Dispersion and the Lorentz Oscillator Model
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You should be comfortable with the equation of motion for a damped harmonic oscillator and the basic definition of refractive index from Electromagnetism I.
Why this matters
The Lorentz oscillator model explains why prisms split white light and why materials absorb specific colours.
In the previous lecture, we treated conductors as media with a constant complex permittivity. However, in dielectrics, the response of bound electrons is frequency-dependent. This variation leads to dispersion, where the speed of light in a medium depends on the frequency of the wave. Understanding this microscopic origin is crucial for interpreting spectroscopy, designing optical filters, and understanding the limits of signal propagation in fibres.
The Microscopic Picture
Imagine the electrons in a dielectric material as tiny balls attached to springs. When an electromagnetic wave passes through, its electric field exerts a force on these electrons, pulling them away from their equilibrium positions. Because they are bound to the atoms, they oscillate back and forth. This oscillation creates a secondary electromagnetic wave that interferes with the original wave. The key insight is that the "stiffness" of the spring (the natural frequency) and the "friction" (damping) determine how much the electrons lag behind the driving field. This lag causes the phase of the material's response to shift relative to the driving field, which is the fundamental mechanism behind dispersion and absorption.
Deriving the Complex Susceptibility
We model a bound electron of mass \(m\) and charge \(-e\) as a damped harmonic oscillator. The equation of motion under a driving electric field \(\vec{E}=\vec{E}_0e^{-i\omega t}\) is \(m\ddot{x} + m\gamma\dot{x} + m\omega_0^2 x = -eE_0e^{-i\omega t}\). Solving for the displacement amplitude \(x_0\), we find the induced dipole moment per electron is \(p = -ex_0\). Summing over \(N\) electrons per unit volume gives the polarization \(\vec{P} = N\vec{p}\). The electric susceptibility \(\chi(\omega)\) is defined by \(\vec{P} = \varepsilon_0 \chi(\omega) \vec{E}\). Substituting the solution for \(x_0\) yields the complex susceptibility shown in the derivation card below.
Here, \(\omega_p^2 = Ne^2/(\varepsilon_0 m)\) is the plasma frequency, which sets the scale of the material's response. The imaginary part of \(\chi\) corresponds to energy loss (absorption), while the real part corresponds to the storage of energy (dispersion).
Normal and Anomalous Dispersion
The refractive index is given by \(\tilde{n} = \sqrt{\varepsilon_r(\omega)} = \sqrt{1+\chi(\omega)}\). For weakly dispersive media where \(\chi \ll 1\), we can approximate \(n \approx 1 + \chi/2\). The real part of the refractive index, \(n(\omega)\), determines the phase velocity \(v_p = c/n\). Away from the resonance frequency \(\omega_0\), \(n\) increases with \(\omega\). This is called normal dispersion. However, in the narrow region around \(\omega_0\), where absorption is strong, the slope \(dn/d\omega\) becomes negative. This is anomalous dispersion. It is a direct consequence of the Kramers-Kronig relations, which link the real and imaginary parts of the susceptibility. You cannot have absorption without dispersion, and vice versa.
Notice the numerator \(\omega_0^2-\omega^2\). When \(\omega < \omega_0\), this is positive, and \(n > 1\). When \(\omega > \omega_0\), this is negative, and \(n < 1\) (close to resonance the full complex square root has to be used, because absorption is no longer small). The transition through zero at \(\omega_0\) is the hallmark of the anomalous region.
Absorption and the Lorentzian Line Shape
The imaginary part of the refractive index, \(\kappa\), determines the attenuation of the wave amplitude. The intensity decays as \(e^{-2\kappa \omega z/c}\). The expression for \(\kappa\) is a Lorentzian function of frequency. It peaks exactly at the resonance frequency \(\omega_0\), where the denominator is minimized. The full width at half maximum (FWHM) of this absorption line is approximately \(\gamma\). This width is determined by the damping rate, which in a real solid arises from collisions with phonons or other electrons. The fact that the absorption peak coincides with the region of anomalous dispersion is a critical diagnostic tool in spectroscopy.
At \(\omega = \omega_0\), the absorption is maximal, and the phase velocity is undefined in the simple sense because the wave is heavily attenuated. The group velocity, which represents the speed of the energy envelope, can become very small or even negative in this region, a phenomenon known as slow light.
The derivation
Dispersion from the Lorentz Oscillator Model
Modelling each bound electron in a dielectric as a damped harmonic oscillator of natural frequency \(\omega_0\) and damping rate \(\gamma\), driven by the electric field of a monochromatic wave \(\vec{E}=\vec{E}_0e^{-i\omega t}\), we derive the complex electric susceptibility \(\chi(\omega)=\omega_p^2/(\omega_0^2-\omega^2-i\gamma\omega)\) with plasma frequency \(\omega_p^2=Ne^2/\varepsilon_0 m\), hence the frequency-dependent permittivity \(\varepsilon_r(\omega)=1+\chi(\omega)\) and complex refractive index \(\tilde{n}=n+i\kappa=\sqrt{\varepsilon_r}\). We show that \(n(\omega)\) rises with frequency (normal dispersion) everywhere except inside the absorption line of full width \(\gamma\) centred on \(\omega_0\), where \(dn/d\omega<0\) (anomalous dispersion), and that absorption \(\kappa(\omega)\) is a Lorentzian peaking exactly where the dispersion is anomalous.
