Computational study of wave propagation through a dielectric thin film medium using WKB approximation

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1 AMERICAN JOURNAL OF SCIENTIFIC AND INDUSTRIAL RESEARCH 0, Science Huβ, htt:// ISSN: X doi:0.55/ajsir Comutational study of wave roagation through a dielectric thin film medium using WKB aroximation Emmanuel Ifeanyi Ugwu, Sadik Olaniyi Maliki and Michael Alo Elebe Deartment of Industrial Physics and Deartment of Industrial Mathematics and Alied Statistics; Ebonyi State University, Abakaliki, Nigeria. ABSTRACT Wave roagation in a non-homogeneous dielectric thin film medium using W.K.B aroximation technique is resented. The aroximation is used to solve a scalar wave equation that contains a dielectric erturbation, Δ ε ( by imosing all the W.K.B aroximation conditions in conjunction with other necessary boundary conditions that were alied in the solution. The influence of the erturbation on the roagating wave rofile through the thin film for ultra-violet, otical and infrared regions of electromagnetic wave was analyed. Keywords: Thin film, wave roagation, W.K.B aroximation, dielectric erturbation, wave rofil,e non-homogeneous medium, scalar wave equation INTRODUCTION Different methods and aroach have been used in the study of wave roagation in an inhomogeneous medium such as wave guide, thin film etc [Ugwu, 0, Ong and Meyer, 983] notwithstanding the comlication and rigor involved in aroach, it is interesting because of its alicability in the study of material and nanocsiences. Abele s matrix method which is a scheme used for comutation of the otical effect in a layered media had been studied [Abele s,950]. In addition, geometrical aroximation, hase integral method, generalied geometrical otic aroximation and method of erturbation theory have also been used to study roagation in non-homogeneous media[thyle / n983]this was enhanced because the aroach ermits an understanding and unified treatment of guided radiation mode and also ermits the roagation of a given field in rather general refractive index structure. Aart from ermitting an understanding and assessment of various roerties of the roagated field, the method can also again be used in comuting the roagating constant [Thyle / n et al99; Feit and Fleck 978] Wentel-Kramers- Brillouin, W.K.B aroximation technique has been one of those aroximation that has been used in studying roagation through non-uniform media both classically and quantum mechanically, esecially when it has got to do with scattering roblems[davydov,99;fitartrick,00;ugwu,00 9] Generally, as it is clearly known that when wave roagates either normally or obliquely incident on a surface or interface where the refractive index exhibit a sudden change within the medium, there is generally an areciable observed reflection of wave [Ugwu et al 007] esecially in a situation where W.K.B aroximation technique which is an aroximate solution obtained by considering that the incident roagating wave through the medium exeriences slowly varying refractive index which on the other hand introduces a change in the wavelength of the roagating wave.this change for sure brings about a change in the hase of the wave which the use of an aroximation can be made ossible introducing the concet of Jeffries connection formula which enables one to comute the reflection co-efficient, a concet that favours a roer understanding of the reflection roerties of the material medium of the thin film material [Ugwu and Uduh 005] Though it is certain that many of these aroximation technique being used in the study of wave roagation through an inhomogeneous medium involved formulation which has shortcoming [Thyle / n et al,983] as it always made use of neglecting some factors, it can be imroved uon by alying some certain correction terms to the aroximation [ Feit and Fleck, 980]. In this aer we consider the wave roagation through a dielectric thin film medium using W.K.B aroximation method with view of analying the influence of the dielectric erturbation, ( on the roagating field rofile through the thin film

2 Am. J. Sci. Ind. Res., 0, (4: Reference Medium ε ref Δ ε ( Deosited thin film Fig. Deosited thin film model on a substrate THEORETICAL PROCEDURE Given ψ + ωμε ψ = ( with ( ( ( 0 0 ε ( =Δ ε( + εref ( Where ( is the dielectric erturbation defined on the thin film medium and ε ref the dielectric reference medium associated with an emty sace without thin film. If we substitute equation ( into equation ( we obtain ψ ( + ωμ0 ( ψ( (3 = ωμε ψ Setting k ref ( = ω μ, then we have; ( k ( ( k ( ψ + Δ ε ψ = ε ψ 0 0 ref Defining k ε ( V ( Δ = as a term containing 0 the erturbation, and neglecting the right hand side, equation (3 becomes; ( V ( ψ ( 0 ψ = (4a We will assume equation (4a to have a solution of the form ψ ( = ex( iφ ( (4b Differentiating twice, we obtain ψ ( = iφ ex( iφ ( φ ex( iφ (5 Substituting into equation (4a, we have iφ ex( iφ ( φ ex( iφ (6 V ( ex( iφ = 0 Assume φ to be very small, and since ex( iφ 0 then φ = V φ =± i V ( ( ( ( ( φ =± V d (7 In accordance with WKB aroximation, the condition for φ to be considered very small results from the fact that; V φ V (8 V A second aroximation is found by iteration from equation (7 which gives us φ V ± V (9 substituting into equation (6 gives c ex ( i V ( d ψ ( 4 V ( + c ex ( i V ( d (9a Or ψ ( ( ( ex ( ( c ex V d 4 V ( + c V d (9b 548

