This document presents a study using MathCAD to numerically solve the scalar wave equation for electromagnetic wave propagation through an inhomogeneous material medium with dielectric constant perturbation. The wave equation was transformed into a form suitable for numerical solution in MathCAD. Solutions were obtained for three values of dielectric constant perturbation representing different absorption levels, within the ultraviolet, optical, and near-infrared regions of the electromagnetic spectrum. The results show the correlation between the optical field profile and propagation distance increases with increasing dielectric perturbation. MathCAD provided greater correlation between the field profile and propagation distance compared to theoretical solutions.
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This is the handout version to take notes on.
Bayesian modelling and computation for Raman spectroscopyMatt Moores
Raman spectroscopy can be used to identify molecules by the characteristic scattering of light from a laser. Each Raman-active dye label has a unique spectral signature, comprised by the locations and amplitudes of the peaks. The Raman spectrum is discretised into a multivariate observation that is highly collinear, hence it lends itself to a reduced-rank representation. We introduce a sequential Monte Carlo (SMC) algorithm to separate this signal into a series of peaks plus a smoothly-varying baseline, corrupted by additive white noise. By incorporating this representation into a Bayesian functional regression, we can quantify the relationship between dye concentration and peak intensity. We also estimate the model evidence using SMC to investigate long-range dependence between peaks. These methods have been implemented as an R package, using RcppEigen and OpenMP.
Dual Gravitons in AdS4/CFT3 and the Holographic Cotton TensorSebastian De Haro
Talk given at the workshop "Gravity in Three Dimensions" at the Erwin Schrödinger Institute, Vienna, April 14-24, 2009. I argue that gravity theories in AdS4 are holographically dual to either of two three-dimensional CFT's: the usual Dirichlet CFT1 where the fixed graviton acts as a source for the stress-energy tensor, or a dual CFT2 with a fixed dual graviton which acts as a source for a dual stress-energy tensor. The dual stress-energy tensor is shown to be the Cotton tensor of the Dirichlet CFT. The two CFT's are related by a Legendre transformation generated by a gravitational Chern-Simons coupling. This duality is a gravitational version of electric-magnetic duality valid at any radius r, where the renormalized stress-energy tensor is the electric field and the Cotton tensor is the magnetic field. Generic Robin boundary conditions lead to CFT's coupled to Cotton gravity or topologically massive gravity. Interaction terms with CFT1 lead to a non-zero vev of the stress-energy tensor in CFT2 coupled to gravity even after the source is removed.
Uncountably many problems in life and nature can be expressed in terms of an optimization principle. We look at the process and find a few good examples.
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Study of the impact of dielectric constant perturbation on electromagnetic
1. Chemistry and Materials Research www.iiste.org
ISSN 2224- 3224 (Print) ISSN 2225- 0956 (Online)
Vol 2, No.6, 2012
Study of the impact of dielectric constant perturbation on
electromagnetic wave propagation through material medium:
MathCAD solution
E.I. Ugwu1, Eke V.O.C2 and Elechi Onyekachi O. 2
1. Department of Industrial Physics
2. Department of Computer Science, Ebonyi State University, Abakaliki, Nigeria.
ABSTRACT
We present the study of dielectric constant and its influence on wave propagation through an inhomogeneous
material medium via mathcard approach. Mathcard was used to solve numerically the wave equation involving a
perturbation in term of dielectric constant ε p ( r ) = ∆ε p ( r ) + ε ref imposed on scalar wave equation.
ψ ′′ ( r ) + ω 2 µ0 ε 0 ε p ( r ) = 0; The equation was reduced to a form suitable for numerical solution using
mathcard on which we applied various values of dielectric constant perturbation for three regions or electromagnetic
spectrum VIS UV, optical near infrared region. The correlation between the field profile and the propagation was
analyzed.
Keyword: dielectric constant, perturbation, numerical solution, mathcard, scalar wave equation, electromagnetic
spectrum, correlation, propagation distant, material medium.
INTRODUCTION
Analytic study of optical field propagating through an inhomogeneous material media have been presented only for a
few specific geometries with a number of schemes for further study of the optical effect of layered. Inhomogeneous
medium has been attempted (Wait, 1970; Brekhovakikh, 1980). Such a scheme as Abeles 2 x 2 propagation matrix,
Jones 2 x 2 and Berreman 4 x 4 propagation matrices has been utilized. (Abele, 1950; Jones 1941; Azzum, 1972) for
analytical study of the geometrical optical approximation. Also, generalized geometrical optic approximation, phase
integral and perturbation theory have been applied (Ong, 1993, Budden, 1966, Ugwu, 2010).
