Patent Publication Number: US-9903934-B2

Title: Apparatus and method of measuring six degrees of freedom

Description:
CROSS REFERENCE TO RELATED APPLICATIONS 
     This application claims the benefit of U.S. Provisional Patent Application No. 62/186,447 filed on Jun. 30, 2015 and U.S. Provisional Patent Application No. 62/237,299, filed on Oct. 5, 2015, the entire contents all of which are incorporated herein by reference in their entirety. 
    
    
     BACKGROUND 
     The present disclosure relates to a coordinate measuring device having the ability to determine three orientational degrees of freedom, and in particular to a coordinate measuring device cooperates with a device configured to measure three translational degrees of freedom, thereby enabling determination of the position and orientation of a rigid body in space. 
     Some coordinate measurement devices have the ability to measure the three-dimensional (3D) coordinates of a point (the three translational degrees of freedom of the point) by sending a beam of light to the point. Some such devices send the beam of light onto a retroreflector target in contact with the point. The device determines the coordinates of the point by measuring the distance and the two angles to the target. The distance is measured with a distance-measuring device such as an absolute distance meter (ADM) or an interferometer. The angles are measured with an angle-measuring device such as an angular encoder. The device may include a gimbaled beam-steering mechanism to direct the beam of light to the point of interest. 
     The laser tracker is a particular type of coordinate-measuring device that tracks the retroreflector target by emitting one or more beams of light. A coordinate-measuring device closely related to the laser tracker is the total station. In many cases, the total station, which is most often used in surveying applications, may be used to measure the coordinates of a retroreflector. Hereinafter, the term “laser tracker” is used in a broad sense to include total stations. It is also understood that the laser tracker may use any type of light source and is not restricted to a laser light source. 
     Ordinarily the laser tracker sends a beam of light to a retroreflector target. A common type of retroreflector target is the spherically mounted retroreflector (SMR), which comprises a cube-corner retroreflector embedded within a metal sphere. The cube-corner retroreflector comprises three mutually perpendicular mirrors that intersect in a common vertex point. For the case of a “hollow” SMR having reflecting surface in contact with air, the vertex is located at the center of the sphere. Because of this placement of the cube corner within the sphere, the perpendicular distance from the vertex to a surface on which the SMR rests remains constant, even as the SMR is rotated. Consequently, the laser tracker can measure the 3D coordinates of a surface by following the position of an SMR as it is moved over the surface. Stating this another way, the laser tracker needs to measure only three degrees of freedom (one radial distance and two angles) to fully characterize the 3D coordinates of a surface. 
     One type of laser tracker contains only an interferometer (IFM) without an ADM. If an object blocks the path of the beam of light from one of these trackers, the IFM loses its distance reference. The operator then tracks the retroreflector to a known location to reset to a reference distance before continuing the measurement. A way around this limitation is to also provide an ADM in the tracker. The ADM can measure distance in a point-and-shoot manner, as described in more detail below. Some laser trackers contain only an ADM without an interferometer. U.S. Pat. No. 7,352,446 (&#39;446) to Bridges et al., the contents of which are herein incorporated by reference, describes a laser tracker having only an ADM (and no IFM) that is able to accurately scan a moving target. Prior to the &#39;446 patent, absolute distance meters were too slow to accurately find the position of a moving target. 
     A gimbal mechanism within the laser tracker may be used to direct the beam of light from the tracker to the SMR. Part of the light retroreflected by the SMR enters the laser tracker and passes onto a position detector. A control system within the laser tracker can use the position of the light on the position detector to adjust the rotation angles of the mechanical axes of the laser tracker to keep the laser beam centered on the SMR. In this way, the tracker is able to follow (track) an SMR that is moved over the surface of an object of interest. The gimbal mechanism used for a laser tracker may be used for a variety of other applications. As a simple example, the laser tracker may be used in a gimbal steering device having a visible pointer beam but no distance meter to steer a light beam to series of retroreflector targets and measure the angles of each of the targets. 
     Angle measuring devices such as angular encoders are attached to the mechanical axes of the tracker. The one distance measurement and two angle measurements performed by the laser tracker are sufficient to determine the three-dimensional location of the SMR. 
     Several laser trackers are available or have been proposed for measuring six, rather than the ordinary three, degrees of freedom. Such laser trackers combine measurement of three orientational degrees of freedom with measurement of three translational degrees of freedom to obtain measurement of six degrees of freedom. 
     A variety of methods have been used or proposed for measuring six degrees of freedom with a laser tracker. These methods usually include measuring three degrees of a retroreflector target by determining a distance and two angles to the retroreflector. In one approach, the three orientational degrees of freedom are determined by measuring the positions of points of light using a camera affixed to the laser tracker. In another approach, an inclinometer pendulum is used in combination with a “leaky” retroreflector to determine the three orientational degrees of freedom. In another approach, marks on a cube-corner retroreflector are imaged by a camera affixed to the laser tracker to determine the three orientational degrees of freedom. 
     Although each of these methods of measuring six degrees of freedom with are laser tracker are suitable for the intended purpose, each has certain shortcomings in terms of product cost and flexibility of operation. What is needed is a method of measuring six degrees of freedom with a laser tracker that overcomes these limitations. 
     SUMMARY OF THE INVENTION 
     According to one embodiment of the invention, a system includes: a dimensional measuring device having a light projecting unit, a steering unit, and a tracking unit, the light projecting unit configured to project a first beam of light having a first wavelength and a second beam of light having a second wavelength, the first beam of light being collinear with the first beam of light, the tracking unit configured to steer the first beam of light to a retroreflector; a remote probe having the retroreflector, a pitch/yaw sensor, and a body, the retroreflector and the pitch/yaw sensor coupled to the body, the pitch/yaw sensor including an aperture, a lens, and a position sensor, the pitch/yaw sensor configured to receive a first portion of the second beam of light through the aperture and to pass the first portion through the lens onto the position sensor, the position sensor being an optical detector configured to produce a first electrical signal indicative of position of received light on the detector; and a processor configured to determine a pitch angle and a yaw angle based at least in part on the first electrical signal. 
     According to another embodiment of the invention, a method is provided. The method comprising: A method comprising: providing a dimensional measuring device, a remote probe, and a processor, the dimensional measuring device configured to project a first beam of light having a first wavelength and a second beam of light having a second wavelength, the second beam of light being collinear to the first beam of light, the remote probe having a body coupled to a retroreflector and a pitch/yaw sensor, the pitch/yaw sensor having an aperture, a lens and an position sensor, the position sensor being a first optical detector configured to produce an electrical signal indicative of position of received light on the first optical detector; projecting the first beam of light onto the retroreflector and the second beam of light onto the pitch/yaw sensor; passing a portion of the second beam of light through the aperture and the lens onto the position sensor and generating a first electrical signal in response; and determining with the processor a pitch angle and a yaw angle of the remote probe based at least in part on the first electrical signal. 
     These and other advantages and features will become more apparent from the following description taken in conjunction with the drawings. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       Referring now to the drawings, exemplary embodiments are shown which should not be construed to be limiting regarding the entire scope of the disclosure, and wherein the elements are numbered alike in several FIGURES: 
         FIG. 1  is a perspective view of a laser tracker system with a retroreflector target in accordance with an embodiment of the present invention; 
         FIG. 2  is a perspective view of a laser tracker system with a six-DOF target in accordance with an embodiment of the present invention; 
         FIG. 3  is a block diagram describing elements of laser tracker optics and electronics in accordance with an embodiment of the present invention; 
         FIG. 4A  and  FIG. 4B  shows two types of prior art afocal beam expanders; 
         FIG. 5  shows a prior art fiber-optic beam launch; 
         FIG. 6  is a block diagram of electrical and electro-optical elements within a prior art ADM; 
         FIG. 7  is a schematic figure showing fiber-optic elements within a prior art fiber-optic network; 
         FIG. 8  is an exploded view of a prior art laser tracker; 
         FIG. 9  is a cross-sectional view of a prior art laser tracker; 
         FIG. 10  is a block diagram of the computing and communication elements of a laser tracker in accordance with an embodiment of the present invention; 
         FIG. 11  is a block diagram of elements in a laser tracker with six-DOF capability according to an embodiment of the present invention; 
         FIG. 12  is a schematic representation of a six-DOF tactile probe according to an embodiment; 
         FIG. 13  is a schematic representation of a six-DOF triangulation scanner according to an embodiment; 
         FIG. 14  is a schematic representation of a six-DOF triangulation scanner further including a tactile probe according to an embodiment; 
         FIG. 15  is a schematic representation of a six-DOF indicator according to an embodiment; 
         FIG. 16  is a schematic representation of a six-DOF projector according to an embodiment; 
         FIG. 17  is a schematic representation of a first type of six-DOF sensor according to an embodiment; 
         FIG. 18  is a schematic representation of a second type of six-DOF sensor according to an embodiment; 
         FIG. 19A  and  FIG. 19B  illustrate the geometry of a glass cube-corner retroreflector; 
         FIG. 20A  and  FIG. 20B  are a cross-section of the glass cube corner and a perspective view of the glass cube corner, respectively; 
         FIG. 21  is a drawing showing a method of evaluation optical reflections; 
         FIG. 22  is a cross sectional view of light passing through glass cube-corner cross sections in two opposing octants; 
         FIG. 23  is a diagram showing how a top view of front faces in two opposing octants indicates a reflected pattern; 
         FIG. 24  illustrate the effect of tilting a cube-corner retroreflector; 
         FIG. 25A  and  FIG. 25B  show a ring placed around the periphery of a cube-corner prism and the appearance of the ring when the retroreflector is tilted, respectively; 
         FIG. 26  illustrates the meaning of the term roll angle and illustrates two alternative ways of describing the other two orientational degrees of freedom: pitch/yaw angles and fold/tilt angles; 
         FIG. 27  illustrates a laser tracker and target cooperating to determine six degrees of freedom of the target by performing separate measurements of the roll angle and the pitch/yaw angles according to an embodiment; 
         FIGS. 28A, 28B  is a schematic drawing of a target that passes light to a position detector for determining pitch/yaw angles; 
         FIGS. 29A, 29B  show front and side views, respectively, that schematically illustrate determining pitch/yaw angles using three position detectors peripheral to the retroreflector according to an embodiment; 
         FIGS. 30A, 30B, and 30C  illustrate three types of position detectors that may be used according to an embodiment; 
         FIG. 31A  and  FIG. 31B  are front and side views, respectively, that schematically represent using a lens to focus light from an aperture onto a position detector to reduce noise according to an embodiment; 
         FIG. 32  illustrates using a lens to measure light through an aperture according to an embodiment; 
         FIG. 33A  and  FIG. 33B  are a schematic representation of a method of removing the influence of unwanted background light by modulating and demodulating light projected from the laser tracker according to an embodiment; 
         FIG. 34A  is a schematic representation showing a method of producing rotating linearly polarized light by sending polarized light through a rotating half-wave plate according to an embodiment; 
         FIG. 34B  is a schematic representation showing a method of producing rotating linearly polarized light by sending unpolarized (randomly polarized) light through a rotating polarizer according to an embodiment; 
         FIG. 35  is a schematic representation of a system that determines a reference phase of a rotating polarization in the laser tracker and a measurement phase of a rotating polarization in a probe according to an embodiment; 
         FIG. 36  is a schematic representation showing a method of determining phase by means of encoder synchronization and noise removal according to an embodiment; 
         FIG. 37  is a schematic representation of a light directly projected onto a polarizer or quarter wave plate rotated by a motor according to an embodiment; 
         FIG. 38  is a schematic representation showing the locking an optical modulation signal to a rotating disk to improve accuracy and remove noise according to an embodiment; 
         FIG. 39A  and  FIG. 39B  are schematic representations of an electrooptic modulator configured to produce rotating linearly polarized light according to an embodiment; 
         FIG. 40  is a schematic representation of a system that applies synchronous modulation and demodulation to rotating polarized light according to an embodiment; 
         FIG. 41A  and  FIG. 41B  are schematic representations of rotating linearly polarized light passing through a cube corner prism and polarizer onto a detector to determine roll angle of the prism according to an embodiment; 
         FIG. 42A  and  FIG. 42B  show a cube corner retroreflector surrounded by three polarization-roll detectors used to measure roll angle according to an embodiment; 
         FIG. 43A  is a schematic representation of a retroreflector surrounded by three roll sensors, three polarization sensors, and a modulation detector according to an embodiment; 
         FIG. 43B  and  FIG. 43C  are cross-sectional schematic representations of the apparatus of  FIG. 43A  having tilted pitch/yaw and polarization roll sensors, respectively, according to an embodiment; 
         FIG. 43D  shows overlap among the angular coverage of three tilted sensors according to an embodiment; 
         FIG. 44  illustrates three sensors, either polarization roll sensors or pitch/yaw sensors surrounding a retroreflector combined with the complementary pitch/yaw sensor or polarization roll sensor positioned to capture light passed through the retroreflector according to an embodiment; 
         FIG. 45A  and  FIG. 45B  illustrates a retroreflector assembly configured to pass light to a single position sensor configured to determine roll using polarization-roll detection and pitch/yaw using position on the position sensor according to an embodiment; 
         FIG. 46  is an exploded, orthographic view of a six-DOF tactile probe according to an embodiment; 
         FIG. 47  is an exploded, orthographic view of components internal to a six-DOF laser tracker according to an embodiment; 
         FIG. 48A  and  FIG. 48B  are top and cross-sectional views, respectively, of components internal to a six-DOF laser tracker according to an embodiment; 
         FIG. 49A  and  FIG. 49B  are top and side views, respectively, of schematic representations of anamorphic prisms included in components internal to a six-DOF laser tracker according to an embodiment; 
         FIGS. 50A, 50B, and 50C  are schematic representations of a six-DOF retroreflector assembly that combines a cube beam splitter with a cube-corner retroreflector made of high-index glass to measure roll and pitch/yaw angles according to an embodiment; 
         FIG. 51  is a cross-sectional view of a schematic representation of an assembly that includes a retroreflector surrounded by a lens ring that projects light onto a position sensor according to an embodiment; 
         FIG. 52  shows the pattern formed on the sensor for the assembly of  FIG. 51  tilted to different angles according to an embodiment; 
         FIG. 53A  illustrates the separation produced in reflected light for two different wavelengths of light incident on a retroreflector according to an embodiment; 
         FIG. 53B  illustrates a mathematically equivalent path followed by the two different wavelengths of light; 
         FIG. 54A  is a graph of the index of refraction of zinc sulfide as a function of wavelength; 
         FIG. 54B  is a graph of lateral separation of the return beams of two different wavelengths of light for zinc sulfide material according to an embodiment; 
         FIG. 55  is an assembly within a six-DOF laser tracker for measuring lateral separation of the return beams according to an embodiment; 
         FIG. 56  illustrates a method for imaging a light source to determine roll angle when combined with information provided by a pitch/yaw sensor according to an embodiment; 
         FIG. 57  illustrates a method of using a single six-DOF sensor assembly in combination with a tactile probe attached to the assembly with two rotating mechanisms and angular encoders according to an embodiment; 
         FIG. 58  illustrates a method of using a six-DOF sensor assembly in combination with a triangulation scanner attached to the assembly with two rotating mechanisms and angular encoders according to an embodiment; 
         FIG. 59A  is a schematic representation of a retroreflector  5910  in combination with a roll sensor  5920 A according to an embodiment; 
         FIGS. 59B-59F  are schematic representations of roll sensors, each including linear polarizers in a prescribed arrangement according to embodiments; 
         FIG. 60  is a cross-sectional representation of optical elements used to launch beams of light from a laser tracker according to an embodiment; 
         FIG. 61A  and  FIG. 61B  are top and side views, respectively, of elements in a secondary optical path of a laser tracker according to an embodiment; 
         FIG. 62A  and  FIG. 62B  are front views of six-DOF sensors that include a retroreflector and roll sensors according to an embodiment; 
         FIG. 62C  is a cross-sectional view of a roll-sensor component according to an embodiment; 
         FIG. 63A  is a schematic illustration of electro-optical components in a secondary optical path of a laser tracker according to an embodiment; 
         FIG. 63B  is another embodiment of optical and electrical components in the roll sensors of a six-DOF device according to an embodiment; 
         FIG. 63C  is a block diagram showing components of a synchronous demodulator according to an embodiment; 
         FIGS. 64A, 64B  are electrical signals produced by a first roll sensor and a second roll sensor according to an embodiment; 
         FIG. 65A  illustrates s- and p-polarizations of incident and transmitted light at an air-glass interface; and 
         FIG. 65B  shows two directions of polarizations of a polarizer at a plane of incidence at an air-polarizer interface. 
     
    
    
     DETAILED DESCRIPTION 
     An exemplary laser tracker system  5  illustrated in  FIG. 1  includes a laser tracker  10 , a retroreflector target  26 , an optional auxiliary unit processor  50 , and an optional auxiliary computer  60 . An exemplary gimbaled beam-steering mechanism  12  of laser tracker  10  comprises a zenith carriage  14  mounted on an azimuth base  16  and rotated about an azimuth axis  20 . A payload  15  is mounted on the zenith carriage  14  and rotated about a zenith axis  18 . Zenith axis  18  and azimuth axis  20  intersect orthogonally, internally to tracker  10 , at gimbal point  22 , which is typically the origin for distance measurements. A beam of light  46  virtually passes through the gimbal point  22  and is pointed orthogonal to zenith axis  18 . In other words, beam of light  46  lies in a plane approximately perpendicular to the zenith axis  18  and that passes through the azimuth axis  20 . Outgoing beam of light  46  is pointed in the desired direction by rotation of payload  15  about zenith axis  18  and by rotation of zenith carriage  14  about azimuth axis  20 . Motors, described in more detail in reference to  FIGS. 8 and 9 , steer the outgoing light beam by rotating tracker components about the azimuth and zenith axes. A zenith angular encoder, internal to the tracker, is attached to a zenith mechanical axis aligned to the zenith axis  18 . An azimuth angular encoder, internal to the tracker, is attached to an azimuth mechanical axis aligned to the azimuth axis  20 . The zenith and azimuth angular encoders measure the zenith and azimuth angles of rotation to relatively high accuracy. Outgoing beam of light  46  travels to the retroreflector target  26 , which might be, for example, an SMR as described above. By measuring the radial distance between gimbal point  22  and retroreflector  26 , the rotation angle about the zenith axis  18 , and the rotation angle about the azimuth axis  20 , the position of retroreflector  26  is found within the spherical coordinate system of the tracker. 
     Outgoing beam of light  46  may include one or more wavelengths, as described hereinafter. For the sake of clarity and simplicity, a steering mechanism of the sort shown in  FIG. 1  is assumed in the following discussion. However, other types of steering mechanisms are possible. For example, it is possible to reflect a laser beam off a mirror rotated about the azimuth and zenith axes. The techniques described herein are applicable, regardless of the type of steering mechanism. 
     Magnetic nests  17  may be included on the laser tracker for resetting the laser tracker to a “home” position for different sized SMRs—for example, 1.5, ⅞, and ½ inch SMRs. An on-tracker retroreflector  19  may be used to reset the tracker to a reference distance. In addition, an on-tracker mirror, not visible from the view of  FIG. 1 , may be used in combination with the on-tracker retroreflector to enable performance of a self-compensation, as described in U.S. Pat. No. 7,327,446, the contents of which are incorporated by reference. 
       FIG. 2  shows an exemplary laser tracker system  7  that is similar to the laser tracker system  5  of  FIG. 1  except that retroreflector target  26  is replaced with a six-DOF probe  1000 . In  FIG. 1 , other types of retroreflector targets may be used. For example, a cateye retroreflector, which is a glass retroreflector in which light focuses to a small spot of light on a reflective rear surface of the glass structure, is sometimes used. 
     In an embodiment illustrated in  FIG. 3 , a laser tracker includes an opto-electronic system  500  that emits a single wavelength of visible light form the tracker. Other embodiments, including some embodiments shown herein below, emit multiple wavelengths of light, some outside the visible spectrum. In an embodiment, the opto-electronic system  500  includes a visible light source  110 , an isolator  115 , a fiber network  420 , ADM electronics  530 , a fiber launch  170 , a beam splitter  145 , and a position detector assembly  150 . The visible light source  110  might be, for example, a red or green diode laser, a vertical cavity surface emitting laser (VCSEL), or a superluminescent diode. The isolator might be a Faraday isolator, an attenuator, or any other device capable of sufficiently reducing the amount of light fed back into the light source to prevent instability in the light source  110 . The light from the isolator  115  travels into the fiber network  420 , which in an embodiment is the fiber network  420 A of  FIG. 7 . 
