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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
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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
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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
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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.
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For Erin
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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
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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
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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
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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
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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
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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
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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
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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,
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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
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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.
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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
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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 .
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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= ∫∫
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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,
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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
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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
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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,
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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
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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
the square root of the relative permittivity as
( )28 0.8(28) 5.6506 1.9821rn j jε= = + = + , (2.57)
and that the absorption coefficient can be given by the imaginary part of the index of
refraction as
( )4 Ima nπµλ
= (2.58)
where the total field strength will decay exponentially according to this absorption
coefficient. Computationally modeling the same phenomenon involves the media
parameter sigma which is derived above in the explanation of how the FDTD model
handles loss. Comparing methods results in a close match in field intensity decay over
space as shown below.
42
0 5 10 15 20 25 30 35 40-0.02
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18Comparative Attenuation Plot - Computational (blue) and Analytic (black)
Spatial Units into the lossy dielectric (δ = λ/20)
Am
plitu
de o
f the
E-fi
eld
(V/m
)
Figure 8: Attenuation as a function of the distance traveled into and absorbing media.
The discrepancies in the above fields could be due to an overshooting of the field
as it approaches zero. Whereas the analytic model’s can never go to zero, we see the
computational model goes negative for a short time. Successive subtractions in the
computational model can cause such behavior if the discrete steps are larger than the
desired change in field amplitude. If this is the case then it is not unexpected for such a
discrepancy to occur. The total error is still small, MSE=0.0037, and the model produces
good steady-state results.
III. MEMS Vibrometer
The application of the above FDTD model to various optical devices is a natural
extension of its designed versatility. The first physical system that was tested using the
43
computational model was that of a vibrometer constructed by Patricia Nieva [21] using a
MEMS fabrication process. This process relies upon the deposition of various layers of
material in such a way that a three dimensional structure capable of some mechanical feat
is created. Specifically, here the structure desired is a singly fixed cantilever beam
composed of silicon nitride (SiN) resting atop a substrate of silicon (Si). This structure’s
purpose is to react in an analytically predictable way to external mechanical vibration.
Specifically, the reaction that takes place should be a bending of the cantilever such that
the spacing between the cantilever and the substrate is altered without permanently
deforming the cantilever beam.
Since the forces which cause the deformation are the vibration we seek to
measure, great effort has been placed in the mechanical model which specifies exactly
how the dynamic forces of vibration will cause the beam to deflect over time. The
measurement of this beam deflection, given the work already completed in mechanically
modeling the beam, can yield the stimulus vibration. Therefore, a means of obtaining the
information about beam displacement over time would yield the desired data on the
specific dynamic forces acting on the MEMS device, completing the job of the vibration
sensor. A schematic showing a top-down and side view of the beam is below.
44
Figure 9: Top down and side views of the MEMS cantilever beam structure with measurements
Figure Courtesy of Patricia Nieva
The beam under consideration is constructed of silicon nitride, and at rest is
placed 1.5um above a silicon substrate. The indices of refraction for the materials
concerned are 2.0737 for SiN, and 3.841+0.0167i for the Si substrate [21]. Mechanical
vibrations under working conditions deflect the beam ±30nm. By measuring the
reflected optical power which falls on a detector, the user of this device can observe the
resulting constructive or destructive interference caused by beam deflection. Further, the
vibration creates a periodically increasing and decreasing optical signal whose amplitude
and frequency can be related to those of the vibration [21].
The method of interrogation must additionally be insensitive to high temperatures
since the intended application of this vibration sensor is to test the vibration of jet engines
45
which can operate at temperatures in excess of several hundred degrees Celsius [21].
Therefore, the decision to optically interrogate the device was chosen because of its lack
of need for on-device connections or sensors that would be vulnerable to the extreme heat.
A beam of coherent light will reflect a different amount of light from the device
depending on the total optical path length of the spacing between the cantilever beam and
the substrate.
