Introduction
A linearly polarized uniform plane wave traveling in free space is incident normally upon a flat dielectric surface is a fundamental scenario in electromagnetics, relevant to a wide range of applications including antenna design, microwave engineering, optical coatings, and remote sensing. Understanding the behavior of such waves as they encounter dielectric interfaces is crucial for predicting reflection, transmission, and absorption phenomena. This article provides a comprehensive analysis of the physics involved, the boundary conditions at the dielectric interface, and the resulting electromagnetic fields. We will explore the wave's propagation characteristics, the nature of the reflected and transmitted waves, and the factors influencing their amplitudes and phases.
Fundamentals of Plane Wave Propagation in Free Space
Characteristics of a Uniform Plane Wave
- Definition: A uniform plane wave is a wave with electric and magnetic fields that are uniform in planes perpendicular to the direction of propagation.
- Propagation Direction: The wave propagates in a specific direction, here taken as the z-axis for simplicity.
- Polarization: The electric field vector oscillates linearly in a fixed plane, which we assume to be the x-direction for this analysis.
- Wave Equation: In free space, the wave satisfies the homogeneous wave equation, with phase velocity \(c\) (speed of light in vacuum).
Mathematical Representation in Free Space
The incident plane wave can be expressed as:
\[
\mathbf{E}i(z, t) = \mathbf{E}0 e^{j(k z - \omega t)} \hat{x}
\]
\[
\mathbf{H}i(z, t) = \frac{1}{\eta0} \mathbf{E}_0 e^{j(k z - \omega t)} \hat{y}
\]
where:
- \(\mathbf{E}_0\): Amplitude of the electric field.
- \(\eta_0 \approx 377\, \Omega\): Intrinsic impedance of free space.
- \(k = \frac{\omega}{c}\): Wavenumber in free space.
- \(\omega\): Angular frequency.
Interface Between Free Space and Dielectric
Properties of the Dielectric Material
- Permittivity: \(\varepsilonr \varepsilon0\), where \(\varepsilon_r\) is the relative permittivity of the dielectric.
- Permeability: Assumed to be \(\mu_0\) (non-magnetic dielectric).
- Refractive index: \(n = \sqrt{\varepsilon_r}\).
Geometry of Incidence
The wave strikes the dielectric interface at \(z=0\) with normal incidence, meaning the wave vector is perpendicular to the surface. The incident wave propagates along the positive z-axis, with the interface at \(z=0\), the dielectric occupying \(z > 0\), and free space for \(z < 0\).
Boundary Conditions at the Dielectric Interface
Maxwell's Boundary Conditions
At the interface \(z=0\), the tangential components of the electric and magnetic fields must be continuous:
- \(\hat{z} \times (\mathbf{E}\text{above} - \mathbf{E}\text{below}) = 0\)
- \(\hat{z} \times (\mathbf{H}\text{above} - \mathbf{H}\text{below}) = \mathbf{J}_s\)
Since there are no free surface currents (\(\mathbf{J}_s=0\)), the tangential fields are continuous across the boundary:
\[
\mathbf{E}{t,\text{free}} = \mathbf{E}{t,\text{dielectric}}
\]
\[
\mathbf{H}{t,\text{free}} = \mathbf{H}{t,\text{dielectric}}
\]
\end{pre>
Implications for the Fields
- The electric field component in the x-direction remains continuous across the interface.
- The magnetic field component in the y-direction also remains continuous.
Reflected and Transmitted Waves
Form of the Fields
At \(z=0\), the total fields are a superposition of incident and reflected waves in free space, and a transmitted wave in the dielectric:
\[
\mathbf{E}\text{total} = \mathbf{E}i + \mathbf{E}_r \quad \text{(in free space, }z<0\text{)}
\]
\[
\mathbf{E}_t \quad \text{(in dielectric, }z>0\text{)}
\]
\end{pre>
Similarly for magnetic fields:
\[
\mathbf{H}\text{total} = \mathbf{H}i + \mathbf{H}_r \quad \text{(in free space)}
\]
\[
\mathbf{H}_t \quad \text{(in dielectric)}
\]
\end{pre>
Expression for the Fields
- Incident wave (\(z<0\)): \(\mathbf{E}i = E0 e^{j(k_0 z - \omega t)} \hat{x}\)
- Reflected wave (\(z<0\)): \(\mathbf{E}r = Er e^{j(-k_0 z - \omega t)} \hat{x}\)
- Transmitted wave (\(z>0\)): \(\mathbf{E}t = Et e^{j(k_1 z - \omega t)} \hat{x}\)
Where \(k0 = \frac{\omega}{c}\) in free space, and \(k1 = \frac{\omega}{vd} = \frac{\omega}{c/\sqrt{\varepsilonr}} = \frac{\omega \sqrt{\varepsilon_r}}{c}\).
Reflection and Transmission Coefficients
Derivation of Coefficients
The continuity of tangential fields at \(z=0\) leads to the Fresnel equations for normal incidence:
\[
r = \frac{Er}{E0} = \frac{\eta1 - \eta2}{\eta1 + \eta2}
\]
\[
t = \frac{Et}{E0} = \frac{2 \eta2}{\eta1 + \eta_2}
\]
where:
- \(\eta1 = \eta0 = 377\, \Omega\): Intrinsic impedance of free space.
- \(\eta2 = \frac{\eta0}{\sqrt{\varepsilon_r}}\): Intrinsic impedance of the dielectric.
Reflection and Transmission Coefficients in Terms of Material Properties
- Reflection coefficient magnitude: \(\left| r \right| = \left| \frac{\eta1 - \eta2}{\eta1 + \eta2} \right|\)
- Transmission coefficient magnitude: \(\left| t \right| = \left| \frac{2 \eta2}{\eta1 + \eta_2} \right|\)
Field Distributions in Space
Region \(z<0\) (Free Space)
- Electric field: \(\mathbf{E}(z, t) = E0 e^{j(k0 z - \omega t)} \hat{x} + r E0 e^{j(-k0 z - \omega t)} \hat{x}\)
- Magnetic field: \(\mathbf{H}(z, t) = \frac{E0}{\eta0} e^{j(k0 z - \omega t)} \hat{y} - r \frac{E0}{\eta0} e^{j(-k0 z - \omega t)} \hat{y}\)
Region \(z>0\) (Dielectric)
- Electric field: \(\mathbf{E}(z, t) = t E_0 e
