When electromagnetic waves travel through matter rather than a vacuum, they interact with the atoms and molecules of the medium. This interaction leads to phenomena such as slowing down, dispersion, and energy absorption, which can be mathematically modeled by treating the material as a homogeneous, isotropic, and linear dielectric. In this post, we provide an exhaustive derivation of the damped wave equation and explore how conductivity leads to the concepts of skin depth and complex refractive indices.
Electromagnetic Waves in a Dielectric Medium: Theory and Derivations
1. Nature of the Medium and Constitutive Relations
To analyze wave propagation in matter, we simplify the material properties. We assume the medium is Homogeneous (properties do not change with position), Isotropic (properties are the same in all directions), and Linear (the response of the material is directly proportional to the applied field).
The interaction of fields with matter is governed by the constitutive relations:
Where \(\epsilon\) is the permittivity and \(\mu\) is the permeability of the medium. For most dielectric materials at optical frequencies, we assume they are non-magnetic, so \(\mu \approx \mu_0\).
Electrical Conductivity and Ohm's Law
In a conducting dielectric, the movement of free charges constitutes a current density \(\mathbf{J}\). According to Ohm’s law:
\[ \mathbf{J} = \sigma \mathbf{E} \]Here, \(\sigma\) is the electrical conductivity. In real materials, \(\sigma\) may depend on the frequency \(\omega\), leading to absorption bands.
2. Maxwell’s Equations in the Medium
We assume there are no free charges (\(\rho_f = 0\)). Maxwell's equations in the medium become:
\[ \begin{aligned} (1)\;& \nabla \cdot \mathbf{E} = 0 \\ (2)\;& \nabla \cdot \mathbf{B} = 0 \\ (3)\;& \nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t} \\ (4)\;& \nabla \times \mathbf{H} = \mathbf{J} + \frac{\partial \mathbf{D}}{\partial t} \end{aligned} \]Substituting the constitutive relations and Ohm's law into equation (4):
\[ \nabla \times \left( \frac{\mathbf{B}}{\mu} \right) = \sigma \mathbf{E} + \epsilon \frac{\partial \mathbf{E}}{\partial t} \] \[ \nabla \times \mathbf{B} = \mu \sigma \mathbf{E} + \mu \epsilon \frac{\partial \mathbf{E}}{\partial t} \]3. Derivation of the Damped Wave Equation
We decouple the fields by taking the curl of Faraday’s Law (Equation 3):
\[ \nabla \times (\nabla \times \mathbf{E}) = -\frac{\partial}{\partial t} (\nabla \times \mathbf{B}) \]Using the vector identity \(\nabla \times (\nabla \times \mathbf{E}) = \nabla(\nabla \cdot \mathbf{E}) - \nabla^2 \mathbf{E}\) and knowing \(\nabla \cdot \mathbf{E} = 0\):
\[ -\nabla^2 \mathbf{E} = -\frac{\partial}{\partial t} (\nabla \times \mathbf{B}) \]Substitute the expression for \(\nabla \times \mathbf{B}\) from Section 2:
\[ -\nabla^2 \mathbf{E} = -\frac{\partial}{\partial t} \left( \mu \sigma \mathbf{E} + \mu \epsilon \frac{\partial \mathbf{E}}{\partial t} \right) \] \[ -\nabla^2 \mathbf{E} = -\mu \sigma \frac{\partial \mathbf{E}}{\partial t} - \mu \epsilon \frac{\partial^2 \mathbf{E}}{\partial t^2} \]Multiplying by \(-1\), we obtain the Damped Wave Equation:
The first term on the right represents the damping (attenuation) due to conductivity, while the second term represents the oscillation (propagation).