However, some factors complicate the Fabry-Perot model beyond the standard
analytic solution [9]. First, instead of a sheet, the SiN cantilever beam used for one of the
resonator walls has several holes etched into it to facilitate the release of the
microstructure by acid wash after fabrication. Without such holes, the wash would not
uniformly remove the SiO2 from between the fabrication layers. However, the fact that
the holes are somewhat beveled and lie in the interrogation beam path makes them
difficult to ignore from or add to the analytic model. By solving using the analytic model
and the computational model separately, a quantitative measurement of how the etch
holes affect the total reflectance of the cantilever beam can be obtained.
The spacing between the cantilever and the substrate differs in index of refraction
from the silicon nitride beam and the silicon substrate. Therefore, light entering the
spacing between these two dielectric interfaces will tend to both reflect back and transmit
out of the spacing according to Fresnel reflection and transmission coefficients. This is
essentially the same behavior as a resonator cavity, and obeys the equation governing
such structures and their modes of operation
2Fcqd
ν = , (3.1)
46
where νF is a resonant freqequency and q=1,2,3… such that for each of these resonant
frequencies the spacing between the walls of the resonator d occur where the field is at a
null allowing successive trips to constructively add in its so-called eigenmode. However,
off resonance, there is not a repetition of the field over successive trips and so the
summation will not always be constructive. The challenge is to obtain the reflection from
this resonator, or that part which in steady state is observed before the first surface, and
the transmitted part that is observed to leave the last surface.
The reflectivity of each of the two bordering interfaces of the cavity determines
the finesse of the cavity
0.5
1r=
rπ−
F , (3.2)
where r is the Fresnel reflection from each of the walls of the cavity. This in turn
contributes to the total reflection from the front of the cavity given by
( )
2
2
2 2
11 1
1 (2 / ) sin ( / )F
tr
R Tπ πν ν
−= − = −
+ F (3.3)
where t is the Fresnel transmission coefficient through the second interface [9].
Using these values will then yield the total reflection and transmission
coefficients where there is a sameness of material outside the cavity and a different
material inside, for a total of two interfaces to consider. However, in systems where there
are multiple reflective surfaces to consider, the previously mentioned matrix method of
successive reflections is necessary. The easiest way to derive this method is to use the
Fresnel coefficients in a system of linear equations specifying the contributions to the
field at each layer in the multilayer system. Such a set of equations can be generalized
47
into matrix form, such that an entire multilayer system can be expressed by an equivalent
matrix. This so called general system matrix for multilayer reflection modeling was
given as
cos( ) sin( )
sin( ) cos( )
iA B ks ksn
C D in ks ks
− = −
(3.4)
where the multilayer system can be specified by a set of equations relying on the
reflection coefficients of each layer for contributions as in the figure below.
Figure 10: Multilayer dielectric with signals of interest
The resulting equations from this model will be that the reflected field has a component
due to the reflection U0 and the transmission of U1. Written as a system of linear
equations they can then be solved yielding expressions like (3.3). From (3.3) individual
layers can be mathematically manipulated into matrix form such that the cascading
matrix of (3.4) is obtained. A similar method of analysis is sometimes used in RF
network analysis and results in the same equation form [27].
The initial problem is defined by a geometry that is discretized into a rectangular
grid pattern where each point in the grid is assigned a set of electrical parameters η and σ.
48
For each cantilever separation distance d, this grid must be reassigned a different value.
One example output of this grid configuration is shown for d=1.50um displaying η and
the field loss coefficient
Wave impedance (ohms).
50 100150 200
100
200
300
400
500
600
700
800
900
1000 100
150
200
250
300
350
Field loss coefficient.
