − = + − ∆ +. (2.21)
In order to achieve a more significant reduction in field strength there are other
methods of attenuating fields that are independent of their direction of propagation. One
such method which is employed frequently in FDTD lattice terminations is the Perfectly
Matched Layer (PML) introduced by J.P. Berenger [12,13]. Its ability to attenuate fields
that propagate perpendicular and parallel to the boundary is derived from its formulation
of separation of E into components “due to” specific contributions of the curl of H . For
example, in our case, the curl of H will result in a single component of E in z , but the
PML would then separate that field into those rising from the change in xH and yH
so that the resulting fields can be attenuated individually.
Additionally, Berenger’s theory uses a non-standard form of Maxwell’s Equations
which includes a magnetic current term. Using the magnetic conductivity, Berenger
proved that with mσ σµ ε= there will be a perfect impedance match between a medium of
arbitrary impedance µεη = . This is known as the Perfectly Matched Layer condition
(PML). Without loss of generalization we can assume 0η η= or 377 ohms at the border
26
between PML and the rest of the discrete space. The FDTD algorithms for a non-
standard set of Maxwell’s laws will then govern the application of the PML algorithm:
m
m m
BE Jt
J Hσ
∂∇× = − −
∂=
(2.22)
DH Jt
J Eσ
∂∇× = +
∂= .
(2.23)
PML additionally states that only yH will be attenuated due to mσ , and so only the part
of zE which gives rise to yH will be attenuated byσ . This in turn causes a separation in
both Faraday’s and Ampére’s laws into 2 components:
ˆ : z xy tx E Hµ∂ ∂∂ ∂= − (2.24)
ˆ : z y m yx ty E H Hµ σ∂ ∂∂ ∂= + (2.25)
x xy z zx tH E Eε σ∂ ∂
∂ ∂= + (2.26)
yx zy tH Eε∂ ∂
∂ ∂− = (2.27)
Even though (2.16) and (2.17) are both in the z direction, we separate them
where x yz z zE E E= + , and x
zE is the term which gives rise to yH , and so is the value
which we strive to attenuate. Adding, we can get an expression for the unseparated
Ampére’s law, and then solve such that the E terms combine, giving rise to the
following equation in the frequency domain
11 y x zx yj H H j Eσ
ωεωε∂ ∂
∂ ∂− − =, (2.28)
which can then be rewritten in terms of an intermediate variable H
27
11y yxj Hσ
ωε
∂∂−
=H (2.29)
y y yj j Hωε σ ωε+ =H H . (2.30)
This intermediate variable is the source of additional storage in this method of field
attenuation. The need is obvious now that there will be a 33% increase in total storage
using this method when a 2D FDTD program is considered. Using equations for
Faraday’s Law, the combined equation for Ampére’s Law, and the auxiliary equation
obtained from (3.12), we can write four equations which completely define the
Maxwell’s Equations for a TM wave which prepares us for derivation of FDTD of the
PML
z xy tE Hµ∂ ∂∂ ∂= − (2.31)
z y m yx tE H Hµ σ∂ ∂∂ ∂= + (2.32)
y x zx y H j Eωε∂ ∂∂ ∂− =H (2.33)
y y yt t Hε σ ε∂ ∂∂ ∂+ =H H ; (2.34)
These 4 equations can then be discretized similarly to the standard FDTD equations
above (2.5-2.7) using the relations for discretization (1.5-1.6).
Solving 4.1-4.4 for the PML FDTD equations
1 12 2
1 12 2
, 1 ,, , [ ]n n n ntx x j k j kj k j kH H E Eµ
+ − ∆+∆+ += − − (2.35)
1 12 21 12 2
[1 ]1, ,, ,[1 ] [1 ]
[ ]tm t
t tm m
n n n ny y j k j kj k j kH H E E
σµ µ
σ σµ µ
∆ ∆∆
∆ ∆
−+ −++ ++ −
= + − (2.36)
1 1 1 12 2 2 21 1 1 12 2 2 2
1, , , , , ,[ ]n n n nn n t
j k j k y y x xj k j k j k j kE E H Hε+ + + ++ ∆
∆ + − + −= + − − +H H (2.37)
1 1 1 12 2 2 21 1 1 12 2 2 2
[1 ] 1, , , ,[1 ] [1 ] [ ]
t
t tn n n n
y y y yj k j k j k j kH Hσεσ σε ε
∆
∆ ∆
−+ − + −+ + + ++ +
= + −H H; (2.38)
28
Values for σ and mσ are given by data collected by Rappaport as [25]:
2( ) ; ; 0.021; 3.7pnimN n pµ
εσ σ σ∆= = = = , (2.39)
where i is the index into the dissipation layer, and N is the total number of dissipation
layers. This has shown 100dB attenuation of field at N=8 layers.
Using (2.4) and (4.9), the equations (4.5-4.8) are reduced to a more simple form:
1 12 2
1 12 2
, 1 ,, , [ ]n n n nrx x j k j kj k j kH H E Eη
+ −++ += − − (2.40)
1 12 21 12 2
[1 ]1, ,[ ], ,[1 ]
[ ]rm
rm m
n n n nry y j k j krj k j kH H E E
ση
ση
η σ
∆
∆
−+ −++ ∆+ ++
= + − (2.41)
1 1 1 12 2 2 21 1 1 12 2 2 2
1, , , , , ,[H H ]n n n nn n
j k j k x xj k j k j k j kE E r H Hη + + + +++ − + −= + − − + (2.42)
1 1 1 12 2 2 21 1 1 12 2 2 2
[1 ] 1[1 ] [1 ], , , ,[ ]n n n nr
y y y yr rj k j k j k j kH Hη ση σ η σ
+ − + −− ∆+ ∆ + ∆+ + + += + −H H
; (2.43)
The choice to proceed using the Mür method is based primarily on the memory
storage requirements. In addition to the storage required to propagate the wave, namely
the three fields for two time steps, there are auxiliary values, and actual recorded data that
must be stored. The largest of the auxiliary values are the media parameters of wave
impedance and electric conductivity. Additional values include global constants and
index values which take comparatively little memory space. However, the media
parameters must be the entire size of the space mesh matrix, and so contribute
significantly to the total memory storage. Finally, the objective of the simulation is not
simply to propagate the wave but also to make measurements of the field at various
planes over time. Only by capturing a field value over several complete periods while the
system is in steady state can the RMS and power values be obtained. Therefore, though a
plane will only use as much memory as one of the space variables, it will need to be
29
saved over the entire time variable’s space. A summary of the total space requirements
for a test confocal system 1580 by 1975 space units is shown in Table 1.
Table 1: Summary of memory requirements – FDTD Code for use in the confocal model
That the space requirements for even the Mür case would be so great is an indication of
the need to conserve what space remains. Adding an additional 47.62MB of space for the
variable H will further tax the memory requirements of the system. This will be
discussed further in the confocal results and conclusions sections of this document.
Additionally, little of the scatter observed when using the Mür method is due to unwanted
boundary reflection in the cases considered since most of the field power is directed
perpendicular to the right boundary excepting a minimal amount from scattering and
diffraction.
C) Excitation Functions
The FDTD formulation and boundary conditions function to set the stage for
wave propagation and scattering modeling. The gridspace conditions already discussed
dictate the mesh spacing requirements that form the computational domain in which a
solution is found. However, the actual propagation requires some impetus in order to
simulate the desired phenomenon. These excitation functions are crafted to simulate a
specific type of incident field that will impinge upon the scattering geometry. The
30
standard types of incident field considered are mathematical simplifications of actual
fields encountered that the model attempts to approximate.
Plane waves are the simplest approximations to an extremely distant point source.
In the very far field this excitation condition would be appropriate for certain modeling
problems. Its main advantage and use however is the simplicity of its implementation.
Since it is simple to perform analytic calculations with plane waves, they become the
natural choice for modeling by computational methods since it is relatively easy to verify
in general terms their results. It is also a good approximation for many other applications.
