Laser-based Chaos Tools

An X-Y monitor can be used to display the phase space of a chaotic process occurring, for example, in an electronic circuit. Typically, a selected circuit voltage is applied to the monitor's x-axis, and a voltage from a different part of the circuit is applied to the y-axis. However, there is more to phase space than this. In some cases, where only one parameter is measurable or of interest, that parameter is applied to one axis and the very same parameter delayed (lagged) in time can be applied to the other axis. This is better done under computer control and usually involves time-series statistics. Also, the selected parameter can be differentiated, for example, in real time using an op amp differentiator. The resulting constantly changing derivatives can be applied to one axis while the undifferentiated values are applied to the other. Most importantly, attractors—be they fixed, periodic, or chaotic—live in phase space. (Chaotic attractors are often referred to as strange attractors.) Attractors are truly where the action is, and we will examine them during some actual experiments! But first, a look at some optical tools employing lasers that should prove useful for the experiments.

3D Waveform Display

For something really different, I dredged up a design I made back in the 1970s to display 3D Lissajous patterns using a laser and rotating vanes. This allows introducing the dimension of time into an otherwise 2D display to reveal how the wave is progressing in time. The problem here is to slow down the waveforms sufficiently to enable galvanometer-controlled mirrors (galvos) to scan both the x-axis and the y-axis. Inexpensive galvos, of the type used to make low-cost laser light show projectors, usually will not work properly if operated at much over 100 Hz. On the other hand, persistence of vision enables us to see the waveform, so a compromise in scanning speed must be achieved.

In order to slow down the waveforms produced by the Chua circuit, the inductance must be changed to a very high value (measured in Henries). This will allow chaos signals to appear at very low frequencies. (Forget about finding an inductor measuring, say, 20 Henries, with an internal resistance of a few ohms. It would weigh a ton and cost a bundle.) This slowdown can be neatly achieved with an inductorless Chua circuit that employs op amps in a gyrator circuit. The results are adjustable, slow-speed scans that display waveforms weaving their way through three-dimensional space! (A gyrator circuit employs a capacitor to simulate an inductor in an op amp circuit and is really a remarkable invention.) The pictures that follow try to represent the effect. For these photos, I used dual signal generators to produce Lissajous figures. Later, we will see the galvo system applied to real-world chaotic signals.

At upper left is a view of the 3D setup showing the rotating vanes in motion. (If you insert the mouse pointer into that view, all will be revealed!)

At upper right is a view nearly head-on of the setup. The figure "8" in the background is a 2D projection of the Lissajous pattern on a white screen and resembles an oscilloscope trace. The blurred luminous image is due to the rotating vanes seen in action during the brief exposure. The motor shaft is seen at the top left of center and the galvos at the bottom left.

At left is an image of the waveform viewed from the side. The x-y galvos are positioned at the extreme left of the waveform. In a completely darkened room, only the 3D image of the wave is visible.You can see the waveform changing in time as the Lissajous forms. As you walk around the floating trace, different 3D views are visible, including the figure eight projection seen in the photo at upper right.

Laser-Doppler Velocimeter (LDV)

The Doppler effect should be very familiar to students of science. When a moving train blows its whistle, the sound is heard to rise in pitch as the train first approaches and then drop as the train zips on by. Doppler heart monitors are used routinely to record the cardiac health of the fetus in utero. When an astronomer looks at the spectrum of a star or a galaxy, the spectral lines will be seen shifted either toward the red or the blue end of the spectrum relative to a static setup in the lab. The direction of the shift depends on whether the object is receding from or approaching the observer, and the amount of the shift depends on the object's velocity.

The astronomer is able to use an optical spectrum analyzer, such as a spectroscope or a spectrograph, because the observed objects are moving at significant fractions of the speed of light. Such is usually not the case in the lab, so other techniques must be used. Whereas light frequencies are measured in Petahertz , the Doppler signals obtained from moving particles in the lab fall into the Megahertz to Gigahertz range. This difference governs the type of measuring equipment that can be employed.