     In the configuration illustrated in  FIG. 3 , a lens  176  in the fiber launch  170  of  FIG. 5  is used to collimate light  250  emitted by the fiber optic ferrule  174  to produce collimated light  252 . In other embodiments, the light emitted from a fiber launch or emitted directly from a light source may be collimated by a lens and then sent later in the path through a beam expander. Examples of two types of beam expanders  140 A and  140 B are shown in  FIGS. 4A and 4B , respectively.  FIG. 4A  illustrates a beam expander  140 A in which a negative lens  141 A is followed by a positive lens  142 A arranged so as to expand a relatively small collimated beam  220 A of light into a relatively larger beam of light  230 A.  FIG. 4B  illustrates a beam expander  140 B that includes two positive lenses  141 B,  142 B. The two positive lenses are arranged so as to expand a relatively small collimated beam of light  220 B into a relatively larger beam of light  230 B. 
     In an embodiment, the ADM  160  includes a light source  110 , ADM electronics  530 , a fiber network  420 , an interconnecting electrical cable  532 , and interconnecting optical fibers  422 ,  423 , and  424 . ADM electronics send electrical modulation and bias voltages to light source  110 , which may, for example, be a semiconductor laser that operates at 635 nm. In an embodiment, the fiber network  420  may be the prior art fiber-optic network  420 A shown in  FIG. 7 . In this embodiment, light from the light source  110  in  FIG. 3  travels over the optical fiber  580 , which is equivalent to the optical fiber  432  in  FIG. 7 . 
     In an embodiment, the fiber network  420 A of  FIG. 7  includes a first fiber coupler  430 , a second fiber coupler  436 , and low-reflectance terminators  435 ,  440 . It should be understood that many types of fiber networks may be constructed and the fiber network  420  is not limited to the fiber network  420 A of  FIG. 7 . The light in fiber network  420 A travels through the first fiber coupler  430  and splits between two paths, the first path through optical fiber  433  to the second fiber coupler  436  and the second path through optical fiber  422  and fiber length equalizer  423 . Fiber length equalizer  423  connects to fiber length  422  in  FIG. 3 , which travels to the reference channel of the ADM electronics  530 . The purpose of fiber length equalizer  423  is to match the length of optical fibers traversed by light in the reference channel to the length of optical fibers traversed by light in the measure channel. Matching the fiber lengths in this way reduces ADM errors caused by changes in the ambient temperature. Such errors may arise because the effective optical path length of an optical fiber is equal to the average index of refraction of the optical fiber times the length of the fiber. Since the index of refraction of the optical fibers depends on the temperature of the fiber, a change in the temperature of the optical fibers causes changes in the effective optical path lengths of the measure and reference channels. If the effective optical path length of the optical fiber in the measure channel changes relative to the effective optical path length of the optical fiber in the reference channel, the result will be an apparent shift in the position of the retroreflector target  90 , even if the retroreflector target  90  is kept stationary. To get around this problem, two steps are performed. First, the length of the fiber in the reference channel is matched, as nearly as possible, to the length of the fiber in the measure channel. Second, the measure and reference fibers are routed side by side to the extent possible to ensure that the optical fibers in the two channels see nearly the same changes in temperature. 
     In  FIG. 7 , some light travels through the second fiber optic coupler  436  and splits into two paths, the first path to the low-reflection fiber terminator  440  and the second path to optical fiber  438 , from which it travels to optical fiber  186  in  FIG. 3 . The light on optical fiber  186  travels through to the second fiber launch  170 . 
     In an embodiment, fiber launch  170  is shown in prior art  FIG. 5 . The light from optical fiber  438  of  FIG. 7  goes to fiber  172  in  FIG. 5 . The fiber launch  170  includes optical fiber  172 , ferrule  174 , and lens  176 . The optical fiber  172  is attached to ferrule  174 , which is stably attached to a structure within the laser tracker  10 . If desired, the end of the optical fiber may be polished at an angle to reduce back reflections. The light  250  emerges from the core of the fiber, which may be a single mode optical fiber with a diameter of between 4 and 12 micrometers, depending on the wavelength of the light being used and the particular type of optical fiber. The light  250  diverges at an angle and intercepts lens  176 , which collimates it. 
     The beam of light  584  travels out of the laser tracker to retroreflector  90  as a first beam, which returns a second portion of the light as a second beam  586 . The second portion of the light returns to the fiber launch  170 , which couples the light back into the optical fiber  172 . 
     In an embodiment, the optical fiber  172  corresponds to the optical fiber  438  in  FIG. 7 . The returning light travels from optical fiber  438  through the second fiber coupler  436  and splits between two paths. A first path leads to optical fiber  424  that, in an embodiment, leads to the measure channel of the ADM electronics  530  in  FIG. 3 . A second path leads to optical fiber  433  and then to the first fiber coupler  430 . The light leaving the first fiber coupler  430  splits between two paths, a first path to the optical fiber  432  and a second path to the low reflectance termination  435 . In an embodiment, optical fiber  432  in  FIG. 7  leads through an isolator  115  to the light source  110  in  FIG. 3 . 
     The light from the fiber network  420  enters ADM electronics  530  through optical fibers  168 ,  169 . An embodiment of prior art ADM electronics is shown in  FIG. 6  Optical fiber  168  in  FIG. 3  corresponds to optical fiber  3232  in  FIG. 6 , and optical fiber  169  in  FIG. 3  corresponds to optical fiber  3230  in  FIG. 6 . Referring now to  FIG. 6 , ADM electronics  3300  includes a frequency reference  3302 , a synthesizer  3304 , a measure detector  3306 , a reference detector  3308 , a measure mixer  3310 , a reference mixer  3312 , conditioning electronics  3314 ,  3316 ,  3318 ,  3320 , a divide-by-N prescaler  3324 , and an analog-to-digital converter (ADC)  3322 . The frequency reference, which might be an oven-controlled crystal oscillator (OCXO), for example, sends a reference frequency f REF , which might be 10 MHz, for example, to the synthesizer, which generates two electrical signals—one signal at a frequency f RF  and two signals at frequency f LO . The signal f RF  goes to the light source  3102 , which corresponds to the light source  162  in  FIG. 3 . The two signals at frequency f LO  go to the measure mixer  3310  and the reference mixer  3312 . The light from optical fibers  168 ,  169  in  FIG. 3  appear on fibers  3232 ,  3230  in  FIG. 6 , respectively, and enter the reference and measure channels, respectively. Reference detector  3308  and measure detector  3306  convert the optical signals into electrical signals. These signals are conditioned by electrical components  3316 ,  3314 , respectively, and are sent to mixers  3312 ,  3310 , respectively. The mixers produce a frequency f IF  equal to the absolute value of f LO −f RF . The signal f RF  may be a relatively high frequency, for example, 2 GHz, while the signal f IF  may have a relatively low frequency, for example, 10 kHz. 
     The reference frequency f REF  is sent to the prescaler  3324 , which divides the frequency by an integer value. For example, a frequency of 10 MHz might be divided by 40 to obtain an output frequency of 250 kHz. In this example, the 10 kHz signals entering the ADC  3322  would be sampled at a rate of 250 kHz, thereby producing 25 samples per cycle. The signals from the ADC  3322  are sent to a data processor  3400 , which might, for example, be one or more digital signal processor (DSP) units located in ADM electronics  530  of  FIG. 3 . 
     The method for extracting a distance is based on the calculation of phase of the ADC signals for the reference and measure channels. This method is described in detail in U.S. Pat. No. 7,701,559 (&#39;559) to Bridges et al., the contents of which are herein incorporated by reference. Calculation includes use of equations (1)-(8) of patent &#39;559. In addition, when the ADM first begins to measure a retroreflector, the frequencies generated by the synthesizer are changed some number of times (for example, three times), and the possible ADM distances calculated in each case. By comparing the possible ADM distances for each of the selected frequencies, an ambiguity in the ADM measurement is removed. The equations (1)-(8) of patent &#39;559 combined with synchronization methods described with respect to FIG. 5 of patent &#39;559 and the Kalman filter methods described in patent &#39;559 enable the ADM to measure a moving target. In other embodiments, other methods of obtaining absolute distance measurements, for example, by using pulsed time-of-flight rather than phase differences, may be used. 
     Referring to  FIG. 3 , in an embodiment the return light beam  586  arrives at the beam splitter  145 , which sends part of the return light  582  to the fiber launch  170  and another part of the light  588  to the position detector assembly  150 . The light  584  emerging from the laser tracker  10  may be referred to as a first beam and the portion of that light  586  reflecting off the retroreflector  90  as a second beam. Portions of the second beam are sent to different functional elements of the opto-electronic system  500 . For example, a first portion may be sent to a distance meter such as an ADM of  FIG. 3 . A second portion  588  may be sent to a position detector assembly  150 . In the example of  FIG. 3 , the beam splitter  145  reflects the light  588  onto the position detector assembly and transmits the light  581  into the fiber launch  170  and ADM. However, the function of the beam splitter could have been reversed, transmitting light to the position detector assembly and reflecting it to the fiber launch and ADM. 
     The position detector assembly  150  includes a position detector  151  and optional conditioning elements  158 . In one embodiment, the position detector is an analog detector designed to indicate position according to the location the return beam of light strikes the detector surface. Examples of such analog detectors include lateral effect detectors and quadrant detectors. In another embodiment, the position detector is a photosensitive array designed to indicate position by calculating a central position of the return light. Such a calculation might be based, for example, on calculating with a processor a centroid value of the return light. Such a photosensitive array might be, for example, a high speed CMOS array. In an embodiment, the position detector further includes one or more conditioning elements such as an aperture or a diffusor. An aperture  158  may be used to help block unwanted ghost beams. A diffuser may help improve uniformity of the beam, for example, to reduce speckle and diffraction effects. 
     A retroreflector of the sort discussed here, a cube corner or a cateye retroreflector, for example, has the property of reflecting a ray of light that enters the retroreflector in a direction parallel to the incident ray. In addition, the incident and reflected rays are symmetrically placed about the point of symmetry of the retroreflector. For example, in an open-air cube corner retroreflector, the point of symmetry of the retroreflector is the vertex of the cube corner. In a glass cube corner retroreflector, the point of symmetry is also the vertex, but one may consider the bending of the light at the glass-air interface in this case. In a cateye retroreflector having an index of refraction of 2.0, the point of symmetry is the center of the sphere. In a cateye retroreflector made of two glass hemispheres symmetrically seated on a common plane, the point of symmetry is a point lying on the plane and at the spherical center of each hemisphere. For the type of retroreflectors ordinarily used with laser trackers, the light returned by a retroreflector to the tracker is shifted to the other side of the vertex relative to the incident laser beam. 
     This behavior of a retroreflector  90  in  FIG. 3  is the basis for the tracking of the retroreflector by the laser tracker. The position sensor has on its surface an ideal retrace point. The ideal retrace point is the point at which a laser beam sent to the point of symmetry of a retroreflector (e.g., the vertex of the cube corner retroreflector in an SMR) will return. Usually the retrace point is near the center of the position sensor. If the laser beam is sent to one side of the retroreflector, it reflects back on the other side and appears off the retrace point on the position sensor. By noting the position of the returning beam of light on the position sensor, the control system of the laser tracker  10  can cause the motors to move the light beam toward the point of symmetry of the retroreflector. 
     If the retroreflector moves transverse to the tracker light at a constant velocity, the light beam at the retroreflector will strike the retroreflector (after transients have settled) a fixed offset distance from the point of symmetry of the retroreflector. The laser tracker makes a correction to account for this offset distance at the retroreflector based on scale factor obtained from controlled measurements. The position detector performs two functions—enabling tracking and correcting measurements to account for the movement of the retroreflector. 
       FIGS. 8 and 9  show exploded and cross sectional views, respectively, of a prior art laser tracker  2100 , which is depicted in FIGS. 2 and 3 of U.S. Pat. No. 8,525,983 to Bridges et al., which is incorporated by reference herein. Azimuth assembly  2110  includes post housing  2112 , azimuth encoder assembly  2120 , lower and upper azimuth bearings  2114 A,  2114 B, azimuth motor assembly  2125 , azimuth slip ring assembly  2130 , and azimuth circuit boards  2135 . 
     The purpose of azimuth encoder assembly  2120  is to accurately measure the angle of rotation of yoke  2142  with respect to the post housing  2112 . Azimuth encoder assembly  2120  includes encoder disk  2121  and read-head assembly  2122 . Encoder disk  2121  is attached to the shaft of yoke housing  2142 , and read head assembly  2122  is attached to post assembly  2110 . Read head assembly  2122  comprises a circuit board onto which one or more read heads are fastened. Laser light sent from read heads reflect off fine grating lines on encoder disk  2121 . Reflected light picked up by detectors on encoder read head(s) is processed to find the angle of the rotating encoder disk in relation to the fixed read heads. 
     Azimuth motor assembly  2125  includes azimuth motor rotor  2126  and azimuth motor stator  2127 . Azimuth motor rotor comprises permanent magnets attached directly to the shaft of yoke housing  2142 . Azimuth motor stator  2127  comprises field windings that generate a prescribed magnetic field. This magnetic field interacts with the magnets of azimuth motor rotor  2126  to produce the desired rotary motion. Azimuth motor stator  2127  is attached to post frame  2112 . 
     Azimuth circuit boards  2135  represent one or more circuit boards that provide electrical functions required by azimuth components such as the encoder and motor. Azimuth slip ring assembly  2130  includes outer part  2131  and inner part  2132 . In an embodiment, wire bundle  2138  emerges from auxiliary unit processor  50 . Wire bundle  2138  may carry power to the tracker or signals to and from the tracker. Some of the wires of wire bundle  2138  may be directed to connectors on circuit boards. In the example shown in  FIG. 9 , wires are routed to azimuth circuit board  2135 , encoder read head assembly  2122 , and azimuth motor assembly  2125 . Other wires are routed to inner part  2132  of slip ring assembly  2130 . Inner part  2132  is attached to post assembly  2110  and consequently remains stationary. Outer part  2131  is attached to yoke assembly  2140  and consequently rotates with respect to inner part  2132 . Slip ring assembly  2130  is designed to permit low impedance electrical contact as outer part  2131  rotates with respect to the inner part  2132 . 
     Zenith assembly  2140  comprises yoke housing  2142 , zenith encoder assembly  2150 , left and right zenith bearings  2144 A,  2144 B, zenith motor assembly  2155 , zenith slip ring assembly  2160 , and zenith circuit board  2165 . 
     The zenith encoder assembly  2150  accurately measures the angle of rotation of payload frame  2172  with respect to yoke housing  2142 . Zenith encoder assembly  2150  comprises zenith encoder disk  2151  and zenith read-head assembly  2152 . Encoder disk  2151  is attached to payload housing  2142 , and read head assembly  2152  is attached to yoke housing  2142 . Zenith read head assembly  2152  comprises a circuit board onto which one or more read heads are fastened. Laser light sent from read heads reflect off fine grating lines on encoder disk  2151 . Reflected light picked up by detectors on encoder read head(s) is processed to find the angle of the rotating encoder disk in relation to the fixed read heads. 
     Zenith motor assembly  2155  comprises azimuth motor rotor  2156  and azimuth motor stator  2157 . Zenith motor rotor  2156  comprises permanent magnets attached directly to the shaft of payload frame  2172 . Zenith motor stator  2157  comprises field windings that generate a prescribed magnetic field. This magnetic field interacts with the rotor magnets to produce the desired rotary motion. Zenith motor stator  2157  is attached to yoke frame  2142 . 
     Zenith circuit board  2165  represents one or more circuit boards that provide electrical functions required by zenith components such as the encoder and motor. Zenith slip ring assembly  2160  comprises outer part  2161  and inner part  2162 . Wire bundle  2168  emerges from azimuth outer slip ring  2131  and may carry power or signals. Some of the wires of wire bundle  2168  may be directed to connectors on circuit board. In the example shown in  FIG. 9 , wires are routed to zenith circuit board  2165 , zenith motor assembly  2150 , and encoder read head assembly  2152 . Other wires are routed to inner part  2162  of slip ring assembly  2160 . Inner part  2162  is attached to yoke frame  2142  and consequently rotates in azimuth angle, but not in zenith angle. Outer part  2161  is attached to payload frame  2172  and consequently rotates in both zenith and azimuth angles. Slip ring assembly  2160  is designed to permit low impedance electrical contact as outer part  2161  rotates with respect to the inner part  2162 . Payload assembly  2170  includes a main optics assembly  2180  and a secondary optics assembly  2190 . 
       FIG. 10  is a block diagram depicting a dimensional measurement electronics processing system  1500  that includes a laser tracker electronics processing system  1510 , processing systems of peripheral elements  1582 ,  1584 ,  1586 , computer  1590 , and other networked components  1600 , represented here as a cloud. Exemplary laser tracker electronics processing system  1510  includes a master processor  1520 , payload functions electronics  1530 , azimuth encoder electronics  1540 , zenith encoder electronics  1550 , display and user interface (UI) electronics  1560 , removable storage hardware  1565 , radio frequency identification (RFID) electronics, and an antenna  1572 . The payload functions electronics  1530  includes a number of subfunctions including the six-DOF electronics  1531 , the camera electronics  1532 , the ADM electronics  1533 , the position detector (PSD) electronics  1534 , and the level electronics  1535 . Most of the subfunctions have at least one processor unit, which might be a digital signal processor (DSP) or field programmable gate array (FPGA), for example. The electronics units  1530 ,  1540 , and  1550  are separated as shown because of their location within the laser tracker. In an embodiment, the payload functions  1530  are located in the payload  2170  of  FIGS. 8, 9 , while the azimuth encoder electronics  1540  is located in the azimuth assembly  2110  and the zenith encoder electronics  1550  is located in the zenith assembly  2140 . 
     Many types of peripheral devices are possible, but here three such devices are shown: a temperature sensor  1582 , a six-DOF probe  1584 , and a personal digital assistant,  1586 , which might be a smart phone, for example. The laser tracker may communicate with peripheral devices in a variety of means, including wireless communication over the antenna  1572 , by means of a vision system such as a camera, and by means of distance and angular readings of the laser tracker to a cooperative target such as the six-DOF probe  1584 . Peripheral devices may contain processors. The six-DOF accessories may include six-DOF probing systems, six-DOF scanners, six-DOF projectors, six-DOF sensors, and six-DOF indicators. The processors in these six-DOF devices may be used in conjunction with processing devices in the laser tracker as well as an external computer and cloud processing resources. Generally, when the term laser tracker processor or measurement device processor is used, it is meant to include possible external computer and cloud support. 
     In an embodiment, a separate communications bus goes from the master processor  1520  to each of the electronics units  1530 ,  1540 ,  1550 ,  1560 ,  1565 , and  1570 . Each communications line may have, for example, three serial lines that include the data line, clock line, and frame line. The frame line indicates whether or not the electronics unit should pay attention to the clock line. If it indicates that attention should be given, the electronics unit reads the current value of the data line at each clock signal. The clock-signal may correspond, for example, to a rising edge of a clock pulse. In an embodiment, information is transmitted over the data line in the form of a packet. In an embodiment, each packet includes an address, a numeric value, a data message, and a checksum. The address indicates where, within the electronics unit, the data message is to be directed. The location may, for example, correspond to a processor subroutine within the electronics unit. The numeric value indicates the length of the data message. The data message contains data or instructions for the electronics unit to carry out. The checksum is a numeric value that is used to minimize the chance that errors are transmitted over the communications line. 
     In an embodiment, the master processor  1520  sends packets of information over bus  1610  to payload functions electronics  1530 , over bus  1611  to azimuth encoder electronics  1540 , over bus  1612  to zenith encoder electronics  1550 , over bus  1613  to display and UI electronics  1560 , over bus  1614  to removable storage hardware  1565 , and over bus  1616  to RFID and wireless electronics  1570 . 
     In an embodiment, master processor  1520  also sends a synch (synchronization) pulse over the synch bus  1630  to each of the electronics units at the same time. The synch pulse provides a way of synchronizing values collected by the measurement functions of the laser tracker. For example, the azimuth encoder electronics  1540  and the zenith electronics  1550  latch their encoder values as soon as the synch pulse is received. Similarly, the payload functions electronics  1530  latch the data collected by the electronics contained within the payload. The six-DOF, ADM, and position detector all latch data when the synch pulse is given. In most cases, the camera and inclinometer collect data at a slower rate than the synch pulse rate but may latch data at multiples of the synch pulse period. 
