*dparker@nrao.edutThe National Radio Astronomy Observatory is operated

by Associated Universities, Inc. under cooperative agreementwith the National Science Foundation

Multidirectional retroreflector assemblya common virtual reflection point

using four-mirror retroreflectors

David H. Parker*The National Radio Astronomy Observatory, Green Bank, WV1

A brief survey of retroreflector designs and appli-cations is presented. A novel multidirectional (asopposed to omnidirectional) retroreflector concept,which uses a four-mirror retroreflector subassem-bly with a common virtual reflection point (thuseliminating the Abbe error), is described. Appli-cations include multilateration with interferometers,laser trackers, and electronic distance measurementsurveying instruments—as well as other radiationsources, e.g., microwaves and acoustics. Exampleconfigurations are given.

Keywords: retroreflector; multilateration; lasertracker; large-scale metrology; coordinate measure-ment machine calibration; four-mirror retroreflector;electronic distance measurement

1 IntroductionA number of laser interferometer and electronic dis-tance measurement (EDM) applications desire wideangle of acceptance, or multiple retroreflectors[l , 2, 3,4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14]. The primary prob-lem using multiple retroreflectors is due to the mea-surement axes not passing through the measurementpoint, and thus making the measurements sensitiveto rotations of the object and/or angle between theinstrument and object, i.e., the Abbe error [15, 16].

Surveying equipment manufacturers have assem-bled solid glass retroreflectors, such as the Leica[17}GRZ4 360 degree prism, but the glass offset is a func-tion of the incident angle[2, 18, 191, and coverageoverlaps between adjacent retroreflectors, so there isa significant Abbe error (several mm for the GRZ4).

Goldman designed a "triplet" assembly consistingof a cat's-eye retroreflector midway between two solidglass corner cubes directed to the rear of the cat's-eye[20]. The center of the cat's-eye and the optimal"optical center" of the corner cubes are colinearlymounted on a rigid beam assembley. One group ofEDM instruments ranging on the cat's-eye and an-other group of EDMs ranging on the pair of cornercubes can be tied together through the triplet benchmark. In order to avoid crosstalk between the twocorner cubes, it is necessary to physically space themapart by several beam diameters. While the angle tothe cat's-eye can in general be anywhere in the fieldof view, the angle to the corner cubes must be limitedto minimize the Abbe error.

Laser interferometers are typically calibrated in aback-to-back retroreflector configuration, where therotation of the retroreflectors is constrained. Forexample, NIST has built a Laser Rail CalibrationSystem (Larcs) for calibrating laser trackers{21, 22]against an interferometer on a linear rail[231. Larcsuses two spherically mounted retroreflectors (SMRs)(described in more detail below), in a back-to-backconfiguration on a carriage, to build a bidirectionalretroreflector assembly, i.e., one direction fixed forthe reference interferometer parallel to the rail, andthe other free to rotate in a nest to accommodate thelaser tracker under test.

The Abbe error is minimized by mounting the tworetroreflectors as close as practical and constrainingthe carriage to a rail system to minimize rotations ofthe assembly. However, for portable rails, the uncer-tainty due to the Abbe error is estimated to be a sig-nificant part of the total error budget. We have alsoused a similar technique at NRAO, but used an ad-ditional mirror on the carriage for an autocollimatortarget and tweek the carriage mechanical alignmentbefore taking readings.

NASA has built custom hollow retroreflector as-

semblies with a common physical reflection point [24,25], and thus eliminated the Abbe error. These solvethe Abbe error problem for some classes of measure-ments. However, since these retroreflectors share a,common physical point, they sacrifice part of the cen-ter aperture, are difficult to build, are difficult to ref-erence to an outside mechanical point, are expensivefor routine applications, and the directions are notadjustable.

Gelbart and Laberge describe an "omnidirectionalretroreflector" pair combined with a fixed probe[9,101. By multilaterating on the pair of retroreflec-tors, the probe coordinate is calculated. The omnidi-rectional retroreflector, described in the '091 patent,"consist of two concentric spheres made of transpar-ent material and having the refractive index of theinner sphere higher than the refractive index of theouter sphere, the outside sphere coated with a par-tially reflective coating." A prototype of this designwas built by CREO Products Inc., Burnaby, B.C.,Canada; but is not commercially available. Gel-bart suggested that an even better design could beachieved by using three concentric spheres[261.

Recently, ideal omnidirectional spherical retrore-flectors have been built from high index of refractionN=2 glass[13, 14, 27]. Unfortunately, the glass is dif-ficult to work, expensive, and the return power islow-due to the spherical aberation and small workingaperture, as well as the low reflection coefficient of theglass/air interface on the back side of the sphere, i.e.,most of the power is transmitted through the sphere.While these problems will hopefully be overcome byadvances in materials and manufacturing techniques,only a few of these highly coveted retroreflectors havebeen built since being introduced in 1994.

