Theoretical Study of Field Propagation Through a Nonhomogeneous Thin Film Medium Using Lippmann-Schwinger Equation

ABSTRACT A computational technique was used to study the field propagation through an inhomogeneous thin film using Lippmann-Schwinger equation. The iterative formalism was used which was constructed based on the principle of Born approximation method due to the difficulty in using direct numerical resolution as a result of implicit nature of the function. The computed field value, Ψ through the thin film with variation of the propagation distance was analyzed within the ultraviolet, visible and near of electromagnetic wave and the absorption of the propagated wave by the thin film studied. The energy band gap was also estimated from the computed absorption coefficient obtained from our formalism.


INTRODUCTION
Various tools have been employed in studying and computing beam or field propagation in a medium with variation of small refractive index Feit and (Fleck, et al, 1976) (Fleck, 1978) (Ugwu et al, 2007) some researchers had employed beam propagation method based on diagonalization of the Hermetician operator that generates the solution of the Helmholtz equation in media with real refractive indices (Thylen and Lee, 1992), while some had used 2 × 2 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 (Ugwu, 2005) and we observed the behaviour of such a material. The effect of variation of refractive index of F e S 2 had also been carried out (Ugwu, 2005).
The parameters of the film that were paramount in this work are dielectric constants 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 µm to 0.7 µm [100 nm to 500 nm] based on the experimentally measured value, at the wavelength, 450 µm (Cox, 1978) (Lee and Brook, 1978).
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 Theoretical Study of Field Propagation Through a Nonhomogeneous Thin Film Medium Using Lippmann-Schwinger Equation using a specific Hamiltonian from where the field function ψ k (z) was obtained. From this, it was observed that to ease out the solution of the Lippmann-Schwinger equation, it was 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-to-point observation of the behaviour 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 PROCEDURE
We set out find the field, Ψ of a scalar wave equation (1) Propagated through a thin film.
The assumption here II that the dielectric function is split into two parts compressing reference homogeneous medium, ε ref and perturbed medium representing the deposited film medium.
Hence the dielectric function of the system is defined thus First Principle Green's function for the problem. The plane wave equation as defined by equation 3.1 depicts our model.
Where ψ (z) represents scalar field which we intend to know its behavior as it propagating through the thin film µ o and ε o are permeability and permittivity of the free space respectively (3) Is the dielectric function which is arbitrarily complex in nature and comprises ε ref representing the dielectric of the reference dielectric homogeneous medium ε ref while ∆ε p (z) is the perturbed dielectric part which signifies the deposited thin film medium. Figure 1 Deposited thin film in a glass substrate.
Substituting this into equation (1) we have (4) Let and being the function that describes the perturbation.
We now seek the Green's function satisfying (5) With the boundary condition that (6) Using the method of finite Fourier sine transform in which we multiply both side of equation (5) by and integrate over (0, z 1 ) to obtain (7) With γ n (z) being the Fourier coefficient of G(z, z′), it follows that and Here the symmetry property of Green's function is obvious while the formula fails if as Green's function has no solution or does not exist at that point.
If we now assume as a variable parameter of G(z, z′) on which it depends on such that the solution is possible only within − ≤ − ≥ l l λ π n z Green's function possesses simple pole at where n = 1, 2, 3… This means that the solution of the Green's Function relies on analyticity of the denominator.
However, where G(z, z′) doesn't exist, there is resonance. The solution of equation 3.1 becomes (10) With this equation, we can now spell out the total field propagating through the thin film as (11) Which is simplified as? 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: (14) Where ∆ kk' is determined by variation of thickness of the thin film medium and the variation of the refractive index (Ugwu, et al. 2007) 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. (Blatt, 1968)

ITERATIVE APPLICATION
Lippman-Schwinger equation can be written as (15) Where G o (z, z') is associated with the homogeneous reference system, (Yaghjian 1980) (Hanson, 1996) (Gao et al, 2006) (Gao and Kong 1983) The function (16) define the perturbation Where (17) The integration domain of equation (13) is limited to the perturbation. Thus we observe that equation (13) 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 that is the thickness. We assume that the discretized system contains ∆ kk , defined by T/N.
Where T is thickness and N is integer (N = 1, 2, 3, N − 1). The discretized form of equation (13) leads to large system of linear equation for the field;

RESULTS AND DISCUSSION
In this case the field behaviour was obtained using born approximation technique both equations in parallel considering the variation of field with the film thickness and mesh size which is given as z / N (Nmax) for wavelength in electromagnetic spectrum of regions in ultraviolet, visible/optical and near infrared. FORTRAN program was used in facilitating the computation with Nmax = 10,50 and 100. As observed from the graphs of the filed profiles, the field in the ultraviolet region has a higher absorbance for all the mesh sizes. It is also seen that for a longer wavelength within the near infrared region, the profile of the field exhibited negative trend within the film medium as shown in fig. 3 to fig. 6. Close observation shows that at about 0.05 m just as the field penetrates the film medium, the absorbance appears to be very small, but tends to rise sharply with a slight variation from that point to the maximum specified propagation distance which is just the thickness of the thin film. At just about the end of the size of the specified film thickness, the filed fall sharply again. This peculiar case of sharp/sudden rise in the field profile tends to suggest the influence of scattering and reflection on the propagating field which is as a result of the particles of thin film medium in accordance with Brillouin and Agarwal [Brillouin, 1963 andAgarwal, 1977]. From the graph of absorbance plotted as a function of wavelength, as shown in fig. 6 it is seen that the absorbance is generally low within the optical region to near infrared in our computation this goes on to confirm the profile behaviour in computed relative amplitude in fig. 3 to fig. 6 310 Theoretical Study of Field Propagation Through a Nonhomogeneous Thin Film Medium Using Lippmann-Schwinger Equation    where the cases of nonabsorbing. Limited absorption and strong absorbing thin film were considered. The values of the band gap were obtained by extrapolating the graphs of (αhν) 2 against (hν) to meet at hν axis. The points on the graphs as in fig. 7 fig. 8 and fig.9 where the extrapolated lines met the photon energy axis is where (αhν) 2 is zero signify the band gaps. They are 2.35eV, 3.25eV and 15.25eV for the three cases we considered. From these energy band value, it is seen that the third one that represents that of the strongly absorbing case has the highest energy band gap, (15.25eV) indicating that it offers the largest impedance to photon energy transmission follows by that of the limited absorbing case with 3.25eV band gap. Non-absorbing case has the last band gap of 2.35eV. This therefore confirms high optical impedance of strongly absorbing case.

312
Theoretical Study of Field Propagation Through a Nonhomogeneous Thin Film Medium Using Lippmann-Schwinger Equation    Figure 7 Graph of (αhν) 2 against photon Energy for limited absorbing case.

CONCLUSION
• Lippmann-Schwinger equation in conjunction with Born Approximation had been used in analyzing the propagation of field through thin film. An 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 obtained showed that variation of the field with thickness depended on the wavelength of the photon energy. The mesh size had no significant contribution in the behaviour of the propagated wave pattern through the thin film. This study helped in determining the absorption behavior. The energy band gap and the probable applications of the thin film.  Figure 9 Graph of (αhν) 2 against photon Energy for strongly absorbing case.