     The azimuth encoder electronics  1540  and zenith encoder electronics  1550  are separated from one another and from the payload electronics  1530  by the slip rings  2130 ,  2160  shown in  FIG. 8  and  FIG. 9 . As such, the bus lines  1610 ,  1611 , and  1612  are depicted as separate bus line in  FIG. 10 . 
     The laser tracker electronics processing system  1510  may communicate with an external computer  1590 , or it may provide computation, display, and UI functions within the laser tracker. The laser tracker communicates with computer  1590  over communications link  1606 , which might be, for example, an Ethernet line or a wireless connection. The laser tracker may also communicate with other elements  1600 , represented by the cloud, over communications link  1602 , which might include one or more electrical cables, such as Ethernet cables, and one or more wireless connections. An example of an element  1600  is another three dimensional test instrument—for example, an articulated arm CMM, which may be relocated by the laser tracker. A communication link  1604  between the computer  1590  and the elements  1600  may be wired (e.g., Ethernet) or wireless. An operator sitting on a remote computer  1590  may make a connection to the Internet, represented by the cloud  1600 , over an Ethernet or wireless line, which in turn connects to the master processor  1520  over an Ethernet or wireless line. In this way, a user may control the action of a remote laser tracker. 
       FIG. 11  is a block diagram of a six-DOF laser tracker  1100  used in combination with a six-DOF target  2000 . In an embodiment, the six-DOF laser tracker  1100  includes the elements of  FIG. 3  and in addition includes optional tracker six-DOF elements  1101  needed to measure three orientational degrees of freedom of the six-DOF target  2000 . In an embodiment, the six-DOF target  2000  includes optional target six-DOF elements  91  in addition to a retroreflector  90 . The elements  91  may assist in determining three orientation degrees of freedom of the six-DOF target  2000  and in providing local probing capability—for example, tactile measurement capability, triangulation scanner capability (line scanner or area scanner), indicator capability, projector capability, or sensor capability. 
       FIG. 12  shows an embodiment of a six-DOF tactile probe  2000  used in conjunction with laser tracker  1100 . The six-DOF probe  2000 A includes a body  2014 , one or more six-DOF assemblies  2002 , a probe extension assembly  2050 , an optional electrical cable  2046 , an optional battery  2044 , an interface component  2012 , an identifier element  2049 , actuator buttons  2016 , an antenna  2048 , and an electronics circuit board  2042 . Each six-DOF assembly  2002  includes a retroreflector, which may be a cube corner retroreflector with a hollow (air) core or a glass core. Several embodiments of six-DOF assemblies  2002  are described herein below. The retroreflector may be a cube-corner retroreflector, a cateye retroreflector, or any other type of retroreflector. The probe extension assembly  2050  includes a probe extension  2052  and a probe tip  2054 . The probe tip  2054  is brought into contact with the object under test. Although the probe tip  2054  is separated from the retroreflector, it is possible for a six-DOF laser tracker to determine the three dimensional coordinates of the probe tip  2054  at a point hidden from the line of sight of the light beam  784  from the laser tracker. This is why a six-DOF tactile probe is sometimes referred to as a hidden-point probe. 
     In an embodiment, electric power may be provided over an optional electrical cable  2046  or by an optional battery  2044 . Electric power is provided to an electronics circuit board  2042  and to the antenna  2048 , which may communicate with the laser tracker or an external computer, and to actuator buttons  2016 . The actuator buttons  2016  provide the user a convenient way of communicating with the laser tracker or external computer. Electric power may also be provided to a light emitting diode (LED), a material temperature sensor (not shown), an air temperature sensor (not shown), an inertial sensor (not shown) or inclinometer (not shown). The interface component  2012  may be, for example, a light source (such as an LED), a small retroreflector, a region of reflective material, or a reference mark. The optional interface component  2012  may be used to establish a coarse orientation of the six-DOF assemblies. The identifier element  2049  is used to provide the laser tracker with parameters or with a serial number for the six-DOF probe. The identifier element may be, for example, a bar code, radio frequency identification (RFID) tag, or near-field communication (NFC) tag. 
     In an embodiment, the six-DOF laser tracker  1100  may provide the light beam  784  to any of a plurality of retroreflectors within a corresponding plurality of six-DOF assemblies  2002 . By providing the light beam  784  to any of a plurality of retroreflectors, the six-DOF probe  2000 A may be oriented in a wide variety of directions while probing with the probing extension assembly  2050 . 
     The six degrees of freedom measured by the laser tracker  1100  may be considered to include three translational degrees of freedom and three orientational degrees of freedom. The three translational degrees of freedom may include a radial distance measurement, a first angular measurement, and a second angular measurement. The radial distance measurement may be made with an IFM or an ADM. The first angular measurement may be made with an azimuth angular measurement device, such as an azimuth angular encoder, and the second angular measurement made with a zenith angular measurement device. In another embodiment, the first angular measurement device may be the zenith angular measurement device and the second angular measurement device may be the azimuth angular measurement device. The radial distance, first angular measurement, and second angular measurement constitute three coordinates in a spherical coordinate system, which can be transformed into three coordinates in a Cartesian coordinate system or another coordinate system. 
     The three orientational degrees of freedom may be determined using a variety of methods, as described herein below. The three translational degrees of freedom and the three orientational degrees of freedom fully define the position and orientation of the six-DOF assembly  2002  in space. It is possible to have systems in which the six degrees of freedom are not independent, but the term six degrees of freedom, as used herein, means that the six degrees of freedom are sufficient to fully define the position and orientation of the six-DOF assemblies  2002  in space. Similarly this information is sufficient to determine the position and orientation of the six-DOF probes  2000 A,  2000 B,  2000 C,  2000 D,  2000 E,  2000 F, and  2000 G in  FIGS. 12, 13, 14, 15, 16, 17, and 18 , respectively. The term “translational set” means three degrees of translational freedom of a six-DOF accessory (such as a six-DOF probe) in the tracker frame-of-reference (or device frame of reference). The term “orientational set” means three orientational degrees of freedom of a six-DOF accessory in a tracker frame of reference. The term “surface set” means three-dimensional coordinates of a point on the object surface in a device frame of reference. 
       FIG. 13  shows an embodiment of a six-DOF triangulation scanner  2000 B used in conjunction with six-DOF tracker  1100 . The six-DOF scanner  2000 B may also be referred to as a “target scanner.” The six-DOF scanner  2000 B includes a body  2514 , one or more six-DOF assemblies  2002 , a scanner camera  2530 , a scanner light projector  2520 , an optional electrical cable  2546 , an optional battery  2444 , an interface component  2512 , an identifier element  2549 , actuator buttons  2516 , an antenna  2548 , and an electronics circuit board  2542 . Each six-DOF assembly  2002  includes a retroreflector. The six-DOF assembly  2002 , the optional electrical cable  2546 , the optional battery  2544 , the interface component  2512 , the identifier element  2549 , the actuator buttons  2516 , the antenna  2548 , and the electronics circuit board  2542  in  FIG. 13  correspond to the six-DOF assemblies  2002 , the optional electrical cable  2046 , the optional battery  2044 , the interface component  2012 , the identifier element  2049 , actuator buttons  2016 , the antenna  2048 , and the electronics circuit board  2042 , respectively, in  FIG. 12 . The descriptions for these corresponding elements are the same as discussed in reference to  FIG. 12 . Together, the scanner projector  2520  and the scanner camera  2530  are used to measure the three dimensional coordinates of a workpiece  2528 . The camera  2530  includes a camera lens system  2532  and a photosensitive array  2534 . The photosensitive array  2534  may be a CCD or CMOS array, for example. The scanner projector  2520  includes a projector lens system  2523  and a source pattern of light  2524 . The source pattern of light may emit a point of light, a line of light, or a structured (two dimensional) pattern of light. If the scanner light source emits a point of light, the point may be scanned, for example, with a moving mirror, to produce a line or an array of lines. If the scanner light source emits a line of light, the line may be scanned, for example, with a moving mirror, to produce an array of lines. In an embodiment, the source pattern of light might be an LED, laser, or other light source reflected off a digital micromirror device (DMD) such as a digital light projector (DLP) from Texas Instruments, an liquid crystal device (LCD) or liquid crystal on silicon (LCOS) device, or it may be a similar device used in transmission mode rather than reflection mode. The source pattern of light might also be a slide pattern, for example, a chrome-on-glass slide, which might have a single pattern or multiple patterns, the slides moved in and out of position as needed. 
     The six-DOF scanner  2000 B may be held by hand or mounted, for example, on a tripod, an instrument stand, a motorized carriage, or a robot end effector. The three dimensional coordinates of the workpiece  2528  are measured by the scanner camera  2530  by using the principles of triangulation. There are several ways that the triangulation measurement may be implemented, depending on the pattern of light emitted by the scanner light source  2520  and the type of photosensitive array  2534 . For example, if the pattern of light emitted by the scanner light source  2520  is a line of light or a point of light scanned into the shape of a line and if the photosensitive array  2534  is a two dimensional array, then one dimension of the two dimensional array  2534  corresponds to a direction of a point  2526  on the surface of the workpiece  2528 . The other dimension of the two dimensional array  2534  corresponds to the distance of the point  2526  from the scanner light source  2520 . Hence the three dimensional coordinates of each point  2526  along the line of light emitted by scanner light source  2520  is known relative to the local frame of reference of the six-DOF scanner  2000 B. From the six degrees of freedom determined by the tracker  2000  in cooperation with the six-DOF target  2000 B, the three dimensional coordinates of the scanned line of light may be found in the tracker frame of reference, which in turn may be converted into the frame of reference of the workpiece  2528  through the measurement by the laser tracker of three points on the workpiece, for example. 
     If the six-DOF scanner  2000 B is held by hand, a line of laser light emitted by the scanner light source  2520  may be moved in such a way as to “paint” the surface of the workpiece  2528 , thereby obtaining the three dimensional coordinates for the entire surface. It is also possible to “paint” the surface of a workpiece using a scanner light source  2520  that emits a structured pattern of light. Alternatively, when using a scanner  2500  that emits a structured pattern of light, more accurate measurements may be made by mounting the 6-DOF scanner on a tripod or instrument stand. The structured light pattern emitted by the scanner light source  2520  might, for example, include a pattern of fringes, each fringe having an irradiance that varies sinusoidally over the surface of the workpiece  2528 . In an embodiment, the sinusoids are shifted by three or more phase values. The amplitude level recorded by each pixel of the camera  2530  for each of the three or more phase values is used to provide the position of each pixel on the sinusoid. This information is used to help determine the three dimensional coordinates of each point  2526 . In another embodiment, the structured light may be in the form of a coded pattern that may be evaluated to determine three-dimensional coordinates based on single, rather than multiple, image frames collected by the camera  2530 . Use of a coded pattern may enable relatively accurate measurements while the 6-DOF scanner  2000 B is moved by hand at a reasonable speed. 
     Projecting a structured light pattern, as opposed to a line of light, has some advantages. In a line of light projected from a handheld six-DOF scanner  2000 B, the density of points may be high along the line but much less between the lines. With a structured light pattern, the spacing of points is usually about the same in each of the two orthogonal directions. In addition, in some modes of operation, the three-dimensional points calculated with a structured light pattern may be more accurate than other methods. For example, by fixing the six-DOF scanner  2000 B in place, for example, by attaching it to a stationary stand or mount, a sequence of structured light patterns may be emitted that enable a more accurate calculation than would be possible other methods in which a single pattern was captured (i.e., a single-shot method). An example of a sequence of structured light patterns is one in which a pattern having a first spatial frequency is projected onto the object. In an embodiment, the projected pattern is pattern of stripes that vary sinusoidally in optical power. In an embodiment, the phase of the sinusoidally varying pattern is shifted, thereby causing the stripes to shift to the side. For example, the pattern may be made to be projected with three phase angles, each shifted by 120 degrees relative to the previous pattern. This sequence of projections provides enough information to enable relatively accurate determination of the phase of each point of the pattern, independent of the background light. This can be done on a point by point basis without considering adjacent points on the object surface. 
     Although the procedure above determines a phase for each point with phases running from 0 to 360 degrees between two adjacent lines, there may still be a question about the identification of the lines. One way to identify the lines is to repeat the sequence of phases, as described above, but using a sinusoidal pattern with a different spatial frequency (i.e., a different fringe pitch). In some cases, the same approach needs to be repeated for three or four different fringe pitches. The method of removing ambiguity using this method is well known in the art and is not discussed further here. 
     To obtain a higher level of accuracy using a sequential projection method such as the sinusoidal phase-shift method described above, it may be advantageous to minimize the movement of the six-DOF scanner. Although the position and orientation of the six-DOF scanner are known from the six-DOF measurements made by the laser tracker and although corrections can be made for movements of a handheld six-DOF scanner, the resulting noise will be somewhat higher than it would have been if the scanner were kept stationary by placing it on a stationary mount, stand, or fixture. 
     In general, 3D coordinates obtained through the use of triangulation require knowledge of the distance between perspective centers of a projector lens and a camera lens or a distance between two camera lenses. The distance between these perspective centers is referred to as the baseline distance. In addition, knowledge of the relative orientation of the projector and camera or first camera and second camera are required to complete a triangulation calculation. Some measurement is usually carried out, perhaps by a manufacture or perhaps by a user, to determine the baseline distance and relative orientations. 
     One limitation in the accuracy of scanners may be present for certain types of objects. For example, some features such as holes or recesses may be difficult to scan effectively. The edges of objects or holes may be difficult to obtain as smoothly as might be desired. Some types of materials may not return as much light as desired or may have a large penetration depth for the light. In other cases, light may reflect off more than one surface (multipath interference) before returning to the scanner so that the observed light is “corrupted,” thereby leading to measurement errors. In any of these cases, it may be advantageous to measure the difficult regions using a six-DOF scanner  2000 C shown in  FIG. 14  that includes a tactile probe that includes the probe tip  2554 , which is part of the probe extension assembly  2550 . After it has been determined that it would be advantageous to measure with a tactile probe, the projector  2520  may send a laser beam to illuminate the region to be measured. In  FIG. 14 , a projected ray of beam of light  2522  is illuminating a point  2527  on an object  2528 , indicating that this point is to be measured by the probe extension assembly  2550 . In some cases, the tactile probe may be moved outside the field of projection of the projector  2550  so as to avoid reducing the measurement region of the scanner. In this case, the beam  2522  from the projector may illuminate a region that the operator may view. The operator can then move the tactile probe  2550  into position to measure the prescribed region. In other cases, the region to be measured may be outside the projection range of the scanner. In this case, the scanner may point the beam  2522  to the extent of its range in the direction to be measured or it may move the beam  2522  in a pattern indicating the direction to which the beam should be placed. Another possibility is to present a CAD model or collected data on a display monitor and then highlight on the display those regions of the CAD model or collected data that should be re-measured. It is also possible to measure highlighted regions using other tools, for example, a spherically mounted retroreflector or a six-DOF probe under control of a laser tracker. 
       FIG. 15  shows an embodiment of a six-DOF indicator  2000 D used in conjunction with the six-DOF laser tracker  1100 . The six-DOF indicator  2000 D includes a body  2814 , one or more six-DOF assemblies  2002 , a mount  2890 , an optional electrical cable  2836 , an optional battery  2834 , an interface component  2812 , an identifier element  2839 , actuator buttons  2816 , an antenna  2838 , and an electronics circuit board  2832 . Each of the one or more six-DOF assemblies  2002  includes a retroreflector. The mount  2890  may be attached to a moving element, thereby enabling the laser tracker to measure the six degrees of moving element. The moving element may be a robotic end effector, a machine tool, or a tool on an assembly (e.g., an assembly line carriage). 
       FIG. 16  shows an embodiment of a six-DOF projector  2000 E used with a six-DOF laser tracker  1100 . The six-DOF projector  2000 E includes a body  2614 , one or more six-DOF assemblies  2002 , a projector  2620 , an optional electrical cable  2636 , an optional battery  2634 , an interface component  2612 , an identifier element  2639 , actuator buttons  2616 , an antenna  2638 , and an electronics circuit board  2632 . Each of the six-DOF assemblies  2002  includes a retroreflector. The descriptions for these elements are the same as discussed for corresponding elements described hereinabove and are not repeated. The six-DOF projector  2000 E may include a light source, a light source and a steering mirror, a MEMS micromirror, a liquid crystal projector, or any other device capable of projecting a pattern of light onto a workpiece  2600 . From the six degrees of freedom, the three dimensional coordinates of the projected pattern of light may be found in the tracker frame of reference, which in turn may be converted into the frame of reference of the workpiece through the measurement by the laser tracker of three points on the workpiece, for example. 
     With the projected pattern of light  2640  on the surface of the workpiece  2660  known in the frame of reference of the workpiece, a variety of useful capabilities can be obtained. As a first example, the projected pattern may indicate where an operator should drill holes or perform other operations to enable the affixing of components onto the workpiece  2660 . For example, gauges may be attached to the cockpit of an aircraft. Such a method of in-situ assembly can be cost effective in many cases. As a second example, the projected pattern may indicate where material needs to be added to or removed from a tool through the use of contour patterns, color coded tolerance patterns, or other graphical means. An operator may use a tool to abrade unwanted material or use a filler material to fill in an area. Because the laser tracker or an external computer attached to the laser tracker may know the details of the CAD model, the six-DOF projector can provide a relatively fast and simple method for modifying a tool to meet CAD tolerances. Other assembly operations might include scribing, applying adhesive, applying a coating, applying a label, and cleaning. As a third example, the projected pattern may indicate hidden components. For example, tubing or electrical cables may be routed behind a surface and hidden from view. The location of these components may be projected onto the workpiece, thereby enabling the operator to avoid them in performing assembly or repair operations. 
     To project light from the projector scanner into the frame of reference of the workpiece, it the frame of reference of the workpiece may be determined in the frame of reference of the laser tracker. One way to do this is to measure three points on the surface of the workpiece with the laser tracker. Then a CAD model or previously measured data may be used to establish a relationship between a workpiece and a laser tracker. 
     When an operator performs assembly operations with the assistance of a six-DOF projector, a useful process is to mount the six-DOF projector on a stationary stand or mount, thereby enabling the operator to perform assembly operations with both hands free. A useful mode of the laser tracker and six-DOF projector is to have the six-DOF projector continue to project a pattern of light even after the laser tracker ceases to track the retroreflector on the six-DOF scanner. In this way, the operator may use the laser tracker to perform measurements, for example, with an SMR, a six-DOF probe, or a six-DOF scanner while the projector continues to display the pattern of light that indicates the assembly operations to be performed. In a similar manner, the tracker may be used to set up two or more scanner projectors that continue to project patterns after the tracker has stopped tracking the retroreflector on each scanner projector. Hence high levels of detail may be projected onto relatively large areas, enabling assistance to several operators simultaneously. It is also possible in a mode to enable the six-DOF scanner to project any of several alternative patterns, thereby enabling the operator to perform assembly operations is a prescribed sequence. 
     In addition to assisting with assembly operations, the projector scanner can also assist in carrying out inspection procedures. In some cases, an inspection procedure may call for an operator to perform a sequence of measurements in a particular order. The six-DOF scanner may point to the positions at which the operator is to make a measurement at each step. The six-DOF scanner may demarcate a region over which a measurement is to be made. For example, by drawing a box, the six-DOF scanner may indicate that the operator is to perform a scanning measurement over the region inside the box, perhaps to determine the flatness of the regions or maybe as part of a longer measurement sequence. Because the projector can continue the sequence of steps with the six-DOF retroreflector being tracked by the laser tracker, the operator may continue an inspection sequence using the tracker or using other tools. If the tracker is performing the measurements, it will know when measurements have been successfully completed and may move onto the next step. The projector scanner may also provide information to the operator in the form of written messages or audio messages. The operator may signal commands to the laser tracker using gestures that may be picked up by the tracker cameras or by other means. 
     The six-DOF projector may use patterns of light, perhaps applied dynamically, to convey information. For example, the six-DOF projector may use a back and forth motion to indicate a direction to which an SMR is to be moved. The six-DOF projector may draw other patterns to give messages that may be interpreted by an operator according to a set of rules, the rules which may be available to the user in written or displayed form. 
     The six-DOF projector may also be used to convey information to the user about the nature of an object under investigation. For example, if dimensional measurements have been performed, the six-DOF projector might project a color coded pattern indicating regions of error associated in the surface coordinates of the object under test. In one embodiment, the projector may display regions or values that are out of tolerance. It may, for example, highlight a region for which the surface profile is outside the tolerance. In another embodiment, the projector may draw a line to indicate a length measured between two points and then write a message on the part indicating the amount of error associated with that distance. 