The form of the plane wave is given for waves traveling in the positive x direction as
0
0
ˆ ˆ
ˆ ˆ
i xz
i xy
E E z E e zEH H y e y
β
β
η
= =
−= =
;
. (2.44)
This can be implemented in the FDTD code by rewriting the formulas in their time
domain form and recognizing that specifying only the electric field is sufficient since
there is an absorbing boundary condition at one side of the field so that it will only
propagate in the other direction. This is analogous to specifying a current source at that
point in space since we can not actually have only a time varying E component without
an accompanying H . The time domain version of the above electric field is
0ˆ ˆcos( )zE E z E t x zω β= = − , (2.45)
and the discretized version of this equation will simply substitute for the values of t and x
the appropriate indexed values of displacement and time
0ˆ ˆcos( )zE E z E n t j zω β δ= = ∆ − . (2.46)
The specification of this boundary condition can be made as a constant along the entire
31
y-axis. This is in fact the definition of the plane wave as being a constant value in a given
plane perpendicular to the direction of propagation at a given point in time.
Increasing the relative complexity, most coherent light sources do not produce
waves that have constant amplitude in the plane perpendicular to the propagation
direction. In fact, the majority of laser light sources operate in one or more of the
Hermite-Gaussian modes, or the Gaussian beam profile which is Hermite-Gaussian mode
[00] [28]. This mode of operation is preferred because of its superior performance in
minimizing the effects of diffraction. Further, it is the preferred Eigenmode of most laser
cavities due to the prevalence of spherical mirrors, which are economical to produce.
Since a coherent source must be used so that the spectral components of the signal are
well defined and easily mathematically expressed, and the overwhelming majority of
coherent sources have Gaussian profiles, it is of great importance that a Gaussian profile
be able to be used as a stimulus source in FDTD. This is easily accomplished as an
extension of the plane wave stimulus model however, and is given by the equation for a
Gaussian wave at its center
2
0
0 ˆcos( )y y
zE E E e t x zσ ω β− −
= = − , (2.47)
which is simply a time harmonic oscillation as the one above with a Gaussian envelope.
The discretized form of this formula will again replace the time and space variables by
their discrete index values, but will vary over the y axis as
20
w2
m
0 ˆcos( )k
zE E E e n t j zδ
ω β δ −
− = = ∆ − , (2.48)
where m0 is the location of the Gaussian center, and w is the width of the Gaussian at the
maximum multiplied by 1e− , where the units of these values will be in gridspace units δ.
32
Finally, where the optical system modeled uses a lens and the computational
domain contains a focus, we may wish to show the focusing of a Gaussian beam to its
waist. In order to do this, we can start with a field that has a specified curvature given by
a spherical wave. That spherical wave will then be governed by the discrete form of
Maxwell’s Equations shown above in the definition of the equations for FDTD such that
the minimum waist radius associated with the diffraction limit will be observed. In this
way a relatively simple expression can effectively model the focusing of a field to a
Gaussian beam center. The formula is most easily expressed in spherical coordinates as
01ˆ ˆ
4i r
zE E z E e zr
β
π= = , (2.49)
where 2 2r x y= + and we have made the assumption that z=0 such that the field exists
only on the x-y plane in our 2D model. Though extended to three dimensions this
formulation would represent a cylindrical wave it is equivalent to the cross-section of a
spherical wave at the plane z=0. Rewriting this using the discretized variables that are
used in the FDTD model the expression becomes
( ) ( )
( ) ( )2 2
0 2 2
1ˆ ˆ4
c ci j j k kz
c c
E E z E e zj j k k
βδ
πδ
− + −= =− + −
, (2.50)
where it is important to additionally note that the change of variable includes the location
of the focal point at cx j δ= and cy k δ= . Finally, at this focal point, the expression
above would yield an infinite field. This is consistent with the specification of a spherical
wave and its impulse centerpoint, but this will not exist as such in the FDTD model since
the model since the diffraction effects that are inherent in the model do not allow focus to
a point.
33
D) Analysis of Time Domain Data and Test Cases
Specifying these various excitation conditions provides a versatile set of sources
to impinge upon the media specified. The computational domain which contains the
media and uses the FDTD propagation algorithm to model the scattering of these
excitation conditions will, after the model reaches steady state conditions, show the fields
from the chosen excitation condition. However, to extract from the literally billions of
data values the desired information requires some knowledge of the analytic behavior of
the fields, and some goal data to guide the analysis. In the case of field scatter models the
most important metric that one could hope to obtain would be scattered field at the source
plane. By specifying a plane of interest, it is possible to save every field value at every
time index at that plane location as the model executes by creating a separate variable for
this purpose with one time and one space dimension. The fact that a plane is specified by
one space dimension is due to the total spatial dimension reduction and the assumed
sameness of all fields over z . This data is shown in Table 1 as “Data Plane 1”. This
would be the total field in the plane, which is separable since the fields are linear, by
subtracting the excitation field as seen by the free space media parameter. This requires
that either the field be known analytically for the source for all space and time, or that the
model be run once with the scatterers present in the media parameter and once without
scatterers. The latter method provides a more elegant solution since any abnormality in
the source due to edge reflection or discretization error will be subtracted out as well.
Once the data is acquired in the plane of interest several post-processing
algorithms might be run on it depending on what the ultimate goal data happens to be.
34
For real optical systems the actual field at any point within the object is not something
that can be measured. Therefore, though useful for analysis of point spread functions and
the concentration of field for nonlinear effects, a focus field plane is not a useful real
world measurement. An actual receiving system would instead be capturing intensity
data such as irradiance or total power. To generate these values from a given data plane
we use the expressions
2 21 ( , )MAX
MAX PER
T
RMS ScatT T TPER
E E j TT = −
= ∑ , (2.51)
where TPER is the period of the signal, TMAX is the final time step of the simulation, and
2
1
MAXYRMS
Scatj
EP yη=
= ∂∫ (2.52)
where YMAX is the last space step in the y-direction. By computing the power in the
incident field in a similar manner and taking the ratio of scattered to incident power, the
reflection coefficient is obtainable.
For the above calculations it is necessary to determine the steady state period. To
make the code robust and avoid the need to perform calculations for every source
frequency chosen, the analysis code might make use of a fast Fourier transform (FFT)
algorithm as well to determine what period should be used in the averaging specified
above. This is a programmed function in most mathematical analysis tools, and the
details of such an algorithm does not need to be discussed here. However, knowing the
magnitude of the spectrum will allow one to easily isolate the period of the scattered field
by inversion
35
1per
max
Tf
= , (2.53)
where maxf is the frequency of the maximum magnitude component in the FFT signal.
The integration used for these discrete variables is the standard trapezoidal
method. By connecting all points in the variable of integration by straight lines and
dropping another line from each point to the horizontal, the created “cells” are trapezoidal
in shape. To find the total area and accomplish the integral, the areas just need be
summed. This is a standard algorithm for most mathematical analysis tools.
Determining steady state conditions is a sometimes difficult and precise definition
of such a condition can save a considerable amount of processing time if only the
necessary amount of indexed time is run in the computational model. We can get a
simple approximation for such an expression from the successive reflection model for a
system of planes of varying index of refraction. This model relies on each contribution to
the total field given by the successive Fresnel coefficients multiplied by themselves at
each interaction with a dielectric boundary. The power, which is the conserved quantity
is given by the field squared for reflection and transmission as
( )
22 2 1
1,2 1,22 1
22 1
1,2 1,22 1
21
n nRn n
nTn n
ρ
ρ
−= = +
= − = + .