50 100 150 200
100
200
300
400
500
600
700
800
900
1000 0
1
2
3
4
5
6
7x 10
-4
Figure 11: The media parameters that define the MEMS cantilever with etch holes for modeling reflections
In this figure, the x and y axis are in units of ∆, the space step, which here is one tenth of
the smallest wavelength or, λSi which is equal λ0/nSi=0.165um making ∆=16.5nm. This is
the minimum spatial resolution of the model. It could be made more precise, however, in
a 2D model the cost is the cube of the model size in processing time since 2 spatial
dimensions and 1 time dimension must be accounted for. In order to have an adequate
number of trips through the interferometer, a computation time of approximately 1.5
hours was required for each cantilever spacing. It was elected that several free spectral
ranges of the cavity would be tested to minimize the precision loss due to our rather large
49
gridspacing. Therefore, for the purposes of modeling, a total cantilever beam
displacement of 2um was tested and the results for power reflection were taken by matrix
method, FDTD method using the above geometry, and an FDTD method without etch
holes such that the Matrix method could be directly compared for consistency. The
results appear below:
1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2
x 10-6
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Pow
er R
efle
ctio
n C
oeffi
cien
t
Cantilever Height (m)
Matrix method (black), FDTD without etch holes (red x) and with (blue *):
Figure 12: Comparison between the FDTD model with and without etch holes to the matrix method
The measure of agreement used is the mean squared error which is given by:
[ ]2
1
1 ( ) ( )N
nMSE f n g n
N =
= −∑. (3.5)
50
The error between the reflection coefficients with no etch holes FDTD and those
calculated using the matrix method data was 0.0029, while the error between the matrix
method data and the FDTD set with etch holes was 0.0066.
In each of the data points in the graph above, a complete simulation taking 1.5
hours approximately was conducted to determine the scattered field resulting from the
interaction with the cantilever beam, spacing and substrate. The method of source
subtraction was then used to remove the source from the total field leaving the scattered
field only. This was then squared, time averaged over one period and it’s square root was
taken to obtain the RMS field, which was then squared and integrated to find the power.
The ratio of this power to the source power was taken to present the final power
reflection coefficient, which was compared to the matrix method above.
IV. QUASI-MONOSTATIC CONFOCAL COHERENT MODELING
The second case considered herein was that of the confocal microscope.
Specifically, we consider such a device whose detector and transmitter are not collocated.
For reasons pertaining to size and mobility it is not always possible to create a device
whose transmitter and receiver planes are conjugate one another as in Figures 1 and 2.
Specifically as shown in Figure 2, it would be necessary to add a beamsplitter, and two
objective lenses. Further, a mirror would be required to make the detector coherent. For
this reason, the more complicated problem of modeling a system with two distinct planes
for transmission and detection needs to be carried out in full. However, being able to
predict fields at the focus for an arbitrary source greatly reduces the otherwise
complicated nature of this problem.
51
Specifically, in this case, there are two waves which converge on the same point
in the absence of scatters but when placed above a scattering media they tend to drift
from the focus in differing ways based on the unique scattering media they each
encounter. Therefore, and individual computation for each of the sources must be carried
out, and their spatially collocated products must be taken to find the resulting detection
field. This process is illustrated below graphically to show the different locations of
source and detector and how each path travels through a unique set of scattering material.
Figure 13: Image of a quasi-monostatic confocal system which uses one objective lens
The previous assertion that the spatially collocated product will yield the detected field
arises as a result of the equation coherent detection where the back-propagated local
oscillator and signal at the target are multiplied and integrated over the target volume to
obtain the total power. However, in this case, the back propagated local oscillator is
entirely different from the transmitter field propagated to the target since the BPLO will
travel from the detector to the target, and that path is in fact one traversed by the detector
52
field. In essence, instead of the field being the square of the field at the focus as in the
monostatic case, we will have two fields with some collocated, and perhaps identical
values, but only where they overlap will this squaring effect be observed. Therefore, in
the absence of unique scattering paths only will the field be squared at the detector. In
other cases a minimal squaring effect depending on the spatial coherence of the two
independent scattering paths will result. Mathematically we can start from equation
(1.15), making the substitutions for path dependence evident with a superscript ‘t’ for
transmitter path and ‘r’ for receiver path (labeled in figure above)
*( ) ( ) ( ) ( , )LO
r tMix d t s t r
Target
P r U r U r dVµ ρ θ θ= ∫∫∫ . (4.1)
Further, as we have already stated, the BPLO should be the signal at the focus with the
path dependence of the returning receiver such that we can approximate the coherent
quasi-static confocal case by
( ) ( )( ) ( , )t
r tMix d t s t r
Target
P r U r U r dVµ ρ θ θ= ∫∫∫ , (4.2)
where it is seen that the two fields differ in path and that their locations in common are
multiplied resulting in the partially squared field described above.
