Dielectric function parametric model, and method of use

Novel dielectric function parametric model oscillator structures comprised of finite order polynomials and/or essentially zero-width finite magnitude discontinuities in appropriate sequences, which novel oscillator structures are suitable for application in a Kronig-Kramer consistent dielectric function oscillator structure based mathematical model, are disclosed. The present invention method of application enables production of one-dimensional normalized dependent variable vs. independent variable evaluating look-up tables by application of convolution integration effected oscillator structure Gaussian broadening, as applied to finite order polynomials, without the requirement that numerical derivatives or integrations be performed. In use, addition of contributions from one or more said present invention oscillator structures allows determination of dependent variable values given independent variable values, without requiring subtraction of relatively large numbers. In addition only a relatively small set of essentially uncorrelated present invention oscillator structure finite order polynomial term coefficient and finite magnitude discontinuity magnitude defining mathematical model coefficients are required. The present invention dielectric function model oscillator structure mathematical model allows user determinable degrees of freedom which allow essentially any plot of dielectric function dependent vs. independent variable data to be modeled with a high degree of mathematical model coefficients.

TECHNICAL FIELD 
The present invention comprises novel Parametric Model Oscillator 
Structures and a method of their application; which Parametric Model, in 
use, requires determination of a relatively small set of essentially 
uncorrelated coefficients to accurately model, for instance, empirically 
obtained real and/or imaginary dependent variable components of the 
refractive index, (or dielectric function) vs. independent variable 
wavelength, (or equivalently the photon energy of an electromagnetic beam 
used in investigation). The present invention also allows convenient 
Gaussian Broadening of the novel Parametric Model Oscillator Structures 
involving a single normalized variable Convolution Integration, which 
results in a one-dimensional normalized variable look-up table, without 
the requirement that numerical derivatives or integrals be found. In 
addition, the method of application of the present invention does not 
require that error introducing differences of large numbers be found to 
evaluate much smaller dependent variable values. 
BACKGROUND 
Empirical investigation of substrate systems allows determination of the 
thicknesses of, and associated optical property component values, (eg. 
refractive index (n) and extinction coefficient (k), or real (e1) and 
imaginary (e2) complex dielectric function values), for films thereon. 
Optical property component values for substrate systems can be empirically 
obtained by many methods, such as those utilizing substrate system film 
effected changes caused in reflected and/or transmitted light beam 
intensities, and those utilizing substrate system film effected changes in 
ellipsometric PSI and DELTA values. However obtained, resulting optical 
property component values, when plotted as a function of the wavelength, 
(ie. photon energy, which comprises a beam of electromagnetic radiation 
utilized in the investigation), typically present as a rather irregular 
plot, which irregular plot can not be accurately represented by a simple 
analytical mathematical function over the range of wavelengths, (ie. 
photon energies), typically of interest, (eg. zero (0.1) ev to six (6.0) 
ev, and above when determined by other than ellipsometer systems). 
To understand the present invention, it is necessary to realize that the 
index of refraction of a substrate system film represents the ratio of the 
speed electromagnetic radiation traveling in vacuum to that in said film 
material, and that it is typically a complex number, (N=n+ik), where N is 
the Index of Refraction, n is termed the refractive index and k is termed 
the extinction coefficient. An alternative representation can be arrived 
at by the relation (e=N 2), and is termed the Dielectric Function, (ie. 
e=e1+ie2), where (e1=n 2-k 2) and (e2=2.times.n.times.k). It is also noted 
that the photon energy of a beam of electromagnetic radiation is related 
to the frequency thereof by the equation (E=h.times.F), where E is energy, 
(typically in electron volts ev), h is Plank's Constant, and F is 
Frequency. As well, Frequency, in electromagnetic radiation, is related to 
Wavelength by the relationship (C=F.times.W), where C is the speed of 
light and W is wavelength, hence (E=h.times.C/W). In addition, electrical 
properties of a material are related to the optical properties by known 
relationships. For instance, electrical conductivity is found to be 
proportional to e2. 
it is also important to understand that the real and imaginary components 
of the Refractive Index (ie. n and k), are not independent of one another. 
A principal of causality applies, known as the Kronig-Kramer, (K-K), 
relationship, and theoretically, allows determining k(E) when n(E) is 
known, and vice-versa. (Said (K-K) relationship relates the real part of a 
complex dielectric function to the imaginary part thereof by mathematical 
integration). Thus, assuming the Kronig-Kramer (K-K) relationship to be 
valid, it will be appreciated that empirical measurement of either n(E) or 
k(E) allows finding k(E) or n(E) respectively. And as stated infra, when 
k(E) and n(E) are known, e1(E) and e2(E) can be found by application of 
definite mathematical relationships, hence, the (K-K) relationship also 
relates (e1) and (e2). 
Continuing, other Dielectric Function Models have been developed for 
semiconductors. Adachi has developed a (K-K) consistent model suitable for 
below energy band-gap calculations that has been used to describe ternary 
alloys by interpolating parameters between binary end-point, (see, S. 
Adachi, "Optical dispersion relations for GaP, GaAs, GaSb, InAs InSb, 
Al(x)Ga(1-x)As, and In(1-x)As(y)P(1-y)", Appl. Phys., Vol. 66, p. 6030, 
1989). The ability to interpolate parameters produces much more physically 
realistic dielectric models than simply averaging dielectric functions of 
the end-point binary materials. For ternary alloy models, the strong 
Critical Point (CP) structure of semiconductors causes simple averaging 
schemes to produce doubled (CP) structures at the energies of binary 
endpoints, (see P. G. Snyder, J. A. Woollam, S. A. Alterovitz and B. Johs, 
"Modeling Al(x)Ga(1-x)As Optical Constants as Functions of Composition", 
J. Appl. Phys., Vol. 68, p. 5925, 1990). Forouhi and Bloomer have provided 
a dielectric model for semiconductors which is (K-K) consistent and has a 
very small set of coefficients, (see A. R. Forouhi and I. Bloomer, 
"Optical Properties of Crystalline Semiconductors and Dielectrics", Phys. 
Rev. B, vol. 38, p. 1865, 1988 and U.S. Pat. No. 5,905,170). The Forouhi 
and Bloomer model, however, has been found to have insufficient 
flexibility to fit existing dielectric functions accurately enough for 
ellipsometric modeling. Oscillator ensembles, (eg. harmonic and Lorentz), 
have been used to describe the above energy-gap behavior of some 
semiconductors and the AlGaAs alloy system, (see a paper by M. Erman, J. 
B. Theeten, P. Chambon, S. M. Kelso and D. E. Aspnes, titles "Optical 
Properties and Damage Analysis of GaAs Single Crystals partly Amorphized 
by Ion Implantation", J. Appl. Phys. vol. 56, p. 2664, 1984; and a paper 
by H. D. Yao, P. G. Snyder and J. A. Woollam, titled "Temperature 
Dependence of Optical Properties of GaAs", J. Appl. Phys., Vol. 70, p. 
3261, 1991; and a paper by F. Terry Jr., "A Modified Harmonic Oscillator 
Approximation Scheme for the Dielectric Constants of Al(x)Ga(1-x)As", J. 
Appl. Vol. 70, p. 409, 1991). These models have been used to fit measured 
ellipsometric data, however, they are incapable of describing 
direct-energy-band-gap spectral regions, and they require extra fictitious 
oscillators to fill in the absorption between (CP's). Kim and Garland et 
al. have developed a (K-K) consistent model that can adequately describe a 
semiconductor dielectric function above, below and through the fundamental 
direct energy-gap, and this model has been applied to the AlGaAs alloy 
system, (see C. C. Kim, J. W. Garland, H. Abad and P. M. Raccah, "Modeling 
the Optical Dielectric Function of Semiconductors: Extension of the 
Critical-Point-Parabolic-Band Approximation", Phys Rev. B, Vol. 45, p. 
11749, 1992; C. C. Kim, J. W. Garland, H. Abad and P. M. Raccah, "Modeling 
the Optical Dielectric Function of the Alloy System Al(x)Ga(1-x)As", Phys. 
Rev. B, Vol. 47, p. 1876, 1993). This model can accurately describe the 
dielectric function and higher order derivatives. However, to determine 
required internal coefficients a two stage fitting process is used. First 
(CP) energies and broadening are determined by fitting derivatives of the 
dielectric function, and then the remaining internal coefficients are 
determined with the energies and broadenings fixed. It will be appreciated 
that this model then requires that the dielectric functions exist before 
the model can be fitted. Furthermore, attempts at fitting all coefficients 
simultaneously (as necessary for direct ellipsometer data fitting), are 
unlikely to succeed because of the highly correlated nature of the 
functions internal to the model. Over part of the spectral range, the 
modeled imaginary part of the dielectric function results from the 
difference of internal function values one-hundred (100) times larger than 
the final value. The internal coefficients are delicately balanced to 
produce the proper output, and small changes in (CP) energies can cause 
large deviations in the model output. In addition, Lorentzian Broadening, 
which is known to be wrong for elements and compounds, is utilized in by 
Kim & Garland et al. work. While the Kim & Garland et al. work is very 
interesting, there remains need for an improved Parametric Model which: 
a. allows relatively mathematically simple application of Gaussian 
Broadening Factors; 
b. allows reduction of Correlation between Model Coefficients; and 
c. does not require the calculation of a difference between two relatively 
large numbers to arrive at a much smaller Parametric Model result. 
Continuing, as applied in an ellipsometry based environment, a Dielectric 
Parametric Function Model can be characterized by Requirements, some 
essential, some desirable and some which are preferable, as follows: 
1. First, a Dielectric Parametric Function Model must have sufficient 
flexibility to describe measured ellipsometric data such that it can fit 
previously available dielectric functions determined by spectroscopic 
ellipsometry, reflectivity and by any other technique. 
2. Second, a Dielectric Parametric Function Model must be able to fit model 
coefficients when extracting optical constants from measured PSI and DELTA 
data, including layered samples. That is, the model must work when optical 
constants are not known a-priori, and optical constant derivatives are not 
available. 
3. Third, a Dielectric Parametric Function Model should be internally (K-K) 
consistent to satisfy causality considerations. 
4. Fourth, a Dielectric Parametric Function Model should be based on 
analytical functions or loop-up tables, (preferably one-dimensional), to 
enhance speed of operation. That is, the model should not require use of 
numerical derivatives and integrations, which often make it prohibitively 
slow to calculate new Dielectric Function values each time a parameter is 
adjusted. 