(2.54)
Shown schematically below, a single beam of unity amplitude sent through a system of
varying index of refraction materials will have contributions from the multiple sources of
reflection, and depending on the dielectric at each layer, the interface reflection will have
a different contribution. Setting a limit on the minimum significant order, such as 0.01 or
36
0.001 of the original amplitude (here unity for purposes of demonstration), will allow one
to limit the number of reflections that are accounted for by the computational model. The
time index necessary can then be found by the Courrrant condition r and the spatial
relationships of the given dielectric boundaries.
Figure 4: Multilayer dielectric slab with labeled Fresnel reflection and transmission coefficients
The above power reflection and transmission coefficients show that the great majority of
power is transmitted in this case since the difference in the index of refraction is small in
this case (similar to the differences seen in human tissue). The primary reflection will be
R01, and the next will be T01T12R23T21T10. The successive values will all involve the outer
reflections R23 and R10 since the index of refraction difference of the two center values
are so small. In cases where coefficient of reflection values are small we can assume that
successive values will be less significant in general. Plotting the number of round trips
through the second and third layer vs. the total contribution of the last trip, we can see
that the decay is rapid and only four round trips will yield a 10-4P0 decay. The number of
round trips can be changed to a time index value by
37
( )( ) ( ), , 1max max res min res max minn j j a j j= − − + − (2.55)
where nmax is the calculated highest time index, jmax,res is the high-side border of the
resonator region, jmin,res is the low-side border of the resonator, jmax is the maximum space
index in the direction of propagation and jmin is the minimum space index (or detection
plane) in the direction of propagation.
1 2 3 4 5 6 710
-7
10-6
10-5
10-4
10-3
10-2
10-1
100
Number of round trips
Pow
er C
ontri
butio
n C
oeffi
cien
t
Successive Reflection Contributions
Figure 5: This graph shows the attenuation that occurs each round trip through the multilayer system
Of course a change in the values of index, specifically the differences between adjacent
layers, will change the number of round trips required to reach this level of attenuation.
However, the fact that small differences cause steady state to be reached much more
quickly is evident. This knowledge can be used in determining the number of round trips
necessary for the MEMS and confocal models to follow. Further, the actual final
scattered field can not be found by summation since there are phase delays that affect the
38
result. However, it is not the intent of this method to yield the actual scattered field, but
to determine the magnitude contribution of each successive reflection to limit the number
of reflections to allow the computational model to consider at a reasonable limit.
Of all cases, the simplest to determine the reflection is a single dielectric
boundary, forming a so-called dielectric half-space where there is no need for a multiple
reflection. All power reflected should come from the center dielectric mismatch, and no
power should be reflected from the boundaries since they are specified with absorbing
conditions. Further, limiting the case to a real dielectric mismatch as well eliminates the
complication of absorption in the model. Therefore it serves as a good test for the model
overall. The results of this test with n0=1 and n1=1.5, chosen to create a readily
observable reflection like that from glass in air. Using the FDTD computational model as
described, and analysis code was used to determine the power reflection coefficient.
Analytically, the method of Fresnel coefficients would predict
2 2
1 0
1 0
0.5 0.042.5
n nRn n
− = = = + , (2.56)
or a 4% power reflection coefficient. Simulated using the FDTD model a reflection
coefficient of 0.035 was obtained using plane waves and 0.0409 was obtained using a
Gaussian with a small width. The fact that the Gaussian in general has better diffraction
parameters and that it had room to expand transverse to the direction of propagation
could both lead to an ability to capture more of the reflected power than in the case of the
plane wave. Viewing the center of the Gaussian beam as a function of time, the
reflection of the wave can be seen in Figure 6.
39
Direction of Propagation (δ = λ/20)
Dis
cret
e Ti
me
Uni
ts (O
ne U
nit =
0.5
c/δ)
E-field resulting from a halfspace dielectric, Colorbar in Volts/Meter
20 40 60 80 100 120 140 160 180 200
100
200
300
400
500
600
700
800
900
1000-1.5
-1
-0.5
0
0.5
1
1.5
Figure 6: False color map showing the field intensity as a function of time and space showing reflection at
the halfspace dielectric boundary
Further, at the detection plane, which was chosen at gridspace location 40 shown in the
above figure, the entire plane was saved for all time resulting in a capturing of the total
field. Doing the same without the dielectric boundary, one can find the portion of the
total field due to the scatter (subtracting the source). This scattered field is shown in
figure 7.
40
Transverse to Direction of Propagation (δ = λ/20)
Dis
cret
e Ti
me
Uni
ts (O
ne U
nit =
0.5
c/δ)
Field at detector as a function of time, Colorbar in Volts/Meter
10 20 30 40 50 60 70 80 90 100
100
200
300
400
500
600
700
800
900
1000-0.25
-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
0.2
0.25
Figure 7: False color map showing the scattered field after the source has been subtracted from the total
field captured at the detection plane.
The scattered field can then be averaged over a period to find the RMS value and
integrated to obtain power as described above. The test of the computational model to
account for changes in real index of refraction was successful as it returned a value
within 2.25% of the analytic value.
Further complicating the demands on the model could involve the additional
concern for attenuation of field by complex index of refraction. One example of a lossy
dielectic material are AlN composites for use as microwave device packaging materials.
Such a material might have a typical loss tangent of 0.7 or more with dielectric constants
in the range of 20-40 at 8-12GHz. A specific doped composite we will model as εr=28 at
a loss tangent of 0.8 operating at 8GHz [26]. This is obviously not an optical component,
41
but will suffice for the testing of field attenuation by lossy dielectric. Loss tangent is the
ratio of imaginary to real dielectric constant. To find the index of refraction we can take
3. C.A. DiMarzio and S.C. Lindberg, “Quantitative analysis of coherent detection in a turbid medium,” SPIE. vol. 2676, pp. 121-129, (1996).
4. Rajadhyaksha, Milind; Gregg M.; Gonzalez, Salvador; Zavislan, James M.;
Dwyer, Peter J., “Confocal cross–polarized imaging of skin cancers to potentially guide mohs micrographic surgery,” Optics and Photonics News. vol. 12, no. 12, pp. 30, (2001).
5. Dunn, Andrew and Kortum-Richards, Rebecca. “Three-Dimensional
Computation of Light Scattering from Cells,” IEEE Journal of Selected Topics in Quantum Electronics. vol. 2, no. 4, pp. 898-905 (1996).
6. Taflove, Allen and Brodwin, Morris E. “Numerical Solution of Steady-State
Electromagnetic Scattering Problems Using the Time-Dependent Maxwell’s Equations,” IEEE Trans. MTT. vol. MTT-23, no. 8, pp. 623-630, (1975).
7. Luebbers, Raymond J. “A Finite-Difference Time-Domain Near Zone to Far
Zone Transformation,” IEEE Transactions on Antennas and Propagation, vol. 39, no. 4, pp. 429-433 (1991).
8. G. Mür, “Absorbing boundary conditions for the finite-difference approximation
of the time-domain electromagnetic field equations,” IEEE Trans. Electromag. Compat., vol. EMC-23, pp. 377-382, (1981).
9. Saleh B. and Teich M, Fundamentals of Photonics. New York: John Wiley &
Sons, Inc., 1991.
10. Sadiku, Matthew, Numerical Techniques in Electromagnetics. New York: CRC Press, 2001. pp. 158-218.
11. Balanis, Constantine, Advanced Engineering Electromagnetics. New York: John
Wiley & Sons, Inc., 1989.
12. J.P. Berenger, “Perfectly matched layer for the FDTD solution of wave-sturcture interaction problems,” IEEE Trans. Ant. Prop., vol. 44, no. 1, pp. 110-117. (1996)
69
13. J.P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” Jour. Comp. Phys., vol. 114, pp. 185-200. (1994).
14. M. Rajadhyaksha, M. Grossman, D. Esterwitz, R. Webb, and R. Anderson “In-
vivo confocal scanning laser microscope of human skin: Melanin provides strong contrast,” J. Invest. Dermatol., vol. 104, pp. 946-952, 1995.
