The Classical Electron Oscillator

Table of Contents:

  1. Review of Real Atoms
  2. Analysis of Classical Electron Oscillator (CEO) Model
  3. Damping and Oscillation Energy Decay
  4. Radiative Decay Rates
  5. Microscopic Dipole Moments and Macroscopic Polarization

REVIEW OF REAL ATOMS

First, a note that commonly in foundational descriptions of lasers, the word “atoms” to describe the operation of a laser is used very generally, and could be modified in its specifics depending on the type of laser you are talking about. For example, in gas lasers, “atom” could be used to describe free atoms, ions, or molecules; in solid or liquid lasers, “atom” could mean individual laser atoms, ions, or molecules; and finally, in semiconductor lasers, “atoms” could simply refer to valence or conduction electrons. I know I apologize on behalf of my kind a lot on this website, but I would like to do so again to make up for the laziness in terminology here. Anyhow, we will still stick with our description of “atoms” and proceed to describe their properties:

  • Simplified greatly, atoms consist of a nucleus with protons and neutrons and a surrounding electron cloud. This is important to note for when we get to talking about how an atom might be polarized.
  • Atoms have sharp resonances in both their spontaneous emission (which arises when pumped or even, in smaller quantities, through interaction with the free space electric field) and in their stimulated emission when excited by applied signal. These are also usually just harmonic responses and do not need to be described by the addition of other frequencies to a fundamental frequency.
  • Most atoms in materials (though not all) respond to the electric field rather than the magnetic field of an incident beam of light. This is called the electric dipole moment of the atoms, and it is typically the strongest type of response to light. There can be magnetic dipole responses, or electric quadrupole, or even higher order responses by some atoms, but we will not discuss those here. If you are completely bored and looking for something to do, though, check out lasers based on these transitions!

ANALYSIS OF CEO MODEL

Note, this is also sometimes called the Lorentz model of the atom, so don’t be confused if you see that in the literature instead of the CEO model! The CEO model starts by modeling the nucleus of the atom attached to an electron or electron cloud by a spring, as shown in the figure below.

Now, if you all have already had Physics I, then you know that if you displace a spring from its resting position, there will be a force to pull it back into place. Actually, if you were ever a child, then you likely played with a slinky at least at one point during that time. As you displace it, it swings back into position. That’s all that’s at work with the spring, too! In Physics I, we called this force the restoring force, and we described the restoring force using Hooke’s Law $F = -kx$, where the “-” indicates that the force acts in the negative x direction (opposite the direction of the displacement) due to the definition of our direction for $+x$. Now, for our CEO model, we can make the analogy that a “displacement” would mean movement of the electron relative to the nucleus, whether negative or positive, through an applied electric field signal $\epsilon_x(t)$. Keeping with our spring analogy, the time-depending perturbation from the electric field will give rise to a restoring force. If we model the restoring force by a time-dependent change in position $x(t)$, we can also label the “constant” that determines the amount of force by $K$.

The two forces at play here are thus the applied signal, given as a force by $e\epsilon(t)$ where “e” is the charge of the electron, and the “restoring force”, $-Kx(t)$. If these two forces are equal, they will cancel each other and the electron will remain stationary since they act in opposite directions. However, if either one “wins”, then we will have a net force given by the rate of change of position multiplied by the mass of the electron. The full force equation is thus:

\begin{equation} m\frac{d^x(t)}{dt^2} = -Kx(t) – e\epsilon_x(t) \end{equation}

or, more generally,

\begin{equation} \frac{d^2x(t)}{dt^2} + \omega_a^2x(t) = -(e/m)\epsilon_x(t) \end{equation}

The term “$\omega_a$” here thus describes the classical oscillator’s resonance frequency. We can equate this with real atoms by denoting the resonance frequency from the CEO model as the transition frequency $\omega_{21} = (E_2 – E_1)/\hbar$ of a real atomic transition. The “$E_2$” and “$E_1$” terms here refer to the second and first energy levels, or orbitals, of the electron respectively. By looking at the units for Planck’s constant as well, we can see that the equality does indeed yield equivalent units of rad/s!

Damping and Oscillation Energy Decay

Like all real oscillatory systems, there must be a damping term since everything eventually “runs out” of energy in time. We can thus add a damping term to the equation of motion as:

\begin{equation} \frac{d^2x(t)}{dt^2} + \gamma\frac{dx(t)}{dt} + \omega_a^2x(t) = -\frac{e}{m}\epsilon_x(t) \end{equation}

where “$\gamma$” is the damping rate or coefficient for the oscillator. Without an applied signal, the equation of motion will thus decay in the fashion given by the equation:

\begin{equation} x(t) = x(t_0)e^{-(\gamma/2)(t-t_0) + j\omega_a'(t-t_0)} \end{equation}

where $\omega_a’$ is the exact optical frequency transition, given as:

\begin{equation} \omega_a’ = \sqrt{\omega_a^2-(\gamma/2)^2} \end{equation}

which means that the frequency of the optical transition is modified by the decay rate. In reality, however, normally the damping rate is much lower than the optical frequency, so the difference between $\omega_a’$ and $\omega_a$ is negligible. Then, since force is the time derivative of energy, we can use our equation of force to find the energy associated with the oscillation and its decay as:

\begin{equation} U_a(t) = \frac{1}{2}Kx^2(t) + \frac{1}{2}mv_x^2(t) = U_a(t_0)e^{-\gamma(t-t_0)} \end{equation}

The physical reason for this energy decay is due to two factors: radiative and nonradiative decay. Radiative decay includes spontanteous emission and fluorescence, where spontaneous emission is a crazy phenomenon that basically means the interaction of the quantized electric field of free space with electrons. To truly understand this, we will use quantum mechanics (to be covered in a couple of sections), but for now just be amazed that this phenomenon of spontaneous radiation of light actually occurs in all fields of science, just under different names! There is also some evidence that humans produce spontaneous emission, although that is such a crazy result that we will only mention it here.

The other method of decay, nonradiative decay, occurs due to collisions with other atoms, heat vibrations into the lattice, or a number of other crazy physical phenomena. As the name implies, there is no emission of light in these loss mechanisms, and we would thus like to minimize the amount of loss due to nonradiative decay in our choice of lasing material.

The total decay rate is thus given as:

\begin{equation} \gamma = \frac{1}{U_a}\frac{dU_a}{dt} = \gamma_{rad} + \gamma_{nr} \end{equation}

RADIATIVE DECAY RATES

The decay rate for a classical electron oscillator embedded in a medium with permittivity $\epsilon$ is given as:

\begin{equation} \gamma_{rad, CEO} = \frac{e^2\omega_a^2}{6\pi\epsilon{m}c^2} \end{equation}

Note that the real decay rate could only be calculated from quantum mechanics, and thus the one given here is only for the classical approximation. In many cases, though, this is a satisfactory approximation!

MICROSCOPIC DIPOLE MOMENTS AND MACROSCOPIC POLARIZATION

The next step to making this a useful analysis for lasers is to go from microscopic polarization of a single oscillator to macroscopic polarization of each of the excited oscillators in the material. To do this, we should first start with what the individual dipole moments would be given as. This is given as:

\begin{equation} \mu_x(t) = [charge] \times [displacement] = -ex(t) \end{equation}

A diagram for this is shown below:

Homework 11 (Due. Apr 3)

The macroscopic polarization for the volume is then given as:

\begin{equation} p_x(r,t) = V^{-1}\sum_{i=1}^{NV}\mu_{xi}(t) \end{equation}

A heuristic picture of the macroscopic polarization is shown below.

Classical Oscillator model and electric dipole transitions in real atoms
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