5. Fifth, a Dielectric Parametric function Model should have sensible 
ranges for internal parameters such that the model remains stable. For 
example, the model should not require the difference of two large numbers 
be calculated. 
6. Sixth, a Dielectric Parametric Function Model might provide some 
physically relevant parameters so that a user can have some way to compare 
models and set up models for new materials based upon models for similar 
materials. 
7. Seventh, a Dielectric Parametric function Model might be suitable for 
application to a family of semiconductor dielectric functions related by 
composition (ternary alloys), temperature and doping etc. 
As regards the identified prior art models discussed infra, Adachi meets 
Requirements 3, 4, 5, 6, and 7 above and Requirement 1 below a fundamental 
energy band-gap only. The Forouhi and Bloomer model meets Requirement 2, 
3, 4, 5, 6 and 7. Oscillator based models meet similar Requirements, and 
in addition meet Requirement 1 above a fundamental energy band-gap only. 
The Kim-Garland et al. model meets Requirement 1, 3, 4, 6 and 7. 
Need exists for a dielectric Parametric Function Model which addresses all 
seven of the Requirements recited above, which model is highly suitable 
for utilization in an ellipsometry based environment. In particular, where 
a model of a substrate system is provided, and values for one or more film 
thicknesses and the optical parameters thereof are to be simultaneously 
determined by a regression approach, there is a definite need for a 
convenient means to determine Dielectric Function values. 
The purposes of the present invention are summarized by stating that the 
present invention meets the identified need for a Dielectric Parametric 
function Model, and to various degrees addresses the above recited 
Requirements. 
DISCLOSURE OF THE INVENTION 
The basis of the present invention is that tabulated empirically obtained 
numerical data relating various dependent and independent numerical 
parameters to one another can be represented by graphs and mathematical 
relationships, and it is often more desirable to evaluate a mathematical 
relationship to arrive at a required dependent numerical value, which 
corresponds to an independent numerical value, than it is to refer to a 
table of data which inherently presents related dependent and independent 
variable data, but might require an interpolation between specifically 
provided independent variable data points. For instance, complex 
dielectric function data can be stored in tabular form such at real (e1) 
and imaginary (e2) components are provided as inherently related to 
independent variable photon energy. Over a range of six (6) electron 
volts, for instance, (a reasonable range where semiconductors are 
involved), anywhere from hundreds to thousands of independent variable 
photon energy data points might be present, for each (e1) and (e2), to 
provide a desired level of accuracy. If the data is assumed to be 
Kronig-Kramer (K-K) consistent, a table of data points of one thereof, 
(ie. (e1) or (e2)), will be sufficient to allow determination of both (e1) 
and (e2) at a given photon energy, thus reducing by half the number of 
tabulated data points necessary to allow finding both (e1) and (e2) as a 
function of energy level. This is because (K-K) consistent (e1) and (e2) 
data are related to one another by a mathematical Integration 
relationship. However, over a range of six (6) electron volts, for 
instance, there is still need to evaluate half the original number of 
hundreds to thousands of data points in tabular data storage is utilized, 
and need remains to perform interpolation, to arrive a specific values of 
(e1) and (e2) at a specific photon energies. (Note, theoretically, data is 
necessary over all photon energies from zero to infinity to apply the 
Kronig-Kramer (K-K) consistency relationship. In practice, however, a 
reasonable analytical function is utilized to model unknown data). 
It should be appreciated that a mathematical relationship which allows 
calculation of (e1) and/or (e2) at a photon energy utilizing a relatively 
small number of stored numerical coefficients, would be of utility. As 
alluded to in the Background Section, various researchers have forwarded 
approaches to providing such mathematical relationships. However, none 
have been able to provide a mathematical model which is generally 
adaptable to describe essentially any dependent vs. independent variable 
data set, such as dielectric function components of materials over a 
reasonable range of photon energies, which range of photon energies can 
include both direct and indirect semiconductor bandgap regions, and 
regions therebelow and thereabove. As best appreciated by reference to 
graphed dielectric function component data, (see detailed Description 
Section and accompanying Drawings), (e2) varies in a complex manner with 
energy, with dependent variable "Peak" and Peak Interconnecting regions 
present over a range of independent variable photon energies. No simple 
analytical mathematical equation can provide a good fit thereto over a 
full range of photon energies based upon, for instance, a "Mean Square 
Error" (MSE) criteria. (See NUMERICAL RECIPES IN C, Cambridge University 
press, 1988 for a description of a (MSE) approach to data fitting. Said 
text is incorporated by reference into this Disclosure). 
One approach to "Modeling" (e1) and/or (e2) data as a function of energy 
has focused upon the use of a plurality of Lorentzian or Gaussian 
"Oscillator Structures" centered at various photon energy levels, each 
such Oscillator Structure having associated therewith an Amplitude and a 
Broadening Factor. (See "OPTICAL PROPERTIES OF SOLIDS", by Wooten, 
Academic Press, 1972, p. 47-52 for a description of Lorentzian Oscillator 
Structures, which reference is incorporated by reference hereinto). By 
summing contributions from various Oscillator Structures present at a 
photon energy, one can then arrive at a value of, for instance, (e2) at 
said photon energy. It will be appreciated that only three (3) parameters 
must be stored to fully define each Lorentzian Oscillator Structure, (eg. 
Amplitude, Broadening Factor and Central Energy), thus the number of 
parameters which must be stored to allow arriving at a value of (e2) at a 
given photon energy can be greatly less than where data is stored in, for 
instance, a tabular form. 
The problem with prior modeling approaches which utilize the summation of 
the effects of a plurality of Oscillator Structures to provide approximate 
values for (e1) and/or (e2) as a function of photon energy, is that they 
do not provide accurate values of (e1) and/or (e2) at all values of photon 
energy. While a Lorentzian or Gaussian Oscillator Structure might be 
capable of providing a good (MSE) fit to a "Peak" region in a plot of (e2) 
with respect to photon energy, for instance, unless a "Tail" region 
projection from said "Peak" region in said plot is precisely 
Mathematically Lorentzian or Gaussian, data points removed from the region 
of the Peak will not be well approximated thereby. That is, the shape of 
the "Tail" regions of the Lorentzian or Gaussian Oscillator Structure is 
dictated by the shape of the peak region because of a limited number of 
degrees of freedom available. In some cases the sum of a plurality of 
Lorentzian or Gaussian Oscillator Structures centered at various central 
energies, with various amplitudes and broadenings can be utilized to 
provide improved fit over a range of photon energies. However, even where 
this approach is utilized, the lack of a sufficient number of degrees of 
freedom in an Oscillator Structure based Model to allow shaping the Peak 
region of an Oscillator Structure independently of the shape of the Tail 
regions, normally prevents achieving a good (MSE) fit over a relatively 
large range of photon energies. 
To overcome the identified shortcomings in prior Oscillator structure based 
Parametric Models for use in calculating Dielectric Function Component 
values, given a photon energy, and to met the First Requirement recited in 
the Background Section, (ie. that, a Dielectric Parametric Function Model 
provide sufficient flexibility to fit essentially any empirically derived 
data set), the present invention teaches that Oscillator Structures should 
be defined which allow sufficient degrees of freedom to allow "Peak" and 
"Tail" regions thereof to be mathematically modeled essentially 
independently. The preferred embodiment of the present invention provides 
that Finite Order Polynomials, (providing terms of any Order), and 
possibly Finite Magnitude Discontinuities, should be combined to form 
present invention "Oscillator Structures". A typical present invention 
Oscillator Structure being comprised of two Finite Order Polynomials (F1 
and F2), to the left of a Center Point region, said two Finite Order 
Polynomials, (F1 and F2), being sequentially continuous in zero (0), first 
(1) and preferably second (2) derivatives at their point of merger; said 
two polynomials, (F1 and F2), being possibly followed by a Finite 
Magnitude Discontinuity; with the resulting foregoing described two 
additional Finite Order Polynomials, (F3 and F4), to the right of the 
Center Point Region, which Finite Order Polynomials, (F3 and F4), are 
sequentially continuous in zero (0), first (1) and preferably second (2) 
derivatives at their point of merger. It is also mentioned that a 
variation of a basic present invention Oscillator Structure provides that 
some of the Finite Order Polynomials can be eliminated or made to contain 
only various Order Terms, (eg. that is contain any of zero, first, second, 
third etc. Order terms, and possibly not others), and still provide a 
present invention Oscillator Structure. For instance, there might be only 
one Finite Order Polynomial on one or both sides of a present invention 
Oscillator Structure Center Point, and a Finite Magnitude Discontinuity 
might, or might not, be present and might, or might not, be preceded or 
followed by a Finite Order Polynomial, for instance. In any case, each 
present invention Oscillator Structure, (assuming more than one such 
Oscillator Structure is utilized), begins and ends at dependent variable 
(e1) and/or (e2) values of zero (0), at an independent variable photon 
energy which is that corresponding to the Center Point of a present 
invention Oscillator Structure, which can be that of the same, or another, 
present invention Oscillator Structure. (Note if both the beginning and 
end of a present invention Oscillator Structure are present at the Center 
Point of the same Oscillator Structure, a Zero-Width Finite Magnitude 
present invention Oscillator Structure "Pole" results, which is 
mathematically equivalent to a Finite Magnitude Discontinuity which begins 
at zero of a dependent variable scale at the Center Energy of the 
independent variable. This equivalency allows various approaches to 
computer programming implementation of the present invention). 
it is noted that Finite Order Polynomials and Finite Magnitude 
Discontinuities are selected as components of present invention Oscillator 
Structures because said mathematical entities can be subjected to a 
Convolution Integration, (a procedure involving, in the present case, the 
product of such a Finite Order Polynomial, with or without an associated 
Finite Magnitude Discontinuity; and a Broadening Factor, (preferably 
Gaussian), which involves Error Functions and/or Exponentials, see supra). 
If only Finite Order Polynomials and Finite Magnitude Discontinuities are 
utilized to construct present invention non-Zero-Width Oscillator 
Structures, analytic functions and published Integral Tables, (see the 
"Handbook of Mathematical Functions", Abromowitz and Stegun, dover 
Publications, 1972 and "Table of Integrals, Series and Products", 
Gradhteyn and Ryzhik, Academic Press, 1980, which references are 
incorporated by reference herein), are available which allow relatively 
easy evaluation of required present invention Oscillator Structure Model 
Convolution Integrals which arise from the application of a present 
invention Oscillator Structure "Broadening Factor", (see supra). This is 
important as regards the Forth Requirement for a Dielectric Parametric 
Function Model recite in the Background Section of this Disclosure, (ie. 
that being that the use of numerically calculated derivatives and/or 
integrations should not be required, but rather that analytic expressions 
or look-up tables should be available to enhance calculations). 