15. K. Yee, “Numerical solutions of intial boundary value problems involving
Maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propagat., vol. AP-14, pp. 302-307, (1966).
16. Dunn, Andrew. “Light Scattering Properties of Cells,” Dissertation.
17. J. Maier, S.Walker, S. Fantini, M. Franceschini, and E. Gratton, “Possible
correlation between blood glucose concentration and the reduced scattering coefficient of tissues in the near infrared,” Opt. Lett. Vol. 19, pp. 2062-2064, 1994.
18. A. Brunsting and P. Mullaney, “Differential light scattering from spherical
mammalian cells,” Biophys. J., vol. 14, pp. 439-453, 1974.
22. R. Gunther. Modern Optics. New York: John Wiley & Sons. 1990.
23. B. Engquist and A. Majda, “Absorbing boundary conditions for the numerical
simulation of waves,” Math. Comput., vol. 31, pp. 629-651, 1977.
24. Light Tec. http://www.lighttec.fr/pages/pfdtd.htm. “OptiFDTD.” Last Visited: 3/24/04.
25. C. Rappaport. Interview. Northeastern University. March, 2003.
26. Sienna Technologies. “Lossy Dielectrics.”
http://www.siennatech.com/ST100.aspx. Last Visited: 3/24/04.
27. R. Ludwig and P. Bretchko, RF Circuit Design. Upper Saddle River, NJ: Prentice Hall, 2000.
28. Kogelnik and Li, Proc. IEEE 54, 1312, 1966.
70
CODE APPENDICIES: Simulation Code for the Dielectric Halfspace Test Case: %Chris Foster %10-31-03 %Testing Code for a Dielectric Halfspace clc; clear; close all; clear mex; pack; load ffile; XN=200; YN=100; E=zeros(XN,YN); Hx=E; Hy=E; Enew=E; Hxnew=E; Hynew=E; u0=4e-7*pi; %permeability of free space e0=8.854e-12; %permittivity of free space Eta0=(u0/e0)^0.5; %Ohms in free space. c=1/(u0*e0)^0.5; %m/s free space phase velocity. lambda=632.8E-9; %Laser wavelength in free space f=c/lambda; nsm=1.5; %index of refraction for silicon (substrate) lambdasm=lambda/nsm; %smallest wavelength (for use in delta gridspacing) del=lambdasm/20; %Space Step r=0.5; %Courrant Condition delt=r*del/c; %Time Step w=2*pi*c/lambda; %Angluar Frequency k=2*pi/lambda; %Wavenumber T=1/f; %period dt=T/delt; %number of discrete steps per period %Gaussian beam statistics: m=YN/2; %mean v=400; %Points of interest: sp=10; %source location dp=40; %detection location xcp=XMAX/2; ycp=YMAX/2; %For all time: tstep=100;
71
for tick=0:9 tick*100 for tock=1:tstep n=tick*100+tock; %Number of time steps. %STIMULUS: Y=1:YN; delay=0.5*(1+erf((n-20)/(5*2^0.5))); Es=delay*cos(w*n*delt).*exp(-((Y-m)./v).^2); %Gaussian Wave dx=(xcp-sp)*del; E(sp,Y)=E(sp,Y)+Es; %Update H's: X=2:XN-1; Y=2:YN-1; Hxnew(X,Y)=Hx(X,Y)-r./Eta(X,Y).*(E(X,Y+1)-E(X,Y)); Hynew(X,Y)=Hy(X,Y)+r./Eta(X,Y).*(E(X+1,Y)-E(X,Y)); %Update E's: X=3:XN-2; Y=3:YN-2; Enew(X,Y)=(1-sigE(X,Y)).*E(X,Y)+r.*Eta(X,Y).*(Hynew(X,Y)-Hynew(X-1,Y)-Hxnew(X,Y)+Hxnew(X,Y-1)); %Mür Absorbing Boundary Conditions on E: LF=(c*delt-del)/(c*delt+del); LFM=(u0*c)/(2*(c*delt+del)); Enew(X,2)=E(X,3)+LF.*(Enew(X,3)-E(X,2)); Enew(X,YN-1)=E(X,YN-2)+LF.*(Enew(X,YN-2)-E(X,YN-1)); Enew(2,Y)=E(3,Y)+LF.*(Enew(3,Y)-E(2,Y)); Enew(XN-1,Y)=E(XN-2,Y)+LF.*(Enew(XN-2,Y)-E(XN-1,Y)); %Save Updated Fields: E=Enew; Hx=Hxnew; Hy=Hynew; Edp(n,1:YN)=squeeze(E(dp,1:YN)); Ecp(n,1:XN)=squeeze(E(1:XN,YN/2))'; end end imagesc(Ecp) CMRmap colorbar xlabel('Direction of Propagation (\delta = \lambda/20)') ylabel('Discrete Time Units (One Unit = 0.5c/\delta)') title('E-field resulting from a halfspace dielectric, Colorbar in Volts/Meter')
72
Media Parameter Code for the Dielectric Halfspace Test Case: %Create a dielectric halfspace ffile to determine functionality. clear; close all; clc; e0=8.854e-12; u0=4*pi*1e-7; Eta0=(u0/e0)^0.5; n1=1; n2=1.5; XMAX=200; YMAX=100; Eta=ones(XMAX,YMAX).*Eta0; sigE=zeros(XMAX,YMAX); for x=XMAX/2:XMAX for y=1:YMAX Eta(x,y)=Eta0./n2; end end imagesc(Eta) colorbar R=((n2-n1)/(n2+n1))^2
73
Analysis Code for the Dielectric Halfspace Test Case: %Chris Foster %8-22-03 %Program to compute the power reflection coefficient: clear; close all; clc; pack; load scatter; Eout=Edp; load freespace; Eblank=Edp; YN=100; %last y-direction index Escat=Eout-Eblank; figure(1) imagesc(Escat) colormap(bone) colorbar title('Field at detector as a function of time, Colorbar in Volts/Meter') xlabel('Transverse to Direction of Propagation (\delta = \lambda/20)') ylabel('Discrete Time Units (One Unit = 0.5c/\delta)') Tmax=1000; %last time index %Determine the period of the signal: signal=squeeze(Escat(Tmax/2:Tmax,YN/2)); N=length(signal); Y=fft(signal,N); P=Y.*abs(Y)./N; f=(0:N/2)./N; [y,i]=max(P(1:round(N/2)+1)); Tper=round(5*1/f(i)); %5 periods. %Tper=120*2; %Find ERMS and Irradiance scattered back: Esquared=squeeze(Escat(Tmax-Tper:Tmax,:)).^2; Eaveraged=zeros(YN,1); for j=1:YN for k=1:Tper Eaveraged(j)=Eaveraged(j)+Esquared(k,j); end end Eaveraged=Eaveraged./Tper; ERMS=Eaveraged.^(0.5);
74
Eta0=(4*pi*1e-7/8.854e-12)^0.5; Irradiance=ERMS.^2./Eta0; %Calculate the PRMS (per meter) scattered back: PRMS=trapz(Irradiance); %Normalize by the power of the incident wave (per meter): Esquared_source=squeeze(Eblank(Tmax-Tper:Tmax,:)).^2; Eaveraged_source=zeros(YN,1); for j=1:YN for k=1:Tper Eaveraged_source(j)=Eaveraged_source(j)+Esquared_source(k,j); end end Eaveraged_source=Eaveraged_source./Tper; ERMS_source=Eaveraged_source.^(0.5); Irradiance_source=ERMS_source.^2./Eta0; PRMS_source=trapz(Irradiance_source); %Power reflectance: R=PRMS/PRMS_source;
75
Simulation Code for the MEMS Cantilever Case: %Chris Foster %10-31-03 %MEMS cantilever modeling using plane waves clc; clear; close all; clear mex; pack; load ffile; Etam=Eta; sigEm=sigE; XN=225; YN=1000; E=zeros(XN,YN); Hx=E; Hy=E; Enew=E; Hxnew=E; Hynew=E; u0=4e-7*pi; %permeability of free space e0=8.854e-12; %permittivity of free space Eta0=(u0/e0)^0.5; %Ohms in free space. c=1/(u0*e0)^0.5; %m/s free space phase velocity. lambda=632.8E-9; %Laser wavelength in free space nsi=3.841; %index of refraction for silicon (substrate) lambdasi=lambda/nsi; %smallest wavelength (for use in delta gridspacing) del=lambdasi/10; %Space Step r=0.5; %Courrant Condition delt=r*del/c; %Time Step w=2*pi*c/lambda; %Angluar Frequency %Gaussian beam statistics: m=YN/2; %mean v=5.5e-6/del; %variance %Points of interest: sp=10; %source location dp=20; %detection location for count=1:5 count Eta=squeeze(Etam(count,:,:)); sigE=squeeze(sigEm(count,:,:)); %For all time: for tick=0:39 tick*100
