Thus, continuity of the tangential component of the electric field across the boundary requires \(\widetilde{\bf E}_1(0)=\widetilde{\bf E}_2(0)\), and therefore, \[\widetilde{\bf E}^i(0) + \widetilde{\bf E}^r(0) = \widetilde{\bf E}^t(0) %~\mbox{, and}\]. behindthesciences | September 21, 2016 | Microwaves Propagation | 1 Comment. �,@S�7[C�tǫ {'�v���q ��_�.&�]fJ��-[%!a��'�l�u�3��$�+��µ��ّ*';g��\9��A{a�2W�R���y-(ۑ�K�B�����Xn�C3^�k�cI�)Y���&r|NkO��*�Eʍ��"�E/l7��Q�V�S}� �j-����/������x���� �}��:$u���,�{,,����Е{I���2k��]_x��Ч}I#*� �������� � � ���c&-�Y 0�l�� �g�@ `� �3Bt0u4���j�P}PX��P5�yqiWiV � �7G�_LoCX�61�00p3�Nwb�5Y6)H�d�>H3q/�0p=��3� Q۹ This convention simplifies writing considerably. where \(\epsilon_{r1}\) and \(\epsilon_{r2}\) are the relative permittivities in Regions 1 and 2, respectively. While it is true that we did not explicitly account for the possibility of a perfect conductor in Region 2, let’s see what the present analysis has to say about this case. In this section, we consider the scenario of a uniform plane wave which is normally incident on the planar boundary between two semi-infinite material regions. The following image shows the different wave components (incidence, reflection and transmission) that we are going to study in this post: Reflection and Transmission Coefficients. A 2*2 matrix method is developed and used to calculate the reflection and transmission amplitudes for normal-incidence reflection and transmission by uniaxial crystals and crystal slabs. The general expression for reflectivity is derivable from Fresnel's Equations.For purposes such as the calculation of reflection losses from optical instruments, it is usually sufficient to have the reflectivity at normal incidence. %%EOF First, note that this may seem at first glance to be a violation of the “lossless” assumption made at the beginning of this section. It may be helpful to note the very strong analogy between reflection of the electric field component of a plane wave from a planar boundary and the reflection of a voltage wave in a transmission line from a terminating impedance. We also acknowledge previous National Science Foundation support under grant numbers 1246120, 1525057, and 1413739. The symmetry of the problem also precludes a change of polarization, so the reflected wave should have no \(\hat{\bf y}\) component. We shall assume that the media are “simple” and lossless (i.e., the imaginary component of permittivity \(\epsilon''\) is equal to zero) and therefore the media are completely defined by a real-valued permittivity and a real-valued permeability. Figure \(\PageIndex{1}\) shows the wave incident on the boundary, which is located at the \(z=0\) plane. In this case, we are concerned about what fraction of the power density is transmitted into Region 2 and what fraction of the power density is reflected from the boundary. We may now solve for \(C\) by substituting Equation \ref{m0161_eB} into Equation \ref{m0161_eBCE}. The basic equations that have been derived in the study of one-dimensional sound propagation are well suited to solve a multitude of problems concerning reflection and sound transmission. In this case, \(\eta_2=\eta_1\), so \(\Gamma_{12}=0\), and we obtain the expected result. For more information contact us at info@libretexts.org or check out our status page at https://status.libretexts.org. Recall that any discontinuity in the tangential component of the total magnetic field intensity must be supported by a current flowing on the surface. In the case of transmission lines, we were concerned about what fraction of the power was delivered into a load and what fraction of the power was reflected from a load. 1. Reflection and transmission at normal incidence onto air-saturated porous materials and direct measurements based on parametric demodulated ultrasonic waves. Now employing Equations \ref{m0161_eHi}, ref{m0161_eHr}, and \ref{m0161_eHt}, we obtain: \[\frac{E_0^i}{\eta_1} - \frac{B}{\eta_1} = \frac{C}{\eta_2} \label{m0161_eBCH}\]. The reflection coefficient from medium n1 to medium n2 (refraction indexes) can be … Not affiliated © 2020 Springer Nature Switzerland AG. Am . Therefore, the tangential components of the magnetic field must be continuous across the boundary. Thus, \(\Gamma_{12}\sim-0.27\). In fact, all other contributions to the total field may be expressed in terms of \(E_0^i\). 31 488 . This equation can be obtained by enforcing the boundary condition on the magnetic field. θ=π1 /2, the refraction angle reaches a limit value θ=c arcsin /()nn12, called the critical angle. [ "article:topic-guide", "license:ccbysa", "authorname:swellingson", "showtoc:no", "program:virginiatech" ], Associate Professor (Electrical and Computer Engineering), 5.1: Plane Waves at Normal Incidence on a Planar Boundary, Virginia Polytechnic Institute and State University, Virginia Tech Libraries' Open Education Initiative. Since the wave impedance \(\eta=\sqrt{\mu/\epsilon}\) in lossless media, and since most lossless materials are non-magnetic (i.e., exhibit \(\mu\approx\mu_0\)), it is possible to express \(\Gamma_{12}\) purely in terms of permittivity for these media. [40 pts) Problem 4. Normal Reflection Coefficient The reflectivity of light from a surface depends upon the angle of incidence and upon the plane of polarization of the light. Since the direction of propagation for the reflected wave is \(-\hat{\bf z}\), we have from the plane wave relationships that, \[\widetilde{\bf H}^r(z) = -\hat{\bf y} \frac{B}{\eta_1} e^{+j\beta_1 z}~\mbox{,}~~z \le 0 \label{m0161_eHr}\].

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