Continuing, much as a Lorentzian or Gaussian Oscillator can be "Broadened" 
by application of larger "full Width at Half Maximum" (FWHM), or Standard 
Deviation Broadening Factors, so can an Oscillator Structure of the 
present invention be "Broadened". In the present invention, however, a 
separate Convolution Integration must be applied to each present invention 
Oscillator Structure Finite Order Polynomial, (and adjacent Finite 
Magnitude Discontinuity if present), separately, over the range of photon 
Energy in which such is applicable. That is, Broadening is not a simple 
matter of applying a single larger (FWHM), for instance, over the full 
range of photon energy over which an Oscillator Structure applies, as is 
the case where Lorentzian or Gaussian Oscillators are utilized. However, 
attendant with the additional complications are additional present 
invention Oscillator Structure Degrees of Freedom, which additional 
Degrees of Freedom allow excellent mathematical modeling of, for instance, 
an (e2) vs. photon energy plot, over a range of photon energies, via a 
summation of contributions from a plurality of Oscillator Structures, at 
each photon energy. In particular, Oscillator Structures of the present 
invention allow essentially independent control of Peak Region, and Tail 
Region shaping. As can be appreciated by reference to the mathematics 
involved, presented supra, application of a "Broadening Factor", in 
combination with a Maximum Amplitude value, will be found to have the 
greatest effect in fitting Peak Regions, while Tail Regions will be found 
to be primarily affected by evaluation of Polynomial Coefficients. 
Practice of the present invention will typically involve inspection of 
Dielectric Function (e2) vs. photon energy to determine the location(s) at 
which "Dependent Variable Peaks" exist. Use of plotted Dependent vs. 
Independent Variable data is "Center Point" of a present invention 
Oscillator Structure is then located at the photon energy corresponding to 
each Peak, with Polynomial Modeled Tails projecting to one or both sides 
thereof. A present invention Oscillator Structure Finite Magnitude 
Discontinuity might also be positioned at a Center Point. (In the case 
where a Dependent Variable Peak corresponds to the Band-Gap Edge in a 
Semiconductor, Dielectric Function Component Magnitudes often demonstrate 
essentially Discontinuous "Jumps" over very small changes in Energy, at a 
photon energy corresponding to the edge of said Band-Gap. The inclusion of 
finite Magnitude Discontinuities in the present invention Model speaks to 
Sixth (6) Requirement recited in the Background Section of this Disclosure 
in which it is stated that a Model might provide physically relevant 
parameters therein, eg. a Bandgap absorption edge). Also, each present 
invention Finite Width Oscillator Structure so positioned will begin and 
end at an independent variable (e2) value of zero (0), which zero (0) 
values are, (assuming more than one present invention Oscillator Structure 
is present), located at the Center Point Energy of a positioned present 
invention Oscillator Structure, which, as alluded to above, can be the 
same or another present invention Oscillator Structure. This approach to 
locating the beginnings and ends of present invention Oscillator 
Structures positioned on a plot of (e20 vs. photon energy, at the Center 
Point Energy of other present invention Oscillator Structures, allows 
moving a Center Point Energy position in relation to other Center Point 
energy positions without causing drastic effects in Model produced results 
at distal photon energies. This approach to locating lower and upper 
energy end-points of Oscillator Structure is considered new, novel and 
useful, though not absolutely essential to application of the present 
invention. 
With an "appropriately" estimated number of appropriately constructed 
present invention Oscillator Structure(s) positioned along a photon energy 
axis of an (e2) vs. photon energy plot, a Finite Order Polynomial 
Coefficient and Finite Magnitude Discontinuity Magnitude defining 
Oscillator Structure Mathematical Model Parameter evaluation routine, 
including applied Broadening Factors, can be applied to simultaneously 
provide values for all Oscillator Structure Mathematical Model 
Coefficients, based upon, for instance, a Mean-Square-Error (MSE) 
criteria. (A typical Coefficient evaluation routine might utilize a 
Levenberg-Marquard approach). Use of said Finite Order Polynomial 
Coefficients and Finite Discontinuity Magnitudes in constructing present 
invention Oscillator Structures then allows evaluation of a present 
invention Oscillator Structure Mathematical Model Coefficients, in view of 
Convolution Integration applied Broadening Factors, which Mathematical 
Model Coefficients are utilized in evaluation of dependent variable, such 
as (e2) values, provided an independent variable value, such as photon 
energy. It will be appreciated that present invention Oscillator Structure 
Mathematical Model Coefficient values for each present invention 
Oscillator Structure utilized are all that have to then be available to 
allow calculation of a value of, for instance, (e2), given a value of, for 
instance, photon energy, when required. If, for example, the number of 
present invention Oscillator Structures utilized is seven (7), and twelve 
(12) variables are found necessary to define each, then only eight-four 
(84) Coefficient values must be stores. Thereafter, when a value of 
dependent variable (e2) for a given independent variable photon Energy 
level (ES) is desired, the various Finite Order Polynomials which are 
present in a range of photon energies in which the specific photon energy 
(ES) of interest is present are produced and evaluated at said photon 
energy (ES) value of interest, and the contributions of each Finite Order 
Polynomial are added together by a summation procedure. If necessary the 
value of a Finite Magnitude Discontinuity, (be it additive or 
subtractive), is included in the summation. 
Now, in use, one can try a number of "Mathematical Models". that is, it 
might be found that while seven (7) present invention Oscillator 
Structures, each providing four (4) Polynomials, each of second (2) order, 
provide a good (MSE) fit over the range of Energies involved, some of the 
attendant Coefficients of the Mathematical Model for present invention 
oscillator Structures are strongly "Correlated", such that change in one 
thereof is attended by a directly off-setting change in one or more of the 
others. That would be indicative of (e2) vs. photon energy Data which are 
not "strong" enough to justify the number of Coefficients requiring 
evaluation assumed into the Mathematical Model. One might then try 
eliminating one or more present Oscillator Structures or possibly one or 
more Finite Order Polynomials from one or more of the present invention 
Oscillator Structures, or perhaps one might try reducing the number of 
Polynomial Order Terms in one or more Finite Order Polynomials present, 
and rerunning the Mathematical Model Coefficient evaluating program. If an 
equally good (MSE) fit is obtained, and less Mathematical Model 
Coefficieint Correlation is present, the "Modified Mathematical Model" 
would be considered superior. In use it has been found that far less than 
the above mentioned eighty-four (84) parameters are required to adequately 
characterize, for instance, the (e2) vs. photon energy Function of a 
Gallium-Arsonide Sample System. (Note, a highly correlated model is 
capable of providing equally as accurate Dielectric Function values as a 
non-correlated model. Correlation problems, however, are troublesome when 
arriving at the model, and when, for instance, attempting to apply it to 
alloy materials etc.) 
Insight to the present invention can be gleamed by consideration of 
mathematics involved therein. 
As per the Third Requirement for a Dielectric Parametric Function Model 
recited in the Background Section, (ie, that the Model should be 
Kronig-Kramer (K-K) consistent), the following Equation 1 is assumed to 
describe a Complex Dielectric Functions: 
##EQU1## 
where W(E) represents a polynomial and B(hw,E) represents a Broadening 
Factor. The factor "hw" represents the Energy level of a beam of light 
used to achieved dependent vs. independent variable data, and "E" is 
simply a variable of integration. 
Note that Eq. 1 provides that the Complex Dielectric Function is a 
summation of Real and Imaginary parts, and that the Imaginary part 
involves an Integral, while the Real part does not. This incorporates the 
(K-K) criteria, as per the Third (3) recited Requirement in the Background 
Section. 
The Broadening Factor, B(hw,E), is typically Lorentzian or Gaussian and is 
provided by: 
##EQU2## 
where: .gamma.=.GAMMA. for lorentzian Broadening; and 
.gamma.=2.sigma..sup.2.sub.5 for Gaussian Broadening. 
The preferred invention Mathematical Model assumes Broadening to be 
Gaussian because analytical approximation equations, and Tables of 
Evaluated Integrals, (see previously cited Gradshteyn and Ryzhik, and 
Abromowitz and Stegun references), are available. 
Next, assuming Gaussian Broadening and recognizing that Finite Order 
Polynomials in a present invention Oscillator Structure are restricted to 
be effective over a limited range of Energy levels, and assuming that 
"Zero-Width" present invention Oscillator Structures, (ie. Poles), can be 
present, Eq. 1 can be generalized to: 
##EQU3## 
are normalized Energy parameters, specific to each Finite Order Polynomial 
range in an Oscillator Structure. It is to be fully appreciated that use 
of said "normalized" parameters addresses Requirement 4 recited in the 
Background Section, as it allows a single variable dependant result after 
integration of products of Finite Order Polynomials and the Gaussian 
Broadening Factor. This is considered extremely important in practice of 
the present invention Method of Application as it enables evaluation of 
dependent variable values associated with independent variables in a 
Gaussian Broadened Oscillator Structure, by use of a single, rather than a 
multiple, variable Look-up Table, (because only a single variable remains 
after integration of the product of Finite Order Polynomial terms and a 
Gaussian Broadening Factor). In addition, known integrals aide in arriving 
at said One-dimensional "Look-up table" and use thereof in evaluation 
Dielectric Function values. If two variables remained after integration, 
(as is the case where normalized parameters are not introduced), a more 
complicated Two-dimensional Dielectric Function evaluating Look-up table 
would result, which would greatly complicate, if not practically 
precluding practice of the present invention where Gaussian Broadening is 
utilized. It is specifically noted that the use of normalized variables to 
provide a single variable dependence after the Convolution Integration of 
a product of Finite Order Polynomial terms and a Gaussian Broadening 
Factor, and which enables production of an easily utilized One-dimensional 
Look-up table for evaluating Gaussian Broadened Dielectric Function values 
as a function of photon energy, is felt to be a Patentable, utility 
providing, distinction with respect to all known prior art. 