This will hold for the coherent modeling case where we actually have a mixing power.
However, in the case where there is no local oscillator the mixing term goes to zero and
the incoherent power that we derived in (1.16) becomes the detected power. Again there
is now a path dependence such that we must substitute the superscript ‘t’ and ‘r’ for
transmitted and receive path respectively
*( ) ( ) ( ) ( , )scat t
r tOpt d s t r
Target
P r U r U r dVµ ρ θ θ= ∫∫∫ . (4.3)
53
However, here unlike in the coherent power, the signals are not to obtainable
simply using the computational model. The method of obtaining the scattered field above
would necessitate three trips through the scattering (one of which is carried out in the
second term as well) when in fact the simpler expression
( ) 2( ) ( , )
scat
trOpt d r s t r
Target
P r U r dVµ ρ θ θ = ∫∫∫ , (4.4)
where the field squared above is actually traced through the entire optical system, first
from the transmitter to the target, and then from the target to the receiver using the two
separate paths. Obtaining an expression for the field at the focal point is not difficult as
the FDTD program need only be given the input field along the transmit side and steady
state be reached by the system. Then a period of field at the focus can be saved so that
the RMS field can be obtained. However, directing the field back towards the detector
along a new plane is somewhat more difficult. First that RMS field must be transformed
into its frequency domain counterpart and the direction and curvature altered such that
the field will propagate towards the receiver location. This will not be explored further
since it is outside the scope of this FDTD model’s capability but will be discussed further
in the conclusions.
The incoherent model then requires the specification of the odd quasi-static
source conditions that will carry the signal along either the transmitter path to the focus
or the receiver path to that same location. The basic form of excitation source that will be
used is a Gaussian beam that is converging to a waist of diameter 2.32um. The
wavelength and Rayleigh range suffice to further specify the beam at 830nm and 5.08um
respectively. These values are determined by a known diameter before the Gaussian
encounters the lens (it is collimated prior to this) and the numerical aperture of the lens.
54
Some comment on numerical aperture is appropriate here since the lens is actually being
used by the transmitting and receiving paths, half of its focusing diameter used for each.
This means that since only half the lens is used, the numerical aperture of the lens is
effectively halved, and even less due to an obstruction to prevent the superposition of the
two paths’ field components. These effects combine to yield the relatively low
numerical aperture of <0.4 for the lens system.
To model the skin’s histology accurately, a brief discussion of the skin’s
components is appropriate. The layers of the skin are several, the top layer being
composed of dead skin cells which flake off and leave the body. These cells are large
and have a high index of refraction (≈1.5) leading to a large reflection. Below this, the
skin cells are alive and have an index much closer to that of water. This is the epidermal
layer and is the first layer that is modeled by the program. Because of the fact that the
cells are larger than the field of view of the computational domain and the scattering
would be so great, the dead skin layer was ignored for purposes of simplification.
Epidermal cells vary in size and their diameters decrease as the depth into tissue increases.
A smooth undulating boundary exists between the epidermal and dermal layers at which
a large change in index of refraction is also encountered (≈1.34 to ≈1.4). We consider
reflections after this boundary to be unimportant since the extinction of photons at this
depth in human tissue is great. The image below was commissioned and annotated by
Milind Rajadhyaksha and drawn by Borysenko in 1979. It illustrates well the layers of
human skin and how they are viewed by confocal microscopy. The indicies of refraction
are also given in the annotation.