76
for tock=1:100 n=tick*100+tock; %Number of time steps. %STIMULUS: Y=1:YN; delay=0.5*(1+erf((n-20)/(5*2^0.5))); Es=delay*cos(w*n*delt).*exp(-((Y-m)./v).^2); E(sp,Y)=E(sp,Y)+Es; Edp(count,n,:)=squeeze(E(dp,:)); %Update H's: X=2:XN-1; Y=2:YN-1; Hxnew(X,Y)=Hx(X,Y)-r./Eta(X,Y).*(E(X,Y+1)-E(X,Y)); Hynew(X,Y)=Hy(X,Y)+r./Eta(X,Y).*(E(X+1,Y)-E(X,Y)); %Update E's: X=3:XN-2; Y=3:YN-2; Enew(X,Y)=(1-sigE(X,Y)).*E(X,Y)+r.*Eta(X,Y).*(Hynew(X,Y)-Hynew(X-1,Y)-Hxnew(X,Y)+Hxnew(X,Y-1)); %Mür Absorbing Boundary Conditions on E: LF=(c*delt-del)/(c*delt+del); LFM=(u0*c)/(2*(c*delt+del)); Enew(X,2)=E(X,3)+LF.*(Enew(X,3)-E(X,2)); Enew(X,YN-1)=E(X,YN-2)+LF.*(Enew(X,YN-2)-E(X,YN-1)); Enew(2,Y)=E(3,Y)+LF.*(Enew(3,Y)-E(2,Y)); Enew(XN-1,Y)=E(XN-2,Y)+LF.*(Enew(XN-2,Y)-E(XN-1,Y)); %Save Updated Fields: E=Enew; Hx=Hxnew; Hy=Hynew; end end end
77
MEMS Media Case for the Cantilever with Etch Holes: clear; close all; clc; pack; %Define Constants: e0=8.854e-12; %F/m u0=(4*pi)*1e-7; %H/m Eta0=(u0/e0)^0.5; nsin=2.0737; %index of refraction for silicon nitride (cantilever) nsi=3.841+i*0.0167; %index of refraction for silicon (substrate) c=1/(e0*u0)^0.5; lambda=632.8E-9; lambdasi=lambda/real(nsi); r0=0.5; %Courrant Condition del=lambdasi/10; %Space Step delt=r0*del/c; %Time Step w=2*pi*c/lambda; esi=nsi^2; sigsi=imag(esi)*w/real(esi)*delt; XMAX=225; YMAX=1000; gap=1500.*1e-9; for count=1:length(gap) %Define Geometry: f=zeros(XMAX,YMAX); for x=round(XMAX/4-0.45e-6/(del*2)):round(XMAX/4+0.45e-6/(del*2)) for y=1:round(YMAX/2-3e-6/del-2*0.2e-6/del-4e-6/del) f(x,y)=1; end for y=round(YMAX/2-3e-6/del):round(YMAX/2+3e-6/del) f(x,y)=1; end for y=round(YMAX/2+3e-6/del+2*0.2e-6/del+4e-6/del):YMAX f(x,y)=1; end for y=round(YMAX/2-3e-6/del-2*0.2e-6/del-4e-6/del):round(YMAX/2-3e-6/del-0.2e-6/del-4e-6/del)
78
if y<0.2/0.45*x+round(YMAX/2-3e-6/del-2*0.2e-6/del-4e-6/del)-0.2*round(XMAX/4-0.45e-6/(del*2))/0.45 f(x,y)=1; end end for y=round(YMAX/2-3e-6/del-0.2e-6/del):round(YMAX/2-3e-6/del) if y>-0.2/0.45*x+round(YMAX/2-3e-6/del)+0.2*round(XMAX/4-0.45e-6/(del*2))/0.45 f(x,y)=1; end end for y=round(YMAX/2+3e-6/del):round(YMAX/2+3e-6/del+0.2e-6/del) if y<0.2/0.45*x+round(YMAX/2+3e-6/del)-0.2*round(XMAX/4-0.45e-6/(del*2))/0.45 f(x,y)=1; end end for y=round(YMAX/2+3e-6/del+4e-6/del+0.2e-6/del):round(YMAX/2+3e-6/del+4e-6/del+2*0.2e-6/del) if y>-0.2/0.45*x+(round(YMAX/2+3e-6/del+4e-6/del+2*0.2e-6/del)+0.2*round(XMAX/4-0.45e-6/(del*2))/0.45) f(x,y)=1; end end end for x=round(XMAX/4)+round(0.45e-6/2/del)+round(gap(count)/del):XMAX for y=1:YMAX f(x,y)=2; end end figure(1) imagesc(f') axis image title('1 unit = lambda_S_i/10. Color differentiates boundaries. X and Y axis are gridspace.') %Calculate a matrix for Eta throughout the defined geometry: Eta=ones(XMAX,YMAX).*Eta0; for x=1:XMAX for y=1:YMAX
79
if(f(x,y)) if(f(x,y)==1) Eta(x,y)=Eta0/nsin; end if(f(x,y)==2) Eta(x,y)=Eta0/real(nsi); end end end end figure(2) imagesc(Eta') axis image colorbar title('Wave impedance.') xlabel('Color scale in Ohms') %Calculate a matrix for Sigma throughout the defined geometry: sigE=zeros(XMAX,YMAX); for x=1:XMAX for y=1:YMAX if(f(x,y)==2) sigE(x,y)=sigsi; end end end figure(3) imagesc(sigE') axis image colorbar title('Loss tangent times angular frequency times time step.') xlabel('1 unit = lambda_S_i/10. X and Y axis are gridspace.') figure(4) subplot(1,2,1); imagesc(Eta'); axis image; colorbar; title('Wave impedance (ohms).'); subplot(1,2,2); imagesc(sigE'); axis image; colorbar;
80
title('Field loss coefficient.'); Etam(count,:,:)=squeeze(Eta); sigEm(count,:,:)=squeeze(sigE); end Eta=Etam; sigE=sigEm;
81
MEMS Media Case for the Cantilever without Etch Holes: clear; close all; clc; %Define Constants: e0=8.854e-12; %F/m u0=(4*pi)*1e-7; %H/m Eta0=(u0/e0)^0.5; nsin=2.0737; %index of refraction for silicon nitride (cantilever) nsi=3.8411+i*0.0167; %index of refraction for silicon (substrate) c=1/(e0*u0)^0.5; lambda=632.8E-9; lambdasi=lambda/real(nsi); r0=0.5; %Courrant Condition del=lambdasi/10; %Space Step delt=r0*del/c; %Time Step w=2*pi*c/lambda; esi=nsi^2; sigsi=imag(esi)*w/real(esi)*delt; XMAX=225; YMAX=1000; gap=[1250].*1e-9; for count=1:length(gap) %Define Geometry: f(:,:,count)=zeros(XMAX,YMAX); for x=round(XMAX/4-0.45e-6/(del*2)):round(XMAX/4+0.45e-6/(del*2)) for y=1:YMAX f(x,y,count)=1; end end for x=round(XMAX/4)+round(0.45e-6/2/del)+round(gap(count)/del):XMAX for y=1:YMAX f(x,y,count)=2; end end figure(1) imagesc(squeeze(f(:,:,count))')
82
axis image title('1 unit = \lambda_S_i/10. Color differentiates boundaries. X and Y axis are gridspace.') colorbar %Calculate a matrix for Eta throughout the defined geometry: Eta(:,:,count)=ones(XMAX,YMAX).*Eta0; for x=1:XMAX for y=1:YMAX if(f(x,y,count)) if(f(x,y,count)==1) Eta(x,y,count)=Eta0/nsin; end if(f(x,y,count)==2) Eta(x,y,count)=Eta0/real(nsi); end end end end figure(2) imagesc(squeeze(Eta(:,:,count))') axis image colorbar title('1 unit = \lambda_S_i/40. Color indicates wave impedance. X and Y axis are gridspace.') xlabel('Color scale in Ohms') colorbar %Calculate a matrix for Sigma throughout the defined geometry: sigE(:,:,count)=zeros(XMAX,YMAX); for x=1:XMAX for y=1:YMAX if(f(x,y,count)==2) sigE(x,y,count)=sigsi; end end end figure(3) imagesc(squeeze(sigE(:,:,count))') axis image colorbar title('Loss tangent times angular frequency times time step.') xlabel('1 unit = \lambda_S_i/10. X and Y axis are gridspace.')