Continuing, as per the Forth (4) Requirement for a Dielectric Parametric 
function Model recited in the Background Section, (ie, that the Model 
should be based on analytic functions or look-up tables to enhance the 
speed of calculation in use), it is noted that the Gaussian based 
Broadening Factor term: 
EQU e.sup.-y2 erf(iy) EQ. 4 
can be approximated by the analytical functions: 
##EQU4## 
and, as mentioned supra, tables of Integrals are available which provide 
evaluations of products of Finite Order Polynomial terms and Gaussian 
Broadening Factor Values between zero (0) and some upper limit: 
##EQU5## 
That a Finite Order Polynomial in the present invention Oscillator 
Structure is generally positioned between some lower Energy and Upper 
Energy limit is handled by the relationship: 
##EQU6## 
In view of the above, an equation for a Complex Dielectric Parametric 
Function including both Broadened present invention Oscillator Structures, 
and Unbroadened Poles, can be derived and is generally presented, in terms 
of normalized Energies in EQ. 8: 
##EQU7## 
where "j" identifies a specific Polynomial or Pole, and "k" is the order 
of a Polynomial. 
it will be appreciated that the present invention allows evaluation of 
"q1(j,k)", "q2(j,k)" and "Aj" Coefficients by performing only summation, 
analytical function evaluation and Look-up table operations. With values 
of said "q1(j,k)" and "q2(j,k)" and "Aj" Coefficients determined, values 
for the present invention Oscillator Structure Mathematical Model, which 
are typically not simply said Coefficients, as presented the Detailed 
Description Section of this Disclosure, can be easily calculated. 
it is mentioned that, regarding Requirement 2 recited in the Background 
Section, concerning a Parametric Dielectric Function Model's ability to 
extract optical constants from measured PSI and DELTA without the need to 
perform numerical differentiation and integrations, is met in the present 
invention, by the convenience of application of the mathematics as just 
described. 
the Computer Program Print-Out at the end of the Detailed Description 
Section serves to provide full disclosure of the mathematics, and the 
applications thereof, involved in the implementing the present invention. 
The defining coefficieints of a present invention Oscillator Structure 
Mathematical Model alluded to infra, which are provided by the Method of 
use of the present invention, are best described graphically. With this in 
mind it should be appreciated that the present invention will be better 
understood by reference to the Detailed Description Section, in 
coordination with the accompanying Drawings.

DETAILED DESCRIPTION 
Turning now to the Drawings, there is shown in FIG. 1, a plot of data 
corresponding to the Imaginary Component (e2) of a presumed Dielectric 
Function vs. Energy, which plot is representative of that which might be 
obtained from investigation of a semiconductor material. Note the presence 
of two dependent variable "Peak" regions, located at "E1" and "E2" on the 
independent variable "X" axis. Next, FIG. 2 shows the general shape of a 
Lorentzian Oscillator Structure centered about a point labeled "EC" on the 
independent variable "X" axis. (While not shown, Gaussian, and other 
Mathematical "Oscillator Structures", could also be utilized for the 
purposes of this demonstration). Note that the Lorentzian Oscillator 
Structure in FIG. 2 presents with a Maximum dependent variable Magnitude 
"A" and is shown with a "Full-Width-Half-Maximum" (FWHM) Broadening "B", 
(which Broadening can extend from Zero (0) to, theoretically, any finite 
value). FIG. 3 shows two such Lorentzian Oscillator Structures positioned 
on the (e2) vs. photon energy plot of FIG. 1. One said Lorentzian 
Oscillator Structure is shown with a Maximum Magnitude of "A1", a (FWHM) 
Broadening of "B1", and it is centered about a point on the independent 
variable "X" axis labeled "E1". the second Lorentzian Oscillator Structure 
is shown with a Maximum Magnitude of "A2", a (FWHM) Broadening of "B2", 
and it is centered about a point on the independent variable "X" axis 
labeled "E2". Note that the independent variable "X" axis points "E1" and 
"E2" in FIG. 1 and FIG. 2 plots correspond to one another. FIG. 3 also 
provides, in bold, a line representing the summation of the contributions 
from the two present Lorentzian Oscillator Structures, at each energy, 
which bold line very roughly corresponds to the (e2) vs. photon energy 
plot in FIG. 1. Note that the plot in FIG. 1 is, however, very different 
from the bold line plot in FIG. 3, particularly in the "Tail" regions 
beyond each "Peak" region. This demonstrates the shortcomings of an 
attempt to use Lorentzian Oscillator Structures to model the Imaginary 
Component (e2) of a Dielectric Function vs. Photon Energy. The Three (3) 
degrees of freedom, (ie. Maximum Magnitude, Broadening and Center Energy), 
which define each Lorentzian Oscillator Structure, are simply typically 
not sufficient to allow achieving a "good" Model fit in both Peak and Tail 
regions of most (e2) vs. Energy plots. However, note that sufficient data 
is contained in six (6) Lorentzian Oscillator Structure defining 
parameters to provide a rough approximation to the (e2) vs. Energy data. 
This compares with hundreds, or even thousands of data points were the 
plot in FIG. 1 is determined and characterized as a table of data. Of 
course, additional Lorentzian Oscillator Structures could be added to the 
plot in FIG. 3, in an attempt to improve the approximation result, but 
this approach has not been found capable of providing results sufficient 
for use in Ellipsometer System settings, emphasis added. 
The forgoing then, demonstrates the concept of utilizing a summation of 
contributions from a plurality of Mathematical Structures to attempt to 
Model a plot of, for instance, (e2) vs. Energy data, in a manner which 
requires determination of a reduced number of parameter values, (as 
compared to an approach utilizing, for instance, tabulated data), and the 
foregoing also identifies the shortcomings of the demonstrated approach. 
The present invention serves to overcome the identified shortcomings by 
defining novel Oscillator Structures which provide more available degrees 
of freedom than provided by, for instance, the Lorentzian Oscillator 
Structure. This is accomplished by present invention utilization of 
defined novel Oscillator Structure(s), which present invention novel 
Oscillator Structures allow shaping the "Peak" and "Tail" regions thereof, 
(which "Tail" Regions project to the right and left of a "Peak" Region 
Center Point thereof), in a very non-correlated manner. The use of a 
"Broadening Factor" applied over the full range of a present invention 
Oscillator Structure, conceptually similar to that described infra with 
respect to the Lorentzian Oscillator Structure, is retained, but 
application thereof to a present invention Oscillator Structure is 
accomplished via Mathematical Convolution Integrations applied 
simultaneously, but independently, to various regions of a present 
invention Oscillator Structure. This is required, because as described 
directly herein, different independent variable regions of a present 
invention Oscillator Structure are defined, for instance, by contributions 
from different Finite Order Polynomials and/or Discontinuities. 
Turning now to FIG. 4a, there is shown a general representation of a novel 
present invention Oscillator Structure. Note that Four (4) Finite Order 
Polynomial defined regions, and one Finite Magnitude Discontinuity are 
shown. The Four (4) defining Polynomials are identified as F1, F2, F3 and 
F4, with F1 beginning at coordinates (0,EL), and merging with Polynomial 
F2 at coordinate (ALM,ELM). Polynomial F2 is terminated at coordinate 
(AL,EC) by the lower end of the LDISC region of the Discontinuity, (shown 
as two dimensionless regions, labeled UDISC and LDISC, above and below 
coordinates (A,EC)) respectively. (Note that the magnitudes of UDISC and 
LDISC are typically selected to be of equal finite Dimensionless 
Magnitudes). The upper end of the UDISC region ends at coordinates 
(AU,EC), whereat Polynomial F3 begins. Polynomial F3 extends to a merger 
with Polynomial F4 at coordinates (AUM,EUM), and Polynomial F4 intersects 
the independent variable "X" axis at coordinate (0,EU). 
As required Oscillator Structure defining Mathematical Model Coefficients 
for the above described present invention Oscillator Structure(s), the 
following, underlined, Nine (9) Coefficients have been chosen for 
evaluation and application in the practice of the Method of use of the 
present invention: 
A; 
EL; 
EC; 
EU; 
AI=A*(1+UDISC); 
AL=A*(1-LDISC); (Note UDISC and LDISC are of equal magnitude) 
AUM=AU*Uamp; 
ALM=AL*Lamp; 
EUM=EC+(EU-EC)*Upos; 
ELM=EC-(EC-EL)*Lpos. 
In addition, as described in the Disclosure of the Invention Section of 
this Disclosure, a Broadening Factor, Sigma (.sigma.), assumed to be, but 
not necessarily, Gaussian, is also evaluated to determine a defined 
present invention Oscillator Structure. (Note, Gaussian Broadening is 
preferred because convenient, and novelly applied, Mathematical aides are 
available when such is the case, as described elsewhere in this 
Disclosure. It is to be understood that other Broadening criteria can be 
utilized and remain within the scope of the present invention.) 
The "Order" of the various Finite Order Polynomials must also be 
identified, and such is accomplished with the use of two (2) additional 
parameters as follows: 
For EL&lt;E&lt;ELM: 
F1=ALM*((1-L2d)*y1+L2d*y1 2) 
For ELM&lt;E&lt;EC 
F2=(ALM+(1-ALM)*(1-cl-dl)*y2+cl*y2 2+dl*y2 4)) 
For EC&lt;E&lt;EUM 
F3=(AUM+(1-AUM)*(1-cu-du)*y4+cu*y4 2+du*y4 4)) 
For EUM&lt;E&lt;EU 
F4=AUM*((1-U2d)*y3+U2d*y3 2) 
where: 
cl=L2d*(ALM/(1-ALM))*((EC-ELM)/(ELM-EL)) 2 
dl=((1/(1-ALM))*(1-((EC-ELM)/(ELM-EL))*ALM*(EC-EL)* 
((L2d/(ELM-EL)+(1/(EC-ELM)) 
cu=U2d*(AUM/(1-AUM))*((EC-EUM)/(EUM-EU)) 2 
du=((1/(1-AUM))*(1-((EC-EUM)/(EUM-EU))*AUM*(EC-EU)* 
((U2d/(EUM-EU)+(1/(EC-EUM)) 
and where: 
y1=(E-EL)/((ELM-EL); 
y2=(E-ELM)/(EC-ELM); 
y3=(EU-E)/(EU-EUM); and 
y4=(EUM-E)/(EUM-EC); 
are normalized energy terms. (Note that the y1 and y2 as used here are not 
the same as the y1 and y2 as defined with respect to EQ. 3 in the 
Disclosure of the Invention Section herein. The y1 and y2 in the 
disclosure of the Invention Section are related to a Gaussian Broadening 
Factor). 
The significance of the use of "normalized" energy terms is particularly 
critical in application of the present invention where Gaussian Broadening 
of present invention Oscillator Structures is utilized. Briefly, use of 
normalized energy terms allows Convolution integration of a product of a 
Finite Order Polynomial and a Gaussian Broadening Factor to provide 
one-dimension, normalized, independent variable based Look-up Tables to 
allow convenient evaluation of corresponding dependent variables. Said 
significance is described more fully in the Disclosure of the Invention 
Section herein. 
