Wave propagation in materials like waveguide, crystals such as thin film had been studied by many researchers. Since the original research of Feit and Fleek (1979a, b and 1980) significant move has been mode in the area of beam propagation method. Having assessed its applicability, this method has certainly been most widely used as modeling tools for integrated optics (Roey et al., 1981).
Electromagnetic wave incident normally or obliquely of the surface of a thin film in the (r) direction experiences attenuation as the field penetrate the film (Ugwu and Uduh, 2005). Further studies has been conducted recently on determination of the refractive index inside the thin film medium as wave propagates on it (Ugwu, 2005).
This study looked at the influence of refractive index on the wave propagating
on the film and specially considered a case where the variations in refractive
index are small with reference to scalar field in which a scalar wave equation
can be derived for the Transverse Electric (TE) or Transversed Magnetic (TM)
modes separately. The change the index introduced as a perturbation to the propagating
wave in the film. The solution of the fields φ was obtained using series
expansion. Solution of creens function (Schiff, 1955) and the results
written as sources of two fields with one due to the perturbation term. The
result was assigned first order differential co-efficient ∂φ1/∂z
and operators to decompose the fields φ (Roey et al., 1981). The
second term in ∂φ1/ ∂z = A+φ1
+ B+φ2 was considered as a correction term
due to the change on the refractive index.
Finally, the observation showed that the smoothly change in the refractive index indicated clearly the sensitivity to polarization being as a result of interface and hence the need to decompose the wave into TE and T.M, Greens function was used in conjunction with the appropriate boundary conditions to analyse the impedance imposed by the thin film medium on the propagating wave.
WAVE EQUATION AND PERTURBATION IN REFRACTIVE INDEX
We consider the propagation of a high frequency beam through an inhomogeneous FeS2 medium. The beam propagation method was considered and the cases in which the variations in refractive index are small or in which a scalar wave equation can be derived for the TE and TM modes separately looked at.
We start from the wave equation:
Where φ represents the scalar field, n (r) the refractive index and k,
the wave number in vacuum, in this equation n2(r) = n2(r)
+ Δn2(r) where no(r) represents the unperturbed part
and Δn2(r) is the perturbed part. This equation 1 can be written
Where the right hand of Eq. (2) is considered to be a source
function. The refractive index no2(r) is chosen in such
a way that the wave equation
Together with the radiation conditions at infinity can be solved. If the solution
φ for Z = ZO is shown, the field φ and its derivative ∂φ1/
∂z can be obtained for all values of Z by means of an operator A+.
Where the operator A+ acts with respect to transverse co-ordinates
(x, y) only. The function G(r, r), for our problem can be determined by
direct construction of Greens function, i.e., by joining the solutions
of the homogeneous problem in equation 3 at r = a after which we now write the
random integral Eq. as
Where GO is the free space Greens function which is the determinant function while the change in refractive index Δn2(r) and the surface Greens function G(r,r1) are the random function (Ishimaru et al., 2000)
SERIES SOLUTION OF GREENS FUNCTION
The Greens can be obtained using different technique, but here we use series expansion technique
The equation can be written
The Greens function becomes where. f (r1) = K2Δn2(r)φ
With boundary condition that
As G (r, r) vanishes at the end of the interval (0, z), we can expand the expression
in a suitably chosen orthogonal functions such as fourier sine series.
Where the expansion co-efficient Υm depends on the parameter
r∞. Differentiating Eq. 9
soloving for ∂(r, r) and Δm (r), γm (r) can be obtained
subitituting Eq. 13 into 9 for the value
of γm (r) we obtain
The solution of the inhomogeneous equation becomes (Butkov, 1968)
Which leads to
By fourier series as we assume Δn to be periodic
ANALYTICAL SOLUTION OF THE PROPAGATING WAVE WITH STEP-INDEX AND THE DISCUSSION
If we consider the beam propagating towards increasing Z with no assumed paraxiality,
we split the field φ into a part φ1 generated by the sources
in the region where r<r and a part φ2 assigned to the sources
where r>r i.e., perturbated term, we can write
Again if the propagation in the unperturbed medium is assigned an operator
A+ and another operator B+ defined on φ2
with respect to the transverse co-ordinate (x,y) only, we can write Eq.
if we neglect the influence of the reflected field on φ1 we
could use φ2instead of φ1in the equation 20,
Equation 21 is an important approximation, though it restricts
the use of the beam propagation method in analyzing the structures of matters
for which the influence of the reflected waves would have on the forward-propagating
wave. However this excludes the use of the method in cases where the refractive
index changes abruptly as a function of r or in which reflections add up coherently.