4. Plane Wave Solution and Complex Wave Number
We seek a monochromatic plane wave solution of the form:
\[ \mathbf{E}(\mathbf{r},t) = \mathbf{E}_0 e^{i(\mathbf{k}\cdot\mathbf{r} - \omega t)} \]Applying the operators:
- \(\nabla^2 \rightarrow -k^2\)
- \(\frac{\partial}{\partial t} \rightarrow -i\omega\)
- \(\frac{\partial^2}{\partial t^2} \rightarrow -\omega^2\)
Substituting into the wave equation:
\[ -k^2 \mathbf{E} = \mu \sigma (-i\omega) \mathbf{E} + \mu \epsilon (-\omega^2) \mathbf{E} \] \[ k^2 = \mu \epsilon \omega^2 + i\mu \sigma \omega \]Because the right side is complex, the wave number \(k\) must also be complex. Let \(k = \beta + i\alpha\). Squaring this gives:
\[ k^2 = (\beta^2 - \alpha^2) + 2i\alpha\beta \]Equating real and imaginary parts:
- \(\beta^2 - \alpha^2 = \mu \epsilon \omega^2\) (Phase propagation)
- \(2\alpha\beta = \mu \sigma \omega\) (Attenuation/Absorption)
Solving these by direct substitution (e.g., \(\alpha = \dots / \beta\)) leads to a complicated quartic equation. The elegant method is to use algebraic identities involving squares.
5. Finding the Sum of Squares
We need to find the value of \( \beta^2 + \alpha^2 \). We use the algebraic identity:
\[ (A + B)^2 = (A - B)^2 + 4AB \]Let \( A = \beta^2 \) and \( B = \alpha^2 \). Then:
\[ (\beta^2 + \alpha^2)^2 = (\beta^2 - \alpha^2)^2 + 4\beta^2\alpha^2 \] \[ (\beta^2 + \alpha^2)^2 = (\beta^2 - \alpha^2)^2 + (2\alpha\beta)^2 \]Now, substitute the physical quantities from Equations (1) and (2) into the right-hand side:
\[ (\beta^2 + \alpha^2)^2 = (\mu\epsilon\omega^2)^2 + (\mu\sigma\omega)^2 \]To simplify, factor out \( (\mu\epsilon\omega^2)^2 \):
\[ (\beta^2 + \alpha^2)^2 = (\mu\epsilon\omega^2)^2 \left[ 1 + \frac{(\mu\sigma\omega)^2}{(\mu\epsilon\omega^2)^2} \right] \]Simplifying the fraction inside the bracket:
\[ \frac{\mu^2\sigma^2\omega^2}{\mu^2\epsilon^2\omega^4} = \frac{\sigma^2}{\epsilon^2\omega^2} = \left(\frac{\sigma}{\epsilon\omega}\right)^2 \]So:
\[ (\beta^2 + \alpha^2)^2 = (\mu\epsilon\omega^2)^2 \left[ 1 + \left(\frac{\sigma}{\epsilon\omega}\right)^2 \right] \]Taking the square root of both sides gives us a crucial intermediate equation (Equation 3):
6. Solving for the Phase Constant (\(\beta\))
To isolate \(\beta\), add Equation (1) and Equation (3):
\[ (\beta^2 - \alpha^2) + (\beta^2 + \alpha^2) = \mu\epsilon\omega^2 + \mu\epsilon\omega^2 \sqrt{1 + \left(\frac{\sigma}{\epsilon\omega}\right)^2} \]The \(\alpha^2\) terms cancel out:
\[ 2\beta^2 = \mu\epsilon\omega^2 \left[ 1 + \sqrt{1 + \left(\frac{\sigma}{\epsilon\omega}\right)^2} \right] \]Divide by 2 and take the square root:
\[ \beta = \omega \sqrt{\frac{\mu\epsilon}{2} \left[ \sqrt{1 + \left(\frac{\sigma}{\epsilon\omega}\right)^2} + 1 \right]} \]Physical Interpretation: This value determines the wavelength in the medium (\(\lambda = 2\pi/\beta\)) and the phase velocity (\(v_p = \omega/\beta\)).
7. Solving for the Attenuation Constant (\(\alpha\))
To isolate \(\alpha\), subtract Equation (1) from Equation (3):
\[ (\beta^2 + \alpha^2) - (\beta^2 - \alpha^2) = \mu\epsilon\omega^2 \sqrt{1 + \left(\frac{\sigma}{\epsilon\omega}\right)^2} - \mu\epsilon\omega^2 \]The \(\beta^2\) terms cancel out:
\[ 2\alpha^2 = \mu\epsilon\omega^2 \left[ \sqrt{1 + \left(\frac{\sigma}{\epsilon\omega}\right)^2} - 1 \right] \]Divide by 2 and take the square root:
\[ \alpha = \omega \sqrt{\frac{\mu\epsilon}{2} \left[ \sqrt{1 + \left(\frac{\sigma}{\epsilon\omega}\right)^2} - 1 \right]} \]Physical Interpretation: This value determines the rate of energy loss. The wave amplitude decays as \(e^{-\alpha z}\).