FIG. 4b shows another present invention Oscillator Structure formed from 
the same basic components as that in FIG. 4a, (eg. Finite Order 
Polynomials and Finite Magnitude Discontinuities), but in which Polynomial 
F2 and the upper end of UDISC meet at coordinates (AU,EC) and Polynomial 
F3 begins at coordinates (AL,EC) and ends at coordinates (0,EU). Note that 
no Polynomial F4 is present. This is demonstrative of the fact that some 
or all of the Four Polynomials shown in FIG. 4a can be absent while still 
defining a present invention Oscillator Structure. In that light note that 
FIG. 4e shows the case where no F1-F4 Polynomials are present, with the 
result being a "Zero-Width-Finite-Magnitude Oscillator Structure", or 
"Pole", of Zero-Width, (ie. (EL), (EC) and (EU) are coincident), but with 
a Finite Magnitude (AU) above the independent variable "X" coordinate 
(EC). (Note that FIG. 4e can alternatively be considered to show a 
Discontinuity of Finite Magnitude (AU) above the dependent variable zero 
(0) level, present at photon energy (EC), said Finite Magnitude Zero-Width 
Oscillator and Finite Magnitude Discontinuity interpretations being 
equivalent mathematically). FIG. 4c shows a case wherein Polynomials F1, 
F2, F3 and F4 are present, but no Discontinuity DISC, (ie. UDISC and 
LDISC, which are equal Finite Magnitude), is present. Polynomials F2 and 
F3 merge at coordinates (A,EC), which correspond to a "Peak" in this FIG. 
FIG. 4d shows a case in which a Discontinuity is present at "X" axis 
coordinate (EC), and in which Polynomials F3 and F4 are present, but 
Polynomials F1 and F2 are absent. It is to be understood that an analogous 
case wherein a Finite Magnitude Discontinuity and Finite Order Polynomials 
F1 and F2 are present to the left thereof, but where Finite Order 
Polynomials F3 and F4 are absent can also be constructed and such 
constitutes a present invention Oscillator Structure. It should also be 
appreciated that a single Finite Order Polynomial with comprising Terms 
thereof of opposite signs can serve to provide a peaked Dependent Variable 
vs. Independent Variable Plot, which begins at an dependent Variable value 
of zero (0.0), and returns to a dependent Variable value of zero (0.0) at 
some Dependent Variable value. For instance the Finite Order Polynomial 
(Y=3*X-X 2) begins at a "Y" value of zero (0.0) at "X"=(0.0) and returns 
to a "Y"=(0.0) at "X"=(3.0). Such a Single Finite Order Polynomial can 
serve as an additional present invention Oscillator Structure, (over a 
range of (0.0) to (3.0)), although coordination with other present 
invention Oscillator Structures and application of Convolution Integral 
effected Broadening thereto, presents difficulties. 
It is to be noted that where two Polynomials merge, (eg. F1 and F2 , F3 and 
F4, or, possibly, when present without a Finite Magnitude Zero-Width 
Discontinuity therebetween, F2 and F3), they do so with continuous zero, 
first and preferably second (2) derivatives at the point of their merger. 
This is handled by the present invention Mathematical Model Coefficient 
evaluation programming involved in implementing the present invention, 
(see included computer program print-out). 
It should then be appreciated that the present invention provides that 
Oscillator Structures be constructed from one or more Finite Order 
Polynomial(s) and Finite Magnitude Discontinuities in various sequences, 
and that in the presently preferred embodiment of the present invention, a 
typical, non-limiting, preferred present invention Oscillator Structure 
can include from zero (0) to Four (4), Finite Order Polynomials, each 
present Finite Order Polynomial including terms of any Finite Order, (eg. 
zero and/or first and/or second and/or third etc.), and being positioned 
to the right or to the left of a Center Point; and that present invention 
Oscillator Structures can include Finite Magnitude Discontinuities 
therein, present at Center Points thereof. It is also to be understood 
that acceptable present invention Oscillators include those of Zero (0) 
width, (ie. Poles) in which no Polynomials are present, said Poles being 
mathematically equivalent to a single Zero_Width Finite Magnitude 
Discontinuity, beginning at a dependent variable value of zero and 
positioned at a specified independent location. 
In use, various constructed present invention Oscillator Structures are 
positioned, by a user, at appropriate locations in regions along an 
independent variable "X" axis, which locations correspond to "Peak" 
locations regions in a plot of, for instance, the dependent variable 
Imaginary Component of a Dielectric Function (e2) vs. Photon Energy, and 
via Mathematical Convolution Integration, appropriate constructed present 
invention Oscillator Structures are subjected to Gaussian Broadening. 
The Method of Use of the present invention can then involve inspecting a 
plot of Dependent Variable vs Independent Variable data to be modeled, 
constructing present invention Oscillator Structure(s) which appear to be 
appropriate to fit each dependent variable "Peak" region in said plot, 
providing for application of Convolution Integral imposed Gaussian 
Broadening Factors, and then allowing a Mathematical Model Coefficient 
evaluating procedure, (eg. Levenberg-Marquard), operate and provide values 
for Finite Order Polynomial and Finite Discontinuity Magnitude defining 
Mathematical Model Coefficients, as determined acceptable by, for 
instance, a Square-Error minimizing (eg. MSE), criteria. It should also be 
mentioned that it is common practice to position a present invention 
Oscillator Structure of, for instance, "Zero-Width" and Finite Magnitude 
at a location to the right of all available data in a plot, which 
"Zero-Width" Finite Magnitude present invention Oscillator Structure 
serves to account for all Dependent Variable effects beyond the 
independent variable range of known measured data. The Method of Use of 
the present invention then provides that said obtained Polynomial 
Coefficient and Discontinuity Magnitude defining, FIG. 4 identified 
Mathematical Model Coefficient, (eg. A, EL, EC, EU, DISC , Uamp, Lamp, 
Upos and Lpos), as well as Finite Order Polynomial Order Terms describing 
parameters L23d and U2d, see supra), be utilized to provide easily 
calculated dependent variable (e2) values, given an independent variable 
photon energy, by means of summing contributions from all present 
invention Oscillator Structures present at said photon energy. (It is to 
be understood that the FIG. 4a Mathematical Model Defining Coefficients as 
identified above by underlining, are fully determinative of Oscillator 
Structure two (2) Term F1, F2, F3 and F4 Finite Order Polynomials 
Coefficients, and Zero-Width Discontinuity Magnitudes present in an 
Oscillator Structure). It is also to be understood that while a convenient 
aide, dependent variable vs. independent variable data need not be 
actually plotted to practice the Method of the present invention. That is, 
practice of the Method of the present invention can involve identifying 
independent variable levels corresponding to dependent variable indicated 
Peak regions, by purely mathematical means without the requirement that an 
actual plot of dependent variable vs. independent variables be constructed 
and visually observed. In addition, evaluation of present invention 
Oscillator Structure Mathematical Model Coefficients, (eg. A, EL, EC, EU, 
DISC, Uamp, Lamp, Upos and Lpos), can, but need not involve intermediate 
direct calculation of present Finite Order Polynomial Coefficients. What 
is required is that the various Finite Order Polynomial Coefficient Terms, 
(and Finite Magnitude Discontinuity Magnitudes), can be arrived at by 
application of present invention Oscillator Structure Mathematical Model 
Coefficients, as required during practice of the Method of the present 
invention, to allow calculating dependent variable values given 
independent variables. 
Turning now to FIG. 5, there is shown a plot of an actual Gallium-Arsonide 
Sample System Imaginary Dielectric Function (e2) Component vs. energy. 
FIG. 6 shows the plot of FIG. 5 approximated by the Summation of the 
contributions from seven (7) present invention Oscillator Structures, (one 
of which, (ie. Oscillator Structure #1 in following Table 1), is not 
readily visible, but is present at the left side of the plot, with an "X" 
coordinate (EC) of 1.785 ev). Note that each of the Seven (7) present 
invention Oscillator Structures is applicable over only portions of the 
full Energy Range in the FIG. 6 plot. The Finite Order Polynomial 
Coefficient and Finite Magnitude Discontinuity defining Mathematical Model 
Coefficients, (see infra for definition thereof), which serve to define 
the seven (7) present invention Oscillator Structures utilized in 
constructing the Mathematical Model represented in FIG. 6 are provided in 
Table 
TABLE 1 
__________________________________________________________________________ 
# EL 
EU 
EC A SIGMA 
DISC 
Lpos 
Lamp 
L2d 
Upos 
Uamp 
U2d 
__________________________________________________________________________ 
0 0 2 1.414 
0.43 
20 0.0 
0.0 
0.0 
0 0.4 
0.75 
0 
1 1 2 1.765 
0.093 
40 0.0 
0.0 
0.0 
0 0.4 
0.75 
0 
2 0 4 2.896 
11.98 
33.1 
0.2 
0.75 
0.196 
1 0.8 
0.4 0 
3 0 4 3.17 
13.19 
45.28 
0.1 
0.3 
0.03 
0 0.1 
0.045 
0 
4 3 6 4.64 
22.38 
154.1 
0.36 
0.4 
0.136 
0 0.6 
0.4 0 
5 3 6 4.69 
22.22 
164.7 
0.72 
0.5 
0.06 
0 0.1 
0.02 
0 
6 5 6 6.5 
5.8 200.0 
0.0 
0.5 
0.5 
0 0.0 
0.0 0 
7 7 7 7.195 
50.75 
0.0 0.0 
0.0 
0.0 
0 0.0 
0.0 0 
__________________________________________________________________________ 
The Zero-Width Finite Magnitude Pole with a Magnitude of 50.75, present at 
7.195 ev, (beyond the range of Energy levels shown in FIG. 6), tabulated 
in the seventh row of Table 1, was assumed as present to account for all 
effects caused by unknown data to the right of 6.5 ev. The value of the 
Magnitude thereof being determined by the parameter evaluation routine 
which also provided values for the various Finite Order Polynomial 
Coefficient and Finite Magnitude Discontinuity Magnitudes defining 
Mathematical Model Coefficients. (Note, a mathematically equivalent result 
could be effected by use of a Finite Magnitude Discontinuity of 50.75, 
beginning at at coordinates (0.0, 7.195), rather than by the use of a 
Zero-Width present invention Oscillator Structure, (ie. a "Pole"), as 
shown in Table 1). 
