THE ARRAY of dark and light bands known as a moiré pattern appears when light is viewed through overlapping window screens or similar structures consisting of repetitive elements. The repetitive elements need not be straight lines, such as the wires in window screening. Two sets of concentric circles that overlap generate moiré patterns in the form of radial lines. Concentric circles that overlap a grid of comparable spacing generate ellipses, parabolas or hyperbolas, depending on the angle of inclination between the plane of the concentric circles and the plane of the grid.

A moiré pattern can represent the solution of a mathematical equation. For example, the curves produced by superposing a grid (which can be regarded as the projection of a plane) and concentric circles (which can similarly be regarded as the projection of a cone) generate patterns that are solutions of equations that express the conic sections. Many other interesting properties of moiré patterns and some of their practical applications were investigated 10 years ago by Gerald Oster and Yasunori Nishijima [see "Moiré Patterns," by Gerald Oster and Yasunori Nishijima; SCIENTIFIC AMERICAN, May, 1963].

Among other techniques for generating moiré patterns, Oster and Nishijim suggested that the figures could be observed by looking through a periodi structure at its own shadow. For example, a pattern can be seen by casting the shadow of a window screen on a wall. Depending on the observer's point of view, some lines of the screen appear to be superposed exactly on top of some lines of the shadow. The pattern appears light in those regions. In other regions the lines of the screen fall between lines of the shadow. The pattern appears dark in those regions. The number of dark and light zones that are generated and the location of the zones within the field of view depend on the distance and shape of the object on which the shadow falls. For example, the shadow of a grid that falls on a flat surface generates a moiré pattern that also has the form of a grid. The spacing of the lines in this pattern varies inversely with the angle made by the plane with respect to the point of view. In contrast, the shadow of a grid that falls on a curved surface generates a pattern of dark and light bands that represent contour lines of the surface. Oster suggested that this effect, which is extraordinarily sensitive, might be used for making a contour map of the moon. A grid of known dimensions would be projected from the earth onto the moon's surface. The resulting shadow would be photographed through the grid. The moiré pattern would disclose surface features of the moon that escape not only telescopes but also cameras in close lunar orbit. This heroic scheme has not been tried for making lunar measurements, but a series of experiments that validate the concept has been made by Bill Lyon (3000 Fillmore Avenue, El Paso, Tex. 79930), who writes:

"My apparatus consists of a projection lamp, a grid of opaque lines, specimens of various shapes and a camera for recording the moiré patterns. The projection lamp is placed at an arbitrary distance from the grid. Specimens occupy an arbitrary position beyond the grid. The camera is put next to the lamp and at an arbitrary distance from it.

"In a typical experiment the lamp could be placed roughly four feet from the grid, the specimen two feet behind the grid and the camera a foot or less from the lamp [see illustration lower left]. I was interested more in the general form of the patterns than in the sharpness or resolution of the bands that make them. For this reason I made a vertical slit for the lamp instead of employing a more effective but costly projection lens. The slit, which functions something like a cylindrical lens, casts reasonably sharp shadows of the grid on the specimen. The quality of the final image was further improved by providing a slit for the camera.

"The lamp housing was made from a discarded coffee can. A porcelain socket was mounted inside the can with a pair of machine screws. A pair of wood screws that passed through the bottom of the can fastened the lamp assembly to a wood base. The lamp cord was brought out of the can through a rubber grommet in the side. The grommet prevents the sharp edge of the metal from cutting through the insulation of the cord and short-circuiting the power line.

"The lid of the can was replaced by sheet-metal top in the form of a slit. The sheet metal was obtained by cutting the top and bottom from a second tin can and flattening the side. Two rectangular pieces were cut from the strip, each somewhat wider than the radius of the can. They were soldered to clips of the same material to form a slit approximately 1/16 inch in diameter. Four 90degree brackets of sheet metal were then soldered in a circular array to the rear surface of the slit. The brackets serve as clamps for attaching the slit to the housing.

"The grid consists of black strings approximately 1/32 inch in diameter stretched in a diagonal array between opposite sides of a wood frame. Uniform spacing between the strings is achieved by supporting each strand, near its ends, in the thread of a long screw that is mounted at right angles on each side of the frame. The screws are prethreaded iron rod that is available in 36-inch lengths from dealers in hardware as stock from which steel bolts are cut. The dimensions of the grid are not crucial. It can be made two feet square or larger, depending on the space available to the experimenter.

"The frame, to which the screws are attached, can be made of one-by-two-inch lumber. I beveled the edges of the frame that are adjacent to the screws at an angle of about 45 degrees. Small brads spaced l /8 inch apart in two staggered rows were driven into the beveled face. I was then ready to lace the frame by passing a loop of string around each brad and through the adjacent thread of the screw [see illustration at left].

"The photographs were made with a 35-millimeter camera of the single-lens reflex type. Most of the photographs were made at a camera-grid distance ranging from two to three feet. The optimum distance must be determined experimentally. It will vary with the focal length of the lens. The mechanical slit for the camera can be made by the technique shown in the accompanying illustration [at right].