55
Figure 14: X-Z sectioning with biological naming and properties of skin cells with X-Y annotations
concerning the relevance to confocal imaging
Applying the above model to the computational domain, a working discretized version
was generated. Using the diameters and indicies of refraction listed in the above figure
along with other sources [5,17,18,20], a composite model was generated. The results are
shown below where we have used n=1.34 for the extracellular fluid separating the
epidermal cells, n=1.37 for the cytoplasm/cell composite in the epidermis and n=1.4 for
the region below the epidermal/dermal boundary.
56
500 1000 1500 2000
200
400
600
800
1000
1200
1400
Transverse to Direction of Propagation (δ = λ/20)P
ropa
gatio
n D
irect
ion
( δ =
λ/1
0)
False Color Impedance (Ω) Image of Epidermis and Dermis
500 1000 1500 20000
0.2
0.4
0.6
0.8
1Normalized Power for each trial
Trial Location (δ = λ/10)
Nor
mal
ized
Pow
er (u
nitle
ss)
Figure 15: X-Z cross-sectioning of human skin epidermal and dermal layers for use by FDTD
and power at each detection location (normalized by freespace case)
In figure 14 on the top, the black lines intersect in nine locations. It is at these
locations that the beam was focused for confocal point detection for each case.
The resulting mathematical product of the detector and transmitter path focal
plane fields has the majority of its power concentrated at the center of the focal
plane because of the effect of the lens and the squaring of fields where there is an
overlapping of field values spatially due to the two beam paths. The
accompanying lower plot in Figure 15 is the power received at the detector (found
by integrating the calculated irradiance) divided by that found in a free space trial.
In this way the ratio of the powers can be expressed as
57
( ) ( )( ) ( )
,
,
r tt t
Target Perturbationsnormalized r t
t tTarget FreeSpace
E r E rP
E r E r=
∫∫∫
∫∫∫ (4.5)
where the irradiances in the numerator include the skin cell and DE layer boundary index
perturbations and the denominator is free of these perturbations. The resulting ratio of
powers gives a method of calculating the resulting detector power by additional scatterers
whose contributions can be determined in closed form individually. By then finding that
scatterer power and multiplying by the ratio above the total detector power can be
calculated. Figure 16 shows one example of the irradiance that leads to the powers
computed above.
0 200 400 600 800 1000 1200 1400 16000
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6x 10
-3 Detected Signal Irradiance
Irrad
ianc
e (W
/m2 )
Transverse Direction (δ=λ/10)
Figure 16: Irradiance measured at the detector by use of BPLO multipath modeling without skin
58
Finally since we expect to obtain some information by the total power returned to the
detector in each case about the material seen at that point we can compare the value of
the index of refraction at the focus to the power value located at the same point. This has
been done and is again shown graphically below.
1.3 1.32 1.34 1.36 1.38 1.4 1.42 1.44 1.46 1.48 1.50.1
0.102
0.104
0.106
0.108
0.11
0.112
0.114
0.116
0.118
Normalized Power at the Detector vs. Index of Refraction at Focus
Index of Refraction (unitless)
Pow
erde
tect
ed /
Pow
ertra
nsm
itted
Figure 17: Power obtained at the detector and normalized by free space power and their relation to the
index of refraction at the focus of the system
The three regions of skin detailed in the model are encountered at the focal point and a
power is returned for each of the nine focal points chosen. One focal point fell upon
extracellular fluid in the epidermis, four in the epidermal cells, and four in the dermis.
The power shown is normalized by the power detected the skin model is replaced by free
space.
59
V. CONCLUSIONS / DISCUSSION
The use of a computational means of generating field values in a complex
geometry of scattering media provides a powerful tool for analysis of real problems. The
application of the specific computational model of the finite difference time domain
approximation of Maxwell’s Equations allowed calculations to be carried out in an
iterative manner such that the fields could be generated over a discretized time and space
parameter in such a way as to completely specify the field behavior within the
computational domain. This in turn allowed researchers to apply this rigorous method of
solution to many problems that are difficult or impossible to solve analytically without
simplifications whose limits of error can not always be tested. The concept of being able
to handle any arbitrary geometry with a relatively simple iterative algorithm is one whose
simplicity of application is unparalleled by its analytic counterparts. The choice to
proceed in this manner to analyze the important problems that one encounters in field
scattering is one of ease and indeed efficiency as a very detailed solution is obtained with
minimal effort.