If a monochromatic light source such as a laser illuminates an object in motion, the light reflected from the object will be shifted in wavelength (or frequency) depending on the velocity of the object relative to the observer. The amount of that shift is proportional to the velocity of the moving object. However, there is usually no indication as to whether the object is approaching or receding unless additional optical information is made available. (Of course, the actual direction of travel for an object is not always required and, for some experiments, measuring its velocity is sufficient.) For the experiments in chaos, light reflected back from opaque particles in motion will be detected, and directional information will prove useful.

To properly observe the wavelength shift of the relatively weak backscattered light, a photomultiplier tube (PMT) is used. The output of the PMT is amplified and fed to an electronic spectrum analyzer (or a computerized data-collection system) to display the frequency of the detected waveform associated with the motion of the particles. To remove the directional ambiguity of an object relative to the observer, an acousto-optic modulator (AOM) is used to frequency modulate the laser light and produce a shifting fringe pattern that is crisscrossed by the object in motion. Light reflected in this setup can now be associated with a sign. We will call it positive if the object is moving in the same direction as the moving fringes or negative if it's heading in the opposite direction. Both velocity and direction can now be determined. In the LDV experiment, backscattered rather than forward-scattered light will be collected. The necessity for the backscatter arrangement will be made clear in the write-up for the experiment.

One use for the laser-Doppler velocimeter (also referred to as a laser-Doppler anemometer) is to measure the velocity and direction of fluids circulating in a rotating system. In fact, such a system using a pair of concentric cylinders (the outer cylinder being transparent) with a liquid between them with one or the other (or both) cylinders in rotation can exhibit chaos. This arrangement of counterrotating concentric cylinders, often employed under various guises in the field of fluid dynamics, is referred to as a Taylor-Couette Cell. It will be the subject of an experiment to be described later on. To render the flow visible to probing laser light, the liquid will be seeded with a tracer composed of specially treated mica flakes. Laser light allows for non-invasive probing of fluid motion. Similarly, tracers need to be size selected to adroitly follow flow patterns without altering them. Tracers need to be small enough to move easily yet large enough to be highly reflective.

A conceptual setup employing the LDV is shown here. Note that the laser beam is first split and then recombined to produce a probing volume crossed by interference fringes. The fringes are produced when coherent, monochromatic light from the split laser beams interfere constructively and destructively. The number of fringes produced is a function of both the angle between the reuniting beams and the laser's wavelength. When this probing region is traversed by an object in motion (we will assume a one-dimensional movement strictly at right angles to the detection equipment) the motion of the object relative to the fringes is measured.

Imagine a shiny, metallic sphere passing through the fringes reflecting the bright and dark bands in rapid succession. That (sinusoidal) pattern is what the PMT detects and translates into the Doppler signal. Also, due to the Gaussian cross section of the laser beams, the most intense reflections occur at the central part of the probe volume with the intensity grading off at the periphery. Whether the object moves up or down the page at a given rate, the measured Doppler frequency is the same, and that's why the direction is ambiguous.

However, when the fringes are given a constant upward or downward motion using an acousto-optic modulating crystal (shown as a Bragg Cell in the diagram) that alters the wavelength of only one of the interfering light beams, then the object's motion will be either in the same direction as the shifting fringes or in the opposite direction. That directionality is detectable as either an increase or a decrease in a given Doppler signal. Thus, acousto-optic modulation serves to supply us with the missing directional information, as well as the velocity.

Though an electronic spectrum analyzer is shown here displaying the Doppler frequency, meaningful research requires computer-aided data collection so that velocity and direction information can be mathematically analyzed. This analysis will be necessary for experiments in chaotic fluid dynamics.

This concludes a look at some of the tools I am currently developing to work with nonlinear dynamics and chaos. Up next is a look at the most fundamental chaotic phenomenon which lies at the heart of the science of chaos: the butterfly effect.