"To photograph moiré patterns place an object six inches or more in height and diameter behind the grid and illuminate it through the grid with the projection lamp. Put the camera adjacent to the lamp, focus it on the specimen and set the lens at the minimum f number. Turn on the projection lamp. The moiré pattern cannot be seen by eye because of the faint illumination. Turn off the room light and make an experimental time exposure of approximately two minutes. A moiré pattern should be visible on the developed emulsion. Alter the exposure time as may be required for optimum photographic quality.

"As I have mentioned, the bands of the moiré pattern represent contour lines of the specimen. It has been suggested that the scheme could be used for making sculptured portraits by means of the photographic process. It should be possible to rough out a model of a three-dimensional object by photographing its moiré pattern and using the bands as contour lines to guide a pantographic carving machine. Machines of this kind are currently used by die makers for sinking rough contours of models into blocks of steel. A skilled carver completes the die by hand. In my own case the experiments were of interest as aids in grasping the fundamentals of making and reading contour maps "

AN amateur who builds a radio set, a small computer or a high-fidelity sound system routinely measures voltages, current and resistances with inexpensive meters. The measurements enable him to improve the performance of his apparatus and thus to increase the satisfactions of his avocation. In contrast, someone who undertakes to construct an electrostatic device such as a Van de Graaff generator, an electret motor or an ultrasonic microphone rarely makes measurements. The cost of instruments that work reliably in electrostatic circuits has been beyond the reach of most amateurs.

Recent advances in solid-state technology have altered the basic design of such meters in ways that reduce their cost dramatically. For example, a relatively inexpensive microammeter that is specifically intended for high-voltage electrostatic circuits has been designed expressly for amateur construction by R. H. Kaufmann (2208 Dean Street, Schenectady, N.Y. 12309), a retired electronics engineer. The instrument measures current from 25 millionths of an ampere to five billionths of an ampere in circuits that carry potentials of up to 500,000 volts. Kaufmann discusses the design and calibration of the meter:

"The meter was constructed for measuring minute currents in various parts of electrostatic generators and other devices, particularly currents that leak into the atmosphere by the phenomenon known as corona discharge. Essentially the instrument consists of a zero-centered meter of the d'Arsonval type that is actuated by a pair of matched field effect transistors connected to operate as a differential amplifier [see illustration at left]. The sensitivity of the instrument can be adjusted by a four-position switch.

"Before undertaking the project I wrecked two ordinary microammeters during attempts to measure current in electrostatic circuits. The delicate coils that turned on finely wrought pivots were damaged by transient surges of current that accompanied the sparkover of high voltages. Accordingly, when I was designing the new meter, I shielded all working parts in a Faraday cage that consisted of a metal housing lined with copper screening. The plastic face of the meter was also covered with screening that was bonded electrically to the housing. Finally, all shielding was connected through a transfer switch to one or the other of the two input terminals. The terminals consist of 1/4-inch rods about four inches long that terminate in aluminum spheres S/S inch in diameter [see illustration lower right].

"The zero-centered scale of the meter is graduated to indicate +25 microamperes. The resistance of the meter, which is 1,200 ohms, must be taken into account when the amplifier is designed. To measure the resistance of the coil in any meter without damaging the instrument, divide 1.S by the maximum current for which the meter is calibrated. The result is equal to the value in ohms of a resistor that, when it is connected in series with the meter and with a dry cell of 1.S volts, will cause the pointer of the meter to move to the full-scale graduation. For example, a meter that is calibrated to measure maximum current of 2S microamperes will indicate full scale when its terminals are connected to a l.5-volt battery in series with a resistor of 60,000 ohms (1.5/25 x 10 = 60,000).

"With the meter thus indicating full scale, connect a variable resistor of, say, 5,000 ohms across the terminals. Gradually reduce the value of the variable resistor. The pointer of the meter will simultaneously drift toward zero. Continue to reduce the value of the variable resistor until the pointer indicates exactly half of the maximum reading. The value of the variable resistor at this adjustment is exactly equal to the resistance of the coil in the meter.

"The electrostatic microammeter works equally well with meter coils of any reasonable resistance, but if the value differs substantially from 1,200 ohms, the shunt resistors of the differential amplifier (R, R, R, and R) must be altered from the values indicated by the accompanying circuit diagram. The procedure for determining the appropriate values will be explained.

"The heart of the differential amplifier is a matched pair of alloy-junction, field effect transistors potted in a single housing (No. U234). The devices are available from Semiconductor Specialists, Inc.(P.O. Box 66125, O'Hare International Airport, Chicago, Ill .60666). The list price of the device is $3.90 , but the distributor accepts only orders in the minimum amount of $5. (Orders from amateurs should be directed to the attention of Richard K. Dahlem.) The characteristics of the device vary somewhat from unit to unit. For this reason I found it necessary as well as interesting to measure the characteristics of each device I bought.

