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Chaotic Cavities

May 15, 2013

Resonators can store energy. An obvious mechanical example is a pendulum. As an electrical example, we have the quartz crystal, which will be set into resonance if we ping it with a voltage pulse. The crystal's resonance decays according to the well-known function
A(t) = Ao e-λt cos(t)
where A(t) is the amplitude as a function of time, Ao is the initial amplitude, and lambda (λ), called the log decrement, sets the rate of decay. I wrote about such damped harmonic oscillators in a previous article (Graphene Resonators, June 24, 2011). In a quartz crystal oscillator, the degree of decay, or damping, is related to the crystal's Q-factor; viz., Q = 2π(Energy Stored/Energy Lost per cycle).

Amplitude of a damped harmonic oscillator
Amplitude of a damped harmonic oscillator. In this case, the log decrement is 0.01, giving the equation shown on the plot. (Graph rendered with Gnumeric.)

Harmonic oscillators are extremely useful, and there must be dozens of electronic oscillators in your house right now, clocking microprocessors in things such as cellphones and television remotes. There's another type of oscillator that's not as useful, but much more interesting. This is the chaotic oscillator, the most common example of which is Chua's circuit,[1-2] named after its inventor, Leon Chua, a professor of Electrical Engineering.

Chua circuit double scroll attractorOne representation of the chaos implicit in the Chua oscillator.

This graph of its double scroll attractor is produced by plotting three different voltages in a simulation of the circuit.

(Via Wikimedia Commons.)[3]

Chaotic oscillators would have been just a curiosity of physics without an application were it not for a discovery made by scientists at the US Naval Research Laboratory in 1990, one of whom was a classmate of mine in graduate school. They discovered that it's possible to synchronize chaotic oscillators remotely.[4] Their report on this research was named a Physical Review Letters Milestone for 1990.

How useful are two coupled chaotic oscillators? The voltage output of a chaotic oscillator is very dependent on the values of its components and its initial state. Small changes in any of these for a chaotic oscillator lead to large changes in the time series of its voltage values.

This means that you can conveniently generate the same sequence of random numbers simultaneously at two different places, and even you won't know what number follows another in the sequence. Such a physical random number generator can be used to send coded messages using the unbreakable cryptographic technique known as the one-time pad. Aside from this application, chaos has been mostly descriptive and rarely proscriptive.[5]

An international team of scientists and engineers from the King Abdullah University of Science and Technology (Saudi Arabia), the University of St Andrews (UK), University of Bologna (Italy) and the University of York (UK) have found another practical use for chaos. They've demonstrated that imposition of chaos causes a six-fold increase of the stored energy in a resonant optical cavity as compared to its classical counterpart of the same volume.[6-7]

This discovery goes against the idea that chaos always diminishes the performance of devices.[7] The research involved ab initio simulations and experiments on photonic-crystal resonators which store light energy by bouncing it between mirrors. In the chaotic case, the mirrors were deformed to disrupt the normal path of the light rays. The demonstrated principle is that increasing the number of degrees of freedom of a system allows more energy storage in the equipartition of energy among the states.[6-7]

Figure caption
Patterns of light distribution in a normal cavity resonator (left) and a chaotic cavity resonator (right). The extra resonant modes for the chaotic cavity are apparent.(York University image, Fratalocchi et al./Nature Photonics, used with permission.)[7]

Even simple cavities, such as glass microspheres and polystyrene microspheres, showed more stored energy when they were deformed.[6-7] Says Thomas F. Krauss, a professor of physics at York University and a coauthor of the study,
"Besides the obvious implications at the fundamental level, where we demonstrate the existence of a fundamental principle of thermodynamics in the framework of Photonics, our results also have real-world practical implications."[7]
One practical application would be to increase the interaction time of solar radiation in photovoltaic cells to increase the energy conversion efficiency. York university is already pursuing a program relating to commercial devices.[7-8]


  1. Bertie Genade, "The Chua chaotic oscillator".
  2. A Tamaševicius, G Mykolaitis, V Pyragas and K Pyragas, "A simple chaotic oscillator for educational purposes," European Journal Of Physics, vol. 26 (November 3, 2004), pp. 61-63.
  3. Chua Circuits Web Site by V. Siderskiy.
  4. Louis M. Pecora and Thomas L. Carroll, "Synchronization in chaotic systems," Phys. Rev. Lett., vol. 64, no. 8 (February 19, 1990) pp. 821-824. A PDF copy is available here.
  5. Jaroslav Stark and Kate Hardy, "Chaos: Useful at Last?" Science, vol. 301, no. 5637 (August 29, 2003) pp. 1192-1193.
  6. C. Liu, A. Di Falco, D. Molinari, Y. Khan, B. S. Ooi, T. F. Krauss and A. Fratalocchi, "Enhanced energy storage in chaotic optical resonators," Nature Photonics, Advance Online Publication, May 5, 2013, doi:10.1038/nphoton.2013.108.
  7. Chaos proves superior to order, University of York Press Release, May 7, 2013.
  8. PrimaLight Research Web Site.

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Linked Keywords: Resonator; energy; mechanics; mechanical; pendulum; electronics; electrical; quartz crystal; voltage; pulse; function; log decrement; damped harmonic oscillator; damping; Q-factor; amplitude; equation; Gnumeric; electronic oscillator; microprocessor; cellphone; television remote; chaos theory; chaotic; Chua's circuit; Leon Chua; professor; Electrical Engineering; Chua oscillator; double scroll attractor; simulation; Wikimedia Commons; physics; scientist; United States Naval Research Laboratory; graduate school; synchronization of chaos; synchronize chaotic oscillators; Physical Review Letters Milestone; random sequence; sequence of random numbers; physical random number generator; code; cryptography; cryptographic; one-time pad; engineer; King Abdullah University of Science and Technology; Saudi Arabia; University of St Andrews; United Kingdom; UK; University of Bologna; Italy; University of York; energy; optical cavity; classical physics; volume; research; first principles; ab initio; experiment; photonic-crystal; mirror; light ray; degrees of freedom; equipartition theorem; equipartition of energy; glass; microsphere; polystyrene; sunlight; solar radiation; solar cell; photovoltaic cell; energy conversion efficiency; Louis M. Pecora and Thomas L. Carroll, "Synchronization in chaotic systems," Phys. Rev. Lett., vol. 64, no. 8 (February 19, 1990) pp. 821-824.

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