     The six-DOF projector may also display information about measured characteristics besides dimensional characteristics, wherein the characteristics are tied to coordinate positions on the object. Such characteristics of an object under test may include temperature values, ultrasound values, microwave values, millimeter-wave values, X-ray values, radiological values, chemical sensing values, and many other types of values. Such object characteristics may be measured and matching to three-dimensional coordinates on an object using a six-DOF scanner, as discussed hereinafter. In an embodiment, characteristics of an object may be measured on the object using a separate measurement device, with the data correlated in some way to dimensional coordinates of the object surface with an object frame of reference. Then by matching the frame of reference of the object to the frame of reference of the laser tracker or the six-DOF projector, information about the object characteristics may be displayed on the object, for example, in graphical form. For example, temperature values of an object surface may be measured using a thermal array. Each of the temperatures may be represented by a color code projected onto the object surface. 
     A six-DOF projector may also project modeled data onto an object surface. For example, it might project the results of a thermal finite element analysis (FEA) onto the object surface and then allow the operator to select which of two displays—FEA or measured thermal data—is displayed at any one time. Because both sets of data are projected onto the object at the actual positions where the characteristic is found—for example, the positions at which particular temperatures have been measured or predicted to exist, the user is provided with a clear and immediate understanding of the physical effects affecting the object. The six-DOF projector may also be attached to a moveable carrier such as a robot or machine tool. 
       FIG. 17  shows an embodiment of a six-DOF sensor  2000 F used in conjunction with a six-DOF laser tracker  1100 . In an embodiment, the six-DOF sensor  2000 F includes a body  4914 , one or more six-DOF assemblies  2002 , a sensor  4920 , an optional source  4950 , an optional electrical cable  4936 , an optional battery  4934 , an interface component  4912 , an identifier element  4939 , actuator buttons  4916 , an antenna  4938 , and an electronics circuit board  4932 . The descriptions for these elements are the same as discussed for corresponding elements described herein above and are not repeated. A plurality of six-DOF assemblies  2002 , each including a retroreflector, may be used to enable the laser tracker  1100  to track the six-DOF sensor from a variety of directions, thereby giving greater flexibility in the directions to which an object may be sensed by the six-DOF sensor  2000 F. 
     The sensor  2000 F may be of a variety of types. For example, it may respond to optical energy in the infrared region of the spectrum, the light having wavelengths from 0.7 to 20 micrometers, thereby enabling determination of a temperature of an object surface at a point  4924 . The sensor  2000 F is configured to collect infrared energy emitted by the object  4960  over a field of view  4940 , which is generally centered about an axis  4922 . The three-dimensional coordinates of the point on the object surface corresponding to the measured surface temperature may be found by projecting the axis  4922  onto the object  4960  and finding the point of intersection  4924 . To determine the point of intersection, the relationship between the object frame of reference and the device (tracker) frame of reference needs to be known. In an embodiment, the relationship between the object frame of reference and the six-DOF sensor frame of reference may be known since the relationship between the tracker frame of reference and the sensor frame of reference is already known. In another embodiment, the relationship between the object frame of reference and the six-DOF sensor frame of reference may be known since the relationship between the tracker frame of reference and the six-DOF sensor is already known from measurements performed by the tracker on the six-DOF sensor. One way to determine the relationship between the object frame of reference and the tracker frame of reference is to measure the three-dimensional coordinates of three points on the surface of the object. By having information about the object in relation to the three measured points, all points on the surface of the object will be known. Information on the object in relation to the three measured points may be obtained, for example, from CAD drawings or from previous measurements made by any type of coordinate measurement device. 
     Besides measuring emitted infrared energy, the electromagnetic spectrum may be measured (sensed) over a wide range of wavelengths, or equivalently frequencies. For example, electromagnetic energy may be in the optical region and may include visible, ultraviolet, infrared, and terahertz regions. Some characteristics, such as the thermal energy emitted by the object according to the temperature of the object, are inherent in the properties of the object and do not require external illumination. Other characteristics, such as the color of an object, depend on background illumination and the sensed results may change according to the characteristics of the illumination, for example, in the amount of optical power available in each of the wavelengths of the illumination. Measured optical characteristics may include optical power received by an optical detector, and may integrate the energy a variety of wavelengths to produce an electrical response according to the responsivity of the optical detector at each wavelength. 
     In some embodiments, the illumination may be intentionally applied to the object by a source  4950 . If an experiment is being carried out in which it is desired that the applied illumination be distinguished from the background illumination, the applied light may be modulated, for example, by a sine wave or a square wave. A lock-in amplifier or similar method can then be used in conjunction with the optical detector in the sensor  4920  to extract just the applied light. 
     Other examples of the sensing of electromagnetic radiation by the sensor  4940  include the sensing of X-rays at wavelengths shorter than those present in ultraviolet light and the sensing of millimeter-wave, microwaves, RF wave, and so forth are examples of wavelengths longer than those present in terahertz waves and other optical waves. X-rays may be used to penetrate materials to obtain information about interior characteristics of object, for example, the presence of defects or the presence of more than one type of material. The source  4950  may be used to emit X-rays to illuminate the object  4960 . By moving the six-DOF sensor  2000 F and observing the presence of a defect or material interface from a plurality of views, it is possible to determine the three-dimensional coordinates of the defect or material interface within the material. Furthermore, if a sensor  2000 F is combined with a projector such as the projector  2720  in  FIG. 16 , a pattern may be projected onto an object surface that indicates where repair work needs to be carried out to repair the defect. 
     In an embodiment, the source  4950  provides electromagnetic energy in the electrical region of the spectrum—millimeter-wave, microwave, or RF wave. The waves from the source illuminate the object  4960 , and the reflected or scattered waves are picked up by the sensor  4920 . In an embodiment, the electrical waves are used to penetrate behind walls or other objects. For example, such a device might be used to detect the presence of RFID tags. In this way, the six-DOF sensor  2000 F may be used to determine the position of RFID tags located throughout a factory. Other objects besides RFID tags may also be located. For example, a source of RF waves or microwaves such as a welding apparatus emitting high levels of broadband electromagnetic energy that is interfering with computers or other electrical devices may be located using a six-DOF scanner. 
     In an embodiment, the source  4950  provides ultrasonic waves and the sensor  4920  is an ultrasonic sensor. Ultrasonic sensors may have an advantage over optical sensors when sensing clear objects, liquid levels, or highly reflective or metallic surfaces. In a medical context, ultrasonic sensors may be used to localize the position of viewed features in relation to a patient&#39;s body. The sensor  4920  may be a chemical sensor configured to detect trace chemical constituents and provide a chemical signature for the detected chemical constituents. The sensor  4920  may be configured to sense the presence of radioactive decay, thereby indicating whether an object poses a risk for human exposure. The sensor  4920  may be configured to measure surface texture such as surface roughness, waviness, and lay. The sensor may be a profilometer, an interferometer, a confocal microscope, a capacitance meter, or similar device. A six-DOF scanner may also be used for measure surface texture. Other object characteristics can be measured using other types of sensors not mentioned hereinabove. 
       FIG. 18  shows an embodiment of a six-DOF sensor  2000 G that is like the six-DOF sensor  2000 F of  FIG. 17  except that the sensor  4922  of the six-DOF sensor  2000 G includes a lens  4923  and a photosensitive array  4924 . An emitted or reflected ray of energy  4925  from within a field of view  4940  of the six-DOF sensor arises at a point  4926  on the object surface  4960 , passes through a perspective center  4927  of sensor lens  4923  to arrive at a point  4928  on the photosensitive array  4924 . A source  4950  may illuminate a region of the object surface  4960 , thereby producing a response on the photosensitive array. Each point is associated with three-dimensional coordinates of the sensed characteristic on the object surface, each three-dimensional point determined by the three orientational degrees of freedom, the three translational degrees of freedom, the geometry of the camera and projector within the sensor assembly, and the position on the photosensitive array corresponding to the point on the object surface. An example of sensor  4922  is a thermal array sensor that responds by providing a temperature at a variety of pixels, each characteristic sensor value associated with a three-dimensional surface coordinate. 
       FIG. 19A  shows a cube-corner retroreflector in a 3D Cartesian frame of reference. A first octant  1905  of a 3D Cartesian frame of reference extends from the origin  1910  in the positive x, y, z directions. The three planes x-y, y-z, and z-x are mutually perpendicular and serve as sides of the cube-corner retroreflector. The sides extending from the vertex (origin)  1910  are of equal length, forming glass cube-corner prism. The fourth face of the cube corner, which is not in contact with the vertex  1910 , is the front face  1920 . A vector r  1925  extending in a perpendicular direction from the vertex to the front surface of the prism is symmetric with respect to the axes x, y, z. In most cases, such prisms are formed into a cylindrical shape  1915  by grinding away a portion of the glass to produce cylindrical glass cube-corner prism  1930 . 
       FIG. 19B  shows the octant  1955  directly opposite the octant  1905  of  FIG. 19A . The octant  1955  occupies a volume extending from the origin  1910  in the −x, −y, −z directions. A cylindrical cube-corner prism  1980  is formed in the same manner as the prism  1930  in  FIG. 19A  and sits directly opposite the cube-corner prism  1930 . A vector −r  1975  extending in a perpendicular from the vertex to the front surface of the prism  1980  is symmetric with respect to the axes −x, −y, −z. 
       FIG. 20A  shows a slice taken through the x-r plane of  FIG. 19A . The diameter of the cylinder  1915  is taken, for scaling purposes, to be 1. A perpendicular drawn from the vertex  1910  to the front face  1920  has an altitude h equal to 0.707. The altitude falls directly in the center of the cylinder. The grinding away of the glass in the prism goes half way down the cylinder at the x axis to a height of 0.707/2. The ground away portion is marked  2320 . The portion of the x axis on the prism is the intersection line segment  2310 . As can be seen from  FIG. 19A , the line  2330  goes through the x-r plane and is opposite the x axis. The line  2330  bisects they and z axes on the y-z plane and, if the prism is not ground into the cylindrical shape, the line  2330  extends all the way to the front face  1920 . The grinding of a cube corner into a cylindrical shape produces the scalloped effect of the prism in  FIG. 20B . 
     In a cube-corner retroreflector, light that enters the front face of the prism reflects off three different reflector surfaces before exiting the front face of the prism, thereafter traveling in a direction opposite that of the incoming light. A method from geometrical optics that may be used to analyze the reflection of light off a single surface is shown in  FIG. 21 . An incident ray of light  2420  strikes a reflecting surface  2410  and reflects at an angle in a ray  2430 . It can be shown that this reflection is equivalent to the light continuing to travel straight through the reflective surface  2410  as long as a reflection of the light  2440  is performed afterwards to obtain the actual reflected light  2430 . 
       FIG. 22  shows a cross-section  2560  of the glass prisms  1930 ,  1980  in the quadrants  1905 ,  1955  in  FIGS. 19A, 19B . The cross section is taken through the axes x, x, r, −r. Light  2562  enters the front face of the cube-corner prism  2300 . For the purposes of mathematical modeling, light  2566  continues through a second cube-corner prism  2564  and exits the front face as light  2568 . The light  2562  that enters the front face of the prism  2300  at the surface point  2570  passes through the vertex  1910  and exits the front face of the prism  2564  at the point  2572 . The points  2570  and  2572  are the same distance from the center of the front faces through which they pass. Only those rays of light that pass through the front face of the prisms  2300  and  2564  may be seen by the camera. 
     In  FIG. 23 , the circle  2582  represents a top view of the cube corner prism  2300 . The curved left edge of the front surface of this prism is illuminated by the light  2562 . The light that illuminates the rightmost part of the prism  2300  is lost because it does not pass through the front face of the prism  2564 . For example, the ray of light  2574  on the edge of the front face of the prism  2300  passes through a central region of the front face of the prism  2564  and hence represents a ray that will be reflected. The reflected ray, represented the point  2576  on the front face of prism  2300 , lies on opposite side of the point  2610 . The distance from the point  2574  to the point  2625  is equal to the distance from the point  2610  to the point  2615 . The situation is similar for the ray passing through the point  2576 , which passes the edge of the front face of the prism  2564  at the point  2635 . 
     The resulting region of cube-corner illumination, which will be viewed by an observed or a camera as a bright region, is the eye-shaped region  2578 . The front face of the prism  2300  is an illuminated circle viewed at an angle by an observer. It can be shown that a tilted circle is an ellipse. Hence an observer or a camera aligned with the direction of the light  2562  will see the front face of the prism  2300  as an ellipse. The front face of the prism  2564  is represented by the circle  2580 , which when viewed by an observer at an angle, appears as an ellipse. The point  2610  is in the center of the “eye” shaped region that encompasses the overlap of the circles  2580  and  2582 . When viewed from an angle, the two segments at the periphery of the eye-shaped region are elliptical segments rather than circular segments. 
     The upper portion of  FIG. 24  shows an ellipse  2760  corresponding to the view of the front surface of the prism  2300  that will be seen by an observer for the prism  2300  tilted at an angle. A cross-sectional side view of the tilted prism in the lower portion of  FIG. 24 . The ray of light  2762  enters the prism at a point  2735  on the front face. The ray of light  2762  has an angle of incidence  2725 , which is taken with respect to a normal  2710  to the front surface. Entering the glass, the refracted ray of light  2715  bends toward the normal to an angle  2730 . The angle of bending of the light may be determined using Snell&#39;s Law, which in one form states that for a glass having an index of refraction n and an angle of incidence a, the angle of refraction b is equal to b=arcsin(sin(a)/n). In this instance, the angle of incidence is a=15 degrees. If the index of refraction of the glass is n=1.78, the angle of refraction is b=8.4 degrees. Because of refraction, the ray of light that intersects the vertex  1910  crosses the front face of prism  2300  at the point  2735 , referred to as the central intersection point. The dashed lines in the ellipse  2760  represent the lines of intersection of the reflector planes as projected perpendicular to the front face. These lines converge at the center  2764  of the front face. 
     The change from a circle to an ellipse in the top view of  FIG. 24  is small and perhaps difficult to detect by eye. However, the change in the position of the center  2764  of the front face relative to the central intersection point  2735  is much larger and may be easily seen by eye. This size of the ellipse along the direction of its minor axis changes from the diameter value by only 1−cos(15°)=0.034, or about 3% of the diameter. In contrast, for an altitude h and diameter D, the central intersection point moves by an amount equal to h sin(15°)=0.707D sin(15°)=0.18D, or about 18 percent of the diameter. The separation parameter  2745  is defined as the distance from the center  2764  of the front face to the central intersection point  2735 . 
       FIGS. 25A and 25B  show a beam of light  3820 , for example from a laser tracker, intersecting the front face of a cube corner retroreflector at an intersection point  3830 .  FIG. 25A  shows the appearance  3800  of the front face from a face-on view.  FIG. 25B  shows the shape  3850  of the front face as viewed from the source of the light  3820 . From this view, the front face has an elliptical shape  3805 . The portion of the ellipse that appears illuminated to an observer is the eye region  3860 , which is bounded by two elliptical segments, one segment being  3870  and the other segment being a portion of the ellipse  3805 . The two elliptical segments an axis of symmetry  3840 , which is referred to as the fold axis. The fold axis  3840  is perpendicular to the direction of the light  3820  and includes the intersection point  3830 . 
     The elliptical edges of an illuminated cube-corner retroreflector are always present in any camera image as long as the front face is fully illuminated and the camera field of view is large enough. One way to describe the amount and direction of tilt is using a fold angle and a tilt angle. Here, the fold angle is defined as an angle of the fold axis and the tilt angle is the amount of rotation of the retroreflector about the fold axis. In the example of  FIG. 25B , the fold angle is 45 degrees and the tilt angle is 40 degrees. 
     Another way to describe the amount and direction of tilt of the front face is in terms of a pitch and yaw angle. As illustrated by  FIG. 26 , the pitch and yaw angles are defined in terms of x and y axes of a specified coordinate system (i.e., frame of reference). The pitch angle is an angle of rotation  3950  about an x axis, and the yaw angle is an angle of rotation  3960  about a y axis. Because rigid body rotation is not commutative, it is necessary to state whether the pitch rotation is occurs before or after the yaw rotation. The fold axis  3940  is an axis about which the front face  3910  is tilted. The fold angle β is given with respect to a reference axis  3930 . The tilt angle γ is an amount of rotation of the front face  3910  about the fold axis  3940 . The vector r, taken from  FIGS. 19A and 20A  is perpendicular to the front face and is distinct from the intersection point  3830  of  FIGS. 25A and 25B  since the definitions described by  FIG. 26  is independent of whether the retroreflector  3900  is illuminated or not. 
     The pitch and yaw angles or the fold and tilt angles are sufficient to fully describe the amount and direction of tilt of the front face, but they do not describe the point of attachment of a probe stylus  3920  and probing element  3922  to the retroreflector  3900 . Such a point of attachment may be described by a roll angle α. This roll angle is given with respect to a reference axis, which in  FIG. 26  is the axis  3930 , but may be any axis, including an axis different than the one used to define the fold angle. Note that the eye pattern  3860  of  FIG. 25B  is determined entirely by the fold and tilt angles or, equivalently, by the pitch and yaw angles. The eye pattern  3860  is independent of the roll angle. 
     Other angular descriptions may replace pitch/yaw angles or fold/tilt angles. One example is the use of direction cosines. Let a beam of light from a laser tracker be along a z axis and the zenith axis along an x axis. Let they axis be perpendicular to the x and z axes. In an embodiment, the direction cosines are three values obtained by taking the cosine of the angle of the x, y, z axes of the laser tracker in relation to r vector of the cube corner prism. However, if the three direction cosines are a, b, c, it can be shown that a 2 +b 2 +c 2 =1, so that only two of the three direction cosines are independent. 
     Other mathematical descriptors that may be used as alternatives to pitch/yaw/roll angles or fold/tilt/roll angles include quaternions and Euler angles. Many other mathematical descriptors may be used to represent the three orientational degrees of freedom. 
     In the remainder of the present application, the term pitch/yaw is used to represent any descriptor that describes the direction and amount of tilt a retroreflector, including fold/tilt or other similar descriptors. 
     To determine three orientational degrees of freedom, it is frequently useful to determine one of the orientational degrees of freedom—namely, the roll angle—separately from the two additional orientational degrees of freedom—namely, pitch/yaw angles (or equivalent). The apparatus and methods for measuring roll angle separately from pitch/yaw angles are described herein. 
       FIG. 27  illustrates a method of separately measuring roll angle and pitch/yaw angles (or equivalent). A device  12700  is configured to send one or more beams of light  12702  to a probe  12750 . In an embodiment, the probe  12750  includes a retroreflector  12755  that returns the beam of light to the probe  12750 . In an embodiment, the device  12700  is a laser tracker that measures a distance and two angles to the retroreflector  12755 , enabling a processor  12740  to determine three translational degrees of freedom, which might be, for example, coordinates in a Cartesian (rectangular) or spherical coordinate system. In an embodiment, the device  12700  provides components  12710  that include a roll conditioner  12712  and roll analyzer  12714 . The roll conditioner generates or conditions light emitted as a part of the light  12702  and received by probe components  12760 . In an embodiment, a portion of the light received by components  12760  is returned in the beam of light  12702  and is received by the roll analyzer  12714  for determining the roll angle of the probe  12750 . In other embodiments, the roll angle is determined without the use of a roll analyzer  12714 . 
     In an embodiment, the device  12700  further provides pitch/yaw conditioner components  12720  that include a pitch/yaw conditioner  12722  and pitch/yaw analyzer  12724 . The pitch/yaw conditioner generates or conditions light emitted as a part of the light  12702  and received by probe components  12770 . In an embodiment, a portion of the light received by components  12770  is returned in the beam of light  12702  and is received by the pitch/yaw analyzer  12724  for determining the pitch/yaw angles (or equivalent) of the probe  12750 . In other embodiments, the pitch/yaw angles are determined without the use of an pitch/yaw analyzer  12724 . In an embodiment, pitch/yaw and roll conditioning is applied to a single beam of light that is combined into the beam of light  12702 . In an embodiment, communication between the device  12700  and the probe  12750  takes place over wired channels  12742  or wireless channels  12744 . 