83
figure(4) subplot(1,2,1); imagesc(squeeze(Eta(:,:,count))'); axis image; colorbar; title('1 unit = \lambda_S_i/40. Color is impedance (in ohms).'); subplot(1,2,2); imagesc(squeeze(sigE(:,:,count))'); axis image; colorbar; title('Field loss coefficient.'); xlabel('1 unit = \lambda_S_i/10. X and Y axis are gridspace.'); end
84
Analysis Code for the MEMS Cantilever Case: %Chris Foster %8-22-03 %Program to compute the power reflection coefficient: clear; close all; clc; pack; load holes_1800_2000; Eoutm=Edp; load Eblank; Eblank=Edp; YN=1000; %last y-direction index for count=1:5 count Eout=squeeze(Eoutm(count,:,:)); Escat=Eout-Eblank; figure(1) imagesc(Escat) title('The scattered field at the detection point as a function of time') xlabel('Detector Width (Lambda_S_i/10)') ylabel('Time Steps') Tmax=4000; %last time index %Determine the period of the signal: signal=squeeze(Escat(Tmax/2:Tmax,YN/2)); N=length(signal); Y=fft(signal,N); P=Y.*abs(Y)./N; f=(0:N/2)./N; [y,i]=max(P(1:round(N/2)+1)); Tper=round(5*1/f(i)); %5 periods. %Find ERMS and Irradiance scattered back: Esquared=squeeze(Escat(Tmax-Tper:Tmax,:)).^2; Eaveraged=zeros(YN,1); for j=1:YN for k=1:Tper Eaveraged(j)=Eaveraged(j)+Esquared(k,j); end end Eaveraged=Eaveraged./Tper;
85
ERMS=Eaveraged.^(0.5); Eta0=(4*pi*1e-7/8.854e-12)^0.5; Irradiance=ERMS.^2./Eta0; %Calculate the PRMS (per meter) scattered back: PRMS=trapz(Irradiance); %Normalize by the power of the incident wave (per meter): Esquared_source=squeeze(Eblank(Tmax-Tper:Tmax,:)).^2; Eaveraged_source=zeros(YN,1); for j=1:YN for k=1:Tper Eaveraged_source(j)=Eaveraged_source(j)+Esquared_source(k,j); end end Eaveraged_source=Eaveraged_source./Tper; ERMS_source=Eaveraged_source.^(0.5); Irradiance_source=ERMS_source.^2./Eta0; PRMS_source=trapz(Irradiance_source); %Power reflectance: R=PRMS/PRMS_source; Rm(count)=R; end
86
Results Code for the MEMS Cantilever Case: %File that will assemble all the results into one file. clear; close all; clc; pack; load Rm_1000_1350; Rmt=Rm; load Rm_1400_1750; for n=1:length(Rm) Rmt(length(Rmt)+1)=Rm(n); end load Rm_1800_2000; for n=1:length(Rm) Rmt(length(Rmt)+1)=Rm(n); end xs=[1000:50:2000].*1e-9; del=632.8e-9/3.8411/10; xs=round(xs./del).*del; R(1,:)=squeeze(xs); R(2,:)=squeeze(Rmt); figure(1) plot(xs,Rmt,'.') hold on; plot(R(1,:),R(2,:),'r+')
87
Additional Results Code for the MEMS Cantilever Case: clear; close all; clc; pack; lambda=632.8e-9; nsi=3.8411; lambdasi=lambda/nsi; del=lambdasi/10; gap=[1000:2000].*1e-9; for count=1:length(gap) n0=1; nt=3.8411+i*0.0167; n1=2.0737; n2=1; s1=round(0.45e-6/del)*del/n1; s2=gap(count)/n2; term1=2*pi*n1*s1/lambda; term2=2*pi*n2*s2/lambda; m1=[cos(term1),-i/n1*sin(term1);-i*n1*sin(term1),cos(term1)]; m2=[cos(term2),-i/n2*sin(term2);-i*n2*sin(term2),cos(term2)]; m=m1*m2; rho(count)=(n0*m(1,1)+n0*nt*m(1,2)-m(2,1)-nt*m(2,2))/... (n0*m(1,1)+n0*nt*m(1,2)+m(2,1)+nt*m(2,2)); end figure(1) hold on; plot(gap,abs(rho).^2,'k') ylabel('Power Reflection Coefficient') xlabel('Cantilever Height (m)') axis([1e-6,2e-6,0,1]) load R_total plot(R(1,:),R(2,:),'rx') load R_holes plot(R(1,:),R(2,:),'b*') title('Matrix method (black), FDTD without etch holes (red x) and with (blue *):')
88
Matrix Method Code for the MEMS Cantilever Case: clear; close all; clc; pack; lambda=632.8e-9; nsi=3.8411; lambdasi=lambda/nsi; del=lambdasi/10; gap=[1000:2000].*1e-9; for count=1:length(gap) n0=1; nt=3.8411+i*0.0167; n1=2.0737; n2=1; s1=round(0.45e-6/del)*del/n1; s2=gap(count)/n2; term1=2*pi*n1*s1/lambda; term2=2*pi*n2*s2/lambda; m1=[cos(term1),-i/n1*sin(term1);-i*n1*sin(term1),cos(term1)]; m2=[cos(term2),-i/n2*sin(term2);-i*n2*sin(term2),cos(term2)]; m=m1*m2; rho(count)=(n0*m(1,1)+n0*nt*m(1,2)-m(2,1)-nt*m(2,2))/... (n0*m(1,1)+n0*nt*m(1,2)+m(2,1)+nt*m(2,2)); end figure(1) hold on; plot(gap,abs(rho).^2,'k') ylabel('Power Reflection Coefficient') xlabel('Cantilever Height (m)') axis([1e-6,2e-6,0,1]) load R_total plot(R(1,:),R(2,:),'rx') load R_holes plot(R(1,:),R(2,:),'b*') title('Matrix method (black), FDTD without etch holes (red x) and with (blue *):')
89
Skin Model Code for Confocal Modeling: clear; close all; clc; warning off MATLAB:divideByZero; e0=8.854e-12; u0=4e-7*pi; eta0=(u0/e0)^0.5; c=1/(u0*e0)^0.5; lambda=830e-9; del=lambda/10; XMAX=round(200e-6/del); YMAX=round(125e-6/del); board=zeros(XMAX,YMAX); reps=1.6; A=25e-6/del; T=XMAX/reps; x=1:XMAX; y=1:YMAX; D=A.*sin(2*pi*1/T.*(1:2*XMAX))-A/2.*sin(2*pi*1/(T/2).*(1:2*XMAX))+75e-6/del; for x=1:XMAX for y=1:YMAX if y<D(x) board(x,y)=1; else board(x,y)=2; end end end a=round(((rand*10+20)*1e-6)/4/del); b=round(((rand*10+20)*1e-6)/4/del); var=max(a,b); x0=2*a; y0=2*b; while y0<50e-6/del while x0<XMAX var=max(a,b); phi=rand*360; for x=x0-var:x0+var
90
