INTRODUCTION
Various tools have been employed in studying and computing beam or field propagation
in a medium with variation of small refractive index (Fleck
et al., 1978; Feit and Fleck, 1979; Ugwu
et al., 2007), some researchers had employed beam propagation method
based on diagonalization of the Hermitian operator that generates the solution
of the Helmholtz equation in media with real refractive indices (Thylen
and Lee, 1992) while, some had used 2x2 propagation matrix formalism for
finding the obliquely propagated electromagnetic fields in layered inhomogeneous
un-axial structure (Ong, 1993).
Recently, we have looked at the propagation of electromagnetic field through a conducting surface and we observed the behaviour of such a material. The effect of variation of refractive index of FeS2 had also been carried out.
The parameters of the film that were paramount in this work are dielectric
constant (Blatt, 1968) and the thickness of the thin film.
The dielectric constants were obtained from a computation using pseudo-dielectric
function in conjunction with experimentally measured extinction co-efficient
and the refractive indices of the film and the thickened of the film which was
assumed to range from 0.1 to 0.7 μm [100 to 500 nm] based on the experimentally
measured value, at the wavelength, 450 μm (Cox, 1978;
Lee and Kong, 1983).
This work is based on a method that involves propagating an input field over
a small distance through the thin film medium and then, iterating the computation
over and over through the propagation distance using Lippmann-schwinger equation
and its counterpart, Dyson’s equation (Economou, 1979)
here, we first derived Lippmann-schwinger equation using a specific Hamiltonian
from where, the field function Ψk (z) was obtained. From this,
it was observed that to ease out the solution of the Lippmann-schwinger equation,
it has ti be discretized. After this, Born approximation was applied in order
to obtain the solution. The formalism is logically built up step-by-step, which
allowed point-t-point observation of the behavior of the field propagating through
the film. The advantage of this approach area, such as field in a medium with
variation of dielectric constant, refractive index becomes apparent and above
all our method requires no resolution of a system of equations and can accommodate
multiple layers easily.
THEORETICAL METHOD
Lippman-schwinger equation is associated with the Hamiltonian H which goes
with H0+V where, H0 is the Hamiltonian before the field
penetrates the thin field and V is the interaction:
The eigenstate of H0+V is the solution of:
where, z is the propagation distance as defined in the problem:
where, η is the boundary condition placed on the Green’s function (Ek-H0)-1. Since, energy is conserved, the propagation field component of the solutions will have energy En with the boundary conditions that only handle the singularity when the eigenvalue of H0 is equal to Ek. Thus, we write:
as the Lippman-schwinger equation without singularity; where õ
is a positively infinitesimal, is the propagating field in the film while is
the reflected. With the above Eq. 3 and 4
one can calculate the matrix elements with (z) and insert a complete set of
z and μ states as shown in Eq. 5:
is the Green’s function for the problem, which is simplified as:
When õ≈0 is substituted in Eq. 6, we have:
The perturbated term of the propagated field due to the inhomogeneous nature of the film occasioned by the solid-state properties of the film is:
where, Δkk is determined by variation of thickness of the thin
film medium and the variation of the refractive Eq. 3 at various
boundary of propagation distance. As, the field passes through the layers of
the propagation distance, reflection and absorption of the field occurs thereby
leading to the attenuation of the propagating field on the film medium.
ITERATIVE APPLICATION
Lippman-Schwinger equation can be written as:
where, Go(z, z’) is associated with the homogeneous reference
system (Yaghjian, 1980; Hanson,
1996; Gao et al., 2005; Lee
and Kong, 1983).
| |
The function  define
the perturbation. |
Where:
The integration domain of Eq. 9 is limited to the perturbation. Thus, we observe that Eq. 9 is implicit in nature for all points located inside the perturbation. Once the field inside the perturbation is computed, it can be generated explicitly for any point of the reference medium. This can be done by defining a grid over the propagation distance of the film, which is the thickness. We assumed that the discretized system contains Δkk defined by T/N.
where, T is thickness and N is integer.
| |
(N = 1, 2, 3, N-1). The discretized form of Eq.