1.1 Multidirectional applicationsLaser trackers incorporate a laser interferometer withan automated mirror system to track a retroreflec-tor. The interferometer measures differential rangevery accurately, with the fundamental limitation be-ing the uncertainty of the index of refraction—whichis typically in the 1 ppm range.

The angle measurements are somewhat less ac-curate. The fundamental limitation is atmosphericturbulence and temperature gradients bending thebeam. There are also practical limitations with theencoders, mechanical system, beam quality, gravi-tational reference, etc. Nakamura et al[13] pointsout that for a distance measurement uncertainty ofSr, in an ideal orthogonal trilateration measurement,the uncertainty volume is (60 3 . For two angles anda distance measurement, the uncertainty volume is

(r60) 2 6r. For example, for a typical distance mea-surement uncertainty of 1 ppm (6r/r = 10 -6) andan angle uncertainty of one arc second (R-, 5x10-6radians), the trilateration uncertainty volume wouldbe

Sv = r310-1-8 (1)

whereas the uncertainty volume for two angles and adistance would be

(5v = 25r3 10-18 , (2)

or 25 times greater than the trilateration uncertaintyvolume—hence the inherent potential improvementin accuracy by using multiple distance measurements.In practice, there are two primary obstacles to achiev-ing this huge improvement. Conventional retroreflec-tors do not support simultaneous measurements inthe three orthogonal directions, and in actual fieldconditions it can be hard to mount an instrument ona stable tower or structure.

Laser trackers typically use spherically mountedretroreflector (SMR) targets. These are typically hol-low or cat's-eye type[28, 29, 30} retroreflectors, withthe optical centers carefully located in the centerof the spherical mounting—thus allowing the opticalmeasurements to be related to the physical center ofthe sphere. Hollow SMRs, such as those built by PLXInc.[311 are more economical than cat's-eye SMRs,but have a reduced angle of acceptance and thus aremore susceptible to dropping the laser interferometerbeam while tracking.

An obvious improvement in the accuracy of thelaser tracker (or EDM) is to use multiple instru-ments and/or augment with additional information,e.g., known artifacts, stable bench marks, hydrostaticleveling, or other constraints. For three or moreinstruments, oriented in the proper baselines, theless accurate angle measurements can be neglectedor weighted less in a least squares, or more sophis-ticated, reduction. While crosstalk is not a prob-lem using multiple laser trackers on a common SMR,the relatively small angle limitation of even the cat's-eyes makes the instrument baselines unfavorable forhigh accuracy multilateration measurements, and ofcourse the Abbe error is the limitation for conven-tional assemblies of SMRs.

The Robert C. Byrd Green Bank Telescope large-scale metrology system was designed to operate as amultilateration system employing 18 laser ranging in-struments measuring ranges to cardinal points on themoving telescope[5]. While some paths are physicallyblocked by the structure, it behooves the designers touse multidirectional retroreflectors in order to maxi-mize the number of independent measurements, and

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thus strengthen the calculation of cardinal point co-ordinates.

The increasing interest in multilateration, usinglaser trackers or other EDMs, has created a needfor less expensive and more practical multidirectionalretroreflectors with zero Abbe error.

The four-mirror retroreflec-tor

Laser tracker manufacturers use a clever four-mirrorretroreflector design disclosed by Brown in US Patent5,530,549{32, 33], and a variation of the same con-cept in US Patent 5,861,956[34]. See the Faro Corp"RetroProbe" [35j, and the Leica, Corp. "SurfaceRetroreflector" [171 for commercial examples. Browndescribes a system of a retroreflector (built of threemirrors), probe tip, and additional first surface mir-ror.

The principle is simple and elegant. If the lineconnecting a hollow retroreflector and probe tip isbisected by a first surface mirror, as shown in Figure1, then a beam directed at the probe tip which inter-sects the "forth mirror" is reflected by the mirror tothe retroreflector, i.e., the virtual point of the probeis at the retroreflector. The optical path length, andangle, to the image of the retroreflector, is identicalto the optical path length, and angle, to the probetip. A similar system is used to focus photographicenlargers by reflecting the projected image from amirror, at a fixed offset from the photo paper plane,onto a reticle which is viewed through a magnifier.

By using a hollow retroreflector, it is insensitiveto the orientation of the four-mirror retroreflector,i.e., no glass offset. Using this system, laser trackersare used to probe locations inaccessible to the moreconventional SRMs.

2.1 Cat's-eyes

It should be pointed out that a cat's-eye retroreflec-tor could also be used in the same way as a hollowretroreflector, for applications that need larger ac-ceptance angles. The center of the cat's-eye wouldreplace the apex of the hollow retroreflector. Themaximum aperture of the fourth mirror is determinedby the retroreflector acceptance angle and distanceto the mirror. For EDM measurements, the glass off-set(s) would have to be corrected, but it would be aconstant and independent of the viewing angle. Forinterferometer measurements, the glass offset wouldbe absorbed into the interferometer initialization.