Laser-based Chaos Tools An X-Y monitor can be used to display the phase space of a chaotic process occurring, for example, in an electronic circuit. Typically, a selected circuit voltage is applied to the monitor's x-axis, and a voltage from a different part of the circuit is applied to the y-axis. However, there is more to phase space than this. In some cases, where only one parameter is measurable or of interest, that parameter is applied to one axis and the very same parameter delayed (lagged) in time can be applied to the other axis. This is better done under computer control and usually involves time-series statistics. Also, the selected parameter can be differentiated, for example, in real time using an op amp differentiator. The resulting constantly changing derivatives can be applied to one axis while the undifferentiated values are applied to the other. Most importantly, attractors—be they fixed, periodic, or chaotic—live in phase space. (Chaotic attractors are often referred to as strange attractors.) Attractors are truly where the action is, and we will examine them during some actual experiments! But first, a look at some optical tools employing lasers that should prove useful for the experiments. 3D Waveform Display For something really different, I dredged up a design I made back in the 1970s to display 3D Lissajous patterns using a laser and rotating vanes. This allows introducing the dimension of time into an otherwise 2D display to reveal how the wave is progressing in time. The problem here is to slow down the waveforms sufficiently to enable galvanometer-controlled mirrors (galvos) to scan both the x-axis and the y-axis. Inexpensive galvos, of the type used to make low-cost laser light show projectors, usually will not work properly if operated at much over 100 Hz. On the other hand, persistence of vision enables us to see the waveform, so a compromise in scanning speed must be achieved. In order to slow down the waveforms produced by the Chua circuit, the inductance must be changed to a very high value (measured in Henries). This will allow chaos signals to appear at very low frequencies. (Forget about finding an inductor measuring, say, 20 Henries, with an internal resistance of a few ohms. It would weigh a ton and cost a bundle.) This slowdown can be neatly achieved with an inductorless Chua circuit that employs op amps in a gyrator circuit. The results are adjustable, slow-speed scans that display waveforms weaving their way through three-dimensional space! (A gyrator circuit employs a capacitor to simulate an inductor in an op amp circuit and is really a remarkable invention.) The pictures that follow try to represent the effect. For these photos, I used dual signal generators to produce Lissajous figures. Later, we will see the galvo system applied to real-world chaotic signals.