9 leads to large system of linear equation for the field: |
However, the direct numerical resolution of Eq. 10 is time
consuming and difficult due to singular behaviour of 
.
As a result, we use iterative scheme of Dayson’s equation, which is the
counter part of Lippman-schwinger equation to obtain 
.
Equation 10 is easily solved by using Born approximation,
which consists of taking the incident field in place of the total field as the
driving field at each point of the propagation distance. With this, the propagated
field through the film thickness was computed and analyzed.
RESULTS AND DISCUSSION
From the result obtained using this formalism, the field behaviour over a finite
distance was contained and analyzed by applying Born approximation method in
Lippman-Schwinger equation involving step by step process. The result yielded
reasonable values in relation to the experimental result of the absorption behavior
of the thin film (Ugwu et al., 2001).
The splitting of the thickness into more finite size had not much affected on the behavior of the field as regarded the absorption trends.
The trend of the graph obtained from the result indicated that the field behavior have the same pattern for all mesh size used in the computation. Though, there is slight fall in absorption within the optical region, the trend of the graph look alike when the thickness is 1.0 μm with minimum absorption occurring when the thickness is 0.5 μm. within the near infrared range and ultraviolet range, (0.25 μm) the absorption rose sharply, reaching a maximum of 1.48 and 1.42, respectively when thickness is 1.0 μm having value greater than unity.
From the behaviour of the propagated field for the specified region, UV, visible
and near infrared (Cody et al., 1982) the propagation
characteristic within the optical and near infrared regions is lower when compared
to UV region counter part irrespective of the mesh size and the number of points
the thickness is divided. The field behaviour is unique within the thin film
as observed in Fig. 3 and 4 for wavelength
1.2 and 1.35 μm and Fig. 1 and depict the field trend
when the mesh size is 10 for wave length = 0.40, 0.70 and 0.90 μm while
Fig. 2 shows the propagated field 50 mesh size for wavelength
= 0.25, 0.70 and 0.90 μm. Figure 5 shows the picture
of the absorbance characteristics of the film.
We also observed in each case that the initial value of the propagation distance Z μm, initial valve of the propagating field is low, but increase sharply as the propagation distance increases within the medium suggesting the influence of scattering and reflection of propagating field produced by the particles of the thin film as it propagates.
Again, as high absorption is observed within the ultraviolet (UV) range, the
thin film can be used as UV filter on any system the film is coated with on
the other hand, it has low absorption within the optical (VIS) and near infrared
(NIR) regions of solar radiation.
|
| Fig. 1: |
The field behaviour as it propagates through the film thickness
Z μm for mesh size = 10 when λ = 0.4, 0.7 and 0.9 μm |
|
| Fig. 2: |
The field behaviour as it propagates through the film thickness
Z μm for mesh size = 50 when λ = 0.25, 0.7 and 0.9 μm |
|
| Fig. 3: |
The field behaviour as it propagates through the film thickness
Z µm for mesh size= 50 when l = 0.25, 0.7 and 0.9 µm |
|
| Fig. 4: |
The field behaviour as it propagates through the film thickness
Z µm for mesh size = 100 when l = 0.25, 0.8 and 1.35 |
|
| Fig. 5: |
The filed absorbance as a function wavelength |
CONCLUSION
Lippmann-schwinger equation in conjunction with Born Approximation had been used in analyzing the propagation of field through FeS2 thin film. Iterative scheme was applied to the formalism by building up step-by-step solution and observation that enabled us to determine the behaviour of the propagated filed. The results showed that variation of the field with thickness depended on the wavelength of the photon energy not on the T/N, even the mesh size did not have any significant change in the behaviour. This study helped in determining the absorption behaviour and the probable applications of the thin film.
ACKNOWLEDGMENTS
Authors humbly acknowledge the contribution of Dr M.I Echi of Department of Physics, University of Agric. Makurdi, Benue State for his effort in developing the program used for this work. Also Engr Onu of Department of Computer Sciences, Ebony State University and Abakaliki is left out for his help in facilitating the running of the program.
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