After reading the math section on this site, you will begin to see that the images created in Laser Light Math are constructed by mixing various sine and cosine functions together at various frequencies, shapes, and amplitudes. On the Internet, many examples describe how this can be done. Of course, you probably know that the creation of images in this fashion falls into the mathematical process of making “roulette” or “spirograph” curves. From your childhood, you might recall working with a “spirograph” toy where you could place one geared wheel within the circumference of another larger wheel. After inserting your drawing pen, you rotated the smaller gear around or within the circumference of the larger gear, and, voila, a graphed representation emerged of a certain mathematical curve in the roulette family (such as the epitrochoid, hypotrochoid, epicycloid, hypocycloid, and or perhaps even the beautiful rose family). The spirograph, of course, is just a simple mechanical plotting device. Performing this same trick with a laser beam and its tiny dot of light calls for a substantially modified approach (more about this later).

As you move through this narrative, you might want to view on occasion a Master Diagram of the system. This diagram represents all of the hardware in the Laser Light Math projection system. When you visit the diagram, you can click on various elements to drill deeper into the system design. Hitting the "back" button on your browser bar will eventually bring you back to this page, but, if you are deeper into the diagram drawings, you can hit the "previous" button on the bottom of the page to bring you back here directly.

Unlike working with a basic spirograph pen plotter, in Laser Light Math we often extend our palette of shapes beyond traditional sine waves. Indeed, we occasionally mix various combinations of square waves, triangle waves, and even saw tooth waveforms. In every case, however, the same mathematical relationship is maintained that creates a sine versus a cosine function. Specifically, mixing a sine wave with a cosine wave of the same amplitude and frequency generates or graphs a circle. In engineering courses, this phenomenon is often demonstrated on oscilloscopes to portray the mathematical relationship that exists between a sine and cosine functions.

For our purposes, however, it is especially important to understand one essential idea regarding sine and cosine math: in a sine/cosine relationship, where two waves are both sinusoidal in shape, you can create a circle pattern when the two waveforms remain exactly ninety degrees out of phase. In electronics, this is referred to as a “quadrature” relationship. Oscillations are in quadrature relationship when they are separated in phase by 90° (π/2, or ?/4). The wave shapes produced in Laser Light Math are heavily dependent upon maintaining a quadrature relationship.

To gain greater artistic flexibility in Laser Light Math, you mix not only sine waves in quadrature relationship, but also other waveforms (saw, triangle, square, and saw tooth). You can produce some unusual effects this way. If you mix two square waves of equal frequency and amplitude in a quadrature relationship, you will get a nicely defined square. Similarly, two triangle waves in quadrature relationship (one on the X axis and the other on the Y axis) will produce a diamond image. Thus, by using sinusoidal waves, square waves, triangle waves, and saw tooth waves, it is possible to create a diverse range of unusual and beautiful images that cannot be generated through the typical or traditional spirograph plotter technique (which uses only sinusoidal functions). By possessing this increased image making capacity, the Laser Light Math system opens up new image making possibilities for laser artists.

Beyond the math, there are still difficult engineering challenges in trying to get the little laser “dot” to form a complete projected pattern(s) of any mathematical curve. Why? Well, the reason is that you must “scan” that dot rapidly through a complete image path at least sixteen times a second before the image will look or appear solid. This effect is known as “persistence of vision.” Because the eye’s retina does not work quite as fast in processing light changes, you can create from a single dot moving though a complete pattern sixteen or more times per second the appearance of a solid image. This effect is essentially the basis of how motion pictures work.

In Laser Light Math, projected patterns are usually variations of sine/cosine functions (epicycloids, hypocycloids, epitrochoids, and so forth). Each image is created by moving a laser dot (sometimes up to three dots) rapidly and repeatedly through the same pattern--anywhere from sixteen to six hundred times per second. For the human eye, the resulting effect appears to be a graph of our selected mathematical curve(s).

For a better understanding of this process, try creating a similar effect by taking a hand-held laser pointer and moving it back and forth quite rapidly. Try it. Moving the laser pointer from left to right will create for the eye the appearance of a horizontal line (assuming that you can move it back and forth more that sixteen times a second). Similarly, moving the pointer rapidly in a circle will create the appearance of a projected circle in light. In reality, what you create by moving a dot of laser light so rapidly is the illusion of a solid line or pattern (thanks to the persistence of vision effect). Essentially, this is the secret to creating any laser image. Of course, it is important to remember that instead of moving the laser itself you must use some “hardware” to “reflect” the laser dot rapidly through its defined pattern. In Laser Light Math, the patterns are all related to various mathematical curves (epitrochoid, hpyococloid, and so forth).