However, the nature of computational solutions to solve complicated field
scattering or wave interaction problems in a way that requires little or no thought to the
geometry of the media concerned save to specify it in the computational domain is
sometimes misleading and can lead to erroneous results. That is why great efforts have
been made to not only generate these computational solutions by also to verify them
using established analytic methods. In the case of the MEMS cantilever for example, we
have used the well-known matrix reflection coefficient model for multiple beam
interference modeling to generate a competing set of data. It was not known, upon the
60
generation of these data whether or not the etch holes in the cantilever beam would in fact
create a large difference in the reflected field. However, observing the solid beam media
data below we can see that in fact a one dimensional model can be more than sufficient in
showing how the multiple beam interference would create a reflection. That is why it
was used in addition to the etch hole model in the comparison. Instead then of simply
using the hole data and disregarding previously accepted means of computing the
reflection coefficient a meaningful application for the existing means was found and a
comparison drawn to show exactly how close the computational model did come to
predicting the correct values for reflected power.
Wave impedance (ohms).
50 100150 200
100
200
300
400
500
600
700
800
900
1000 100
150
200
250
300
350
Field loss coefficient.
50 100150 200
100
200
300
400
500
600
700
800
900
1000 0
1
2
3
4
5
6
7x 10
-4
Figure 18: Discrete media for the MEMS case with no etch holes
61
The behavior of the analytic model is considered to be representative of the “true”
case observed using the FDTD simulation. Only a 12% decrease in peak power is
observed due to the reduced first reflection term (from losses due to the absence of the
bar). Further, the fact that the 2D cross-section chosen was at the hole diameter means
that the effective slit aperture modeled was a worst case scenario. Assuming that the
incident light beam is centered on the point between etch holes as above, their effect is
minimal. It can be finally concluded that since the form of the closed-form resonance
effect remains the same, the entire FDTD modeling simulation is not necessary.
This concept of being able to use an analytic method to check some aspect of the
computational model does not fall merely on the results but is used throughout the
development of the computational model as a means of verification that the model is
behaving as it should. In fact the rigor of the model works to the advantage of one who
wishes to inspect it since in fact any given optical property can be observed in the FDTD
model and any such behavior’s absence is a warning to the user that an error exists in the
system. Examples of this, some of which have been covered in the testing section of the
development of the model, and others of which have simply been used in the generation
of the results as visual checks include but are not limited to
• A finite focal point dictated by wavelength
• Diffraction effects that vary based on beam profile
• Loss of field based on the conductivity of the region
• Lack of spurious fields originating at boundary conditions
• Expected field behavior for free space (i.e. source field specified over region)
• Correct reflections from simple geometries given known Fresnel coefficients
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• Expected propagation time (specified by Courrant condition)
• Fields settle to steady state in allocated calculated time (can be an indication of
spurious fields generated by incorrect absorbing boundary conditions)
All of these checks and more have been useful in the testing and generation of results
phases of the development of the computational model. Often criticism arises when
results are generated using a numerical approximation to Maxwell’s equations whether it
be the FDTD algorithm that has been used here, or perhaps the frequency domain model
(FDFD) because such a rigorous exploration of the solutions is obtained with relatively
little work. Indeed it is quite a lot of data that is generated by the FDTD solution to the
problem of the MEMS cantilever, when for all appearances all that was required was a
single number from each trial. Each trial however generated 1000X225X3X4000 values.
A total of 2.7 billion numbers were generated for every one used. Viewed in this way,
and in the light of all of the several observations (of which my list has but a few) that
must be made to assure that this mass of data is correct, the algorithm does not seem the
short cut it once did.