"Field-effect transistors include three terminals known as (1) the source, S, (2) the drain, D, and (3) the gate, G. To test the U234 I connected each of the two transistors sequentially in a simple circuit that included a 0-3 voltmeter and a 0-3 milliammeter [see illustration at left]. A fixed potential of approximately 20 volts was applied across the source-drain (1-2) terminals of the transistor. Similarly, a potential that was gradually increased from zero to three volts by means of a five-kilohm potentiometer was applied between the source and the gate (1-3) of the transistor. This potential, designated V was tabulated. Simultaneously, corresponding current in the drain circuit, I, was observed and also tabulated. The resulting data were then plotted as a graph that displayed the drain current, I, against the corresponding gate-source voltage, V [see illustration at right].

"Note that the shape of the graph approaches a straight line within an interval that corresponds to a gate-source potential of between-1 volt and-2 volts. In other words, through this interval current in the drain circuit of the transistor varies in direct proportion to the gate-source voltage. The operation of the amplifier is said to be linear within this range.

"The center of the linear portion of the graph corresponds, approximately, to a V potential of 1.6 volts. At this potential the drain circuit of the device draws one milliampere. Excursions of V from 1.6 volts cause directly proportional excursions of current, I, in the drain circuit. The potential of 1.6 volts is known as the operating point of the transistor. The operating point differs from one No. U234 device to another and must be determined experimentally for each device.

"Next I assumed a value of 1,000 ohms for the resistor identified as R. On the basis of this assumption and the data determined by the preceding test I calculated the value of resistor R. The arithmetical procedure is specified in simple, abbreviated form by the equation R = . After the corresponding numbers have been substituted the equation becomes (1.6/.002)- (1,000/4) = 800 - 250 = 550 = R. These resistors, R and R, establish a bias voltage on the gates of the field-effect transistors of such value that the meter can be set to zero when no voltage is present across the input terminals of the instrument.

"To produce meter deflection from this steady-state zero position a differential voltage must be applied to the gates of the two transistors. The required voltage originates across the shunt resistors: R, R, R and R. The value of these resistors is determined by a relatively simple procedure. Before it can be undertaken, however, one must wire the remaining portions of the circuit, including the calibration potentiometer, the drain resistors, the meter and the protective diodes. The diodes are not essential. In effect, they act as fuses to protect the meter. Those who undertake the construction, however, are reminded that diodes are much less expensive to replace than the meter.

"Use a pair of matched 4,000-ohm resistors for R. Operate the four-position selector switch to connect the outboard ends of the resistors to the gates of the transistors. Set the calibration resistor, R, to its maximum value. Apply an adjustable source of low voltage to the input terminals and hence across the RS resistors Increase the voltage until the pointer stands at full-scale deflection. With any accurate microammeter measure the current in the R resistors. If 4,000 ohms is approximately correct for R, the voltage across the gates of the transistors (and across R) is equal to the current through the R pair multiplied by 8 x 10.

"Assume that the current amounts to 25 X 10 ampere, a typical value. The voltage across the transistors is equal to the product of the current multiplied by the resistance, or 25 X 10 X 2 X 4 X 103, or .2, volt. (The transistors operate satisfactorily in this application with any potential across the gates between the limits of .5 and .1 volt.) Had the measured voltage fallen outside this limit, the value of RS would have been increased or decreased from 4,000 ohms as necessary to develop .2 volt. With RS equal to 4,000 ohms in this specific circuit, the meter indicates a full-scale deflection of 25 microamperes.

"Next I determine the value of the succeeding shunt resistors that control the gain of the amplifier and thus fix the scale of the meter. Assume that the next more sensitive full-scale value is to be five microamperes. The controlling resistor, R, is equal to in which I, is the current (5 X 10 ampere) that is desired for full-scale deflection and has been previously determined. These values, when substituted in the equation, determine the value of R: = 16,000 ohms. Similarly, the value of R that would be required for the still greater sensitivity of 2.5-microampere full-scale deflection is found by dividing .2 by the product of two times 2.5 microamperes and subtracting the sum of R plus R Thus R = kilohms.

"The diodes and the capacitor that function as protective devices must be of excellent quality to avoid the leakage of current that would impair the accuracy of the instrument The forward conduction of the diodes should not exceed 10 billionths of an ampere at the normal working voltage of the instrument. Leakage through the capacitor should not exceed 10 ampere at an impressed potential of 10 volts. With the exception of the indicating meter and the matched pair of field-emission transistors, I constructed the instrument entirely of parts salvaged from surplus electronic assemblies."

Bibliography

INTERFERENCE SYSTEMS OF CROSSED DIFFRACTION GRATING: THEORY OF MOIRÉ FRINGES, J. Guild. Oxford University Press, 1956.

ELECTROSTATICS: EXPLORING, CONTROLLING AND USING STATIC ELECTRICITY, A. D. Moore. Doubleday & Company, Inc., 1968.

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