The major engineering challenge in Laser Light Math is this: to create accurate visual representation of any waveform(s), you must move a laser dot very precisely and rapidly in a fashion that represents the selected mathematical curve(s) you hope to project. In this project, the technique for generating visual representations of chosen waveforms (sine, square, triangle, or saw tooth) requires a mathematical “lookup table” for each waveform. Every table must hold enough data points (graphing points) to create or graph the selected laser image. Data points are forwarded to a digital-to-analog converter. This circuit converts each data point or element into an “electrical” signal that is a proportional representation of the data point’s numerical value (for example, a 255 might produce a 3 volt signal while a 128 value would create a 1.5 volt signal). These low level signals are input into a series of powerful direct current (DC) amplifiers (with absolutely no capacitive coupling) where the smaller low level signals, say 1 to 3 volts, can be translated into high current high amplitude output signals (about 0 to 20 volts at up to 4 amperes of current).

DC amplifier signal outputs are then fed into a series of “galvonometers” (or “galvos” as they are called in the industry). Galvos are amazing and rather expensive little electro-mechanical devices that rotate a small shaft, with a first surface mirror attached to it, in one direction or another (depending upon the polarity of the input signal). Rotation is usually limited to + and – 20 degrees, or somewhere in that range. High quality galvos provide a feedback signal that lets the DC amplifier know what the actual position of the mirror is. When a difference between desired position and actual position exists, the amplifier compensates for the discrepancy by adding or reducing current flow. Popular galvo devices include those built by Cambridge Devices and General Scanning Systems. The amount a mirror rotates is precisely relative to the amount of current being sent through the galvo.

If you send a single sine wave signal into a galvo through a DC amplifier and reflect a laser beam off of that galvo’s mirror, you will project the appearance of straight line with the moving laser dot (assuming you move it faster than 16 times per second). Similarly, if you send a sine wave into one “X” direction galvo and cosine wave into its partner “Y” direction galvo, as you might do with the X/Y galvo mount illustrated on this site, you will project a circular pattern (assuming both signals are the same frequency and amplitude). Waveforms created in his manner can be very precise. In Laser Light Math, one complete cycle of a sine wave image contains (in a lookup table stored inside an EPROM or electrically programmable read only memory chip) 512 points of mathematical data.

Thus, to create an actual waveform image, the system must rapidly look up (or clock through the EPROM table) and send each data point to a digital-to-analog (DAC) converter circuit. Creating waveforms in this manner is called “digital synthesis.” The DAC itself is a relatively simple circuit that takes a binary number and converts it into a proportional electrical signal. Still, the entire conversion process can be fairly demanding in terms of “clocking” rates. To illustrate, consider that each complete sine wave possesses 512 points of data, and, to shape properly, it must be clocked through repeatedly and rapidly. To produce a 600 Hz sine wave, for example, requires that you clock through the sine wave look table at a rate of 307,200 Hz. Examine the “synthesizer” block diagram and you will see how this process is accomplished for both the X- and Y-axis signals simultaneously. Remember, creating a simple circle requires that the system mix both sine and cosine waveforms in quadrature relationship, which means the laser dot must move simultaneously through combined X- and Y-axis data. Trust me, doing it right requires the math to work in a quick and integrated fashion.

To expand from these explanations of how sine and cosine waves can be combined to create simple laser images, take a closer look at the overall projection system that has been created for Laser Light Math. As an aid in this process, I have included a series of system diagrams intended to help explain the general flow of electronics and supporting logic. The all-important first diagram presents an overview of the entire system (Fig. 1). Here you can see each element of the system, such as the starting point which is a high-speed laptop computer running Windows XP (the software also works with Windows 98B and Windows 2K).

The laptop becomes the control console through an original piece of software written in Visual Basic. This software, detailed in the software section of the Laser Light Math site, allows the laser artist to extract primary images of selected mathematical curves from a library. The artist can also create customized images and store them in the library. Multiple libraries can be created. Special screens can be used to control each laser. Once an image is complete and running the way the laser artist wants it to work, the image can be saved to a library. Also, since the system uses three laser color lines, red, blue, and green, you can store related images for each color. After the appropriate images have been collected, the laser artist selects another screen and builds his/her show. Once the show is complete and saved, the artist can play it back repeatedly and watch the beauty and precision of laser light in motion.