Note, in Table 1, that the listings under (EL) and (EU) refer to the (EC) 
value of the indicated present invention Oscillator Structure. For 
instance, present invention Oscillator Structure #2 is centered at and 
(EC) of 2.896 ev, and begins at an (EL) of 1.414 and ends at an EU of 
4.64, the EC values of present invention Oscillator Structure #0 and #4 
respectively. The use of Center Point Energies (EC) of present invention 
Oscillator Structures as the Upper Point (EU) and Lower Point (EL) Energy 
levels for the same, and for other present invention Oscillator Structures 
utilized, provides utility in that a slight modification in the position 
of a Center Point Energy (EC), as might be found necessary during practice 
of the Method of Use of the present invention to provide an overall better 
Model fit, will not have a drastic effect on the Modeling of distal Energy 
level regions. 
Continuing, note that present invention Oscillator Structure 0 is centered 
at 1.414 ev, and has (EL) and (EU) values of 1.414 and 2.896 respectively. 
That is, no F1 and F2 Polynomials are present as the (EL) and (EC) values 
are the same. Also note that while DISC values are provided for present 
invention Oscillator Structures 2 through 5, the plot in FIG. 6 does not 
seem to indicate the presence thereof. Application of the Broadening 
Factors, (ie. The Sigmas, (.sigma.)), via Convolution Integrations, mask 
the initial DISC presence, (see the Disclosure of the Invention Section of 
this Disclosure for a description of said Convolution Integrations). 
Comparison of FIGS. 5 and 6 show that the Mathematical Model arrived at by 
practice of the Method of Use of the present invention as applied to the 
plotted data in FIG. 5, provides a very close replication of said data, as 
shown in FIG. 6. With that in mind, it is noted that only present 
invention Oscillator Structure 2, (see Table 1), provides second order 
Polynomials, (ie. L2d is 1), and that present invention Oscillator 
Structures #0 and #1 have no associated F1 and F2 Polynomials present. 
Note, Requirement 5 recited in the Background Section, concerning Model 
stability is always achieved if both L2d and U2d are bounded by (+/-1). If 
that is the case then all Oscillator Structures will provide only positive 
values and a calculated Dependent Variable value can never involve a 
subtraction between values of similar magnitude, (see comparative 
discussion of the Kim and Garland et al. Parametric Model in the 
Background Section of this Disclosure). This is considered an important 
point. In addition, it has been found in practice that it is not always 
necessary to let all present coefficients be variable. That is, it has 
been found that for a family of similar materials, many of the variables 
in an Oscillator Structure Mathematical Model can be arrived at once, and 
set as constants, leaving a relatively small set of remaining variables to 
be modified in Mathematically Modeling similar materials. (This result, it 
should be realized, is responsive to the seventh Requirement recited in 
the Background Section of this Disclosure, which, provides that a 
Parameterized Dielectric Function Model should be suitable for application 
to a family of semiconductor dielectric functions related by composition, 
temperature and/or doping etc.). It should be appreciated then that while 
the present invention provides for Oscillator Structures with essentially 
unlimited degrees of freedom, often times the required degrees of freedom 
required are surprisingly few to provide very good Mathematical Model 
Coefficient (MSE) determined acceptable dependent variable vs. independent 
variable data fit. This is of benefit as little computational time is then 
required to provide values for Mathematical Model Coefficients in use. 
It should be appreciated that the present invention Oscillator Structure 
Mathematical Model Coefficients identified as determined in Table 1 and 
FIG. 4a, are arbitrary, and that any Coefficients which allow evaluating 
present Finite Order Polynomial Coefficients and Finite Discontinuity 
Magnitudes, consistent with application of Convolution Integral effected 
Oscillator Structure Broadening, could be selected as appropriate for 
determination and use in calculation of dependent variable values given 
independent variable values, which in the above provided example were (e2) 
vs. Photon Energy. For purposes of Claim interpretation, the terminology 
"Mathematical Model Coefficients" are to be understood to include any 
functionally appropriate Coefficients. 
It is to be understood that Dielectric Function vs. Photon Energy data used 
in the practice of the Method of the present invention can be obtained by 
Ellipsometry, and by any other applicable technique, such as Transmission 
and Reflectance. 
it is also noted that the terminology "Finite Magnitude Discontinuity", in 
the strict mathematical sense implies a finite change in a magnitude over 
a Zero-Width. The Terminology "Essentially Zero-Width" will be utilized in 
the Claims when referring to Discontinuities as they are used in this 
invention, in recognition of the fact that physical universe variable do 
not demonstrate true absolute "discontinuity", and to imply that a 
physically meaningful Magnitude change over a relatively Small-Width is to 
be included within the Rights Claimed. It is also noted that a Zero-Width 
present invention Oscillator Structure is provided by essentially setting 
(EL)=(EU), at the location of (EC) in any of the FIGS. 4a-4d, while 
maintaining a Magnitude (A) or (AU) as appropriate. As discussed infra, 
this result is represented in FIG. 4e. It should also be appreciated 
however, that a Zero-Width Discontinuity, such as shown in FIGS. 4a and 
4b, which does not originate at a lower dependent variable value of zero 
(0), however, is not strictly a Zero-Width present invention Oscillator 
Structure. As well, it is to be understood that the terminology "Finite 
Width" refers to a case where (EL) is not equal to (EU). 
In addition, the terminology "Finite Order Polynomial" is to be understood 
as not limited to cases where any particular term, or series of terms is 
present. That is, a simple constant identity, and any equation in which a 
constant and/or some additional term or terms are present, rather or not 
all powers of a dependent variable are represented, and rather or not all 
terms have the same or opposite sign, are to be considered as included by 
said terminology. As well, any Mathematical function which can be 
represented by a Finite Term Power, etc. Series, to an acceptable level of 
accuracy, is to be considered as a "Finite Order Polynomial". For 
instance, a Finite Order Polynomial resulting from Truncating an Infinite 
Series, can accurately represent an Exponential Logarithmic Function, a 
Trignometric Function, a Hyperbolic Function, Exponentials of Trignometric 
or of Hyperbolic Functions, Inverse Trignometric or Hyperbolic Functions 
etc. A Truncated Fourier Series with Trignometric Cos and Sin Functions 
approximated by Truncated Power Series expansion terms also provides a 
"Finite Order Polynomial". In general, any series of terms of various 
powers of an independent variable is to be considered within the bounds of 
a "Finite Order Polynomial" for the purposes of Claim interpretation. 
While the present invention is applicable to Modeling Dielectric function 
Components as a function of Photon Energy, it should be understood that it 
can also be utilized in providing a Mathematical Model for any other data 
which can be presented in a graphed format. As such, while independent 
variable (EL), (EC) and (EU) have been referred to as photon energy 
values, it is to be understood that they can be referred to more generally 
as Lower, Center and Upper Points. It should then be understood for the 
purposes of Claim construction that Lower Point, Central Point and Upper 
Point are simply more generalized language to identify points on a graph 
of plotted dependent vs. independent data, analogous to the demonstrated 
plots of FIGS. 5 and 6 described infra in which photon energies were 
presented. 
FIG. 7 shows a Flow Chart of the Method of the present invention. Said 
method has as its purpose the enabling of the calculation of Dielectric 
Function dependent variable numerical values given independent variable 
numerical values, (eg. Photon Energies or Wavelengths). 
the first step is shown to consist of: 
A. Obtaining Dielectric Function dependent vs. independent numerical 
variable data and identifying independent variable locations of one or 
more dependent variable peak regions thereon. 
The second step is shown to consist of: 
B. For each of at least one of said identified dependent variable peak 
regions proposing a Finite Width Oscillator Structure for application in 
mathematical modeling of said numerical values which demonstrate a 
dependent-independent variable relationship, (eg. that between an 
Imaginary Component (e2) of a Dielectric Function vs. Photon Energy), each 
of said one or more proposed Finite Width Oscillator Structure(s) 
comprising at least one component selected from the group consisting of: 
a. a Finite-Width combination of a Finite Order Polynomial and an 
essentially Zero-Width, Finite Magnitude Discontinuity; 
b. a Finite-Width combination of a plurality of Finite Order Polynomials; 
and 
c. an essentially Zero-Width, Finite Magnitude Discontinuity. 
It is noted that typical constructed Finite Width Oscillator Structures 
are: 
1. A Finite-Width combination of a Finite Order Polynomial and an 
essentially Zero-Width Finite Magnitude Discontinuity to the right 
thereof; 
2. a Finite-Width combination of a Finite Order Polynomial and an 
essentially Zero-Width Finite Magnitude Discontinuity to the left thereof; 
3. a Finite-Width combination of two adjacent Finite Order Polynomials and 
an essentially Zero-Width Finite Magnitude Discontinuity to the right 
thereof, said two adjacent Finite Order Polynomials being merged with 
continuous zero and at least one higher order derivative(s); 
4. a Finite-Width combination of two adjacent Finite Order Polynomials and 
an essentially Zero-Width Finite Magnitude Discontinuity to the left 
thereof, said two adjacent Finite Order Polynomials being merged with 
continuous zero and at least one higher order derivative(s); 
5. a Finite-Width combination of two adjacent Finite Order Polynomials and 
an essentially Zero-Width Finite Magnitude Discontinuity to the left 
thereof, in combination with two adjacent Finite Order Polynomials to the 
right of said Essentially Zero-Width Finite Magnitude Discontinuity, with 
adjacent finite Order Polynomials being merged with continuous zero and at 
least one higher order derivative(s); 
6. a Finite-Width combination of one Finite Order Polynomial and an 
essentially Zero-Width Finite Magnitude Discontinuity to the left thereof, 
in combination with two adjacent Finite Order Polynomials to the right of 
said essentially Zero-Width Finite Magnitude Discontinuity, with adjacent 
Finite Order Polynomials being merged with continuous zero and at least 
one higher order derivative(s); 
7. a Finite-Width combination of two adjacent finite Order Polynomials and 
an essentially Zero-Width Finite Magnitude Discontinuity to the left 
thereof, in combination with one adjacent Finite Order Polynomials to the 
right of said essentially Zero-Width Finite Magnitude Discontinuity, with 
adjacent Finite Order Polynomials being merged with continuous zero and at 
least one higher order derivative(s); 
8. a Finite-Width combination of a plurality of adjacent Finite Order 
Polynomials, with adjacent Finite Order Polynomials being, at points of 
merger therebetween, merged with continuous zero and at least one higher 
order derivative(s); 
9. an essentially Zero-Width, Finite Magnitude Discontinuity; and 
10. a Finite Order Polynomial comprised of terms of opposite sign; 
It is noted that Finite Order Polynomial Coefficient(s) and any essentially 
Zero-Width Finite Magnitude Discontinuity Magnitude serve to define Finite 
Width Oscillator Structure Mathematical Model Coefficients which can be 
evaluated in use, and said Finite Width Oscillator Structure Mathematical 
Model Coefficients at least partially define a relationship between 
dependent and independent numerical variables over a range of said 
independent variable, said range corresponding to the width of a Finite 
Width Oscillator Structure. Said Finite Width Oscillator Structure 
Mathematical Model Coefficient evaluation is typically based upon a 
Square-Error reducing criteria. 
the third step is shown to be: 
C. Assigning each said proposed Finite Width Oscillator Structure to an 
appropriate peak in said Dielectric Function Component dependent variable 
vs. independent variable data. 
