Once these values are known, the problem is completely solved. interface at normal incidence Our intention here is to find the transmission and reflection coefficients Let us rewrite the fields in the two regions: region I (z<0) and region II (z>0) (see Table 6.1) and Apply the boundary conditions to obtain the two unknown 21 Electromagnetic Field Theory by R. S. Kshetrimayum 3/25/2014 coefficients The constant \(C\), like \(B\), is a complex-valued constant that remains to be determined. Thumbnail: Sinusoidal traveling plane wave entering a region of lower wave velocity at an angle, illustrating the decrease in wavelength and change of direction (refraction) that results. The reflection results take the form r=r0 cos2 phi +rcsin2 phi , r'=(r0-re)cos phi sin phi , where phi is the angle…, Bounds and zeros in reflection and refraction by uniaxial crystals, Electromagnetic Wave Reflectance, Transmittance, and Absorption in a Graphene-Covered Uniaxial Crystal Slab, Reflection and transmission of an electromagnetic wave in a strongly anisotropic medium: application to polarizers and antireflection layers on a conductive plane, Normal incidence transmission ellipsometry of anisotropic layers, Optical properties of an isotropic layer on a uniaxial crystal substrate. where \(\beta_1=\omega\sqrt{\mu_1 \epsilon_1}\) is the phase propagation constant in Region 1 and \(E_0^i\) is a complex-valued constant. Making the substitution above and eliminating all extraneous factors of \(\epsilon_0\), we find: \[\Gamma_{12} = \frac{\sqrt{\epsilon_{r1}}-\sqrt{\epsilon_{r2}}}{\sqrt{\epsilon_{r1}}+\sqrt{\epsilon_{r2}}} \label{m0161_eGer}\]. Reflection and transmission at normal incidence onto air-saturated porous materials and direct measurements based on parametric demodulated ultrasonic waves. The symmetry of the problem precludes any direction of propagation other than \(+{\bf z}\), and with no possibility of a wave traveling in the \(-\hat{\bf z}\) direction for \(z>0\). ScienceDirect ® is a registered trademark of Elsevier B.V. ScienceDirect ® is a registered trademark of Elsevier B.V. %%EOF
Equations \ref{m0161_eErf} and \ref{m0161_eEtf} are the reflected and transmitted fields, respectively, in response to the incident field given in Equation \ref{m0161_eEi} in the normal incidence scenario shown in Figure \(\PageIndex{1}\). Similarly, we infer the existence of a “transmitted” plane wave propagating on the \(z>0\) side of the boundary. This equation can be obtained by enforcing the boundary condition on the magnetic field. Have questions or comments? The LibreTexts libraries are Powered by MindTouch® and are supported by the Department of Education Open Textbook Pilot Project, the UC Davis Office of the Provost, the UC Davis Library, the California State University Affordable Learning Solutions Program, and Merlot. �tq�X)I)B>==����
�ȉ��9. From the plane wave relationships, we determine that the associated magnetic field intensity is, \[\widetilde{\bf H}^i(z) = \hat{\bf y} \frac{E_0^i}{\eta_1} e^{-j\beta_1 z}~\mbox{,}~~z \le 0 \label{m0161_eHi}\]. Normal Incidence Reflection Seismogram¶ The principles of the normal incidence reflection seismogram are illustrated in the diagrams below. 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. At radio frequencies, a wavelength is many orders of magnitude less than the diameter of the Moon, so we are justified in treating the Moon’s surface as a planar boundary between free space and a semi-infinite region of lossy dielectric material. When a plane wave encounters a discontinuity in media, reflection from the discontinuity and transmission into the second medium is possible. Thus, we obtained the correct answer because we were able to independently determine that \(\eta_2=0\) in a perfect conductor. Singularities in polarization resolved angular patterns: transmittance of nematic liquid crystal cells. Since the direction of propagation for the reflected wave is \(-\hat{\bf z}\), we have from the plane wave relationships that, \[\widetilde{\bf H}^r(z) = -\hat{\bf y} \frac{B}{\eta_1} e^{+j\beta_1 z}~\mbox{,}~~z \le 0 \label{m0161_eHr}\]. Now employing Equations \ref{m0161_eEi}, \ref{m0161_eEr}, and \ref{m0161_eEt}, we obtain: Clearly a second equation is required to determine both \(B\) and \(C\). Expressions for \(\widetilde{\bf H}^r\) and \(\widetilde{\bf H}^t\) may be obtained by applying the plane wave relationships to the preceding expressions. Comparisons between theoretical modeling and experimental data are provided and prospective industrial applications are discussed. Copyright © 2020 Elsevier B.V. or its licensors or contributors.