So far we have developed the mathematical model of the driven oscillator by thinking of mechanical oscillations and mechanical resonance. As we mentioned in the introduction there are many other examples of resonance from non-mechanical oscillator systems. The electrical response of a tuned circuit or the resonant response of a tuned cavity or an atom or nucleus to an electromagnetic wave are cases in point. The damping is caused by any process through which energy can be transferred out of the vibrational system, often through heating. For example, in an electrical circuit the dissipation arises from the heating effect of the current as it flows through any circuit resistance. While these other systems are not always amenable to the simple linearly damped mathematical model introduced here, the model enables us to appreciate the processes involved.
We can describe all these systems in terms of resonance, with an appropriate Q-factor implied by the frequency response shown on the power absorption curve. In some cases these systems have Q-factors which are very much larger than those of the mechanical systems considered thus far and so the frequency response is a great deal narrower and more selective. If the frequency of an oscillator is narrowly defined then the period of the oscillation is also narrowly defined and the oscillator can be used as a clock. For this reason, resonant systems find widespread application in the maintenance of frequency or time standards. The higher the Q-factor of the oscillator the greater the potential for using the resonant oscillator as a well-defined frequency standard. Here we cannot describe these systems in any detail but it is worth mentioning a few important examples and quoting the Q-factors that typify them.
We begin with a simple familiar mechanical oscillator, the pendulum clock.
The pendulum oscillations are sustained by energy input to compensate for dissipative processes, such as friction. The damping of the system is minimized and an estimate for the Q-factor achieved can be found from Equation 26,
resonance absorption bandwidth ∆f = ω2 −ω1
2π = γ2π = ω 0 2πQ =
f0
Q (Eqn 26)
knowing the typical performance of a good mechanical clock. If the clock is to gain or lose by no more than 101s a day then this is a fractional time error of 101s in 24 × 36001s, or about 10−4. The fractional frequency error is the same as the fractional time error and if we take a typical frequency error to be the full width of the power curve at half height
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(i.e. ∆0f1) then Equation 26 gives the required Q-value as f0/∆f = 104.Our next example is a quartz crystal oscillator, such as is used in a watch. This is another mechanical oscillation, but on a smaller size scale and at a much higher frequency (a few tens of kHz). Some materials (quartz, for example) have the property that an applied electric field causes mechanical stresses in the material. They are said to be piezoelectric materials and an oscillating voltage applied across them causes them to vibrate at the driver frequency, rather like a loudspeaker, but piezoelectric crystals can respond at much higher driver frequencies than is possible for a conventional loudspeaker. In the quartz crystal oscillator the mechanical oscillations of the crystal are maintained by electrical oscillations in a circuit which drives the crystal. The crystal resonance stabilizes the frequency of these oscillations and makes the resonance of the tuned circuit much sharper. The oscillations are then counted and this is converted into a time display. Such watches are capable of an order of magnitude better time-keeping than the pendulum clock and so the required Q-value is about 105. High quality quartz-controlled clocks can do rather better than this but our next example shows a spectacular improvement on this.
The caesium atomic clock is so precise that its oscillations are used to establish and maintain our fundamental unit of time, the second. The oscillations here are rather far removed from mechanical oscillations; they correspond to a characteristic internal transition frequency within single caesium atoms. This is not the place to go into details of this process but suffice to say that caesium atoms can absorb electromagnetic radiation at a frequency of around 91921MHz, in a resonant process.
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For our purpose here the important point is that the sharpness of the atomic resonance can be made so great that the full resonance absorption bandwidth is as low as 11Hz. With a basic resonance frequency of 9192 MHz this gives a Q-factor of about 1010!Sophisticated technology is used to stabilize the frequency to within a small fraction of this bandwidth and the frequency of such clocks can then be made reproducible to about 1 part in 1013. Two such clocks will keep the same time to within about 3 seconds in a million years! Unfortunately, these clocks are of a size which would fill a medium-sized room and you would not want one strapped around your wrist. Their use is confined to establishing the international time standard and in calibrating other more portable clocks. Such high precision is needed in certain scientific experiments. For example, caesium clocks flown in aircraft have enabled the direct observation of the tiny time changes for moving clocks, predicted by Einstein’s special theory of relativity.
Even the caesium clock could be made to look rather crude when compared to future clocks that may be based on frequency-stabilized lasers. In a laser the oscillators are effectively an ensemble of atoms in a cavity, rather than individual atoms, and the driver is an electromagnetic wave (light), trapped within the same cavity, between mirrors. The atoms absorb and re-emit the wave within the cavity but because they are all in communication with the same wave they all absorb and emit with well-defined phases, rather than with random phases as would be the case if they were acting individually. Such phase-organized light is called coherent light. The process by which an incoming wave drives an atom to emit is known as stimulated emission and it is the counterpart to absorption. If the stimulated emission exceeds the absorption then this will give rise to laser action, or Light
Amplification by Stimulated Emission of Radiation. Within the cavity of a laser oscillator there is a single giant
amplitude oscillation with a well-defined phase and an extremely well-defined frequency. The frequency output bandwidth for the laser is very much narrower than would be the case if the same atoms were placed outside the cavity, as in an ordinary lamp.
Lasers have many advantages over conventional light sources but the one that concerns us here is this very narrow bandwidth. It gives rise to an enormous value for the Q-factor of the oscillator. The full resonance bandwidth of a laser oscillator can be made to be only a few Hz, even when the laser is operating in the visible spectrum, at frequencies of around 5 × 1014 Hz. If such a laser could also have its frequency stabilized sufficiently to capitalize on this low bandwidth then the resulting frequency-stabilized laser could have a
Q-factor of around 1014, with a potential improvement factor of 104 over the caesium clock. The technology for this is under development and stabilized lasers could well be the reference clocks of the future.