A common misconception is that \(n < 1\) implies the phase velocity exceeds \(c\), violating relativity. In the anomalous dispersion region near \(\omega_0\), \(n\) can indeed be less than 1, but the wave is strongly absorbed. The signal velocity, which is the speed at which information is transmitted, is always less than or equal to \(c\). The phase velocity is a property of the monochromatic component, not the information carrier. C
Worked example
Problem. A dielectric material has a plasma frequency \(\omega_p = 1.0 \times 10^{14}\ \mathrm{rad/s}\), a resonance frequency \(\omega_0 = 5.0 \times 10^{14}\ \mathrm{rad/s}\), and a damping rate \(\gamma = 1.0 \times 10^{13}\ \mathrm{rad/s}\). Calculate the real part of the refractive index \(n\) and the extinction coefficient \(\kappa\) at a frequency \(\omega = 4.0 \times 10^{14}\ \mathrm{rad/s}\). Assume \(\chi \ll 1\) for the approximation.
- Calculate the term \(\Delta = \omega_0^2 - \omega^2\). \(\omega_0^2 = (5.0 \times 10^{14})^2 = 2.5 \times 10^{29}\ \mathrm{rad^2/s^2}\). \(\omega^2 = (4.0 \times 10^{14})^2 = 1.6 \times 10^{29}\ \mathrm{rad^2/s^2}\). \(\Delta = 0.9 \times 10^{29}\ \mathrm{rad^2/s^2}\).
- Calculate the denominator term \(D = (\omega_0^2 - \omega^2)^2 + \gamma^2\omega^2\). \((\Delta)^2 = (0.9 \times 10^{29})^2 = 0.81 \times 10^{58}\). \(\gamma^2\omega^2 = (1.0 \times 10^{13})^2 (1.6 \times 10^{29}) = 1.6 \times 10^{55} = 0.0016 \times 10^{58}\). \(D \approx 0.8116 \times 10^{58}\ \mathrm{rad^4/s^4}\).
- Calculate the prefactor \(A = \omega_p^2 / 2\). \(\omega_p^2 = (1.0 \times 10^{14})^2 = 1.0 \times 10^{28}\ \mathrm{rad^2/s^2}\). \(A = 0.5 \times 10^{28}\ \mathrm{rad^2/s^2}\).
- Calculate \(n - 1\). \(n - 1 = A \frac{\Delta}{D} = (0.5 \times 10^{28}) \frac{0.9 \times 10^{29}}{0.8116 \times 10^{58}}\). \(n - 1 = \frac{0.45 \times 10^{57}}{0.8116 \times 10^{58}} = \frac{0.045}{0.8116} \approx 0.0554\).
- Calculate \(\kappa\). \(\kappa = A \frac{\gamma\omega}{D} = (0.5 \times 10^{28}) \frac{(1.0 \times 10^{13})(4.0 \times 10^{14})}{0.8116 \times 10^{58}}\). \(\kappa = \frac{0.5 \times 10^{28} \times 4.0 \times 10^{27}}{0.8116 \times 10^{58}} = \frac{2.0 \times 10^{55}}{0.8116 \times 10^{58}} \approx 2.46 \times 10^{-3}\).
Units check. The ratio \(\Delta/D\) is \(\mathrm{rad^2/s^2} / \mathrm{rad^4/s^4} = \mathrm{s^2/rad^2}\). Multiplying by \(A\) (\(\mathrm{rad^2/s^2}\)) yields a dimensionless quantity, as required for \(n\) and \(\kappa\).
Check the approximation: \(\chi\) itself is about \(0.11\), small enough for \(n \approx 1 + \chi/2\); the exact complex square root gives \(n = 1.054\). With a plasma frequency of \(10^{15}\ \mathrm{rad\,s^{-1}}\) instead, \(\chi\) would be about 11 and the small-\(\chi\) formula would be useless.
Three quick tests
- What is the physical origin of the damping rate \(\gamma\) in a solid dielectric?
Answer
Collisions with lattice vibrations (phonons) and other electrons, which dissipate energy as heat. - In which region of the spectrum does anomalous dispersion occur?
Answer
In the narrow frequency band around the resonance frequency \(\omega_0\), where absorption is significant. - If \(\omega \gg \omega_0\), what happens to the refractive index \(n\)?
Answer
\(n\) approaches 1 from below (in the simple model), indicating the material becomes transparent and non-dispersive at high frequencies.
Problem set for PU-204 →
Each derivation in this unit ends with five graded problems with full solutions.
Lecture notes checked line by line by the editor; every number in the worked example was recomputed.