Indeed, by embarking on the task of creating a solution to a scattering problem
using FDTD one is not simply forgoing the analytic solution and opting for an easy
method of solution, but the statement is rather that the problem presents an obstacle not
easily accomplished by other means. The problem in fact may have solutions, but there
is doubt to whether or not the simplifying assumptions in these cases are justified, and the
ultimate judge of this is to painstakingly obtain the solution computationally so that the
questionable assumptions can be verified. That is what is meant by a practical
application of the computational modeling outlined herein. If by a reasonable method of
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analysis a problem is deemed solvable with dubious simplifications, it becomes a
candidate for analysis by computational methods. Further, if this problem is of an optical
nature, it may be suitable for modeling by FDTD.
It so happens that in the case of the cantilevered beam in MEMS our conclusion is
that there is not a significant contribution to the power reflected caused by the etch holes
such that a simple matrix model would suffice. This is based on the fact that the total
change between the solid beam cantilever and that with etch holes makes only a 12%
difference in power at the peak. Since the intent is to use the Fabry-Perot effect as a way
to judge cantilever height dynamically, and the oscillation in power is observable with
holes, there is no need to detail the loss due to the addition of said holes. Therefore, the
need for further computational modeling in these cases is lessened such that unless the
holes are significantly larger than these with respect to the total size of the structure, or
unless the materials are changed, it would not be necessary or even suggested. The great
amount of time invested in the computational modeling is simply not worth the small
fraction of error that is given above for the differences between the solid cantilever, the
matrix method, and the etch hole data. What is going to be observed is the relative
change over time of the reflection coefficient, and it is readily observable that the general
form of the data is the same in all three cases. The actual breakdown of power we would
expect is that in the case with holes some would be reflected away from the detector due
to the beveled edges, and some would be insensitive of cantilever position due to the fact
that it would reflect off the substrate alone (due to etch holes). Aside from this, the great
majority of power would reflect in the same way. Indeed a model of these three cases
could be done analytically and their composite could serve as a better analytic model than
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the standard matrix method. However, none of this would be known, and the error
percentages would not be limited unless this computational model was created. In this,
the model was successful.
In the case of the confocal microscope, conclusive results to confirm the behavior of
existing optical devices were not found. As previously mentioned there were three
distinct values of index of refraction used in this model. n=1.34 for extracellular fluid in
the epidermis, n=1.37 for epidermal cells and n=1.4 for the dermal layer. Figure 16
shows that dermal layer cells tend to accurately predict the power received at a value of
about 0.115P0 where P0 is the detected power in the absence of scatter. This value is
repeated within 0.005P0 for four focal points in the dermal layer. That gives a fairly
uniform result when compared to the range of power fractions received from the
epidermal layers from 0.102P0 to 0.117P0. This may be due to the increased irregularity
of the model in that region and the fact that the power is actually coming from a source
that exists over multiple discretized points. Further study of the impact of the
surrounding index of refraction may better reveal a trend in this data, however, because
of the length of time involved in obtaining data points a more powerful computing system
may be required.
However this variation in power is not actually significant. The ratio of the smallest
to largest normalized power obtained here was about 88%, and so the variation in the
received power is less pronounced than expected. In the laboratory such a similar
confocal system is prone to experience signal dropouts where the total power received is
much less than expected. Here that behavior is not observed as almost all of the values
fell within a very small range. Further changing the skin model used in the FDTD
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simulation to include more of the features of skin that may steer or scatter the two
(transmitter and receiver) beams apart from each other may create that dropout effect and
more closely match the data seen in the laboratory. Specifically, the Stratum Corneum
which has a high index of refraction and appears as angular planes on the scale of the
beam diameter could steer the beams by refraction.
In the case of skin modeling there exists other work that has detailed the behavior of
skin on a cellular level that reflects the behavior of organelles as scatterers to optical
wavelength light [16]. However, as a whole, the tissue has not been modeled in this way.