However, one concept is paramount for its application. This is the concept of definition of generalized field vector in
propagation matrix and perturbation term that depends on the dielectric parameters and other optical and solid state
parameters of the material medium. The propagation distance of the medium depicting the thickness of the material
ingestion and the propagation vector of the incident field is taken into consideration unless where approximation
value is necessary (Ugwu, 2007: 2010).
In all, beam propagation method has remained a very good technique for analytical study of wave propagation
through various material medium since it provides a unified treatment of various guided and radiation field problems
subject to paraxiality and spatial frequency content of the index profile (Thylen, 1986).
It can also be used to analyze non linear directional coupler operation for various combinations of non linear
materials and initially mismatched guide. (Steyeman, 1985; Jensen, 1982). This is because the technique involves
propagating the input beam over small distance through homogeneous space in the material medium and then
analyzing the influence of variation of the solid state properties of the material such as refractive index and dielectric
constant on the propagating wave (Fleck et al., 1976; Feit, 1979; Ugwu, 2011, Yeh, 1979). A number of derivations
of the method adopted amenable to specific problems have been made possible using beam propagation method
(Roay, 1981, Ugwu et al., 2011, Yevick et al., 1985, Kim et al., 1990, Brykhovetskii et al., 1985, Tatarskii, 1979,
Rytov et al., 1987).
THEORETICAL FRAMEWORK
1
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We begin by defining scalar wave equation
Given
ψ ′′ ( r ) + ω 2 µ0 ε 0 ε p ( r ) ψ ( r ) = 0
Which is a second order differential equation represent a scalar wave equation with a perturbation due to its
propagation through a medium with a dielectric function,
ε p ( r ) = ∆ε p ( r ) + ε ref .
∴ψ ′′ ( r ) + ω 2 µ0 ε 0 ( ∆ε p ( r ) + ε ref )ψ ( r ) = 0
⇒ ψ ′′ ( r ) + ω 2 µ0 ε 0 ∆ε p ( r )ψ ( r ) = − ω 2 µ0 ε 0 ε ref ψ ( r )
2
2π
Assuming ε ref = 0 and − V ( r ) = k02 ∆ε p ( r ) , k02 = ω 2 µ0ε 0 =
λ
ψ ′′ ( r ) − V ( r )ψ ( r ) = 0
Hence V = k02 ∆ε p ( r ) with the perturbed term contained in V, we transform the equation in the form that makes
suitable for Mathcard solution.
dx1
Let ′′ ′
x1 = ψ ⇒ x1 − Vx1 = 0, x1 = .
dr
Let ′
x1 = x2 , the differential equation becomes;
x1′ = x2 x1 ′ 0 x2 x1
⇒ =
′
x2 = Vx1 x2 Vx1 0 x2
The form of this equation is suitable for numerical solution by MathCAD.
In the solution, we considered three different values associated with the value of dielectric perturbation ∆ε p :
2
3. Chemistry and Materials Research www.iiste.org
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Vol 2, No.6, 2012
∆ε p = 0.5; 3.5 and 10.50 which represents negligible, limited and strong absorbing medium. In the mathcard
numerical solution as used here we considered the ultraviolet, optical and near infrared region of electromagnetic
wave spectrum.