The fourth step is shown to be: 
D. Causing a procedure to be executed such that present Finite Width 
Oscillator Structure Mathematical Model Coefficients determined by Finite 
order Term Polynomial Coefficients and/or any essentially finite Magnitude 
Zero-Width Discontinuity Magnitude, are for each of said one or more 
Finite Width Oscillator Structures, evaluated. This step can include 
application of Convolution Integration effected Gaussian Broadening 
Factors to at least one present non-Zero-Width Finite Magnitude Oscillator 
Structure(s) which contain Finite Order Polynomial(s). 
The fifth step is shown to be: 
E. Utilizing said determined Mathematical Model Coefficients to, for each 
finite Width Oscillator Structure, calculate dependent variable values 
which correspond to independent variable values, and combining 
contributions from each Finite Width Oscillator Structure at each 
independent variable. 
Said method of the present invention can further comprise: 
F. Inspecting the evaluated Finite Width Oscillator Structure Mathematical 
Model Coefficients for Correlation therebetween, and inspecting the 
results provided from practice of step E. thereof for goodness of fit 
between corresponding calculated Dependent Variable values and Actual 
Dielectric Function Dependent Variable values, and if said inspections 
disclose less than user determined desired quality, repeating steps B. 
through E. 
It is also noted that practice of the Method of the present invention 
ultimately depends on the skill of a user in constructing and applying 
present invention Oscillator Structures to data to be modeled. That is, a 
user of the present invention must become proficient in selecting, 
combining and positioning Polynomial and Discontinuity elements to provide 
present invention Oscillator Structure(s) for instance on a plot of data, 
which Oscillator Structure(s) provide sufficient, but not redundant, 
degrees of freedom to accurately model said plotted data. A user will 
typically have to try various numbers of and various combinations of 
Polynomials, (of various Finite Orders), and possibly include 
Discontinuities, to provide appropriate constructed present invention 
Oscillator Structure(s), and try various number of, and various 
positioning thereof on plots of dependent variable vs. independent 
variable data, to achieve good Model fit to said plotted data, by, for 
instance, a Square-Error, (eg. MSE), reducing criteria. As well, attention 
to avoidance of providing too many degrees of freedom in view of the data 
to be Modeled, as evidenced by high Correlation between various Polynomial 
Coefficients and/or Discontinuity Magnitude defining Mathematical Model 
parameters is necessary. However, with a bit of practice, when presented 
with data, a user can become demonstratably proficient in proposing and 
positioning a reasonable number of present invention Oscillator 
Structures, of appropriate constructions, to allow a good Square-Error 
reducing Uncorrelated Mathematical Model Coefficieint fit therefore to be 
achieved, which Mathematical Model requires evaluation of relatively few 
coefficieints to allow its application. The present invention then, is 
found in the definition of novel Oscillator Structures and the Method of 
application thereof in Modeling data. The utility of the present invention 
is found in the fact that by appropriate Construction and Broadening of, 
and application of, novel present invention Oscillator Structure(s) which 
are comprised of Finite Order Polynomials and/or possibly finite Magnitude 
Discontinuities, a Mathematical Model for essentially any data can be 
developed which can be made to be acceptable under essentially arbitrarily 
user defined (MSE) and Mathematical Model Coefficient Correlation 
criteria, which Mathematical Model requires determination of but a 
relatively few Parameters. 
The present invention also, and importantly, identifies an independent 
variable normalization which allows convolution Integration effected 
Gaussian broadening of present invention Finite Order Polynomial based 
Oscillator Structures, which normalization enables convenient provision of 
a result involving only a single independent variable, after said 
Convolution Integration. This enables production of "One-dimensional 
Look-up" Tables to evaluate dependent variables, given independent 
variables. In addition, it is not necessary to perform numerical 
derivatives or integrations to effect said Gaussian Broadening. Also, as 
stated infra, practice of the present invention does not require finding 
the difference between two relatively large numbers to arrive at an error 
prone, much smaller number. No known prior art provides such capability 
and convenience. 
To assure full disclosure, the following computer Program Print-Out is 
provided. Said Computer Program is the actual Code utilized in practice of 
the present invention and serves to evaluate the Coefficients of the 
Mathematical Model described infra, including the Gaussian Broadening 
procedure and formation of One-Dimensional Look-Up Tables. 
__________________________________________________________________________ 
struct PSEMI { 
double Am20!,Br20!,En20!,Asym20!; 
double FL20!,LP120!,LP220!,FR20!,RP120!,RP220!; 
double PolePos,PoleMag,PolePos2,PoleMag2; 
int ConnectLeft20!,ConnectRight20!,Selected20!; 
}; 
const int NUM.sub.-- LO.sub.-- POINTS=1000; 
const double BREAK.sub.-- POINT=20.; 
complex *int.sub.-- gauss.sub.-- low5!; 
double int.sub.-- factor11!={1./3.,4./3.,2./3.,4./3.,2./3.,4/3.,2/3.,4/3. 
,2./3.,4./3.,1/3.}; 
int GaussCalculated=0; 
complex GaussOsc(double y){ 
int j,start,finish,inv=0; double x,t; complex c; 
if(y&lt;0){inv=1; y=-y;} 
x=0; 
j=1, do{t=y+.5*j; t=exp(-t*t)/j; x=t; j++;} while(t)&gt;1e-12); 
start=2*y-12; if(start&lt;1) start=1; finish=start+24; 
for(j=start;j&lt;=finish;j++) {t=y-5*j; t=exp(t*t)/j; x+=t;} 
c.i=exp(-y*y); c.r=-(y*c.i+x)/PI; if(inv)c.r=c.r; 
return(c); 
void GaussInterp(double.y,complex .quadrature., int num){ 
int i,j,inv=0; double r2,i2,c1,c2,c3,c4,d,y1,y2,y3,y4,x12,x13,x14,x23,x24, 
x34; 
double sc1,sc2; 
double xm,xr,x10; 
if(y&lt;0) {inv=1;y=-y;} 
if(y&lt;BREAK.sub.-- POINT){ 
y*=NUM.sub.-- LO.sub.-- POINTS/BREAK.sub.-- POINT; i=y; if(i&lt;1) i=1; 
y1=y(i-1); y2=y-i; y3=y(i+1);y4=y(i+2); 
c1=-1./6.*y2*y3*y4; c2=1./2.*y1*y3*y4; 
c3=-1./2. *y1*y2*y4; c4=1./6.*y1*y2*y3; 
//y*=NUM.sub.-- LO.sub.-- POINTS/BREAK.sub.-- POINT; i=y;sc1=y-i;sc2=-1.-s 
c1; TABLE CALCULATION 
for(j=0;j&lt;num; j++){ 
// aj!.r=sc2*int.sub.-- gauss.sub.-- lowj!i!.r+sc1*int.sub.-- gauss.sub 
.-- lowj!i+1!.r; 
// aj!.i=sc2*int.sub.-- gauss.sub.-- lowj!i!.i+sc1*int.sub.-- gauss.sub 
.-- lowj!i+1!.i; 
aj!.r=c1*int.sub.-- gauss.sub.-- lowj!i-1!.r+c2*int.sub.-- gauss.sub 
.-- lowj!i!.r 
+c3*int.sub.-- gauss.sub.-- lowj!i+1!.r+c4*int.sub.-- gauss.sub.-- 
lowj!i+2!.r; 
aj!.i=c1*int.sub.-- gauss.sub.-- lowj!i-1!.i+c2*int.sub.-- gauss.sub 
.-- lowj!i!.i 
+c3*int.sub.-- gauss.sub.-- lowj!i+1!.i+c4*int.sub.-- gauss.sub.-- 
lowj!i+2!.i; 
ifinv){ 
if(j|=1 &&j|=3) aj!.i=-aj!.i; 
else, aj!.r=aj!.r; 
} 
} 
} else { 
xm=1; x10=1; 
for(j=0;j&lt;num=;j++){ 
switch(j){ 
case 0: xr=in(BREAK.sub.-- POINT)-in(y); break; 
case 1: xr=x10-xm; break; 
case 2: xr=.5*(x10-xm); break; 
case 3: xr=1./3.*(x10-xm); break; 
case 4: xr=.25*(x10-xm); break; 
} 
xm*=y; x10*=BREAK.sub.-- POINT; 
aj!.r=xr*0.564189584+int.sub.-- gauss.sub.-- lowj!NUM.sub.-- LO.sub.-- 
POINTS!.r; 
aj!.i=int.sub.-- gauss.sub.-- lowj!NUM.sub.-- LO.sub.-- POINTS!.i; 
if(inv){ 
if(j|=1 &&j|=3) aj!.i=aj!.i; 
else aj!.r=-aj!.r; 
} 
} 
} 
} 
complex psemi.sub.-- nk(PSEMI &p double w){ 
int ij; 
double a1,b1,c1,d1,EC,EL,ER,EM,dECM,dEML,AM,AM1,t1,t2,dECM2; 
complex Cp,Cm,C4,x,Ap1,Ap2,Ap4,Am1,Am2,Am4,tx,A,wELp,wELm,wECp,wECm; 
double w2,w3,w4,G,G2,G3,G4,AA,B,C,D,x1,x2; 
complex am5!,ap5!,mm5!,mp5!,cm5!,cp5!; 
w2=w*w; w3=w*w2; w4=w2*w2; 
x.r=1.0+p.PoleMag/(p.PolePos*p.PolePos-w2)+p.PoleMag2/(p.PolePos2*p.PolePo 
s2=w2-w2); x.i=0; STORE TABLES 
if(|GaussCalculated){ 
int i,j,k,m; double x,xx,int.sub.-- const; complex s,t; 
GaussCalculated=1; 
int.sub.-- gauss.sub.-- low0!=(complex*)malloc(sizeoff(complex)*(NUM.sub. 