According to Eq. 21, the propagation of the field φ1is
given by the term describing the propagation in an unperturbed medium and the
correction term representing the influence of Δn2(r). Equation
21 is also first order differential equation which made it easy for one
to determine the field φI1 for z>zo
starting from the input beam on the plane z = zo as the transversal
variations of Δn2(r). φI1 are considered
to be very slow, than the second term in Eq. 21 becomes
Eq. 21 becomes now
and by the boundary condition imposed this equation, ∂ε,
if the field φ1 is known at z = z0 and if Δn2 is zero, Eq. 21 reduces to
where ε (x, y, zO) = φ1 (x, y, zO+) is given and ε representing the field propagating in a medium with a refractive index no(r).
As the beam is propagated through the thin film showing a large step in refractive
index of an imperfectly homogeneous thin film (Fig. 1) this
condition presents the enabling provisions for the use of a constant refractive
index no of the thin film. One can then choose arbitrarily two different
refractive indices n1 and n2 at the two sides of the step
for all x with smoothly changing refractive index at both sides of the step
(Fig. 1) we assume that the sensitivity to polarization is
due mainly to interface and hence in propagating a field φ through such
a medium, one has to decompose the field into TE and TM polarized field for which we neglect the
coupling between the E and H fields due to the small index variation (n-n0).
when the interface conditions that φm and were continuous
at x = 0 were satisfied, the TE field could be propagated by the virtue of these
decomposition similarly, TM field were also propagated by considering that φm
continuous at x = 0. When we use a set or discrete mode, different sets of φm
can be obtained by the application of the discrete Fourier transform because
of the periodic extension of the field in the Fig. 2 Also
to obtain a square wave function for n0(x) as in Fig.
2 obtained by the application of the discrete Fourier transform because
of the periodic extension of the field in the Fig. 2.
|| Refractive index profile showing a step
||Periodic extension of the refractive index profile of thin
film used in this description (Roey et al., 1981)
Also to obtain a square wave function for n0(x) as in Fig.
2, no has to be considered period we were primarily interested.
In the field guided at the interface x = x1. The field radiated away from the interface was assumed not influence the field in the adjacent region because of the presence of suitable absorber at x =-x1 and x = x x,. The correction operator B+ contains the perturbation term Δn2 and as we considered it to be periodic function without any constant part as in Eq. 18. The phase variation of the correction term is the same such that the correction term provided a coupling between the two waves.
The Greens function as obtained in Eq. 17 satisfied
at the source point and satisfied the impedance boundary condition that.
Is the free space characteristic impedance and ∂/∂n is the normal
derivative. The impedance zs offered to the propagating wave by the
thin film is given by
= Zo/n (1 -2πλ-2/ λ-1o)
From the same condition.
Where n is the average refractive index of the film
λs = wavelength of the wave in the thin film
λs = wavelength of the wave in free space from Eq.
29, for every given wave with a wavelength say λs in the
film the appropriate Δn and the impedance Zs of the film calculated.
In this study we analysed the effect of change in refractive index of FeS2 thin film medium on propagating electromagnetic wave. With a change in refractive index, which is periodic, was identified as a perturbation. The propagating field φ used in the analysis was obtained by series expansion solution of the Greens function and splitted into two with the second term considered as a correction term factor due to change in the refractive index. We also observed that smoothly change in refractive index at both sides of the step according to the model is sensitive to polarization, which was assumed to be as a result of interface. However, the field was decomposed into Transverse Electric mode (TE) and Transverse Magnetic mode (TM) and with the refractive index being periodic, the phase variation of the correction terms were the same. Again, with the satisfaction of the boundary conditions at the some point and at impedance boundary condition, the expression for the calculation of the impedance of the film was given in Eq. 25.