     In an embodiment illustrated in  FIG. 28A  and  FIG. 28B , a pitch/yaw analyzer is a pass-through retroreflector assembly  12800  that includes a retroreflector  12810  and a position detector  12820 . In an embodiment, the retroreflector  12810  is a cube-corner retroreflector  12810  having a front face  12812 , three mutually perpendicular reflecting surfaces  12814 , and a truncated region  12816  near a virtual vertex  12818  of the retroreflector. A beam of light  12830  from a laser tracker or similar device intersects the front face of the retroreflector in a spot of light  12840 . It refracts at the surface of the retroreflector and travels as a beam of light  12832  toward the virtual vertex  12818 . In an embodiment, the beam of light  12830  continues to track the virtual vertex  12818  as the retroreflector assembly  12800  is moved. The beam of light  12832  refracts at the truncated region as the beam of light  12834  and intersects the position detector  12820  in the spot of light  12842 . The position detector  12820  is configured to determine the position of the light  12834  on the surface of the position detector  12820 . The light received by the position detector  12820  may be processed electrically according to methods described herein below and the pitch/yaw angle of the retroreflector  12810  determined by a processor based at least in part on the signal provided by the position detector  12820 . 
     In an embodiment illustrated in  FIG. 29A  and  FIG. 29B , a pitch/yaw analyzer is a retroreflector assembly  12900  that includes a retroreflector  12910  and one or more pitch/yaw sensors. In an embodiment, the retroreflector  12910  is a glass cube-corner prism having a front face  12912 , three mutually perpendicular reflecting surfaces  12914 , and a vertex  12918  at which the reflecting surfaces intersect. In an embodiment, there are three pitch/yaw sensors  12920 A,  12920 B, and  12920 C. Each pitch/yaw sensor includes a position detector and an aperture. The pitch/yaw sensors  12920 A,  12920 B,  12920 C include apertures  12950 A,  12950 B,  12950 C, respectively, and position detectors  12922 A,  12922 B,  12922 C, respectively. A beam of light from a laser tracker or similar device projects one or more beams of light onto the retroreflector  12910 . In an embodiment, the one or more beams of light includes a relatively small beam of light  12930  centered on the front surface of the retroreflector  12910  and a relatively larger beam of light  12960 , also centered on the retroreflector. In an embodiment, the beam of light  12930  that travels to the vertex  12918  and intersects the front face  12912  at the point  12934 . For each of the three pitch/yaw sensors, a portion of light  12960  passes through an aperture before passing to the position detector. In the pitch/yaw sensor  12920 A, a light portion  12962 A passes through the aperture  12950 A and strikes the position detector  12922 A at position  12940 A. In the pitch/yaw sensor  12920 B, a light portion  12962 B passes through the aperture  12950 B and strikes the position detector  12922 B at position  12940 B. In the pitch/yaw sensor  12920 C, a light portion  12962 C passes through the aperture  12950 C and strikes the position detector  12922 C at position  12940 C. The light received by the position detectors  12922 A,  12922 B, and  12922 C may be processed electrically according to methods described herein below and the pitch/yaw angles of the assembly  12900  determined by a processor based at least in part on the signals provided by the position detectors  12922 A,  12922 B, and  12922 C. 
       FIGS. 30A, 30B, and 30C  show three common types of position detectors. Other types of position detectors may also be used.  FIG. 30A  is a drawing of a tetra-lateral position sensitive detector (PSD)  3000 , which includes an active PIN diode detector region  3002  and four electrodes  3004 ,  3005 ,  3006 , and  3007  arranged in a square outside the active area. The tetra-lateral PSD  3000  provides the ability to determine a position in two dimensions of a beam of light striking any position on the active region  3002 . The overall power striking the active region  3002  is usually proportional to the sum of electrical currents received by the four electrodes  3004 ,  3005 ,  3006 , and  3007 . The position in x and y is found in terms of sums and differences in the currents received by the electrodes according to formulas provided by the manufacturer of the detector  3000 .  FIG. 30B  is a drawing of a quadrant detector  3010 , which is a detector that includes four separated detector regions  3012 ,  3013 ,  3014 , and  3015 . Such detectors provide a position of a beam intercepted by all four separated detector regions. Such detectors are usually less linear than tetra-lateral PSDs and less able to measure small spots of light.  FIG. 30C  is a two-dimensional photosensitive array  3020  that includes an array of individual detector elements (pixels)  3022 . Examples of photosensitive arrays are charge-coupled devices (CCDs) and complimentary metal-oxide semiconductor (CMOS) devices. 
     A type of noise present in each of the types of position sensors of  FIGS. 30A, 30B, and 30C  is thermal noise. For a spot of light striking a position sensor, the thermal noise will depend on the resistance of the position sensor and the bandwidth of the measurement system. Considering only thermal noise, the signal-to-noise ratio of a uniform beam of light passing through an aperture will increase as the size of the aperture increases since a larger aperture increases the signal received by the position sensor. Another type of noise seen in factory environments is scintillation noise resulting from atmospheric turbulence. Scintillation noise is caused by atmospheric turbulence for the case in which the cell size is smaller than the aperture. A turbulent atmosphere can be modeled as a collection of cells of differing sizes, each cell having a different index of refraction. A beam of light propagating through a collection of cells will be bent in different directions as it travels. In consequence, the beam of light will be non-homogeneous when it arrives at the aperture. If cells in an environment are relatively small compared to an aperture, the optical power passing through aperture can be observed to vary significantly over time. If the cells are relatively large compared to an aperture, the fluctuations are greatly reduced. In a typical situation, fluctuations are seen to be relatively small for an aperture having a diameter of 0.5 mm and relatively large for an aperture having a diameter of 2.0 mm. However, the signal-to-noise ratio is reduced for the smaller aperture. For example, the signal-to-noise ratio is reduced by a factor of 16 by using an aperture of 0.5 mm rather than 2.0 mm. 
     A way to maintain relatively high signal-to-noise ratio while minimizing the fluctuations resulting from atmospheric turbulence is to use a relatively large aperture in combination with a lens. In an embodiment shown in  FIG. 31A  and  FIG. 31B , a glass cube-corner prism  3100  includes a front face  3112 , three reflecting surfaces  3114 , and a truncated aperture  3116  as shown in  FIG. 31A  and  FIG. 31B . A beam of light  3140  arrives at the front face  3112  as a spot  3152  that travels collimated in a bundle of rays  3142  and refracts to the bundle of rays  3144 . A portion of the bundle of rays  3144  passes through the truncated aperture  3116 , refracts again as the collimated bundle  3146  and is focused by the lens  3130  to the converging rays  3147  before creating the spot  3154  on a position sensor  3148 . By making the spot  3154  smaller than the truncated aperture  3116 , the fluctuations in optical power over the aperture are reduced on the surface of the position detector, thereby reducing the fluctuations in the calculated position of the beam on the position sensor  3100 . 
       FIG. 32  shows an example of a lens  3230  that might be used with an aperture plate  3120  and a position detector  3220 .  FIG. 32  illustrates the path taken by three different beams  3240 A,  3240 B, and  3240 C that may arrive at the aperture at three different angles: +45, 0, and −45 degrees. As illustrated in the figure, each of the three beams pass through the lens  3230  to form a different focused spot  3242 A,  3242 B, or  3242 C, respectively. 
     When determining pitch/yaw angles using an apparatus having an aperture and position detector, results may be contaminated by background lights—for example, overhead lights or light coming through windows. Such background light may pass through an aperture and strike the position detector, thereby leading to an incorrect determination of pitch/yaw angle. One means of alleviating this issue is to modulate the emitted light at the laser tracker or similar device and demodulate the light at the probe. In the demodulation process, the background light may be discarded using methods that are now described. 
     In an embodiment shown in  FIG. 33A , a laser tracker or similar device  13300  includes a light source  13302  that emits a beam of light  13308 . The light source is electrically modulated by a signal  13305 , which in an embodiment is a sine wave but might be a square wave or other waveform. The light emitted by the light source  13302  is modulated at the same rate. In an embodiment, the modulation causes a modulation of the optical power of emitted by the light source  13302 . The emitted beam of light may be combined with other light from the device  13300  in a beam  13310  that travels to the probe  13320 . The probe includes a position detector (i.e., position sensor) that is an analog detector  13322  such as the tetra-lateral PSD  3000  or the quadrant detector  3010 . The optical detector  13322  is configured to receive the light  13328  and convert it into an electrical signal, which it sends to an electrical circuit  13324 . The electrical circuit  13324  extracts the modulated signal and rejects background noise such as unmodulated background light. Detector  13322  is a two-dimensional position detector that receives the desired signal  13332  and in addition unwanted background light  13334 . The electrical circuit rejects the background light  13334  and uses the position of the transmitted light  13332  to determine the pitch/yaw angles of the probe. In an embodiment, the electrical circuit is configured to receive a signal  13325  having the same frequency as the electrical signal  13305 . In an embodiment, a lock-in amplifier or alternative filtering method is used to extract the desired signal  13332  in the presence of background light. The frequency of modulation of the signals  13305  and  13325  may be any of a wide range of frequencies but, in an embodiment, is 10 kHz. 
     In an embodiment shown in  FIG. 33B , a laser tracker or similar device  13340  includes a light source  13342  that emits a beam of light  13348 . The light source is electrically modulated by a signal  13345  from an electrical circuit  13344 . In an embodiment, the signal  13345  is a pulsed rectangular wave but might be a different waveform. The emitted beam of light may be combined with other light from the device  13340  in a beam  13310  that travels to the probe  13350 . The probe includes a position detector  13352  that is a digital detector such as a CCD or CMOS array. The optical detector  13352  is configured to receive the light  13358  and convert it into an electrical signal, which it sends to an electrical circuit  13354 . Element  13352  is a two-dimensional position detector that shows the presence of the desired signal emitted by the device  13300  and in addition unwanted background light  13334 . The electrical circuit rejects the background light  13334  and uses the position of the transmitted light  13332  to determine the pitch/yaw angles of the probe. In an embodiment, the electrical circuit  13354  is configured to synchronize the electrical signal received from the position sensor  13352  with the modulated signal  13345  through the provided signal  13355 . In this way, the influence of the background light  13334  is reduced in comparison to the transmitted light  13332 . 
     The embodiments of  FIGS. 28-33B  have mainly concerned the measurement of a first two orientational degrees of freedom involving the direction and tilt of the retroreflector, which may be described by pitch and yaw angles, by fold and tilt angles, or by a variety of other angular measures. Attention is now turned to the measurement the third orientational degree of freedom, which is usually quantified by a roll angle. 
       FIG. 34A  and  FIG. 34B  describe apparatus for generating rotating linearly polarized light. Such an apparatus may be included in a laser tracker or similar device such as the device  12700  of  FIG. 27 . In  FIG. 34A , an apparatus  3401  includes a source of light  3402 , a lens  3406 , a polarizer  3408 , and a rotating half-wave retarder (i.e., half waveplate) assembly  3410 . In an embodiment, the light source  3402  may be an LED, a laser, or a superluminescent diode. If the light source is an LED, the light may be emitted into a multimode fiber butt coupled to the LED chip. The light source emits a beam of light  3404 , which may be collimated by a lens  3406 . The light passes through a polarizer  3408  that produces linearly polarized light. The linearly polarized light passes through a hollow shaft  3412  to which is attached a half waveplate  3414 . The hollow shaft and half waveplate are turned by a motor  3416 . The hollow shaft may be mounted on mechanical bearings. The linearly polarized light is sent through the rotating half waveplate  3414  to produce rotating linearly polarized light. 
     In an alternative embodiment shown in  FIG. 34B , an apparatus  3420  includes a source of light  3422 , a lens  3426 , and a rotating polarizer assembly  3430 . In an embodiment, the light source  3422  is generates randomly polarized light by coupling light from an LED into a multimode fiber polarization. In an embodiment, the light source  2422  sends the beam of light  3424  through a collimation lens  3426 . The light passes through a hollow shaft  3433  to which is attached a linear polarizer  3434 . The hollow shaft and linear polarizer are turned by a motor  3436 . The hollow shaft may be mounted on mechanical bearings. The effect of sending randomly polarized light through a rotating polarizer is to produce rotating linearly polarized light. 
       FIG. 35  is a schematic representation illustrating a system for measuring a roll angle of a probe through the use of a polarization-roll generator  3500  with a polarization-roll sensor  3550 . The polarization-roll generator  3500  may be placed in a laser tracker or other device such as device  12700 . The polarization-roll sensor  3550  may be placed in a six-DOF probe such as the probe  12750 . In an embodiment, the polarization-roll generator  3500  combines the apparatus of  FIG. 34A  with a beam splitter  3510 , a polarizer  3520 , an optical detector  3522 , an electrical circuit  3524 , and a beam projector  3540 . The polarization-roll sensor  3550  includes a polarizer  3555 , an optical detector  3560 , and an electrical circuit  3562 . In an embodiment, the polarizer  3555  is a nanoparticle thin-film polarizer having a contrast ratio of better than 100,000 at acceptance angles of up to +/−20 degrees for wavelengths from 850 to 1600 nm. The polarization-roll generator  3500  and polarization-roll sensor  3550  cooperate to determine the roll angle of the probe. In other embodiments, the polarization-roll sensor uses the apparatus of  FIG. 34B  in place of the apparatus of  FIG. 34A . 
     In an embodiment, the beam splitter  3510  sends a part  3512  of light having rotating linear polarization through the polarizer  3520  and into the optical detector  3522 , which produces an electrical signal  3530  that varies sinusoidally with time. An electrical circuit  3524  measures a phase of the sinusoidal signal at an instant in time  3532 . The phase of the signal  3530  at the time  3532  is referred to as the reference phase. A variety of methods may be used to measure the reference phase. For example, the sinusoidal signal may be sampled with an analog-to-digital converter (ADC) and the samples analyzed numerically to determine the phase using algorithms that are well known in the art. For example, in one method, two summations are formed. A first summation S is equal to the sum of product of the measured samples and the values of a sinusoidal function at the frequency to be extracted (the modulation frequency), where the summation is taken over an integral number of cycles of the modulation frequency. A second summation C is equal to the sum of product of the measured samples and the values of the cosinusoidal function at the frequency to be extracted (the modulation frequency), where the summation is taken over an integral number of cycles of the modulation frequency. The phase is found as the arc tangent of C/S. 
     In an embodiment, the beam of light  3542  sent out of the beam projector  3540  is relatively large, and in another embodiment, the beam of light is relatively small. Various methods of projecting and receiving the beam of light are discussed herein below. In an embodiment, the light  3542  passes through a polarizer  3555  and into an optical detector  3560  that produces an electrical waveform  3565 . In an embodiment, an electrical circuit  3562  determines a phase of a sinusoidal signal  3565  at a time  3566 . This determined phase is referred to as the measure phase. The measure phase may be determined by the ADC sampling method described above or by any of the other methods known in the art. The measure phase minus a reference phase, which is taken at a time  3432 , is the phase difference. A time difference  3567  exists between measurements of the reference phase and the measure phase. A processor in the system is used to determine the roll angle of the polarization-roll sensor  3550  based at least in part on the determined phase difference. In an embodiment, the processor is included in the electrical circuit  3562 . In an embodiment, the reference phase of the probe at the time  3432  is synchronized to the reference phase that occurs at the time  3532  in the polarization-roll generator  3500 . This synchronization may be done, for example, through wired or wireless communication channels using coordinated timing signals. Other methods may also be used to obtain synchronization. 
     A first potential limitation of the method of the apparatus of  FIG. 35  is non-uniform angular rotation rate of the hollow shaft  3412 . A second potential limitation is contamination of the optical signal at the detector  3560  by (ambient) background light. The apparatus of  FIG. 36  overcomes these potential limitations.  FIG. 36  shows an apparatus that measures roll angle of a probe by using a polarization-roll generator  3600  and a polarization-roll sensor  3620 . The polarization-roll generator  3600  may be placed in a laser tracker or other device such as device  12700 . The polarization-roll sensor  3620  may be placed in a six-DOF probe such as the probe  12750 . In an embodiment, the polarization-roll generator  3600  combines the apparatus  3401  of  FIG. 34A  with a beam projector  3540 , an angular encoder disk  3602 , one or more read heads  3604 , and an electrical circuit  3610 . The encoder disk  3602  includes a plurality of markings  3603  that are read by the one or more read heads  3604  to determine the angle of rotation of the disk. The encoder disk is attached to the hollow shaft  3412  and rotates with it. One or more of the markings on the read head may be an index mark that provides the angular position of each mark in a frame of reference of the disk. The read head may send light through the marks or reflect light off the marks to detect the marks. The electrical circuit  3610  is configured to generate an electrical signal  3612  synchronized to the plurality of markings on the encoder disk  3602 . In an embodiment, the generated signal is a sinusoidal signal  3612 . Another type of signal  3612  that may be generated is a square wave or a pulsed wave, but any type of synchronized electrical signal may be provided by the electrical circuit  3610 . The signal  3612  is sent to the light source  3402  to modulate the optical power of the beam of light  3404 . The light may be generated by an LED, superluminescent diode, diode laser, or other light source. In an embodiment, the encoder disk  3602  includes 960 fine marks and one index mark. In an embodiment, the hollow shaft  3412  rotates at 30 cycles per second. For the configuration shown in  FIG. 36 , the linearly polarized light rotates at twice this rate, or 60 Hz. The frequency of the optical power passing through the polarizer  3622  is doubled again to produce an effective rotation rate of 120 Hz. In an embodiment, the electrical modulation  3612  provided by the electrical circuit  3610  to the light source  3402  is tied to the rotation of the encoder in such a way that each polarization cycle always includes the same number of fast modulation cycles—in this case, one rotation of the half waveplate  3414  for each passing of 16 fine marks on the encoder disk  3602 . 
     The polarization-roll sensor  3620  includes a first detector channel  3641  and a second detector channel  3643 . The first detector channel  3641  includes a polarizer  3622 , a first optical detector  3624 , and a first electrical circuit  3640 . The second detector channel  3643  includes a second optical detector  3634  and a second electrical circuit  3636 . The first electrical signal  3625  from the first electrical circuit  3640  has a relatively slow modulation resulting from the rotating linearly polarized light being converted into optical power variations by the polarizer  3622 . The first electrical signal  3625  is further modulated at a relatively fast rate by the signal  3612 . For the case of there being 16 cycles of optical power for each single rotation of the half waveplate  3414 , there will be half this number of cycles, or 8 cycles, in each cycle of detected optical power by the optical detector  3522 , as shown in the resulting waveform, which contains 8 cycles of sinusoidally varying optical power for each main power cycle, as shown in the exemplary electrical waveform  3625 . 
     The second optical detector  3634  receives an optical signal that includes only the modulation from the signal  3612 . The second electrical circuit  3636  receives the electrical signal  3639  from which it extracts the desired signal  3642 . Because the light from the polarization-roll generator  3600  is modulated by the electrical circuit  3610  and the background (ambient) light is not, the electrical waveform may be used to remove the effects of background light by providing a synchronization signal to assist in demodulation of the received electrical waveform  2525  to obtain the waveform  3654 . The unwanted signals from background light may be eliminated by an electrical filtering process, for example, by using a lock-in amplifier. Many types of lock-in amplifiers are possible including analog and digital lock-in amplifiers. Other filtering techniques may also be used. One way to remove the relatively high speed modulation in the signal  3625  from the relatively lower speed modulation in the signal  3642  is to directly sample and fit the lower speed signal so as to correctly account for nonuniformities in the rotation rate of the rotating shaft  3412 . A reference phase  2652  is compared to a measure phase  2650  to determine an angle of rotation of the polarization-roll sensor  3620  relative to the polarization-roll generator  3600 . As explained herein above, the reference phase and measure phase may be calculated using any of the methods known in the art, for example, the arc tangent method described herein above. 
     For all of the polarization-roll methods described herein above and herein below, an additional effect may be considered. When light strikes a glass surface, which might for example be a glass surface of a polarizer, an optical detector, a glass retroreflector, or a cube beam splitter, the light splits into s and p polarization components according to the tilt of the surface relative to the incoming beam of light. Each of the s and p polarization components has a different transmittance through the glass surface that may be calculated using Fresnel equations, as is well known in the art. According to the Fresnel equations, the amount of transmittance through a glass surface for s and p polarization depends on the angle of incidence of the light on the glass surface and on the index of refraction of the glass. If the surface of the glass is uncoated, the effect of passing on linearly polarized light through a glass surface is to slightly change the direction of polarization of the linearly polarized light. The effect of such a change in polarization on the calculated roll angle can be accounted for by first determining the pitch and yaw (or equivalent) angles of tilt target assembly and then using these values to correct for the change in the polarization state. To facilitate such a correction, compensation procedures may be carried out at the factory to determine how the linear polarization state changes with the pitch and yaw (or equivalent) angles of the target assembly. 
