3.3.1 The ideal spring
A ubiquitous concept within physics is that of the simple harmonic oscilla-tor. This type of oscillator occurs, for instance, whenever some physical displacement in a field or material medium is resisted by a restoring force that is proportional to the displacement. Another way of putting this, which is of general significance, is that this restoring force is linear in the displace-ment.
Although modern physics informs us that light is not a displacement in a material medium (it was originally supposed that this was the case, the medium being called the ‘ether’), we may still view light as being a displace-ment of a field. For the time being, and through much of this book, we shall refer to this as the ‘optical field’. We may then apply much of the same intuition, and indeed mathematics, that we glean the elastic displacement of a physical material in mechanics.
3.3. THE SIMPLE HARMONIC OSCILLATOR 49 A good deal of insight may be initially gained from a mechanical exam-ple. An elementary but universal concept is that of the ideal spring. By definition, an ideal spring is a mathematical model of a real spring to which Hooke’s law applies. This just says that the restoring force is proportional to the extension of the spring. The constant of proportionality is then known as the spring constant, which we shall denote by K. The force exerted by the spring is then proportional to the displacement u
F = −Ku. (3.1)
Applying Newton’s second law of motion, we have md2u
dt2 = −Ku. (3.2)
This has the general solution
u (t) = A sin (ωt + φ) , (3.3) is the angular frequency and φ is a phase factor determining the position at time t = 0. Note that the angular frequency of Eq. (3.4) is characteristic of the system, being given in terms of the spring constant K and the mass m. In fact, this is the resonant angular frequency of the system and will usually be denoted by ω = ω0. This, of course, may be expressed in terms of frequency f0 and the oscillation period T0
ω0 = 2πf0 = 2π T0
. (3.5)
3.3.2 Optical sources as harmonic oscillators
So how does the concept of the simple harmonic oscillator relate to optics?
As we shall see in Chapter 6, light may be understood as an electromag-netic oscillation propagating through space. A possible source of such a disturbance is an oscillating electric dipole, in which we have two charges of opposite sign separated by some distance z. Although this is not the place for a detailed treatment of electric dipoles, we wish to argue for an approximate model of a dipole that may be treated as a harmonic oscillator.
Classically, the dipole is visualized as two point charges displaced from one another. In reality, however, in the absence of any other restraints, two such classical charges, q and −q, would be drawn together with an increasing force, given by Coulomb’s Law
F = − q2
4π0z2, (3.6)
where the constant 0 is known as the permittivity of free space and z is, as defined above, the distance between the charges (we assume, for sim-plicity, that the material medium has a relative permittivity of unity). This is symptomatic of a general problem of classical physics in that it fails to ex-plain the stability of matter composed, as it is, of equal amounts of negative and positive charge.
This problem is solved very elegantly in quantum theory, where the con-cept of the point-like particle evaporates and is replaced with a wave de-scription of particles. For our present purposes, this means that we may imagine a negatively charged electron, not as a point particle, but rather a ‘cloud’ of charge, with the highest charge densities spread out over sur-faces in space. For instance, an ‘s-type’ electronic state can be imagined approximately as a sphere of charge.
The positively charged protons will also be smeared out in space, al-though, due to there greater mass, the uncertainty principle will allow them to be far more localized. We will therefore continue to imagine a proton as a point-like particle.
Figure 3.1: An electric dipole modelled as a sphere of radius R and con-stant negative charge density displaced from a positive point charge by u.
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3.3. THE SIMPLE HARMONIC OSCILLATOR 51 Let us imagine a somewhat fictionalized (but mathematically tractable) form for an electron state in which the negative charge −q is distributed uniformly over a sphere of radius R. In equilibrium, this cloud of charge is centred on a positive point charge +q at the origin. The sphere then becomes displaced from the positive charge by u (see Fig. 3.1). Now the point charge only sees a field due to the charge within the sphere centred on u. Thus, if the negative charge density is ρ, the charge contributing to the Coulombic force is
Q = −4πρu3
3 . (3.7)
The force between the positive charge and electron cloud is therefore F = qQ
4π0u2 = −qρ
30u. (3.8)
Hence, this model of an electric dipole yields the same force law as the ideal spring with K = qρ/ (30).
3.3.3 Energy in a simple harmonic oscillator
In the case of the mass on an ideal spring, the mass will have its maximum kinetic energy as it passes through its minimum of potential energy, at which point there is no force acting on it. Thereafter, the restoring force acts to reduce the kinetic energy, converting it to the potential energy stored in the spring. For a displacement u, the potential energy U is given by
U = − where U0 is the potential energy at u = 0. Since this is arbitrary, we are free to set U0 = 0. Note that this parabolic form for the potential energy is just another way of specifying a harmonic oscillator.
At the maximum displacement, all the energy of the oscillator is stored as potential energy. Since the total energy of the system is a constant, we must have
= 12KA2, (3.10)
where A is the amplitude of the oscillation. Thus, the energy of the oscil-lator is proportional to square of the amplitude. An analogous result holds quite generally for linear oscillations, i.e. oscillations in which force is pro-portional to a displacement. Whilst the particular physics of the optical field are quite different, we shall find that the energy of waves in this field is also proportional to the square of the amplitude.