Study On The Reflection And Transmission Of Inhomogenous Wave

Since a continuous varying index profile is nothing more than a stack of thin layers which have homogeneous refractive indeices when the thickness of layers is very very small while the number of layers is very very large, in the present investigation, instead of the numerical calculations with transfer matrix approach, the analytical transfer matrix method is employed to determine the reflection coefficient from an inhomogeneous stratified media. With the help of the definition of general wavenumber, which contains both the main waves and the subwaves propagated in the structure with continuously varying index profile, an explicit and exact expression with clear physical insight is obtained without introducing any approximation.The result shows that, besides the ambience and the refraction indices of both front-surface and back-surface of the structure, the unique dependence of the reflection coefficient is the total phase shift accumulated by the main wave and the subwaves.The tranditional WKBJ approach is an approximation in which the subwaves refrected from the interior of the structure are neglected while the result presented here is exact.The reflection from one-layer planar structure, two-layer planaer structure and three-layer structure is studied using the expression. Even more, the refractive index profile is an exponential function, or Gauss function, or periodi sine functions is calculated and the results are discussed. In fact the expression obtained in the paper can be applied to arbitrary refractive index profile.The study on the refrection and transmission of lights can be adopted in many application, such as high sensitive sensor or measurement instructions. Especially it can be used in the control of thickness of materials selection in the thin film process, the reflection coefficient will keep unchange only if the change in the thickness or refractive index satisfied the derived condition presented here. The conclusions are very useful for the hot application in the infra-red thin film or solar thin film today.In the alaysis of reflection and transmission probability of the particle wave from inhomogeneous potential field, since the Shrodinger equation is similar with the light waveguide equation, such as the time-dependent Schrodinger equation is similar with Fresnel equation while the time-independent Schrodinger equation is similar with Helmhoz equation, it can be regarded that the potential filed equates to refractive index profile, energy eigenvalues equates to light propagation constant, high refractive index section to potential well while low refractive index section to potential barrier. So the analytical transfer matrix method can be also applied to analyzing the Schrodinger equation. Without solving the Schrodinger equation, and with the help of defining the general wavenumber, an exact and general expression for the transmission and reflection probabilities are presented in a very explicit way.Different from the WKB method and its refined versions, subwaves, which inherently exist in a inhomogeneous systemand is always neglected in the semiclassical approaches, is taken into account, results in the total phase shift of a quantum particle across an arbitrary potential barrier. Moreever, it is not subject to the requirement of the de Broglie wavelength and the range of the particle energy.As a consequence, the expression obtained here may extensively be applied to many basic quantum phenomena, such as, quantum tunneling, quantum reflection, the time related to a tunneling particle and the resonant tunneling. The parabolic barrier, double barrier with a rectangular well, double barrier with a parabolic well structure and the potential barrier placed in an external field are discussed here.The reflection and the transmission of light and quantum particle are unified in the expressions, obviously it is very important in physics.