       FIG. 37  shows an apparatus  3720  that provides an alternative method for generating rotating linearly polarized light. A light source  3702  sends light  3706  through a lens  3704  to an optical element  3708  that is mounted on a mirror  3709  rotated by a shaft  3712  of a motor  3710 . The optical element  3708  may be a linear polarizer or a quarter waveplate. If the optical element is a linear polarizer, the light provided by the light source  3702  needs to have a random (and rapidly changing polarization). This type of light may be obtained in some cases from a multi-mode fiber butt coupled to an LED. In this case, the light double passes the linear polarizer and emerges as linear polarized light that rotates as the polarizer rotates. If the optical element is a quarter waveplate, the light provided by the light source  3702  needs to be linearly polarized. Such light may be obtained in some cases from polarized light source or from a polarization maintaining fiber attached to a light source. In other cases, such linearly polarized light may be obtained by sending the light through a linear polarizer, for example, following the lens  3704 . The light  3706  strikes the optical element  3708  at an angle of incidence, which is the angle between normal vector of the element optical element  3708  and the direction of the light. The light  3706  reflects off the mirror in a beam  3714  at an angle equal to the angle of incidence to produce the beam  3714 . 
     The embodiment illustrated in  FIG. 38  adds elements to the apparatus illustrated in  FIG. 37  to overcome the two potential limitations described earlier—non-uniform rotation rate and contamination from background light. The polarization-roll generator  3720  adds to the elements  3700  of  FIG. 37  an encoder disk  3722 , one or more read heads  3724 , and an electrical circuit  3728 . Electrical signals from marks on the encoder disk are sent over line  3726  to the electrical circuit, resulting in a signal  3729  that is sent to the light source  3702  to provide modulation to the signal sent from the projector  3540  to the polarization-roll sensor  3620 . The operation of the polarization roll sensor  3620  has already been discussed herein above in reference to  FIG. 36 . 
       FIGS. 39A and 39B  are schematic representations of a polarization-roll generator  4000  that replaces mechanically rotating elements described herein above with an electro-optic (EO) modulator configured to produce rotating linearly polarized light. Light source  4002  sends light  4004  through a collimating lens  4006  and a linear polarizer.  FIG. 39B  is a schematic representation  2000  of the crystal axes of the crystal in the EO modulator. Light passes into the EO modulator  4010 , which includes an electro-optic crystal having each of two mutually perpendicular crystal axes  4014 A and  4014 B, each oriented at 45 degrees with respect to the direction of polarization of the incoming light  4014 C. An electrical circuit  4012  applies an electrical signal to the first axis  4014 B to produce a change in refractive index along that axis. As a result of this action, the light along the first axis  4014 B is delayed or advanced relative to the other axis. In this manner, the polarization of the light passing through the EO modulator may be varied over all possible states of elliptical polarization from linear vertical to circular to linear horizontal and to all elliptical polarization states in between. A quarter waveplate  4018  is placed at the output of the EO modulator with the axis of the quarter waveplate  4018  placed along the direction of incoming light  4014 C of the initial polarization of the light. When no electrical signal is applied to the EO modulator by the electrical circuit  4012 , the light retains its linearly polarized state. As sinusoidally modulated light is applied by the EO modulator, the polarization changes to an elliptical state according to the level of the peak voltage applied. As the elliptical polarized light passes through the quarter waveplate  4018  is changes to a linear state. By increasing or decreasing the amplitude of the sinusoidally modulated electrical signal from the electrical circuit  4012 , the linearly polarized light leaving the quarter waveplate  4018  can be made to rotate. 
     In  FIG. 40 , a polarization-roll generator  4040  has the elements of the polarization-roll generator  4000  of  FIG. 39A  and in addition includes a beam splitter  4022 , a polarizer  4024 , an optical detector  4026 , an electrical circuit  4028 , and an electrical circuit  4030 . The electrical circuit  4030  applies a first electrical signal to the electrical circuit  4012  and a second electrical faster signal to modulate the optical power emitted by the light source  4002 . In an embodiment, the electrical circuit  4012  applies a 1 kHz sinusoidal signal to the EO modulator, and the electrical circuit  4030  produces a synchronized electrical signal at 16 kHz. The signal  4027  received by the optical detector  4026  includes a relatively fast modulation within a more slowly modulated signal  4029 . The electrical circuit  4028  extracts the signal  4029  with the slower modulation and measures a phase  3652  at an instant in time. It sends this phase measurement to the polarization-roll sensor  3620  in  FIG. 40 . The polarization-roll sensor works as described herein above with respect to  FIG. 36 . 
     In an embodiment illustrated in  FIGS. 41A and 41B , a roll analyzer is a pass-through retroreflector assembly  4100  that includes a retroreflector  4110 , a polarizer  4134 , and a detector  4120 . The general appearance of the roll analyzer  4100  is similar to that of the pass-through retroreflector assembly or pitch/yaw sensor  12800  of  FIGS. 28A, 28B , but the roll analyzer  4100  is intended to measure roll, while the pass-through retroreflector assembly or pitch/yaw analyzer  12800  is intended to measure pitch and yaw. In an embodiment, the roll analyzer  4100  cooperates with a polarization-roll generator such as one of the polarization-roll generators illustrated in  FIGS. 34A, 34B, 35, 36, 37, 38, 39, 40 . In general, it is not necessary that the detector  4120  be a position detector; a detector capable of measuring optical power but without the capability of measuring position is suitable. However, the use of a position detector may be advantageous in determining the roll angle as the position of the spot of light on the detector  4120  may be used to determine the pitch and yaw angles, which in turn may be used to determine the p- and s-polarization components of the light passing through the roll analyzer  4100 . This may be advantageous as the amount of transmittance of light through glass surfaces may be different for s and p polarizations. The use of a position detector as the detector  4120  may also enable the roll analyzer  4100  to be used as a combination pitch/yaw sensor and polarization-roll sensor. 
     In an embodiment, the retroreflector  4110  is a cube-corner retroreflector  4110  having a front face  4112 , three mutually perpendicular reflecting surfaces  4114 , and a truncated region  4116  near a virtual vertex  4118  of the retroreflector. A beam of light  4130  from a laser tracker or similar device intersects the front face  4112  of the retroreflector in a spot of light  4140 . It refracts at the surface of the retroreflector and travels as a beam of light  4132  toward the virtual vertex  4118 . In an embodiment, the beam of light  4130  continues to track the virtual vertex  4118  as the roll analyzer  4100  is moved. The beam of light  4132  refracts at the truncated region as the beam  4132  and intersects the detector  4120  in the spot of light  4142 . The method for determining the roll angle according to a sequence of optical powers has already been discussed herein above. 
     In an embodiment illustrated in  FIG. 42A  and  FIG. 42B , a roll analyzer is a retroreflector assembly  4200  that includes a retroreflector  4210  and one or more polarization-roll sensors. In an embodiment, the retroreflector  4210  is a glass cube-corner prism having a front face  4212 , three mutually perpendicular reflecting surfaces  4214 , and a vertex  4218  at which the reflecting surfaces intersect. In an embodiment, there are three polarization-roll sensors  4220 A,  4220 B, and  4220 C. Each polarization-roll sensor includes a position detector, a polarizer, and an aperture. The polarization-roll  4220 A,  4220 B,  4220 C include apertures  4250 A,  4250 B,  4250 C, respectively, polarizers  4221 A,  4221 B,  4221 C, respectively, and position detectors  4222 A,  4222 B,  4222 C, respectively. A beam of light from a laser tracker or similar device such as the device  12700  of  FIG. 27  projects one or more beams of light onto the roll analyzer  4210 . In an embodiment, the one or more beams of light includes a relatively small beam of light  4230  centered on the front surface of the retroreflector  4210  and a relatively large beam of light  4260 , also centered on the point of intersection  4234  of the light into the retroreflector. In an embodiment, the beam of light  4230  that travels to the vertex  4218  and intersects the front face  4212  at the point  4234 . For each of the three polarization-roll sensors, a portion of light  4260  passes through an aperture before passing through a polarizer and into a position detector. In the polarization-roll sensor  4220 A, a light portion  4262 A passes through the aperture  4250 A and polarizer  4221 A before striking the position detector  4222 A at position  4240 A. In the polarization-roll sensor  4220 B, a light portion  4262 B passes through the aperture  4250 B and polarizer  4221 B before striking the position detector  4222 B at position  4940 B. In the polarization-roll sensor  4920 C, a light portion  4962 C passes through the aperture  4950 C and polarizer  4221 C before striking the position detector  4922 C at position  4240 C. The light received by the position detectors  4222 A,  4222 B, and  4222 C may be processed electrically according to methods described herein below and the roll angle of the assembly  4200  determined by a processor based at least in part on the signals provided by the position detectors  4222 A,  4222 B, and  4222 C. 
       FIG. 43A  is a schematic representation of a six-DOF assembly  4300  that includes a retroreflector  4302 , three pitch/yaw sensors  4310 A,  4310 B,  4310 C, three polarization-roll sensors  4320 A,  4320 B,  4320 C, and an optical detector  4330 .  FIG. 43B  and  FIG. 43C  are cross-section A-A and cross-section B-B, respectively. In an embodiment, pitch/yaw sensors  4310 A,  4310 B,  4310 C include apertures  4312 A,  4312 B,  4312 C, respectively, lenses  4314 A,  4314 B,  4314 C, respectively, and position detectors  4316 A,  4316 B,  4316 C, respectively. In an embodiment, polarization-roll sensors  4320 A,  4320 B,  4320 C include apertures  4322 A,  4322 B,  4322 C, respectively, polarizers  4324 A,  4324 B,  4324 C, respectively, and optical detectors  4326 A,  4326 B,  4326 C, respectively. Optical detector  4330  measures the optical power of the light received from the laser tracker or other device. 
     In an embodiment, the optical detector  4330  provides the function of the optical detector  3634  in  FIG. 36 . In this way, it eliminates the need to provide the detector  3634  in each of the three polarization-roll sensors  4320 A,  4320 B, and  4320 C. Similarly the optical detector  4330  may provide an electrical signal corresponding to the total optical power to the electrical circuit  13324  in  FIG. 33A . For the case in which the position detectors  4316 A,  4316 B, and  4316 C are tetra-lateral PSDs, the total optical power may be found by adding the electrical signals from each of the four legs of the PSD. Similarly, it is possible using other position sensors, such as the positions sensors of  FIGS. 30B and 30C  to determine the optical power. However, it is simpler to use the electrical signal received from a single optical detector  4330  to extract the modulation signal corresponding to the optical power variations in the emitted light. 
     In an embodiment illustrated in  FIG. 43B  and  FIG. 43C , the front side of each of the sensor assemblies,  4310 A-C and  4320 A-C, is tilted away from the retroreflector  4302 . In an embodiment shown in  FIG. 43B  and  FIG. 43C , the angle of tilt away from the retroreflector is 25 degrees. To understand the advantage provided by tilting the sensor assemblies away from the retroreflector, note that in  FIGS. 31A, 31B, 41A, and 41B , the beam from the laser tracker or other device strikes the front surface of the retroreflector off the center of the retroreflector, with the position of the center on the retroreflector moved in the direction of the beam of light. Note also in these figures that the light strikes the optical detector  3148  in  FIGS. 31A, 31B  and the detector  4120  in  FIGS. 41A, 41B  off center in the direction opposite that at which the light strikes the front surface of the retroreflector. By tilting the sensor assemblies,  4310 A-C and  4320 A-C, away from the retroreflector by 25 degrees, when the light strikes the retroreflector at an angle of incidence of 25 degrees, the sensor assembly nearest the direction of the light is at normal incidence to the received light. This is also the condition at which this same sensor assembly receives a relatively large amount of light since the beam is shifted in the direction of the sensor assembly. By tilting each of the sensor assemblies away from the retroreflector  4302  by 25 degrees, that sensor assembly may cover a retroreflector tilt of 0 to 50 degrees for a sensor assembly having a field of view of +/−30 degrees. For the case of a pitch/yaw assembly that uses a lens, as in  FIG. 32 , a relaxation in the required field of view simplifies the design and improves performance. For the case of a polarization-roll assembly, a relaxation in the required field of view improves the contrast ratio, resulting in better performance. That a field of view of +/−30 degrees is adequate to provide coverage for tilt angles of 0 to 50 degrees is illustrated in  FIG. 43D , which is a graph showing the angular coverage in the x, y, and z directions for sensor assemblies tilted at 25 degrees and having fields of view of +/−30 degrees. The overlap of the three angular regions indicates that all angles up to +/−50 degrees are covered for any orientation of the assembly  4300 . 
       FIG. 44  illustrates an assembly  4400  that includes three sensors  4406 A,  4406 B,  4406 C, which may be either polarization-roll sensors or pitch/yaw sensors as described herein above. The sensors  4406 A,  4406 B,  4406 C surround a retroreflector  4407  having a region  4408  that includes a truncated portion of the glass cube corner and in addition has elements needed to obtain polarization-roll sensors such as  4320 A,  4320 B,  4320 C or pitch/yaw sensors such as  4310 A,  4310 B,  4310 C as described herein above. In an embodiment, the region  4408  is designed so as to complement the three sensors  4406 A,  4406 B, and  4406 C, thereby providing the assembly  4400  with the ability to be used in determining all six degrees-of-freedom. 
       FIG. 45A  and  FIG. 45B  show front and cross-sectional views, respectively, of a retroreflector assembly  4420  configured to measure pitch/yaw or roll or both. Light from an external device such as a laser tracker travels in a light bundle  3142  and refracts at the front face  3112  in a light bundle  3144  before passing through the truncated aperture  3116 . The light  3146  passing through the truncated aperture  3116  passes through lens  3130  and optionally through polarizer  4422  before being focused in converging rays  3147  onto optical detector  3148 , which may be a position sensor. 
       FIG. 46  is an exploded view of a tactile six-DOF probe  4600  according to an embodiment. A probe tip  4606  having a spherical surface is attached to a stylus shaft  4604 , which is affixed to a front housing  4602 . At the center of the front face of the front housing  4602  is a circular opening  4608  sized to accept a retroreflector  4644 . In an embodiment, the retroreflector is a hollow-core (air) cube corner retroreflector  4644 . In an embodiment, the retroreflector  4644  is a glass prism having three mutually perpendicular reflecting surfaces. The front housing includes three apertures  4610 A,  4610 B,  4610 C that transmit received light to pitch/yaw assemblies  4612 A,  4612 B,  4612 C, respectively, each pitch/yaw assembly including a lens and a position detector. The front housing also includes three polarizers  4620 A,  4620 B, and  4620 C that are above optical detectors (not visible), the polarizers sitting next to reference detectors  4622 A,  4622 B, and  4622 C, respectively. The polarizers, detectors (not visible), and reference detectors, along with associated electronics, comprise the polarization-roll sensors. An electronics board  4630  is provided to support the pitch/yaw sensors and polarization-roll sensors. A circular hole  4632  at the center of the electronics board  4630  is sized to pass the retroreflector  4644  and shaft  4642  on the back-plate assembly  4640 . A battery  4646  may also be provided. 
     In other embodiments, the probe tip  4606  and stylus shaft  4604  are replaced with other measurement devices or accessories such as a triangulation scanner, an indicator, a sensor, or a projector. These devices and accessories were discussed herein above with reference to  FIGS. 12-18 . In some embodiments, a plurality of six-DOF orientation sensors capable of measuring three orientational degrees of freedom (i.e., pitch, yaw, and roll) are mounted together on a probe to enable rapid measurement of the probe from a variety of different directions with the measurement devices or accessories likewise oriented in a variety of different directions. In other embodiments, a six-DOF orientation sensor, such as the sensor represented by the front housing  4602 , are attached to the measurement devices or accessories with one or more rotation joints that have an angular encoder or other angle measuring device. With this approach, the operator steers the retroreflector to point at the beam of light from the laser tracker or other device and with the rotation joints adjusts the position and orientation of the measurement device or accessory as needed. 
       FIG. 47  is an exploded view of a payload assembly  4700  of a laser tracker according to an embodiment. The payload assembly  4700  is similar to the payload assembly  15  of  FIG. 1 . The payload assembly  4700  includes a main optics path  4710  having a light source  4712  that produces a beam of light included in the launch beam  4714 . The launch beam  4714  may be used to track a retroreflector target and measure the distance and two angles to the target. The launch beam  4714  is directed by steering a motorized shaft  4720  about a zenith axis  4722 , which is equivalent to the zenith axis  18  of  FIG. 1 . The payload assembly is also steered about an azimuth axis, which is not shown in  FIG. 47  but is similarly arranged as the azimuth axis  20  illustrated in  FIG. 1 . In an embodiment, the payload assembly  4700  also includes a secondary optics path  4730  that includes a light source  4732  that sends light through a polarization-roll generator  4734 , which generates rotating polarized light using a rotating motor or an EO modulator, for example, as described herein above. In an embodiment, the light with the rotating linear polarization is used for the pitch/yaw sensor as well as the polarization-roll sensor. The light passes through beam expander lens elements to produce a collimated beam of light that, in an embodiment, has a diameter of 31 mm. The beam is folded down by mirror  4736  and reflected off a beam splitter  4738 . This light is combined with the beam from the light source  4712  to form a single composite launch beam  4714  from co-propagating beams of light. A top view of the payload assembly  4700  is shown in  FIG. 48A . A cross-section along A-A is shown in  FIG. 48B . This cross-sectional view shows more clearly the lens elements in the secondary path, including lenses  4740 ,  4742 ,  4744 , and  4746 . 
     A potential problem that may be encountered when using curved lens elements to expand a beam is that the polarization state of the beam may change slightly over the diameter of the beam. The reason for this is that, at each curved point on surface, a ray of light may be divided into s and p polarization states relative to the curved surface. These s and p polarization states may each encounter a slightly different phase shift and attenuation in passing through the surface. The result of this effect is that a curved surface tends to change linearly polarized light into elliptically polarized light, with the state of polarization changing with the distance from the beam center. 
     One means for addressing this issue is to obtain beam expansion using anamorphic prisms rather than curved lens elements such as the lens elements  4740 ,  4742 ,  4744 , and  4746 .  FIG. 49A  and  FIG. 49B  show side and top views, respectively, of a collection of anamorphic prisms that produce the same expansion of the beam as the curved lens elements of  FIG. 48B  without affecting the linear polarization state of the beam of light generated in the secondary optics path.  FIG. 49A  and  FIG. 49B  show four anamorphic prisms,  4802 ,  4804 ,  4808 ,  4810 , and one turning prism  4806 .  FIG. 49A  shows how the anamorphic prisms  4808  and  4810  expand the incoming beam of light  4612  in one dimension, which is apparent in the side view. The beam of light  4612  strikes the first surface of the anamorphic prism  4808  obliquely. If the angle of incidence of the beam  4612  is θ, the size of the beam on the first surface of the prism (as seen in the side view) is increased over the beam diameter by a factor of 1/cos(θ). By configuring the prism so that the second surface is perpendicular to the refracted beam of light in the prism  4808 , the full size of the expanded beam is retained as it exits the prism  4808 . Hence the beam  4814  is larger than the beam  4812  (in the side view). The beam  4814  is similarly expanded to the beam  4816  by the anamorphic prism  4810 . In a similar manner, the anamorphic prisms  4802  and  4802  expand the size of the beam in the top view. The beam  4810  is smaller than the beam  4811 , which is smaller than the beam  4812 . The prism  4808  does not change the size of the beam but steers it in a desired direction. 
     Another means for addressing the issue with variation in polarization state over the beam width is to use only the center part of a larger beam for the polarization-roll sensor. This method is possible with the arrangements illustrated in  FIGS. 41A, 41B, 44, 45A , and  FIG. 45B . For the case of the  FIG. 45A  and  FIG. 45B , the beam generated by the secondary path may be relatively small—such as, approximately the same size as the beam produced in the primary path. In this case, it may be possible in some embodiments to eliminate the secondary beam altogether and to use a beam having rotating linear polarization for tracking and measuring the three translation degrees of freedom as well as determining the three orientational degrees of freedom of the six-DOF probe. 