MathCAD Solution
r0 = 0 r1 =150 Solution interval endpoints (5)
0
ic = Initial condition vector (6)
1
N =1500 Number of solution values on (t0, t1) (7)
X1
D ( r, X ) = Derivative function
4. π .ε . X
2 (8)
λ2
0
Solution matrix:
S = rkfixed (ie, r0, N D)
r = S <0> Independent variable values
ψ = S <1> Solution function values
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Fig;1 Field,Ψ VS Propagation distant for Δεp= 0.5 within UV region
200
150
ψ 100
50
0 20 40 60 80 100 120 140
r
Fig;2. Field,Ψ VS Propagation distant for Δεp= 0.5 within optical region
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Vol 2, No.6, 2012
200
150
ψ 100
50
0 20 40 60 80 100 120 140
r
Fig;3. Field,Ψ VS Propagation distant for Δεp= 0.5 within near infrared region
200
150
ψ 100
50
0 20 40 60 80 100 120 140
r
Fig.4 Field,Ψ VS Propagation distant for Δεp= 3.5 within UV region
5
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Vol 2, No.6, 2012
200
150
ψ 100
50
0 20 40 60 80 100 120 140
r
Fig;5. Field,Ψ VS Propagation distant for Δεp= 3.5 within near UV region
4
1.5 10
4
1 10
ψ
5000
0 20 40 60 80 100 120 140
r
Fig.6 Field,Ψ VS Propagation distant for Δεp= 3.5 within near infrared region
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Vol 2, No.6, 2012
λ 650 ε 3.5
300
200
ψ
100
0 20 40 60 80 100 120 140
r
Fig.7 Field,Ψ VS Propagation distant for Δεp= 10.5 within UV region
λ 950 ε 3.5
300
200
ψ
100
0 20 40 60 80 100 120 140
r
Fig.7 Field,Ψ VS Propagation distant for Δεp= 10.5 within UV region
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Vol 2, No.6, 2012
6
1.5 10
6
1 10
ψ
5
5 10
0 20 40 60 80 100 120 140
r
Fig;9 Field,Ψ VS Propagation distant for Δεp= 10.5 within near infrared region
RESULT OF DISCUSSION
Figure 1 to figure 9 depict the behaviour of propagated wave profile through the medium against the propagation
distance presented in accordance with the values of dielectric perturbation.
In fig 1, the field profile when ∆ε p = 0.5 within UV region appear parabolic in nature but tends to a straight-line
with the optical and near infrared region as in fig 2 and 3.
When the dielectric perturbation increases to 3.5, the field behaviour within the UV region remained constant up to
49nm of the propagation distant after which it rise up sharply to maximum of 1.3 x 104 within 140nm of the
propagation distance as in fig. 4. In fig 5 and 6 depict the field profile within optical and near infrared region
respectively, the correlation between the field and the propagation distant tends to increase for ∆ε p = 10.5 which
can note the strong absorption value, the field behaviour within the UV increases sharply after 129nm of the
propagation distant and begins to experience a slight increment on correlation between the field profile and
propagation distant for both optical and near infrared region. comparing the work with the one obtained using
theoretical solution of the scalar wave equation, one observed that the wave profile exhibited a defined behaviour in
relation, the propagation distance unlike the results from the theoretical solution technology and W. K. B
approximation which only depicted the oscillatory pattern of the wave as it propagates through the medium (Ugwu et
al., 2011, Ugwu and Maliki, 2011).
Another interest phenomenon is the correlation between the field profile within the UV region for ∆ε p , = 0.5, 3.5
and 10.5 . One observes that the correlation increases with the increase in the value of perturbed dielectric constant.
Unlike the field profile behaviour in the case of the theoretical result whereby the oscillatory wave pattern is
dampened as ∆ε p increase especially within the near infrared region (Ugwu and Maliki 2011).
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9. Chemistry and Materials Research www.iiste.org
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Vol 2, No.6, 2012
CONCLUSION
The study of this work, we have presented the propagation behaviour of wave through a medium with variation of
dielectric perturbation using numerical solution by Mathcard. Though other method such as theoretical solution of
scalar wave equation and W. K B approximation teeth: we has been used, all exhibited different wave profile
∆ε ,p
uniquely in relation to the propagation distant for various values of However, numerical solution by mathcard
exhibited greater correlation between the wave profile and propagation distant than the theoretical solution of scalar
wave equation and W. K. B approximation technique. This formalism allowed the understanding of how the
dielectric constant perturbation influences the correlation between the filed profile and propagation distance which
signifies the thickness of the medium through which the wave propagate.
ACKNOWLEGDEMENT
We humbly acknowledge the contribution of Dr. M.I. Echi of Department of Physics, University of Agric. Makurdi,
and Benue State for his effort in developing the program used for this work. Also Dr. S.O.Maliki of Industrial
Mathematics and Applied Statistics Department, Ebony State University, Abakaliki is not left out for his help in
facilitating the running of the program.
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10. Chemistry and Materials Research www.iiste.org
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Ugwu, E.I., Uduh P.C. and Agbo G.A. 2007. “The Effect of change in Refractive Index on Wave Propagation
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10
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