Confocal microscopy is well suited to the task of imaging this kind of tissue since it is
able to minimize the amount of light received that does not come from the focus. In this
way a skin tissue sample can be imaged at various imaging depths resulting in a skin
image that is sensitive to the boundaries between the epidermis and dermal layers where
skin cancers tend to form.
The application of an FDTD simulation to this problem, if done correctly, creates
a diagnostic tool that researchers using confocal microscopy on skin can benefit from
when they are testing their equipment. Assuming that the model of skin used here is
close enough to actual skin of patients in its distribution of the index of refraction and
location of various key features like the epidermal-dermal layers, then knowing the
detected field from such an instrument is instructive not only in knowing what the
detected field of the instrument should be expected to be, but in placing test cases within
the model as a means of determining, without the need for real samples, exactly how the
system will react to different media.
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This part of the research begs for greater development since the confocal system
modeled is currently limited to coherent detection schemes. In actuality the devices
being constructed by members of the Optical Science Laboratory at Northeastern,
specifically Peter Dwyer, rely on incoherent detection methods for their simplicity and
compact design. In order to achieve an incoherent modeling scheme in the FDTD code
used here additional functionality in the form of frequency transform implementation for
isolated focus fields, as well as a means of rotation of direction of fields once transformed.
The rotated fields could then be sent back through the system resulting in the squared
effect that is observed in incoherent detection.
The analysis of field propagation through coherent and incoherent detection systems
analytically has been explored thoroughly. In this way a reemphasis on the analysis of
systems as opposed to the blind application of computational methods has been asserted.
The analysis in fact dictated the application of specific methods of computation which
minimized the amount of time necessary to obtain detected field results! By a careful
analysis of the system using the technique of the BPLO and coherent detection methods a
simplification of the fields expected at the detector was obtained. In fact, there was no
need to actually find the far field approximation for these cases as was done in Dunn’s
paper on the same topic. Though application was limited to only those systems with
mixing power and coherent detection, this simplified result reduces the total
computational time by a factor of two over the incoherent model.
In considering how a composite model might benefit the MEMS case a similar
thought occurred that might benefit the confocal modeling case. Using the results of
Dunn’s cellular level model a composite of uniform scatter due to small organelles and a
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simple predictive model based on the results of this FDTD simulation might be possible.
Specifically, a point current source of appropriate amplitude might mimic the
omnidirectional sources of melanin and other small organelles within cells, while nucleus
level structure could be implemented even in the given constraints of the model as it
stands as a 2D FDTD simulation of PC implementable scale with respect to
computational complexity.
Finally, these two models as they exist in the computational domain have shown
likeness to their analytic counterparts where available, and present an accurate report on
the behavior of fields as they would exist in the presence of these scatters. The diversity
of application of FDTD is great, and the value of its results is as well. However, through
the development of the model and search for applicable problems, a certain amount of
discretion was learned. A computational tool is only a useful tool when used
appropriately, and one can only use it appropriately when a deep understanding of the
underlying problem is obtained prior to working with the discretized version. Though it
is possible, and it has been observed, that learning about the nuances of the problem can
occur as a result of observing solutions it is incredibly difficult to make sense of 2.7
billion values unless some knowledge of the problem has already been internalized. An
FDTD model then becomes a product of an analytic solution, and one that finalizes
knowledge of a particular problem and opens doors to the solution.
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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
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correlation between blood glucose concentration and the reduced scattering coefficient of tissues in the near infrared,” Opt. Lett. Vol. 19, pp. 2062-2064, 1994.
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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;
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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')
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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
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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);
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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;
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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
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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
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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)
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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
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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;
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title('Field loss coefficient.'); Etam(count,:,:)=squeeze(Eta); sigEm(count,:,:)=squeeze(sigE); end Eta=Etam; sigE=sigEm;
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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))')
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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.')
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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
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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;
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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
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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+')
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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 *):')
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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 *):')
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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
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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;
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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;
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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;
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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
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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;
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%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
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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
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%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;
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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])
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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
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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')
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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)')