for y=y0:y0+var r=((x-x0)^2+(y-y0)^2)^0.5; if r==0 board(x,y)=3; else theta=atan((x-x0)/(y-y0))-phi; if r<1/(cos(theta)^2/b^2+sin(theta)^2/a^2)^0.5-0.01; board(x,y)=3; end end end end for x=x0-var:x0+var for y=y0-var:y0 r=((x-x0)^2+(y-y0)^2)^0.5; if r==0 board(x,y)=3; else theta=atan((x-x0)/(y-y0))-phi; if r<1/(cos(theta)^2/b^2+sin(theta)^2/a^2)^0.5+0.01; board(x,y)=3; end end end end a=round(((rand*10+20)*1e-6)/4/del); var=max(a,b); x0=x0+2*a; end a=round(((rand*10+20)*1e-6)/4/del); b=round(((rand*10+20)*1e-6)/4/del); var=max(a,b); x0=2*a; y0=y0+2*b; end a=round(((rand*10+10)*1e-6)/4/del); b=round(((rand*10+10)*1e-6)/4/del); var=max(a,b); x0=2*a; y0=round(50e-6/del); while y0<YMAX while x0<XMAX var=max(a,b); phi=rand*360;
91
if y0<D(x0) for x=x0-var:x0+var for y=y0:y0+var r=((x-x0)^2+(y-y0)^2)^0.5; if r==0 board(x,y)=3; else theta=atan((x-x0)/(y-y0))-phi; if r<1/(cos(theta)^2/b^2+sin(theta)^2/a^2)^0.5-0.01; board(x,y)=3; end end end end for x=x0-var:x0+var for y=y0-var:y0 r=((x-x0)^2+(y-y0)^2)^0.5; if r==0 board(x,y)=3; else theta=atan((x-x0)/(y-y0))-phi; if r<1/(cos(theta)^2/b^2+sin(theta)^2/a^2)^0.5+0.01; board(x,y)=3; end end end end end a=round(((rand*10+10)*1e-6)/4/del); x0=x0+2*a; end a=round(((rand*10+10)*1e-6)/4/del); x0=2*a; b=round(((rand*10+10)*1e-6)/4/del); y0=y0+2*b; end figure(1) imagesc(board); axis image; board2=board(1:XMAX,1:YMAX); figure(2) imagesc(board2) axis image; nepidermis=1.34;
92
ncytoplasm=1.37; ndermis=1.4; for x=1:XMAX for y=1:YMAX if board2(x,y)==1 Eta(x,y)=eta0/nepidermis; end if board2(x,y)==2 Eta(x,y)=eta0/ndermis; end if board2(x,y)==3 Eta(x,y)=eta0/ncytoplasm; end end end sigE=zeros(XMAX,YMAX); figure(3); imagesc(Eta); axis image; xlabel('Propagation Direction, Units in \delta = \lambda/10'); ylabel('Transverse Direction, Units in \delta = \lambda/10'); title('Computational Domain for Confocal Skin Model showing Epidermal and Dermal layers'); Eta=Eta'; figure(4) imagesc(sigE); axis image; xlabel('Propagation Direction, Units in \delta = \lambda/10'); ylabel('Transverse Direction, Units in \delta = \lambda/10'); title('Computational Domain for Confocal Skin Model showing Epidermal and Dermal layers'); sigE=sigE'; save ffile Eta sigE;
93
Simulation Code for Confocal Modeling: %Chris Foster %10-31-03 %Confocal Modeling using an Apodized Gaussian Wave for zm=-400:100:400 for hl=0:1 clc; clear E Hx Hy Enew Hxnew Hynew Eta sigE; close all; clear mex; pack; zm hl load ffile; YN=2410; XN=1506; u0=4e-7*pi; %permeability of free space e0=8.854e-12; %permittivity of free space Eta0=(u0/e0)^0.5; %Ohms in free space. c=1/(u0*e0)^0.5; %m/s free space phase velocity. lambda=830E-9; %Laser wavelength in free space f=c/lambda; nsm=5.6506; %index of refraction for silicon (substrate) lambdasm=lambda/nsm; %smallest wavelength (for use in delta gridspacing) del=lambdasm/10; %Space Step r=0.5; %Courrant Condition delt=r*del/c; %Time Step w=2*pi*c/lambda; %Angluar Frequency k=2*pi/lambda; %Wavenumber %Points of interest: sp=10; %source location dp=40; %detection location NA=0.4; %numerical aperture angle=asin(NA); temp=tan(angle); xc=round(XN/2); yc=round(YN/2)+zm; %Gaussian beam statistics: if hl==0 m=round(yc+temp*(xc-sp)); %mean end if hl==1
94
m=round(yc-temp*(xc-sp)); %mean end %SHORT CODE FOR CALCULATING BEAM DIAMETER n=1.33; %index of refraction in water (water immersion lens) z=(xc-sp)*del; %distance on axis from focus i d0=2.3175e-6; %diameter at focus b=pi/4*d0^2/(lambda/n); %Rayleigh range d=d0*(1+(z/b)^2)^0.5; %beam diameter at source point %----------------------------------------- v=(d/2)/del; %variance xp=z+b^2/z; %radius of curvature ydel=round(abs(m-yc)+3*v); Eta=Eta(1:XN,yc-ydel:yc+ydel); sigE=sigE(1:XN,yc-ydel:yc+ydel); YN=2*ydel; yc=YN/2; if hl==0 m=round(3*v); end if hl==1 m=round(YN-3*v); end E=zeros(XN,YN); Hx=E; Hy=E; Enew=E; Hxnew=E; Hynew=E; %For all time: tstep=1000; for tick=0:2 tick*1000 for tock=1:tstep n=tick*1000+tock; %Number of time steps. %STIMULUS: Y=1:YN; delay=0.5*(1+erf((n-20)/(5*2^0.5))); for Yi=1:YN dy=(yc-Yi)*del; d=(xp^2+dy^2)^0.5; Es(Yi)=delay*cos(w*n*delt+k*d)*exp(-((Yi-m)./v).^2); %Gaussian Apodized Spherical Wave end E(sp,Y)=E(sp,Y)+Es;
95
%Update H's: X=2:XN-1; Y=2:YN-1; Hxnew(X,Y)=Hx(X,Y)-r./Eta(X,Y).*(E(X,Y+1)-E(X,Y)); Hynew(X,Y)=Hy(X,Y)+r./Eta(X,Y).*(E(X+1,Y)-E(X,Y)); %Update E's: X=3:XN-2; Y=3:YN-2; Enew(X,Y)=(1-sigE(X,Y)).*E(X,Y)+r.*Eta(X,Y).*(Hynew(X,Y)-Hynew(X-1,Y)-Hxnew(X,Y)+Hxnew(X,Y-1)); %Mür Absorbing Boundary Conditions on E: LF=(c*delt-del)/(c*delt+del); LFM=(u0*c)/(2*(c*delt+del)); Enew(X,2)=E(X,3)+LF.*(Enew(X,3)-E(X,2)); Enew(X,YN-1)=E(X,YN-2)+LF.*(Enew(X,YN-2)-E(X,YN-1)); Enew(2,Y)=E(3,Y)+LF.*(Enew(3,Y)-E(2,Y)); Enew(XN-1,Y)=E(XN-2,Y)+LF.*(Enew(XN-2,Y)-E(XN-1,Y)); %Save Updated Fields: E=Enew; Hx=Hxnew; Hy=Hynew; Efp(n,1:YN)=squeeze(E(xc,1:YN)); %Capture the field at the focal plane. Edp(n,1:YN)=squeeze(E(dp,1:YN)); %Capture the field at the detection plane. Ecp(n,1:XN)=squeeze(E(1:XN,m))'; %Capture the field at the Gaussian centerpoint for all time. end end imagesc(E) save(strcat('E',num2str(zm),'_',num2str(hl)),'E','Efp','yc') end end
96