     Another embodiment is now described for enabling the use of a small beam from a laser tracker or similar device for determining the six degrees of freedom of a retroreflector target. Types of apparatus supporting a small beam from the laser tracker or other device are shown in  FIG. 50A ,  FIG. 50B , and  FIG. 50C . In each embodiment, a beam splitter is placed over a cube-corner prism. In an embodiment shown in  FIG. 50A ,  FIG. 50B , and  FIG. 50C , the cube-corner retroreflector prism and the beam splitter are made of a high index of refraction transparent material, which may be an amorphous glass or a crystalline material. In an embodiment, the material is clear grade polycrystalline zinc sulfide (ZnS) having an index of refraction of 2.35 at a wavelength of 633 nm. In  FIG. 50A , the orientational sensor includes a ZnS cube corner retroreflector prism  5010  attached to a ZnS beam splitter  5020 . On the side of the beam splitter that projects reflected light, a spacer element  5030  of ZnS is attached. A combination sensor  5040  that includes a functionality of a polarization-roll sensor and pitch/yaw sensor is obtained by placing an aperture  5042 , lens  5044 , polarizer  5046 , and position sensor  5048  as shown in  FIG. 50A . The point  5032  is a virtual image of the cube-corner vertex  5012  of  FIG. 50A . The light from a laser tracker follow (tracks) the virtual image point  5032  in the same way it follows (tracks) the vertex of the cube-corner retroreflector. The behavior of the combination sensor  5040  is therefore similar to the sensor described with respect to  FIG. 45A  and  FIG. 45B  except that the truncated aperture  3116  in the cube-corner retroreflector is replaced with the aperture in element  5042 . 
       FIG. 50B  illustrates an alternative embodiment in which a polarization-roll sensor  5070  is placed in a position to capture light transmitted directly through the truncated vertex  5014 . Other elements in the polarization-roll sensor  5070  include the polarizer  5072  and the optical detector  5074 . The pitch/yaw sensor in  FIG. 50B  receives light through the pitch/yaw sensor  504 B, which is the same as the sensor of  FIG. 5040A  except that the polarizer has been removed. 
       FIG. 50C  shows another embodiment, in which the material block  5030  is replaced with a cube beam splitter  5082  that sends part of the light to a polarization-roll sensor  5081  and another part of the light to a pitch yaw sensor  5085 . In this configuration, the polarization roll sensor  5081  includes an aperture  5083 , polarizer  5084 , and optical detector  5086 . The pitch/yaw sensor  5085  includes an aperture  5087 , lens  5088 , and position detector  5089 . 
       FIG. 51  shows another pitch/yaw sensor  5100  for determining the pitch and yaw angles (or equivalent) of a retroreflector  5112 . In an embodiment, the pitch/yaw sensor includes a retroreflector assembly  5110 , position sensor  5120 , and electrical circuit  5130 . The retroreflector assembly  5110  includes a cube-corner retroreflector  5112  and a lens ring  5118 . In an embodiment, the cube-corner retroreflector is a glass prism having a vertex  5114  and cylindrically ground sides  5113 , with the prism assembly mounted in a cylindrical can  5116 . The lens ring  5118  is a lens that has a central region removed so as to fit around the back side of the can  5116 . A beam of light  5140  illuminates the retroreflector assembly  5110 . One ray  5142  from the beam of light  5140  refracts at the front surface  5115  of the retroreflector  5112  and travels toward the vertex  5114 . If the light  5140  comes from a laser tracker or other device having a tracking system, the beam of light  5140  is centered on the ray  5142 . For a tilted retroreflector, the intersection point  5117  tends to move away from the center of the front surface  5115  in the direction of the nearest retroreflector edge, as illustrated in  FIG. 51 . A collection of rays  5142  from the beam of light  5140  illuminates the side of the lens ring  5118 . Points along illuminated portions of the lens ring refract the light into rays  5144 , which travels to position sensor  5120 . In an embodiment, the lens ring  5118  is darkened except in a transparent ring near positions  5150  around the periphery of the lens ring. The position sensor may be a photosensitive array such as a CMOS or CCD array, or it may be an analog position sensor such as a tetra-lateral PSD. 
     In an embodiment, the diameter of the retroreflector prism  5112  and surrounding can  5116  is 8 mm, the lens ring is obtained from a plano-convex lens made from borosilicate crown glass (such as N-BK7 manufactured by Schott North America, Inc. of Elmsford, N.Y.) having a radius of curvature of 40 mm, and the position sensor is an array having 2048×2048 pixels, each 5 pixel being 5 micrometers on a side (such as that manufactured by CMOSIS America, LLC of Raleigh, N.C. for example). The pattern formed on the photosensitive array  5120  for this embodiment is shown in  FIG. 52  for different angles of rotation about a given axis (e.g., pitch axis or yaw axis). Different angles of rotation are marked along the top of the graph for negative rotations and along the bottom axis for positive rotations. For the beam  5140  striking the retroreflector front surface  5115  at normal incidence, the pattern is circle, indicated in  FIG. 52  by a dashed line. For the case in which there is a combination of pitch rotation and yaw rotation, the pattern of  FIG. 52  is rotated at an angle. In an embodiment, the observed image pattern is fit to the possible curves such as those of  FIG. 52  to determine the pitch/yaw angles (or equivalent). In an embodiment, the method of pulsing the source of light is used, as discussed herein above with reference to  FIG. 33B . In another embodiment, the position sensor is a tetra-lateral PSD. The PSD provides the centroid of the light, which is used to determine the pitch/yaw angles. The methods of background light reduction described herein above in reference to  FIG. 33A  may be used in this case. 
       FIG. 53A  and  FIG. 53B  show another method for determining pitch/yaw (or equivalent) angles based on dispersion of glass.  FIG. 53A  shows a cross section of a cube-corner retroreflector prism  5300  having a front face  5302  and a vertex  5304 . A beam of light  5310  is launched from a laser tracker or other device. The beam  5310  includes two wavelengths of light: a blue wavelength and a red wavelength. Over the path from the laser tracker to the retroreflector  5300 , the two wavelengths of light are co-propagating and collinear. The laser tracker tracks on the red beam by keeping the red beam centered on the vertex  5304 .  FIG. 54A  shows a graphical plot of the index of refraction of a clear ZnS polycrystalline material. The index of refraction for blue light at 405 nm is 2.55. The index of refraction for red light at 637 nm is 2.36. The greater index of blue light in ZnS causes the blue beam to bend inward toward the normal.  FIG. 53A  shows that the red beam travels to the vertex and returns from the retroreflector as beam  5310 . The blue beam reflects to the left of the vertex and returns on the right of the vertex, exiting the retroreflector surface  5302  closer to the center of the front face  5302 . The total distance traveled by the beams of red and blue light as they enter the retroreflector, reflect of three perpendicular surfaces and exit the retroreflector is twice the distance from the center of the front face  5302  to the vertex. The path traveled by the blue and red light is modeled mathematically following two oppositely directed retroreflectors, as discussed herein above with reference to  FIGS. 21-23 . The exit position of the blue beam  5312 B and the red beam  5310 B on the bottom surface of the reversed retroreflector  5320  can be used to obtain the positions of the beams  5310 ,  5312  from the front face  5302  of the retroreflector  5300 . 
       FIG. 55  shows how the red and blue light is generated and evaluated in a laser tracker or other device. A first beam of red light having a wavelength of λ 2  is generated by light source  5502 . A second beam of blue light having a wavelength of λ 1  is generated by light source  5504 . The red and blue lights and combined by dichroic beam splitter  5506  and travel outward from the laser tracker or similar device on beam  5310 . For the retroreflector beam, a red light is returned on beam  5310  and a blue light is returned on beam  5312 . The red light is tracked by the position detector and hence is kept fixed in place. The return blue light reflects off the dichroic beam splitter  5506 , reflects partially off beam splitter  5508 , travels through beam expander  5510 , passes through polarizer  5516 , and into the position detector  5518 . An electrical circuit  5520  processes the received data to determine the tilt (pitch/yaw angles or similar) of the retroreflector target. 
     In an embodiment, the beam expander includes a negative lens and a positive lens that together provide a magnification of 3. The amount of displacement on the position detector  5518  of this beam is shown as a function of pitch/yaw angle in  FIG. 54B . For this situation with a one-inch ZnS retroreflector, the pitch/yaw angle produces a displacement on the position detector of around 250 micrometers, which occurs at the maximum angle of incidence of +/−45 degrees. 
       FIG. 56  shows a laser tracker or similar device  5600  that sends a beam of light  5632  to a retroreflector target included in a sensor  2002  that is a part of a target assembly  5620 . The retroreflector returns the beam of light to the device  5610  as a beam  5634 . In an embodiment, the device  5610  is a camera. In an embodiment, the one or more sensors  2002  measure the pitch/yaw angles. The target assembly further includes one or more light sources  5622  attached to the target assembly  5620 . An image of the light source  5622  is obtained by the camera  5610 , which includes a lens  5612  and a photosensitive array  5614 . A processor  5616 , which may be included in the device  5600  or in an external computer, determines the roll angle of the target assembly  5620  relative to the frame of reference of the device  5600  based on the pitch/yaw angles (or equivalent) measured by sensors  2002  and on the roll angle determined based at least in part on the image obtained on the photosensitive array  5614 . 
       FIG. 57  is a schematic representation of a six-DOF tactile probe  5700 . It includes a six-DOF assembly  2002  having a retroreflector that receives light from and returns light to a laser tracker or other device. It also includes a processor  5744  and an antenna  5745 . In addition, in an embodiment, it includes a first encoder assembly  5701 , a second encoder assembly  5703 , and a tactile probe assembly  5705 . In an embodiment, the first encoder assembly  5701  includes an axle  5712 , two bearings  5714 , a mounting block  5716 , an encoder disk  5718 , one or more read heads  5720 , and electrical wires  5722  running from the read heads to the processor  5744 . In an embodiment, axle  5712  turns on bearings  5714 , which are seated within mounting block  5716 . The encoder disk  5718  is fixedly attached to axle  5712 , and read heads  5720  are fixedly attached to the mounting block  5716 . The encoder disk  5718  rotates about the axis  5724 . The electrical signals from the read heads  5720  are sent over wires  5722  to processor  5744 , which evaluates the signals to determine the angle of rotation of the encoder disk  5718  and axle  5712  about the axis  5724 . 
     In an embodiment, the second encoder assembly includes an axle  5732 , two bearings  5734 , an inner mounting block  5736 , an outer mounting block  5037 , an encoder disk  5738 , one or more read heads  5740 , and electrical wires  5742  running from the read heads to the processor  5744 . The axle  5732  turns on bearings  5734 , which are seated within inner mounting block  5736 , which is affixed to outer mounting block  5737 . The encoder disk  5738  is fixedly attached to axle  5732 , and read heads  5740  are fixedly attached to the mounting block  5736 . In an embodiment, the electrical signals from the read heads  5740  are sent over wires  5742  through hollow axle  5712  to processor  5744 , which evaluates the signals to determine the angle of rotation of the encoder disk  5738  and axle  5732 . 
     In an embodiment, the tactile probe assembly  5705  includes a probe shaft  5752  and a probe tip  5754 . The probe shaft  5752  is attached to inner mounting structure/inner mounting block  5736 . The processor  5744  or one of the processors in the tracker is configured to determine 3D coordinates of the center of the probe tip  5754  based on the three translational degrees of freedom measured by the laser tracker  10  or similar device and by the three orientational degrees of freedom measured by the six-DOF probe  5700 . 
     In an embodiment, a laser tracker  10  may steer a beam of light out of the tracker about the zenith axis  18  with a resulting angle relative to a horizontal plane of approximately −52 degrees to +78 degrees. The retroreflector in the six-DOF assembly  2002  may be pointed in the direction of a laser beam emitted by the laser tracker. Furthermore, the six-DOF tactile probe  5700  may be rotated to any angle about the axis  5760 . In other words, the probe tip  5754  may be rotated about the axis  5760  to point below, above, or to the side of the body  5707 . The additional degree of rotational freedom  5762  provided by the first encoder assembly  5701  permits the probe to be rotated to the front, back, or side of the body  5707  relative to the retroreflector in the six-DOF assembly  2002 . The additional degree of rotational freedom  5764  provided by the second encoder assembly  5703  permits the probe to be rotated in an arc forward, down, or backwards. 
       FIG. 58  is a schematic representation of a six-DOF triangulation scanner  5770 . It includes the elements of six-DOF tactile probe  5700  of  FIG. 57  but replaces the tactile probe assembly  5705  with a triangulation scanner assembly  5780 . In an embodiment, the triangulation scanner includes a projector  5782 , a camera  5783 , and a processor  5784 . The triangulation scanner may be rotated to a wide variety of positions as explained herein above for the case of the six-DOF tactile probe  5700  of  FIG. 57 . Many types of triangulation scanners are available. Some project a line of light, while other scanners project an area of light. Some scanners make multiple sequential measurements, while others measure in single shots as discussed herein above. 
     The six-DOF triangulation scanner  5770  measures 3D coordinates of a workpiece using the principles of triangulation. There are several ways that the triangulation measurement may be implemented, depending on the pattern of light emitted by the scanner light source and the type of photosensitive array. For example, if the pattern of light emitted by the scanner light source is a line of light or a point of light scanned into the shape of a line and if the photosensitive array is a two dimensional array, then one dimension of the two dimensional array corresponds to a direction of a point on the surface of the workpiece. The other dimension of the two dimensional array corresponds to the distance of the point from the scanner light source. Hence the three dimensional coordinates of each point along the line of light emitted by scanner light source is known relative to the local frame of reference of the six-DOF scanner  5770 . 
     For a six-DOF scanner  5770  held by hand, a line of laser light emitted by the scanner light source may be moved in such a way as to “paint” the surface of the workpiece, thereby obtaining the three dimensional coordinates for the entire surface. It is also possible to “paint” the surface of a workpiece using a scanner light source that emits a structured pattern of light over an area. In an embodiment, the structured light may be in the form of a coded pattern that may be evaluated to determine three-dimensional coordinates based on single image frames collected by the camera  5783  of the scanner  5780 . 
     In an embodiment, the roll angle of a six-DOF probe is based on projection of non-rotating linearly polarized light onto a collection of polarizers in front of optical detectors. In an embodiment illustrated in  FIG. 59A , the collection of polarizers and optical detectors are attached to a housing  5902  that also holds a retroreflector  5910 . In an embodiment, the collection of polarizers includes four linear polarizers  5922 A,  5923 A,  5924 A, and  5925 A, each of the detectors oriented at a different angle. In an embodiment, the polarizers are placed over optical detectors  5930 , represented as circles in  FIG. 59A . In an embodiment, the polarizers  5922 A,  5923 A,  5924 A, and  5925 A have polarization directions (with respect to a horizontal axis) of 90, 135, 0, and 45 degrees, respectively. The optical power through these four directions may be analyzed to determine the direction of linear polarization of an incoming beam of light from a laser tracker. A collection of polarization and detector elements sufficient to determination a polarization angle of light striking the assembly may be referred to as a polarization angle detector. In  FIG. 59A , the polarization angle detector is the element  5920 A. In  FIGS. 59B-F , the polarization angle detectors are  5920 B,  5920 C,  5920 D,  5920 E, and  5920 F. In an embodiment, a plurality of polarization angle detectors are placed around the retroreflector  5910  on a housing  5902 . In an embodiment, three polarization angle detectors are placed at intervals of 120 degrees, along with three pitch/yaw sensors, in a manner analogous to that shown in  FIGS. 43A-C . In other embodiments, more or fewer polarization angle sensors are placed around the retroreflector  5910 . In an embodiment, the polarization sensor is tilted at an angle to permit a probe tilt angle of more than 45 degrees, as illustrated in  FIG. 43B  and  FIG. 43C . 
     For polarizer angles shown in  FIG. 59A - FIG. 59F , each of the polarization angle detectors  5920 A- 5920 F is an example of a Stokes polarimeter, which is a device capable of measuring polarization angle by measuring the Stokes parameters of the incoming light. The Stokes parameters, also referred to as the Stokes vectors, may be represented by a matrix having four matrix elements S 0 , S 1 , S 2 , and S 3 . These four matrix elements are alternatively referred to as I, Q, U, and V. Let P x  be the optical power of light polarized in the x direction, and let P y  be the optical power of light polarized in they direction. Let P a  be the optical power of light polarized in at +45 degrees, and let P b  be the optical power of light polarized at −45 degrees. Let P l  be the optical power of light having left circular polarization, and let P r  be light having right circular polarization. For the case considered here, the polarization state of the light is not random. In this case, the Stokes parameter S 0  is the total optical power. In other words, S 0 =P x +P y =P a +P b =P l +P r . The other Stokes parameters are defined as S 1 =P x −P y , S 2 =P a −P a , and S 3 =P l −P r . For the case of linearly polarized light, it can be shown that the angle of linear polarization is AoLP=0.5 arctan(S 2 /S 1 ). Hence, for this case, the angle of linear polarization may be found by measuring the optical power through each of the four polarizers  5922 A,  5923 A,  5924 A,  5925 A onto the optical detectors  5930 , taking the differences in the optical powers to obtain S 2  and S 1 , and then substituting these values into the formula above. 
     In some embodiments, the polarizers  5922 A,  5923 A,  5924 A, and  5925 A are not exactly spaced at increments of 45 degrees. It is still possible to calculate the angle of linear polarization if the angles of the polarizers are known. In this case, an optimization procedure, such as a least squares fit, is performed to determine AoLP. 
     Alternative realizations of a polarization angle detector are illustrated as elements  5920 B,  5920 C,  5920 D,  5920 E, and  5920 F shown in  FIGS. 59B, 59C, 59D, 59E , and  FIG. 59F , respectively.  FIG. 59B  illustrates the four linear polarizers  5922 A,  5923 A,  5924 A, and  5925 A placed over four separated regions of a quadrant detector  5935 , the regions being separated by gap lines  5937 . Electrical signals from the four quadrant regions are evaluated to determine the optical power present on each region. 
     As explained herein below, the optical source of linearly polarized light in the laser tracker is configured to provide uniform optical power over the area of each polarization angle detector. However, optical power per unit area is never exactly uniform, and hence several embodiments described herein below are provided to assist in accounting for the variations in optical power. In an embodiment, a polarization angle detector  5920 C of  FIG. 59C  includes a photosensitive array  5940  having a plurality of pixels. The photosensitive array  5940  is placed beneath the four linear polarizers  5922 A,  5923 A,  5924 A, and  5925 A. The pixels to the sides of each polarizer indicate the variation in optical power. In an embodiment, this variation provides an estimate of the optical power incident on the polarizers  5922 A,  5923 A,  5924 A, and  5925 A. The optical powers received through the polarizers  5922 A,  5923 A,  5924 A, and  5925 A by the photosensitive array are normalized to account for the estimated variations in incident optical power. 
     In an embodiment of  FIG. 59D  incident optical power is measured by detectors  5930  placed about the linear polarizers  5922 A,  5923 A,  5924 A, and  5925 A (e.g above, below, left and right of the linear polarizers when viewed from a position shown in  FIG. 59D ). As in the embodiment of  FIG. 59C , interpolation of received optical power by the detectors  5930  is used to estimate the incident optical power received by linear polarizers  5922 A,  5923 A,  5924 A, and  5925 A. 
     In an embodiment of  FIG. 59E , a polarization angle detector  5920 E includes four linear polarizers  5922 A,  5923 A,  5924 A, and  5925 A placed over four photodetector elements of a 5 by 5 array of photodetectors  5962 . As in the case of the collection of detectors in  FIG. 59D , the detectors  5962  not covered by the linear polarizers provide information on the optical power surrounding each of the linear polarizers. Interpolation is used to estimate the level of optical power incident on each of the polarizers  5922 A,  5923 A,  5924 A, and  5925 A. The electrical die  5960  includes a substrate  5963  having an array of photodetectors  5962 . The substrate  5963  is attached to the chip package by wire bonds  5965  that make electrical connection between die contacts  5964  and package contacts  5966 . 