8. Final General Formulas
Here are the complete, exact solutions valid for any linear, isotropic, homogeneous medium:
Attenuation Constant (Absorption)
\[ \alpha = \omega \sqrt{\frac{\mu\epsilon}{2} \left[ \sqrt{1 + \left(\frac{\sigma}{\epsilon\omega}\right)^2} - 1 \right]} \]Phase Constant (Propagation)
\[ \beta = \omega \sqrt{\frac{\mu\epsilon}{2} \left[ \sqrt{1 + \left(\frac{\sigma}{\epsilon\omega}\right)^2} + 1 \right]} \]9. Analysis of the Loss Tangent
The term inside the square root, \(\frac{\sigma}{\epsilon\omega}\), is often called the "loss tangent" or ratio of conduction current to displacement current.
- Perfect Dielectric (\(\sigma = 0\)):
\(\alpha = 0\) (No absorption).
\(\beta = \omega\sqrt{\mu\epsilon}\) (Standard wave propagation). - Good Conductor (\(\sigma \gg \epsilon\omega\)):
The term \(\frac{\sigma}{\epsilon\omega}\) dominates.
\(\alpha \approx \beta \approx \sqrt{\frac{\omega\mu\sigma}{2}}\).
This results in rapid attenuation (small skin depth).
10. Physical Solution and Skin Depth
Substituting \(k = \beta + i\alpha\) back into the plane wave expression (assuming propagation along \(z\)):
\[ \mathbf{E}(z,t) = \mathbf{E}_0 e^{i((\beta + i\alpha)z - \omega t)} \]The term \(e^{-\alpha z}\) indicates that the amplitude of the field decays exponentially as it penetrates the medium.
Definition: Skin Depth (\(\delta\))
The skin depth is defined as the distance at which the amplitude falls to \(1/e\) (approx. 37%) of its initial value. This occurs when \(\alpha z = 1\):
\[ \delta = \frac{1}{\alpha} \]In highly conductive materials (good conductors), \(\alpha\) is very large, making \(\delta\) very small—this explains why EM waves (like radio) cannot penetrate deep into metals.
11. Complex Refractive Index
In a vacuum, \(k = \frac{\omega}{c}\). In a medium, we define a complex refractive index \(\tilde{n}\) such that:
\[ k = \frac{\omega}{c} \tilde{n} \] \[ \tilde{n} = n + i\kappa \]Where:
- \(n\): The real refractive index (related to phase velocity and dispersion).
- \(\kappa\): The extinction coefficient (related to absorption).
Relating these to our previous parameters:
\[ \beta = \frac{\omega n}{c}, \quad \alpha = \frac{\omega \kappa}{c} \]Substituting these into the field equation:
\[ \mathbf{E} = \mathbf{E}_0 e^{-\frac{\omega\kappa}{c} z} e^{i\left(\frac{\omega n}{c}z - \omega t\right)} \]12. Dispersion and Kramers–Kronig Relations
The real part \(n(\omega)\) and the imaginary part \(\kappa(\omega)\) are not independent. They are connected by the Kramers–Kronig relations, which stem from the principle of causality.
As seen in the characteristic sketches:
- Absorption Peaks: Occur at the resonant frequencies of the material's molecules (\(\kappa\) is maximum).
- Anomalous Dispersion: In the region of high absorption, the refractive index \(n\) can decrease as frequency increases, contrary to normal dispersion.
13. Summary of Parameters
| Parameter | Symbol | Physical Role |
|---|---|---|
| Phase Constant | \(\beta\) | Determines wavelength in the medium |
| Attenuation Constant | \(\alpha\) | Determines the rate of absorption |
| Refractive Index | \(n\) | Determines phase velocity (\(v = c/n\)) |
| Extinction Coefficient | \(\kappa\) | Quantifies the loss of energy to the medium |