Analysis Code for Confocal Modeling: %Analysis of Confocal Data clear; close all; clc; %Calculate the power at the detector at all 9 focal points tested: x=-400:100:400; for n=1:length(x) current0=strcat('E',num2str(x(n)),'_0.mat'); current1=strcat('E',num2str(x(n)),'_1.mat'); load(current0) Efp0=Efp; load(current1) Efp1=Efp; Efp_final=Efp0.*Efp1; YN=1546; Tmax=3000; %last time index %Determine the period of the signal: signal=squeeze(Efp_final(round(2*Tmax/3):Tmax,round(YN/2))); N=length(signal); Y=fft(signal,N); P=Y.*abs(Y)./N; f=(0:N/2)./N; [y,i]=max(P(2:round(N/2)+1)); Tper=round(5*1/f(i)); %5 periods. %Find ERMS and Irradiance scattered back: Esquared=squeeze(Efp_final(Tmax-Tper:Tmax,:)).^2; Eaveraged=zeros(YN,1); for j=1:YN for k=1:Tper Eaveraged(j)=Eaveraged(j)+Esquared(k,j); end end Eaveraged=Eaveraged./Tper; ERMS=Eaveraged.^(0.5); Eta0=(4*pi*1e-7/8.854e-12)^0.5; Irradiance=ERMS.^2./Eta0; %Calculate the PRMS (per meter) scattered back: PRMS=trapz(Irradiance); Irradiancem(:,n)=Irradiance(:); end
97
%Normalize by the power with no scatterer: load E0_blank_0 Efp0=Efp; load E0_blank_1 Efp1=Efp; Efp_final=Efp0.*Efp1; %Determine the period of the signal: signal=squeeze(Efp_final(round(2*Tmax/3):Tmax,round(YN/2))); N=length(signal); Y=fft(signal,N); P=Y.*abs(Y)./N; f=(0:N/2)./N; [y,i]=max(P(2:round(N/2)+1)); Tper=round(5*1/f(i)); %5 periods. %Find ERMS and Irradiance scattered back: Esquared=squeeze(Efp_final(Tmax-Tper:Tmax,:)).^2; Eaveraged=zeros(YN,1); for j=1:YN for k=1:Tper Eaveraged(j)=Eaveraged(j)+Esquared(k,j); end end Eaveraged=Eaveraged./Tper; ERMS=Eaveraged.^(0.5); Eta0=(4*pi*1e-7/8.854e-12)^0.5; Irradiance=ERMS.^2./Eta0; %Calculate the PRMS (per meter) scattered back: PRMS=trapz(Irradiance); for n=1:9 Pm(n)=trapz(Irradiancem(:,n)); end Pm=Pm./PRMS; %normalized power load ffile_final_data; monkey=size(Eta); XN=monkey(1); YN=monkey(2); yc=YN/2; xc=XN/2; hold on; x=-400:100:400;
98
for n=1:length(x) Eta_focus_m(n)=Eta(xc,yc+x(n)); end e0=8.854e-12; u0=4e-7*pi; eta0=(u0/e0)^0.5; plot(1./(Eta_focus_m./eta0),Pm,'b.'); title('Normalized Power at the Detector vs. Index of Refraction at Focus') xlabel('Index of Refraction (unitless)') ylabel('Power_d_e_t_e_c_t_e_d / Power_t_r_a_n_s_m_i_t_t_e_d') axis([1.3,1.5,0.1,0.12])
99
Free Space Normalization Code for Confocal Modeling: load E0_blank_0 Efp0=Efp; load E0_blank_1 Efp1=Efp; Efp_final=Efp0.*Efp1; imagesc(Efp_final) YN=1546; %Detector Width Tmax=3000; %last time index %Determine the period of the signal: signal=squeeze(Efp_final(2*Tmax/3:Tmax,YN/2)); N=length(signal); Y=fft(signal,N); P=Y.*abs(Y)./N; f=(0:N/2)./N; [y,i]=max(P(2:round(N/2)+1)); Tper=round(5*1/f(i)); %5 periods. %Find ERMS and Irradiance scattered back: Esquared=squeeze(Efp_final(Tmax-Tper:Tmax,:)).^2; Eaveraged=zeros(YN,1); for j=1:YN for k=1:Tper Eaveraged(j)=Eaveraged(j)+Esquared(k,j); end end Eaveraged=Eaveraged./Tper; ERMS=Eaveraged.^(0.5); Eta0=(4*pi*1e-7/8.854e-12)^0.5; Irradiance0=ERMS.^2./Eta0; %Calculate the PRMS (per meter) scattered back: PRMS=trapz(Irradiance0); P0=PRMS; save Irradiance_Norm Irradiance0 P0
100
Results Code for Confocal Modeling: clear; close all; clc; load Irradiance.mat hold on; figure(1) for n=1:9 plot(1:1546,Irradiancem(1:1546,n)); Pm(n)=trapz(Irradiancem(1:1546,n)); end figure(2) plot(1:length(Pm),Pm) load ffile_final_data; sizingvar=size(Eta); XN=sizingvar(1); YN=sizingvar(2); yc=YN/2; xc=XN/2; hold on; monkey=-400:100:400; for n=1:length(monkey) Etam(n)=Eta(xc,yc+monkey(n)) end e0=8.854e-12; u0=4e-7*pi; eta0=(u0/e0)^0.5; n=1./(Etam./eta0); figure(3) plot(n,Pm,'bd') xlabel('Index of Refraction (unitless)') ylabel('Power_d_e_t_e_c_t_e_d/Power_t_r_a_n_s_m_i_t_t_e_d') title('Normalized Power at the Detector vs. Index of Refraction at Focus') load Irradiance_Norm figure(4) Pm_norm=Pm./P0; plot(n,Pm_norm,'bd') xlabel('Index of Refraction (unitless)') ylabel('Power_d_e_t_e_c_t_e_d/Power_t_r_a_n_s_m_i_t_t_e_d') title('Normalized Power at the Detector vs. Index of Refraction at Focus')
101
figure(5) subplot(2,1,1) hold on imagesc(Eta) plot(1:YN,xc,'k') offset=-400:100:400; for n=1:length(offset) plot(yc+offset(n),1:XN,'k') end axis([1,YN,1,XN]) axis image title('False Color Impedance (\Omega) Image of Epidermis and Dermis') xlabel('Transverse to Direction of Propagation (\delta = \lambda/20)') ylabel('Propagation Direction (\delta = \lambda/10)') subplot(2,1,2); plot(yc+[-400:100:400],Pm_norm,'-b',yc+[-400:100:400],Pm_norm,'bX') axis([1,YN,0,1]) grid on grid minor title('Normalized Power for each trial') xlabel('Trial Location (\delta = \lambda/10)') ylabel('Normalized Power (unitless)') figure(6) hold on imagesc(Eta) plot(1:YN,xc,'k') offset=-400:100:400; for n=1:length(offset) plot(yc+offset(n),1:XN,'k') end axis([1,YN,1,XN]) axis image title('False Color Impedance (\Omega) Image of Epidermis and Dermis') xlabel('Transverse to Direction of Propagation (\delta = \lambda/20)') ylabel('Propagation Direction (\delta = \lambda/10)')