Figure 1: Principles of the four-mirror retroreflector.

Extensions of the four-mirror

There are a number of practical extensions of theprinciples used in the four-mirror retroreflector forbidirectional and multidirectional retroreflectors--both with and without a probe; and with fixed oradjustable directions l .

3.1 Bidirectional retroreflector withvirtual point at one retroreflector

By replacing the probe tip with a second retroreflec-tor, with the apex at the center of the former probetip location—as shown in Figure 2—a system is con-structed whereby measurements to both retroreflec-tors are made to the same virtual point, i.e., the for-mer probe center.

Note that the probe tip replacement retroreflectorcan be oriented in any direction, e.g., 180 degrees tothe four-mirror retroreflector path (for back-to-backmeasurements) or orthogonal (including into and outof the paper) to the four-mirror retroreflector pathfor X-Y measurements, etc. Moreover, the probetip replacement retroreflector direction can be fixedor adjustable, and captive or separable (such as aSMR/nest configuration). For example, this couldbe used to make simultaneous ED1VI measurements,sequential EDM measurements from different direc-tions without turning the retroreflector, bring twolaser trackers into coincidence, etc.

1Patent pending

MIRROR

Figure 2: Bidirectional retroreflector.

Figure 3: Multidirectional retroreflectors.

3.2 Multidirectional retroreflectorswith virtual point at one retrore-flector

Since the four-mirror retroreflector does not physi-cally intersect the virtual reflection point, the bidi-rectional assembly can also be extended to multiplemirrored retroreflectors, as shown in Figure 3. Theonly restriction is that the mirrored retroreflectorscan't block the visibility of other retroreflectors.

3.3 Multidirectional retroreflectorsusing all mirrored retroreflectors

Of course, the multidirectional retroreflectors, withthe virtual point at one retroreflector, can be ex-tended to a cluster of four-mirror retroreflectors with-out a directly illuminated retroreflector, or the probetip could be reintroduced.

3.4 Other applicationsWhile the immediate application is directed at laserbeams, the same concepts could easily be adapted for

Figure 4: Cubic manifold.

other radiation sources, e.g., microwaves and acous-tics.

4 Example configurations

In US Patent 5,335,111[28], Bleier describes how toconstruct hollow retroreflectors in a hard mount as-sembly. One-piece replicated retroreflectors are alsoavailable from manufacturers such as Opticon, whichthe author has used with success in hostile outdoorenvironments.

In a practical application of the four-mirror system,the problem is to mechanically bisect the line betweenthe retroreflector and virtual point with a first surfacemirror. Mechanically locating the apex of a hollowretroreflector is traditionally accomplished by care-fully inserting a sphere into the retroreflector. Thecenter of the sphere is located at a height

h = rvrd (3)

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Figure 5: Cylinder manifold.

Figure 6: Spherical manifold.

from the apex. Of course, the Abbe error, due tothe distance between the center of the sphere and theapex, is a problem. The orientation of the retrore-flector can be established by autocollimating on eachof the retroreflector mirrors, or by building a fixtureto fix the orientation. At NRAO, we have used asolid glass retroreflector mounted in a fixture to ori-ent hollow retroreflectors, i.e., gently drop the hol-low retroreflector over the inverted glass retroreflec-tor. Differential optical measurements could be madeby EDM or an interferometer using a SMR.

While fixturing could be built to construct everydesired configuration, there may be merit to simplybuilding the four-mirror retroreflector assembly andthen configuring the assemblies as needed. For ex-ample, by building four-mirror retrorefiectors, witha uniform offset and reference footing, any numberof configurations would be practical. The foot couldbe a fixed attachment or magnetic base to facilitaterapid reconfiguration and adjustment.

Flat footBy constructing a four-mirror retroreflector with aflat, three ball, or Kelvin mount foot, one could con-struct any number of configurations, in an orthogonalcoordinate system, around a simple (and inexpensive)cubic manifold, as shown in Figure 4

4.2 "V" footBy constructing a four-mirror retroreflector with a"V" foot, one could construct an array that wouldcover a circular pattern around a simple cylindermanifold, as shown in Figure 5.

4.3 Nest footThree-dimensional coverage could easily be achievedby building four-mirror retroreflectors with a three-point, or conical, nest foot that would attach to aspherical manifold, as shown in Figure 6.

4.4 Articulated assembliesDue to the fact that the Abbe error is eliminated, anyof the suggested examples could be articulated. Forexample, an assembly could be mounted on a coordi-nate measurement machine, rotated about an axis bya motor; or maintained in a fixed orientation, with re-spect to gravity, by a pendulum counterweight, e.g.,to compensate for the rotation in elevation of a radiotelescope.

5 SummaryUntil true omnidirectional retroreflectors become eco-nomical and commercially available, the four-mirrorretroreflector offers a partial solution by building mul-tidirectional retroreflectors.

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