     In some embodiments, the linear polarizers in the polarization angle detector are made of a thin glass containing nanoparticles. An example of such a material is the Polarcor™ polarizer made by Corning Incorporated of Corning, N.Y. This type of polarizer may be diced into square sections and placed over detector elements. For example, the polarizers  5922 A,  5923 A,  5924 A, and  5925 A in  FIG. 59A - FIG. 59E  may be Polarcor™ polarizers. In another embodiment, illustrated in  FIG. 59F , the polarizer may be a wiregrid polarizer. In an embodiment, a polarization angle sensor  5920 F includes a glass substrate  5970  on which are placed lithographically constructed metal gratings having sub-wavelength spacing. In an embodiment, the metal gratings  5972 ,  5973 ,  5974 , and  5975  are constructed to produce the same directions of linear polarization as the polarizers  5922 A,  5923 A,  5924 A, and  5925 A, respectively. In an embodiment, the four metal gratings are sized to fit over a quadrant detector such as the SPOT-4D detector manufactured by OSI Optronics of Hawthorne, Calif. Each of the four metal gratings is large enough to cover a corresponding photosensitive region of the quadrant detector. Regions of solid metal  5971  are placed around the collection of four gratings  5972 ,  5973 ,  5974 , and  5975  and in the gaps between the metal gratings. In other embodiment, the wiregrid polarizers  5972 ,  5973 ,  5974 , and  5975  are placed over other types of optical detectors. 
       FIG. 60  is a cross-sectional representation of optical elements in an exemplary beam launching system  6000 . In an embodiment, the beam launch that provides light used to determine orientational degrees of freedom such as pitch, roll, and yaw angles is a secondary launch used in combination with a main optics assembly used to launch a beam of light to determine three translational degrees of freedom such as x, y, and z. In a previous embodiment illustrated in  FIG. 47 , a secondary optics path  4730  was used to generate a beam of light having a rotating linear polarization state. In the embodiment of  FIG. 60 , a beam of light  6018  generated by a secondary optical path  6020  has a fixed linear polarization, which is combined with a beam of light  6016  from a main optics assembly  6010 . In an embodiment, the second optical path  6020  includes an optical fiber  6022 , a ferrule  6024 , a mirror  6026 , a lens  6028 , a linear polarizer  6030 , a dichroic beam splitter  6012 , and an exit window  6014 . In an embodiment, the optical fiber carries light produced by an LED at a wavelength of 940 nm. In an embodiment, the light is a multimode optical fiber that launches light from the ferrule  6024  in a diverging top hat pattern. Such a pattern is has a nearly flat wavefront and optical irradiance over a cylindrical cross section. In an embodiment, the launched light overfills the mirror  6026  and the lens  6028 . In an embodiment, the lens  6028  is an aspheric lens configured to minimize aberrations, including spherical aberration. The aspheric lens collimates the light and sends it through the polarizer  6030 , which produces linearly polarized light. In an embodiment, the linearly polarized light strikes the dichroic beam splitter  6012  in an s-polarized state and exits the laser tracker in a horizontal polarization direction. In an embodiment, the dichroic beam splitter  6012  includes a coating that reflects light at a wavelength of 940 nm but transmits red light  6016  from the main optics assembly  6010 . In other embodiments, other wavelengths are used in the main and secondary optical paths. In an embodiment, the beam of light  6016  has a diameter of 8 millimeters and the beam of light  6018  has a beam diameter of 32 millimeters. 
     A potential limitation with the roll-measuring systems illustrated in  FIGS. 59A, 59B, 59C, 59D, 59E, 59F  may result if the calculated roll angle depends, not only on the polarization state of the projected light, but also on the non-uniformity of the projected beam of light received by the four polarizers. A way around this potential limitation is now described in reference to  FIGS. 61-65 .  FIG. 61A  shows, in a top view, components of a secondary optical path  6100  configured to replace the secondary optical path  4730  shown in  FIG. 47 . In an embodiment, light is provided alternately through optical fibers  6102 A,  6102 B to corresponding collimating lenses  6104 A,  6104 B. In an embodiment, light  6106 B from the collimating lens  6104 B passes through a Risley prism pair  6108 B and then through a polarizing beam splitter  6112 . Each Risley prism in the pair is a window of glass having a slight wedge angle. The two Risley prisms in the Risley prism pair  6108 B are rotated to adjust the pointing direction of the light  6106 B. In an embodiment, the light  6106 B entering the polarizing beam splitter  6112  is a nearly flat-top beam having a random polarization. In an embodiment, the polarizing beam splitter transmits only the p-polarization state of the light  6106 B so that the light  6106 B emerges from the polarizing beam splitter  6112  polarized in the plane of the paper of  FIG. 61A . 
     In an embodiment, light  6106 A from the collimating lens  6104 A passes through a Risley prism pair  6108 A to adjust the pointing direction of the light  6106 A. In an embodiment, the light  6106 A reflects off a right-angle prism  6110  and a glass extender  6111  before it enters the polarizing beam splitter  6112 , which causes the reflected light to be linearly polarized perpendicular to the polarization of the light  6106 B. In an embodiment, the polarizing beam splitter  6112  reflects only the s-polarization state of the light  6106 A so that the light  6106 A merges from the polarizing beam splitter  6112  being polarized perpendicular to the plane of the paper of  FIG. 61A . Upon exiting the polarizing beam splitter  6112 , the beams of light  6106 A and  6016 B are together the beam of light  6114 . In an embodiment, the light  6106 A is turned on when the light  6106 B is turned off and vice versa. Consequently the beam of light  6114  alternately changes its polarization state from a first linearly polarized state to a second linearly polarized state perpendicular to the first linearly polarized state. In an embodiment, additional polarizing elements are added to improve the extinction ratio of the undesired linear polarization state to the desired linear polarization state for each of the two polarization states in the beam of light  6114 . In an embodiment, the extinction ratio is at least 40 dB. In a further embodiment, an optical detector  6113  is used to measure the optical power of the light beam  6106 A transmitted by the polarizing beam splitter  6112  and the optical power of the light beam  6106 B reflected by the polarizing beam splitter  6112 . These optical powers are sent to the optical detector  6113  in a beam of light  6115 . The optical powers in the two polarization states in the beam of light  6115  are proportional to the optical powers in the orthogonal polarization states of the beam of light  6114 . Hence the optical powers received by the optical detector  6113  may be used to correct for relative differences in optical powers delivered sent through the two optical fibers  6102 A and  6102 B, as explained further herein below. The output of the optical detector  6113  is sent to a processor for use in calculations of the roll angle of a remote six-DOF sensor, as further described in herein below in reference to  FIG. 62 . 
       FIG. 61B  shows, in a side view, the secondary optical path  6100 . The beam of light  6114  travels through negative lens element  6116 , reflects off mirror  6118 , travels through positive lens assembly  6120 , and reflects off dichroic beam splitter  6122 . The dichroic beam splitter  6122  is configured to reflect light having the wavelengths of the beam of light  6114 , while transmitting light having the wavelengths of beam of light  6124  arriving from the main optics path. The beams of light  6114  and  6124  are combined into a composite beam of light  6126 . In an embodiment, the beam of light  6124  is red light having a wavelength of around 637 nm, while the beam of light  6114  is near-infrared light having a wavelength of around 940 nm. The composite beam of light  6126  travels to a six-DOF target. Possible embodiments of the six-DOF target elements are illustrated schematically in  FIGS. 62A, 62B, 62C . 
       FIG. 62A  shows elements on the front face of a six-DOF device  6200 A such as a six-DOF tactile probe or six-DOF scanner. The elements include a body  6202 , a retroreflector  6204 , and a roll sensor  6205 . The roll sensor  6205  includes a first sensor  6210  and a second sensor  6220 . The first sensor  6210  includes a polarizer/optical filter  6212 , an optical detector  6214 , and electronics discussed herein below with reference to  FIGS. 63A-63C . The optical filter in the polarizer/optical filter  6212  helps block out unwanted background light. In an embodiment, the optical filter is constructed as a thin film multi-layer dielectric coating placed on top of the polarizer element. The polarizer transmits linearly polarized light in a preferred direction and blocks light polarized perpendicular to the preferred direction. In an embodiment, the polarizer is a relatively thin sodium-silicate glass in which are embedded spherical ellipsoidal nanoparticles. In an embodiment, the polarizer has an extinction ratio of −50 dB at the wavelength of the light  6114  from the secondary channel of  FIGS. 61A, 61B . 
     According to an embodiment, elements of the first sensor  6210  and the second sensor  6220  are shown in profile in  FIG. 62C . A beam of light  6230  arrives at a front surface of the sensor  6240 , which might be the first sensor  6210  or the second sensor  6220 . The light  6230  enters a thin-film dielectric coating  6241  that blocks light except at the desired wavelength, which in an embodiment is 940 nm. The thin-film dielectric coating  6241  is applied to the polarizer  6242 , which in an embodiment is a sodium-silicate glass having embedded spherical ellipsoidal nanoparticles. In an embodiment, the relatively thin polarizer element  6242  is glued on a substrate  6244  by a thin, flat glue layer  6243 . The substrate  6244  is glued onto an optical detector  6246  by a thin, flat glue layer  6245 . In an embodiment, to minimize reflections, the indexes of refraction of the glue layers  6243 ,  6245  are matched closely as possible to the indexes of refraction of the polarizer  6242 , substrate  6244 , and detector  6246 . A wire  6248  sends the electrical signal from the optical detector  6246  to an electrical circuit in communication with a processor. 
     The second sensor  6220  includes a polarizer/optical filter  6222  and an optical detector  6224 . In an embodiment, the polarizer transmits light that is linearly polarized at 45 degrees relative to the polarization direction of the polarizer/optical filter  6212 . 
       FIG. 62B  shows elements on the front face of a six-DOF device  6200 B. The device  6200 B includes a retroreflector  6204  and three roll sensors  6205 A,  6205 B, and  6205 C, each similar to the roll sensor  6205  of  FIG. 62A . In an embodiment, the roll sensors are tilted at an angle, as illustrated in  FIGS. 43A, 43B, 43C . In an embodiment, the specifications for extinction ratio of the polarizer in the polarizer/optical filter  6212  are valid up to a tilt angle of 25 degrees. By providing three roll sensors  6205 A,  6205 B,  6205 C and tilting the sensors as illustrated in  FIGS. 43A, 43B, 43C , the six-DOF tactile probe is capable of operating over an extended range of tilt angles. In further embodiments, pitch/yaw sensors such as those discussed herein above may be added to the front-face of the six-DOF device,  6200 A or  6200 B. 
       FIG. 63A  is a block diagram showing the main elements in electro-optic system  6300 . The electro-optic system  6300  includes a reference clock  6302 , a modulator  6304 , a first light source  6306 A, and a second light source  6306 B. The light sources  6306 A,  6306 B send modulated light to optical fibers  6102 A,  6102 B, respectively. The reference clock  6302  provides a stable clock signal to the modulator  6304 , which has two channels A and B. The modulator  6304  provides a first modulation signal from the A channel to the first light source  6306 A and a second modulation signal from the B channel to the second light source  6306 B. One aspect of the A and B modulation signals is that the light source  6306 B is turned off when the light source  6306 A is turned on and vice versa. This alternating electrical ON-OFF modulation provides an alternating polarization state in the beam of light  6126 . In an embodiment, a further modulation is provided in the modulation ON state. In the illustration of  FIG. 63A , the further modulation is square-wave modulation between a maximum and a minimum value, with multiple square-wave cycles provided for each polarization state. In other embodiments, another type of modulation such as sinusoidal modulation is provided instead of square-wave modulation. 
     The two orthogonal polarizations of light in the light beam  6126  strike the sensors  6210 ,  6220 , which in response send electrical signals  6312 ,  6314 , respectively, to an electrical circuit  6310 . In an embodiment, the electrical circuit  6310  includes a processor  6316  that determines the roll angle of the sensor based on methods discussed herein below with respect to  FIGS. 64, 65 . In an embodiment, the calculated roll angles are sent from the six-DOF device  6200 A over a wired or wireless channel  6320  to the laser tracker or other device that projects the light  6126  onto the six-DOF device  6200 A. In a further embodiment, the laser tracker or other device also sends the six-DOF device  6200 A synchronization signals for modulation of channels A and B over the wired or wireless channel  6320 . In an alternative embodiment, the electronics circuit  6316  sends signals to a processor outside the six-DOF device  6200 A for determination of the roll angle of the six-DOF device  6200 A. 
     In another embodiment illustrated in  FIG. 63B , the first sensor  6210  and the second sensor  6220  rely on the received optical signals (e.g. light  6126 ) to reconstruct synchronization signals for modulation channels A and B. The dashed line  6340  indicates that the received optical signals provide their own synchronization. In a further embodiment, a clock  6302 B is used to assist in establishing the synchronization. In an embodiment, the electronics circuit  6310 B includes a processor  6316 B that determines the roll angle of the sensor based on methods discussed herein below with respect to  FIGS. 64, 65 . In an alternative embodiment, the electronics circuit  6316 B sends signals to a processor outside the six-DOF device  6200 A for determination of the roll angle of the six-DOF device  6200 A. 
     The electrical circuit  6310  or  6310 B may process the received signals  6312 ,  6314  to reduce background light or electrical noise. Background (ambient) light will usually have a detected optical time signature that is mainly DC or that varies at twice the frequency of the electrical mains. The light  6326  on the other hand is modulated with a square wave or sine wave at a much higher frequency, for example, 1 kHz or higher. One way to exclude the unwanted background light, as well as unwanted electrical noise, is to use a digital lock-in amplifier or analog lock-in amplifier. The principles of a digital lock-in amplifier were discussed herein above.  FIG. 63C  shows an example of an analog lock-in amplifier  6350 , also known as a synchronous demodulator. In an embodiment, electrical signals  6312 ,  6314  are sent to a bandpass filter  6352 , which is centered at the modulation frequency f mod  of the relatively rapid square-wave modulation shown in  FIG. 63A  or other modulation such as sinusoidal modulation. For example, if this modulation frequency is f mod =8 kHz, then the modulation of the background (ambient) light, which is typically at DC or twice the line frequency will be eliminated, as will electrical noise outside the passband. The filtered electrical signal is applied to a mixer  6354  having the modulation frequency f mod  applied to the local oscillator port of the mixer  6354 , which causes the modulated signal to be down converted to DC. A low-pass filter  6356  filters unwanted high frequency signals from the mixer and provides a DC level that may be read by an analog-to-digital converter to determine a digital value  6358  corresponding to the noisier input signal  6312  or  6314 . 
     The graph of  FIG. 64A  indicates on the vertical axis the optical power received by the optical detector of the second sensor  6220 . The polarization state of the light  6126  alternates between two orthogonal polarization states in a first half-cycle and a second half-cycle as explained herein above. The horizontal axis of  FIG. 64A  indicates the angle of polarization of the light  6126  during the first half-cycle relative to the x axis (shown in  FIG. 62A ) of the six-DOF device  6200 A. The solid line shows the detected optical power during the first half-cycle, and the dashed line shows the detected optical power during the second half-cycle. If the sensor  6220  is oriented normal to the light  6126  and the optical power of the beam of light  6126  is the same in the first and second half-cycles, then the solid and dashed lines of  FIG. 64A  are also the same, only shifted by 90 degrees. Furthermore the sum of the values of the solid and dashed curves is constant at all angles, here set to a normalized value of 1. This enables the roll angle of the six-DOF device to be determined from the horizontal-axis angle corresponding to the measured value of the solid line. For the sensor  6220 , the horizontal-axis angle may be determined without ambiguity within a range of 90 degrees since the pattern repeats in inverted form every 90 degrees. 
     A way to extend the ambiguity in the roll angle to 180 degrees is to add a sensor  6210 . The response of the sensor  6210  to the beam of light  6126  is shown in  FIG. 64B . The solid and dashed curves are shifted by 45 degrees relative to the solid and dashed curves in  FIG. 64A . The ambiguity near 90 degrees in  FIG. 64A  is removed in  FIG. 64A  since the solid and dashed curves are either increasing or decreasing in this range of angles. 
     Another advantage of using two sensors  6310 ,  6320  is to increase the sensitivity of the roll measurement. In  FIG. 64A , the measurement of roll angle by the sensor  6220  is most sensitive near an angle of 45 degrees since this is the angle at which the level of detected optical power changes most rapidly with angle. The measurement of roll angle is least sensitive near an angle of 90 degrees since this is the angle at which the level of detected optical power changes least rapidly with angle. By adding the sensor  6210 , the overall sensitivity is improved since this sensor is most sensitive near 45 degrees and least sensitive near 90 degrees. 
     A potential problem may occur if the optical power in the first half-cycle is different than the optical power in the second half-cycle. In an embodiment, this potential problem is corrected by measuring the optical power in the first and second half-cycles with the optical detector  6113  in  FIG. 61A . The optical powers detected by the optical detector  6113  in the first and second half-cycles are proportional to the corresponding optical powers in the first and second half cycles of the first and second half cycles of the beam of light  6126  at the desired wavelength. By measuring the optical powers at the desired wavelength in the first and second half cycles of the beam of light  6126  as it leaves the tracker (or similar device), the proportionality constant may be determined for the first and second half cycles. In an embodiment, the powers measured by the optical detector  6113  in the first and second half-cycles are multiplied by the determined proportionality constants to obtain the optical power of the beam of light  6126  in the first and second half cycles (at the desired wavelength). 
     In general, the six-DOF device  6200 A may vary in roll, pitch, and yaw angles. A method for accounting for the effects of pitch and yaw angles on the determined roll angle is now discussed.  FIG. 65A  shows the interface between air and glass, with the interface between the air and the glass having a normal unit vector n. Light arrives in a direction given by a propagation unit vector k. The vectors n and k define a plane of incidence that intersects the air-glass interface in a unit vector u. 
     The amount of light reflected and transmitted through an interface depends on whether the light is in an s-polarization state or a p-polarization state.  FIG. 65A  shows that the incident electric field E I  has an s-polarized component (E I ) S  perpendicular to the plane of incidence and a p-polarized component (E I ) P  perpendicular to the s-polarized component and to the propagation unit vector k. The s- and p-polarized components for the transmitted light E T  is defined in an analogous way. The amount of p-polarized light transmitted into the glass is given by (E T ) P =T P ·(E I ) P  and the amount of s-polarized light transmitted into the glass is (E T ) S =T S ·(E I ) S , where T P  and T S  are transmission coefficients for s- and p-polarized light at this air-glass interface. These equations assume that the glass does not have a preferred polarization direction. For the case in which the glass is a polarizer, as shown in  FIG. 65B , we need to decompose the unit vector for the direction of polarization f into its component parts along the s- and p-directions as f S  and f P . The fraction of the incident s-polarized light that aligns with f S  passes into the polarizer after applying the formula (E T ) S =T S ·(E I ) S . The fraction of the incident p-polarized light that lies in the plane of incidence (that also contains f P ) passes into the polarizer after applying the formula (E T ) P =T P ·(E I ) P . 
     In an embodiment, for the generally complicated geometry of the sensor  6240  as illustrated in  FIG. 62C , the transmission coefficients T S , T P  are experimentally determined as a function of the angle θ of the propagation unit vector k relative to the unit vector n. These transmission coefficients are determined as a function of θ for both the direction of polarization of the polarizer (such as the polarizer  6212 ,  6222 ) and the direction perpendicular to the polarization of the polarizer. 
     In an embodiment, the pitch and yaw angles are provided by a pitch/yaw sensor. These angles are provided to the processor  6316  or to an external processor configured to determine the roll angle of the six-DOF device  6200 A. The pitch and yaw angles determine the angle θ in  FIG. 65A . The processor extracts the transmission coefficients T S (θ) and T P (θ) from a look-up table or formula based on previous experimental results as discussed herein above. In an embodiment, an assumed initial roll angle is used to obtain the components of the polarization of the sensor  6210 ,  6220  that lie in the s- and p-directions. These polarization components are applied to the corresponding transmission equations to obtain the resulting optical power received by each of the sensors  6210 ,  6220  for each of the polarization states in the first and second half-cycles. An optimization routine is performed to iteratively adjust the roll angle to match the observed powers to the calculated powers as well as possible. The final roll angle obtained by the iteration is included with the pitch and yaw angles measured by a pitch/yaw sensor and with the three translational coordinates of the retroreflector measured by the laser tracker (or similar device) to obtain the full six degrees-of-freedom of a six-DOF probe such as  6200 A or  6200 B. 
     While the invention has been described with reference to example embodiments, it will be understood by those skilled in the art that various changes may be made and equivalents may be substituted for elements thereof without departing from the scope of the invention. In addition, many modifications may be made to adapt a particular situation or material to the teachings of the invention without departing from the essential scope thereof. Therefore, it is intended that the invention not be limited to the particular embodiment disclosed as the best mode contemplated for carrying out this invention, but that the invention will include all embodiments falling within the scope of the appended claims. Moreover, the use of the terms first, second, etc. do not denote any order or importance, but rather the terms first, second, etc. are used to distinguish one element from another. Furthermore, the use of the terms a, an, etc. do not denote a limitation of quantity, but rather denote the presence of at least one of the referenced item.