The process of moving the program images to the laser projection device is done the following way: the laser show data is first sent to the linear and digital logic portions of the system via a high-speed serial interface that converts a 115Khz baud rate RS232 signal to an industry standard DMX512 serial data transfer device. The nature of the DMX signal is outside the realm of this discussion, but a quick search on the Internet will produce a wealth of knowledge concerning how DMX512 works. In any case, the DMX512 signal transfers its serial digital data to a DMX conversion board that creates 15 analog (0 to 10 volt dc) control outputs and one eight bit parallel data port. The analog signals control the system’s frequency and amplitude functions while the digital port selects waveform data. These combined digital and analog data are sent to the all important synthesizer module(s). Frequency signals are accurate to sixteen bits of resolution. Amplitude signals are accurate to eight bits. Each synthesizer unit contains the wave lookup tables, the digital to analog conversion circuits, and the high speed clocking circuits required to sequence the lookup data table addresses. Every synthesizer module possesses three X/Y control paths, allowing for the creation of remarkably complex images (done by mixing numerous waveforms into a single X /Y output).

The complete Laser Light Math system contains three waveform synthesizer modules, one for laser color. The three output images are layered one on the other. When done properly, they all assume the same center point. The “multi-color images” on this site exemplify this coordinated approach. Of course, even small alterations in one laser clocking frequency will create a subtle shift in phase, a shift that causes the image to rotate right or left. Look at the multi-color moving images (MOV files) to see this effect in action. It is quite beautiful.

Once the control signals have been processed by the synthesizer module(s), the resultant X and Y output signals (all low level) are sent to high power DC amplifier units. These translate low-level electrical signals into high-level current signals, which is what the galvonometers require to perform properly. And, as previously indicated, these amplifiers are responsible for processing the position feedback signals from the galvos in ways that compensate for any position error issues. Precision scanning requires that each galvo mirror achieve the position that the control signal presents. When a position error occurs, the amplifier compensates and brings the galvo into its correct alignment.

Controlling actual laser light output level is difficult. Lasers, at least gas lasers like those used in this project, are not easily dimmed. Controlling beam intensity requires a Polychromatic Acoustic-Optical Modulator (PCAOM), a small optical device that contains a precision quartz crystal. The laser light beam passes through the crystal and emerges in two forms, both a diffracted line and an un-diffracted line.

Through the PCAOM, the light beam can be diffracted (or reflected in a different direction) when a particular radio frequency is applied to the crystal from an amplifier. Laser Light Math uses an eight channel PCAOM manufactured by MVM Electronics. Applying small dc signals to respective driver amplifier inputs controls the amplitudes of differing laser frequencies.

The Laser Light Math optics layout uses the diffracted beam for image making light. The un-diffracted beam is “dumped” or wasted, which is one reason why such systems are relatively inefficient. It takes a lot of light to make this process work properly. Despite its inefficiency, however, the PCAOM is still an especially handsome technology that makes it possible to control the intensity of any laser beam. Additionally, it is possible to switch the light output on and off rapidly. This “on and off” switching action is known as “chopping” the light beam. In Laser Light Math, the PCAOM can chop the laser signal anywhere from 0 to 10,000 times a second. Additionally, the shape of the output pulse or signal can be altered significantly, making it a variable width square wave (anywhere from 0 to 100% in width). Changing pulse width is referred to as “pulse-width-modulation.” For laser artists, these capabilities add a multitude of possibilities to the composing of projected images. Some of the images on this site include “chopping.”

A special note: signals to control the PCAOM module are first generated in the synthesizer module(s). To work properly in relationship to the PCAOM control amplifier, however, these signals must be shaped through the actions of a voltage-controlled amplifier stage. This stage controls, via 0 to 10 volt DC input signals, the final amplitude of the square wave chopper signals. And, coorespondingly, the amplitude of a chopper signal controls the light level of its respective laser beam.

Finally, to help illustrate my description of the hardware utilized in the Laser Light Math project, I have included some JPEG images of the system and its various components—entire system, optical table, PCAOM control unit, DC amplifier, and wave synthesizer module. It has been a long but enjoyable journey.


Master Diagram of the system.