3 Am. J. Sci. Ind. Res., 0, (4: The factor will be taken simly as a 4 V ( constant and therefore can be embedded in the constants c and c to give new constants A and B. Thus the aroximate solution can be written ψ ( Aex ( V ( d (0 + B ex V d Now ( ( ( V d = k0 d π = i Hence the required solution is π ψ ( Aex i π + B ex i ( For large values of the term; π ex i will cause the solution to be unbounded. Hence the correct aroximate solution is π ψ ( Acos π + A sin ( With ψ ( 0 = 0, and ψ ( 0 = we get; A = 0, π ψ ( A sin Now π π ψ ( = A cos Hence ψ ( 0 = A π Δ ε = A = π It then follows that π ψ ( sin π (3 gives WKB aroximate solution. RESULT AND DISCUSSION The study of the field roagation through a dielectric thin film in this work emanated from the solution of the scalar wave equation on thin film considered to have dielectric erturbation with roagation distance in accordance with the equation(as defined in fig. on which WKB aroximation technique and other necessary boundary conditions were alied to obtain the solution of the field rofile as it roagate through the thin film as in equation (3 The grahs as obtained from equation (3 are shown in fig. deicts the wave rofile for three regions of electromagnetic wave comrising ultraviolet,otical and near infrared region when the dielectric erturbation, Δ ε ( =0.50, ψ (, ψ ( andψ ( reresent the rofile in each 3 case resectively. From this, it is observed that just for a small roagation distance between 0 to 0 0 nm from the thin film, the field rofile aear to exhibit the same attern after which uv rofile, ψ ( rises sharly and eaked at 80x 0 nm and falls while the otical field rofile ( and that of infrared ψ ( 3 tend to oscillate out of hase as they roagate through the film. Fig.3 resents the rofile when Δ ε ( = 3.50 for the same three defined electromagnetic wave regions. In this case, the rofile seemed to maintain the same attern as in the first case but after the roagation distance of 0 0 nmψ ( assumes a more linear characteristic while the 0tical and infrared rofiles ceases and reduces oscillation resectively. In fig. 4 were shown in the rofile when, Δ ε ( =0.5 for the same regions in question. Here, the linearity of the uv ψ 549

4 Am. J. Sci. Ind. Res., 0, (4: rofile becomes more ronounced while ψ ( tend to assume a straight line with ceasure of ψ 3 ( oscillation. However, this difference in the rofiles of the electromagnetic wave regions is due to their difference in resonse to the influence of scattering and reflection of the roagating wave through the thin film articles. This is because for some of the regions, the small scattered art of their comonents may combine constructively with the rimary incident wave resulting in the hase change that is tantamount to change of the wave velocity roagating through the thin film medium. On the other hand the comonents may be lost as a result of destructive interference and hence leads to decrease in characteristic wave rofile [Brykhovestkii and Tigrov, 985] 00 ψ ( ψ ( 00 ψ 3 ( Fig. Field rofile in the thin film as it roagates through the thin film thickness ( when =50nm, 650nm and 950nm 0 nm for ε ( Δ = ψ ( ψ ( 00 ψ 3 ( Fig.3 Field rofile in the thin film as it roagates through the thin film thickness ( =50nm, 650nm and 950nm 0 nm for Δ ε ( =3.50 when 550

5 Am. J. Sci. Ind. Res., 0, (4: ψ ( ψ ( ψ 3 ( Fig.4 Field rofile in the thin film as it roagates through the thin film thickness (x when =50nm, 650nm and 950nm 0 nm for ε ( Δ =0.50 CONCLUSION In this work we have studied the roagation of wave through a thin film medium in which dielectric erturbation was introduced as a factor affecting the roagating field rofile was analyed.w.k.b aroximation technique with all the necessary boundary conditions were used as a recursor in obtaining the value of the roagating field from the defined scalar wave equation. Also considered was the effect of the variation in the value of the dielectric erturbation, ( on uv, otical and infrared region of the roagating electromagnetic wave through the thin film. REFERENCE [] Ugwu, E.I 0 Comutation method of studying the field roagation through an inhomogeneous thin film medium using Limann-Schwinger Equation: Trend in Alied Science Research 6( [] Ong H.L and Meyer.R.B 983 Electromagnetic Wave roagation in a eriodically bent Nematic liquid Crystal J.Ot Soc Am73: [3] Abele / s F 950 Investigations on the roagation of Sinusoidal Electromagnetic wave in stratified media. Alication to Thin films AnnPhy. (Daris. 5; [4] Thyle / n, L 983 The beam Proagation Method; An analysis of its alicability. Ot.Quantum Electron 5: [5] Thyle / n L and C. M Lee 99 Beam Proagation method on matrix Diagonaliation, J. Ot. Soc. Am. A 9(. [6] Feit M.D and J.A Fleck 978 light Proagation in graded index Otical fibers A.Otics 7; [7] Feit M.D and Fleck, J.A 980 Comutation of eigenfunctions in graded index otical fibers by the roagating Beam method A. Otic [8] Davydov A.S 99 Quantum Mechanics. Pergamon [9] Fitatrik, R 005 Electromagnetic wave roagationidielectric;htt://farside.h. utexa.edu/teaching/jkl/lectures/node79.html. [0] Ugwu E.I 009 Analytical study of electromagnetic wave scattering behavior using Limann-Schwinger equation IJPS Vl.4( [] Ugwu, E.I and G.A Agbo 007 Wave Proagation in non-homogeneous thin films of slowly varying refractive index W.K.B solution model PJST 8( [] Ugwu.E.I and P. C Uduh 005 Determination of Refractive index, Reflection /Transmittance of iron sulhide thin film deosited by solution growth technique (S.G.T JICCOTECH Maiden edition.pp.07-. [3] Brykhovestskii A.S and Tigrov, M. 985; Effective Imedance Tension of the statically Rough ideally conductive Surface. Radio quantum. Electron