	At upper left is a view of the 3D setup showing the rotating vanes in motion. (If you insert the mouse pointer into that view, all will be revealed!) At upper right is a view nearly head-on of the setup. The figure "8" in the background is a 2D projection of the Lissajous pattern on a white screen and resembles an oscilloscope trace. The blurred luminous image is due to the rotating vanes seen in action during the brief exposure. The motor shaft is seen at the top left of center and the galvos at the bottom left. At left is an image of the waveform viewed from the side. The x-y galvos are positioned at the extreme left of the waveform. In a completely darkened room, only the 3D image of the wave is visible.You can see the waveform changing in time as the Lissajous forms. As you walk around the floating trace, different 3D views are visible, including the figure eight projection seen in the photo at upper right.
Laser-Doppler Velocimeter (LDV) The Doppler effect should be very familiar to students of science. When a moving train blows its whistle, the sound is heard to rise in pitch as the train first approaches and then drop as the train zips on by. Doppler heart monitors are used routinely to record the cardiac health of the fetus in utero. When an astronomer looks at the spectrum of a star or a galaxy, the spectral lines will be seen shifted either toward the red or the blue end of the spectrum relative to a static setup in the lab. The direction of the shift depends on whether the object is receding from or approaching the observer, and the amount of the shift depends on the object's velocity. The astronomer is able to use an optical spectrum analyzer, such as a spectroscope or a spectrograph, because the observed objects are moving at significant fractions of the speed of light. Such is usually not the case in the lab, so other techniques must be used. Whereas light frequencies are measured in Petahertz , the Doppler signals obtained from moving particles in the lab fall into the Megahertz to Gigahertz range. This difference governs the type of measuring equipment that can be employed. If a monochromatic light source such as a laser illuminates an object in motion, the light reflected from the object will be shifted in wavelength (or frequency) depending on the velocity of the object relative to the observer. The amount of that shift is proportional to the velocity of the moving object. However, there is usually no indication as to whether the object is approaching or receding unless additional optical information is made available. (Of course, the actual direction of travel for an object is not always required and, for some experiments, measuring its velocity is sufficient.) For the experiments in chaos, light reflected back from opaque particles in motion will be detected, and directional information will prove useful. To properly observe the wavelength shift of the relatively weak backscattered light, a photomultiplier tube (PMT) is used. The output of the PMT is amplified and fed to an electronic spectrum analyzer (or a computerized data-collection system) to display the frequency of the detected waveform associated with the motion of the particles. To remove the directional ambiguity of an object relative to the observer, an acousto-optic modulator (AOM) is used to frequency modulate the laser light and produce a shifting fringe pattern that is crisscrossed by the object in motion. Light reflected in this setup can now be associated with a sign. We will call it positive if the object is moving in the same direction as the moving fringes or negative if it's heading in the opposite direction. Both velocity and direction can now be determined. In the LDV experiment, backscattered rather than forward-scattered light will be collected. The necessity for the backscatter arrangement will be made clear in the write-up for the experiment. One use for the laser-Doppler velocimeter (also referred to as a laser-Doppler anemometer) is to measure the velocity and direction of fluids circulating in a rotating system. In fact, such a system using a pair of concentric cylinders (the outer cylinder being transparent) with a liquid between them with one or the other (or both) cylinders in rotation can exhibit chaos. This arrangement of counterrotating concentric cylinders, often employed under various guises in the field of fluid dynamics, is referred to as a Taylor-Couette Cell. It will be the subject of an experiment to be described later on. To render the flow visible to probing laser light, the liquid will be seeded with a tracer composed of specially treated mica flakes. Laser light allows for non-invasive probing of fluid motion. Similarly, tracers need to be size selected to adroitly follow flow patterns without altering them. Tracers need to be small enough to move easily yet large enough to be highly reflective.
A conceptual setup employing the LDV is shown here. Note that the laser beam is first split and then recombined to produce a probing volume crossed by interference fringes. The fringes are produced when coherent, monochromatic light from the split laser beams interfere constructively and destructively. The number of fringes produced is a function of both the angle between the reuniting beams and the laser's wavelength. When this probing region is traversed by an object in motion (we will assume a one-dimensional movement strictly at right angles to the detection equipment) the motion of the object relative to the fringes is measured. Imagine a shiny, metallic sphere passing through the fringes reflecting the bright and dark bands in rapid succession. That (sinusoidal) pattern is what the PMT detects and translates into the Doppler signal. Also, due to the Gaussian cross section of the laser beams, the most intense reflections occur at the central part of the probe volume with the intensity grading off at the periphery. Whether the object moves up or down the page at a given rate, the measured Doppler frequency is the same, and that's why the direction is ambiguous. However, when the fringes are given a constant upward or downward motion using an acousto-optic modulating crystal (shown as a Bragg Cell in the diagram) that alters the wavelength of only one of the interfering light beams, then the object's motion will be either in the same direction as the shifting fringes or in the opposite direction. That directionality is detectable as either an increase or a decrease in a given Doppler signal. Thus, acousto-optic modulation serves to supply us with the missing directional information, as well as the velocity. Though an electronic spectrum analyzer is shown here displaying the Doppler frequency, meaningful research requires computer-aided data collection so that velocity and direction information can be mathematically analyzed. This analysis will be necessary for experiments in chaotic fluid dynamics.

This concludes a look at some of the tools I am currently developing to work with nonlinear dynamics and chaos. Up next is a look at the most fundamental chaotic phenomenon which lies at the heart of the science of chaos: the butterfly effect.

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