-- LO.sub.-- POINTS+5)); 
int.sub.-- gauss.sub.-- low1!=(complex*)malloc(sizeoff(complex)*(NUM.sub. 
-- LO.sub.-- POINTS+5)); 
int.sub.-- gauss.sub.-- low2!=(complex*)malloc(sizeoff(complex)*(NUM.sub. 
-- LO.sub.-- POINTS+5)); 
int.sub.-- gauss.sub.-- low3!=(complex*)malloc(sizeoff(complex)*(NUM.sub. 
-- LO.sub.-- POINTS+5)); 
int.sub.-- gauss.sub.-- low4!=(complex*)malloc(sizeoff(complex)*(NUM.sub. 
-- LO.sub.-- POINTS+5)); 
for(k=0; k&lt;5; k++){ 
s,r=s,i=0; 
int.sub.-- const=.1*BREAK.sub.-- POINT/NUM.sub.-- LO.sub.-- POINTS; 
for(i=0; i&lt;NUM.sub.-- LO.sub.-- POINTS+5; i++){ 
int.sub.-- gauss.sub.-- lowk!i!.r=s.r*int.sub.-- const; int.sub.-- 
gauss.sub.-- lowk!i!.i=s.i*int.sub.-- const; 
for(j=0;j&lt;=10;j++){ 
x=(i*10+j)*int.sub.-- const; 
t=GaussOsc(x); 
if(k&gt;0){ 
xx=x; 
for(m=1; m&lt;k;m++) xx*=x; 
tr*=xx; ti*=xx; 
} 
s.r+=t.r*int.sub.-- factorj!;s.i+=t,i*int.sub.-- factorj!; 
} 
} 
} 
} 
for(i=0; i&lt;20; i++){ 
if(|p.Selectedi!) continue; 
//Gaussian Broadening 
EC=p.Eni!; EL=p.Enp.ConnectLefti!!; ER=p.Enp.ConnectRighti!!; 
if(EL&gt;=EC && ER&lt;=EC), continue; 
G=p.Bri!*.001*1.2; G2=G*G; G3=G*G2; G4=G2*G2; 
GaussInterp((w+EC)/G,cp,5); GaussInterp((w-EC)/G,cm,5); MODEL 
CALCULATIONS 
//// 
x=x+(GaussOsc((w-EC)/G)-GaussOsc((w+EC)/G)) * RP2i!; 
if(EL&lt;EC){ 
EM=EL+p.FLi!*(EC-EL); AM=p.LP1i!; b1=p.LP2i!; 
AM1=1/(1-AM); dEML=EM-EL; 
GaussInterp((w+EM)/G,mp,5); GaussInterp((w-EM)/G,mm,5); 
GaussInterp((w+EL)/G,ap,3); GaussInterp((w-EL)/G,am,3); 
x1=(w+EL)*(b1*(w+EM)-dEML); x2=(w-EL)*(b1*(w-EM)+dEML); 
txr=(ap0!.r-mp0!.r)*x1+(am0!.r)mm0!.r)*x2; tx.i=(ap0!.i-mp0!.i)*x1+( 
am0!.i-mm0!.i)*x2; 
x1=G*(dEML-2*b1*w-b1*(EM+EL)); x2=G*(b1*(EM+EL)-dEML-2*b1*w); 
tx.r+=(ap1!.r-mp1!.r)*x1+(am1!.r-mm1!.r)*x2; tx.i+=(ap1!.i-mp1!.i)*x 
1+(am1!.i-mm1!.i)*x2; 
x1=b1*G2; 
tx.r+=(ap2!,r-mp2!.r+am2!.r-mm2!.r)*x1; tx,i+=(ap2!.i-mp2!.i+am2!.i 
-mm2!.i)*x1; 
x=x+AM*p.AMi!*(1-p.Asymi!)*0.564189584/(dEML*dEML)*tx; 
dECM=(EC-EM); dECM2=dECM*dECM; 
d1=c1=dECM/dEML; c1*=c1*b1*AM*AM1; 
d1=AM1*(1-d1*AM*(EC-EL)*(b1/dEML+1./dECM); 
D=-4*d1*EM; C=6*d1*EM*EM+c1*dECM2; B=dECM2*(dECM*(1-cl)-(EC+EM)*c1)-4*EM*E 
M*EM*d1; 
AA=EC*EM*(dECM2*(c1+EM/EC-1)+(EC*EC-3*EM*dECM)*d1)+dECM2*dECM2*AM1*AM; 
x1=AA-B*w-D*w3+C*w2+d1*w4; x2=AA+B*w+D*w3+C*w2+d1*w4; 
tx.r=(mp0!.r-cp0!.r)*x1+(mm0!.r-cm0!.r)*x2; tx.i=(mp0!.i-cp0!.i)*x1+ 
(mm0!.i-cm0!.i)*x2; 
x1=-G*(2*C*w-3*D*w2-B+4*d1*w3); x2=-G*(2*C*w+3*D*w2+B+4*d1*w3); 
tx.r+-(mp1!.r-cp1!.r)*x1+(mm1!.r-cm1!.r)*x2; tx,i+=(mp1!.i-cp1!.i)*x 
1+(mm1!.i-cm1!.i)*x2; 
x1=G2*(C-3*D*w+6*d1*w2); x2=G2*(C+3*D*w+6*d1*w2); 
tx.r+=(mp2!.r-cp2!.r)*x1+(mm2!.r-cm2!.r)*x2;tx,i+=(mp2!.i-cp2!i*x1+( 
mm2!.i-cm2!.i)*x2; 
x1=G3*(D-4*d1*w); x2=-G3*(D+4*d1*w); 
tx.r+=(mp3!.r-cp3!.r)*x1+(mm3!.r-cm3!.r)*x2;tx.i+=(mp3!.i-cp3!.i)*x1 
+(mm3!.i-cm3!.i)*x2; 
x1=d1*G4; 
tx.r+=(mp4!.r-cp4!.r+mm4!.r-cm4!.r)*x1; tx,i+=(mp4!.i-cp4!.i+mm4!.i 
-cm4!.i)*x1; 
x=x+(1-AM)*p.Ami!*(1.p.Asymi!)*0.564189584/(dECM2*dECM2)*tx; 
} 
if(ER&gt;EC){ 
EL=ER; 
EM=EL+p.FRi!*(EC-EL); AM=p.RP1i!; b1=p.RP2i!; //b1=0; 
AM1=1/(1-AM); dEML=EM-EL; 
GaussInterp((w+EM)/G,mp,5); GaussInterp((w-EM)/G,mm,5); 
GaussInterp((w+EL)/G,ap,3); GaussInterp((w-EL)/G,am,3); 
x1=(w+EL)*(b1*(w+EM)-dEML); 
x2=(w-EL)*(b1*(w-EM)+dEML); 
tx.r=(ap0!.r-mp0!.r)*x1+(am0!.r-mm0!.r)*x2; 
tx.i=(ap0!.i-mp0!.i*x1+(am0!.i-mm0!.i)*x2; 
x1=G*(dEML-2*b1*w-b1*(EM+EL)); x2=G*(b1*(EM+EL)-dEML-2*b1*(w); 
tx.r+=(ap1!,r-mp1!.r)*x1+(am1!.r-mm1!.r)*x2; tx.i+=(ap1!.i-mp1!.i)*x 
1+*(am1!.i-mm1!.i)*x2; 
x1=b1*G2; 
tx.r+=(ap2!,r-mp2!.r+am2!.r-mm2!.r)*x1; tx,i+=(ap2!,i-mp2!.i+am2!.i 
-mm2!.i)*x1; 
x=x-AM*p.Ami!*(1+p.Asymi!)*0.564189584/(dEML*dEML)*tx; 
dECM=(EC-EM); dECM2=dECM*dECM; 
d1=ci=dECM/dEML; c1*=c1*b1*AM*AM1; 
d1=AM1*(1-d1*AM*(EC-EL)*(b1/dEML+1/dECM)). 
D=4*d1*EM; C=6*d1*EM*EM+c1*dECM2; B=dECM2*(dECM*(1-d1)-(EC+EM)*c1)-4*EM*EM 
*EM*d1; 
AA=EC*EM*(dECM2*(c1+EM/EC-1)+(EC*EC-3*EM*dECM)*d1)+dECM2*dECM2*AM1*AM;. 
x1=AA-B*w-D*w3+C*w2+d1*w4; x2=AA+B*w+D*w3+C*w2+d1*w4; 
tx.r=(mp0!,r-cp0!r)*x1+(mm0!.r)*x2; tx.i=(mp0!.i-cp0!.i)*x1+(mm0!.i- 
cm0!.i)*x2; 
x1=-G*(2*C*w-3*D*w2-B+4*d1*w3); x2=-G*(2*C*w+3*D*w2+B+4*d1*w3); 
tx.r=(mp1!r-cp1!.r)*x1+(mm1!.r-cm1!.r)*x2; tx.i+=(mp1!,i-cp1!.i)*x1+ 
(mm1!.i-cm1!.i)*x2; 
x1=G2*(C-3*D*w+6*d1*w2); x2=G2*(C+3*D*w+6*d1*w2); 
tx.r+=(mp2!.r-cp2!.r)*x1+(mm2!.r-cm2!r)*x2; tx.i+=(mp2!.i-cp2!.i)*x1 
+(mm2!.i-cm2!.i)*x2; 
x1=G3*(D-4*d1*w); x2=-G3*(D+4*d1*w); 
tx.r+=(mp3!.r-cp3!.r)*x1+(mm3!.r-cm3!.r)*x2,i+=(mp3!.i-cp3!.i)*x1+(m 
m3!.i-cm3!.i)*x2; 
x1=d1*G4; 
tx.r+=(mp4!.r-cp4!.r+mm4!.r-cm4!r)*x1; tx,i+=(mp4!.i-cp4!,i+mm4!.i- 
cm4!.i)*x1; 
x=x-(1-AM)*p.Ami!*(1+p.Asymi!)*0.564189584/(dECM2*dECM2)*tx; 
} 
} 
if(x.i&lt;1e-6)x,i=0; 
return(x); 
} 
__________________________________________________________________________ 
Having hereby disclosed the subject matter of the present invention, it 
should be obvious that many modifications, substitutions, and variations 
of the present invention are possible in light of the teachings. It is to 
be understood that the invention can be practiced other than as 
specifically described, and should be limited in breadth and scope only by 
the Claims.